Handbook of Spallation Research: Theory, Experiments and Applications

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Detlef Filges and Frank Goldenbaum Handbook of Spallation Research

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Detlef Filges and Frank Goldenbaum

Handbook of Spallation Research Theory, Experiments and Applications

The Authors Prof. Dr. Detlef Filges Institut für Kernphysik Forschungszentrum Jülich 52425 Jülich PD Dr. Frank Goldenbaum Institut für Kernphysik Forschungszentrum Jülich 52425 Jülich

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.: applied for British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Bibliographic information published by the Deutsche Nationalbibliothek Die Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.d-nb.de.  2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Cover Typesetting Laserwords Private Limited, Chennai, India Printing and Binding Strauss GmbH, Mörlenbach Printed in the Federal Republic of Germany Printed on acid-free paper ISBN: 978-3-527-40714-9

‘‘For a successful technology, reality must take precedence over public relations, for Nature cannot be fooled.’’ Richard P. Feynman (1921–1988)

VII

Contents

Preface

XVII

Part I

Principles 1

1 1.1 1.2 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.3.5 1.3.5.1 1.3.5.2 1.3.5.3 1.3.5.4 1.3.5.5

The Spallation Process 3 Historic Remarks and Definitions 3 Spallation by Cosmic Ray-Induced Reactions 5 Physics of the Spallation Process 12 Introduction 12 The Fission Process 15 Spallation and Fission 17 The Logical Scheme of Spallation Reactions 21 Particle Interaction Mechanisms 23 The Elementary Forces and Particles 23 Feynman Diagrams 26 Resonance Decay, Pion Absorption, and Pion Charge-Exchange 27 Kinetic Energy, Total Energy, and Momentum 30 Cross Sections, Absorption Length, Collision Length, and Mean Free Path 31 Electromagnetic and Atomic Interactions 33 Energy Loss of Heavy Particles by Ionization and Excitation – the Bethe or Bethe–Bloch Formula 33 Coulomb Scattering 37 Bremsstrahlung 39 Energy Loss by Direct Pair Production and by Photonuclear Interaction 40 Total Energy Loss 40 High-Energy Hadronic Cascades and Nuclear Interactions 41 Qualitative Features of Hadron–Nucleus Collisions 41 Characteristics of Hadron Cascades in Thick Targets 42 Spatial Propagation of Hadron Cascades and Particle Production 43 Total Cross Sections in Nucleon–Nucleus Collisions 44

1.3.6 1.3.6.1 1.3.6.2 1.3.6.3 1.3.6.4 1.3.6.5 1.3.7 1.3.7.1 1.3.7.2 1.3.7.3 1.3.7.4

Handbook of Spallation Research: Theory, Experiments and Applications. Detlef Filges and Frank Goldenbaum Copyright  2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40714-9

VIII

Contents

1.3.7.5 1.3.7.6 1.3.8

Total Reaction Cross Sections in Nucleus–Nucleus Reactions 49 Differential Cross Sections 54 Hadronic–Electromagnetic Cascade Coupling 57

2 2.1 2.2 2.2.1 2.2.2 2.2.2.1 2.2.2.2 2.2.2.3 2.2.2.4 2.2.3 2.2.3.1 2.2.3.2 2.2.4 2.2.5

The Intranuclear Cascade Models 63 Introduction 63 The BERTINI Approach 64 Features of the BERTINI Nuclear Model 64 The Nuclear Model 65 Nucleon Density Distribution Inside the Nucleus 65 Momentum Distribution of Nucleons Inside the Nucleus 68 Potential Energy Distribution Inside the Nucleus 70 Application of the Pauli Exclusion Principle 73 The Cross-Section Data 74 Nucleon–Nucleon Cross-Section Data 76 Pion Production and Pion Nucleon Reactions 77 Method of Computation 85 Assumptions, Limits, and Constraints on the Energy and Application Regime 87 The Cugnon INCL Approach 88 Features of the Model 90 Participants and Spectators 90 Nuclear Surface 91 Meson–Nucleon Cross Sections 92 Elastic Nucleon–Nucleon Cross Sections 93 Angular Distributions 94 Dynamic Pauli Blocking 97 Cutoff Criteria – Stopping Time of the Cascade 98 Light Clusters as Incident Particles 103 Surface Percolation Procedure for Emission of Light Charged Clusters 103 Assumptions, Limits, and Constraints on the Energy and Application Regime 109 The ISABEL Model 110 Features of the Model 110 The Nuclear Model 111 The Time-Like Basis Cascading of Particle–Nucleus Interactions 112 Nucleus–Nucleus Collisions 113 Assumptions, Limits, and Constraints on the Energy and Application Regime 114 The CEM (Cascade-Exciton Model) Approach 114 Features of the CEM Model 115 Assumptions, Limits, and Constraints on the Energy and Application Regime 117

2.3 2.3.1 2.3.1.1 2.3.1.2 2.3.1.3 2.3.1.4 2.3.1.5 2.3.1.6 2.3.1.7 2.3.1.8 2.3.1.9 2.3.2 2.4 2.4.1 2.4.1.1 2.4.1.2 2.4.1.3 2.4.2 2.5 2.5.1 2.5.2

Contents

2.6 2.6.1 2.6.2 2.6.3 2.7 2.7.1 2.7.1.1 2.7.1.2 2.7.1.3

Other Intranuclear-Cascade Models 119 The Dubna Models 119 The Hänssgen–Ranft Model 122 The MICRES Model 123 Alternative Models 127 The Quantum-Molecular-Dynamic (QMD) Model 127 The Equation of Motion and the Reaction Treatment 128 The Cutoff Criteria – the Stopping Time of the QMD Process Selected Examples of QMD Simulations 132

3 3.1 3.2 3.3 3.3.1 3.3.2 3.4 3.4.1 3.4.2 3.4.3 3.4.4 3.4.5 3.4.6 3.5 3.5.1 3.5.2 3.5.3 3.6 3.7 3.7.1 3.7.2 3.7.3 3.7.3.1 3.8 3.9 3.10

Evaporation and High-Energy Fission 135 Introduction 135 The Statistical Model in its Standard Form 136 The Evaporation Model EVAP of Dostrovsky–Dresner 139 The Level Density Parameter in the EVAP Model 140 The Inverse Cross Sections 142 The Generalized Evaporation Model (GEM) 143 Evaporation Model in GEM 144 The Decay Width in the GEM Model 145 Parameter of the Inverse Cross Sections and the Coulomb Barrier 146 The Level Density Parameter GCCI in the GEM Model 147 The High-Energy Fission Process in the GEM Model 148 Method of Computation 151 The GSI ABLA Model 154 Time-Dependent Fission Width 157 The Simultaneous Breakup Stage 158 Conclusion of ABLA 159 The GEMINI Model 159 High-Energy Fission Models 165 The Dynamics of the Fission Process 166 Basic Features of Fission Models 170 The RAL Model of Atchison 171 Postfission Parameters of the RAL Model 173 Fermi Breakup for Light Nuclei 176 Photon Evaporation and Gamma Ray Production 178 Vaporization and Multifragmentation 181

4 4.1 4.2 4.3 4.3.1 4.3.2 4.4

The Particle Transport in Matter 185 Introduction 185 Hadronic and Electromagnetic Showers 185 The General Transport Equation 191 The Angular Flux, Fluence, Current, and Energy Spectra 195 Monte Carlo Estimation of Particle Fluxes and Reaction Rates 197 Range Straggling 200

130

IX

X

Contents

4.5 4.6

The Elastic Scattering of Protons and Neutrons 201 The Treatment of Pion Transport in Matter 205

5 5.1 5.2 5.2.1 5.2.2

Particle Transport Simulation Code Systems 207 Introduction 207 Particle Transport Code Systems and Event Generators 208 Particle Transport Systems and Event Generators 209 The Three-Dimensional Geometry Systems of Particle Transport Code Systems 210

6

Materials Damage by High-Energy Neutrons and Charged Particles 215 Introduction 215 Displacement of Lattice Atoms 216 Damage Energy and Displacements 220 Hydrogen and Helium Production 224 Cross Section Examples 227 Radiation Damage Effects of High-Intensity Proton Beams in the GeV Range 231

6.1 6.2 6.2.1 6.3 6.4 6.5

7 7.1 7.2 7.2.1 7.2.2 7.2.3 7.3 7.3.1 7.3.2 7.4

Shielding Issues 233 Introduction 233 The Attenuation Length and the Moyer Model 234 Accelerator Shielding and the Generalized Moyer Model 235 The Moyer Model Parameters 238 Attenuation Lengths 241 Advanced Shielding Methods for Spallation Sources 242 Monte Carlo Discrete Ordinates Coupling 246 Monte Carlo Techniques and Deep Penetration 249 Sky- and Groundshine Phenomena 254

8 8.1 8.2 8.2.1 8.2.2 8.2.3 8.2.4 8.2.4.1 8.2.5 8.2.6 8.3 8.3.1 8.3.2 8.3.3 8.3.4

The Basic Parameters of Spallation Neutron Sources 257 Introduction 257 Parameter Regime for Spallation Neutron Sources 257 The Particle Type 257 The Kinetic Energy 259 The Target Material 262 The Neutron Production 263 Spatial Leakage Distribution of Neutrons and Target Shape The Target Heating 266 The Induced Radioactivity 267 The Spallation Neutron Source Facility 269 Continuous Spallation Neutron Sources 272 Short-Pulse Spallation Neutron Sources 272 Long-Pulse Spallation Neutron Sources 272 Scattering of Neutrons by Matter 273

264

Contents

Part II

Experiments 277

9 9.1 9.2 9.3 9.4

Why Spallation Physics Experiments? 279 Introduction 279 Application-Driven Motivation 279 Space Science and Astrophysics-Driven Motivation Nuclear Physics Driven Motivation 281

10

Proton-Nucleus-Induced Secondary Particle Production – The ‘‘Thin’’ Target Experiments 287 Introduction 287 Neutron, Pion, and Proton Double Differential Measurements and Experiments 288 The Double-Differential Neutron-Production Measurements at the LAMPF-WNR Facility 289 The Time-of-Flight Experiment at the LAMPF-WNR Facility 290 The Experimental Results of Double-Differential Neutron-Production Cross-Section Measurements 299 The Double-Differential Neutron-Production Measurements at the SATURNE Facility 303 The Experimental Apparatus at the SATURNE Accelerator 305 The Experimental Results of Double-Differential Neutron-Production Cross Section Measurements 310 The Double-Differential Neutron-Production Measurements at the KEK Facility 312 The Time-of-Flight Method with a Short Flight Path 312 The Experimental Results of Double-Differential Neutron-Production Cross Section Measurements 316 The Double Differential Pion Production Measurements 318 The Double Differential Pion Production Measurements at the Berkley Cyclotron with Incident Protons of 730 MeV 319 The Double Differential Pion Production Measurements at the Cyclotron of the Paul Scherer Institut (PSI) 325 The ‘‘Thin’’ Target Particle Production Measurements at the 2.5 GeV Proton Cooler Synchrotron COSY at Jülich 327 The COoler SYnchrotron COSY 331 The NESSI Experiment 333 Experimental Setup of NESSI 335 Experimental Results of NESSI 341 The PISA Experiment 355 Experimental Setup of PISA 356 Experimental Results of PISA 361 Data Library of H and He in Proton-Induced Reactions 369 Production of Residual Nuclides at Various Proton Energies 369

10.1 10.2 10.2.1 10.2.1.1 10.2.1.2 10.2.2 10.2.2.1 10.2.2.2 10.2.3 10.2.3.1 10.2.3.2 10.2.4 10.2.4.1 10.2.4.2 10.3

10.3.1 10.3.2 10.3.2.1 10.3.2.2 10.3.3 10.3.3.1 10.3.3.2 10.3.3.3 10.4

280

XI

XII

Contents

10.4.1 10.4.2 10.4.2.1

Excitation Functions and Production Cross Sections 370 Isotopic and Mass Distributions of Residual Nuclides 373 Inverse Kinematics Measurements at GSI 373

11

Proton-Matter-Induced Secondary Particle Production–The ‘‘Thick’’ Target Experiments 379 Introduction 379 Proton-Induced Thick Target Experiments in the Energy Range 0.1–2.5 GeV 381 Experiments to Measure the Neutron Yield of ‘‘Thick’’ Targets 381 The Brookhaven Cosmotron Experiments 381 The Fertile-to-Fissile Conversion Experiments 382 The PSI Thick Target Lead/Bismuth Experiments 388 Neutron Multiplicities Measured with a 4π Detector at the COSY Accelerator at Jülich 390 Thick Target Results of the NESSI Experiment at the COSY Accelerator 394 A Summary of Neutron Yield Experimental Data 402 Neutron Leakage Spectra Distributions of Thick Targets 407 The LANL-WNR Thick Target Experiments 407 The KEK Time-of-Flight Thick Pb Target Experiments 408 The LANL SUNNYSIDE Experiments 409 Energy Deposition Experiments with Thick Mercury Targets 413 The Energy Deposition Thick Mercury Target Experiment at COSY Jülich 414 The Thick Mercury Target Experiment ASTE at the AGS Accelerator at BNL 419

11.1 11.2 11.2.1 11.2.1.1 11.2.1.2 11.2.1.3 11.2.2 11.2.2.1 11.2.2.2 11.2.3 11.2.3.1 11.2.3.2 11.2.4 11.2.5 11.2.5.1 11.2.5.2

12

Neutron Production by Proton, Antiproton, Deuteron, Pion, and Kaon Projectiles 425

13

Experiments to Study the Performance of Spallation Neutron Sources 431 Introduction 431 The Target–Moderator–Reflector Issue 431 Target–Moderator–Reflector Experiments with Complex Geometries and Realistic Material Compositions 437 The Early Experiments 437 Neutron Performance Studies at Reflected Target–Moderator Systems 438 Neutron Studies of a Reflected ‘‘T’’-Shape Moderator at LANL-WNR 438 The SNQ Target–Moderator–Reflector Experiments at the PSI Accelerator 439

13.1 13.2 13.3 13.3.1 13.3.2 13.3.2.1 13.3.2.2

Contents

13.3.3 13.3.3.1 13.3.3.2 13.3.3.3

14 14.1 14.1.1 14.1.2 14.1.2.1 14.1.2.2

15 15.1 15.2 15.3

Experiments of Short-Pulsed Target–Moderator–Reflector Systems 445 Target–Moderator–Reflector Experiments at the Hokkaido Electron Linear Accelerator 446 Target–Moderator–Reflector Experiments at the Jülich Proton Synchrotron COSY 454 Target–Moderator–Reflector Experiments at the Brookhaven Alternating Gradient Synchrotron AGS 466 Experiments on Radiation Damage in a Spallation Environment 471 Introduction 471 Irradiation Conditions and Studied Materials 472 Experimental Results 473 Microhardness and Three-Point Bending Tests 473 Tensile Strength, Scanning- and Transmission Electron Microscopy 475 Experiments to Shield High-Energy Neutrons of Spallation Sources 481 Shielding Experiments at the Los Alamos WNR Facility’s Spallation Target 482 Shielding Experiments at the ISIS Spallation Source 486 Shielding Experiments at the ASTE–AGS Target–Moderator–Reflector Assembly 491

Part III

Technology and Applications 495

16 16.1 16.2

Proton Drivers for Particle Production 497 An Introduction 497 Proton Drivers for Spallation Neutron Sources and Secondary Particle Production 498 Synchrotron-Based vs. Linear-Accelerator-Based Spallation Neutron Source 500 The Beam Loss Issue at High-Intensity Proton Accelerators 502

16.2.1 16.2.2 17 17.1 17.2 17.3 17.3.1 17.3.2 17.3.3

The Accelerator-Based Neutron Spallation Sources 505 Introduction 505 Research Reactors or Continuous/Pulsed Spallation Sources? 507 Spallation Neutron Sources 510 The LANL Spallation Neutron Source MLNSC and the Los Alamos Neutron Science Center LANSCE 510 The Rutherford and Appleton Laboratory Short-Pulsed Spallation Neutron Source ISIS 514 The PSI Continuous Spallation Neutron Source SINQ 519

XIII

XIV

Contents

17.3.4 17.3.5 17.3.6 17.3.7 17.3.8

18 18.1 18.2 18.2.1

The European Spallation Neutron Source Project ESS 524 The ORNL Spallation Neutron Source SNS 538 The Japanese Spallation Source and the Accelerator Complex J-PARC 548 Safety Aspects and Radiation Protection 554 Parameter Overview of the Existing, Commissioned and Planned Spallation Neutron Sources with a Beam Power Above 0.1 MW 558

18.2.2.3 18.2.2.4 18.2.3

Target Engineering 561 Introduction 561 Spallation Source Neutron-Generating Targets 562 Spallation Source Neutron-Generating Targets at Beam Power Levels of About Some 100 kW 562 Spallation Source Neutron-Generating Targets at Beam Power Levels in the MW Range 564 The SINQ Target Systems 565 The Mercury Targets for the Pulsed Spallation Neutron Sources ESS, SNS, and JSNS 569 Rotating High-Power Targets 579 Windowless Liquid Metal Targets 582 Materials for Accelerators and Targets of Spallation Sources 583

19 19.1 19.1.1 19.1.2 19.1.3 19.1.4 19.1.5 19.1.6 19.1.7 19.1.8

Research with Neutrons 585 Introduction 585 Solid-State Physics 587 Materials Science and Engineering 588 Chemical Structure, Kinetics, and Dynamics 588 Soft Condensed Matter 589 Biology and Biotechnology 589 Earth and Environmental Science 589 Fundamental Neutron Physics 590 Muons as Probes for Condensed Matter 590

20 20.1 20.2 20.2.1 20.2.2 20.3 20.3.1 20.4 20.4.1 20.5 20.6

Accelerator Transmutation of Nuclear Waste – ATW 593 Introduction 593 The Concepts of Transmutation 594 Balance Criteria for ADTT Systems 596 The Accelerator Issue of ATW/ADTT Systems 597 The Spent Reactor Fuel and Transmutation 600 An Example of a Transmutation Complex 602 Partitioning 605 EU Projects on Partitioning and Transmutation 606 Advances in Accelerator Breeding of Fissile Material 607 Accelerator Production of Tritium-APT 609

18.2.2 18.2.2.1 18.2.2.2

Contents

21 21.1 21.2 21.3 21.3.1 21.4 22 22.1 22.1.1 22.1.2 22.1.3 22.2 22.2.1 22.2.2 22.2.3 22.2.4 22.3 22.3.1 22.3.2 22.4 22.4.1 22.4.2 22.4.3 22.4.4 22.4.5 22.4.5.1 22.4.5.2 22.4.5.3 22.4.5.4 22.4.5.5

23 23.1 23.2 23.2.1 23.2.2

23.2.2.1

Accelerator Production of Electrical Energy – ‘‘Energy Amplifying’’ 613 Introduction 613 Principle of the ‘‘Energy-Amplifier’’ 613 Feasibility of the ‘‘Energy Amplifier’’ 615 An Energy Amplifier Test Experiment 618 Advantages and Disadvantages of the EA Concept 619 Advanced Applications of Spallation Physics 623 Proton and Light Heavy Ion Cancer Therapy 623 History of Hadron Therapy 624 What is Radiation Therapy and How Does it Work? 626 Differences Between Protons, Heavy Ion, and X-ray Therapy 627 High-Energy Physics Calorimeter 631 The Physics of Particle Calorimetry 631 Electromagnetic Showers 633 Hadronic Showers 634 Combined Electromagnetic/Hadronic Calorimeters 635 Neutrino Factories and Neutrino Super/Beta Beams 638 Some Words on Research with Neutrinos 638 International Scoping Study of ν Factories and Superbeam Facilities 639 Ultracold Neutrons 642 History 642 Properties of UCNs 642 Reflecting Materials 644 UCN Production 645 Experiments with UCN 647 Measurement of the Neutron Lifetime 648 Measurement of the Neutron Electric Dipole Moment 648 Observation of the Gravitational Interactions of the Neutron 649 Measurement of the Neutron-Mirror Neutron Oscillation Time 650 Measurement of the a-Coefficient of the Neutron Beta Decay Correlation 651 Space Missions and Radiation in Space 653 Introduction 653 Galactic Cosmic Ray (GCR) Induced Reactions in Moon and MARS Soil 654 GCR-Induced Neutron Flux Density in the Lunar Soil During Apollo Missions 654 The Mars Observer Orbiter Mission to Measure the Spallation-Induced Gamma Flux Return of GCR on the Martian Surface 657 The Aim of the Mars Observer Mission 657

XV

XVI

Contents

23.2.2.2 23.3 23.3.1 23.3.2 23.3.3

Application to the Martian Surface 659 The Space Experiment LDEF (Long Duration Exposure Facility) 663 The Aim of the LDEF Mission 663 The LDEF Orbiter System and Results on Induced Radioactivity of LDEF Materials 665 Hazard Radiations in Space 669

Appendix A Values of Fundamental Physical Constants and Relations 675 Appendix B Basic Definitions in Nuclear Technology Concerning the Fuel Cycle 679 Appendix C Material Properties of Structure and Target Materials 683 Appendix D Moderator and Reflector Materials Appendix E Shielding Materials 689 References 691 Index 757

687

XVII

Preface Spallation research is based on nuclear reaction mechanism called spallation as a source of an enormous radiation of many different particles such as neutrons, protons, pions, muons, electrons, photons, charged particles, and neutrinos. Originally the occurrence of spallation reactions was first recognized in astrophysics. Most of the early systematic research on spallation processes was done at highenergy accelerators. Cosmic ray physicists refer usually to the spallation reactions as fragmentation. But for all practical applications, spallation reactions are inelastic nuclear reactions in which at least one of the two reaction partners is a complex nucleus. Spallation research has become popular, funding and industrial participation were increasing with the advent of different spallation research projects and with the study and construction of high-intensity neutron spallation sources all over the world. This was also a result of a continued progress in accelerator physics and research. Applications will provide new opportunities in particle physics, solid-state physics, and life- and material science. A recent renascence of interest for energetic proton-induced production of neutrons originates therefore largely from the inception of projects of intense spallation neutron sources in Europe, USA, Japan, and China. In particular, the new powerful spallation sources could provide a basis for various potentially important future applications as neutrino factories and accelerator-driven transmutation technologies. Such facilities may be used to achieve the incineration and transmutation of long-lived radioactive waste or in combination with fission to produce energy. The objective of this book is not only to gain insights into the complex spallation reaction mechanisms itself, but also to summarize and identify the essential applications. The purpose of this handbook is to provide a description of methods, problems, and issues in spallation research, which will be useful to those readers entering the field and to those already engaged in spallation research, but more specializing in one area. The covered particle energy range in this book leads to a description of hadronic and electromagnetic phenomena over twelve orders of magnitude ranging from incident particle energies of several GeV down to the energy of subthermal meV neutrons of about 10−12 GeV, one of the important domains of spallation neutron sources for the investigation of the structural and dynamical properties of matter. Handbook of Spallation Research: Theory, Experiments and Applications. Detlef Filges and Frank Goldenbaum Copyright  2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40714-9

XVIII

Preface

The handbook has three main parts: Part 1 (Principles); Part 2 (Experiments); and Part 3 (Technology and Applications). Part 1, Principles, describes the theory of spallation reaction mechanisms in terms of the nuclear physics processes involved. For the first stage of the reactions – the fast phase – the emphasis is put on the description of intranuclearcascade models. Alternative models are also explained. For the second stage – the slow phase – evaporation models including high-energy fission, subsequent deexcitation, and fragmentation are incorporated. The state-of-the-art used models and codes are addressed with their different features, parameters, and constraints. The propagation of particle showers induced by the internuclear-cascade-evaporation phenomena and the particle transport in thick target systems is explained. The important features of simulation code systems for particle transport in matter are considered. Theoretical issues on material and shielding problems in spallation research are presented. The basic design parameters for neutron spallation sources are reviewed. Part 2, Experiments, describes the proton-nucleus ‘‘thin’’ target experiments in terms of secondary particle production such as hadrons, pions, light and intermediate masses, and isotope production performed at various accelerators. A variety of proton-matter induced ‘‘thick’’ target experiments on neutron yield, neutron leakage and multiplicity distributions as well as heating and energy deposition measurements are illustrated. The neutron production is also discussed for primary incident particles other than protons. Target–moderator–reflector experiments to study the neutron flux performance of spallation neutron sources, radiation damage investigations in a spallation environment, and high-energy neutron shielding experiments are characterized and specified. Many of the experiments have been associated with studies, investigations, and the design layout of spallation neutron sources until today. Part 3, Technology and Applications, describes the various issues associated with applications in spallation research. Proton drivers for secondary particle production, the features and parameters of high-intensity pulsed and continuous spallation neutron sources, e.g., LANSCE (USA), ISIS (UK), ESS (Europe), SNS (USA), J-PARC (Japan), and SINQ (Switzerland) as well as the concepts of neutron producing targets at beam power levels in the megawatt range are shown in detail. An overview about the research with neutrons is also given. The studies and projects on accelerator transmutation of radioactive waste from nuclear reactors, and the ideas of the ‘‘energy amplifying’’ using spallation reactions are considered. A broad spectrum of examples of advanced applications of spallation research as proton and light heavy ion therapy, high-energy physics calorimetry, neutrino factories, ultracold neutrons as well as the activation of space vehicles and hazard radiations in space are exemplified. In the appendix, useful values of fundamental physical constants and relations, and material properties of target, structure, moderator, reflector, and shielding materials are summarized. The idea of this book is to give a detailed overview on spallation research. Most of the work and research in this field is and was published in single articles

Preface

in journals and in reports of international or national institutions or presented at conferences during the last twenty years. A large amount of information was collected in national and international project reports of institutions working in spallation research, development, and applications in connection with construction and technical design of high-intensity spallation sources in the megawatt power range. Therefore, the authors of this book set a high value on describing the stateof-the-art of spallation research. This is also accomplished by a comprehensive bibliography. In general, we have tried to make the individual chapters independent of each other, even at the risk of repetition. Concerning the references, we have adopted the philosophy of quoting the original or the recent publications that we believe contained sufficient details to be useful for the uninitiated reader. We have preferred this method because the referenced material usually illustrates the current knowledge. The reader is well advised to use the references in the cited literature if supplementary details are required. We would like to acknowledge for permission to reproduce various photographs and figures cited in the text to the following laboratories: Los Alamos National Laboratory, Los Alamos, USA, Paul Scherrer Institut, Villigen, Switzerland, Oak Ridge National Laboratory, Oak Ridge, USA, Rutherford and Appleton Laboratory, Didcot, United Kingdom, and the High Energy Accelerator Research Organization of the Japan Atomic Energy Agency, Tokai-mura, Japan. We also owe a great debt to Prof. Dr. Hans Ullmaier from the Forschungszentrum Jülich GmbH, Germany, for many suggestions and improvements during the preparation of the manuscript. Last but not least, we acknowledge the excellent co-operation with Dr. Christoph v. Friedeburg as well as Nina Stadthaus and Esther Dörring of the publisher WILEY-VCH Weinheim, who contributed considerably to the success. Jülich, April 2009

Detlef Filges and Frank Goldenbaum

XIX

Part I Principles

3

1 The Spallation Process 1.1 Historic Remarks and Definitions

There is no generally accepted definition of the term ‘‘spallation reaction’’ although this type of nuclear reaction is observed in astrophysics, geophysics, radiotherapy, radiobiology, and at all applications together with accelerators. Cosmic ray physicists refer still to such reactions induced by cosmic rays as ‘‘fragmentation.’’ But for all practical purposes, spallation refers to nonelastic nuclear reactions that occur when energetic particles, for example, protons, neutrons, or pions interact with an atomic nucleus. So at least one of the two collision partners is a complex nucleus in which the available energy exceeds the interaction energy between nucleons in the nucleus. Thus, a nucleon–nucleus or pion–nucleus or nucleus–nucleus reaction, in which the incident energy exceeds some 10th of MeV per a.m.u. is referred to as spallation or spallation reaction. The term comes from the verb ‘‘to spall,’’ meaning to chip with a hammer [1]. There is no clear separation of spallation from the lower energy nuclear reactions. One type may merge into the other as the energy of the incident particle increases. (a) A definition found in Encyclopedia Britannica: ‘‘Spallation is a high-energy nuclear reaction in which a target nucleus struck by an incident (bombarding) particle of energy greater than about 50 million electron volts (50 MeV) ejects numerous lighter particles and becomes a product nucleus correspondingly lighter than the original nucleus. The light ejected particles may be neutrons, protons, or various composite particles equivalent.’’ (b) A definition for spallation to be specified in the context of accelerator driven systems or high-intensity neutron sources is ‘‘the disintegration of a nucleus by means of high energetic proton-induced reactions. Typically approximately 20 neutrons are created per incident GeV proton.’’ (c) A much shorter definition: ‘‘Spallation is a nonelastic nuclear interaction induced by a high-energy particle ≥50 MeV producing numerous secondary particles.’’ The following terminology is used here for clarification to describe the highenergy nuclear processes. The terms ‘‘inelastic,’’ ‘‘quasielastic,’’ ‘‘absorption,’’ and ‘‘nonelastic’’ describe usually reactions which are not pure elastic scattering from the nucleus as a whole. Absorption is in general used to describe all nonelastic Handbook of Spallation Research: Theory, Experiments and Applications. Detlef Filges and Frank Goldenbaum Copyright  2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40714-9

4

1 The Spallation Process

events when an optical analysis is made. The term inelastic is used in describing low-energy phenomena where scattered particles hold their identity but leave the nucleus in an excited state. The term ‘‘nonelastic’’ is used to describe all events which are not elastic scattering with the nucleus as a whole (see also page 44). Historically the idea of the basic reaction mechanism – the intranuclear cascade – was first proposed by Serber [2]. He suggested that at energies of about 100 MeV, the deBroglie wavelength of the incident particle becomes comparable to, or shorter than, the average internucleon distance within the nucleus 10−13 cm. That is the collision time between the incident particle and a nucleon inside the nucleus becomes short compared to the time between nucleon collisions inside the nucleus. Goldberger [3] was the first to perform calculations using Serbers’s model approach. (It was reported that two people working for 2 weeks were able to generate 100 Monte Carlo particle histories by hand calculations.) Metropolis et al. [4, 5] were the first to report extensive calculations using computers and generated large amounts of data for incident particles 2, and 2% electrons and positrons [16]. At high energies above several GeV the spectra shown in Figure 1.2 could be well represented within interstellar space by a power-law energy dependence: φi ∝ 1.8(E/A)−γ [nucleons/(cm2 s sr GeV)],

(1.2)

with φi the differential flux of element i at kinetic energy per nucleon (E/A) including rest mass energy and γ is in the range 2.5 to 2.7. The relative fractions of the primary and secondary incident nuclei are listed in Table 1.2. The spectra distributions at 1 AU of particles with A ≤ 4 are quite similar if the energies are taken per nucleon (E/A). They are characterized by a broad maximum of the differential spectrum between 100 and 1 GeV/A, and decrease by a power law for higher energies. Below ∼10 GeV/nucleon the spectra are effected by solar modulation or the solar magnetic field [17]. Galactic radiation arriving within the heliosphere is isotropic. The spectra of nuclei extend to energies in excess of about 1020 eV. Figure 1.4 shows the vertical fluxes and spectra of the major cosmic ray components in the atmosphere. Except for protons and electrons near the top of the atmosphere, all particles are produced in interactions of the primary cosmic rays with the air. Muons and neutrinos are products of the decay of charged mesons, while electrons and photons originate in decays of neutral mesons. Most of the measurements are made at ground level or near the top of the atmosphere.

7

8


Primary cosmic particles

Nuclear interactions with the Earth's atmosphere

p+ p−

k± k0

µ±

Hadronic cascade

µ+ nµ nµ µ− Muonic component and neutrinos

n

p p± k±

p±

k± k0

p0

µ±

g

e +e −

e +e −

e +g e −g

e +g e −g

Hadronic component nuclear fragments

Electromagnetic component

g

e +e −

Cherenkov and fluorescence radiation

Fig. 1.3 The development of the components of cosmic rays propagating in the earth’s atmosphere with kaon branches shown. Ratios of relative abundances of cosmic ray nuclei at 10.6 (GeV/nucleon) normalized to oxygen (≡1) with an oxygen flux of 3.26 × 10−6 (cm−2 s−1 sr−1 (MeV/nucleon)−1 ) (see Ref. [20]). Tab. 1.2

Z

Element

Abundance ratio

1 2 3–5 6–8 9–10 11–12

Ha Hea Li–Be C–O F–Ne Na–Mg

540 26 0.40 2.20 0.30 0.22

a

Z

Element

13–14 15–16 17–18 19–20 21–25 26–28

Al–Si P–S Cl–Ar K–Ca Sc–Mn Fe–Ni

Abundance ratio 0.19 0.03 0.01 0.02 0.05 0.12

The hydrogen and helium abundances are from [21].

According to [26] the modulated differential GCR proton flux can be described by the following formula:

(Ep , M) = Cp ·

Ep (Ep + 2mp c2 )(Ep + m + M)−2.65 −2 −2 cm s MeV−1 (1.3) (Ep + M)(Ep + 2mc2 + M)

1.2 Spallation by Cosmic Ray-Induced Reactions

Altitude [km] 15

104

10

5

3

2

1

0

Altitude flux spectra [m−2 s−1 sr−1]

103 nµ + n−µ µ+ + µ−

2

10

101

p+n

100

e+ + e− p+ + p −

10−1

10−2

0

200

400

600

800

1000

−2

Atmospheric depth [g cm ] Fig. 1.4 Vertical fluxes and spectra of cosmic rays in the atmosphere with energy E > 1 GeV estimated from the nucleon flux of formula 1.2 (see also [19]). The symbols show measurements of negative muons with energy E > 1 GeV [22–25].

with m = 780 · exp(−2.5 × 10−4 · Ep ), and the normalization factor Cp = 1.24 × 106 (cm−2 s−1 MeV−1 ), mp = 938.3(MeV c−2 ) the proton rest mass, c the velocity of light, Ep the proton energy in (MeV), and M (MeV) the modulation parameter of the sun. The solar modulation parameter M varies with the 11-year solar cycle. As shown in [26], small modulation parameters are typical for spectra during times of a quiet sun – 1965 M = 450 MeV and 1977 M = 300 MeV–, large modulation parameters for an active sun – 1969 M = 900 MeV. For the simulation of present-day irradiation processes, the arithmetic mean GCR spectrum from solar minimum (1965) with M = 450 MeV and solar maximum (1969) with M = 900 MeV may be adequate [27]. This spectrum is equivalent to a GCR spectrum with M = 675 MeV. An example for different values of M is shown in Figure 1.5 using formula (1.3) of [26]. The elemental composition of cosmic rays is an active area of research since the relative abundances provide tests of theories on the origin of the elements, or nucleosynthesis. Figure 1.6 shows the relative abundances of all elements at a typical energy of 2 GeV/nucleon. In radiation effect assessments, it is common to consider the cosmic ray ions in three groups:

9


M = 675

1.0 ×10−3 Proton flux distribution [cm−2 s−2 MeV−1]

10

1965 solar minimum at 1 AU

M = 450 1969 solar maximum at 1 AU 1.0 ×10−4

1.0 ×10−5 0.01

M = 900

0.1 1.0 Proton energy [MeV]

10.0

Fig. 1.5 Comparison of GCR cosmic ray spectra with different modulation parameters M = 450, 675, 900 MeV at 1 AU. (1 AU ≡ 1 astronomical unit = 1.49 × 1013 cm, the mean distance between the earth and the sun).

• H and He, • HZE ions (i.e., high charge number Z, and high-energy loss E, which includes ions having Z ≤ 30), • ultraheavy cosmic rays Z > 30. Since energy deposition by ionization is proportional to Z2 , the most important HZE ions in terms of the specific energy deposition density or specific dose density in materials are Fe (Z = 26) and to a lesser extent ions in the Z = 8–14 range. The idea of introducing spallation processes into astrophysics arose in 1970 [28], when it was realized that abundances of some elements in the cosmic rays are significantly different than those observed in the solar system. For isotopes of Li, Be, B, F, Cl, K, Sc, Ti, V, Cr, Mn the cosmic ray abundances exceed the solar ones by at least one order of magnitude and in an extreme case of Li, Be, B – so-called LiBeB puzzle – it is even six orders of magnitude. This is seen in Figure 1.7, where the production of Li, Be, and B by high-energy spallation reactions between cosmic rays and the interstellar medium induced on C, N, and O is compared to the abundances of products of the solar system (see [15, 29–31]). The underestimation of the abundances of elements with atomic mass numbers from 20 to 25 by three to four orders of magnitude is also observed. The most possible formation of the LiBeB production is nowadays thought to be interactions of galactic comic rays with the interstellar medium, mainly carbon, nitrogen, and oxygen nuclei, where the most abundant high-energy cosmic ray particles are protons and α-particles. Other possible origins are primordial, stellar, or supernova neutrino spallation (see Refs. [32, 33]). Stable and radioactive nuclides are produced by the interaction of solar and galactic cosmic ray particles with terrestrial and extraterrestrial matter. In extraterrestrial materials, such as meteorites and lunar surface material samples, these product nuclides are called cosmogenic nuclides. A limited number of cosmogenic nuclides is observable in the earth’s atmosphere due to its relatively simple chemical composition. In extraterrestrial matter more than 30 cosmogenic nuclides have been

1.2 Spallation by Cosmic Ray-Induced Reactions

Relative cosmic ray flux (normalized to Si = 106)

1010 H

109

C

107

O Si

106

Fe

105 104 103 102 Zr

101

Ba

Pt

100

Pb

10−1

Individual elements

10−2

Fig. 1.6

Abundances relative to silicon = 106

Nuclear composition of galactic cosmic rays (~2 GeV/nucleon)

He

108

Even-Z elements

Element groups

10 20 30 40 50 60 70 80 90 100 Atomic charge number [Z]

Relative particle flux of cosmic rays as a function of the nuclear charge Z.

He

109

O

C

107

Ne

Mg Si

N

Fe

S Ar Ca Ti

105

Cr

Ni

Na Al

103

B

F

P Cl

Mn

K

Co V

Li

101

Sc 70–280 MeV/nucleon cosmic rays (Simpson, 1983)

1

Be

0

5

Solar system elements (Lang, 1980)

10 15 20 25 Nuclear charge number [Z]

30

Fig. 1.7 Comparison of elemental abundances of 70–280 MeV/nucleon for cosmic rays (from Simpson [15]) and to the solar system abundances (from Cameron [29] and Lang [30]) normalized to (Si = 106 ).

observed. The production cross section of these nuclides with galactic cosmic or solar rays is an actual problem of modern cosmo-chemistry. To analyze the formation of the isotopic content of cosmic matter and its evolution under different radiation conditions, the knowledge of isotopic excitation functions is required. The observed

11

1 The Spallation Process 16O(p,x)7Be

14

Cross section [mb]

12

N(p,x)7Be

101

100

0.01

Exp. Mashnik et al. 1998

Exp. Mashnik et al. 1998

Bernas et al.1967

Bernas et al.1967

0.1

1

10 0.01

0.1

1

10

Proton energy [GeV]

Fig. 1.8 Comparison of cross sections used in astrophysical simulations (Mashnik et al. [41] and Bernas et al. [40]).

concentrations of cosmogenic nuclides in extraterrestrial matter could be used to determine the exposure history of the irradiated specimen or the history of the cosmic radiation itself (Michel et al. [34–36]). Theoretical and experimental investigations of cosmogenic nuclides in extraterrestrial matter yield information about spectral distribution, composition, and intensity of solar and galactic cosmic rays over time scales of up to 109 a. By no other means than by cosmic nuclides this information about the history of the solar system may be obtained. A large number of experiments were performed in the past to demonstrate the importance of excitation functions for the production of cosmogenic nuclides (see [37–39] and also Chapter 23 on page 653). In Figure 1.8, two examples of cross sections used in astrophysical simulations (solid lines) are compared with presently available data (open circles) [40, 41]. Meanwhile, many other reliable measurements were performed [38] and compared with simulations. But experiments are unable to provide all the necessary information. Some semiempirical systematics of nuclear the spallation reaction were evaluated in the 1960s called Rudstam systematics [42]. The Rudstam systematics were than further developed by Silberberg and Tsao [43–47], taking the validity of Serber’s model [2] into account. With the progress in spallation physics and the development of Monte Carlo particle reaction mechanisms and particle transport in matter during the recent years, the simulation of these reaction processes by using basic principles is possible nowadays. Experimental investigations, measurements, and validations, will be discussed and demonstrated in Part 2 on page 287.

1.3 Physics of the Spallation Process 1.3.1 Introduction

The production of secondary particles by bombarding the atomic nucleus by energetic particles was first demonstrated by Rutherford in 1919 [48], who was using α-particles on 14 N causing the reaction 14 N +4 He −→ 17 O + p. In 1934 Curie and

1.3 Physics of the Spallation Process Tab. 1.3

Examples for some neutron-producing mechanisms.

Nuclear process

400 keV D on T absorbed in titanium 35 MeV D on liquid Li 100 MeV e− on 238 U 235 U(n,f) 1000 MeV protons on thick Hg target Laser-ion beam imploded pellet

Technical application

Neutronproduction yield

Energy deposition (MeV/n)

D, T solid target

4 × 10−5 (n/d)

10.000

Deuteron stripping Electron bremsstrahlung Thermal fission spallation

3 × 10−3 (n/d) 5 × 10−2 (n/e) 3 (n/fission) 30 (n/proton)

10.000 2000 190 55

(D, T) CTR fusion

1 (n/fusion)

3

Joliot [49] produced the first artificial radioactivity using α-particles by the reaction 27 Al +4 He −→ 30 P + n. In 1932, the invention of the cyclotron by Lawrence and Livingston [50] and in 1939 the discovery of the nuclear fission by Hahn and Strassmann [51, 52] opened new possibilities for secondary particle production as high-intensity neutrons, radioactive isotopes, gammas, pions, and muons. In addition to fission, nuclear fusion of light elements produces secondary particles such as neutrons, produces energy, and radioactive species. The fusion of light nuclei (hydrogen isotopes) was first observed by Oliphant in 1933 [53]. The cycle of nuclear fusion in stars and the subsequent reactions were worked out by von Weizsäcker [54] and Bethe [55] in 1935/1939. Candidates for terrestrial reactions for neutron production are D + T −→ 4 He + n + 17.59 MeV and D + D −→ 3 He + n + 3.27 MeV or in the abbreviated form T(d, n)4 He and D(d, n)3 He, where the energy of the produced neutrons is 14.1 MeV and 2.45 MeV, respectively. It may be therefore very useful to compare several of the above-mentioned methods from the standpoint of possible mechanisms of neutron production. Features of several processes are shown in Table 1.3. Each of the processes, except the last one, has been used for the production of neutrons for certain purposes. Since power density limits must be considered, the energy deposition per neutron is a very important parameter. The usefulness of different reactions depends not only on the values of the given table, but also on other factors, such as the energy and angular distribution of the neutrons, the range of the charged particle production in charged particle reactions, the availability of suitable accelerators, and critical constraints in the case of fission reactors [56, 57]. In the case of the controlled thermonuclear reaction (CTR) the feasibility has still to be demonstrated. Together with one of the first projects to design and build a high-intensity neutron spallation source ‘‘ING’’ (Intense Neutron Generator) [58] experiments on thick targets were done to measure neutron production yields [59] and to evaluate the energy- or heat depositions. In Figure 1.9 these results for cylindrical thick targets of uranium and lead are shown.

13

Energy deposition/ neutron [MeV]


80

Uranium

60 40

Lead

20 0

Neutrons per incident proton

14

40 30

Uranium

20 Lead

10 0.0

0.2

0.4

0.6

0.8

1.0

Incident proton energy [GeV] Fig. 1.9 ING measurements on neutron production yields and evaluated heat production for cylindrical targets of uranium and lead of diameter = 10 cm, and length = 60 cm.

At an incident proton energy of 1 GeV the ratio of the neutron yields and the ratio of the heat per neutron for the targets of lead and uranium (diameter = 10 cm, length = 60 cm) are • yielduranium /yieldlead = 36.9/20.8 = 1.77, • heaturanium /heatlead = 50.1/25.7 = 1.95. A part of the higher neutron yield for uranium comes from the ‘‘extra’’-neutronsproduced fast fission processes in uranium. The increase in heat production per neutron for the uranium target is due to fast fission. For a proton beam power of 1 MW at 1 GeV incident proton energy, the neutron source strength and the heat deposition for these targets are given in Table 1.4. For a neutron source strength of about 1017 neutrons per second and a proton beam power of 1 MW, a lead target would produce about 0.33 MW and a uranium target 1.2 MW. Carpenter [56] used the measurements of Figure 1.9 to determine a semiempirical relation for the neutron yield y(E) production for incident proton beam energies 0.2 GeV ≤ E ≤ 1.5 GeV as follows: y(E) = 0.1(A + 20)(EGeV − 0.12) for 9 ≤ A ≤238 U,

(1.4)

y(E) = 50(EGeV − 0.12) for

(1.5)

and 238

U,

where A is the atomic weight (g/mole) and EGeV is the incident proton energy in GeV. These semiempirical formulas (1.4) and (1.5) are only valid for targets of diameter 10 cm and length 60 cm. Other target geometries and material compositions will

1.3 Physics of the Spallation Process Estimated source strength and energy deposition of ‘‘ING’’ targets of Figure 1.9 on the facing page.

Tab. 1.4

Lead target

Uranium target

Source strengtha Number of neutrons

1.3 × 1017

2.3 × 1017

Energy deposition (MeV) (MW)

3.3 × 1018 0.33

1.2 × 1019 1.2

a

Calculated for an incident proton beam energy of 1 GeV and 1 MW beam power = 1 GeV mA, where 1 mA ≡ 6.24 × 1015 protons/s and 1 W ≡ 1 J/s, and 1 eV ≡ 1.6 × 10−19 J.

produce different neutron yields, which will arise from secondary processes such as production or absorption, of particle leakage and may be dependent on the full development of the so-called hadronic cascade inside the target. 1.3.2 The Fission Process

Because fission is a well-known process in physics and technology, it may be interesting to discuss some similarities and different basic features between spallation and fission. In the case of fission, slow neutrons of thermal energy (En ≈ 0.024 eV), interact with metastable 235 U. The excited nucleus decays in a cascade of fission products of smaller atomic weight. These fragments, or fission products, are about equal to half the original mass. This so-called binary splitting of an excited nucleus into two approximately equal parts is still considered as one of the most interesting phenomena of collective motion of nuclear matter and as an excellent example of the nuclear multiparticle problem. According to the ‘‘liquid drop model’’ [60], regarding the nucleus as an incompressible liquid, a surface tension is responsible for the inner forces on all surface nucleons, which in turn makes the nucleus unstable against deformation. The nucleus breaks apart when the Coulomb repulsion between the two halves of the deformed nucleus equals the restoring force of the nuclear surface tension. Neutrons will occasionally be freed during the scission process, but are mainly released by evaporation from the fission fragments. The fragments are neutron rich and have an excess of internal energy. Details may seen in Refs. [61] and [62]. On average about 2.5 neutrons are produced by the fission of one 235 U nucleus. The sum of the masses of these fission fragments is less than the original mass. This ‘‘missing’’ mass (about 0.1% of the original mass) is converted into energy according to Einstein’s equation E = mc2 . The so-called produced fission neutrons possess very elevated energies of about 2 MeV and are thus unsuitable for inducing further fission processes. With the help of moderators

15

16


Neutron Nucleus U-235

Neutron with thermal energy

Fission product

Neutrons

Proton

Slow neutron Fig. 1.10

Fission product

Neutron

Neutron

Fission

Chain reaction

Schematics of fission processes.

the fast neutrons are slowed down to meV energies. These slow neutrons sustain the chain reaction in a nuclear reactor. A chain reaction refers to a process in which neutrons released in fission produce an additional fission in at least one further nucleus. This nucleus in turn again produces neutrons, and the process repeats. If each neutron releases two more neutrons, then the number of fissions doubles each generation. In that case, in 10 generations there are about 1024 fissions and in 80 generations about 6 × 1023 fissions, which is a mole of fissions. In Figure 1.10, the schematic of the fission process is shown. One should refer to Sections 1.3.3, 3.4.5, 3.7, and 3.7.1 for more details of fission and spallation and fission. The spectrum of neutrons obtained in a reactor has a broad energy distribution, reaching from 0.001 eV to over 10 MeV. It is characterized by the spectrum (E). (E)dE is the flux of neutrons between E and E + dE. Within this spectrum three different regions are distinguished: • E > 0.5 MeV – the region of fast neutrons. Here (E) (formula (1.6)) approximately follows the energy distribution of the neutrons produced in fission. The fission spectrum can be approximated by formula (1.7) (details may be seen also in Refs. [61, 63]) and is shown in Figure 1.11. • 0.2 eV < E < 0.5 MeV – the region of epithermal or resonance neutrons (formula (1.8)). In this energy range, the spectrum is mainly predominated by neutrons being slowed down by elastic collisions with the nuclei of the moderator substance. • E < 0.2 eV – the region of thermal neutrons. Thermal neutrons are nearly in thermodynamic equilibrium with the thermal motion of the moderator atoms. Their energy distribution can frequently be approximated by a Maxwell distribution (formula (1.9)). ∞ f =

(E)dE

(E > 0.5 MeV)

(1.6)

0.5 MeV

n(E) = 0.484 · e−E · sinh

√

2E

(MeV)

(1.7)

Neutron production n(E) [arbitrary units]

1.3 Physics of the Spallation Process

0.3

0.2

0.1

0.0 0.0

2.0

4.0 6.0 8.0 Energy of fission neutrons [MeV]

10.0

Fig. 1.11 The energy spectrum of neutrons produced in thermal neutron fission of 235 U using formula (1.7).

(E)dE =

epi dE E

(E)dE = th ·

(0.2 eV < E < 0.5 MeV)

E dE · e−E/kT · kT kT

(E < 0.2 eV).

(1.8) (1.9)

Here T is the absolute temperature and k is the Boltzmann constant, which is in fact a conversion factor between energy and temperature units. In terms of eV energy units it has a value of 8.617 × 10−5 (eV/K) with k = 1.38 × 10−23 (J/K) and 1 J 6.25 × 1018 (eV). The spectrum of fission neutrons could also be evaluated fairly well using the Maxwell distribution in the following form: n(E) =

2 · E 1/2 3/2

π 1/2 · ET

· e−E/ET .

(1.10)

Here ET is the kinetic energy and kT is corresponding to a certain ‘‘temperature’’ of the nucleons in the nucleus. Values for ET are 1.29 MeV for 235 U fission and 1.33 MeV for 239 Pu fission corresponding to temperatures of about 1.14 × 108 K and 1.18 × 108 K, respectively. The mean kinetic energy of the neutron distribution is E = (3/2) × ET here. The distribution of the energy released to the reaction products in nuclear fission is summarized in Table 1.5. 1.3.3 Spallation and Fission

In contrast to fission, the spallation process is not an exothermal process – energetic particles are necessary to enable the process. As mentioned in Section 1.1, spallation refers to nuclear nonelastic or inelastic reactions that occur when energetic

17

18

1 The Spallation Process Energy release of reaction products in nuclear fission. The fission reaction is therefore an exothermal process.

Tab. 1.5

Reaction product Fission fragments Neutrons Prompt γ s β-particles Decay γ s Neutrinos Sum

Released energy (MeV) 167 ± 5 5 6±1 8 ± 1.5 6±1 12 ± 2.5 204

subatomic particles – protons, deuterons, neutrons, pions, muons, etc. – interact with an atomic nucleus, which is usually referred to as ‘‘target’’ nucleus. In this context, energetic means kinetic energies of the primary incident particles on the target nucleus larger than several tens of MeV and more accurately, within the meaning of the validity of theoretical physical models, an incident energy of about 100–150 MeV. At these energies the deBroglie wavelength λ [64–66], e.g., of the proton, is only ≈ 10−13 cm (see Eq. (1.11)): λ = h/ 2 · mp · Ep (cm) = (h · c)/ 2 · mp · c2 · Ep ) (cm)

(1.11)

with mp = 938.2 (MeV/c2 ) the rest mass of the proton, Ep the energy of the proton in MeV, h = 6.626 × 10−34 (J s) = 4.136 × 10−21 (MeV s) the Planck constant, and h · c ≈ 1240 × 10−7 (eV cm). • For proton energies Ep = 100– 150 MeV, λ is about λ ≈ 2.8 × 10−13 –2.3 × 10−13 (cm), which is smaller as the size of the nucleus with about ≈ 10−12 (cm). • At an incident energy of Ep = 1000 MeV, λ is about λ ≈ 9.0 × 10−14 (cm). • For lower incident energies, e.g., Ep = 10 MeV, λ is ≈ 10−12 (cm), which is the size of the nucleus. In this case a nucleon does not interact with individual nucleons, but with the whole nucleus. At 100–150 MeV, the deBroglie wavelength λ is therefore short enough to allow the incident particle to interact with the individual nucleons inside the nucleus. As is discussed in more detail in Chapter 2, this is an important prerequisite for the application of particle nucleus collision models limiting their validity above some tens of MeV. Therefore, it is no longer correct that a spallation reaction is proceeding in the formation of a compound nucleus. The initial collision between the incident particle and the target nucleus leads to a series of direct reactions – the so-called intranuclear cascade – where individual nucleons or small groups of nucleons are ejected from the nucleus. Above a few GeV per incident nucleon fragmentation of the target nucleus may also occur. After the ‘‘intranuclear cascade’’ phase of


Target nucleus

Internuclear cascade

Intranuclear cascade

n

n

p

Incident particle

p a

≥ 150 MeV

n n d

High-energy fission n p n

n

g

Evaporation or de-excitation

n

Proton Neutron

Fig. 1.12

(n, xn), (n, f) Nuclear (n, g), (n, n p) reactions (n, n′a) etc.

The principal scheme of spallation.

the reaction the nucleus is left in an excited state. This is shown in principle in Figure 1.12. During the first stage – cascade/pre-equilibrium stage – the incident particle undergoes a series of direct reactions with the nucleons–neutrons and protons – inside the target nucleus, where high-energy secondary particles such as protons, neutrons, and pions from 20 MeV up to the energy of the incident particle are created in an intranuclear cascade inside the nucleus. From the intranuclear cascade some of these high-energy hadrons escape as secondary particles. Also low-energy pre-equilibrium particles in the low MeV energy range are ejected from the nucleus leaving the nucleus in a highly excited state. In the second stage nuclear de-excitation or evaporation takes place, when the excited nucleus relaxes by emitting low-energy ( 5. This evaluation is based mainly on Ref. [90]. This formula as given by Eq. (1.47) is used as a basis for the hadronic package GEANT4 [100]. For the most important isotopes in accelerator-driven technologies (ADT) such as iron, tungsten, and lead, Prael and Chadwick evaluated the total-nonelastic cross section for neutrons and protons up to incident energies of 2 GeV [110–112]. These cross-section evaluations are applied in the LAHET code [113] and validate also the

45

1 The Spallation Process 2600

2400 Nonelastic cross section [mb]

46

2200 Uranium 2000 Lead 1800

Mercury Tungsten

1600

1400 200

400

600

800 1000 1200 1400 Incident proton energy [MeV]

1600

1800

2000

Fig. 1.26 Calculated semiempirical total-nonelastic cross sections for W, Hg, Pb, and U using formula (1.46). Tab. 1.14

Errors in the energy-dependent formula (1.46).a

Energy range

Estimated maximum error (%)

E > 1 GeV 300 MeV < E ≤ 1 GeV 100 MeV < E ≤ 300 MeV 40 MeV < E ≤ 100 MeV 10 MeV < E ≤ 40 MeV a

2 5 10 20 ∼ a factor of 2

Mean errors are about 0.5 times the maximum errors

semiempirical formula of [91]. σtotal-nonelastic ∼ σreaction = fcorr · π · r02 · ln N · [1 + A1/3 − b0 (1 − A−1/3 ), (1.47) with N the number of neutrons in the target, A the atomic mass, and b0 and r0 defined as published in [90]: b0

=

2.247 − 0.915 · (1 + A1/3 ),

r0

=

1.36 fm = 1.36 × 10−13 cm,

fcorr

=

1+0.15·exp(−Ekin ) . 1+0.0007·A

(1.48)


Uranium

Tungsten

1

1

10−3

10−2

10−1

1

10−3

10−2

Cross section [barn]

Lead

1

Mercury

1

10−3

10−1

1

10−2

10−1

1

10−3

10−2

10−1

1

Aluminum

Iron 1 1 − snonelastic

− snonelastic − selastic

10−1

− selastic 10−1 10−3

10−2

10−1

1

10−3

Proton energy [GeV]

10−2

10−1

1

Proton energy [GeV]

Fig. 1.27 Representative examples of total-elastic and totalnonelastic proton nucleus cross sections for Al, Fe, W, Hg, Pb, and U as a function of the incident proton energy in the MeV–GeV range calculated by GEANT4 [100].

In Figure 1.27 examples of total-elastic and total-nonelastic cross sections as determined by GEANT4 for Al, Fe, W, Hg, Pb, and U target materials of relevance for spallation reaction applications are depicted. A very useful source of evaluated experimental values as total cross sections, elastic cross sections, and nonelastic cross sections of interaction of neutrons and protons for many different target materials from deuterium to uranium nuclei is given in Refs. [87, 108, 109]. Only published data have been included. There are clearly gaps in energy and in target material in the data. Representative examples of the data are shown in Figures 1.28, 1.29, 1.30. The uncertainty of the measured data is mostly about 1σ = ±10% at incident proton energies E ≥ 200 MeV and is within the size of the plotted symbols. Measurements of the total reaction cross section in connection with the development of high power spallation sources were recently

47


Total cross section [mb]

104

103

- U - 238 - Pb - 207.8 - Fe - 55.9 - Al - 27 - C - 12

102 102

103 Incident proton energy [MeV]

104

Fig. 1.28 Examples of experimental values – uncertainty about 1σ = ±10% – of proton-induced total cross sections from [108, 109].

Total nonelastic cross section [mb]

48

103 9 ×102 8 ×102 7 ×102 6 ×102 5 ×102 4 ×102 3 ×102 2 ×102 Fe - 55.8 Al - 27 C - 12 102 100

101 103 102 Incident proton energy [MeV]

104

Fig. 1.29 Examples of experimental values – uncertainty about 1σ = ±10% – of proton-induced total-nonelastic cross sections of C, Al, and Fe from [108, 109].

reported in [114, 115]. These measurements give similar results as summarized in [87, 108, 109] (Table 1.15). As an example, the elastic scattering cross sections for protons are shown for the materials C, Al, Fe, and Pb in Figure 1.31 evaluated from Cloth et al. [88] (cf. elastic scattering of neutrons and protons in Chapter 4, Section 4.5).


Total nonelastic cross section [mb]

104

103

102

U - 238 Pb - 207.8 Ta - 180.9

101 100


104

Fig. 1.30 Examples of experimental values – uncertainty about 1σ = ±10% – of proton-induced total-nonelastic cross sections of Ta, Pb, and U from [108, 109]. Tab. 1.15 Comparison of measured and theoretical evaluated nonelastic cross sections at an incident proton energy of E = 1.2 GeV.

References

Barashenkov [108] Enke et al. [114] Letourneau et al. [115] Letawa et al. [107] Wilsonb et al. [116] Cugnon model INCL 4.3 [117]

a b

Target element

Total reaction cross section/ Total nonelastic cross section (mb) ± 1σ (mb)

Pb U Pb U Pb U Pb U Pb Pb U

1713 ± 103 1975 ± 119 1780 ± 125 1730 ± 121 1650 ± 30 − 1850 ± 16 2090 ± 45 1693 ± 150 1801 ± 16 2013 ± 45

Semiempirical formula. Calculations by means of a high-energy heavy-ion model.

1.3.7.5 Total Reaction Cross Sections in Nucleus–Nucleus Reactions The transportation of light and heavy ions in matter is of much interest in science. The total reaction cross sections have been studied both experimentally and theoretically. Several empirical parameterizations as derived for nucleus–nucleus

49


104 Total nonelastic cross section [mb]

50

103

102

Pb - 208 Fe - 56 Al - 27 C - 12

101 101


105

Fig. 1.31 Examples of proton-induced total elastic cross sections – uncertainty about 1σ = ±10% – for C, Al, Fe, and Pb evaluated in [88, 116] in the energy range from 0.1 to 22.5 GeV.

collisions (cf. Section 1.3.7.4) are given in the references by Sihver et al. [90], Kox et al. [118], and Tripathi et al. [119–121] and the citations therein. As given in Section 1.3.7.4, the total reaction cross section σ (N, N)R is defined as σ (N, N)R = σtotal − σelastic

(1.49)

with σtotal the total cross section and σelastic the elastic cross section. The Sihver formula The formula of Sihver et al. [90] is in its simplest form similar to Eq. (1.46), the formula of Letaw et al. [107]:

2 1/3 1/3 −1/3 −1/3 , σ (N, N)R = π · r02 Aprojectile + Atarget − b0 Aprojectile + Atarget

(1.50)

where Aprojectile and Atarget are the atomic mass numbers of the projectile and the target nuclei, respectively, r0 = 1.36 fm and b0 is given by

−1/3 −1/3 b0 = 1.581 − 0.876 Aprojectile + Atarget . 1/3

(1.51) 1/3

The formula consists of a geometrical term (Aprojectile + Atarget ) and a transparency parameter b0 for the nucleons in the nucleus. It is assumed that the cross section is independent of incident energies ≥ 100 MeV/nucleon. The formula gives a good agreement with experimental data for most of the collision systems, which have been studied (see Ref. [90]). In the case of nucleon–nucleus interactions, formula (1.50) is almost identical for proton–nucleus (with Ztarget ≤ 26) if Aprojectile = 1 is a proton and the parameter b0

1.3 Physics of the Spallation Process −1/3

is expressed as a polynomial function of the first order in (1 + Atarget ). The formula then becomes 1/3 −1/3 2 σ (p, N)R = π · r02 1 + Atarget − b0 [1 + Atarget ]

−1/3 (1.52) b0 = 2.247 − 0.915 1 + Atarget for incident proton energies Eproton ≥ 200 MeV and r0 = 1.36 fm. The Kox formula The Kox formula in Ref. [118] is based on the strong absorption model to describe low-energy nuclear reactions. The formula is based on the analysis of total reaction cross sections for heavy ion collisions in the intermediate-energy range of about 10–300 MeV/nucleon. The formula determines the total reaction cross section σ (N, N)R in terms of the interaction radius IR, the nucleus–nucleus interaction barrier BC , and the center-of-mass energy Ec.m. of the colliding system. In this framework it is considered that a reaction occurs whenever a substantial contact occurs between nuclear matter. The formula is given in its general form as

σ (N, N)R = π(IR)2 [1 − BC /Ec.m. ].

(1.53)

The parameter BC is the Coulomb barrier of the projectile–target system. It is given by

1/3 1/3 BC = Zprojectile · Ztarget · e2 / rC (Aprojectile + Atarget ) , (1.54) where rC = 1.3 fm, e is the electron charge, Zprojectile and Ztarget are the atomic numbers of the projectile and the target nuclei, and Aprojectile and Atarget are the atomic masses of the projectile and the target nuclei, respectively. The interaction radius IR is divided in the Kox formula into volume and surface terms, Rvol and Rsurf : IR = Rvol + Rsurf .

(1.55)

The terms Rvol and Rsurf correspond to energy-independent and energy-dependent components of the reactions, respectively. Collisions with smaller impact parameters are independent of energy and mass number and are characterized as a volume component of the interaction radius and therefore depends only on the volume of the projectile and the target nuclei. It is given by 1/3

1/3

Rvol = r0 (Aprojectile + Atarget ).

(1.56)

The second term of the interaction radius IR in Eq. (1.55) is a nuclear surface contribution and is parameterized by   1/3 1/3 Aprojectile · Atarget Rsurf = r0 a · 1/3 − c + D. (1.57) 1/3 Aprojectile + Atarget

51

52


The first term in the brackets is the mass asymmetry term, which is related to the volume overlap of the projectile and the target. The second term c is an energydependent parameter, which takes into account the increasing surface transparency as the projectile energy increases. The parameter D is the neutron excess which is important in collisions with heavy or neutron-rich targets. It is given by D=

5(Atarget · Ztarget )Zprojectile . Aprojectile · Atarget

(1.58)

The parameters r0 = 1.1 fm and a = 1.85 are fixed values where the parameter c is a function of the beam energy per nucleon which is a simple analytical function derived from Ref. [118] and used in GEANT4 [100] to 10 c = − 5 + 2.0 x

10 x 3 c = − 5 + 2.0 · 1.5 1.5 x = log(Ekin ),

for x ≥ 1.5, for x < 1.5, (1.59)

where Ekin is the kinetic projectile energy in units MeV/nucleon in the laboratory system. The Tripathi formula For nucleon–nucleon and nucleus–nucleus interactions Triphati et al. [119–121] have proposed alternative algorithms for calculating the interaction cross sections. The formula is a simple universal parameterization of the total reaction cross section for any system of colliding nuclei valid for the entire energy range from a few A MeV to a few A GeV. This approach treats the proton–nucleus collision as a special case of the nucleus–nucleus collision, where the projectile has charge and mass number 1. Parameters to treat the Coulomb interaction at lower energies and modifications of the reaction cross section at higher energies due to the Pauli blocking are also taken into account:

2 1/3 1/3 σ (N, N)R = πr02 · Aprojectile + Atarget + δE · 1 −

B Ec.m.

,

(1.60)

where r0 = 1.1 fm. In formula (1.60) the parameter B, the energy-dependent Coulomb barrier, and R are given by B = 1.44 ·

Zprojectile · Ztarget , R

R = rprojectile + rtarget +

1/3

1/3

1.2 · Aprojectile + Atarget 1/3

Ec.m.

(1.61)

,

(1.62)

where ri is the equivalent sphere radius and is related to the rrms,i radius by ri = 1.29 · rrms,i with i = projectile, target), and Ec.m. , the center of mass energy, is given in MeV.


There is an energy dependence of the reaction cross section at intermediate and higher energies mainly due to two effects, the transparency and the Pauli blocking [119]. This is represented by the energy-dependent term δE in formula (1.60) given as δE = 1.85 · S1 + 0.16 1/3

S1 = S2 =

S1 1/3

Ec.m.

− CE + 0.91 · S2

1/3

Aprojectile · Atarget 1/3

1/3

Aprojectile + Atarget (Atarget − 2Ztarget )Zprojectile , Aprojectile · Atarget

(1.63)

where S1 is the mass asymmetry term and is related to the volume overlap of the collision system and the term S2 in the formula is related to the isotope dependence of the reaction cross section. The term CE in formula (1.63) is related to the transparency and the Pauli blocking and is given by E E CE = DPauli · 1 − exp − − 0, 292 · exp − · cos 0.229 · E 0.453 . 40 792 (1.64) The parameter DPauli in formula (1.64) takes into account the density dependence of the colliding system, scaled with respect to the density of the (12 C +12 C) colliding system. DPauli = 1.75 · (ρAprojectile + ρAtarget )/(ρAC + ρAC ),

(1.65)

where the density of a nucleus using the hard sphere model [122] with a given nucleus of mass number Ai is given by ρAi = Ai /

3 π · ri3 , 4

(1.66)

where the radius of the nucleus ri is defined above with the root-mean-square radius, (ri )rms , obtained directly from experimental data [123]. With the parameter DPauli simulates the modifications of the reaction cross section caused by the Pauli blocking which was introduced by Tripathi et al. [119] in the parameterization formula for the first time. At lower energies where the overlap of interacting nuclei is small and Coulomb interaction modifies the cross sections significantly, the influence of the Pauli blocking is small. The modification of the reaction cross section due to Pauli blocking plays an important role at energies above 100 Mev/nucleon in nucleus–nucleus collisions which lead to higher densities. For proton–nucleus collisions where the compression effect is low, a single constant value DPauli = 2.05 gives good results for all proton–nucleus collisions [119].

53

1 The Spallation Process 2500

1000 800 600 400 200 0 (a)

p+

27

p + 13 Al sabsorption [mb]

sabsorption [mb]

54

2000 1500 1000 500 0

1

10

100

1000

Proton energy [MeV]

208 82 Pb

1

(b)

10

100

1000

Proton energy [MeV]

Fig. 1.32 Absorption cross section for proton–aluminum collisions (a) and proton–lead collisions (b) as a function of the incident proton energy. The experimental data points are from de Vries et al. [123]. The solid lines are the results of calculations given in Ref. [120] (after Tripathi et al. [120]).

For alpha–nucleus and lithium–nucleus collisions, where there is also little compression, the following terms for the DPauli parameter are useful: DPauli (α) = 2.77 − (8.0 × 10−3 · Atarget ) + (1.8 × 10−5 · A2target ) − 0.8/[1 + exp[(250 − E)/75]] DPauli (lithium) = DPauli /3

(see Eq. (1.65)).

(1.67)

Figure 1.32 shows as an example the results of calculations for the reaction cross sections for protons on Al and Pb targets of Ref. [120]. The experimental data have been taken from the compilation of Ref. [123]. The Tripathi formalism gives a very good agreement with the experimental data also for other collision systems for the entire energy range1) . 1.3.7.6 Differential Cross Sections As mentioned earlier, the cross section is an extremely important concept in describing the interaction of particles with matter. By definition the angular differential cross section dσ/d is the derivative of σ to the solid angle and is proportional to the probability of a reaction of an incident particle with a nucleus by emitting a secondary particle into a cone d. The differential cross section has the dimension barns/steradian (b/sr). Note the condition (dσ/d)d = σ , which makes it clear why dσ/d is called the angular differential cross section. Another condition is to consider the incident particle to have a certain energy E and particles after interaction to have energy in an interval dE about E . This defines in a similar way as above an energy dependent differential cross section, dσ/dE , which is a measure of the probability that an incident particle with incident energy E will have as a result of the interaction an outgoing energy E . Both dσ/d and dσ/dE are distribution functions; the former is a distribution in the variable , the solid angle, whereas the latter is a distribution in E , the energy after scattering. Their dimensions are barns per steradian and barns per unit energy, respectively. 1) See comment in Section 2.5.1 on page 115.


Combining the two extensions above from the cross sections to differential cross sections, it follows a further extension to the so-called double differential cross section: d2 σ/d dE(b sr−1 MeV−1 ),

(1.68)

which is a quantity that has been extensively studied in particle scattering. This cross section contains the most fundamental information about the structure and dynamics of particle–nucleus or particle–matter interactions and collisions. Measurements of double differential production cross sections provide a very sensitive observable for the validity of the spallation physics models. In the past systematic studies and measurements were investigated as part of the development of intense spallation neutron sources, for a new generation of meson factories, and for the theoretical understanding of medium energy physics in the energy range up to several GeV [124–132]. For ADS (accelerator-driven systems) projects [133] in references on the interaction of protons and neutrons mainly for lead targets are summarized. The experiments are discussed in Part 2 on page 277. There is still interest in medium-energy particle reaction data as double differential secondary production cross sections of charged particles, neutrons, pions, radioactive isotopes, etc., mainly for incident particles above 100 MeV. Some examples of semiempirical systematics on neutron spectra measurements [129, 130] from high-energy proton bombardment by 318, 590, and 800 MeV protons on eight targets in the mass range A = 12 to 238 were considered by Pearlstein [134] to illustrate the general characteristics of the neutron emission spectra. In Figures 1.33 and 1.34, two examples of measurements of double differential neutron production cross sections at the Los Alamos WNR facility by Amian et al. [129, 130] (in angles 30◦ , 60◦ , 120◦ , and 150◦ ) by 800 MeV protons on targets of Fe and Pb are shown. More details of these experiments are given in Part 2. The cross-section curves are seen to be very much alike, showing a broad maximum at ∼2–3 MeV, followed by a rapid fall of about above ∼20 MeV. The two different main components in the spectra are attributed to evaporation neutrons dominating the range below ∼20 MeV, and the cascade neutrons governing the shape above ∼20 MeV. The double differential cross sections are increasing with increasing target mass over the entire energy range and for all neutron emission angles. It can be seen from these examples that the cross sections below about 20 MeV for all angles widely coincide, indicating an almost isotropic neutron emission. In contrast to the evaporation region, the neutron production cross sections in the cascade region are strongly angular dependent, and forward peaking is evident. While the cross sections around 20 MeV are still of about the same order for all angles, the cascade component tails off rapidly, increasing with increasing emission angle. The general features of the angular dependent cross sections – their anisotropy – are qualitatively very similar for most of the target masses. The total uncertainties of the experimental results presented here are determined by ±1σ = 10–15% and thus are of the order of the data point sizes. The aboveshown experimental results are based on the so-called time-of-flight method (TOF), which will be described in more detail in Part 2 on page 289.

55


d2s /dΩdE cross section [b sr−1 MeV−1]

100 10−1 10−2 10−3 Fe Ep = 800 MeV - 30°

10−4

Fe Ep = 800 MeV - 60°

10−5

Fe Ep = 800 MeV - 120°

10−6

Fe Ep = 800 MeV - 150°

10−7 10−8 10−1

100

101 102 Neutron energy [MeV]

103

Fig. 1.33 Examples of proton (800 MeV) + Fe-induced experimental double differential neutron production cross sections at 30◦ , 60◦ , 120◦ , and 150◦ – measurements of Amian et al. [129, 130].

100 d2s / dΩdE cross section [b sr−1 MeV−1]

56

10−1 10−2 10−3 Pb Ep = 800 MeV - 30°

10−4

Pb Ep = 800 MeV - 60°

10−5

Pb Ep = 800 MeV - 120°

10−6

Pb Ep = 800 MeV - 150°

10−7 10−8 10−1

100


103

Fig. 1.34 Examples of proton (800 MeV) + Pb-induced experimental double differential neutron production cross sections at 30◦ , 60◦ , 120◦ , and 150◦ – measurements of Amian et al. [129, 130].

Although the neutron production channel in spallation collisions is the dominant one, other secondary particle production channels as the production of secondary protons, pions, or light charged, intermediate mass fragments (IMFs), and heavy residuals (cf. Figure 1.12 on page 19) are at least also important in model predictions and applications (cf. Part 2 on page 277 and Part 3 on page 495).


1.3.8 Hadronic–Electromagnetic Cascade Coupling

It should be noted that often both hadronic and electromagnetic cascades are important in the same problem whether the source is hadrons – protons or neutrons – or photons, electrons, and positrons. These processes are mainly a function of secondary particle production and their decay during the propagation of an intra- or internuclear-cascade in the target nucleus or in a so-called thick target, respectively. The ‘‘coupling’’ of hadronic and electromagnetic cascades may occur in the following way. The process may be initiated for example by either hadrons or leptons as incident particles on a target nucleus or a thick target: starting with hadrons, e.g., protons or neutrons: proton/neutron

+

nucleus/target −→

π 0 −→ 2 · γ

−→

π ± −→ µ± −→ e±

−→

k0 , k± −→ µ± , π ± , π 0 −→ e± , γ (1.69)

starting with leptons, e.g., electrons or positrons: electron

+


γ , e± +

γ , e+ , e−


hadron–meson cascade

(1.70)

The decay scheme and the particle properties have already been discussed in Section 1.2 in Eqs. (1.1) and in Table 1.1. A substantial fraction of the energy deposited in a hadron cascade is a direct result of electromagnetic cascades produced by the decay of neutral pions π 0 (mean life ∼ 10−16 s) into two photons, by low-energy photons produced during the intra- and internuclear evaporation process or by the decay of charged pions and muons. There is a large class of hadron cascade applications and issues which require the handling of electromagnetic cascades, their shower propagation, their energy deposition, and the production of hadrons and other secondary particles. The principal production processes during the development of electromagnetic showers (bremsstrahlung by production of electrons and positrons, pair production for photons) are well described by quantum electro dynamics (QED) theory. It should be mentioned that at higher incident hadron energies the hadron production rate, e.g., for protons and neutrons, deviates from the linear rule due to the increase of π 0 production and the subsequent 2γ decay into the electromagnetic channel the so-called electromagnetic drain of the hadron cascade. As mentioned above, the mean half-life of π 0 is with ∼10−16 s very short; therefore, this fast

57


decay does not allow π 0 to take part in the internuclear cascade whereas the π ± do. The π ± decay time of 26 × 10−9 s is sufficiently long to allow further hadronic interactions once the π ± are created. At higher incident energies above 10 GeV, other meson production channels will be opened which in addition deplete the cascade of energy. Applications where such showers are observed are e.g., high-intensity spallation sources, high-energy physics calorimeters, shielding of accelerators, or the impact of space radiation on space vehicles. Details are given in Part 3 on page 495. Besides the general way of spallation reactions induced by high-energy protons, hadron production, mainly neutrons, will also take place in target materials by electrons or bremsstrahlung photons. It should be noted that it is the photon interacting with the target material that releases the hadrons, rather than the direct interaction of the electrons. Several types of photonuclear interactions result in hadron production such as the so-called giant resonance production with photons of energy Eγ ≈ 10–30 MeV, photodisintegration with photons of energy Eγ ≈ 50–100 MeV, and photopion absorption with photons of Eγ about ≥ 140 MeV. Figure 1.35 shows for small incident photon energies the giant-resonance peaks for Cu and Pb and above the pion production threshold of about 200 MeV the photo–pion production cross section as the average of proton and neutron. Above 140 MeV, the cross section for photons on nuclei rises again, opening the channels for photopion production. The cross section is characterized by a number of resonance peaks below and above of about 1 GeV which are caused by nucleon isobar formation (see Section 1.3.5.3). Between these two energies, e.g., 10 and 101 Production cross section per nucleon [mb/nucleon]

58

Giant resonance s(g, n)

Pb Total neutrons

100

Cu Ta 10−1

s (g, p) (Z = 13 - 47) Pb

s QD(L = 5) Cu

10−2

101

103 102 Incident photon energy [MeV]

Fig. 1.35 A qualitative picture of the photoneutron production mechanism given as the cross section per target nucleon as a function of the incident photon energy (after Swanson [135]).

104


100 MeV, the behavior for medium Z-targets is shown as hachures [136], and a calculation for Ta [137] based on a intranuclear cascade model. Also shown for this energy region is the photoneutron cross section derived from a simple quasideuteron model, assuming N = Z = A/2, L = 5, where A, Z are for the target nucleus, and L is a dimensionless coefficient (Eq. (1.71)). The quasideuteron effect above the giant resonance is the remaining dominant neutron production mechanism in which the photon interacts initially with a neutron–proton pair within the nucleus, rather than with the nucleus as a whole, hence the name quasideuteron. As Figure 1.35 shows, this cross section is an order of magnitude below the giant-resonance peak. The cross section is related to the deuteron photodisintegration cross section σD (E), qualitatively given by [138] σQD (E) L ·

NZ · σD (E), A

(1.71)

where N, A, and Z refer to the target nucleus, and L (between 3 and 13) is a dimensionless coefficient as a measure of the probability that a neutron–proton pair is within a suitable interaction distance relative to the deuteron [139]. The overall effect is to add a tail of higher energy neutrons to the giant-resonance spectrum. Electrons and positrons lose their energy by ionization as charged particles do. However, because of their small mass, they have significant losses also due to the production of radiation. For lead, the fractional energy loss due to bremsstrahlung exceeds that due to ionization for electron energies above 10 MeV, which is shown in Figure 1.36. Other significant energy loss mechanisms are elastic scattering described by the Bhabha differential cross section for electrons and the Møller formula for positrons, and positron annihilation. The dominant energy losses for high-energy electrons are bremsstrahlung and pair production which lead to the production of electromagnetic radiation or electromagnetic showers by the motion of electrons and positrons through matter. The well-known physical processes are described in detail standard textbooks such as, e.g., in Ref. [140]. Electrons and positrons have similar electromagnetic interactions in matter. Figure 1.36 shows the fractional energy loss per radiation length in Pb as a function of electron or positron energy. Electron and positron scattering is considered as ionization when the energy loss per collision is below 2.55 MeV, and as Møller or Bhabha scattering when it is above. At low energies electrons and positrons lose energy essentially by ionization, although other processes (Møller and Bhabha scattering, electron annihilation) contribute as shown in Figure 1.36. The ionization losses decrease exponentially with energy, bremsstrahlung losses rise nearly linearly (the fractional loss is nearly independent of energy), and dominates above a few tens of MeV in most materials. The radiation length X0 , usually measured in (g/cm2 ), has been calculated by Tsai [141] and is provided by an approximation formula in Ref. [19]. X0 =

716.4 · A g cm−2 , √ Z · (Z + 1) · ln(287/ Z)

(1.72)

59


Positrons

0.20 Lead Z=82

Electrons

0.15

1.0 Bremsstrahlung

Ionization

0.10

[cm2 g−1]

−1

−1/E dE/dx [X0 ]

60

M ller (e−)

0.5

Bhabha (e+)

0.05

Positron annihilation

0

1

10 100 Particle energy [MeV]

1000

Fig. 1.36 Fractional energy loss per radiation length X0 in lead as a function of electron or positron energy (after Yao et al. [19]). Tab. 1.16 Calculated radiation lengths X0 of Be to U using Eq. (1.72).

Medium

Z

A

Radiation length (g cm−2 )

Be C Al Fe Cu Ta W Au Hg Pb U

4 6 13 26 29 73 74 79 80 82 92

9.01 12.01 26.98 55.85 63.55 180.95 183.84 196.97 200.59 207.2 238.03

65.19 42.70 24.01 13.84 12.86 6.83 6.76 6.42 6.39 6.31 6.00

where A and Z for the target material. In Table 1.16 calculated radiation lengths X0 are given for some materials using Eq. (1.72). It should be mentioned that the basic features of the energy loss already discussed for heavy particles in Section 1.3.6 are also valid for electrons and positrons, but the Bethe–Bloch formula (1.27) must be modified for several reasons: Electrons and positrons have a relatively small mass compared to heavy charged particles, at small incident energies the total energy loss is also determined by bremsstrahlung losses, and for electrons and positrons the collisions take place between identical particles, which lead to a indistinguishability. Also positron annihilation must be taken into


account (e+ + e− → γ + γ ). A modified Bethe–Bloch formula for (e+ + e− ) can for example be found in Ref. [140] As already discussed during the 1980s in Ref. [142], high power target systems of spallation sources or high-intensity proton accelerators are at the same time a source of charged pions and muons which decay into neutrinos ν and antineutrinos ν. Thus, such facilities can also provide a source for neutrino research in a so-called neutrino factory. There are currently two kinds of intense neutrino of factories discussed. 1 The neutrino production mechanism from pions and muons: The mesons π ± decay via the weak interaction with an average lifetime of about τ = 2.55 × 10−8 s in leptons:

π + −→ µ+ + νµ or π

+

π

−

−→ e+ + νe

branching ratio = 99.9877%

branching ratio = 1.2 × 10−4 %

and −→ µ− + ν µ

or π − −→ e− + ν e .

(1.73)

Muons µ± have an average lifetime of about τ = 2.2 × 10−6 s and decay via the following scheme: µ− −→ e− + ν e + νµ and µ+ −→ e+ + νe + ν µ

branching ratio ≈ 100% .

(1.74)

2 The decay of stored β-active emitters instead to produce neutrinos via the decay of pions and muons: The neutrino production with such emitters as 62 He and 18 10 Ne is as follows: 6 2 He

−→ 63 Li + e− + ν e , with a half-life and

18 10 Ne

+ −→ 18 9 F + e + νe , with a half-life

T1/2 = 0.8 s

T1/2 = 1.7 s .

(1.75)

The physics applications of the neutrino research are primarily neutrino oscillation physics and CP violation studies, but also measurements of cross sections of neutrino nucleus interactions. An example of a spallation neutron source for neutrino research is the KARMEN experiment at the Rutherford Laboratory (The KArlsruhe Rutherford Medium Energy Neutrino Experiment) [143]. Other projects on neutrino factories can be found in Refs. [144, 145] and references therein. Experiments at existing spallation sources and the production of different parent ions for neutrino beams will be discussed in Part 3 on pages 585 and 638.

61

63

2 The Intranuclear Cascade Models 2.1 Introduction

The most important physics uncertainty associated with predicting the interaction of high-energy hadrons with matter lies in determining the multiplicities, energy, and angular distributions of particles produced in nonelastic collisions. Theoretical models, e.g., the inter/intracascade-evaporation model, were developed over the past 30 years, with considerable success. The basic assumption of most methods employed in the models is that the interactions of high-energy particles with the nucleus can be represented by free particle–particle collisions inside the nucleus, an approach, as already mentioned on page 3, first suggested by Serber [2]. The important justification for this assumption is that the wavelength of the incident particle is of the order of the internucleon distance ∼10−13 cm. Details were given in the previous section and in Refs. [64–66]. Goldberger [3] was the first to perform calculations using Serbers’s model approach. The physical process, approximately described by free particle collisions within the nucleus is called the cascade. The class of models are named as intranuclear-cascade (INC) models. The cascade process is usually followed by subsequent physical processes given in terms of evaporation, pre-equilibrium, fission or de-excitation processes in photons, etc., which will be described in Chapter 3 on page 135. To evaluate the physical observable in phase space the models are well suited to a statistical approach using Monte Carlo techniques and kinematics of high-energy particles [146–150]. There are several methods for generating nonelastic hadronic events in hadron nucleus interactions evaluating and calculating hadron cascades. • Sampling from exclusive cross sections – the exclusive approach: the hadron–nucleus interactions are treated as a series of nucleon–nucleon interactions. The most complete models in the energy from 0.1 GeV to about 5 GeV are the INC models. References of the different existing models are references of the BERTINI model are [3–7, 12, 76, 77, 103, 151–158], references of the DUBNA models are [10, 13, 78, 101, 159, 160], references of the ISABEL model are [11, 79, 161–167], references of the INCL model are [117, 168–172], references of the CEM model are [41, 173–175], and references concerning the Hänssgen–Ranft model are [176–178]. The first model sampling from exclusive cross sections was the model from Sternheimer [76], which was used by Handbook of Spallation Research: Theory, Experiments and Applications. Detlef Filges and Frank Goldenbaum Copyright  2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40714-9

64

2 The Intranuclear Cascade Models

Bertini [77, 153] for INC calculations. This early model of Bertini has been restricted so far to incoming and produced pions and nucleons only, and it is, besides the early investigations of Goldberger [3] and Metropolis et al. [4, 5], the precursor of all modern models. All the INCs models consider energy–momentum conservation and correlation in hadron cascades. • Sampling from single-particle inclusive cross sections – the inclusive approach: the hadron nucleus interactions are treated by means of some model representations, experimental data, and semiempirical single inclusive particle distributions. Such models work best at higher energies where the inclusive particle production cross sections are well understood. But they are unreliable at energies below a few GeV. A summary of the methodology of this approach is given by Ranft in [106, 179, 180] • Sampling from phase space: such models are often used for sampling well-defined exclusive channels and simulating experiments using phase space kinematics. Monte Carlo methods are applied to calculate phase space distributions arising from particle interactions according to the Lorentz-invariant phase space [148, 181, 182]. • Other models are the QMD models (Quantum Molecular Dynamic) [183–186] and the BUU models (Boltzmann–Uehling–Uhlenbeck) [187–190]. The QMD and BUU models are mainly used in the field of heavy ion collisions. A good overview about the applicability in nucleon interactions is given in Ref. [191] and references therein. The purposes of the following sections are (1) to give an overview of the main features of the existing INC models and codes, and (2) to discuss the differences of the models with their assumptions, their limits and constraints on energy and application regime. In Section 2.2, a very detailed description of the BERTINI model is given. Many ideas of this model were still the base of all the existing INC models. In Sections 2.3, 2.4, and 2.5 the state-of-the-art models, such as INCL, ISABEL, and CEM will be discussed and compared with the features used in the BERTINI model. Section 2.6 summarizes some special INC models mainly based on the previous ones and in the final Section 2.7, a short review is given on alternative models for spallation reactions as quantum molecular models (QMD), a quantum extension of the classical molecular dynamics model widely used to analyze various aspects of heavy ion reactions.

2.2 The BERTINI Approach 2.2.1 Features of the BERTINI Nuclear Model

The basic assumptions and representations that have been used in the BERTINI model are summarized in the following. Most of the description may be found

2.2 The BERTINI Approach

in the references of Bertini et al. [12, 77, 154–156] and of Armstrong [105]. Further developments and refinements of the BERTINI model [158], which are not included in the original model, will also be discussed. The description of the BERTINI model here is based on Refs. [12, 105, 154–156]. The BERTINI model has been used successfully in Monte Carlo simulations at intermediate energies. Standard implementations of the model can be found in code systems such as HETC and HERMES [88, 192], LAHET [193], MCNPX [194], CALOR [195], and TIERCE [196], and based on the original ideas of Bertini with advanced physics in FLUKA and GEANT4 [100, 197]. The limitations of the earlier models of Metropolis et al. [4, 5] are mainly in the nuclear models. These are: (a) the use of a uniform density nucleus with a sharp boundary, (b) no pion–nucleon potential was employed, and (c) the neglect of the refraction and reflection of cascade nucleons due to a spatial nonuniformity of the nuclear potential [11]. 2.2.2 The Nuclear Model

The target nucleus is modeled as a three-region approximation to the continuously changing density distribution of the matter within the nucleus. The Fermi gas model, one model to describe the behavior of nucleons in a nucleus, is used. A Fermi–Dirac gas obeys Fermi–Dirac statistics, and therefore obeys the Pauli exclusion principle. More details may be seen in Refs. [69, 70, 198]. The cascade process starts when an incident particle – proton, neutron, pion, etc. – hits a nucleon in the target nucleus and produces secondary particles. The life history of each particle is followed until it either escapes from the nucleus or its energy falls below an arbitrary cutoff energy, which, in general, is taken to be half of the Coulomb barrier of the surface of the nucleus. This energy is different for different particle species and can, e.g., be found in Ref. [88]. At this point the energy conversation of the cascade process is verified. Relativistic kinematics is applied throughout the BERTINI INC model. 2.2.2.1 Nucleon Density Distribution Inside the Nucleus The density distribution of the nucleons (protons and neutrons) is assumed to approximate a nonzero Fermi gas momentum distribution. In the model of BERTINI the nucleus consists of three concentric spheres – a central sphere and two surrounding spherical annuli, each with a uniform density of neutrons and protons. The proton density in each region is made to be proportional to the average value over the same nuclear region of a continuous nonzero Fermi charge distribution function obtained by the approximation of measurements of electron scattering data by Hofstadter [199]. The radial dependence of the charge density in a nucleus can be described by these measurements as

ρ(r) =

ρ0

0 1 + exp( ri −r α )

,

(2.1)

65


where ρ0 is a normalization constant, ri with i = 1, 2, 3 corresponds to the three zones of the nucleus, r0 = 1.07 · A1/3 × 10−13 cm, α is given by 0.545 × 10−13 cm, and A is the mass number of the nucleus. It is further assumed that neutrons are distributed in the same way as protons, therefore, ρ0 can be calculated by ρ0 =

4π A

·

∞

dr · r 2 · ρ(r).

(2.2)

0

As the model assumes a subdivision of the nucleus in three zones Vi , i = 1, 2, 3, the outer radii ri of the three zones of the nucleus are determined such that the distances are at 90%, 20%, and 10% of the central value of Hofstadter’s [199] continuous distribution. This means that the density decreases with increasing radius:    ρ(r1 ) = 0.90 · ρ0  . ri = ρ(r2 ) = 0.20 · ρ0   ρ(r3 ) = 0.10 · ρ0

(2.3)

In Eq. (2.3) the values for the standard nuclear configuration are given. In cases where these values are all equal, the density distribution corresponds to a uniform density distribution as indicated by the dashed-dotted line in Figures 2.1 and 2.2. 1.0 0.9 Nuclear density r(r) [relative units]

66

0.8

Cu-65

r0

0.7 r1 0.6 0.5 0.4 r2 0.3 0.2 0.1 r3 0.0 0.0

2.0

4.0

6.0

8.0 10.0 12.0

Nuclear radius r [cm × 10−13 ]

Fig. 2.1 Comparison of various nucleon density distributions for nucleons inside the nucleus. Solid lines: the standard three zones’ nonuniform nucleon density configuration; dashed-dotted lines: uniform nucleon density distribution; dashed line: experimental curve of Hofstadter [199].

2.2 The BERTINI Approach 1.0 0.9 Au -197 Nuclear density [relative units]

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

2.0

4.0

6.0

8.0 10.0 12.0

Nuclear radius [cm × 10−13]

Fig. 2.2 Nucleon density distributions within the nucleus when the nuclear radius is assumed to be medium. Solid lines: the three zones’ nonuniform nucleon density distribution; dashed-dotted lines: uniform nucleon density distribution; dashed line: experimental curve of Hofstadter [199].

With the definition of Eq. (2.3) the nucleon density ρi of the nucleons Ni in zone Vi is calculated by ρi =

4π · Vi

ri

dr · r 2 · ρ(r),

(2.4)

ri−1

with r0 = 0 and Vi the volume of zone Vi of the nucleus. Since proton and neutron densities are assumed to behave like Eq. (2.1) the fraction of the number of protons and neutrons in each of the three zones is the same as for the nucleus as a whole and is constant: pi

=

(ρi /A) · Z

ni

=

(ρi /A) · (A − Z)

or (ni /pi )Vi

=

((A − Z)/Z)nucleus .

(2.5)

Here pi and ni are the zone-dependent proton and neutron densities, A denotes the mass number of the nucleus, Z is the charge number, and A − Z = N is the number of neutrons. The examples of Figures 2.1 and 2.2 give comparisons of typical nucleon density distributions for nucleons inside the nucleus for 65 Cu and 197 Au showing the

67

68


functional dependence of the nuclear radii ri = r1,2,3 of the density ρi and the difference of Cu and Au. This functional dependence is also depicted in Figures 2.6 and 2.7 on page 72. 2.2.2.2 Momentum Distribution of Nucleons Inside the Nucleus The momentum distribution of the nucleons in the nucleus is assumed to be a mixture of degenerated Fermi gases of protons and neutrons. Considering the nucleus as a collection of free protons and neutrons enclosed in a sphere of radius R or volume V. The protons and neutrons are fermions and thus are two identical particles in different quantum states according to the exclusion principle. The number of states NST corresponding to a momentum smaller than the Fermi momentum PF for protons or for neutrons is given by Segre [198] as

NST =

2 4 · · π · V · PF3 , (2π)3 3

(2.6)

which may be obtained by dividing the phase space into cells of volume (2π)3 and assigning two particles to a cell, to take into account the statistical weight of states of particles of spin = 12 . At a complete degeneracy of the Fermi gas, the Fermi momenta for protons PFp and neutrons PFn for the ground state are given as PFp = (3π 2 )1/3 · · (Z/V)1/3 PFn = (3π 2 )1/3 · · ((A − Z)/V)1/3 .

(2.7)

By the approximation Z = N = A/2 the Fermi momentum is given in a common form [198] as PF = ·

3π 2 · A 2·V

1/3 ,

(2.8)

with V = (4/3) · R3 and R = r0 A1/3 , V being the volume of the nucleus. In the BERTINI model the momentum distribution in each of the three zones Vi , i = 1, 2, 3, of the nucleus follows a Fermi distribution with zero temperature:

PFi (ri )

f (p) dp = Np

or Nn ,

(2.9)

0

where Np and Nn are the number of protons and neutrons, respectively. The above integral is the total number of protons or neutrons in one of the three zones of the nucleus and PFi (ri ) is the radius-dependent momentum of a nucleon corresponding to the radius-dependent Fermi energy (see Eq. (2.11)). In this picture the Fermi momentum PFi (ri ) is given for different zones of the nucleus by PFi (ri ) = ·

3π 2 · ρ(ri ) 2

1/3 .

(2.10)


As the target nucleus is a mixture of degenerated Fermi gases of protons and neutrons, the Fermi energy EF is represented by Eq. (2.11) and can be calculated by a local radius-dependent density approximation as EFi (ri ) =

2/3 2 · 3π 2 · ρp,n (ri ) , 2 · mN

(2.11)

where the subscripts p, n stand for the proton and neutron densities, respectively, and i stands for the three zones i = 1, 2, 3, mN is the nucleon mass, = h/2π is the Planck constant (1.054 × 10−34 J s), and ρp,n (ri ) is the density of protons and neutrons defined by Eqs. (2.1), (2.4) and (2.5), respectively. The momentum PFi (ri ) corresponds to the Fermi energy as follows: EFi (ri ) = (PFi (ri ))2 /2 · mN .

(2.12)

In general, the Fermi gas model considers all nucleons to move as elements of a fermion gas within the nuclear volume V given as V 43 · π · r03 · A. Since the Fermi energy is dependent upon the nucleon density and, therefore, different for each type of nucleon in each zone Vi , the composite momentum distribution for the entire nucleus is not a zero-temperature Fermi distribution. As an example, the composite Fermi distribution and a Maxwell–Boltzmann distribution with a kT value of 15 MeV are shown in Figure 2.3. (× 10−14) 12.0

Fraction per unit momentum

10.0

8.0

6.0 Atomic mass A =120

4.0

2.0

0.0 0.0

0.5

1.0

1.5

2.0

2.5 (× 10−14)

Momentum [g cm s−1]

Fig. 2.3 Fermi momentum distribution of nucleons inside the nucleus. Solid lines: distribution from the three zero-temperature Fermi energy distributions; dashed line: Maxwell–Boltzmann distribution with the kT value of 15 MeV (after Bertini [153]).

69

70


As mentioned earlier and may be seen from Eqs. (2.10), (2.11), and (2.12), the value of the Fermi energy depends rather sensitively on the value of the nuclear radius ri and the assumed density. This requires the use of a radius- and density-dependent nuclear model. 2.2.2.3 Potential Energy Distribution Inside the Nucleus The potential for a nucleon may be written as

V(A, Z, ri ) = EFi (ri ) + B(A, Z) + VC (A, Z),

(2.13)

where EFi (ri ) is the radius-dependent Fermi energy, B(A, Z) is the binding energy of a nucleon, and VC (A, Z) is the Coulomb potential1) . Using expressions (2.11) and (2.12) this results for the proton or neutron potential in zone Vi to the following equation: Vi (A, Z) = EFi (ri ) + B(A, Z) = ()2 · ((3π 2 · ρp,n (ri ))2/3 )/(2mN ) + B(A, Z) = (PFi (ri ))2 /(2mN ) + B(A, Z),

(2.14)

with i = 1, 2, 3. In order to get an idea of the maximum potential well depth Emax in MeV, a simple expression for the Fermi momentum PF is derived using Eqs. (2.7) and (2.8). The proton and neutron momentum may be written as PFproton = /r0 · (9π · Z/4A)1/3 PFneutron = /r0 · (9π · N/4A)1/3 ,

(2.15)

where Z and N are the numbers of protons and neutrons, respectively, and the nuclear volume V is given as V (4/3)π · r03 · A. Based on Eq. (2.15) a simple estimate of the Fermi momenta can be given by considering self-conjugate nuclei [200] with N = Z = A/2, resulting in the following expression: PFproton PFneutron /r0 ·

9π 8

1/3 .

(2.16)

Using the expression · c = 197 (MeV fm) (with = 1.054 × 10−34 (J s) = 6.582 × 10−22 (MeV s), c = 2.9979 × 1010 (cm/s), 1f m = 10−13 (cm)) yields PFproton = PFneutron

297 r0

MeV/c.

1) The Coulomb barrier or potential VC = EC is the energy in the c.m. system, which is needed to overcome the Coulomb repulsion

(2.17) and is given by EC = (Z1 Z2 e2 )/RC , where RC is the Coulomb radius.

2.2 The BERTINI Approach Proton potential

Fermi surface

Binding energy 8 MeV

Fermi sea Nuclear potential depth Fermi energy Emax ~ 42 MeV

EF, proton

Fermi energy ECoulomb EF, neutron ~ 34 MeV

Fig. 2.4 Example of the potential well for protons and neutrons in a nucleus, showing the Fermi energy level of the proton and the neutron, the nucleon binding energy B of the last nucleon, and the maximum potential depth Emax .

The corresponding kinetic Fermi energy F(Emax ) by using a value of r0 = 1.07 fm or 1.07 × 10−13 cm is then estimated as F(Emax ) = (PFproton )2 /(2 · mproton ) = (PFneutron )2 /(2 · mneutron ) 34 MeV. (2.18) This Fermi energy corresponds to the kinetic energy of the highest occupied orbit, e.g., smallest binding energy. By giving a binding energy B = 8 MeV the result of Eq. (2.18) yields a good estimate of the nuclear well depth with Emax = F(Emax ) + B 42 MeV. Figure 2.4 is an example of the potential well for protons and neutrons in a nucleus. The Fermi energy EF of the neutron is here 34 MeV. This is the maximum kinetic energy of a neutron bound in the nucleus. If the binding energy of the most loosely bound nucleon is 8 MeV, the maximum potential energy Emax must then be 42 MeV. Note also the difference between proton and neutron wells due to the Coulomb well. The binding energy of the most loosely bound nucleon is taken to be 8 MeV in the BERTINI model and is assumed to be the same for all the three zones and for all the nuclei. The potential energy in each of the zones of the nucleus is determined by the sum of the zero-temperature Fermi energy of the nucleons in each zone plus the binding energy of the most loosely bound nucleon. The pion potential in each zone is taken to be the same as the potential of the nucleon with which it interacts. This is in contrast to a more refined approach in the INCL-Cugnon [170] INC model as will be described later in Section 2.3 on page 88. Potential values and distributions for a typical case (Cu-65) are illustrated in Figure 2.5. As can be

71


0.0

Nucleon potential [MeV]

Binding energy 8 MeV

−10.0 Protons Neutrons 65Cu

−20.0 −30.0 −40.0 −50.0 0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

Nuclear radius [cm × 10−13] Fig. 2.5 Nucleon potential as a function of the nuclear radius for a typical nucleus Cu-65. The fixed binding energy is 8 MeV of the most loosely bound nucleon (after Bertini [153]).

10 9 Nuclear radii [cm × 10−13]

72

Radius r3

8

Radius r2

7 6

Radius r1

5 4 3 2 1 0 0

20 40 60 80 100 120 140 160 180 200 220 240 Atomic mass number A

Fig. 2.6 The radial boundaries defining the three zones V1,2,3 of the nucleus as a function of the atomic mass number A.

seen, the potentials are constant in the zones limited by the radii given in Eq. (2.3). The effects of the reflection and refraction of the nucleons at the zone boundaries are not considered in the model. Figures 2.6, 2.7, and 2.8 show as a function of the atomic mass number A the radial boundaries defining the three zones V1,2,3 of the nucleus, the nucleon density in each of the three zones V1,2,3 of the nucleus, and the Fermi energy per nucleon in each of the three zones V1,2,3 .

Nuclear densities per atomic mass A [cm−3]


108

107 Radius r1 106 Radius r2 105 Radius r3 104 0


Fig. 2.7 The nucleon density in each of the three zones V1,2,3 of the nucleus as a function of the atomic mass number A.

Fermi energies [MeV]

102

101 Radius r1 Radius r2

100

Radius r3 10−1 0


Fig. 2.8 The Fermi energy per nucleon in each of the three zones V1,2,3 as a function of the atomic mass number.

2.2.2.4 Application of the Pauli Exclusion Principle The only quantum mechanical effect included in the INC model is the Pauli exclusion principle, which is taken into account by introducing the Pauli blocking factor. The Pauli blocking factor accounts for the fact that the nucleons are fermions and therefore obey the Pauli principle. It should be mentioned that in the model only nucleons (protons and neutrons) observe the Pauli exclusion principle, since the densities of other fermionic particles are small. Since collisions are considered inside nuclear matter, the Pauli exclusion principle forbids interaction where

73

74


the collision products would be in occupied states. For a completely degenerate Fermi gas, the levels are filled starting from the lowest level. Therefore, collisions are forbidden in which either very large or very small amount of energy is transferred. The minimum energy allowed for the low-energy product of a collision corresponds to the lowest unfilled level of the system, which is the Fermi energy in the appropriate zone Vi of the nucleus. Since the time scale of INC reactions is rather small (≤10−22 s), it can be supposed that the nucleus will stay in the ground state. Therefore the Fermi momentum PF is well defined by PF = · (3π · ρ(r))1/3 ,

(2.19)

where ρ(r) is the nucleon density as defined in Eq. (2.1). However, because the nucleus is subdivided into three zones Vi , i = 1, 2, 3, Eq. (2.19) is written as PFi = · (3π · ρi )1/3

(2.20)

with ρi from Eq. (2.4). As a consequence the Pauli factor f is given by − → f = (x) · (| P | − PFi ).

(2.21)

(x) is the Heavyside step function, written as (x) =

1 for x > 0 . 0 for x < 0

(2.22)

Thus, the final state particle will be blocked in the case when the absolute value − → of its momentum P is smaller than the Fermi momentum in zone Vi where it is located. In the Monte Carlo calculations the exclusion principle is taken into account by neglecting any collision where the energy of any collision product falls below the Fermi energy. So in practice, the Pauli exclusion principle is taken into account by accepting only secondary nucleons which have an energy EN larger than the Fermi energy EF , EN > EF . 2.2.3 The Cross-Section Data

The greatest advantage of the BERTINI model is that the basic ingredients required are only the free particle cross-section data for nucleon–nucleon elastic and nonelastic, pion–nucleon elastic scattering, charge exchange, and pion production cross sections, which is relatively well known from experimental data [19]. In the BERTINI model, single and double-pion production in pion–nucleon collisions are taken into account. This restricts the upper energy limit of the original BERTINI model [12] to about 3 GeV since at higher energies higher order pion production, other meson production channels or higher baryon resonances become important.

2.2 The BERTINI Approach Tab. 2.1

Cross sections required for the BERTINI INC model.

Cross section

Reaction

n–p p–p π + –p π − –p π − –p π 0 –p π − –p p–p n–p π + –p π − –p

Differential Differential Differential Differential Differential charge exchange Differential Charge exchange Single-pion production Single-pion production Single-pion production Single pion production

Cross section n–p p–p π − −p π 0 –p π 0 –n π + –p π 0 –p p–p n–p π 0 –p π 0 –n

Reaction Elastic Elastic Elastic Elastic elastic Absorption Absorption Double-pion production Double-pion production Single-pion production Single pion production

Twenty-two cross-section sets are only needed with ∼3 × 104 values. In Table 2.1 the used nucleon–nucleon and pion–nucleon cross sections for the BERTINI model are summarized. Making the assumption to use for example particle–nucleus cross sections for the INC model instead of nucleon–nucleon cross sections one would like to know for nonelastic collisions the differential cross sections, for example, d2 σnonelastic − → (Ej , j , hi , Ei , Ak ), dE d

(2.23)

− → in energy E and direction of secondary particles of type j produced when a hadron hi of type i and energy Ei interacts with a target nucleus of mass Ak . This information is presently not known accurately over the wide range of parameters of interest in spallation research and would need about 107 input values for the INC model. This is still the major physics uncertainty in predicting hadronic cascades. Basic approaches are therefore to estimate d2 σ/dE d by using available hadron–hadron data or extending the models to hadron–nucleus collisions by theoretical models in conjunction with accelerator and cosmic ray data to evaluate the semiempirical formula [106, 179]. The energy range to treat nuclear collisions of the BERTINI model is for neutron and proton reactions up to 3.5 GeV and for pion reactions up to 2.5 GeV. The methods used in treating nuclear collisions depend upon • whether the collision is elastic or nonelastic, • whether the struck nucleus is hydrogen, and • whether the energy of the particle is above or below an energy cut Ecut about 3.5 GeV for nucleons or 2.5 GeV for pions, respectively. Hence the upper boundary momenta are 4.34 GeV/c for nucleons and 2.64 GeV/c for pions. The treatment of nuclear collisions of the BERTINI model is given in Table 2.2. In a later extension of the BERTINI model [12], it should be mentioned here, that

75

76

2 The Intranuclear Cascade Models BERTINI model treatment of nuclear collisions in the original version based on Ref. [12] and the extended version on Refs. [202, 205].

Tab. 2.2

A = 1a E < Ecut same n–p, p–p, π –p cross sections as used in INCc

A > 1b E > Ecut Ranft–Borak distributions [201, 202, 206]

E < Ecut INCc model

E > Ecut INCc model with extrapolation model [202, 205]

Elastic collisions A = 1a E < Ecut same p cross sections as used in INCc

A > 1b E > Ecut parametric fits to experimental data [201, 202]

E < Ecut for p and n

E > Ecut INCc model with extrapolation model [202, 205]

= 1 denotes collisions with hydrogen nuclei. > 1 denotes all nonhydrogen target nuclei. Intranuclear-cascade model original version Ref. [12].

aA

bA c

at nucleon energies >3.5 GeV and pion energies >2.5 GeV particle production data for nonelastic collisions are obtained in the BERTINI model from a very approximate model investigated by Gabriel et al. [201, 202]. Basically, this extension of the BERTINI model [12] employs particle production data from the INC model at energies ∼3 GeV and uses energy- and angle-scaling relations to estimate particle production data for higher energy nucleon–nucleus and pion–nucleus collisions. Examples for the validity of the extrapolation model are given in Refs. [158, 203, 204]. In the following the differently used particle–particle cross section data used by the BERTINI model are graphically shown. A short overview of the pion production model – the Sternheimer–Lindenbaum isobar model [76, 152] – is also presented. The BERTINI model is restricted to incoming and produced nucleons and pions only. A discussion about the production of secondary particles in nonelastic collisions via resonance decay was already given in Section 1.3.5.3 on page 27. The BERTINI model data will be also discussed in context with available evaluated data, which are provided by the Particle Data Group given in Ref. [19]. A graphical representation of these data is shown in Figures 2.11, 2.12, 2.14, and 2.15. 2.2.3.1 Nucleon–Nucleon Cross-Section Data The differential cross sections for p–p scattering (dσ/d)pp used in the BERTINI model are assumed to be isotropic in the center-of-mass system for protons up to


0.500 GeV. For energies from 0.44 to 4.4 GeV semiempirical fits used an expression of the form (dσ/d)pp = A + B · µ3 ,

(2.24)

where µ represents the cosine of the scattering angle in the center-of-mass system. The angular distribution is assumed to be symmetric about µ = 0. The differential cross sections for np-scattering (dσ/d)np used in the BERTINI model are treated for forward and backward scattering separately. The semiempirical fits used an expression of the form given below for different energy ranges. For the energy range: • for 0 ≤ Eneutron ≤ 0.3 GeV and 0 ≤ µ ≤ 1 (dσ/d)np = AF + BF · µ3 (dσ/d)np = AB + BB · µ4 .

(2.25)

For the energy range: • for 0.3 GeV ≤ Eneutron ≤ 0.74 GeV and −1 ≤ µ ≤ 0 (dσ/d)np = AF + BF · µ3 (dσ/d)np = AB + BB · µ6 .

(2.26)

And for the energy ranges: • for 0.66 GeV ≤ Eneutron ≤ 3.52 GeV and cross section tabulated for µ intervals, forward scattering µ ≥ 0 • for 0.66 GeV ≤ Eneutron ≤ 3.52 GeV and cross sections tabulated for µ intervals, backward scattering µ ≤ 0 where the index F stands for forward scattering and the index B for backward scattering. The different coefficients A, AB , AF , B, BB , and BF are tabulated in Refs. [77, 88]. The total proton–proton and total neutron–proton scattering cross section that are used in the BERTINI model [12] are shown in Figures 2.9 and 2.10. For comparison proton–proton (pp) and neutron–proton (np) scattering cross-section data evaluated by the Particle Data Group [19] as a function of the laboratory momentum plab in GeV/c are shown in Figures 2.11 and 2.12. The agreement with the original BERTINI data shown in Figures 2.9 and 2.10 is fairly well concerning the energy range up to 3.5 GeV. 2.2.3.2 Pion Production and Pion Nucleon Reactions Single- and double-pion production in nucleon–nucleon collisions For single-pion production the nonelastic reaction channel with the lowest threshold opens around 290 MeV in nucleon–nucleon collisions (pp or np), and arrive a maximum at about 700 MeV. Double-pion production starts at a threshold at about 800 MeV by

77


Cross sections [mb]

100 80 60 pp - total 40 pp-elastic

20 0 0.0

0.4

0.8

1.2

2.0

1.6

2.4

2.8

3.2

3.6

Proton energy [GeV]

Fig. 2.9 Total proton–proton σpp -total and -elastic cross sections as a function of the incident proton energy used in the original BERTINI INC model (after Bertini [12]). 100 Cross sections [mb]

78

80 60 snp -total 40 20 0 0.0

snp -elastic 0.4

0.8

1.2

1.6

2.0

2.4

2.8

3.2

3.6

Neutron energy [GeV]

Fig. 2.10 Total neutron–proton σnp -total and -elastic cross sections as a function of the incident neutron energy used in the original BERTINI INC model (after Bertini [12]).

nucleon–nucleon collisions. The nonelastic reaction channels are dominated by the existence of several resonances. Details are already discussed on page 27. The most prominent one is the (1232) resonance (see Figure 1.17 on page 28), which has a mass of m = 1232 MeV/c2 and width of about = 110–120 MeV/c2 . The resonance decays via the strong force corresponding to a decay time τ 5 × 10−24 s. As already mentioned, this resonance is usually named 33 or (3,3)-resonance. For nucleon–nucleon or pion–nucleon collisions that lead to pion production, the isobar model of Sternheimer–Lindenbaum [76, 152] is used to determine the final states. The model assumes that pions are produced through the decay of isobars N ∗ or of -resonances of the excited nucleon system, which is formed when a nucleon is excited in a collision. All reactions proceed through an intermediate state containing at least one resonance. There are two main classes of reactions, those which form a resonant intermediate state (possible in π –nucleon reactions) and those which contain two particles in the intermediate state. The former exhibits bumps in the cross sections corresponding to the energy of the formed resonances (cf. Figure 1.17 on page 28 and Figure 2.13).


Cross section [mb]

103

pp-total pp-elastic

102

101

100 10−1

100

101

102

103

104

Laboratory beam momentum Plab [GeV/c] Fig. 2.11 Proton–proton σpp -total and -elastic cross sections as a function of the laboratory proton momentum plab MeV/c evaluated by the Particle Data Group (pdg) from Ref. [19].

105

Cross section [mb]

104

np-total np-elastic pn-total

103

102

101

100 10−3

10−2 10−1 100 101 102 Laboratory beam momentum Plab [GeV/c ]

103

Fig. 2.12 Neutron–proton σnp -proton–neutron σpn -total, and neutron–proton σnp -elastic cross sections as a function of the laboratory proton momentum plab MeV/c evaluated by the Particle Data Group (pdg) from Ref. [19].

The models of Sternheimer–Lindenbaum [76, 152] used in the BERTINI intranuclear-cascade allows single and double-pion production in p–p and n–p nuclear collisions and single-pion production in pion–nucleon collisions.

79


225

Cross sections [mb]

80

175 stotal (p+ + p)

125

stotal (p− + p)

75 25 0 0.0

0.4

0.8 1.2 1.6 Pion energy [GeV]

2.0

2.4 2.5

Fig. 2.13 Pion π + + proton and π − + proton total cross sections as a function of the incident pion energy used in the original BERTINI INC model (after Bertini [12]).

The reaction formalism for single- and double-pion production in nucleon– nucleon collisions is given by Eqs. (2.27) and (2.28). • single-pion production close to the threshhold of 290 MeV: ˆ 1 + (1232) −→ N ˆ1 +N ˆ2 +π N1 + N2 −→ N ˆ2 +π with (1232) −→ N

(2.27)

• double-pion production: ˆ 1 + π1 + N ˆ 2 + π2 N1 + N2 −→ 1 (1232) + 2 (1232) −→ N ˆ ˆ with 1 (1232) −→ N1 + π1 , 2 (1232) −→ N2 + π2 .

(2.28)

Single-pion production in pion–nucleon collisions All nonelastic pion–nucleon reactions are assumed to be single π-meson production events. The formalism for pion production in pion–nucleon nonelastic collisions is also based on the Sternheimer–Lindenbaum model [76, 152]. The pion is produced also via the formation and the subsequent decay of the (1232) or 33 which limits the model to produce one pion in pion–nucleon collisions. The reaction formalism for singlepion production in pion–nucleon collisions is given by Eqs. (2.29). Figure 2.13 shows the total pion–proton reaction cross sections of π − p and π + p collisions used by the BERTINI model (see also Figures 2.14–2.17). The resonance structure as indicated in Figure 1.17 on page 28 is clearly seen.

ˆ2 π + N1 −→ (1232) −→ π + N

(2.29)

To determine the final-state momentum distributions, the angular distributions of the isobars or resonances and the decay of the pions must be specified. These


Cross section [mb]

103

p+ p-total p+ p-elastic

102

101

100 10−1

100

101

102

103

Laboratory beam momentum Plab [GeV/c] Fig. 2.14 Pion π + + proton total- and elastic cross sections as a function of the laboratory momentum plab MeV/c evaluated by the Particle Data Group (PDG) from Ref. [19].

102

Cross section [mb]

p− p-total p− p-elastic

101

100 10−1

100

101

102

103

Laboratory beam momentum Plab [GeV/c ] π −+

Fig. 2.15 Pion proton total- and elastic cross sections as a function of the laboratory momentum plab MeV/c evaluated by the Particle Data Group (pdg) from Ref. [19].

distributions are calculated phenomenologically through comparisons with experimental data [156]. These distributions are given for nucleon–nucleon single- and double-pion production in Table 2.3 and for pion–nucleon single-pion production in Table 2.4.

81


25

Cross section [mb]

20

15

10

5

np-s p np-d p

pp-s p, pp-d p,

0 250

1000

2000

3000

4000

Incident proton / neutron energy [MeV] Fig. 2.16 Single (pp), (np)-s π and double-pion (pp), (np)d π pion production cross sections for nucleon–nucleon collisions as a function of the incident proton and neutron energy used in the original BERTINI INC model (Ref. [12]). 35 30 Cross section [mb]

82

25 20 p+ p-single production

15

p0 p-single production

10

p− p-single production p0 p-single production

5 0 100

500

1000 1500 2000 Incident pion energy [MeV]

2500

3000

Fig. 2.17 Single-pion (π + p), (π − p), (π 0 p), and (π 0 n) production cross sections for pion–nucleon collisions as a function of the incident pion energy used in the original BERTINI INC model (Ref. [12]). The (π − p) and (π 0 p) cross section are assumed to be the same.

Pion–nucleon absorption, charge-exchange, and scattering Pion absorption is considered to occur on a two-nucleon cluster [12, 77, 105, 156]. The type of cluster pp, pn, or nn for absorption is determined from the number of each type of particle pairs within the nucleus. Pair types which would violate charge conservation are not considered in the calculation of the absorption probabilities. This means, e.g.,

2.2 The BERTINI Approach Angular distribution in the center-of-mass system for nucleon–nucleon single- and double π -production reactions.

Tab. 2.3

Laboratory kinetic energy range (GeV)

E< 0.5 0.5 ≤ E 1.0 ≤ E 1.3 ≤ E 2.5 ≤ E

< 1.0 < 1.3 < 2.5 < 3.5

All energies

Percentage of angular distribution Isotropic

Forward

Single-pion production 100 75 50 25 0 Double-pion production 0

Backward

0 12.5 25 37.5 50

0 12.5 25 37.5 50

50

50

Angular distribution in the center-of mass-system for pion–nucleon single π -production reactions.

Tab. 2.4

Laboratory kinetic energy range (GeV)

All energies E < 0.5 E ≥ 0.5

Percentage of angular distribution Isotropic π + p and π − n 75 π − p and π + n 80 80 π 0 p and π 0 n assumed to the same as π − p

Forward

0 12.5 20 0

Backward

25 12.5 0 37.5

that π + absorption can take place with np pair and nn-pair clusters only. When π + absorption occurrs, the probability of the pair type being a np-pair is given by the ratio of the number of np-pairs to the sum of np and nn pairs in the nucleus. A short overview about the physics of pion absorption is given on page 27. It should be mentioned here that at the time 1963–1972 when the Bertini model was designed the knowledge in pion physics concerning the -resonance decay was poor, although Brueckner et al. [207] and Gell-Mann and Watson [208] proposed a model that the cross section for pion absorption is related to a deuteron absorption cross section. The cross sections for π − p charge exchange are given in [12, 77]. The cross section for elastic π 0 p scattering is calculated from the relation σelastic (π 0 p) = 1/2[σelastic (π + p) + σelastic (π − p) − σexchange (π − p)].

(2.30)

The π 0 n cross section is set equal to the π 0 p cross section concerning the conservation of the isotopic spin.

83


50

Cross section [mb]

40 p+ p-absorption

30

p0 p-absorption

20

10

0 0

50

100 150 200 250 300 Incident pion energy [MeV]

350

400

Fig. 2.18 Pion–nucleon (π + p) and (π 0 p) absorption cross sections as a function of the incident pion energy used in the original BERTINI INC model [12].

50

40 Cross section [mb]

84

30 p− p- charge exchange 20

10

0 10

500

1000 1500 2000 Incident pion energy [MeV]

2500

3000

Fig. 2.19 Pion–proton (π − p) charge-exchange cross section as a function of the incident pion energy used in the original BERTINI INC model (Ref. [12]).

In Figures 2.18 and 2.19 the pion absorption π + p, π 0 p and the pion chargeexchange π − p cross section data, and in Figures 2.20, 2.21, and 2.22 the proton–proton pp, neutron–proton np, and the pion–nucleon π + p, π − p, π 0− p, π 0 n elastic scattering cross sections used by the BERTINI model are graphically shown.


2× 103

Cross section [mb]

103 np - elastic cross section pp - elastic cross section

102

101 100

101

102

103

4× 103

Energy [MeV] Fig. 2.20 Proton–proton (pp) and neutron–proton (np) elastic scattering cross section as a function of the incident proton and neutron energy used in the original BERTINI INC model [12].

Cross section [mb]

102

101

100 p0p/p0n - elastic cross section p−p-elastic cross section

10−1 5

101

102 Energy [MeV]

103

4× 103

Fig. 2.21 Pion–nucleon (π − p), (π 0− p), and (π 0 n) elastic scattering cross section as a function of the incident pion energy used in the original BERTINI INC model [12]. The (π 0 p) and (π 0 N) cross section are assumed to be the same.

2.2.4 Method of Computation

The method of computation used in the BERTINI INC model is the Monte Carlo technique essentially in an analog fashion that is nonelastic interactions of hadrons with nuclei are simulated. At the beginning the incident particle interacts with a

85


2.5×102 102 Cross section [mb]

86

101 p+ p-elastic cross section

100 5

101

102 Energy [MeV]

103

4×103

Fig. 2.22 Pion–nucleon (π + p) elastic scattering cross section as a function of the incident pion energy used in the original BERTINI INC model [12].

single nucleon inside the nucleus, much as the nucleons were in free space. The collision is not exactly analogous in free space because the Pauli principle will exclude those encounters with a certain momentum transfer (see page 73). Also, at high energies the incident particle can traverse the nucleus without experiencing an interaction, resulting in a nuclear transparency. The locations of the collisions, the momentum of the struck nucleon, scattering angles, etc. can be determined by using a statistical sampling technique together with the free particle (pp, pn, πp, etc.) cross-section data. The basic steps of the Monte Carlo simulation are straightforward, and given below: 1 The spatial point – the impact parameter – where the incident particle enters the nucleus is determined by selecting uniformly over a circular area representing the projected area of the nucleus. This is shown in Figure 2.23. 2 A path length is selected for the distance the incident particle tracks before undergoing a collision, using the total particle–particle cross sections and the radial-zone-dependent nucleon densities (cf. pages 65 and 74). The internuclear distance to a collision point, the type of collision, and the momentum of the struck particle are ‘‘sampled’’ by a rejection technique well known in Monte Carlo particle transport through matter (see Refs. [149, 150]). For example, by sampling from e− T ·z , where T = Np · σp + Nn · σn is the total interaction cross section determining how far the particle travels before interacting. From the ratio i / T , the probability that the ith interaction occurs, where N j=1 j = T. 3 If the particle escapes the nucleus without having a collision, this particle contributes to the ‘‘nuclear transparency.’’ Otherwise, the momentum of the


x Nucleus

Incident particle z z

Projected circular area of spherical nucleus

ri

y

Fig. 2.23 The entry of an incident particle into a nucleus by select uniformly over a projected area.

struck nucleon, the type of reaction, and the energy and direction of the reaction products are determined. 4 If the collision is allowed according to the exclusion principle, and if the kinetic energy of the product is above a predefined cutoff energy, then the simulation chooses again point (2) and the resulting product particles are transported further. 5 After the completion of the intranuclear-cascade for an incident particle, the mass and charge, A∗ and Z∗ , of the residual nucleus are determined from a particle balance, and the residual excitation energy, E ∗ , is determined from an energy balance. The general outcome of the INC model of the BERTINI model is then the double differential particle spectra of cascade particles in energy E and − → laboratory angle , dσp,n,π /dE d and a residual nucleus Nresidual (A∗ , Z∗ , E ∗ ).

2.2.5 Assumptions, Limits, and Constraints on the Energy and Application Regime

The BERTINI INC model describes the target nucleus as a three-zone approximation of the continuously changing density distribution of nuclear matter within the nucleus. The cascade begins when the incident particle strikes a nucleon in the target nucleus and produces secondary particles. The secondaries may in turn interact with other nucleons in the nucleus or be absorbed. The cascade ends when all particles, which are kinematically able to do so, escape the nucleus. At that point conservation is checked. Relativistic kinematics is applied throughout the model. • The BERTINI model is essentially parameter free and provides results with absolute normalization. • The model requires mainly only fundamental particle–particle cross-section data, which are relatively well known. • The model has been shown to be in reasonably good agreement with a wide range of experimental data (see Part 2 on page 277).

87

88


• The model has rather general applicability. (a) All target nuclei with A ≥ 1 are allowed. But it should be noted that the approximation of subdividing the nucleus into several zones is only valid for heavier nuclei. For light nuclei more realistic models, e.g., the shell model should be used, as shown and studied in Ref. [209]. If the target material is hydrogen A = 1, a direct particle–particle interaction is performed, and no nuclear model is required. (b) Different incident particles (protons, neutrons, and charged pions, with recent extensions to light heavy ions [88]) are possible. (c) The model covers a wide energy range of about 0.1–3.0 GeV. The valid energy range is not well defined. The lower energy limit is ∼100–150 MeV. The original BERTINI model [12] is limited to energies below ∼3 GeV because only single- and double-pion production is included. (d) Extensions are made to the BERTINI model [12] version. Extensions to tens or even hundredsof GeV have been made by using approximate scaling relations by Gabriel et al. [202]. Also Barashenkov et al. [10] and Bertini et al. [158] have made modifications to the basic model [12] (e.g., to incorporate nuclear depletion, which takes into account the time dependence of the changing nucleon density in the path of the development of the cascade inside the nucleus and to divide the nucleus into a maximum of 50 spherical zones compared to the three-zone model of the nucleus used in the basic model.) • The model is capable of providing very detailed results on an event-by-event basis. (a) type, energy, and direction of each emitted particle; (b) type, excitation and recoil energy of the residual nuclei; (c) the photo source from π 0 decay and nuclear de-excitation; (d) the low-energy En ≤ 15 MeV neutron production, which is important in many applications. • The BERTINI model is only valid for incident protons, neutrons, and pions and valid for particles treated in the model such as protons, neutrons, pions, nuclear isotopes (residuals), and photons. The necessary condition for the low-energy limit of the model is λdeBroglie /v 1.4 GeV/c, (2.36)

where plab is the laboratory momentum. For resonance absorption reactions an isotropic angular distribution is assumed. The Mandelstam variables [217] are Lorentz invariant variables describing the kinematics of particle reactions via a relation of relativistic momentum-energy

95

96


four vectors. The two-body elastic scattering amplitudes is given in terms of the dispersion relations as functions of two complex variables s and t. Nowadays, Mandelstam variables are also widely used to describe the kinematics of multibody final states seen as two incident and two outgoing systems [85]. By defining the incident particles 1 and 2, the outgoing systems 3 and 4, and using four-momentum conversation, it follows that p1 + p2 − p3 − p4 = 0

and (p1 )2 = mi2 .

The Mandelstam variables can be written as follows: s = (p1 + p2 )2 = (p1 + p2 )2 t = (p1 + p3 )2 = (p2 + p4 )2

(2.37)

u = (p1 + p4 ) = (p2 + p3 ) 2

2

from which it follows s + t + u = constant = m12 + m22 + m32 + m42 . Elastic baryon–baryon collisions Cugnon et al. [215] proposed a simple form for the differential elastic proton–proton cross section, which – with regard to isospin symmetry – is also valid for neutron–neutron scattering. It is similar to Eq. (2.34)

dσ ∼ exp (Bpp · t), d

(2.38)

with Bpp given as a function of the center-of-momentum energy or equivalently of the laboratory momentum plab , Bpp =

(5.5p8lab )/(7.7 + p8lab ) for plab ≤ 2 GeV/c . 5.334 + 0.67 plab − 2 for plab > 2 GeV/c

(2.39)

For elastic neutron–proton scattering the situation is more complicated, since the charge exchange reaction has to be considered as well. In the INCL model the parameterization for the differential behavior in neutron–proton elastic reactions suggested by Cugnon et al. [170] is used. Later on, it was slightly modified by Cugnon et al. in Ref. [216]: el dσnp

d

el = σnp Const eBnp t + aeBnp u + ceαc u ,

(2.40)

where the term within the brackets give the angular distribution in terms of the Mandelstam variables t and u (see Eq. (2.37)). u = −2p2cms (1 + cos θ ),

(2.41)

2.3 The Cugnon INCL Approach

with pcms denoting the center-of-mass momentum. The third term in (2.40) accounts for the charge exchange interaction. The parameter Const in Eq. (2.40) el is fixed by normalization, as the angular integration should yield σnp . The energy, the momentum dependence of the slope parameter Bnp and of the coefficients a, c, and αc are, respectively, given by  −1  (7.16 − 1.63plab )(1.0 + exp(0.45 − plab /0.05)) , plab < 0.8 GeV/c Bnp = 9.87 − 4.88plab , plab < 1.1 GeV/c  3.68 + 0.76plab , plab ≥ 1.1 GeV/c, (2.42) a=

0.8 plab

2 ,

(2.43)

6.23 exp(−1.78plab ) for plab ≤ 1.7 GeV/c 0.3 for plab > 1.7 GeV/c αc = 100. c=

(2.44) (2.45)

In these relations plab should be given in GeV/c. Bnp and αc are provided in units of (GeV/c)2 . Moreover, it should be noted that the parameterization for Bnp for plab < 0.8 GeV/c is directly taken from the INCL2.0 code [169, 170, 215, 216, 218]. For resonance-nucleon elastic scattering experimental data are not available. Hence the same differential scattering behavior as in the case of proton–proton elastic scattering as defined in Eq. (2.39) is used in the model presented here. 2.3.1.6 Dynamic Pauli Blocking − → In Section 2.2.2.4, Eq. (2.21), the Pauli blocking factor f = (x) · (| P | − PFi ) was formally introduced into the collision integral. As discussed in Section 2.2.2.4 on page 72, the term accounts for the fact that the nucleons are Fermions and therefore obey the Pauli principle. Note that in the INC models only nucleons, i.e., neutrons and protons, observe the Pauli exclusion principle, since the density of other fermionic particles (-particles, baryonic resonances) is small. That is why one can safely neglect this effect for resonance states. Nevertheless, Pauli blocking is enforced to nucleons resulting from resonance () decays. Generally the strict Pauli blocking as discussed on page 73 is realized in INC programs. One shortcoming of the strict Pauli blocking is that the depletion of the Fermi sphere due to prior reactions is neglected. Furthermore, collision processes also introduce a temporary depletion of the spacial density. However, as the strict Pauli blocking is implemented in many heavy-ion collision models and in the INCL Liège [170] code, the Pauli blocking should operate in phase space. If two particles i and j suffer → → a collision at a position − r i(j) leading to a final state with the momenta − p i(j) , the phase space occupation probabilities are calculated by counting nearby nucleons in a small phase space volume:

fdyni(j) =

1 1

→ → → → Rr − |− r i−− r k | Rp − |− p i−− p k| , 2 Vr Vp k =i

(2.46)

97

98


where Vr and Vp are spherical volumes in the coordinate or momentum space of the nucleus, respectively. The factor 1/2 is introduced because spins are ignored. The sum in Eq. (2.46) is limited to particles k with the same isospin component as particle i (or j). The collision between participant i and j is allowed or forbidden following the comparison of a random number with the product (1 − fi )(1 − fj ). In the INCL2 model as published in Ref. [170] the radii in coordinate and momentum space were set to Rr = 2 fm and Rp = 200 MeV/c. The parameters Rr and Rp should not be taken too small; otherwise fi is going to be always vanishingly small, nor too large, otherwise the details of the phase-space occupation can be missed. There is no a priori criterion for the appropriate choice of these parameters. In practice Rr and Rp are taken just large enough for results to be roughly insensitive to moderate modifications of their values (see Ref. [170]). In Ref. [172] for the INCL4 model slightly different values of Rr = 3.18 fm and Rp = 200 MeV/c have been chosen, which correspond to a measuring volume of ≈2.3 natural units of phase space. 2.3.1.7 Cutoff Criteria – Stopping Time of the Cascade A special attention is drawn in the INCL model to the stopping time tstop , i.e., the time at which the cascade is stopped to give way to pre-equilibrium and evaporation. The time evolution of various quantities, like the excitation energy E ∗ , the mass number of the residual nucleus Ares , and the number of ejectiles and participants, is discussed in [169, 170, 216] and shown in Figure 2.29. The panels of Figure 2.29 refer, in a clockwise order, starting from the upper left, to the excitation energy E ∗ , the average kinetic energy of the ejectiles T, the asymmetry ζzz of the participant momentum distribution, and the time derivative of the excitation energy dE ∗ /dE. The asymmetry ζzz is a quantity measuring the anisotropy of the momentum content of the participant baryons inside the target volume. In the early time of the collision, ζzz is different from zero because of the motion of the incident particle. It then decreases and tends to zero signaling that the system reaches a high degree of randomization. The results shown in Figure 2.29 correspond to collisions of 1 GeV protons with Pb nuclei with an impact parameter of b = 4 fm. As presented in Figure 2.29, the time evolution of the average value of many physical quantities show, when the cascade is run for a long time, a phase of rapid variations, followed by a phase of much slower variations. In addition, the time of separation between the two phases is roughly the same for most of these physical quantities. These results enable us to define the stopping time more or less precisely as the common separation time of the phases of variation of the physical quantities. As discussed in Refs. [169, 172], the excitation energy E ∗ raises rapidly when the incident particle penetrates the target nucleus, reaches a maximum excitation energy after some fm/c and then decreases rather steeply. However, as shown in Figure 2.29, the slope flattens with increasing times similar to an evaporation process. It is also manifested from Figure 2.29 that the change of slope in the curve of E ∗ is correlated with the change in the curve representing the average kinetic energy of the emitted particles: the latter quantity is rather large in the first phase,


2000

1000 900 800 700 600 500

Average kinetic energy of ejectiles

< T > [MeV]

E* [MeV]

Excitation energy E*

400 300

102

200 10 100 90 80 70

0

50 100 Time [fm/c]

0

150

dE*/dt

100 Time [fm/c]

150

Momentum asymmetry of partcipants

25

102

50

10

xzz [%]

dE*/dt [MeV (fm/c)−1]

20

15

10 1 5

0 0

50

100 Time [fm/c]

150

0

Fig. 2.29 Time variation of the average value of a few physical quantities, within our INC model. The arrows indicate the chosen stopping time. (Figure taken from Boudard et al. [172].)

50

100 Time [fm/c]

150

99


but reaches in the second phase a small and smoothly decreasing value, typical of the evaporative cooling of an equilibrated system. Thus, it is the idea to stop the cascade at this particular time and proceed with an evaporation calculation. This cutoff is not only introduced to save computational time, but it seems to be necessary as the evaporation process is known to depend sensitively on the level density, which is presumably different from the single particle model value, to which INC models correspond. In summary, the collision process of a hadron–nucleus reaction can be divided into two parts. The first one corresponds to a rapid variation of physical quantities and the second one is characterized by slow variations, in which the nucleus is fairly well equilibrated. Unfortunately, this transition is not very sharp. Hence there will be some uncertainty of about 2–5 fm/c in the cutoff time. The INCL model adopts a parameterization of the stopping time tstop presented in [172]. It is given by tstop = fstop t0

A 208

0.16 (fm/c),

(2.47)

where fstop = 1, t0 = 70 fm/c and A is the atomic mass of the target nucleus. In Figure 2.30 the dependence of the stopping time tstop on the mass number of the target nucleus A according to Eq. (2.47) is shown. The INC process will end in the case when the elapsed time exceeds tstop . Moreover, the calculation is also terminated if no particle that was participating in the reaction is able to escape from the nucleus. The time as given by Eq. (2.47) [172] for the INCL4 version is substantially larger than the values adopted in the standard older versions (as described in Refs. [216, 219]), although the criteria are basically the same. This is merely due to the fact that, because of the presence of a diffuse surface, the incident particle is initially situated farther away from the target than

70

Time tstop [fm/c]

100

60 50 40 30 20

0

50

100 150 Atomic mass A

200

Fig. 2.30 Stopping time tstop used in INCL as a function of the mass number A of the target nucleus.

250


p (1200 MeV) + Pb

102 0° 10

10° (10−1) 1 25° (10−2)

10−1

40° (10−3)

d2s/ dΩ dE [mb/sr MeV]

10−2

55° (10−4)

10−3

70° (10−5)

10−4

85° (10−6)

10−5

100° (10−7)

10−6 10−7

115° (10−8)

10−8

130° (10−9)

10−9

145° (10−10)

10−10

160° (10−11)

10−11 10−12 1


103

Fig. 2.31 Comparison of simulated neutron double differential cross section with measurements for proton-induced reactions on a 2-cm-thick Pb target at 1200 MeV and detection angles from 0◦ to 160◦ . The predictions are given by the histograms, and data are as circles from Ref. [220]. The spectra have been multiplied by decreasing powers of 10, except for 0◦ (after Boudard et al. [172]).

in the previous INCL versions. The stopping time is now largely independent of the impact parameter and of the incident energy, for the same reason. The most important observable for applications to spallation reactions in spallation neutron sources or in general in all accelerator-driven systems (ADS) is the neutron production cross section. As examples in Figures 2.31 and 2.32, predictions of double differential neutron production cross sections in proton-induced reactions on Pb and Th targets for 1200 MeV along with measurements at the

101


103 p (1200 MeV) + Th

102 0° 10

10° (10−1) 1 25° (10−2)

10−1

40° (10−3)

10−2 d2s/dΩ dE [mb/sr MeV]

102

55° (10−4)

10−3

70° (10−5)

10−4

85° (10−6)

10−5

100° (10−7)

10−6 10−7

115° (10−8)

10−8

130° (10−9)

10−9

145° (10−10)

10−10

160° (10−11)

10−11 10−12 10−13

Fig. 2.32

1


103

Same as Figure 2.31 for a 2-cm-thick Th target (after Boudard et al. [172]).

SATURNE accelerator [220] are given. The targets used in Ref. [220] had a finite thickness of 2 cm. The data are plotted versus the logarithm of the neutron energy. The comparison with experimental data requires the coupling of the INCL with an evaporation model. The predictions in Figures 2.31 and 2.32 have been done by coupling the INCL4 model with the GSI evaporation model ABLA-KHSv3p [221, 222] also known as the Schmidt model (cf. Section page 154). The overall agreement is rather good although the cross section values are extending over several decades. It is easily seen that the INCL model plus the ABLA model reproduce the neutron production cross sections in the whole phase space satisfactorily. Another feature may be seen in Figures 2.31 and 2.32 is the correctly reproduced cross section of the quasielastic peak at 0◦ . More examples at lower incident proton energies and other target materials are shown in Ref. [172].

2.3 The Cugnon INCL Approach Parameters of the Gaussian forms used to describe radial distance and momentum distributions in light incident clusters.

Tab. 2.5

Incident particle d t 3 He 4 He

r 2 (fm) 1.91 1.8 1.8 1.63

p2 (MeV/c) 77 110 110 153

2.3.1.8 Light Clusters as Incident Particles Besides proton, neutron, π + , π − , or π 0 as types of incident particles, the INCL model has been extended to incident clusters of nucleons (deuteron, triton, 3 He, 4 He) [172]. It is sufficient to generate the initial distribution of nucleons inside the incoming cluster, as the history of all particles is followed in time. For their initial state due to the small size of these clusters, it is not appropriate to use a distribution of the form (2.31). In INCL a Gaussian shape for the spacial distribution is used with a width which is determined by the charge r.m.s. radius. Except for the deuteron, the same is done for the momentum distribution, with widths taken from the literature [223–226] (see Table 2.5). For the deuteron as the loosely bound system, a more careful microscopic treatment is required. Here the modulus squared of the wavefunction in momentum space, as calculated with the Paris potential [172, 227], has been used. There is no nuclear mean field introduced inside the incoming ion. This approximation is reasonable in view of their weak binding (except for 4 He). To correct for the latter in the energy balance, the incident kinetic energy is decreased by an amount equal to the binding energy in order to have the correct total incident energy. 2.3.1.9 Surface Percolation Procedure for Emission of Light Charged Clusters To describe the emission of light charged clusters during the cascade stage in nucleon–nucleus reactions, a surface percolation procedure has recently been developed for the INCL4 Liège INC model [117]. As has been discussed in Sections 1.3.3 and 1.3.4 on page 21, the reaction process can be divided into two stages. The first one is dominated by fast particle emission and in the second one, the residuum of the target releases its remaining excitation energy by evaporation of slow particles (and/or by fission for heavy targets). Not only nucleons, but also light charged clusters (d, t,3 He,4 He) are emitted in two stages, as suggested by the observation of their spectra (see Figures 2.33, 2.34, and spectra shown in Part 2 on page 277). As an example for light charged particle production, double differential energy spectra of 1,2,3 H and 3,4 He ejectiles following 1.2 GeV p-induced reactions on the Ta target as measured by NESSI collaboration at the COSY-Jülich accelerator [228] are shown for different angles in respect to the incident proton in Figure 2.33.

103

2 The Intranuclear Cascade Models 30° 10

75° 1

150° 1

H

1

H

H

1 −1

10

10−2 10

2H

2H

2H

3H

3H

3H

1 10−1 d2s/(dE·dΘ) (mb/(MeV sr))

104

10−2 1 10−1 10−2 10−3 10

4

4

He

4

He

He

1 10−1 10−2 1

3He

3He

3He

10−1 10−2 10−3 0

25

50

75

0

25

50

75

0

25

50

75

Ekin (MeV)

Fig. 2.33 Energy spectra of 1,2,3 H and 3,4 He for 1.2 GeV p+Ta. dots: experimental data, shaded histogram: calculated evaporation spectra, dashed histogram: pre-equilibrium protons as calculated by INCL2.0 [228].

The experimental data shown clearly feature the two components, an evaporation component dominant for all angles and at low kinetic energies and a high-energy component all the more pronounced the smaller the angle of the ejectile in respect to the incident proton is. In Figure 2.33 for theoretical description the INCL2.0 [170] INC code is coupled to the evaporation code GEMINI [229] (cf. Section 3.6) on page 159. Only for protons (upper panels) both components can be well described. Due to the lack of composite particle emission in the early phase of the reaction in the INCL2.0 model, the high-energy tails of the spectra for d,t,3,4 He are not described by the calculations. The shape of the calculated evaporation component (shaded gray histogram in Figure 2.33) however is well reflected also for composite particles.


The description of cluster ejectiles in the first phase of a nuclear reaction, i.e. the high-energy tails of the spectra in Figure 2.33, is a quite important aspect of INCL4, a feature missing in the previous versions of INCL and other codes designed for transport calculations with GeV hadron beams. The deficiency to accommodate emission of light clusters in the evaporation stage only is not a serious problem as far as global particle multiplicities are concerned. The analysis of experimental data has shown that in proton-induced reactions on heavy targets in the GeV range, the ratio of the number of nucleons appearing in the form of clusters emitted during the cascade stage to the total number of emitted nucleons, whatever their origin, lies between 5 and 10%. However, the lack of cascade light cluster emission appears more serious in view of technological applications. Indeed light clusters correspond to gaseous elements (H, He), which are liable to create voids or other damages in materials. Therefore, it is of utmost interest for the designers of (solid) spallation sources to have at their disposal a good model for the production of these elements. Another motivation for the incorporation of light cluster emission in the cascade stage is that for spallation reactions in the energy range under consideration, it is very hard to couple nucleon degrees of freedom and cluster degrees of freedom on a microscopic basis, i.e., to handle the formation of clusters from nucleons (and their possible destruction) via a microscopic and dynamical model involving the explicit effects of nuclear forces. Emission of light charged clusters prior to the eventual evaporation is generally described on a phenomenological basis, either by the standard coalescence model [230, 231] (in momentum space) or by percolation models applied at the end of the cascade stage [232]. In the approach realized in the current INCL4, the cluster emission relies on the microscopic phase space occupancy at the nuclear surface. They are emitted according to the following procedure: (1) When a nucleon hits the surface and satisfies successfully the test for emission, i.e., has sufficient energy and escapes reflection (after the usual test of comparing a random number with the calculated transmission probability through the appropriate barrier including Coulomb potential for the protons), it is tested to see whether it belongs to a possible cluster. Such a cluster is defined as a set of nucleons which are sufficiently close to each other in phase space. Actually, the candidate cluster is constructed, starting from the considered nucleon, by finding a second, then a third, etc., nucleon fulfilling the following condition: ri,[i−1] pi,[i−1] ≤ h0 ,

(2.48)

where ri,[i−1] and pi,[i−1] are the Jacobian coordinates of the ith nucleon, i.e., the relative spatial and momentum coordinates of this nucleon with respect to the subgroup constituted of the first [i − 1] nucleons. The following light clusters are considered: d, t,3 He,4 He. (2) Fast nucleons are checked for emission at R(pF ), in the outer fringes of the nucleus, where the density is very small and where they have little chance of being in a cluster. Particles are checked for emission so far away, since they may undergo collisions before reaching this place. To correct for this, the candidate is moved for

105

106


emission along its trajectory until it is at a distance D outside the sphere of radius R0 , before building the possible clusters. (3) A hierarchy between clusters is established for testing their possible emission. It is evident, from the way clusters are constructed, that if the candidate nucleon belongs to a given cluster, e.g., an α-particle, it also belongs to a lighter cluster. Clusters are checked for emission in the following priority list: 4 He > (3 He or t)> d. In other words, the largest candidate cluster is first tested for emission. The total energy of the cluster, including the potential energy, and corrected by the binding energy of the cluster, should be positive in such a way that a composite with positive kinetic energy can be emitted. Furthermore, it is checked whether the cluster can tunnel through the appropriate Coulomb barrier, comparing a random number with the relevant transmission probability. If these conditions are met, the cluster is emitted in the direction of its c.m. momentum. Energy conservation is fulfilled in this process. If the test for emission fails, then the next cluster candidate in the priority list is checked for emission on the same criteria, and so on. If the test for emission fails for all clusters, the original candidate nucleon is emitted (-resonances are not considered in the cluster formation). Summarized, this simple model appears as a kind of ‘‘surface-coalescence’’ model, compatible with two rather well-established features: the small probability of having pre-existing clusters inside nuclei, at least in medium-heavy and heavy ones, and the necessary dynamical generation of correlated clusters of nucleons near the surface before emission. Clusters can be emitted at any time during the cascade stage. This is to be in contrast with the composite emission in preequilibrium models, in which a cluster can be emitted from an uncorrelated target with a suitable probability. The cluster production model implemented in INCL4 utilizes the microscopic phase space distribution, as generated dynamically by the INC. It however contains some limited amount of phenomenology, since explicit coupling of individual nucleon and composite degrees of freedom is avoided and replaced by a geometrical construction, involving the introduction of two parameters h0 and D. In Ref. [117] which describes the surface-coalescence model implemented in INCL4 in detail, the values have been fixed to h0 = 387 MeV fm/c (=pF × 1.4) and D = 1.75 fm. The value of h0 roughly corresponds to selecting a unit volume of phase space. The value of D is such that the cluster is formed in a region of relatively low density on the average. In Figure 2.34 the INCL4 surface coalescence model coupled with the GEM evaporation model (see Section 3.4.1 on page 144 has been confronted with experimental data of the NESSI collaboration for p, d, t,3,4 He ejectiles [233] emitted in the 2.5 GeV p + Au reaction at different angles (30◦ , 75◦ , 105◦ , 150◦ ) in respect to the incident proton beam. The experimental data range to energies as large as 200 MeV, allowing a meaningful test of cascade emission. The comparison with model calculations of INCL4 coupled to GEM shows an overall agreement being satisfactory. The evaporation contributions are well described for deuterons d and tritons t. However, the 3 He evaporation spectrum at larger angles seems to be underestimated. The 4 He evaporation peak is well reproduced by GEM. Also the production of alpha-particles in the cascade stage is underestimated. The comparison of


10

: q = 30°

1 10

10

p

1

: q = 75°

−1

n

q = 30° q = 75°

−1

10 : q = 105°

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: q = 150°

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q = 105°

10−2

q = 150°

10−3

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10−4 0

50

100

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150 3

d2s/dΩdE [mb/sr Mev]

d

He

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10−2

10−3

10−3

10−4

10−4

10−5

200

0

50

100

10−5 150 200 0 Particle energy [MeV]

Fig. 2.34 Double differential cross sections for proton, neutron, and light charged cluster d, t,3,4 He-production in the 2.5 GeV p + Au reaction (symbols: experimental data [233], histograms: full lines INCL4+GEM model including the coalescence option, dotted lines: without coalescence). Different symbols

50

100

150

200

correspond to different emission angles as indicated. In each panel, the cross sections are given in absolute values for the smallest angle. They are multiplied by 10−1 , 10−2 etc., for the other angles, in increasing order. Note the different vertical scales (after Boudard et al. [117]).

107

2 The Intranuclear Cascade Models 102 Residue mass production cross section s(A) [mb]

108

Pb (1 GeV/A) + proton 10

1

10−1

0

25

50

75 100 125 150 Atomic mass number A

175

200

Fig. 2.35 Comparison of INCL4-ABLA model results for residue mass production in the proton+208 Pb system of 1 GeV energy with experimental results of Enquist et al. [68] with and without cluster production in the INCL4 model (histograms: full lines INCL4+GEM model including the coalescence option, dotted lines: without coalescence, dots: experimental results) (after Boudard et al. [117]).

calculations and experimental data in Figure 2.34 clearly indicates that the cascade production of protons and cluster particles is not negligible. As expected, clusters in the cascade stage are formed at the expense of neutron and proton production. This is particularly noticeable for protons (cf. Figure 2.34, upper-left panel) at small angles in the energy range spanning from 40 to 100 MeV, whose yield is clearly underestimated when the coalescence option is chosen in the INCL4 model. Figure 2.35 of Boudard et al. [117] shows a comparison of INCL4-ABLA model results for residue mass production in the proton+208 Pb system of 1 GeV energy with experimental results of Enquist et al. [68] with and without cluster production in the INCL4 model. There is only a small difference in the residues production. The lack of residue production in the low mass region suffers from the used ABLA model [221, 222] which is not able to evaporate light particles already mentioned in [172] (cf. Section 3.5). A comprehensive discussion and comparison of the results with and without cluster formation in the cascade stage is left for Ref. [117]. Also comparisons of INCL4 coupled to the ABLA evaporation code [221, 222] as an alternative statistical evaporation code are performed. On summarizing the following observations are made: including the coalescence concept, as expected, about 20% less free nucleons are emitted in the cascade stage. However, this is overcompensated by the emission of nucleons bound within the emitted clusters. In the cascade stage, the total multiplicity of the emitted nucleons, free or bound, is increased by 10% for neutrons and 15% for protons. With the cluster scenario activated in the code, the emission of nucleons is facilitated for three reasons:


1 removing a bound system costs less energy than removing all of its nucleons independently; 2 tunneling through the potential + Coulomb barrier favors the emission of clustered nucleons: for instance, the tunneling probability is smaller for an α-particle than for a proton, but the test is applied only once for the latter. The probability of the uncorrelated emission of two protons and two neutrons is equal to the square of the proton emission probability multiplied by the square of the neutron emission probability; 3 inherent to the scenario itself, probably the emission of a group of nucleons is favored, which otherwise would have somehow diverging trajectories. The excitation energy E ∗ at the end of the intranuclear cascade is not really changed when the cluster emission is added. Consequently, the evaporation multiplicities are not really changed either. In sum, the total yield of emitted neutrons (either free or bound) is increased by ≈4% and that of emitted protons by ≈7%. The production of light charged clusters in the cascade stage of spallation reactions opens the possibility of having a unified INC + evaporation model handling cluster emission on the same footing as nucleon emission, filling a long-standing gap. Nonevaporative light cluster emission has often been handled by introducing a so-called pre-equilibrium stage between the cascade and the evaporation stages. In this stage, based ordinarily on exciton models, emission of clusters is usually treated on a phenomenological basis, just by attaching cluster emission probabilities, generally fitted to experiment, to exciton configurations. The method presented here has thus the double advantage of being more microscopically founded and of allowing the emission of composites at any time. It remains the extension of the coalescence model implemented in INCL4 to the production of heavier composites (6,7 Li, 8 Be, 9,10 B, C,. . .) and the extension of the model to low incident energies (less than ≈100 MeV). 2.3.2 Assumptions, Limits, and Constraints on the Energy and Application Regime

The INCL model in its newest form (cf. Ref. [117, 172, 234]) is a well-validated INC model applicable in the energy range from 0.2 to 2.0 GeV for the incident energy per nucleon. As for any INC model the basic physical assumptions apply also for the INCL model as has been outlined in detail in this chapter. As already mentioned, the basic fundamental assumptions of the INCL approach resemble to a large extent the semiclassical microscopic description of a collision between a particle and a nucleus as outlined in detail in Section 2.2 for the BERTINI model. In the INCL model latest physics insights and nuclear data findings and improvements have been incorporated and some new features as described in Section 2.3 have been considered (diffuse nuclear surface, treatment of the Pauli blocking improved, progressive depletion of the Fermi sphere, collisions between moving spectator nucleons explicitly suppressed, rest angular momentum included, history of all

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particles followed as a function of time, self-consistent determination of the stopping time). As incident particles not only proton, neutron, and π + , π − or π 0 are considered, but the INCL model has been extended to light incident clusters of nucleons (deuteron, triton, 3 He, 4 He). The applicable incident particle energy regime in INCL is essentially subject of the same constraints and restrictions as the ones for Bertini, i.e., approximately 100–3000 MeV. In contrast to the BERTINI model, for INCL also versions for antiproton–nucleus interactions [212] had been developed. Unfortunately in latest releases the antiproton–nucleus description has been abandoned, such that the interested user would have to return to older releases. As for the description of high-energy composite ejectiles, a major step has been achieved in 2004 [117]. A surface percolation procedure of the nucleus has been developed which is proposed to describe the emission of light charged clusters during the cascade stage in nucleon–nucleus reactions. Clearly as mentioned earlier, in the family of INC codes the newest version of the INCL model appears to be the most sophisticated and elaborated code including latest hadron and meson reaction data, and reaction mechanisms. It therefore manifests apparently superior to, e.g., the BERTINI approach. The model is tested successfully against a large database (cf. Ref. [234] and references therein).

2.4 The ISABEL Model

The ISABEL INC model was developed by Yariv and Fraenkel [162, 164] during the years 1978–1981 and is implemented as an event generator in the particle transport code LAHET [193] and the system MCNPX [194]. The ISABEL model is an INC model based on the Monte Carlo method for hadron–nucleus and nucleus–nucleus interaction in the energy range from about 0.1 to 1.5 GeV/particle. The model allows in the original version as incident particles hadrons, pions, kaons, and antinucleons. The model is based on the VEGAS model of Chen et al. [11, 235] and on the ISOBAR model of Harp et al. [79, 161]. The model was further developed and extended to take into account heavy-ion reactions [162, 164, 167, 236], including additional options for the nuclear potential density distributions [163], subthreshold π 0 production [166], and a revised Sternheimer–Lindenbaum isobar model for pion production [167, 237, 238]. 2.4.1 Features of the Model

The nucleon–nucleon cross sections used in ISABEL are free nucleon–nucleon cross sections via the parameterization used in the VEGAS model [11, 235]. The differential cross sections for the intranuclear nucleon–nucleon collisions are obtained by interpolation of tabulated values, which give normalized differential cross


sections for elastic scattering, together with the corresponding total cross sections for neutron–proton collisions, for proton–proton collisions, and neutron–neutron collisions [11]. The pion production and pion absorption are included in the ISABEL model via the 33 -resonance formation as a pion–nucleon isobar in nucleon–nucleon scattering as given by Harp [79]. This is very similar to the realization of the BERTINI and INCL models (cf. Sections 2.2 and 2.3). A physics model for antiproton–nucleus interactions to describe antinucleon annihilation is also employed in the ISABEL model (cf. Ref. [165]). 2.4.1.1 The Nuclear Model • The nuclear charge distribution: the original VEGAS model [11] uses nuclear charge distributions as measured by Hofstadter [199, 239] according to a Fermi distribution given by

ρ(r) = ρ0 /[1 + exp(r − c)/a],

(2.49)

where the ‘‘sharp radius’’ is c = 1.07 · A1/3 fm and the so-called skin thickness is a = 0.55 fm. These parameters are identical to the BERTINI model (cf. Section 2.2). The ISABEL model includes some additional options for the nuclear charge density distributions named as the Yukawa-folding distributions [163], which takes into account a reduction of the nuclear potential energy in such cases where the finite range of the nuclear force and the diffuse nuclear surface become important. A step function distribution is used to approximate the nuclear charge distribution. The target nucleus is divided into several concentric zones, e.g., 8 to 16 zones, each of constant density, respectively. The ratio of the proton to the neutron density is assumed to be Z/(A − Z) in all concentric zones of the nucleus (cf. Sections 2.2 and 2.3). The relative densities in the 8–16 constant density zones are fitted to a folded Yukawa sharp-cutoff density distribution [163] with a sharp cutoff radius cISABEL = 1.18 · A1/3 fm different to the VEGAS model. • The momentum distribution: as assumed by the BERTINI and the INCL model, the momentum distribution of the nucleons inside the nucleus is a degenerate Fermi gas distribution with the Fermi energy given as EFi = (2 /2m) · (3π 2 · ρi )2/3 ,

(2.50)

where the subscript i stands for either protons or neutrons, m is the nucleon mass, and ρi is the density of protons and neutrons. • Antiproton–nucleus interactions: for antinucleons of kinetic energy EN GeV, the following parametrizations are used (cf. Ref. [165]):

NN σtotal = exp[4.5485 · exp(−0.060 · ln EN ) (mb), NN σelastic

= exp[4.6052 − 1.0365 · exp(0.380 · ln EN ) (mb),

(2.51) (2.52)

111

112


σ

pp−→nn

charge-exchange

= 4.40/ EN

dσ/dt = exp(b · t),

(mb), and

b = 12.94 + 39.03 · exp(−2.075 · Plab ),

(2.53) (2.54) (2.55)

where t is the four-momentum transfer in (GeV/c2 ) and Plab is the laboratory antiproton momentum in (GeV/c). These parametrizations are valid in an energy range 0.1 GeV < Ep < 3.0 GeV. The Coulomb interactions between the target nucleus and the incident or emitted charged particle is explicitly considered [167]. The refraction of the particles just outside the nuclear boundary is determined by taking into account the Coulomb potential there. 2.4.1.2 The Time-Like Basis Cascading of Particle–Nucleus Interactions An important difference to other INC models compared to the ISABEL model lies in the sequence in which the history of particles taking part in the intranuclear cascade is followed. Previous models, i.e., of Metropolis et al. [4, 5] or of BERTINI [12, 77] follow one particle at a time from its first interaction with another cascade particle or its entrance into the nucleus until either its energy falls below a certain cutoff energy or it escapes from the nucleus. In the ISABEL model the development of the intranuclear is followed on a time-like basis [167, 238]. The method to follow an intranuclear cascade on a time-like basis was originally described and used in the VEGAS model by Chen et al. A short description of the main steps of the method is given below. More details are given in Ref. [11]. • At the beginning of the intranuclear cascade the total nonelastic cross sections σtotal-proton and σtotal-neutron of the incident particle with a stationary proton and neutron in the laboratory system, respectively, are determined and an approximate mean free path λ of the incident particle is then determined as

λ = (A/ρmax )/ Z · σtotal-proton + (A − Z) · σtotal-neutron ,

(2.56)

where ρmax is the total nuclear density in the center of the nucleus. Equation (2.56) gives a sufficient good estimate of the mean free path for the selection of a time interval, if there are no resonances in σtotal-proton and σtotal-neutron . The first time interval τ is then given by τ = λ/nβ,

(2.57)

where β is the velocity of the incident particle. The nucleon distribution is assumed to be that of a degenerated Fermi gas. A good estimate of n is n =20–30 as calculated in Ref. [11]. • An impact parameter is randomly chosen. The momentum from the interaction of the incident particle with a proton or neutron of the Fermi zone of the nucleus is determined.


• The path length is determined according to Eq. (2.57) as λ/n = β · τ . If no interaction takes place, the particle is transported by the path length λ/n = β · τ , a new value of τ is calculated for the next time interval, and a new candidate for the interaction is chosen and the process is repeated until the particle makes an interaction or escapes from the nucleus. • If an interaction takes place in a given time interval, the position of the interaction is determined, the particle is transported to this point, and the energy of the outgoing particle is calculated. Outgoing particles are checked for the Pauli principle violation. • At the end of each time interval a new candidate for an interaction partner is chosen and a new time interval τi is for each particle determined: τi = λi /nβi .

(2.58)

The smallest τi is the one that is chosen for the next time interval τ = min[τi ].

(2.59)

• The Fermi sea is now depleted and the cascade starts again. The whole cascade process stops when all the cascade particles leave the target volume or fall below a certain energy cutoff, which sets the limit between the cascade stage of the intranuclear cascade and the pre-equilibrium and theevaporation stages (cf. Section 1.3.4). The cutoff energy for neutrons must be at least its separation energy EF + EB and the cutoff energy for protons must be the separation energy plus the Coulomb barrier EF + EB + EC , where EF , EB , and EC are the Fermi, the binding, and the Coulomb barrier energies, respectively. The advantage of the time-like basis procedure is the possibility in changing the global properties of the cascade as the cascade proceeds. As the cascade develops the density of the participating Fermi sea is depleted. In the ISABEL model a prescription for this density rearrangement named as slow density rearrangement is foreseen [167]. After each collision a hole of radius rmin is punched in the density distribution of the configuration space of the nuclear system around the actual position of the nucleon–nucleus interaction. No more interactions are allowed in this hole. The best rearrangement prescription was found to be isospin-dependent holes with a radius of rmin,I = 1.1 fm for nucleon–nucleus and nucleus–nucleus collisions. The depletion of the Fermi sea affects the Pauli blocking. Two possibilities of Pauli blocking procedures similar to the INCL model (see Section 2.3.1.6 on page 97) full Pauli blocking and partial Pauli blocking can optionally be chosen in the ISABEL model. 2.4.1.3 Nucleus–Nucleus Collisions The extension of the ISABEL model for nucleus–nucleus collisions uses several assumptions [162, 164]. In the nucleus–nucleus collision one has two Fermi seas interacting with each other.

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• To calculate the interaction between nucleons of the projectile Fermi sea and the nucleons of the target Fermi sea the projectile Fermi sea is assumed to be a gas of discrete particles whose position in space and momenta can be randomly chosen from the appropriate distribution. • For calculating the interaction between the discrete cascade particles and the Fermi sea of the projectile or the target, the latter two distributions are considered continuous. • The above procedure treats the collision in the same manner as in nucleon–nucleus INC calculations. It ensures the equal treatment of projectile and target, i.e., the Lorentz invariance of the calculation. Following the interactions processes on a time-like basis has also in nucleus–nucleus collisions the advantage to include interactions between cascade particles –cascade–cascade interactions (Ref. [164]) – which is neglected in previous versions of the ISABEL model (cf. Ref. [11, 162]). In addition, the ISABEL model considers the reflection and refraction of cascade particles at the zone boundaries of the nucleus. 2.4.2 Assumptions, Limits, and Constraints on the Energy and Application Regime

The important features of the ISABEL model are to follow the intranuclear cascade on a time-like basis and to accommodate nucleus–nucleus interactions. Many other features are similar to the models of VEGAS or BERTINI (cf. Ref. [11, 12, 155, 235]) and especially of the INCL model (cf. Ref. [117, 170–172, 240]). The ISABEL model is only applicable up to incident particle energies of 1.5 GeV per nucleon. Antinucleon–nucleus interactions to calculate the very strong attraction between antiprotons and nuclei have also been employed in the model [165].

2.5 The CEM (Cascade-Exciton Model) Approach

The CEM model (cascade-exciton model) was originally developed at the JINR, Dubna, by Gudima, Mashnik and Toneev [173] during the 1980s and further improved by Mashnik and co-workers [41, 174, 175, 241–249] during the last few years. The CEM model considers the spallation reactions as proceeding through three stages [173]: (i) cascade, (ii) pre-equilibrium, (iii) equilibrium. The last stage considers the compound nucleus evaporation and fission. All these three components contribute to a reaction of a high-energy particle interacting with matter, which has already been discussed in Section 1.3.4. In this respect the CEM model is different to pure cascade models, e.g., the models of BERTINI, INCL or ISABEL. The improved versions of the CEM model are named as CEM03 and LAQGSM03 (Los Alamos quark gluon string model) [175, 247, 249, 250]. The improved versions of the CEM model are also able to treat nucleus–nucleus reactions [249, 251].


The cascade stage of the interaction is based on the standard version of the DUBNA model [10] (see also Section 2.6.1) with further improvements concerning the energy momentum conservation, a better approximation of the total reaction cross sections and using real binding energies for nucleons during the cascade stage [175]. The Los Alamos quark gluon string model LAQGSM is a further development of a model of Amelin et al. (Ref. [252] and references therein) and describes particle- and nucleus-induced reactions up to about 1 TeV/nucleon. The excited residual nucleus remained after the emission of cascade particles determines inside the CEM model the starting point for the second stage, the pre-equilibrium stage of the reaction. The subsequent relaxation of the nuclear excitation is then treated by an extended modified exciton model MEM [253] of the pre-equilibrium decay, which also includes a description of the equilibrium evaporation third stage of the reaction. The up to date version of the CEM03 model [175] includes in addition the evaporation model GEM [254, 255] to describe fission and light fragment production and the sequential binary-decay model GEMINI of Charity [229, 256, 257] (see details in Sections 3.4 and 3.6). 2.5.1 Features of the CEM Model

The three components of the CEMmodel, cascade, pre-equilibrium, and equilibrium, determine the inclusive cross sections of the spallation reactions in the following way [173]: σ (p )dp = σnonelas [Ncascacade (p ) + Npre−eq (p ) + Neq (p )],

(2.60)

where p is the linear momentum of a single particle state specified by its position vector r . The values N represent the total number of particle interactions considered by the cascade, pre-equilibrium, and equilibrium process, respectively. The nonelastic cross section σnonelas is not taken from experimental data, but is calculated within the cascade part of the CEM model by using the geometrical cross section of the target nucleus σgeom σnonelas = σgeom · Nnonelas /(Nnonelas + Nelas ),

(2.61)

where Nnonelas and Nelas are the total nonelastic and elastic simulated events. The value of the geometrical cross section is given as σgeom = π · R2 with R the radius of the last zone of the target nucleus. This approach provides a good agreement with available data at incident particle energies above about 100 GeV, but this is not valid at lower incident particle energies. For all incident protons and neutrons in the latest CEM model [244] the NASA systematics by Tripathi et al. [119–121] is used (cf. Section 1.3.7.5). As shown in Figure 2.36, these systematics describe much better the total reaction cross sections for neutron and proton-induced reactions at incident energies below about 100 MeV and correspondingly any other partial cross sections.

115


1200 n + 27Al

snonelastic [mb]

1000 800 600 400 200 0 10

800 snonelastic [mb]

116

Barashenkov '93 CEM2k CEM97 HMS-ALICE

100 1000 Incident neutron energy

p + 27Al

600 400 Barashenkov '93 CEM2k CEM97 HMS-ALICE

200 0 10

100 1000 Incident protron energy

Fig. 2.36 Total nonelastic reaction cross sections for neutron- and proton-induced reactions on Al calculated by CEM2k and CEM97 compared with experimental data compiled by Barashenkov [108] and calculations from the ALICE model of Blann [258, 259] (after Mashnik and Sierk [244]).

The INC DUBNA model version (see Ref. [10] and Section 2.6.1) inside the CEM model includes assumptions, which are similar to the BERTINI and INCL models. The target nucleus is divided into seven concentric spherical zones different to the BERTINI model which uses only three zones. • The CEM model uses a different approach to decide when a particle is leaving the cascade. For this purpose an effective local optical absorptive potential Wopt.mod. (r) is defined from the local interaction cross section of the particle, including a Pauli blocking procedure. In models such as BERTINI, ISABEL or INCL, fast particles are followed down to some minimal energy, the cutoff energy Ecut (or in INCL to a certain cutoff time). This energy is usually about 7–10 MeV above the nuclear potential, below which particles are considered to be absorbed. The optical potential Wopt.mod. (r) calculated in the cascade model of CEM is related to its experimental value Wopt.exp. (r), which is obtained from analysis of data on particle-nucleus elastic scattering [173]. The difference of these potentials is given by a parameter P: P = |(Wopt.mod. (r) − Wopt.exp. (r))/Wopt.exp. (r)|.

(2.62)


When P increases above an empirically chosen value, the particle leaves the cascade, and is then considered to be an exciton. In the CEM model a fixed value P = 0.3 is used. With the value it was found that the cascade stage in CEM is generally shorter compared to other cascade models. • Recently published better systematics for pp, np, and nn interactions by Boudard and Cugnon et al. of the INCL model [172] and of Duarte et al. of the BRIC code [260, 261] are implemented in the CEM model and used instead of the 30-year-old data of the DUBNA INC model [10]. • When the cascade stage of the reaction is completed the CEM and LAQGSM models also use a coalescence model (cf. Section 2.3.1.9 and Refs. [250, 262]). The high-energy d, t, 3 He, and 4 He ejectiles are created by final state interactions among cascade particles [250]. The subsequent relaxation of the nuclear excitation is treated in terms of an improved version of the modified exciton model of pre-equilibrium decay followed by the equilibrium evaporation-fission stage of the reaction using the models GEM and GEMINI (see Sections 3.4 and 3.6). If residuals after the INC cascade have atomic mass numbers with A ≤ 12, a Fermi break-up model is used [263, 264] (see Section 3.8) to determine their further disintegration instead of using the pre-equilibrium and evaporation models. The Fermi break-up model is much faster and the results are very similar to the more detailed models [249].

Figure 2.37 shows the benchmark examples of calculated neutron production spectra from the interaction of protons with a thin 208 Pb target at incident energies of 0.8 and 1.5 GeV compared with experimental data from Ishibashi et al. [265]. The CEM model describes well neutron spectra for proton–nucleus collisions. Similar results are obtained for other incident energies and for other target nuclei and even for nucleus–nucleus collisions (see Refs. [175, 249, 257]).

2.5.2 Assumptions, Limits, and Constraints on the Energy and Application Regime

As mentioned before, the cascade exciton model (CEM) is not a pure cascade model compared to the BERTINI, INCL, or ISABEL models. Many different physics simulation models are merged in CEM: the Dubna cascade model, the BRIC model, an EXCITON model, the evaporation models GEM and GEMINI, the RAL fission model, a quark gluon string model, a Fermi-breakup model, a Coalescence model, and a model to describe high-energy photonuclear reactions. The CEM model is very well validated although with its monolithic structure it may be difficult to validate the significance of different physics implications. The CEM model covers an energy regime from eV to several TeV. The advantage of the model is lying in the prediction of residual nuclei, fission fragments, low- and high-energy light particle such as d, t, 3 He and 4 He, and product yields and recoil properties from high-energy photonuclear reactions.

117


108 0.8 GeV protons on Pb

107 106 105 Double-differential neutron-production cross section d2s / dE/dΩ [mb/MeV/sr]

118

104 103 102 101

15 deg × 105 30 deg × 104 60 deg × 103 90 deg × 102 120 deg × 10 150 deg × 1 CEM2K CEM97

100 10−1 10−2 10−3

1

10

100

1000

108 1.5 GeV protons on Pb

107 106 105 104 103 102 101

15 deg × 105 30 deg × 104 60 deg × 103 90 deg × 102 120 deg × 10 150 deg × 1 CEM2K CEM97

100 10−1 10−2 10−3

1

10 100 Neutron energy Elab [MeV]

Fig. 2.37 Comparison of the measured double differential neutron production cross sections of Ishibashi et al. [265] from incident protons of 0.8 and 1.5 GeV energy with CEM2k and CEM97 calculations (after Mashnik et al. [175]).

1000

2.6 Other Intranuclear-Cascade Models


The previous sections on pages 64, 88,110, and 114 were devoted to such INC models which in general are currently utilized and implemented in the most important and commonly used Monte Carlo simulation systems for particle transport and interaction with matter (see also the introduction of Chapter 2 on page 63). For completion in the following section, a description and an overview of similar models are given mainly based on the previous ones. These modifications were sometimes undertaken for the purpose of special applications such as, e.g., cosmic ray studies, experimental background investigations, shielding of high-energy particles, and detector development and simulation in medium- and high-energy physics. 2.6.1 The Dubna Models

The INC models, in this section called the Dubna models, were independently developed in Dubna, Russia, during the 1960s and further in the 1970s. The first advanced Dubna INC model developed by Barashenkov [266], similar to the BERTINI model [13] was then further developed by Gudima of the JINR, Dubna, during the years 1968–1969 [267]. First publications only in Russian are internal reports of the Joint Institute for Nuclear Research (JINR), Dubna (see Refs. [10, 78, 101, 266, 268]). The Dubna model was further developed by young people from Barashenkov’s school and also implemented as the INC model in many particle transport codes and models developed in Dubna, e.g., in the SHIELD system of Sobolevsky et al. [269, 270], and in the CEM model of Mashnik [32, 41, 174] which was discussed in Section 2.5. An INC model to describe heavy-ion collisions to study the behavior of hot and dense matter was also developed by Toneev and Gudima [262] during the 1970s based on the DUBNA INC model. Furthermore, INC models considering the trailing effect in intranuclear cascades, i.e., depletion of the nuclear density due to the knocking out of intranuclear nucleons by the cascade particle shower, were studied to determine explicitly the time coordinate during the development of the intranuclear cascade discussed in Ref. [271]. It should be noted here that the time coordinate was first introduced into INC model calculations originally published by Chen et al. [11], where the cascade particle positions and their interaction points inside the nucleus were fixed for a series of discrete time moments separated by a time interval t so small that the results of the calculations are practically independent of a further decrease of the time interval t. The whole process could be considered as continuously changing in time. It was found that the value of t should be smaller than the time which is necessary for a cascade particle to cover about 1/20–1/30 of its mean free path. That means the calculation time needed for cascade calculations would be exceptionally long.

119

120


The approach given in Ref. [271] considers the time coordinates values only at interaction points. Intermediate times ti are included implicitly by the means of the integral t =

2

dS/V(S),

(2.63)

1

which determines the interval between two successive collisions of a cascade particle with intranuclear nucleons. V(S) is the particle velocity changing along a trajectory S as a function of the intranuclear potential variations. If an intranuclear collision is permitted by the Pauli principle, then for each of the created particles an interaction point is sampled and the corresponding moment of time ti = t + ti is calculated. Here t is the time of the particle’s creation and t is determined by Eq. (2.63), where the integral is replaced by the sum

2

dS/V(S) =

1

Sj /V j .

(2.64)

j

In Eq. (2.64) the Sj = Sj+1 − Sj are projections of the internucleon distances on the direction of the cascade particle momentum (Figure 2.38) and V j = 1/2(Vj + Vj+1 ) is the average velocity of this cascade particle in the interval Sj . The summation is performed over all nucleons inside a cylinder with the radius Rint = r + λ, where r 1.3 fm is the radius for the strong interaction and λ is the deBroglie wavelength defined by Eq. (1.11). A similar method of time tracing the intranuclear cascade has been used by Bertini et al. [158], which is not included in the standard BERTINI model [12, 77]. Some of the common features of different Dubna models are as follows: • The matter density is described by a Fermi distribution with parameters taken from electron–nucleus scattering experiments [199, 239]. The nuclear outer radius is assumed to be 1.3 · A1/3 × 10−13 (cm), where A is the mass number of the target nucleus. The radial step distribution – the number of zones – of the nucleus is assumed as a three-step distribution similar to the Bertini model already introduced in Figure 2.6 on page 71. t + ∆ tn

t Rint

S1

S2

S3

Sn

Fig. 2.38 Movement of a cascade particle within a nucleus between successive collisions with intranuclear nucleons at the planes S1 and S2 . The particle is able to interact with any nucleon inside a cylinder with radius Rint .


A newer version of the Dubna model called CASCADE uses a seven-step distribution of the nuclear density [160]. The model takes the Pauli exclusion principle into account. • The hadron interactions are sampled by means of tables of phenomenological parameters [10]. Energy–momentum conservation and the decrease of the internuclear density due to a knock-out of nucleons during the intranuclear cascade which results in a depletion of the nuclear density [271] are determined and the time dependence of the intranuclear cascade is considered. The sampling of the nucleon coordinates inside the nucleus is carried out by the method of random rejection according to the Wood–Saxon nuclear density distribution ρ(r) with parameters from electron scattering experiments. • The interaction of the incident particle with the nucleus is approximated as a series of successive quasifree collisions of nucleons or pions with the nucleus similar to that in the BERTINI model. The model uses elementary nucleon–nucleon and pion–nucleon cross sections given in Ref. [108] simulating angular and momentum distributions of secondary particles by polynomial expressions with energy-dependent coefficients [10]. Pion production and pion absorption processes are treated comparably to the Bertini model. Intercomparison of the Dubna, BERTINI, and VEGAS models A comparison of the Dubna model of JINR, Dubna, Russia, with the Bertini model of ORNL, USA [12, 155], and the VEGAS model of BNL, USA, of Chen et al. [11, 235] was made. Results are given in Ref. [13]. The three models were used to study the interactions of protons with incident energies of 150 and 300 MeV on thin 27 Al and 181 Ta targets. Compared were: (1) energy distributions of emitted protons at angles of 30◦ ± 5◦ and at 80◦ ± 5◦ , (2) excitation energy distributions for various residual nuclei produced in the INC process, and (3) mass yields produced. All of the models represent the nuclear radial density distribution by a series of radial zones inside the nucleus which approximates a Fermi distribution consistent with Hofstadter’s data on the radial variation of the nuclear charge distribution. The number of radial zones is different. The Bertini and the Dubna models use three zones whereas the VEGAS-Chen model uses eight. In all models the radial density distribution determines concentric zones, each having a constant density. The proton and neutron density, and the proton and neutron momentum distributions were determined as described by the BERTINI model on page 64. In the BERTINI and Dubna models the potential acting on the cascade particles in a certain zone is the sum of the proton or neutron Fermi energy in this zone and the average binding energy of a nucleon in the nucleus. The kinetic energy in both models of the cascade nucleon is increased or decreased as it travels from a potential region to another, but the direction of the nucleon is unchanged. Thus in these models refraction or reflection of cascade nucleons at potential boundaries is neglected.

121

122


In the VEGAS-Chen model the potential is energy dependent and of the form V = Vzone · (1 − E/Emax ),

(2.65)

with E < Emax , V = 0 for E ≥ Emax , E is the kinetic energy of the nucleon and Emax is taken to be 100 MeV. The zone-dependent potential Vzone is given as Vzone = −(EF-zone + B)/[(1 − (EF-zone /Emax )],

(2.66)

where EF-zone is the Fermi energy in the particular zone for the same type of particle as the cascade nucleon and B is the average binding energy of a nucleon in the nucleus. The three different INC simulations compared in Ref. [13] give remarkably similar results in the particle spectra. Large discrepancies are observed in the excitation energy distributions of the cascade residual products. 2.6.2 The H¨ anssgen–Ranft Model

Hänssgen and Ranft have chosen another approach to model the intranuclear cascade. A description of the model is given in Refs. [176–178] and some technical details of the program organization can be found in Refs. [272–274]. According to their ansatz, the nuclear cascade below 5 GeV kinetic energy can be split into two processes, namely the emission of cascade nucleons (simulated by the NUCRIN model [273]) and ‘‘one inelastic’’ hadron–nucleon collision of the projectile with ‘‘one’’ of the target nucleons at a reduced energy (simulated by the HADRIN model [274]). The authors justify the assumption of only one inelastic collision at a reduced energy with the experimental fact that the number of π − produced in inelastic hadron–nucleus collisions is less than the corresponding number for hadron–nucleon collisions. The average energy available for the emission of cascade nucleons and for the excitation of the residual nucleus is parameterized as a function of the atomic mass number A of the target nucleus and the kinetic energy E0 of the incident particle or projectile. For every event, the cascade energy Ecascade and excitation energy Eex are sampled from Gaussian distributions at around average values. The remaining energy is used to simulate an inelastic hadron–nucleon collision. The inclusive distribution of the cascade nucleons is assumed to factorize according to d2 Ni /dE d = fi (A, E0 , E) · g(A, E, ).

(2.67)

The index i denotes proton or neutron, E is the kinetic energy, and is the emission angle of the cascade nucleon. After sampling E from the distribution fi , the emission angle is sampled from the normalized function g. This procedure is repeated until the cascade energy is less than the binding energy of the nucleons in the target nucleus. Then, the remaining cascade energy as well as the kinetic


energy of all particles falling below some cut-off value is added to the excitation energy of the residual nucleus. For the modeling of inelastic hadron–nucleon collisions the most important assumption is that all inelastic channels are produced by quasi two-body reactions which produce one or two resonances in the intermediate state according to one of the following equations: h1 + h2 −→ h3 + h∗4 h1 + h2 −→ h∗3 + h∗4 h1 + h2 −→ h∗3 .

(2.68)

Here, h1 and h2 denote the incident particle or projectile and the target nucleon, respectively. The parameters h∗3 and h∗4 denote the produced resonances in the intermediate state. The production cross sections of all resonances contributing more than 2% to the total inelastic cross section are taken into account. The scattering angles of the intermediate state resonances are determined from the four-momentum transfer t, which are sampled from the distribution dσ/dt = N · exp[B(M) · t],

(2.69)

where t is the four-momentum transfer t = (p1 − p3 )2 , N is a normalization constant, and B(M) a mass-dependent slope parameter. After the type and scattering angle of the outgoing resonances are determined, a decay algorithm (DECAY) determines their subsequent decay [272]. Up to three decay particles are considered and the decay channels are sampled from tabulated branching ratios. After the decay which is assumed to be isotropic in the rest system of the decaying particle, the products are Lorentz transformed into the laboratory system. This procedure is repeated until all particles are stable. It should be noted that the cascade particles have no kinematical correlation to each other, since the cascade is not followed in detail but sampled from inclusive distributions. In contrast, in the microscopic models, the BERTINI model [12], the VEGAS -Chen model [11, 235], the ISABEL model [164, 167], and the INC-Liège model [117, 170, 172, 212] energy and momentum conservation is satisfied at each reaction vertex in the nucleus. This also allows for a simple implementation of total momentum conservation, which does not hold for the NUCRIN model [273]. 2.6.3 The MICRES Model

The MICRES model [275–277] was developed to overcome the limitations of INC models in considering all resonances in elementary collisions contributing more than 2% to the total cross section up to kinetic energies of 5 GeV. In addition, angular distributions based on phase shift analysis are used for elastic nucleon–nucleon as well as elastic and charge exchange pion–nucleon scattering.

123

124


Also kaons and antinucleons can be treated as incident particles. A similar model was also designed with a revised pion production model and described in Ref. [278]. For the NUCRIN model of Hänssgen and Ranft, see Section 2.6.2, the way of emitting cascade nucleons is unsatisfactory, since it leads to the loss of the kinematical correlations which is important when treating problems like estimation of backgrounds in exclusive experiments. As shown below, it even cannot account for particle production spectra to the same level of accuracy as more microscopic models. On the other hand, for certain applications the BERTINI or INCL Liège models are limited to relatively low energies due to the fact that only the formation of the 33 -resonance is included. This must not be a limitation for more inclusive problems, e.g., to calculate the energy deposition in a thick absorber or for calorimeter simulations, but may again lead to discrepancies when using the model for background simulations in exclusive experiments. Here, the correct multiplicity in each event is essential. Applying trigger conditions may suppress those final states which make up the major part of the total cross section but do not affect other channels which contribute relatively little to the total cross section. The argument that the latter can be neglected does not hold anymore. This is especially true for light nuclei where larger multiplicities stem only from one- or two-step processes and the emission of additional cascade nucleons is not important. To overcome those shortcomings the positive features of NUCRIN/HADRIN and the BERTINI model are combined, i.e., the inclusion of all resonances contributing more than 2% to the total cross section in hadron–nucleon collisions and the microscopic description of the intranuclear cascade is considered. The precision of the resonance model used inside the HADRIN model is shown in Figure 2.39 where measured and calculated total cross sections for various channels are given for pion as incident particles. For kaon projectiles the accuracy is the same as seen in Ref. [176]. In most cases the data are reproduced within the experimental error. The basic idea for the improvement is to use the more sophisticated HADRIN model for the elementary hadron–nucleon collisions inside the nucleus and to keep the microscopic description of the cascade and the nuclear model of the BERTINI model basically unchanged. The ansatz in the MICRES model is very similar to that given in Ref. [178]. Another advantage of this procedure is the extension of the BERTINI model to all other particles which are accepted by HADRIN, namely antinucleons and kaons. For this aim a model of Cloth et al. [279] was used as a starting version and all necessary modifications were performed to include the HADRIN description for hadron–nucleon collisions. In addition, it was found that the parameterization of the differential cross section equation (2.69) as used in HADRIN is not sufficient for reproducing the elastic nucleon–nucleon collisions inside the nucleus, i.e., the intranuclear cascade, to the same accuracy as in the original INC code. Therefore, normalized angular distributions have been calculated from nucleon–nucleon phase shifts as given by Arndt et al. [280, 281]. For the elastic and the charge exchange scattering of pions the phase shifts of the Helsinki group are used [282]. By assuming the principle of isospin invariance, the following relations hold:


2

p++p exclusive channels total inelastic pp+p0 np+p+ pp+p+p− pp+p+p−p0

Cross section [µb]

104

4

2

103

0.6 0.7 0.8 0.9 1.0

2.0

3.0

4.0

Particle momentum plab [GeV/c ]

Fig. 2.39 Comparison of experimental and calculated exclusive total reaction cross section as given by Hänssgen and Ranft [176].

dσ d (pp

→ pp) =

dσ d (nn

→ nn)

dσ + d (π p

→ π + p) =

dσ − d (π n

→ π − n)

dσ − d (π p

→ π − p) =

dσ + d (π n

→ π + n → π + n)

dσ 0 d (π p

→ π 0 p) =

dσ 0 d (π n

→ π 0 n)

dσ + d (π n

→ π 0 p) =

dσ − d (π p

→ π 0 n)

=

dσ 0 d (π p

→ π + n)

=

dσ 0 d (π n

→ π − p).

(2.70)

The angular distributions for elastic and charge exchange scattering of charged pions and for elastic scattering of nucleons on proton have been incorporated into MICRES by tables in the energy range from 0.05 to 3 GeV/c laboratory momentum of the projectile in bins of 50 MeV/c. A linear interpolation is used in between, and for higher momenta the distribution at 3 GeV/c is used. For all other channels the simple parameterization of Eq. (2.69) is done. Since in the energy range under consideration, pions and nucleons are by far the most important shower particles, there should be no need for a similar precise description of other particles. Also for

125


200

stot [mbarn]

175 150 p+ p

125

p− p

100 75

pp

np

50 25 0

0

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

3.6

Particle momentum plab [GeV/c ] Fig. 2.40 Total cross sections for nucleon–nucleon and pion–nucleon scattering as used in theMICRES model.

30 Total cross sections stotal [mb]

126

pp→pp (elastic) MICRES

25 20 pp→pnp +

15 10

pp→ppp0

5 0

0.5

1

1.5

pp→ppp+p− 2 2.5

3

Energy [GeV] Fig. 2.41 Calculated pp-scattering cross sections with the MICRES model compared with experimental data [214].

the total cross section of pions and nucleons tables were calculated from the phase shifts mentioned above. Here an energy binning of E = 20 MeV/c from 0 to 4 GeV/c is used. Also in this case, for higher momenta the cross section at 4 GeV/c is used. This is justified because there is only a very small momentum dependence of σtot in the GeV energy range (see Figure 2.40). Calculations of pp-scattering cross section [283] with the MICRES model were compared with experimental data of Ref. [214]. Figure 2.41 shows the values of the absolute cross sections of the most important exit channels of the pp-scattering from [214] at energies of 793, 1500, and 2589 MeV. The cross sections are normalized to the total cross section. The elastic scattering cross section and the pnπ + are in good agreement with the experimental results, whereas the cross section ppπ 0 is lower by a factor 3 at lower energies and the cross section ppπ + π − is a factor 2 higher at higher energies. The main goal for the development of the MICRES model was to provide an event generator to support experimentalists in detector design and data analysis of experiments in the medium energy range up to 5 GeV/c. The MICRES model was therefore implemented into the GEANT detector simulation package [100, 284].

2.7 Alternative Models

The EDDA collaboration [285] has performed a great number of simulations to test the validity of the MICRES model, has simulated the EDDA experiment during its design and has estimated the detector performance especially the contributions concerning the background [277, 283]. The essential steps in simulating inelastic hadron–nucleus collision by the MICRES model are described in more detail in Ref. [277].


In addition to the INC models, the presently most commonly used alternative models are microscopic models which follow the time evolution of the onebody phase space distribution and models which are based on n-body molecular dynamics. The microscopic models for the one-body Wigner phase space have different names although they solve the same equation. The models differ in their technical realization, i.e., in the numerical algorithms or the computer programs. The models are named as Boltzmann–Uehling–Uhlenbeck (BUU) as cited in [187–189, 286] and as Vlasov–Uehling–Uhlenbeck (VUU) as cited in [190, 286, 287]. The approach beyond the one-body description is the quantum molecular dynamics (QMD) model as cited in [183, 184, 191, 288–291] which is an n-body theory simulating ion reactions at intermediate energies on an event-by-event basis. The QMD model has the advantage to take into account fluctuations and correlations that means calculating many-body processes by treating the formation of complex fragments. The QMD model is also capable of analyzing on an event-by-event basis ion reactions similar to the methods which are used for the simulation of INC methods [191]. Although the QMD approach was developed for heavy ion collisions, it is interesting to use the QMD approach for spallation reactions on nucleon–nucleus, meson–nucleus, or lightion-nucleus-induced reactions [184–186, 292]. Since QMD and BUU calculations for proton–nucleus reactions generally are CPU consuming models in particular for nuclei with a large number of nucleons, they still do not find routine applications in spallation relevant issues so far albeit the physics implemented is most realistic. A detailed description of the quantum molecular dynamics approach shall not be contents of this book. Therefore, we restrict ourselves to giving some important references and a short overview about QMDs in the next sections. 2.7.1 The Quantum-Molecular-Dynamic (QMD) Model

The QMD theory includes, in a self-consistent way, many important aspects of the nucleon-induced reaction mechanisms of particle interactions at the intermediate energy range [185], i.e., (1) a realistic momentum distribution of the nucleons inside nuclei, (2) entrance and exit channel refraction, (3) Coulomb deflection, (4) multistep processes, (5) multiple pre-equilibrium emission processes

127

128


(MPEs), (6) variation of the mean-field potential due to particle hole excitation and particle emission, (7) transition between unbound and bound states, and (8) energy-dependent, anisotropic N–N elastic and nonelastic scattering including the Pauli blocking effect. All these features make the QMD model a useful tool to determine the systematics of nucleon-induced processes first investigated by Peilert et al. [291]. Niita et al. [184] analyzed (N, xN ) reactions by QMD models plus a statistical decay model, Chiba et al. [185] determined nucleon-induced preequilibrium reactions in terms of quantum molecular dynamics, and Polanski et al. [292] studied QMD models to describe fission and fragment production applying proton–nucleus interactions. Although the QMD method is widely used to study nuclear fragmentation [189], the details of the prescription may differ from author to author. 2.7.1.1 The Equation of Motion and the Reaction Treatment In the following the prescription of a simple QMD standard model, useful for spallation interactions, derived by Niita et al. [184], is given. The model takes into account relativistic kinematics and the relativistic correction for the effective interaction. In addition, the and N ∗ (1440) resonances of the nucleon are treated, and pions with their isospin degrees of freedom in the equation of motion are taken into account. A statistical decay model SDM is incorporated to compare various measured double differential cross sections calculated with the QMD model for proton-induced reactions with incident energies from 0.1 GeV up to 3.0 GeV. In the QMD approach all nucleons are spread out in the phase space with a Gaussian wavefunction of width L [183–185, 191]:

i (r − Ri )2 + r · Pi , φi (r) = 1/(2πL)3/4 · exp − 4L

(2.71)

where Ri and Pi are the centers of the position and the momentum of the ith nucleon, respectively. The total wavefunction is assumed to be a direct product of these wavefunctions. Thus the one-body distribution function is determined by a Wigner transform of the wavefunction, f (r, p) =

fi (r, p),

(2.72)

i

2L(p − Pi )2 (r − Ri )2 − . fi (r, p) = 8 · exp − 4L 2

(2.73)

The time evolution of Ri and Pi is described by Newtonian equations and the stochastic two-body collision term. The Newtonian equations are derived on the basis of the time-dependent variational principles [183] as R˙ i = ∂H/∂Pi ,

P˙ i = −∂H/∂Ri ,

(2.74)


where the Hamiltonian H consists of the single-particle energy including the mass term and the energy of the two-body interaction. Further the Skyrme-type effective N–NMinteraction2) plus the Coulomb, and the symmetry energy terms are taken into account. By using the Gaussian function of the nucleons (cf. Eq. (2.73)), it follows (see Ref. [184]) that H=

B 1 A 1 · ·

ρi +

ρi τ 2 ρ0 1 + τ ρ0τ i i i √ 1 e2 · erf |Ri − Rj |/ 4L + ci · cj · 2 |Ri − Rj | Ei +

i,j( =i)

Cs · (1 − 2|ci − cj |) · ρij , + 2ρ0

(2.75)

i,j( =i)

with Ei = mi2 + P2i , and erf as the error function. In Eq. (2.75) the parameter ci has the value 1 for protons and the value 0 for neutrons. The quantity ρi defines an overlap of the density with other nucleons and is defined as

ρi ≡

ρij ≡

j =i

drρi (r)ρj (r)

j =i

=

(4πL)−3/2 exp[−(Ri − Rj )2 /4L],

(2.76)

j =i

with ρi (r) ≡

dp · fi (r, p) (2π)3

= (2πL)−3/2 · exp[−(r − Ri )2 /2L].

(2.77)

Niita et al. [184] have used the following parameters in Eqs. (2.76) and (2.77) for their QMD simulations to validate spallation reactions in the energy range from 0.1 GeV up to 3.0 GeV. (a) The symmetry energy parameter Cs is chosen to be 25 MeV. (b) The remaining four parameters, the saturation density ρo , and the Skyrme parameters A, B, and τ are ρ0 = 0.168 fm−3 , A = −219.4 MeV, B = 165.3 MeV, and τ = 4/3, respectively. (c) These values give a binding energy/nucleon 16 MeV for infinite nuclear matter at the saturation density ρ0 and a compressibility 237.3 MeV for nuclear matter limit. (d) The only arbitrary parameter in QMD, the width parameter L, is fixed as L = 2.0 fm2 . In the QMD approach of Niita et al. [184] to calculate nuclear cascades, nucleon–nucleon collisions are considered when the impact parameter of two √ nucleons is smaller than a value of σ/π , where σ denotes the energy-dependent 2) The Skyrme-type interaction for nuclear structure calculations is based of the mean effective internucleon potential in the shell

model description of nuclei derived by T.H.R. Skyrme 1958 (for details see Refs. [293, 294]).

129


N–N cross section for elastic or nonelastic N–N collisions. A parametrization of the N–N cross sections is used [184] similar to the INCL model of Cugnon et al. [210] to take into account the in-medium effects which reduce the absolute magnitude and the forward peaking of the N–N cross sections. The Pauli blocking of the final phase space is checked after each collision. In order to treat spallation reactions of particles with higher incident energies in the QMD model, the nucleons N, deltas (1232), N ∗ (1440)’s, and pions with their isospin degrees of freedom are included. In the following, a list is given of all channels included in the collision term: (1) (2) (3) (4) (5) (6) (7) (8)

Bi + Bj −→ Bi + Bj , N + N −→ N + , N + −→ N + N, N + N −→ N + N∗ , N + N∗ −→ N + N, N + π −→ , N + π −→ N∗ , + π −→ N∗ ,

(2.78)

where B denotes a baryon and a N a nucleon. 2.7.1.2 The Cutoff Criteria – the Stopping Time of the QMD Process At the end of the fast stage of a QMD simulation the result yields many fragments, which are normally in highly excited states. Although the QMD models allow us to continue the simulation by applying then the decay process of the fragments, it might be in general only applicable if one can continue the calculation for a long enough time. Therefore, it is practically more useful to stop the fast stage of the 104 208Pb

ds/ dE [mb/MeV]

130

(p,xn) Ep = 1500 MeV tsw = 50 fm/c tsw = 100 fm/c tsw = 150 fm/c

103

102

101

0

20

40

60

80

100

Neutron energy [MeV] Fig. 2.42 The total neutron energy spectrum calculated by a QMD + SDM model with different switching times tsw as parameter (after Niita et al. [184]).

2.7 Alternative Models 104 Pb (p,xn) Ep = 1500 MeV

d2s/dΩdE [mb/sr/MeV]

208

103

Ishibashi et al. QMD +SDM from QMD from SDM

102 101 100 10−1 10−2 100


103

Fig. 2.43 Double differential neutron energy spectrum at 30◦ laboratory angle for the reaction p+208 Pb at Eproton = 1500 MeV. The solid line is the result of the QMD + SDM model simulation with a switching time tsw = 100 fm/c. The experimental data are taken from Ref. [296] (after Niita et al. [184]).

103 102

208

Pb (p,xn) Ep = 1500 MeV

103

15°

d2s / dΩdE (mb/sr MeV)


10 30° (×10−2)

10−1 10

−2

−4

60° (×10 )

10−3 10−4

90° (×10−6)

10−5 10−6 10

−7

QMD + SDM Ishibashi et al.


1

15°

100 10−1

30° (× 10−2)

10−2 10−3 10−4 10

60° (×10−4)

−5

10−6

90° (×10−6)

10−7

10−8 10−9 100


102

101 100

208

10−8 101

102

103

10−9 0

500

En (MeV)

1000

1500

En (MeV)

Fig. 2.44 Double differential neutron energy spectrum at different laboratory angles for the reaction p+208 Pb at an incident beam energy Eproton = 1500 MeV. The x-axis En is plotted in a logarithmic scale on the left-hand side, and in a linear scale on the right-hand side. The experimental data are taken from Ref. [296] (after Niita et al. [184]).

QMD at a certain time and switch to a statistical decay SDM model (cf. Chapter 3). There two reasons for such a coupling: (i) the time scales of the dynamical and statistical processes are quite different, e.g., for the fast stage about 10−23 s to 10−22 s and 10−21 s to 10−15 s, respectively. (ii) A more fundamental reason is that the

131


Fermi statistics which describe the decay process of the fragments may not be treated correctly by QMD simulations [184, 295]. In QMD cascade spallation reaction simulations one may identify fragments together with their excitation energies at about 100–150 fm/c. The dependence of different switching times tsw when the QMD simulations are stopped and switched to a SDM model is depicted in Figure 2.42, where the total neutron energy spectrum determined by the model QMD + SDM is shown as a function of the neutron energy. It was concluded that the results are not sensitive to the switching tsw as long as it is chosen after the time when thermal equilibrium is achieved. A switching time of about 100 fm/c seemed to be sufficient. 2.7.1.3 Selected Examples of QMD Simulations Figure 2.43 shows a typical double differential neutron spectrum at 30◦ laboratory angle for the reaction protons +208 Pb at an incident proton energy Ep = 1500 MeV. The figure shows the spectra obtained by the QMD + SDM model, by the QMD model only, and the SDM model with a switching time of tsw = 100 fm/c. Simulations have been shown that the sum of the two components of the spectrum (QMD and SDM) will be almost unchanged if an adequately long time is chosen for the switching time tsw (cf. Figure 2.42). Figures 2.44 and 2.45 show a comparison of double differential neutron energy spectra determined by the QMD + SDM simulations with experimental data of 208

103

Pb (p,xn)

Ep = 3000 MeV

208


103

15°

102

102

101


101 30° (×10−2)

100



132

10−1 10−2

60° (×10−4)

10−3 10−4

90° (×10−6)

10−5 10−6

10−1

10−3 10−4 60° (×10−4)

10−5 10−6

QMD + SDM

10−7

10−8

Ishibashi et al.

10−8

100

30° (×10−2)

10−2

10−7

10−9

15°

100

101

102 En (MeV)

103

90° (×10−6)

10−9 0

Fig. 2.45 Same as Figure 2.44 for the reaction p+208 Pb at an incident beam energy Eproton = 3000 MeV. The experimental data are taken from Ref. [296] (after Niita et al. [184]).

1000 2000 En (MeV)

3000


Ishibashi et al. [296] for a Pb target with incident proton energies of 1.5 and 3.0 GeV at different laboratory angles. Such data, from 1 MeV up to the incident beam energy, play an important role in the fields of application of accelerators, such as spallation sources, accelerator transmutation systems, and shielding of cosmic rays in space activities. For an efficient comparison the same results are shown in the two figures with the x-axis in a logarithmic and a linear scale, respectively. The present QMD model plus a SDM model of Niita et al. [184] is applied on (N,xN’] reactions for a broad range of incident proton energies from 100 MeV to 3000 MeV on different target masses. The model reproduces the double differential neutron production spectra quite well without changing the used parameter set given above. Although there are many parameters in the model which are not extensively studied, the neutron spectra do not depend very much on them [184].

133

135

3 Evaporation and High-Energy Fission 3.1 Introduction

As mentioned in Chapter 2, in practice for modeling pN or NN reactions intranuclear cascade or equivalent models are applied in order to describe the first stage of a nuclear collision. These INC calculations are stopped once it can be assumed that equilibrium has been achieved (cf. Section on page 98). After equilibration is reached, generally an evaporation model is appended to account for the full nuclear reaction process. This chapter is devoted to models which are needed to continue the de-excitation process after the fast stage of the spallation process (cf. Section 1.3.4, page 21) and describes the modeling of the de-excitation and the associated particle emission of the highly excited nucleus remaining after the intranuclear cascade or the fast stage of the spallation reaction. Most of the models apply the statistical theory of evaporation from the excited compound nucleus, which was originally developed by Weisskopf [8]. All the models are based on similar essential assumptions; however, the individual models differ in their particular algorithms or their implementations. Dostrovsky et al. [7, 297, 298] have described a Monte Carlo model based on this theory which is known as the Dresner EVAP model [299]. The original EVAP model [299] has been further revised by Guthrie [300, 301], by Cloth et al. [88] taking into account, (i) the atomic mass tables of Ref. [302], (ii) the level density parameters with shell corrections by Baba [303], and (iii) by Prael [193]. Prael added to the version of EVAP in LAHET [193, 304] a level density parameter using the energy-dependent formulation of Ignatyuk [305–307]. The latest version of the EVAP model is described in Section 3.3. A generalized evaporation model GEM was recently developed by Furihata et al. [255, 308] and described in Section 3.4. In Sections 3.5 and 3.6 two important evaporation models, the abrasion-abla model ABLA by Schmidt et al. [221, 222] and a binary sequential decay model GEMINI by Charity [229, 256] are discussed. High-energy fission becomes an important nuclear mechanism for nucleon collisions in the spallation energy regime mainly with high atomic mass targets. In this case there is competition between fission and evaporation at each step of the nuclear de-excitation process [309, 310]. Evaporation models such as the EVAP or the GEM models are linked with fission models developed, e.g., by Handbook of Spallation Research: Theory, Experiments and Applications. Detlef Filges and Frank Goldenbaum Copyright  2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40714-9

136

3 Evaporation and High-Energy Fission

Atchison [311–313] and by Alsmiller et al. [314], see Section 3.7. Other evaporation models included their own fission model algorithms (cf. Sections 3.5 and 3.6 on pages 154, 159). In addition, the subsequent de-excitation of the residual nucleus left after the fast stage of the spallation reaction may optionally employ a multistage multistep pre-equilibrium exciton model first proposed by Griffin [315, 316]. If residuals after the fast stage of the spallation cascade have atomic mass numbers with A ≤ 17, a Fermi breakup model may be used [263, 264] (see Section 3.8) to determine their further disintegration instead of using the pre-equilibrium and evaporation models. The Fermi breakup model is much faster and the results are very similar compared to the more detailed models [249]. At the very end of the deexcitation process when the excitation energy falls below some specified cutoff, usually at about 7 MeV, further deexcitation is assumed to occur by evaporation or emission of photons with discrete energies. There exist libraries with a large number of isotopes with experimentally measured excited level energies, spins, parities, and relative transition probabilities [317]. Models for the evaporation of photons, e.g., the Los Alamos PHT model [304] and the FZ-Jülich NDM model [88] are discussed briefly in Section 3.9. Section 3.10 about vaporization and multifragmentation completes this chapter. The interested reader is also referred to the book of Cole [318] which provides in greater detail the theoretical considerations of the statistical models for nuclear decay as elements of equilibrium statistical mechanics, single- and multistep evaporation, sequential binary decay and multifragmentation, statistical models for multifragmentation, e.g., the Copenhagen liquid drop model, the Metropolis algorithm, percolation, caloric curves and vaporization.

3.2 The Statistical Model in its Standard Form

The basic principles of the nuclear reactions are first postulated by Bohr [309, 319] and the theory of the emission of particles from excited compound nuclei is originally due to Weisskopf [8, 320]. An excellent overview is given by LeCoutier [9]. The theory of Weisskopf is based on the following assumptions: • The formation and decay of the compound nucleus are independent. • The total energy and the angular momentum are conserved. • All possible decay channels are occupied with identical probability. • In the case of barriers existing for particular decay channels, the probability for that occupational state is reduced by a transmission coefficient. • The intrinsic degrees of freedom of the compound nucleus are equilibrated and a quasistationary situation of collective degrees of freedom at the saddle point is assumed. In any statistical evaporation model generally an excited compound nucleus decays via fission or by the emission of light particles. For the particle emission the

3.2 The Statistical Model in its Standard Form

partial decay width ν (E ∗ , J) can be expressed as a function of angular-momentumdependent level densities ρ of the initial and final state and the transmission coefficients Tl , J +s

ν (E ∗ , J) =

f ∞ 1 · 2πρ(E ∗ , J)

J+S

l=0 S=|Jf −s| |J−S|

·

E ∗ −E

r (Jf

)−Bν

0

ρ(E ∗ − Bν − Eν , Jf )Tl (Eν ) dEν .

(3.1)

The coupling of the angular momenta of the mother nucleus J and the residual Jf is performed via the angular momentum l of the emitted particle ν with Spin s and energy Eν . E ∗ represents the thermal excitation energy of the parent nucleus. Er (Jf ) is the rotational energy of the residual nucleus, Bν is the particle binding energy and S = Jf + s. The transmission coefficients1) are based on the fit of optical potentials of elastic scattering data, which partly are quite uncertain [321]. In contrast to the channel for particle decay, the fission decay widths do not depend on particle level densities or other statistical properties of residuals (here the fragments in infinite distance), but following the hypothesis of Bohr–Wheeler on the properties of the compound nucleus at which the level density exhibits a minimum between equilibrium location and separated fragments [8, 320]. This point is generally named saddle point configuration; the angular-dependent potential energy (fission barrier Bf (J)) has its maximum for this configuration. Fission is handled analog to the particle decay [309, 322], fBW (E ∗ , J) =

1 2πρ(E ∗ , J)

E ∗ −Bf (J)

ρ(E ∗ − Bf (J) − Ek ) dEk .

(3.2)

0

The transmission coefficient is equal to 1, in the case of the total energy exceeding the fission barrier; else it is set to 0. The fission barrier Bf (J) can be calculated employing the ‘‘Rotating Finite Range Model2) .’’ Very similar to the calculation of intrinsic or thermal excitation energy determining the density of the quantum mechanical states, k represents the kinetic energy at the saddle point and not the asymptotic relative kinetic energy. For hot nuclei with E ∗ Bf + k , Eq. (3.2) can be simplified as fBW =

Bf T exp − . 2π T

(3.3)

In this equation, fBW only depends on temperature and fission barrier. 1) In the statistical model GEMINI the transmission coefficients are taken from the ‘‘incoming wave boundary condition’’ model [229].

2) GEMINI [229] uses the conditional barriers for symmetric fission as calculated by Sierk [229, 323].

137

138


Hot nuclei de-excite via many successive decay steps until after particle emission, γ -emission, or possibly fission only a heavy residual or cold fission fragments remain in a ground state. Generally statistical codes used nowadays follow the decay of an individual compound nucleus of a given ensemble by Monte Carlo methods, until the residual does not emit particles anymore, or as a competing process of particle emission, fission is favored. The precision of any observable predicted in a statistical model depends on the abundance of its appearance and the number of simulated cascades. The modifications necessary in the statistical model for taking into account nuclear dissipation and therefore a delay of the fission process are taken into account, e.g., in the ABLA and GEMINI models in Sections 3.5 and 3.6. Besides the transmission coefficients, the partial emission widths (cf. Eq. (3.1)) are mainly determined by the nuclear level densities. For a given thermal excitation energy E ∗ and considering a nucleus as a Fermi gas of A independent particles, the level density ρ(E ∗ ) given by the number of possibilities E ∗ can be distributed on the different single particle states, e.g., [324] ρ(E ∗ ) =

1√ π a(q)−1/4 (E ∗ )−5/4 exp 2 a(q)E ∗ . 12

(3.4)

As the thermal excitation energy E ∗ of a nucleus increases, excited level stages get closer together in energy. At large E ∗ , the density of excited levels is 1/D with D being the average distance between the levels with the form 1/D ∝ exp(2 · (a · E ∗ )2 ), where a affects the decay width of particles emitted during the evaporation process and called the level density. For the level density parameter a(q) in the literature simple relations such as a = A/x (x = 7 to 14 MeV) are found as well as phenomenological approaches which take the influence of the surface structure and shape of the nuclei into account. Different parameterizations exist [305, 306, 325, 326], partly even considering a dependence on the temperature of the decaying system [327]. The level density parameter is defined by [324] a(q) =

π2 g(q) 6

with

g=

3A 2 · EF

(3.5)

with g(q) being the nucleus shape-dependent single particle level density at the Fermi energy EF and representing the sum of proton- and neutron-level densities. For the validity range of the statistical model, the following shall apply: EF · A−1 E ∗ EF · A1/3 .

(3.6)

The lower limit is given by the condition that E ∗ should be large compared to the energy g −1 of the first excited level of the Fermi gas. With increasing E ∗ , the level density increases or the number of degrees of freedom of single particle states and

3.3 The Evaporation Model EVAP of Dostrovsky–Dresner

therefore the entropy S = ln ρ. In this context the nuclear temperature T can be introduced, T −1 =

∂S ∂E

= V

1 ∂ρ . ρ ∂E

(3.7)

Using the level density ρ(E) given in Eq. (3.4) the temperature can be expressed as a 1/2 5 T −1 = − E −1 + , 4 E

(3.8)

with E EF A−1 and Eq. (3.5) a E −1 and therefore with the Fermi gas expression E = a · T 2.

(3.9)

The following sections are committed to the description of specific models generally employed for modeling the statistical evaporation and high-energy fission processes.


In the following a short outline of the EVAP model is given with the basic assumptions which have been considered in the model. More specific details about the GEM model can be found in Section 3.4. Although the GEM model is based on the classical Weisskopf–Ewing formalism, the GEM model is a significant step in solving light fragment production from medium-energy proton-induced spallation reactions. As mentioned in Section 3.1, the basic theory of evaporation was developed by Weisskopf [8] and Weisskopf–Ewing [320]. The EVAP model originally developed by Dostrovsky et al. [7] has been continuously refined. The different EVAP model versions are summarized in Table 3.1. The revised versions of the EVAP model are implemented, e.g., in the ORNL CALOR [195], the LANL- LAHET, and the MCNPX [113, 193, 194, 304, 328] code systems. The probability Pi (E) that a nucleus excited to the energy U will emit a particle of type i having kinetic energy E is given by Weisskopf as Pi (E) ∝ (2Si + 1) · mi · σinvi (E) · ω(E ∗ ) with i = emitted particle type (n, p, d, t, 3 He, 4 He), E = kinetic energy, Si = spin of the emitted particle, mi = emitted particle mass,

(3.10)

139

140

3 Evaporation and High-Energy Fission Tab. 3.1

The different versions of the EVAP evaporation model.

Model

Comments

Nuclear mass data

EVAP - original model Dostrovsky et al., [7, 297, 298] and Dresner [299]

- 19 types of emitted particles from any compound nucleus: (n,p, d, t,3 He,4 He,6 He,6,7,8,9 Li,7,9,10 Be) and (6,7,8 Li,7,10 Be) with excited statesa

- Tables by Wapstra [329, 330] and Huizenga [331] - semiempirical formula of Cameron [332, 333]

EVAP-NEW - EVAP-2/EVAP-3 Guthrie [300, 301] - and updates by Cloth et al. [88], Prael et al. [113, 193, 304]

- 6 types of evaporated particles: (n,p,d,t,3 He,4 He)b - breakup of 8 Be into two 4 He particles excited recoil nucleus (A, Z ∗ ) - kinetic energy, Erecoil , of recoil nuclei - modified level density parameter: Ignatyuk [305, 306], Gilbert–Cameron–Cook [336, 337] and Baba formalism [303]

- shell plus pairing energy corrections for Z or N < 13 by Peele, Aebersold [334] - tables by Wapstra, Bos [302] Audi, Wapstra [335]

a The evaporated particles (6,7,8 Li,7,10 Be) have excited states lying at 3.56, 0.477, 0.430, 0.970, and 3.370 MeV above the ground state. b Only six types of evaporated particles are considered instead of Dresner’s original 19.

σinvi = the ‘‘inverse’’ cross section (see Eqs. (3.15) and (3.16)), e.g., the compound nucleus formation cross section corresponding to bombarding the residual nucleus with particles of type i and energy E, E ∗ = excitation energy of the residuals (A∗ ,Z∗ ), with E ∗ = U − Q − E, Q = binding energy, U = energy of the excited nucleus, ω(E ∗ ) = density of energy levels of the excited residual nucleus.

3.3.1 The Level Density Parameter in the EVAP Model

The density of states in a nucleus of A = N + Z nucleons at excitation energy E ∗ can be approximately represented in terms of the ‘‘level density parameter’’ a and is evaluated using the level density formula given below. This is the mass and isospin-dependent model which is used in the EVAP model of Dostrovsky et al. [7] and Dresner[299]: ω(E ∗ ) = exp 2 · a · (E ∗ − δ) ,

(3.11)

where δ = pairing energy, with δ = 0 for odd–odd nuclei and δ ≥ 0 for other nuclei. The parameter a = level density is computed as


a = A/B0 ·

1 + Y · 2 A2

(3.12)

with = A − 2Z the neutron excess. The parameters B0 ≈ 8-20 MeV−1 and Y ≈ 1.5 to 1.8 are a kind of ‘‘universal’’ constants. The quantity in parenthesis in Eq. (3.12) differs little from unity, so a realistic approximation of the parameter assumes a linear mass dependence, a∼

A MeV−1 . B0

(3.13)

This relation is used in the EVAP version of Dresner [299]. A commonly used value is a = A/8 MeV−1 . In the latest version of the EVAP model (cf. Table 3.1) two other parameterizations [193] for the level density parameter a are used. This is the energy-dependent formulation for a of Ignatyuk et al. [305, 306] with the provision lim a(E ∗ ) = a0 ,

(3.14)

E ∗ →0

Level density parameter a [MeV−1]

where E ∗ is the excitation energy and a0 is the Gilbert–Cameron–Cook (GCC) level density parameter [336]. The low- and high-excitation limits are given in Figure 3.1. A different approach for the mass-dependent level density parameter a is that of Cloth et al. [88]. It takes into account shell effects by using compiled level densities from neutron resonance measurements for about 200 isotopes of Baba [303] (see Figure 3.2). From experiments on nuclear level spacing it is observed that the level density is strongly influenced by the nuclear shell structure, with densities for 35.0 30.0 25.0 20.0 15.0 10.0 5.0 0.0 0

50

100 150 Atomic mass number A

200

Fig. 3.1 Level density parameters as a function of the atomic mass number A. The points are Gilbert–Cameron–Cook values for atomic masses near the line of stability. The line is the Ignatyuk high-excitation limit. ¨ The circles are the Julich mass-dependent model (after Prael [193]).

250

141


Level density parameter a [MeV−1]

142

35.0 30.0 25.0 20.0 15.0 10.0 5.0 0.0 0

50

100 150 Atomic mass number A

Fig. 3.2 Level density parameters as a function of the atomic mass number A. The thick line gives typical values for the original EVAP mass- and isospin-dependent model for masses near the line of stability ¨ of Dresner [299]. The circles are the Julich

200

250

mass-dependent model. For comparison the dotted line gives the A/8-dependence, the solid line gives the A/10-dependence, and the dashed line gives the A/14-dependence (after Prael [193]).

magic or nearly magic nuclei several orders of magnitude lower, Figure 3.2, than for mid-shell nuclei [303]. This is important on neutron production from spallation targets applying, e.g., lead or lead–bismuth materials [338]. These extensions in EVAP for the level density parameter are adopted into the LAHET code by Prael et al. [304]. Nowadays these new parameterizations are named the Gilbert–Cameron–Cook– Ignatyuk (GCCI) level density parameter (cf. Section 3.4 on page 143 about the physics in the GEM evaporation model). 3.3.2 The Inverse Cross Sections

The inverse cross section σinvi (Eq. (3.10)) in EVAP are those recommended by Dostrovsky et al. The cross section is taken to be the geometric cross section modified by an empirical expression to take into account the energy dependence, and a Coulomb barrier correction for charged particle emission. For neutrons the inverse cross section is given as σinvi -neutron (E) = α · (1 + β/E) · π R2 α = 0.76 + 1.93 · A1/3 α · β = 1.66 · A−2/3 − 0.05 R = 1.7 · A1/3 fm.

For charged particle emission the inverse cross section is given as

for E ≥ ki Vi πR2 · (1 + ci )(1 − kiE·Vi ) σinvi -charge (E) = 0 for E < ki Vi ,

(3.15)

(3.16)

3.4 The Generalized Evaporation Model (GEM) Parameters used in the EVAP model to compute inverse charged particle cross sections as a function of the charge Z of the mother nucleus (Dostrovsky et al. [7] and Cloth et al. [339]).

Tab. 3.2

Z

kproton

cproton

k4

10 20 30 40 50 60 > 70

0.36 0.51 0.60 0.66 0.68 0.69 0.69

0.08 0.00 −0.06 −0.10 −0.10 −0.10 −0.10

0.77 0.81 0.85 0.89 0.93 0.97 1.00

He

where Vi is the Coulomb barrier determined from the classical electrostatic expression (Eq. (3.17) and Section 2.2.2.3) and the factor ki is inserted to take into account approximately barrier penetration effects3) . The numerical values of ci and ki are those given by Dostrovsky et al. [7]. Values for protons and 3 He are given in Table 3.2, whereas values for the deuterons, tritons, and 3 He are determined from Eqs. (3.18): Vi =

Zi · Z · e2 , R + Ri

(3.17)

where e is the electron charge, Ri is zero for protons and 1.2 fm for all other particle types, Z is the parent nucleus, and Zi is the emitted particle: kdeuteron = kproton + 0.06, ktriton = kproton + 0.12, k3 He = k4 He − 0.06,

cproton 2 cproton ctriton = 3 c3 He = c4 He = 0

cdeuteron =

(3.18)

3.4 The Generalized Evaporation Model (GEM)

The generalized evaporation model (GEM) of Furihata [254, 255, 308] is based on the classical Weisskopf–Ewing approach [8, 320] as the EVAP model, which is restricted to the emission of particles up to 4 He only (cf. Section 3.3). It is obvious 3) The empirical expected Coulomb radius RC is larger than the sum of the radii of the mass distributions for R1 , R2 , e.g.,

Rc = R1 + R2 + (3.0 ± 0.5) (fm), and Ri = 1/3 1.12Ai − 0.93A1/3 (fm).

143

144


however [340] that light nuclei heavier than 4 He particles can be emitted from excited nuclei via the evaporation process. In the GEM, 66 nuclides up to Mg are included as ejectiles. The fission model used in GEM is based on Atchison’s model [311, 312] as implemented in LAHET [110–113, 193], often referred to in the literature as the Rutherford Appleton Laboratory4) (RAL) model. The fission model of Atchison is discussed in Section 3.7. For GEM here only the differences and re-evaluated parameters compared to the standard Atchison approach are reviewed.

3.4.1 Evaporation Model in GEM

According to the Weisskopf approach [8], the probability Pj of evaporation of the particle j in its ground state with Aj and Zj from a parent compound nucleus i with Ai and Zi and with a total kinetic energy E in the center-of-mass system between energy E and energy E + dE is defined as Pj (E)dE = gj σinv (E)

ρd (E ∗ − Q − E) EdE, ρi (E ∗ )

(3.19)

where E ∗ is the excitation energy of the parent nucleus i, d denotes a daughter nucleus produced after the emission of ejectile j, and ρi , ρd are the level densities (MeV−1 ) for the parent and daughter nucleus, respectively. Q denotes the Q-value of the reaction for emission of particle j. For the determination the Audi-Wapstra mass table [335] has been used. The statistical and normalization factor gj is defined as gj = (2Sj + 1)mj /π 2 2 ,

(3.20)

where Sj and mj are the spin and the mass of the emitted particle j, respectively. The cross section σinv for the inverse reaction is evaluated from σinv (E) = σg · P(E)

(3.21)

where σg is the geometrical cross section. The evaluation and the parameters are given in Section 3.4.3. The 66 nuclides heavier than helium GEM are considered as ejectiles are not only in their ground state, but also in their excited states. Including the excited states in the particles emitted via the evaporation process enhances significantly the yield of heavy particle emission [341]. The ejectiles listed in Table 3.3 are characterized by the following criteria: (i) isotopes with Zj ≤ 12, (ii) isotopes naturally existing or near the stability line, (iii) isotopes with half-life longer than 1 ms. For the barrier 4) Name originated from the Rutherford Appleton Laboratory, UK, where the code was developed.

3.4 The Generalized Evaporation Model (GEM) Tab. 3.3

The ejectiles taken into consideration in the GEM model.

Zj

Ejectiles

0 1 2 3 4 5 6 7 8 9 10 11 12

n p 3 He 6 Li 7 Be 8B 10 C 12 N 14 O 17 F 18 Ne 21 Na 22 Mg

d

t

4 He

6 He

7 Li

8 Li

9 Li

9 Be

10 Be

11 Be

10 B

11 B

12 B

13 B

11 C

12 C

13 C

14 C

15 C

13 N

14 N

15 N

16 N

17 N

15 O

16 O

17 O

18 O

19 O

20 O

18 F

19 F

20 F

21 F

19 Ne

20 Ne

21 Ne

22 Ne

23 Ne

24 Ne

22 Na

23 Na

24 Na

25 Na

23 Mg

24 Mg

25 Mg

26 Mg

27 Mg

28 Mg

8 He 12 Be 16 C

penetration probability P the form of Ref. [7] is used. The parameters for light particles (n, p, d, t, 3 He, and 4 He) are taken from Dostrovsky et al. [7] whereas those for IMFs (intermediate mass fragments) were adopted from the work of Matsuse et al. [342] (cf. Section 3.4.3). 3.4.2 The Decay Width in the GEM Model

The total decay width j is calculated by integrating Eq. (3.19) using Eq. (3.21) with respect to the total kinetic energy E from the Coulomb barrier V up to the maximum possible value (E ∗ − Q) (Q = binding energy) and is expressed as j =

gj σg ρi (E ∗ )

E ∗ −Q

E · P(E) · ρd · (E ∗ − Q − E)dE.

(3.22)

V

According to the Fermi gas model expression the total level density ρ(E ∗ ) = ρi (E ∗ ) of a nucleus summed over all the possible states with the angular momentum is given by the expression of Ref. [336]: π exp (2 a(E ∗ − δ) 12 a1/4 (E ∗ − δ)5/4 π 1 exp(E ∗ − Eo )/T = 12 T

ρ(E ∗ ) =

for E ∗ ≥ Ex∗ for E ∗ ≤ Ex∗ .

(3.23)

The parameters in Eq. (3.23) are as follows: • a (MeV−1 ) is the level density parameter usually given in its simplest form as a = A/B0 (MeV−1 ), e.g., a = A/8 (cf. Section 3.3.1).

145

146


• δ (MeV) is the pairing energy of the residual of the daughter nucleus. The pairing energy δ is expressed as a sum of separate contributions from neutrons and protons. • Cook et al. [337] calculated Ex∗ for which the level density changes as Ex∗ = Ux + δ where Ux = 2.5 + 150/Ad . Ad is the produced daughter nucleus and U is the energy of the excited nucleus. • T is again the nuclear temperature given by 1/T =

a/Ux − 1.5/Ux .

(3.24)

• To obtain a smooth continuity between the two formulas, the E0 parameter is determined as follows: E0 = Ex∗ − T · log T − 0.25 log a − 1.25 log Ux + 2 aUx .

(3.25)

3.4.3 Parameter of the Inverse Cross Sections and the Coulomb Barrier

The formulas to determine the inverse cross sections and the Coulomb barrier are in principle similar to the ones used in the EVAP model (cf. Section 3.3.2, Eqs. (3.15)–(3.18), and Table 3.2)5) . Because the prescription is slightly different in the GEM model for completion the used ansatz in GEM will be considered below. The cross sections for the inverse reaction σinvi and the Coulomb barrier V are given by

σinvi (E) =

π · R2b · cn (1 + b/E) for neutrons π · R2b · cj (1 + V/E) for charged particles

V = kj ·

Zj · Zd · e2 , Rc

(3.26)

where the indices i and j stand for the incident particle and the emitted particle, respectively, and d is the index for the daughter nucleus. Rb is the radius of the geometric cross section σg = πR2b (fm2 ), kj is defined as in Eq. (3.16), Rc is the Coulomb radius, and e is the electron charge. The GEM model uses two different parameter sets, one is named the ‘‘simple’’ parameter set, and the other is named the ‘‘precise’’ parameter set. • The simple parameter set: using the formulas of Eq. (3.26) the parameters are −1/3 1/3 + Ad ) (fm). So only r0 is needed as cn = cj = kj = 1 and Rb = Rc = r0 · (Aj the parameter for determining the inverse cross section. 5) See the publication of Dostrovsky et al. added as a note in Ref. [7] on page 699.

3.4 The Generalized Evaporation Model (GEM) Parameters used in the GEM model to compute inverse charged particle cross sections (Dostrovsky et al. [7] and (cf. Table 3.2)).

Tab. 3.4

Zd

k

k4 He

≤ 20 30 40 ≥ 50

0.51 0.60 0.66 0.68

0.81 0.85 0.89 0.93

c 0.0 −0.06 −0.10 −0.10

• The ‘‘precise’’ parameter set to estimate the inverse cross section is a combination of the approach of Dostrovsky et al. [7] and that of Matuse et al. [342]. (1) For n, p, d, t, 3 He, and 4 He Dostrovsky’s approach gives the following parameters for Eq. (3.26): −1/3

cn = 0.76 + 1.93Ad −2/3

b = (1.66Ad

−1/3

− 0.05)/(0.76 + 1.93Ad

)

(b = 0 for Ad ≥ 192) cp = 1 + c,

cd = 1 + c/2,

ct = 1 + c/3,

c3 He = c4 He = 0 kp = k, kd = k + 0.06, kt = k + 0.12, k3 He = k4 He − 0.06

(3.27)

where c, k, and k4 He are given in Table 3.4. (2) For protons and neutrons the nuclear distance is given with a radius 1/3 Rb = 1.5 · Ad , and for d, t, 3 He, and 4 He with a radius 1/3 1/3 Rb = 1.5(Ad + Aj ). (3) The nuclear distance for the Coulomb barrier given as Rc = Rd + Rj is for: neutrons and protons −→ Rd = 1.7 · A1/3 (Rj = 0) (fm) d, t, 3 He, and 4 He −→ Rj = 1.2 (fm). (4) IMFs emission parameters were adopted from the work of Matsuse et al. [342], where the critical distance was evaluated from fusion reactions of heavy ion systems. cj = k = 1, Rb = R0 (Aj ) + R0 (Ad ) + 2.85 (fm) Rc = R0 (Aj ) + R0 (Ad ) + 3.75 (fm), where R0 (A) = 1.12 · A1/3 − 0.86 · A−1/3

3.4.4 The Level Density Parameter GCCI in the GEM Model

In the calculation with the precise parameter set of the GEM model, the Gilbert–Cameron–Cook–Ignatyuk (GCCI) level density parameter [113, 193] is

147

148


used, in which pairing corrections and the energy dependence of the level density parameter are taken into account, instead of the simple expression a = Ad /8 (see also the comments about the level density in the EVAP model, Section 3.3.1). The GCCI level density parameter is also employed in the LAHET code [113, 193]. For the calculation of the total decay widths, expressions (3.23) is used. The simple form ρ ∝ exp(2 a(E ∗ − δ)) as used in Dostrovsky’s evaporation model [7] is a good approximation for high excitation energies; however, it is not applicable for residual nuclei with small mass and at low excitation energy. In the GEM model, the GCCI level density parameter a (MeV−1 ) is given for produced residuals for charge numbers Zd ≤ 98 and numbers of neutrons Nd ≤ 150 – practically without limitation a = a1 ·

1 − exp(−u) 1 − exp(−u) + a2 1 − u u

(3.28)

where u = 0.05(E ∗ − δ), and

Zd < 9 or Nd < 9 Ad /8 Ad · (a3 + 0.00917 · S) for all others

a2 = Ad · 0.1375 − 8.36 × 10−5 · Ad . a1 =

(3.29)

For deformed nuclei with 54 ≤ Zd ≤ 98, 86 ≤ Nd ≤ 122, or 130 ≤ Nd ≤ 150, the constants are given by a3 = 0.12 and a3 = 0.142 for all other nuclei. The shell correction S is the sum of the contributions from protons and neutrons, and is given as S = S(Zd ) + S(Nd ).

(3.30)

The shell correction parameters are tabulated in detail in Ref. [255]. 3.4.5 The High-Energy Fission Process in the GEM Model

The initial version of the GEM model only simulates spallation and does not calculate the process of fission, nor does it provide fission fragments and a further possible evaporation of particles from them. To be able to describe nuclide production in the fission region, the model has to be extended by incorporating a model for high-energy fission. In order to account for fission the GEM model is improved with Atchison’s fission or RAL model [311, 312] as implemented in LAHET [110, 111, 193]. There are two choices of parameters for the fission model in the improved version GEM2: one is the original parameter set by Atchison [311, 312], and the other one is a parameter set evaluated by Furihata [255, 308, 343]. The Atchison fission model is designed only to describe fission of nuclei with Z ≥ 70. It assumes that fission competes only with neutron emission,


i.e. from the width j of n,p,d,t,3 He, and 4 He; the RAL code calculates the probability of evaporation of any particle. When a charged particle is selected for evaporation, no fission competition is taken into account. When a neutron is selected for evaporation, the model does not actually simulate its evaporation; instead it considers that fission may compete and chooses either fission or evaporation of a neutron according to the fission probability Pf . The parameters for the fission used in the GEM model and their difference to the original fission of Atchison are discussed. More details about high-energy fission models and about the RAL fission model of Atchison are given in Sections 3.7 and 3.7.3. (1) Mass distribution: (a) Mass distribution σm-GEM in the GEM model of Furihata: The mass of a fission fragment produced by symmetric fission has a Gaussian distribution with mean Ai /2 with the index i as the fragment type and width σm . Furihata [308] assumed for the GEM model that σm based on experimental data from Refs. [344, 345] is given as σm-GEM = C1 · (Zi2 /Ai )2 + C2 · (Zi2 /Ai ) + C3 · (E ∗ − Bf ) + C4 , (3.31) with the coefficients C1 = 0.122, C2 = −7.77, C3 = 3.32 × 10−2 , C4 = 134.0. The fission barrier Bf is given by Myers and Swiatecki [346]. σm-GEM is assumed to be constant for E ∗ − Bf > 450 (MeV). Figure 3.3 shows the Z2 /A dependence of the width σm-GEM of the fission fragment mass distribution.

Mass distribution width sm [MeV]

30.0 25.8 21.7 17.5 13.3 9.2 5.0 20

25

30

35

40

Z2 / A Fig. 3.3 The dependence of the dispersion of the fission fragment mass distribution on Z 2 /A where σm-GEM is determined by setting (E − Bf ) = 0 (cf. Eq. (3.31)) (after Furihata et al. [255]).

45

149

150


(b) Mass distribution σm-RAL in the RAL model of Atchison: The σm-RAL in the original RAL model of Atchison is given as  3.97 + 0.425(E ∗ − Bf ) − 0.00212(E ∗ − Bf )2    for (E ∗ − Bf ) < 100 MeV σm-RAL =  25.27   for (E ∗ −Bf ) ≥ 100 MeV (2) Charge distribution: (a) Charge σZ-GEM in the GEM model of Furihata: The charge distribution of a fission fragment is assumed to be also a Gauss distribution of mean Zf and width σZ-GEM · Zf and is given as Zf =

Zi + Z1 + Z2 2

with Zx =

65.5 · Ax 2/3

131 + Ax

for x = 1, 2

(3.32)

The evaluation of formula (3.32) gives a value for σZ-GEM = 0.75, which gives better results for the RAL model. (b) Charge σZ-RAL in the original RAL model of Atchison: The original RAL model uses a value of σZ-RAL = 2.0. (3) Kinetic energy distribution: (a) The kinetic energy distribution with the parameters f and σf in the GEM model of Furihata: The kinetic energy of fission fragments is determined by a Gaussian distribution with mean Ef and width σEf . The refinement of Furihata for the improved GEM model looks as follows:

Ef =

1/3

1/3

for Zi2 /Ai ≤ 900 0.131Zi2 /Ai 1/3 1/3 2 0.104Zi /Ai + 24.3 for 900 < Zi2 /Ai ≤ 1800 (3.33)

and

σE f =

1/3 C1 Zi2 /Ai − 1000 + C2 C2

1/3

> 1000

1/3 Zi2 /Ai

≤ 1000

for Zi2 /Ai for

(3.34) where C1 = 5.7 × 10−4 and C2 = 86.5. For details refer to [255]. (b) The kinetic energy distribution parameters Ef and σEf in the RAL model of Atchison: The original parameters in the Atchison model are given by 1/3

Ef = 0.133 · Zi2 /Ai

− 11.4

and σEf = 0.084 · Ef

(3.35)


Incident protons of 1 GeV on 208Pb

Cross section [mb]

102

Exp. data / GSI ISABEL / GEM ISABEL / PREQ / EVAP4

101

100

10−1 30

50

70

90 110 130 150 170 190 210 Atomic mass number [A]

Fig. 3.4 Comparison of the experimental mass distribution [68, 347] of nuclides produced in the reaction p(1 GeV) + Pb with different calculations (ISABEL + GEM and ISABEL + Preq + EVAP4).

Figure 3.4 confronts the experimental mass distribution of the inverse kinematics reaction p(1 GeV)+Pb measured at GSI [68, 347] with results found by merging ISABEL + GEM and ISABEL + Preq + EVAP4 [299]), respectively. The solid line shows the results of ISABEL merged to GEM as described in subsection 3.4.1 on page 144, while the dashed line is the ISABEL + Preq + EVAP4 [299] calculation. The combination of ISABEL+GEM reproduces the mass distributions of fission fragments with 50% accuracy, whereas ISABEL + Preq + EVAP4 does not reproduce the width and the heights of the distributions. The use of Furihata’s σm in the fission model improves the predictions [308]. As concerns the mass distributions of the spallation products shown in Figure 3.4, ISABEL + GEM and ISABEL + Preq + EVAP4 produce almost identical cross sections and for both models the isotopes predicted in the spallation region (not too far from the target) agree well with the experimental data [347]. It should be mentioned that the original RAL fission model has modified recently and improved by Atchison [313, 348]. The modifications of [313] however have not yet been taken into account in the improved GEM model. 3.4.6 Method of Computation

In the Monte Carlo simulation, ejectile j is selected according to the probability distribution calculated as pj = j / j j . The total kinetic energy E of the emitted particle j and the daughter nucleus (residual) is chosen according to the probability distribution given by Eq. (3.19). The angular distribution of the motion is randomly selected to be isotropic in the center-of-mass system. The excitation energy of the daughter nucleus (residuum) Ed is calculated as Ed = E ∗ − Q − E.

151


The contribution of the emission of IMFs in a long living excited state is taken into account together with those which decay to the ground state. The condition for the lifetime of excited nuclei considered in GEM is as follows: T1/2 / ln 2 > / j∗ . The value of the decay width of the resonance emission j∗ is defined as the emission width of the decaying ejectile and is calculated in the same way as for the ground state, i.e., by Eq. (3.22). Rather than treating a resonance as an independent particle, the decay width of the ground state is simply enhanced in the following way. The total emission width j of an ejectile is summed over its ground state j0 and all its excited states jn , i.e., j = j0 + n jn . The Q-value for the resonance emission is expressed as Q ∗ = Q + Ej∗ , with Ej∗ being the excitation energy of the resonance. The spin state of the ground state Sj is replaced by the spin state of the resonance S∗j for the calculation of gj in Eq. (3.19). The ground state masses mj are also used for the excited states, since the difference between the masses is negligible. The total kinetic energy distribution of the excited particle emission is assumed to be identical to that of the ground state particle emission. The values for S∗j , Ej∗ , and T1/2 have been taken from the Evaluated Nuclear Structure Data Files (ENSDF) database by the National Nuclear Data Center [317]. The original focus when developing GEM was put to the proper description of 7 Be nuclei produced by proton-induced reactions in the energy range 10 MeV to 3 GeV, because 7 Be is the most intensively measured light fragment produced from various targets, and many experimental data are available for comparison. The excitation functions of 7 Be produced by protons impinging onto the 27 Al target are shown in Figure 3.5 in comparison with different INC/GEM predictions. The open square symbols are experimental data taken from Ref. [340], and the dashed and solid lines are BERTINI and ISABEL [162, 164, 167] INC calculations coupled 102

Cross section [mb]

152

101

100

Exp. data INC/GEM ISABEL/GEM

10−1

10−2 101

103 102 Incident proton energy [MeV]

Fig. 3.5 The excitation function of the production cross section of 7 Be as a function of the proton incident energy on the 27 Al target.


to GEM, respectively. For ISABEL the code implemented in LAHET has been applied for incident energies below 1 GeV. The combination of ISABEL/GEM underestimates the cross sections by a factor of 2 for incident proton energies above 300 MeV. ISABEL/GEM produced less 7 Be than INC-Bertini/GEM does, because the average excitation energy calculated by the ISABEL model is lower than that predicted by the BERTINI model in the medium- to high-incident energy region. For detailed comparison to experimental data and model parameters confer Ref. [254]. Merging the improved GEM by Furihata [254, 255, 308] with the INCL or ISABEL model allows us to describe reasonably well fission and fragmentation/spallation reactions. This overall good agreement lends credibility to the approach; however, there are some drawbacks to be mentioned [175]: (1) Evaporation of up to 66 particles in GEM becomes extremely time consuming, in particular, when calculating reactions with heavy targets at high incident energies. (2) The current improved version of GEM2 suffers some lack of self-consistency, e.g., – using different not physically related parameterizations for inverse cross sections and Coulomb barriers for different particles and fragments; – using different level density parameters for the same compound nuclei when calculating evaporation and fission probability from the widths of neutron evaporation and fission; – different and purely phenomenological treatments of fission for pre-actinide and actinide nuclei; – not taking into account at all the angular momentum of compound and fissioning nuclei; – rough estimations for the fission barriers and level density parameters, etc.

This means that an approach like the improved GEM [255] has the potential, in principle, to be used to describe fission and evaporation of particles and fragments heavier that 4 He after the INC, but it should be considerably improved, striving first to progressively incorporate better physics, and then adjusting the selected data. An example for isotopic distributions of light fragments for He and heavier than He produced from 480 MeV protons on a Ag target is shown in Figure 3.6. The calculated results by Bertini-INC/GEM simulations are compared with experimental data of Ref. [349]. The simulations of He, Li, and Be production rates agree with the experimental data within an accuracy of about 50%, whereas Bertini-INC/GEM underestimate the cross sections for heavier fragment masses. All input parameters are given as standard parameters in the model. GEM is written in Fortran 77 and freely available, i.e., in the LAHET/LCS [113, 193] or MCNPX model packages. For further details of GEM interested readers are directed to Refs. [175, 254, 255, 308].

153

154


103 Production cross section [mb]

He

Li

Be

B

C

N

O

F

Ne

102 101 100 10−1 10−2 10−3 10−4 4

66

8

8 10 8 10 12 10 12 14 12 14 16 Atomic mass number A

16 18 20 18 2018 20 22

Fig. 3.6 Isotope distributions of nuclei produced from 480 MeV protons incident on a nat Ag target, (open squares with dashed lines denote the Bertini-INC/GEM calculations, black circles with the solid lines denote experimental data from Ref. [349], the numbers given at the abscissa are the atomic mass numbers of the isotopes from He to Ne, only the even numbers are printed) (after Furihata et al. [255]).

3.5 The GSI ABLA Model

At GSI, the abrasion–ablation code system ABRABLA [221, 222] was developed and extended to describe also spallation reactions. For this purpose the BURST model [68, 350] developed at GSI was recently inserted in the ABRABLA code [351]. The ABLA code is part of the model ABRABLA describing the evaporation/fission competition in the decay of excited nuclei. Lately ABLA was improved by including a time-dependent fission decay width [352], emission of composite light-charged particles and intermediate mass fragments [350] as well as the simultaneous breakup. The particular strength of the ABLA code is the consideration of the dynamical nature of fission in the fission model taken into account. In the following some general but detailed features of the ABLA code are described although they resemble basically the assumptions of other evaporation/fission codes –though with some differences. ABLA is a dynamical code that describes de-excitation of the compound nucleus through the evaporation of light particles and fission. The probability that a compound nucleus with charge Z, neutron number N, and excitation energy E ∗ decays via channel ν is given by ν (Z, N, E ∗ ) , Pν (E) = ∗ i i (Z, N, E ) (3.36)


where i denotes all the possible decay channels (specifically, neutron emission, proton emission, alpha emission, fission). The particle evaporation is considered in the framework of the Weisskopf formalism. The particle decay widths can be written as [8] 1 4mνR2 2 T · ρd · (E − Sν − Bν ), 2πρc (E) 2

ν (E) =

(3.37)

where mν denotes the particle mass, Sν is the particle separation energy, Bν is the effective Coulomb barrier that takes into account the tunneling through the barrier, R is the radius of the nucleus, T is the temperature of the residual nucleus after particle emission, and ρc and ρd are the level densities of the compound nucleus and the daughter nucleus, respectively. The density of excited states, ρ, is calculated with the well-known Fermi gas formula [353]: √ ρ(E) =

π exp(S) , 12 a1/4 E 5/4 eff

(3.38)

with the entropy S: S=2·

aEcorr = 2 ·

a(Eeff + δUk(Eeff ) + δP · h(Eeff ))

(3.39)

and the asymptotic level density parameter a as given in [305]: a = 0.073 · A + 0.095 · BS · A2/3 ,

(3.40)

where A is the mass of a nucleus, and BS is the ratio of the surface of the deformed nucleus to that of a spherical nucleus. δU is the shell correction, which for the ground state is calculated as the difference between the experimental ground-state mass and the corresponding macroscopic value from the finite-range liquid-drop model [323]. As pointed out already, at the saddle point, shell corrections are assumed to be negligible. The function k(Eeff ) describes the damping of the shell effects with excitation energy, and is calculated according to Ref. [305] as k(Eeff ) = 1 − exp(−γ Eeff ), with the parameter γ determined by γ = a/(0.4 · A4/3 ). The effective pairing energy shift δP is calculated as 1 δP = − 2 · g + 2, 4

(3.41)

√ with an average pairing gap = 12/ A, and the single-particle level density at the Fermi energy g = 6 · a/π 2 . h(Eeff ) parameterizes the superfluid phase transition [354] at the critical energy Ecrit = 10 MeV:

h(Eeff ) =

1− 1− 1

Eeff Ecrit

2

for Eeff < Ecrit for Eeff > Ecrit .

(3.42)

155

156


The effective energy Eeff is shifted with respect to the excitation energy E ∗ to accommodate for different energies of even–even, odd-mass, and odd–odd nuclei: Eeff = E ∗ for odd–odd Eeff = E ∗ − for odd mass Eeff = E ∗ − 2 for even–even

(3.43)

As was shown in Ref. [222], collective excitations can contribute considerably to the nuclear level density. In deformed nuclei, the most important contribution to the collective enhancement of the level density originates from rotational bands, while in spherical nuclei the collective enhancement is caused by vibrational excitations. In ABLA, the contribution of collective excitation to the level density is described in the following way: for nuclei with a quadrupole deformation |β2 | > 0.15, the rotational enhancement factor Krot (Ecorr ) is calculated in terms of the spin-cutoff parameter σ⊥ : (σ⊥2 − 1)f (Ecorr ) + 1 for σ⊥2 > 1 (3.44) Krot (Ecorr ) = 1 for σ⊥2 < 1, with σ⊥2 =

JT , 2

Ecorr − Ec −1 f (Ecorr ) = 1 + exp , dc

(3.45)

where Ecorr is defined in Eq. (3.39), J⊥ = 2/5m0 AR2 (1 + β2 /3) is the rigid-body moment of inertia perpendicular to the symmetry axis, and m0 is the mass unit. The ground-state quadrupole deformation β2 is taken from the finite-range liquid-drop model including microscopic corrections [355], while the saddle point deformation is taken from the liquid-drop model as given in Ref. [356]. The damping of the collective modes with increasing excitation energy is described by a Fermi function f (E) with parameters Ec = 40 MeV and dc = 10 MeV. The vibrational enhancement for spherical nuclei is generally smaller than the rotational enhancement for deformed nuclei. For nuclei with a quadrupole deformation |β2 | < 0.15, the 2 vibrational enhancement factor is calculated as Kvib (Ecorr ) = 50βeff Krot (Ecorr ), where βeff is a dynamical deformation parameter: βeff = 0.022 + 0.003N + 0.005Z. N and Z are the absolute values of the number of neutrons and protons, respectively, above or below the nearest shell closure. More details about collective enhancement can be found in Ref. [222]. Finally, the total level density is calculated as the product of the intrinsic level density given by Eq. (3.38) and Kvib (Ecorr ) and Krot (Ecorr ). To define the fission-decay width, apart from the level densities, the necessary ingredients are also the dissipation effect that will be described in the next subsection and fission barriers. The angular-momentum-dependent fission barriers are taken from the finite-range liquid-drop model predictions of Sierk [323]. In order to describe the fission fragments’ mass and charge distributions, as well as their kinetic energies, the ABLA code is coupled to the semiempirical fission model described below.


157

3.5.1 Time-Dependent Fission Width

The modeling of the fission decay width at high excitation energies requires the treatment of the evolution of the fission degree of freedom as a diffusion process, determined by the interaction of the fission collective degree of freedom with the heat bath formed by the individual nucleons [357]. Such a process can be described by the Fokker–Planck equation (FPE) [358], where the variable is the time- and dissipation-dependent probability distribution W(x, p; t, β) as a function of the deformation in the fission direction x and its canonically conjugate momentum p. β is the reduced dissipation coefficient. The solution of the FPE leads to a time-dependent fission width f (t) . The results for the case of 238 U at a temperature of 3 MeV for different values of reduced dissipation coefficient β are shown in Figure 3.7 with solid lines. However, these numerical calculations are too time consuming to be used in nuclear-reaction codes. Therefore, in most of the model calculations one of the following approximations for the time-dependent fission width f (t) is used: a step function that sets in at time τf :

f (t) =

0 fk

for t < τf for t ≥ τf .

(3.46)

An exponential in-growth function: f (t) = fk (1 − exp (−t/τ )),

(3.47)

where τ = τf /2.3, with τf being the transient time defined as a time in which τf (t) reaches 90% of its asymptotic value given by the Kramers fission width τfk [357]. These approximations strongly deviate from the numerical solution and thus severely influence the results [352, 359]. Therefore, a new highly realistic description of the fission width based on the analytical solution of

l f(t) [1021 s−1]

0.04 0.03 0.02 b=2.0 × 1021 s−1

0.01 0.00

0 (a)

1

2

3

4

5

6

b =1.0 × 1021 s−1 7 0 (b)

1

2

3

4

5

Time t [10−21 s]

Fig. 3.7 Fission rate λf (t) = f (t) / as a function of time for three different values of the reduced dissipation coefficient β for 238 U at T = 3 MeV. The solid line is the numerical solution of the FPE while the dashed line is calculated using an analytical solution as described in Ref. [359].

6

b = 0.5 × 1021 s−1 7 0 (c)

1

2

3

4

5

6

7

158


the FPE when the nuclear potential is approximated by a parabola was developed [359]. The new analytical solution is based on the following assumptions: 1. the shape of the probability distribution at the barrier deformation as a function of the velocity v is constant and only its height varies with time; 2. for Wn (x = xb ; t, β) the solution of the FPE obtained using a parabolic nuclear potential can be used 3. zero deformation and zero velocity are considered as initial conditions. In addition, the zero-point motion was taken into account by shifting the time scale by the time needed to establish the initial shape of the probability distribution. A detailed description can be found in Ref. [359]. As one can see in Figure 3.7, this analytical approximation reproduces the exact solution for the critical damping (β = 2 × 1021 s−1 ) rather well. A similar agreement is reached in the overdamped regime (β > 2 × 1021 s−1 ). The approximation also gives a rather good description of the slightly underdamped motion (β = 1 × 1021 s−1 ). Even in the under-damped case (β = 0.5 × 1021 s−1), the oscillations are reproduced very well, although the absolute magnitude of the fission width is somewhat underestimated. In the previous investigations, it was shown that the total suppression of the fission width for small time values and the gradual increase are the most critical features of a realistic in-growth function. Both features, missing in the previously used descriptions, are well reproduced by the analytical approximation [352, 359]. The actual version of the ABLA code explicitly treats the relaxation process in deformation space and the resulting time-dependent fission-decay width, using the described approximate solution of the FPE. 3.5.2 The Simultaneous Breakup Stage

As for example the analysis [360] of the isotopic distributions of heavy projectile fragments from the reactions of a 238 U beam in a lead target and a titanium target gave evidence that the initial temperature of the last stage of the reaction, the evaporation cascade, is limited to a universal upper value of approximately 5 MeV. This is consistent with results on the caloric curve from multifragmentation experiments [361]. The interpretation of this effect relies on the onset of the simultaneous breakup process for systems whose temperature after the first stage of the reaction (e.g., the intranuclear cascade) is larger than 5 MeV. In ABLA, the simultaneous breakup stage is modeled in the following way: if the temperature after the first stage of reaction exceeds the value of 5 MeV, the additional energy is used for the formation of clusters and the simultaneous emission of these clusters and several nucleons. The number of protons and neutrons emitted is assumed to conserve the N-over-Z ratio of the projectile (or target) spectator, and an amount of about 20 MeV per nucleon emitted is released. The breakup stage is assumed to be very fast, and thus the fission collective degree of freedom is

3.6 The GEMINI Model

not excited. The major fragment left over from the projectile (or target) spectator undergoes the sequential decay. A more elaborate description of the breakup process on the basis of the statistical multifragmentation model (SMM) is given in ref. [362]. In the case of spallation reactions, the breakup stage plays an important role for light targets, while for heavy targets only a small fraction of prefragments in the upper tail of the excitation-energy distribution are formed with temperatures exceeding 5 MeV. As a consequence, the production of intermediate-mass fragments through the simultaneous breakup is more enhanced for light targets (e.g., iron). This could explain the failure of a standard evaporation model to describe the cross section for the production of intermediate-mass fragments (e.g., 7 Be, 14 C). 3.5.3 Conclusion of ABLA

The coupling of an intranuclear cascade code (e.g., INCL) to the ABLA model has shown to give a satisfying agreement with the isotopic distributions of spallation residues in the region of fission and heavy evaporation products. For the deexcitation stage of the reaction, the ABLA model has proved to give a much better reproduction of isotopic distributions and fission yields than other well-known models. The INCL4–ABLA combination has been compared to a large set of available experimental data obtained during the HINDAS project but also to earlier experimental results. An overall good agreement has been found for a broad range of different observable and bombarding energies. Most recently the evaporation part of ABLA was extended to the emission of not only neutrons, protons, tritons, deuterons, 3 He, and alphas but also intermediate mass fragments, i.e., light nuclei with Z > 2 [350]. ABLA has been implemented to high-energy transport codes widely used for the spallation neutron source and ADS design such as LAHET3, MCNPX, HERMES, MC4 and just recently to GEANT4 (cf. Table 5.2 on page 211 and is now available to the community.)


GEMINI is a statistical model code capable of describing the decay of compound nuclei and complex fragment cross sections by sequential binary decays. All possible binary divisions from light-particle emission to symmetric division are considered. The model employs a Monte Carlo technique to follow the decay chains of individual compound nuclei through sequential binary decays until the resulting products are unable to undergo further decay. The decay width for the evaporation of fragments with Z ≤ 2 is calculated using the Hauser–Feshbach formalism [363]. Similar to previously described evaporation models (cf. Eq. (3.1)), for the emission of a light particle (Z1 , A1 ) of spin J1 from

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a system (Z0 , A0 ) of excitation energy E ∗ and spin J0 , leaving the residual system (Z2 , A2 ) with spin J2 , the decay width is given by J2 (Z1 , A1 , Z2 , A2 ) =

2J1 + 1 · 2πρ0

J0 +J2

E ∗ −B−Erot (J2 )

Tl ()ρ2 (U2 , J2 )d,

0

l=|J0 −J2 |

(3.48) where l and are the orbital angular momentum and kinetic energy of the emitted particle, respectively, ρ2 (U2 , J2 ) is the level density of the residual system with thermal excitation energy U2 = E ∗ − B − Erot (J2 ) − , B is the binding energy, Erot (J2 ) is the rotation plus deformation energy of the residual system, and ρ0 is the level density of the initial system. The transmission coefficients Tl () were calculated with the sharp cutoff approximation for a classical system of absorptive radius R, and are given by

Tl () =

0

for > ECoul +

1

for ≤ ECoul +

2l(l+1) 2µR2 2l(l+1) 2µR2 ,

(3.49)

where µ is the reduced mass. The Coulomb barriers ECoul were calculated using the empirical expressions of Vaz and Alexander [364] and the absorptive radius was taken as

1/3 1.16A2 + 2.6 fm for proton and neutron emission R= (3.50) 1/3 1.16A2 + 3.7 fm for alpha particle emission. In calculating the binding energy for heavy systems (A > 12), the masses of the initial and residual systems were obtained from the Yukawa-plus-exponential model of Krappe, Nix, and Sierk [163] without the shell and pairing correction terms. The parameters for this model were taken from the more recent fit to experimental masses of Möller and Nix [365]. These separation energies are expected to be more appropriate at high excitation energies where shell and pairing effects are predicted to wash out. For very light systems (A ≤ 12), binding energies were calculated from the experimental masses. The rotation plus deformation energy Erot of a nucleus was taken from the RFRM calculations of Sierk [323]. For binary divisions corresponding to the emission of heavier fragments, the decay width was calculated using the transition state formalism of Moretto [366]: (Z1 , A1 , Z2 , A2 ) =

1 · 2πρ0

E ∗ −Esad (J0 )

ρsad (Usad , J0 )d,

(3.51)

0

where Usad and ρsad are the thermal energy and level density of the conditional saddle point configuration, respectively, Usad = E ∗ − Esad (J0 ) − . Esad (J0 ) is the deformation plus rotation energy of the saddle point configuration and now is the kinetic energy of the translational degree of freedom. The deformation plus rotation energy was calculated with the RFRM using a two-spheroid


parameterization for the shape of the conditional saddle point configuration. This parameterization results in conditional barriers which are within 2 MeV of saddle point energies calculated with more realistic shape parameterizations [367] for A0 = 110 [323]. Better agreement is obtained for lighter nuclei. To correct for this difference to first order, the two-spheroid saddle point energies were scaled by a constant factor for all mass asymmetries and angular momentum. The scaling factor was chosen so that for symmetric division, the scaled saddle point energy was equal to the value calculated with the more realistic shape parameterization by Sierk [323, 367]. For Z ≤ 6, these RFRM saddle point energies (E RFRM (J0 )) were modified by RFRM (J0 ) − MY+e (Z1 , A1 ) + Mexp (Z1 , A1 ), Esad (J0 ) = Esad

(3.52)

where MY+e is the mass predicted by the Yukawa-plus-exponential model without shell and pairing corrections and Mexp is the experimental mass. This modification is an attempt to introduce shell effects into the saddle point energies for very asymmetric divisions, where one expects them to become more important. For all level densities, the Fermi gas (cf. also Sections 2.2.2.1 and 2.2.2.2 on pages 2.2.2.1 and 2.2.2.2) expression [324, 368] ρ(U, J) = (2J + 1)

2 2J

√ a exp(2 aU) 12 U2

2/3 √

(3.53)

was used, where J is the moment-of-inertia of the residual nucleus or saddle point configuration. The level density parameter was taken as a = A/8.5 MeV−1 for both the residual nucleus and in saddle point configurations. The integrations in Eqs. (3.48) and (3.51) were performed by first expanding the integrand around the lower limit, giving the approximate expression for the decay width for Z1 ≤ 2 as (Z1 , A1 , Z2 , A2 ) =

∞ 2J1 + 1 · 2πρ0

J0 +J2

t2 ρ2 (U2 , J2 ),

(3.54)

J2 =0 l=|J0 −J2 |

where now U2 = E ∗ − B − Erot (J2 ) − ECoul −

2l(l + 1) 2µR2

and the nuclear temperature is approximately t2 = For Z1 > 2 (Z1 , A1 , Z2 , A2 ) =

1 tsad ρsad (Usad , J0 ), 2πρ0

(3.55) √ U2 /a.

(3.56)

where now Usad = E ∗ − Esad (J0 ) and the temperature of the saddle point configura√ tion is approximately tsad = Usad /a . The secondary products formed in the binary

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decay of the initial system, (Z1 , A1 ) and (Z2 , A2 ), were allowed to undergo sequential binary decay. The spin of the residual system was chosen in a Monte Carlo fashion from the calculated partial decay widths J2 (Z1 , A1 , Z2 , A2 ). Its excitation energy was calculated as E2∗ = U2 − 2t2 + Erot (J2 ).

(3.57)

For Z1 > 2, the spin of the fragments was calculated in the sticking limit, i.e., for fragment 1: J1 = (J1 /J )J0 , where J1 is the moment of inertia of the fragment and J is the total moment of inertia of the system. Its excitation energy was derived assuming equal temperatures for the two fragments, as E1∗ =

A1 [Usad − tsad ] + Erot (J1 ). A0

(3.58)

This is strictly valid only when the saddle and scission point configurations are degenerate and this is approximately true for the systems and reactions under consideration in, e.g., Ref. [229]. However, Eq. (3.58) is not applicable for heavier systems. As for the level density parameter a (cf. also Sections 3.2, 3.3.1, and 3.4.4), several model options can be chosen in the GEMINI model: • a = A0 /const • Toke, Swiatecki level density parameters [325]1) • Ignatyuk et al. [305]1) • Gottschalk and Ledergerber [369]1) • Ormand et al. [327]2) • Lestone [370]3) • Fineman et al. [371]3) . It is up to the user to select the most appropriate one for his or her respective system. The model also contains options for enabling or disabling fission and intermediate-mass (IMF) emission. Either a formalism is used which is identical to the one used in the models PACE and CASCADE (Bohr–Wheeler formalism) or all asymmetric divisions may be considered (Moretto’s formalism). Sequential decay from the IMFs and fission fragments is considered. In a simplified way the fission dynamics is taken into account by introducing optionally the Kramers factor [60, 309, 357, 372–377] depending on the viscosity at the saddle point. The Kramers factor scales down the fission decay width at the saddle point by a constant (cf. also detailed description in Section 3.7). Charity [229] has 1) Mass asymmetry dependent. 2) The temperature dependence of the level density parameter is used, however, surface area dependence was neglected.

3) Temperature dependent.


extended this formalism by an additional degree of freedom – the mass asymmetry. The mass asymmetry is defined by m1 − m2 , η = m1 + m2

(3.59)

with m1 and m2 being the two primary fragments of a binary decay process. The value of η ranges from 0 for symmetric fission to 1 for the emission of light particles. Following this definition, and assuming a Gaussian distribution for the mass asymmetry dependence, the delay time τ (η) is expressed as 1 −η 2 , τ (η) = τ (0) exp 2 ση

(3.60)

with τ (0) being the maximum possible delay time for the symmetric splitting of the system. For a chosen width of the distribution of ση = 0.25, the delay time τ (η) for the emission of light particles is significantly smaller than typical times of statistical emission (/ evap ). Complex fragment decays can be very improbable events. Thus in analog Monte Carlo simulation one may have to run a large number events to achieve reasonable statistics for intermediate mass fragments. To solve this problem and reduce CPU time, to some extent, weighting has been introduced in the GEMINI model by modifying the simulation of the event histories in a nonanalog manner. There exist various variance reduction methods used in nonanalog Monte Carlo which are described, e.g., by Lewis and Miller, Jr., in Ref. [378]. The GEMINI model considers the following Monte Carlo procedure in nonanalog fashion. • At the beginning of the nonanalog procedure a parameter Iweight = 1 is set to include weighting. • At the start of an event, Kweight is set to 1. While Kweight = 1, the decay widths for complex fragment emission are increased relative to their calculated values. If in the decay chain, a complex fragment is emitted, then Kweight is set to 0 and the complex fragment decay widths are not weighted for the remainder of the decay chain. • To take into account the increased yield of complex fragments, each event now has a weight. If a complex fragment is emitted the weight is decreased, whereas if no complex fragment is emitted, the weight is increased. • The weight of an event is given as Prob(statistical model) , Prob(simulation)

(3.61)

where Prob(statistical model) is the probability of that event, as calculated by the statistical model, and Prob(simulation) is the probability of GEMINI simulating that event.

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164


• If GEMINI simulates events according to the statistical model probabilities (i.e., Iweight = 0), then the weight of each event is always 1, as is expected. • If Iweight = 1, GEMINI selects the existing channels in the following manner: – The complex fragment decay widths are multiplied by a constant (FACT), so that at the first chance the complex fragment emission probability is at least a value given by a given parameter FACT0 (set to FACT0 = 0.1, which may be changed in GEMINI). – The same value of FACT is used for the second, third, etc. chance complex fragment emission. If the first chance complex fragment decay probability is already greater than FACT0, then the parameter FACT is set to 1. – Also after a complex fragment decay, FACT is set to 1 in simulating the decay of the primary fragments. • Now the probability of GEMINI selecting a light particle decay channels is channel , total−light + FACT ∗ IMF−total

(3.62)

where channel is the decay width of the exit channel, total−light is the total decay width for light particle emission and IMF−total is the total decay width for complex fragment emission. • Now the probability of choosing any complex fragment exit channel is FACT ∗ IMF−total . total−light + FACT ∗ IMF−total

(3.63)

The actual channel is chosen so that all charge splits are equally likely. The mass split is then chosen according to the statistical model decay widths. In the GEMINI model caution must be given for input angular momenta larger than where the fission barrier vanishes. For instance, the subroutine of Sierk to give rotating ground state energies often gives rotation energies which decrease with angular momentum in this range. Sierk never intended his subroutines to be run at these angular momenta. Similar caution must be given concerning the extrapolation of asymmetric barriers. GEMINI principally considers the compound nuclei decay only up to 194 Hg. In this case the fissility of the nucleus is larger than the one of 194 Hg, the asymmetry dependence of the fission barriers of 194 Hg is assumed, and barriers are scaled such that the symmetric barriers for the respective nucleus are correct. This procedure is valid only for nuclei not much heavier than 194 Hg. Furthermore, as for any statistical model, the excitation energies E ∗ shall be considerably smaller than the binding energy of the nucleus. This is one of the fundamental assumptions of statistical models, because before and after the emission of any fragment a statistical equilibrium is required. This prerequisite is fulfilled only, if enough time is left for the nucleus to achieve equilibrium again after the emission of a fragment.

3.7 High-Energy Fission Models

To the best of our knowledge, for GEMINI there exists no published manual or handbook and mostly users of the code have to refer to the original article of Charity et al. [229]. Actually this publication presents a variety of observables, e.g., charge distributions, excitation energy of the compound nuclei, compound-nucleus spin distributions, etc., as calculated with GEMINI and confronts experimental data and complex fragments with 3 < Z ≤ 35 for Nb + Be, C, and Al reactions. As a general tendency cross sections calculated with the GEMINI model code reproduce both the shape and magnitude of the experimental charge distributions rather well as has also been shown in Refs. [114, 228, 379]. Probably the most comprehensive and helpful description of the physics and parameters of GEMINI is found in the well-commented source code of the model and notes therein. The computer code of the GEMINI model by Charity [229] is written in standard Fortran 95 language except that for a number of OPEN statements where the author uses the DEC/Compaq option ‘‘shared.’’ This can be removed and will not affect the program; it does however allow you to sort the data while they are being generated, without it you risk killing GEMINI if you sort an event file which is being written to. Most recent versions of GEMINI are available from http://wunmr.wustl.edu/rc/.


Without fission, spallation collisions can be treated as a two-step process: (i) an intranuclear cascade, described by a series of independent particle collisions inside the nucleus, and (ii) subsequent de-excitation by a series of particle emissions, which are described by evaporation models. For heavy nuclei, there is competition between evaporation and fission at each step of the de-excitation sequence. The probability of fission at some step during de-excitation in high-energy (E ≥ 100 MeV) collisions is proportional to Z2 /A of the target nucleus. For example, the ratio of the fission cross section to the total cross section for 1 GeV protons on lead and uranium targets is σfission /σtotal ≈ 0.05 for lead and is σfission /σtotal ≈ 0.8 for uranium (see for example Ref. [380]). As illustrated in Figure 3.8, the high-energy fission in spallation reactions may be considered as a four-step process. The information which must be determined by fission models is the probability of fission Pf = Pf (E ∗ , A, Z) of the residual nucleus (A, Z) with the excitation energy E ∗ , which is left after the intranuclear cascade, at each step and the parameters of the fission fragments after fission, which are then used as input for a postfission evaporation calculation for each fragment [338], respectively. The four steps shown in Figure 3.8 consider mainly the following reaction mechanisms: (1) production of high-energy particles and an excited residual nucleus, (2) emission of low-energy neutrons, protons, tritons, 3 He, and 4 He particles, (3) fission-evaporation competition at each stage of the de-excitation, (4) deexcitation of fission products by particle emission. After step 4, a fifth step may follow by de-excitation to the ground state of the residual nucleus by a gamma-ray

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166

3 Evaporation and High-Energy Fission Protons

1. Intranuclear cascade

2. Prefission evaporation

3. Fission

4. Postfission evaporation

Fig. 3.8

The principle of the high-energy fission model considered as a four-step process.

cascade. The dynamics of the fission process as a collective motion of nuclear matter is shortly described in Section 3.7.1. A short overview about the fission process was already given in Section 1.3.2 on page 15 and in Section 1.3.3 on page 17. 3.7.1 The Dynamics of the Fission Process

The binary splitting of an excited nucleus into two approximately equal parts is still considered as one of the most interesting phenomena of collective motion of nuclear matter and as an excellent example of the nuclear multiparticle problem [62]. A simple model regarding the nucleus as an incompressible liquid is drawn by the liquid drop model (LDM) [60]. Analog to a liquid drop, a surface tension is responsible for the inner forces acting on all surface nucleons. As demonstrated in Figure 3.9(a), fission is understood as a consequence of the deformation when repelling electrostatic Coulomb forces on the protons overbalance the short-range attractive nuclear forces. A ground-state-deformed nucleus is situated in the minimum of the potential energy which increases with increasing deformation toward the so-called saddle point deformation. Beyond the saddle point the potential energy declines due to the decreasing Coulomb repulsion until the scission point is reached. Then the nucleus is constricted in such a way that fragmentation into two parts is likely. The fission barrier Bf represents the difference of the potential energy at the ground-state and the saddle point deformation. As shown in Figure 3.9(b), there is a maximum of Bf at A 70 and a substantial decline for light and heavy nuclei. Also with increasing E ∗ the fission barrier declines, because due to the expansion of the nucleus at high excitations the surface energy


Potential [MeV]

tF

t

Bf

tss

E* = 0

50

50 MeV

40

0

100 MeV

−200

Saddle

−30

Ggw

30 Scission

Deformation

200 MeV

20 10 0

0

50 100 150 200 250 Atomic mass number

(a)

Potential deformation

(b)

Fission barrier

Fig. 3.9 (a) Potential energy as a function of deformation. The transition time from equilibrium (Ggw) to the saddle point τ and the one from saddle to scission τss are indicated (the figure adapted from [381]). (b) Fission barrier Bf as a function of the atomic number A and E∗ along the β-stability line (the figure adapted from [382]). Bf is given in units of MeV.

decreases faster than the Coulomb energy. The barrier is also a function of the angular momentum of the nucleus, because additional rotational energies disperse the nucleus. These phenomena are considered in the ‘‘rotating LDM’’ [383] which describes the fissility by [384] χ=

Z2/A Ec0 = 2 · Es0 50.883(1 − 1.7826 · I2 )

with

I = (N − Z)/A,

(3.64)

where Ec0 and Es0 being the Coulomb and surface energy of the nucleus in the ground state, and Z, N, and A the atomic, neutron, and mass numbers, respectively. Asymmetric mass splitting at low E ∗ cannot be explained by the LDM. Shell effects have to be taken into account, which however disappear for excitations larger than a few 10 MeV [385–388]. At high E ∗ , fission is in competition to the successive neutron emission of the excited nucleus. The competition between fission and particle emission as well as the characteristics of fission products can be described by statistical models. Under the assumption of thermal equilibrium the probability for fission or neutron emission can be described according to Eq. (3.65) by the ratio of decay widths n and f [389]: 4A2/3 af (E ∗ − Bn ) n = 1/2 f C0 an [2af (E ∗ − Bf )1/2 − 1] 1/2 ∗ 1/2 × exp 2a1/2 − 2af (E ∗ − Bf )1/2 , n (E − Bn )

(3.65)

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168


where Bf , E ∗ , and Bn represent the height of the fission barrier, the excitation energy and the binding energy of the neutron, respectively. The constant C0 is C0 = 2/2mn r0 2 (mn = neutron mass) and af and an are the level density parameter at the saddle point and the ground-state deformation after the emission of a neutron. For small angular momenta and nuclei with fission barriers smaller than their neutron binding energy (Bf < Bn in Eq. (3.65), the fission takes place early, since n / f increases after the emission of each emitted neutron. For high angular momenta n / f decreases with increasing E ∗ and consequently the nucleus fissions at a later stage. A phenomenological approach for fission product yield calculations at intermediate incident particle energy has been proposed by Konobeyev et al. [388]. This approach is based on the Fong statistical model and empirical expressions. A good agreement with experimental data was demonstrated. For a comprehensive description of the dynamics of the fission process the concept of dissipation and the viscosity of nuclear matter refer to review articles [60, 309, 357, 372–375, 377]. Dissipative processes are accompanied by statistical fluctuations which exchange energy between intrinsic and collective degrees of freedom. Latterly collective transport models are discussed which consider these statistical fluctuations [390–393]. Besides the saddle to scission time τss these models require a transient time6) τ . The transient time is a function of dissipation or friction and can be parameterized by the so-called Kramers factor [376]. The total time scale τF of the dynamical fission process is described by the sum of the transient time and the saddle to scission time τF = τ + τss (Figure 3.9(a)). Actually the time shall not be subdivided into pre- and postsaddle, because in reality the saddle point could be passed through several times. In collective transport models the dynamical process of the system from the ground state to the scission is continuously traced. While the characteristic time scales for processes described by intrinsic degrees of freedom are of the order of 10−23 s [394], the ones for collective motions are two orders of magnitude larger. Therefore, a theory is proposed decoupling the Hamiltonian for the total energy of the system into a collective and an intrinsic part. The transport equations are controlled by one or several transport coefficients, such as, e.g., the reduced dissipation coefficient β, which reflects the coupling strength between the collective and intrinsic degrees of freedom. These coefficients can be used for defining the time scale of the process. The total energy of the system in the collective transport model is given by the temperature-dependent function H(q, p, T) [324, 395]: H(q, p, T) = Ekin (q, p) + F(q, T).

(3.66)

Ekin (q, p) and F(q, T) are the kinetic and free energy of the system, respectively, and q ≡ q1 , q2 , . . . , qN denote the N generalized collective coordinates representing 6) The transient time is the time the system needs for passing the saddle point configuration.


the form of the system. pi = Mij (q) q˙ j are the collective generalized momenta. The dynamics of the system and the equations of motion, which contain the effects of dissipation, can be deduced from the Hamiltonian equations [396]: ∂H ∂Ekin = ∂pi ∂pi ∂Ekin ∂F ∂H ∂F p˙ i = + Qi + δXi = + + + δXi . ∂qi S ∂qi T ∂qi ∂ q˙ i q˙ i =

(3.67) (3.68)

The dissipative force Q is expressed by the ‘‘Rayleigh dissipation function’’ F and a fluctuation term δXi . The conservative force ∂H ∂qi S can be described either by the free energy F or by the entropy S according to the thermodynamical relation

∂S ∂H ∂F = = T . ∂qi S ∂qi T ∂qi E

(3.69)

The collective kinetic energy Ekin (q, p) is given by Ekin (q, p) = 1/2Mij (q)˙qi q˙ j [60, 396]. The Rayleigh dissipation function F = 1/2ηij (q)˙qi q˙ j [60, 396] includes a shapedependent dissipation tensor ηij (q), which describes the conversion from collective to single particle energy. For the term representing the fluctuations no history, e.g., no temporal correlations exist (Markovian assumption). Employing the collective energy Ekin and the dissipation function F in Eqs. (3.67) and (3.68) the multidimensional Langevin equation is obtained, which represents in most general form the collective transport model. Due the fluctuation term δX the Langevin equation is a stochastic equation with stochastic variables p and q. The temporal evolution of the function F(q, p, T) in the phase space of collective coordinates and their conjugated momenta is presented by the generalized Fokker–Planck equation [358]. The stochastic description of the Langevin equation is equivalent to the Fokker–Planck equation provided that the Markovian assumption is accepted. Previous investigations which try to describe the fission process using transport equations are based on the Fokker–Planck equation with one collective degree of freedom [390–392]. These studies certainly resulted in some insight into the dynamics of the fission process, but with the addition of latest precise measurements of the fission lifetime as a function of E ∗ using blocking effects in a single crystal, important consequences for the dynamical description of the collective process have been illustrated. This work essentially carried out at GANIL, France, by Goldenbaum [397–399] and Morjean et al. [400] will not be part of the present book. As another approach, besides the application of these transport models, the statistical model can be modified in such a way that pre- and postsaddle time scales are evaluated by the measurement of pre- and post scission particle multiplicities and the magnitude of dissipation is estimated. Measurements of prescission neutron multiplicities [401–403], light charged particles [404, 405] and giant γ resonances [406] allowed for accessing the relative time scales of fission and particle emission. A significant relaxation time of collective degrees of freedom pointing to a large dissipation has been demonstrated. The total time scale of

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170


fission has been limited to τF ≤ 50 × 10−21 s [401]. Lestone [407] confines applying his measurements τF to τF ≤ 30 × 10−21 s. 3.7.2 Basic Features of Fission Models

There are several high-energy fission models developed for use in spallation reaction calculations and in thick-target nucleon meson transport codes (i) the RAL Atchison model of the Rutherford Appelton Laboratory, UK [311–313], (ii) the ORNL model at the Oak Ridge National Laboratory, USA, developed by Alsmiller et al. [314, 408], (iii) the BNL model at the Brookhaven National Laboratory, USA, by Takahashi [409], the JAERI model at the Japan Atomic Energy Research Institute, Japan, by Nakahara [410, 411], and the Barashenkov model at Dubna by Barashenkov [412]. In addition to the more stand-alone fission models there are fission models included into evaporation models as in the ABLA model of Schmidt et al. [221, 222] or in the GEMINI model of Charity et al. [229, 256] (cf. Sections 3.5 and 3.6 on pages 154 and 159). The fundamental basis of all these models is the statistical model of fission developed by Fong [310, 413]. Basically, the assumption is that the fission process is ‘‘slow,’’ i.e., the nucleus exists in an equilibrium state at any time, so the probability of a particular fission mode, the state of the fission fragments, is proportional to the density of the quantum states at the time of splitting. From the theory of Fong the fission mode probability is expressed as a function of 12 variables: • N(A1 , A2 , Z1 , Z2 , E1∗ E2∗ j1 , j2 ) C, D, K, E, where the subscripts denote the two fission fragments; • A, Z, E ∗ , and j denote the mass number, charge number, excitation energy, and angular momentum, respectively; • the remaining parameters are energy variables, C = Coulomb energy, D = deformation energy, K = translational energy, E = total kinetic energy. The high-energy fission models developed differ in the approximations made in arriving at a practical application and implementation. It is not the purpose here to evaluate the various approximations. Therefore as an example only the features of the RAL model of Atchison are discussed in more detail. A comparison of the RAL model with the ORNL model of calculated fission cross sections, neutron production, and residual mass distributions with experimental data is given in Ref. [380]. Indeed, many of the approximations of the above fission models are interdependent, making a judgment of the practical importance of particular assumptions difficult. A simple characterization of the differences is given for different models such as follows: • RAL model: empirical formulas used as far as possible, • ORNL model: heavy reliance on empirically derived constants, fission is neglected for subactinides Z < 91, • BNL model: close simulation of Fong’s statistical model formulas with minimum reliance on experimental data,

3.7 High-Energy Fission Models Tab. 3.5

B0 values used in standard versions of fission models.

Fission model

RAL

ORNL

BNL

Parameter B0

14

10

8

JAERI 8

• JAERI model: similar to the RAL approach, but it is different in detail.

As discussed in the context of the level density in Sections 3.3.1 an 3.4.4 on pages 140, 147 and given by Eq. (3.13), the level density a is in the simplest form depending on the value of the parameter B0 . The different fission models are used in the standard versions values of B0 as shown in Table 3.5. It should be noted that the value of B0 has an important influence on neutron production as shown in Refs. [338, 380]. 3.7.3 The RAL Model of Atchison

The RAL fission model is based on several general assumptions. • Fission competes at all stages of the nuclear de-excitation with fission probabilities that vary with the nuclear state. • Postfission parameters depend only on the state of the nucleus at the time of fission and not on the previous sequence of events leading to the state of the nucleus at the time of fission. • Only binary fission will occur. This means that fission is locked off for fission fragments themselves. Although ternary fission and high-order fission have been observed, e.g., Ref. [414], they occur with probabilities for which most spallation research applications are insignificant. • The mass split is always complete. The two fission fragments will conserve baryons and charge number. Thus, ‘‘fission’’ neutrons will be produced from either evaporation prior to fission or de-excitation of the fission fragments. • Fission for nuclei with Z < 70 is not considered. The fission probability Pf is given by the ratio of evap = n and f for the widths for evaporation and fission at some point in the de-excitation cycle: Pf =

f 1 = . f + n 1 + n / f

(3.70)

Because fission is only considered for heavy nuclei for which evaporation is dominated by neutron emission, the evaporation width is adequately represented

171

172


by the width for neutron emission n . The treatment of the fission probability Pf is differently considered for elements above and below Z = 89. In the subactinide region, Z < 89, results of experiments show that the fission probability Pf is strongly dependent on the excitation energy E ∗ (e.g., [412]). The RAL model [311, 312] uses the Weisskopf–Ewing statistical model [320] with an energy-independent pre-exponential factor for the level density and Dostrovsky’s [7] inverse cross-section. Separate fits are made for neutron emission and fission to allow separate level density parameters. The expressions for the neutron emission or neutron width n for nuclei with 70 ≤ Zj ≤ 88 are: 1/3

2/3

n = 0.352(1.68J0 + 1.93Ai J1 + Ai (0.76J1 − 0.05J0 )),

(3.71)

where J0 and J1 are functions of the level density parameter an and Sn with Sn = 2 an (E ∗ − Q − δ) = 2 an (E ∗ − B ) and (Sn − 1) · exp(Sn ) + 1 2an (2S2n − 6Sn + 6) · exp(Sn ) + S2n − 6 J1 = 8a2n J0 =

(3.72) (3.73)

with E ∗ the excitation energy and B the separation energy minus pairing energy. The expression for the fission width f for nuclei with 70 ≤ Zj ≤ 88 are: f =

((Sf − 1) · exp(Sf ) + 1) , af

(3.74)

where Sf = 2 af (E ∗ − Q − δ) = 2 af (E ∗ − Bf ) and the level density parameter in the fission mode af is fitted by Atchison to describe the measured f / n as af = an (1.08926 + 0.01098(Zi2 /Ai − 31.08551)2 )

(3.75)

with E ∗ the excitation energy and Bf the fission barrier. The fission barriers Bf in [MeV] are estimated as 2 Zi2 Zi2 + 0.218 . Bf = Q + 321.2 − 16.7 Ai Ai

(3.76)

The RAL model uses a fixed value for the level density parameter an , namely, (an = (Ai − 1)/8 (MeV)−1 ). Note that neither the angular momentum nor the excitation energy of the nucleus is taken into account in estimating Bf . The above expressions are used for Z ≤ 88.


The fit parameters for the fission barrier Bf and the level density ratio, af /an , are based on data for Z < 85, so the model fits are used for extrapolation to obtain the fission probability in the Z = 85 to the Z = 88 region. In the actinide region for high-Z nuclei, Zi ≥ 89, the fission barrier is relative constant at about 6 MeV, and the probability for fission, Pf is not strongly dependent on the excitation energy E ∗ (see, e.g., Gavron et al. [415]). Therefore, the RAL model assumes Pf = 0 for E ∗ ≤ 6 (MeV) and Pf independent of E ∗ above 6 MeV. Neither the fission width f is calculated nor Eq. (3.70) is employed for the fission probability Pf . Instead a semiempirical expression is obtained in the RAL model by approximating the experimental values of n / f published by Vandenbosch and Huizenga [389] to calculate the fission probability

log n / f = C(Zi )(Ai − D(Zi )),

(3.77)

where C(Zi ) and D(Zi ) are functions depending on the nuclear charge Zi only. The expression is used for 89 ≤ Zi ≤ 99 with fission not considered for higher Zi . The values of these constants can be found in the current version of LAHET [110, 111, 193]. 3.7.3.1 Postfission Parameters of the RAL Model If fission has occurred for a nucleus of charge Zi , atomic mass Ai , excitation energy Ei∗ , and recoil energy Ei-recoil the corresponding parameters for each of the fission fragments have to be determined (cf. Section 3.4.5). (1) Mass distribution:

(a) The selection of the masses for the fission fragments depends on whether the fission is symmetric (a wide, single peak for the fragment mass distribution) or asymmetric (two peaks). (b) For Zi2 /Ai ≤ 35 only symmetric fission is allowed for a prefission nucleus, and the mass A1 is selected from a Gaussian distribution of mean A1 = Ai /2 with the index i as fragment type and width σm-RAL . σm-RAL = 3.97 + 0.425 · Ui − 2.12 × 10 − 3 · Ui2

(3.78)

with Ui = Ei∗ − Bf as the excitation energy above the fission barrier Bf . This symmetric fission mass distribution is based on the systematics given by Neuzil and Fairhall [416]. (c) If Zi > 88 the fission energy barrier Bf is determined in the RAL model based on the work of Seaborg and Vandenbosch [417, 418], Bf = C − 0.36 with C = 18.8 for odd–odd nuclei, C = 18.1 for even–odd nuclei, C = 18.1 for odd–even nuclei, and C = 18.5 for even–even nuclei.

(3.79)

173

174


(d) For Zi2 /Ai > 35 both symmetric and asymmetric fission are allowed, depending on the excitation energy E ∗ of the fissioning nucleus. For symmetric fission, A1 is selected from a Gaussian distribution of mean Af = Ai /2. With A1 determined , the mass of the other fission fragment is A2 = Ai − A1 . Whether the fission is symmetric or not is determined by the asymmetric fission probability Pasy Pasy =

4870 · exp(−0.36E ∗ ) . 1 + 4870 · exp(−0.36E ∗ )

(3.80)

For asymmetric fission, the mass of one of the postfission fragments A1 is selected from the Gaussian distribution of mean Af = 140 and width σm-RAL = 6.5. Consequently the second mass is A2 = Ai − A1 . (2) Charge distribution: (a) The charge distribution of the fission fragments is assumed to be a Gaussian distribution of mean Zf and width σZ = 2. Zf is expressed as Zf =

Zi − Z1 − Z2 2

with Zi =

65.5Ai 2/3

131 + Ai

,

i = 1, 2.

(3.81)

(3) Energy distribution: (a) The total kinetic energy of the fission fragment in the c.m. system, Etotal , is selected from a Gaussian distribution having a mean kinetic energy Ef and a width σEf , 1/3

Ef = 0.133 · Zi2 /Ai

− 11.4

σEf = 0.084 · Ef .

(3.82)

These relations are based on the evidence of the strong correlation of the fission fragment kinetic energies with the Coulomb repulsion parameter 1/3 Zi2 /Ai of the fissioning nucleus (e.g., [414]). The kinetic energy of each fragment in the c.m. system is given by E1 = A2 · Ef /Ai

and

E2 = Ef − E1 .

(3.83)

Isotropic emission in the c.m. system is then assumed to obtain the kinetic energies in the laboratory system. (b) The excitation energies of the fission fragments are then determined by an energy balance E1∗ = (A1 /Ai ) · (Ei∗ + B1 + B2 − Bi − Ef ) and

E2∗ = E1∗ · A2 /A1 , (3.84)

where B denotes the binding energies.


100

Probability of fission

100 MeV −2

10

20 MeV

10−4 50 MeV

10−6 175

200

225

250

Atomic mass number A Fig. 3.10 Probability of fission simulated using the RAL model for various isotopes arbitrarily selected over the atomic mass range A = 175–250. The top, middle, and bottom curves are for assumed excitation energies of 100, 50, and 20 MeV, respectively.

Figure 3.10 shows the fission probability simulated with the RAL model for various arbitrarily selected isotopes covering the atomic mass range from 175 to 250. The figure shows that the fission probability is very small for masses A ≤ 200, accounting for a peak in this region, which is shown in Figure 3.11. Figure 3.11 illustrates the prediction of mass distributions with and without high-energy fission taken into account for protons of 1.0 GeV incident on a 238 U target. The RAL model predicts three peaks in the mass distribution: The fission fragment peak at a mass about A ≈ 110, the spallation product peak at a mass of A = 238, and an intermediate peak at a mass of about A ≈ 200. This intermediate peak apparently results from spallation products which ‘‘survive’’ de-excitation through the mass region of high-fission probability into a lower mass region where further de-excitation by neutron emission is much more likely than fission. This intermediate peak in the mass distribution is probably most evident at ‘‘medium’’ incident beam energies, e.g., at low beam energies of about ∼100 MeV where there is no sufficient excitation energy to produce many nuclei in the lower mass region of low-fission probability (see Figure 3.10), whereas at high beam energies there is sufficient excitation energy that spallation products can be produced with very low masses which overlap with the higher mass fission fragments.

175


Residual nuclei distribution per mass number per nonelastic collision

176

10−1 1 GeV protons on a 238 U target

With fission Without fission

10−2

10−3

10−4 0.0

40.0

80.0 120.0 160.0 200.0 240.0 Mass number A

Fig. 3.11 Comparison of mass distributions with and without high-energy fission simulated with an EVAP-RAL [380] model combination taken into account for protons of 1.0 GeV incident on a 238 U target.

3.8 Fermi Breakup for Light Nuclei

As mentioned in Sections 3.1 and 3.2, the decay of the compound nucleus is usually treated by statistical models using sequential emission. The models of Weisskopf and Weiskopf–Ewing [8, 320] and of Hauser–Feshbach [363] are useful. However, the level density formalisms employed in evaporation models are, in general, poor approximations for light nuclei. Furthermore, cascade residuals or evaporation residuals may de-excite by a simultaneous statistical breakup into multiparticle exit channels [419–422] rather than by sequential evaporation of light particles. The breakup mechanism was originally proposed by Fermi [263] to describe highenergy hadronic collisions due to the multiple production of particles in NN and in πN collisions. The Fermi breakup mechanism (FBM) is particularly applicable for the de-excitation of light nuclei which have a lower density of states, fewer degrees of freedom of the nucleons, and lower Coulomb barriers. Excited light primary fragments left as residuals after the intranuclear cascade have often values of the excitation energy E ∗ per nucleon comparable with the nucleon binding energy. In this case one can assume that the principal mechanism of de-excitation is an explosive decay of the excited nucleus into several separate clusters and isolated nucleons [422]. Several high-energy particle transport models and codes, e.g., CEM [249], FLUKA [423], GEANT4 [100, 424], MCNPX [194], are replacing the evaporation models by a Fermi breakup model for the disintegration of light nuclei mainly with atomic masses A < 17 to predict the final states (cf. Chapter 5). The condition for the Fermi breakup A breakup decay is allowed if the total kinetic energy E for all fragments of a given channel at the moment of breakup is positive.

3.8 Fermi Breakup for Light Nuclei

The total kinetic energy E for a breakup into n fragments is calculated using the expression E = U + M(A, Z) − EC −

n

(mb + E ∗ ),

(3.85)

b=1

where mb and E ∗ are the masses and excitation energies of the fragments, respectively. EC is the Coulomb barrier for the appropriate channel b and is given by n 3 · e2 1/3 −1/3 2 1/3 2 , EC = · Z /A − Z /Ab (1 + V/V0 ) 5 · r0

(3.86)

b=1

where the relation V/V0 = 1 is normally used. V0 is the volume of the system corresponding to the normal nuclear matter density. The total probability The total probability for the residual nucleus for separation in n components, e.g., nucleons, d, t, 3 He, 4 He, in the final state in channel b is given by the expression

P(E, n) = (V/)n−1 ) · ρn (E),

(3.87)

where V is the decay volume of the system, = (2π)3 is the normalization volume of the system, and ρn (E) is the density of final states. The density can be defined as a product of three factors ρn (E) = Mn (E) · Sn · Gn .

(3.88)

The different terms in Eq. (3.88) are defined as follows: • the phase factor7) Mn (E) Mn (E) =

+∞

−∞

···

+∞

−∞

δ

pb

b=1n

n n 2 2 δ E− p + mb · d3 pb (3.89) b=1

b=1

where pb is the fragment b momentum. • the spin factor Sn : Sn =

n

(2sb + 1),

b=1

which gives the number of states of the fragment b with different spin orientations. 7) In the nonrelativistic case the integration of Eq. (3.89) can be evaluated analytically [425].

(3.90)

177

178


• the permutation factor Gn : Gn =

k 1 , nj !

(3.91)

j=1

which takes into account the identity of the fragments. The value nj is the number of components of the j-type particle and k is defined by n = kj=1 nj . By substituting in Eq. (3.89) the corresponding functions for the density, the phase space factor, the fragment momentum, and the spin orientation and solving the integrals of Eq. (3.89), the probability P(E, n) for a nucleus with energy E disassembling into n fragments with masses mb , where b = 1, 2, 3, n is expressed as [422] P(E, n) = Sn · Gn · ·

V

n−1 · n

1 1

b=1

3/2

b=1 mb

3(n−1)/2

(2π) · E (3n/2)−(5/2) , (3(n − 1)/2)

(3.92)

where is the gamma function. It is important to note that the Fermi breakup model has only one free parameter, the volume V. The volume V is the volume of the decaying system and can be calculated as follows: V = 4π · R3 /3 = 4π · r03 · A/3

(3.93)

where r0 = 1.4 fm may be used. The application of the Fermi breakup model is necessary to accurately describe the de-excitation process of light excited nuclei, e.g., by using the EVAP model. Results show that the GEM evaporation model has a comparable prediction power as a combination of the EVAP model and FERMI breakup model (see [426] and Figure 3.12).

3.9 Photon Evaporation and Gamma Ray Production

There are two main mechanisms for photon or gamma ray production during or after the cascade and the evaporation process in spallation reactions in the GeV energy range. One source of photons is the produced neutral pions, π 0 , which decay immediately after their production in high-energy photons (cf. Section 1.2 on page 6 and Table 1.1). The other source of photons is generated from the decay of residual nuclei left after the cascade and the evaporation. Here the usual assumption is made that all particle decay modes have been exhausted, thus photon or gamma emission does not compete with particle emission.

Cross section [mb]

3.9 Photon Evaporation and Gamma Ray Production

102

102

100

100

14

10−2 101

N(p,X)11C

102

16

10−2 103

101

O(p,X)13N 102

103

Incident proton energy [MeV]

Fig. 3.12 Excitation functions of carbon 11 C and nitrogen 13 N production. (a) The figure shows the reaction of incident protons on a nitrogen target, and (b) shows the reaction of protons on an oxygen target (solid lines are the simulation using an INCE-BERTINI/GEM model, dotted lines are an INCE-BERTINI/Fermi breakup model, and open squares are the experimental data given in Ref. [426]).

There are several models for gamma emission employed together with particle transport codes, e.g., the PHT model in the LAHET code and in the MCNPX system [113, 193, 194, 304, 328], the NDEM model in HERMES [88]. A model with similar physics assumptions is used in GEANT4 [100, 424]. The general assumptions are shortly described. The probability of gamma emission is given by Pγ =

γ γ + i ·i

(3.94)

where γ is the width for γ -emission and the i are the transitions widths for different open channels, e.g., n, p, d, t, 3 He, 4 He, etc., and fission for the emission. Discrete photon evaporation It is assumed that there exist a range of known energy levels given by the Gilbert–Cameron formulas [336]. The discrete photon transitions such as E1 , M1 , and or E2 transitions are used from tabulated isotopic libraries, e.g., the BNL data file [317] or the Lund/LBNL data file [427]. These libraries contain a large number of isotopes with experimentally measured excited level energies, spins, parities, and relative transition probabilities. The most probable transition is the E1 or electric dipole transition which is the main source of photons from highly excited nuclei [428, 429]. For an E1 transition to occur between two states, the change in the angular momentum should be 1, corresponding to the emission of one photon. If the change in angular momentum between the states is very large, then the lower order or most probable transitions such as electric dipole E1 or electric quadrupole E2 or magnetic dipole M1 cannot occur. In this case the difference in angular momentum is so large that the state is very long lived. Although an M2 decay is less probable, it does occur finally. Table 3.6 summarizes the multipolarity types of photon transmissions. The multipole character of photon or gamma radiation is discussed in more detail in standard textbooks, e.g., [430].

179

180

3 Evaporation and High-Energy Fission Tab. 3.6

Types of gamma or photon transitions.

Multipolarity

Electric dipole Magnetic dipole Electric or Magnetic quadrupole Electric or Magnetic octopole Electric or Magnetic 24 -pole

Transition

Change in angular momentum

Change in parity

E1 M1 E2, M2

1 1 2

Yes No Yes, no

E3, M3

3

Yes, no

E4, M4

4

Yes, no

If the excitation energy is above the maximum known level, the level is assumed to be in the continuum. Total probability for photon evaporation The probability of either evaporating or alternatively of emitting photons in the energy interval (Eγ , Eγ + dEγ ) is given by

Pγ (Eγ ) =

1 π 2 (c)3

· σγ (Eγ ) ·

ρ(E ∗ − Eγ ) 2 · Eγ , ρ(E ∗ )

(3.95)

where σγ (Eγ ) is the inverse absorption cross section of the photon, and ρ is a nucleus level density defined as ρ(E) = C · exp 2 a(E ∗ − δ ,

(3.96)

where C is a constant and δ is the pairing energy correction of the daughter nucleus evaluated by Gilbert and Cameron [336]. The probability of a transition between levels is assumed to be proportional to the Weisskopf single-particle estimates (cf. [430] page 414), unless specified by the isotope libraries. The calculation by Weisskopf [431] and Blatt and Weisskopf [432] is based on a nuclear model which assumes such a weak coupling between the constituent nucleons that in a photon or gamma transition only a single nucleon experiences a change in its quantum state. Internal conversion electron emission The GEANT4 [100, 424] particle transport system uses in addition to photon emission another important competitive channel namely the internal conversion electron emission8) . This is done including into 8) Internal conversion is a radioactive decay process where an excited nucleus interacts directly with one or more orbital electrons within the nuclear region. The nucleus imparts its excess energy to the conversion

electron in a one-step process and causes it to be injected from the atom with a comparatively high energy, where Eγ = Ee + EB .

3.10 Vaporization and Multifragmentation

the photon evaporation database internal conversion coefficients such as those evaluated by Rösel [433], and Hager and Seltzer [434]. A review of the reliability these data evaluations is given by Ryˇsavý and Dragoun [435]. As conversion and photon decay compete, the branching ratio provides a measure of the relative probability in a given transition. Therefore, a conversion coefficient CC can be defined as CC =

λe conversion probability N ≡ = e γ -emission probability λγ Nγ

(3.97)

in terms of the decay constants λ, where Ne and Nγ are the number of conversion electrons and photons, respectively. The applicability of spallation reactions of photon evaporation or induced gamma ray spectra (cf. page 179) to analyze planetary surface compositions is shown in Part 3 on page 653.


Depending on the thermal excitation energy E ∗ dissipated into the nucleus, there exists beyond evaporation and fission also more ‘‘exotic’’ decay modes such as, e.g., vaporization and multifragmentation. Although they set in only at higher temperatures, and therefore compete with evaporation or fission only marginally in the E ∗ regime under consideration here, we devote for the matter of completeness this section to these specific decay modes. Definition For the so-called thermal multifragmentation (MF) the formation of intermediate mass fragments(IMFs) at higher temperatures is a consequence of increasing thermal motion linked with increasing mean distance of the nucleons. An excited remnant achieves the thermal equilibrium state and then expands, eventually reaching the freeze-out volume. At this point it fragments into neutrons, light charged particles, and IMFs. Due to the short ranging nuclear forces of nucleons the mean field collapses and IMFs are formed by condensation. This unique phenomenon can be observed only for nuclei. Dynamical fragmentation in contrast is caused by high angular momenta, compression of the nuclear density, large momentum transfer and the formation of noncompact deformed nuclei. In this case the whole system and its parts (projectile and target remnants) never pass through states of thermal equilibrium. As schematically illustrated in Figure 3.13, the multifragmentation (MF) is observed in the E ∗ domain between spallation-evaporation and the ‘‘explosive’’ fragmentation regime. As mentioned in the previous section the global properties of nuclear matter are well described by the LDM for E ∗ /A ≤ 1 MeV/nucleon. In this domain the basic decay mechanism is the successive emission of particles via evaporation from the surface of the compound nucleus (left picture in Figure 3.13) and fission. In the case of evaporation a heavy residual remains which is comparable in size to the original target nucleus.

181

182


Multifragmentation

Excitation energy E*

Fig. 3.13 Schematic sketch of the three decay modes such as spallation-evaporation, multifragmentation, and vaporization. The circle drawn indicates the original size of the target nucleus.

Many models [232, 422, 436–438] predict for nuclei A > 150 the multifragmentation within a range of the excitation energy from 3 to 5 MeV/nucleon. It is open so far whether one may regard the emission of IMFs by the conventional description of the evaporation from an equilibrated source, or whether other simultaneous decay characteristics need to be taken into account [437]. In this intermediate energy region the system fragments into many spallation products, having a size no longer comparable with the target mass (Figure 3.13, center). There is no accurate definition of the IMFs, however for heavy targets around A ≥ 150 fragments of mass 5 ≤ A ≤ 40 or atomic number 3 ≤ Z ≤ 20 are generally considered as IMFs. For E ∗ in the range of the entire binding energy of the target nucleus (7.5 MeV/nucleon) vaporization (right representation in Figure 3.13) begins to set in. Vaporization is a specific case of MF and is defined as a decay in which all reaction products have atomic numbers A ≤ 4. In particular for MF, the time scale of fragment emission in general and the question of sequential or simultaneous decay of highly excited nuclei is the focus of theoretical [439, 440] and experimental [441–443] studies. Models have been developed based on the chemical equilibrium [229, 444], phase transitions with simultaneous evaporation during the dynamical expansion [445] or a dynamic of statistical decay [446]. Alternatively Moretto [442, 447], Tso [448], and Botvina [437] propose as a signature of statistical nature of MF the linear dependence between the natural logarithm of the decay ratios of an n-body decay as a function of (E ∗ )−1/2 ∝ 1/T (Arrhenius plots). However, so far no clear-cut answer could be given to the question whether the MF is subject to a prompt or sequential decay mechanism [379]. Also different model approaches try to find an answer whether the multifragmentation mechanism is dominated by dynamical or statistical decay. Some of them describe the multifragmentation by instabilities in the gas–liquid phase of the nuclear matter [436, 445]. Analog to a Van der Waals liquid, nuclear matter having a density of ∼ 0.17 nucleons/fm3 in the ground state (T = p = 0) can be described by phase diagrams (Figure 3.14).


1.5

1.0 Pressure [MeV fm−3]

T = 20

0.5 Tc = 17.9

16 12

0

0.05

Nuclei

0.10

0.25

Density [fm−3] 8

−0.5

4

Spinodal Fig. 3.14 Solid lines (isotherm) indicate the pressure of the nuclear matter as a function of nuclear density for different fixed temperatures (in MeV). The unstable states of the homogeneous infinitely expanded (and thus idealized) system is represented by the broken area. A nucleus in the ground

state resides in the condition p = T = 0. The dashed curve shows the expansion of a hot, expanding nuclear system (see the text). The line at Tc = 17.9 MeV represents pressure and density for which liquid and gas phases co-exist (figure adapted from [436]).

In the ground state the nucleus resides at p = T = 0 and in the liquid phase. After an initial compression induced, e.g., by a heavy ion reaction the hot nucleus expands. The hot nuclear matter follows the dashed line (isotherm) of Figure 3.14 and ends finally as a diluted system in the spinodal region. In this region the compression coefficient κ is given by ρ κ = (∂p/∂V)T = − V

∂p ∂ρ

,

(3.98)

T

where κ exhibits positive values. Under these conditions (ρ = 0.2–0.5 ρ0 ) density fluctuations may easily lead to fragmentation. The phase diagram, as represented in Figure 3.14, applies only for infinitely expanded homogeneous matter, in which no Coulomb interactions are considered. Therefore, the diagram is at most valid for hot neutron stars, but certainly not for real nuclei. This model is not quantitatively being established in detail and would reveal MF as a sudden phenomenon. The formation of cluster in the region of low nuclear density was studied theoretically also by Chomaz et al. [449]. They showed that – due to the finite range of strong nuclear forces – fluctuations of the system do not have short wavelength components. Consequently the production of fragments of almost

183

184


same intermediate size is more probable than the formation of small fragments. Theoretically the process of MF can also be described by the percolation theory. Percolation models treat the nucleus as a lattice with nucleons located at nodes of the lattice. It has been found that results of perculation calculations depend significantly upon the details of the lattice structure. For reasons of computational convenience, the simple cubic lattice has most frequently been used in MF simulations [450], but several studies [451, 452] have found that the face-centered-cubic lattice more accurately reproduces the experimental distributions of fragment masses and their energy spectra. A detailed analysis on fragmentation of excited remnants using the face-centered-cubic lattice model of nuclear structure is given in Ref. [453]. Finally also statistical models are used for the description of the fragment emission. They are based either on the sequential binary statistical decay [229] or the simultaneous splitting of the system. The breakup of the system strongly depends on the so-called freeze-out volume [439, 444, 445] and shows a rising probability with increasing excitation energy. Decay models describing the simultaneous breakup take as a basis the accessible phase space, which determines the probabilities of different decay channels. These ‘‘phase-space models’’ describe both for low excitation energies the evaporation and for higher energies (based on the assumption of an increasing phase space) the breakup into 3, 4, 5 and more fragments of intermediate mass.

185

4 The Particle Transport in Matter 4.1 Introduction

Particle transport in matter became an increasing importance in many fields of basic science, technology, and applications in the recent years. The investigation of hadronic and electromagnetic cascades is of considerable interest for all aspects of radiation physics around, e.g., • particle sources, e.g., high-intensity neutron spallation sources and neutrino complexes; • particle accelerators; • biological shielding against high-intensity particle radiation for radiation protection for personnel and for components; • study of radiation damage of materials in a high-intensity radiation field; • radiation therapy in cancer treatment with hadron, light ion, electron, gamma, and neutron radiation; • astrophysics, cosmic ray reactions, and space applications; • accelerator transmutation of radioactive waste (ATW); • sustainable energy production by accelerator-driven systems (ADS). As discussed earlier, the simulation of particle transport in matter and their interaction with matter involves a huge variety of particles like protons, neutrons, light and heavy ions, electrons and photons, muons, and neutrinos, etc., with energies starting from meV – the energy of ultracold neutrons – up to the energies of high-energy accelerator collider systems or cosmic rays at about several tens of TeV or even higher.

4.2 Hadronic and Electromagnetic Showers

In spallation and particle physics, a shower is a cascade of secondary particles produced as a result of a high-energy particle interacting with dense matter, e.g., with a heavy metal target as used by spallation neutron sources or by acceleratordriven systems, with the shielding material of iron or concrete to shield particle Handbook of Spallation Research: Theory, Experiments and Applications. Detlef Filges and Frank Goldenbaum Copyright  2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40714-9

186

4 The Particle Transport in Matter

accelerators or with high-energy physics calorimeters. The incoming particle interacts with the target matter and in producing multiple new particles with lesser energy. The process ends if the secondary particles are absorbed and stopped in the matter or escape from the target system. There are two basic types of showers. Hadronic showers are produced by hadrons (i.e., nucleons and other particles made of quarks), and proceed mostly via the strong nuclear force and obey the quantum chromo dynamic (QCD theory). Electromagnetic showers in contrast are produced by a particle that interacts primarily or exclusively via the electromagnetic force, usually a photon or electron, and are well described by quantum electrodynamics (QED theory) over a wide energy range. (1) A hadronic shower is produced by a high-energy hadron such as a nucleon, pion, or atomic nucleus. Some such particles have electric charge, and so produce showers that are partially electromagnetic, but they also interact with nuclei via the strong force. Although the details are more complex for this force, such an interaction involves one hadron interacting with a nucleus and producing several lower energy hadrons. This continues, as with the electromagnetic shower, until all particles are stopped or absorbed in the material. (2) An electromagnetic shower begins when a high-energy electron or photon enters a material. At high energy, photons interact with matter primarily via pair production, that is, they convert into an electron–positron pair, interacting with an atomic nucleus or electron in order to conserve momentum. High-energy electrons and positrons primarily emit photons, a process called bremsstrahlung. These two processes continue in turn, until the remaining particles have lower energy. Electrons and photons then lose energy via scattering until they are absorbed by atoms. The development of electromagnetic showers with their principal production processes – bremsstrahlung for electrons and positrons, pair production for photons, becoming energy dependent above 1 GeV. In order to visualize typical hadronic and electromagnetic shower propagation in nuclear matter, Figures 4.1 and 4.2 exhibit in different colors the tracks of particles generated by 1 and 10 GeV incident energy of protons, photons, and µ− in a lead target (Figure 4.1) and a target of plastic scintillator material (Figure 4.2), respectively. The Monte Carlo simulation has been performed by GEANT4 [100], hadronic interaction, multiple scattering, bremsstrahlung, ionization, pair production, photo-, Compton-effect, electroweak decays, and annihilation have been taken into account. The particle beam is coming from left in both the Figures 4.1 and 4.2. For the purpose of comparability the tracks of 20 primary incident particles in the bulk of the Pb material and of plastic material of target geometry of 100 × 80 × 80 cm3 are represented, respectively. The target materials have been chosen to compare a high density material (Pb with ρPb = 11.35 g/cm3 , with a radiation length of rlPb = 5.612 mm), and a hydrogen containing low density material, (plastic scintillator material1) with ρscin = 1.032 g/cm3 , with a radiation length of rlscin = 42.549 cm). 1) Assumed is a molecule H10 C9 .


(a) Shower events of protons with energy of 1 GeV on a Pb target

(b) Shower events of protons with energy of 10 GeV on a Pb target

(c) Shower events of photons with energy of 1 GeV on a Pb target

(d) Shower events of photons with energy of 10 GeV on a Pb target

(e) Shower events of muons (m−) of energy of 1 GeV on a Pb target

(f) Shower events of muons (m−) with energy of 10 GeV on a Pb target

Fig. 4.1 Particle tracks following a GEANT4 [100] simulation for 20 events of 1 and 10 GeV energy of protons, photons, and µ− -induced spallation reactions on a Pb target with the dimensions 100 × 80 × 80 cm3 . The color coding is as follows: protons are blue, neutrons are yellow, photons are gray, µ− are red, and e− are magenta.

187

188


(a) Shower events of protons with energy of 1 GeV on a target of scintillator material

(b) Shower events of protons with energy of 10 GeV on a target of scintillator material

(c) Shower events of photons with energy of 1 GeV on a target of scintillator material

(d) Shower events of photons with energy of 10 GeV on a target of scintillator material

(e) Shower events of muons (m−) with energy of 1 GeV on a target of scintillator material

(f) Shower events of muons (m−) with energy of 10 GeV on a target of scintillator material

Fig. 4.2 Particle tracks following a GEANT4 [100] simulation for 20 events of 1 and 10 GeV energy of protons, photons, and µ− -induced spallation reactions on a target of scintillator material with the dimensions 100 × 80 × 80 cm3 . The color coding is as follows: protons are blue, neutrons are yellow, photons are gray, µ− are red, and e− are magenta.


As a general tendency the showers penetrate deeper into the material, the higher the incident energy or the smaller the density of the material is. For 1 GeV incident energy charged particles as representatively shown in Figure 4.1 for protons and µ− are stopped in Pb after few cm, but may penetrate the low density plastic scintillator material as it is shown in Figure 4.2. For 10 GeV incident energy protons and µ− are neither stopped in Pb nor in scintillator material via electromagnetic processes. Due to their hadronic interactions protons – but also neutrons, not shown here – produce along their trajectory a huge number of secondary particles mainly neutrons (see Figures 4.1(a) and (b)) including photons, electrons, pions, kaons, and a long list of other mesons and baryons. For identical incident energy the hadronic shower produced by neutrons penetrates slightly deeper into the material, because neutron do not suffer any energy loss as the primary protons do. The electromagnetic shower of photons shown in Figures 4.1 and 4.2 gets quickly extinct even for incident energies as high as 10 GeV in particular in Pb material known as highly efficient photon or γ absorber due to its high charge number Z. At deeper penetrations the photons are inducing an electromagnetic shower starting even only after a few cm, which is observed in the plastic scintillator material (cf. Figures 4.2(c) and (d)). Since as mentioned above at high energy, photons interact with matter primarily via pair production and high-energy electrons and positrons likewise emit photons via bremsstrahlung when passing through matter, the electromagnetic showers caused by electrons and photons resemble. The lepton µ− does not undergo hadronic interactions and as a 200 times heavier relative of the electron much less photons are created via bremsstrahlung processes. As mentioned in the case of extended targets, the reaction scenario includes secondary and higher order reactions induced by the reaction products themselves and, therefore, the calculations must include a three-dimensional simulation of internuclear cascades. Such a three-dimensional description of the propagation of the internuclear cascade and the transport of particles in thick targets is a rather complex problem that involves various boundary conditions. The propagation of protons and neutrons is considered separately in longitudinal and radial directions in the following. The energy losses of high-energy particles (≥ 1 GeV) traveling through matter are mainly determined by the production of secondary particles and not due to electronic stopping which is dominating at lower bombarding energies. Thus, the main feature of the cascade is an initial increase of the particle intensity with depth and time. If the energy of these produced secondary particles is high enough, they in turn knock out additional particles. There exists however a physical limit for the development of further cascades, because the initial energy of the primary particle is distributed among the produced particles. Therefore, the multiplicities tend to decrease again during the cascade process and fade away because the average energy of the cascade particles decreases and a greater fraction of the individual particle energy is now dissipated by ionization losses. At the end of the internuclear cascade process, subsequent emission of many low-energy particles, mainly neutrons, takes place, known as evaporation process [320].

189

4 The Particle Transport in Matter Neutron flux 0.07 p−1 cm−2

Proton flux

6

Ep = 2.5 GeV

4

0.002 p−1 cm−2

2 Target radius R [cm]

190

0 6

0.07 p−1 cm−2

4

Ep = 1.2 GeV 0.002 p−1 cm−2

2 0 Ep = 0.4 GeV

6 Protons

4 0.07 p−1 cm−2

2 0

0 (a)

10

0.002 p−1 cm−2

20 30 0 10 20 Target depth L [cm] (b)

30

Fig. 4.3 Neutron (a) and proton (b) flux (per square centimeter and source proton, cm−2 p−1 ) of a hadronic shower in a cylindrical target of 35 cm × 15 cm lead as a function of incident energy of the protons (2.5, 1.2, and 0.4 GeV, top to bottom). Sequent lines are separated by factors of 1.5. Calculations have been performed using the HERMES simulation package (see the text).

As an example, the simulated propagation of three-dimensional hadronic showers following the bombardment of cylindrical lead targets of length (l = 35 cm) and diameter (d = 15 cm) by 0.4, 1.2, and 2.5 GeV protons is illustrated in the contour plots of Figure 4.3. The simulations have been done with the HERMES particle transport system taking into account the development of all shower processes. (cf. Chapter 5). The complexity and entanglement of all intra- and internuclear cascade and electromagnetic processes finally also causing the production of spallation neutrons requires a complex record keeping of all particles actually participating in terms of energy, direction, and location. In the example of Figure 4.3, the HERMES Monte Carlo particle shower simulations produced the data for Figure 4.3 in a cylindrical target divided into cylindrical zones of 0.5 cm spatial resolution in radial direction and of 1 cm spatial resolution in longitudinal direction. The particle flux for neutrons and protons is estimated by the track-length estimator2) . The symmetry axis of the cylinder is oriented in the z direction and pointing downstream of the proton beam. The 2) The track-length flux estimator is defined in Eq. (4.18) on page 197.

4.3 The General Transport Equation

track-length flux of neutrons (left) and protons (right) reflects the radial and longitudinal propagation of particles involved in the intra- and internuclear cascades inside the target volume. The track-length flux comprises both, cascade and evaporation particles. The primary beam protons are not included for the proton track-length flux in Figure 4.3. Multiplying the proton beam current, measured in protons per second, by the track-length flux specified in Figure 4.3, the particle flux, generally used in units of (particles cm−2 s−1 ), is obtained for neutrons and protons. As also shown for the individual tracks in Figures 4.1 and 4.2 as a general tendency one observes in Figure 4.3 that a deeper and deeper penetration into the target the higher the kinetic energy of the incident proton is. The maximum of the evolution in the radial direction is found after the hadronic cascade has already propagated 5 to 10 cm in the longitudinal direction. Neutrons tend to spread out radially much more than protons do, because especially low-energy protons experience high electronic stopping power and consequently short range. That is also why protons develop along their trajectory in a more narrow cone. Note the difference in the absolute track-length flux of more than one order of magnitude between neutrons and protons. For low incident proton energies (0.4 GeV) it is well shown that the cascade rapidly becomes extinct, since the leading particles are stopped before being able to convert their energy effectively into the production of neutrons or protons. Whereas for large kinetic energies (≈GeV) the range due to the stopping power of protons in the lead target is larger than the dimension of the cylinder in z. Although the presentation of hadronic cascades in the r, z-plane is illustrative to explain phenomenologically the interplay of intra- and internuclear cascade descriptions and the well-known consequences of stopping powers applied to charged particles, a more quantitative analysis would include the study of kinetic energy and multiplicity spectra or angular distributions of particles released. The next section is devoted to the theoretical background of the particle transport.


Phenomena in radiation physics and particle transport in matter of leptons, baryons, mesons, and energetic photons are described by the Boltzmann integrodifferential equation (4.2). It was derived by Ludwig Boltzmann in 1872 to study the properties of gases. The important processes involved in the equation will be briefly described and some solutions are presented to introduce the nomenclature of frequently used important parameters, which are describing the particle transport in various spallation environments. More details are given in the literature for reactor physics and neutron transport [63, 150, 454]. The Boltzmann equation (4.2) is a continuity equation in phase space, which is made up of three space coordinates of euclidian geometry, the kinetic energy and the direction of motion, Ω, of the particle. The vector, Ω, is a unit vector in the particle’s direction of motion. Its three components, in Cartesian form, in terms of

191

192

4 The Particle Transport in Matter x Ω Θ

Y

y

z

Fig. 4.4

The coordinate system of the Boltzmann equation (4.2).

the so-called polar angle and the azimuthal angle are given as x = cos y = sin cos z = sin sin .

(4.1)

The coordinate system of the Boltzmann equation is given in Figure 4.4. The Boltzmann’s equation or transport equation (4.2) is expressed in terms of − → the variable i (r , E, , t), called the angular flux. The angular flux is the number of particles of a given type (nuclei, protons, neutron, leptons, mesons, etc.) in the volume element dxdydz about r in the energy element dE with the direction of the motion dΩ about Ω, multiplied by the velocity of these particles. The velocity is in terms of the scalar v = (p · c)/Etotal , where p is the particle momentum (see 1/2 Eq. (1.19)) with p = (E 2 + 2 · E · mi ) , c is the velocity of light, Etotal is the total particle energy E + mi , and mi is the particle mass (cf. Section 1.3.5.4 on page 30. The Boltzmann equation in its general form is given by 1 · vi

∂i ∂t

= − Ω · grad i +

i

· j (r , EB , Ω , t) −

dΩ · dEB · σi,j (r , EB → E, Ω → Ω )

dΩ dEB

· σi,j (r , E → EB , Ω → Ω ) · j (r , E, Ω, t)

− σi (r , E) · i (r , E, Ω, t) + (∂/∂E)(i (r , E, Ω, t) · S(r , E)) 1 − · i + Yi (r , E, Ω, t). (4.2) λi It follows a discussion of the meaning of different terms of the Boltzmann equation (4.2). The density of the particle radiation in a volume of the phase space may change in five ways:


(1) uniform translation, with changing the spatial coordinates of the vector r with the Cartesian coordinates x, y, z, but the energy-angle coordinates remain unchanged; (2) collisions, with changing the energy-angle coordinates, but the spatial coordinates remain unchanged; (3) continuous slowing down of the particle energy, a process where a uniform translation is combined with a continuous energy loss; (4) decay, with a changing of particles through radioactive transmutation into other particles; (5) external source, the direct emission of particles into the volume of phase space of interest, e.g., a spallation neutron source, a fission source, other radioactive sources as electrons and photons, and cosmic rays, etc. The first process (1) results in a particle current reducing the phase-space density 1 ∂i (r , E, Ω, t) = −div[Ω · i (r , E, Ω, t)] = −Ω · gradi (r , E, Ω, t). vi ∂t (4.3) The second process (2) results in a change in energy, angle, and particle type as a result of the collisions between the particle radiation and the nuclei of the matter 1 ∂i (r , E, Ω, t) (2) = · dΩ · dEB · σi,j (r , EB → E, Ω → Ω) vi ∂t j · j (r , EB , Ω , t) − dΩ dEB · σi,j (r , E → EB , Ω → Ω ) · j (r , E, Ω, t) (1)

− σi (r , E) · i (r , E, Ω, t).

(4.4)

In Eq. (4.4), the term σi,j (r , EB → E, Ω → Ω) represents the cross section which is the probability of producing an i-type particle with the phase-space coordinates (r , E, Ω, t) as a result of a collision with a j-type particle with the phase-space coordinates (r , EB , Ω , t). The energy EB is higher than the energy E, and therefore this first term in the square brackets is called the down scattering integral. The cross-section term in Eq. (4.4) σi,j (r , E → EB , Ω → Ω ) represents the probability of the production of a particle with higher energy at the energy EB from a lower energy particle. This so-called up-scattering integral is important for all moderation processes in matter. Up-scattering is especially significant in treating thermal neutrons with energies low enough to gain energy in colliding with the atoms of matter. Both processes – down/up-scattering – play an important role in the moderation of neutrons in reactor physics and technology. The physics and technology of cryogenic moderator systems are especially considered for the neutronic performance of high-intensity spallation neutron sources. The third process (3) is connected with the stopping power S of particles by traversing matter. Particles will loose energy continuously at a rate S(r , E) per unit path length.

193

194


The densities of particles at an energy EB and on slowing down to the energy E are given as – (a) i (r , EB , Ω , t) · S(r , EB ) and – (b) i (r , E, Ω , t) · S(r , E), respectively. (3)

1 ∂i ∂(i S) = vi ∂t ∂E

(4.5)

As seen from Eq. (4.5), the energy loss of a particle in matter is associated with the motion of the particle through the matter. The fourth process (4) is connected with the decay probability λi of a particle. In the decay process is a simple removal process where the particle i disappears. (4)

1 1 ∂i = − · i , vi ∂t λi

(4.6)

where the decay probability λi is given per unit path-length by λ = τi · c

βi , (1 − βi )1/2

(4.7)

with τi as the mean lifetime of the rest frame of the particle, c as the velocity of light, and βi as the particle velocity relative to the velocity of light. The fifth process (5) describes the application of an external source. This means the direct emission of particles into the volume of phase space of interest, e.g., a spallation neutron source, a fission source, other radioactive sources as electrons and photons, and cosmic rays, etc. (5)

1 ∂i = Y(r , E, Ω, t). vi ∂t

(4.8)

The combination of the five processes given by Eqs. (4.3), (4.4), (4.5), (4.6), (4.7), and (4.8) results in the Boltzmann equation in its general form given by Eq. (4.2). The general form of the Boltzmann equation (4.2) is written in the stationary form with ∂i /∂t = 0. The subscripts and the arguments have been dropped in the following description: Ω · grad + σ · + (/λ) − [∂(S)/∂E] = Q + Y,

(4.9)

where Q is the sum over the up- and down-scattering integrals in the square brackets in Eq. (4.4). Another often used abbreviated form in operator notation is given as B · = Q + Y,

(4.10)

where B = Ω · grad + σ + λ−1 − (∂S/∂E). The general form and also the stationary form of the Boltzmann equation are difficult to solve. Therefore, a large number of numerical methods and special techniques were developed to yield for different particle transport cases useful


results. These numerical techniques are the straight-ahead approximation [455, 456], the method spherical harmonics [457, 458], the method of discrete ordinates, often named SN -method [459, 460], and the Monte Carlo method [146, 149, 150], which is capable of treating very complex three-dimensional configurations of multiparticle transport problems. The Monte Carlo method is a stochastic model in which the expected value of a certain random variable is equivalent to the value of the physical quantity to be determined. The expected value is estimated by the average of many independent samples representing the random variable. Particle tracks or histories3) are generated by simulating the real physical situation. There is not even the need to invoke the transport equation for more elementary operations. Only the complete mathematical description of probability relationships is needed that govern the track length of individual particles between interaction points, the choice of interaction type, the new energies and directions and the possible production of secondary particles. Especially for three-dimensional problems, SN methods and Monte Carlo techniques turned out to be most advantageous. Because the transport of particles through matter obeys stochastic processes, the application of Monte Carlo solutions of particle transport problems in matter are the favorable methods nowadays. 4.3.1 The Angular Flux, Fluence, Current, and Energy Spectra

As mentioned earlier, Eq. (4.2) is a system of coupled transport equations, which is, in general difficult to solve. Solving the equation for hadronic cascades is more difficult than, for instance, for neutrons in the core of a nuclear reactor, because of secondary particle production. Thus the solution involves solving the fluxes for many different particle types. The angular flux or flux spectrum, i (r , E, Ω, t), gives the number of particles per cm2 per MeV per steradian per second of a given kind at a given location at a given time. In the following there are some useful relations determined with the angular flux characterizing the radiation field of particles. • The integral quantity (actually used to define the angular flux i (r , E, Ω, t) in units of (cm−2 s−1 sr−1 MeV−1 ) is the fluence i (r ). i (r ) =

dE

E

i (r , E, Ω, t).

dΩ 4π

(4.11)

t

The official definition of fluence by the International Commission on Radiation Units and Measurements (ICRU), 1993 [461] is based on crossing of a surface and defines the fluence as the quotient of dN by dα, where dN is the number of particles incident on a sphere of cross-sectional area dα, i (r ) = dN/dα. This definition is the source of frequent mistakes. It is not to be interpreted as ‘‘flow’’ or ‘‘flux’’ of particles through a surface, but to be understood as a density of particle path lengths in an infinitesimal volume: i (r ) = limV→0 i si /V 3) The experience a particle undergoes from the time it leaves its source until it is absorbed or

until it leaves the system is called the particle’s history.

195

196


(cm× cm−3 = cm−2 ), where i si is the sum of path-length segments. The fluence is therefore a measure of the concentration of the particle path in an infinitesimal volume element around a space point. If the particle’s path length is measured in units of mean free path λ = 1/σ , the expression of fluence is equivalent to the density of collisions σ · i (r ). The most important fluence estimator is the track-length estimator (see Section 4.3.2), which represents the average fluence in a space region when the sum of track lengths is divided by the volume). Frequently the fluence is calculated because it is proportional to the effect of interest, since many effects can be expressed as volume concentrations of some quantity proportional to the ‘number of collisions’. • The fluence rate or flux density is also referred to as scalar flux is expressed in terms of the sum of path segments transversed within a given volume per time unit:

i (r , t) =

dEi (r , E, Ω, t).

dΩ 4π

(4.12)

E

In Monte Carlo calculations with a source given in units of particles per unit time, the scalar flux represents a fluence quality. • The particle current J or the boundary crossing estimator is

J i (r , t) = 2π

π

sin θ dθ

0

dE cos θ i (r , E, Ω, t).

(4.13)

E

To estimate the average fluence on a boundary, the factor 1/ cos θ for each particle has to be added, where θ is the angle between the particle’s direction and the normal to the surface at the crossing point. Therefore, the current is equal to the fluence only if all particles pass perpendicular to the surface. • The flux density of particles or number of particles per volume element dxdydz is n(r , t) =

dEi (r , E, Ω, t)/v.

dΩ 4π

(4.14)

E

• The energy spectrum of particles can be expressed by i (r , E, t) =

dΩi (r , E, Ω, t).

(4.15)

4π

An often used nomenclature for the flux density of particles having a kinetic energy in the range E and E + dE in radiation fields is written as (E)dE, and the parameter (E) as a function of E is named the differential energy spectrum. The flux density, , over all energies is given by = 0

∞

(E)dE,

(4.16)


and the integral flux density, (>E ), is given by (> E ) =

∞

E

(E)dE.

(4.17)

4.3.2 Monte Carlo Estimation of Particle Fluxes and Reaction Rates

As discussed earlier, the particle flux is a function of the spatial position, the energy, the time, the direction, etc. In many practical problems one only needs the particle flux as a function of a spatial position and energy. There are, e.g., in general four methods to calculate particle fluxes of particles transported through matter using Monte Carlo methods. These are referred to as track length estimation, collision density estimation, boundary crossing estimation, and statistical estimation. The formalisms of estimators are usually named ‘detectors’ by analogy with particle detectors to measure particle fluxes by an experiment or named ‘tallies’ to register the particles traversing the matter (more details can be seen in standard textbooks about Monte Carlo [149, 150] or in manuals of modern Monte Carlo particle transport code systems GEANT4 [100, 424], FLUKA [197, 423, 462], and MCNPX [194, 463]). (1) The track length estimator: The energy-dependent flux per unit source particle is calculated in an arbitrary volume in space by N

(r , E) =

li · wti

i=1

V · E · M

,

(4.18)

where • li is the track length of the ith particle in the volume, • wti is the statistical weight of the ith particle, which is one in analog Monte Carlo, • V is the volume traversed by the particle, • M is the total number of source particles, • N is the total number of track-lengths in the volume V by particles of energy E about E. The volume V can be of any size and shape. Large volumes may be needed to improve the statistics. (2) The collision density estimator: The collision density estimator counts the number of particle collisions within a given volume. But it only approximates the total track length. This detector works well if there is a large number of particle collisions in the volume of interest. The collision density estimation of the particle flux is given by (r , E) =

N i=1

total

wti , (E) · V · E · M

(4.19)

197

198


where • wti is the statistical weight of the ith particle producing the collision, which is one in analog Monte Carlo, • (E) is the total cross section in cm−1 of the matter where the collision takes total

place, • V is the volume traversed by the particle, • M is the total number of source particles, • N is the total number of track-lengths in the volume V by particles of energy E about E. (3) The boundary crossing estimator: The boundary crossing estimator calculates the particle flux through a given surface A in a position of interest in a geometry. The energy- and spatial-dependent particle flux per unit source particle is calculated as (r , E) =

N wti · (|n · Ω|−1 )i i=1

A · E · M

,

(4.20)

where • wti is the statistical weight of the ith particle transverses the surface A, which is one in analog Monte Carlo, • A is the surface area, • n · Ω is the cosine of the angle between the direction of the particle and the normal vector n of the surface area A at the point the particle crosses the surface, • N is the total number of particles crossing the surface area with energy E about E, • M is the total number of source particles. Numerical problems may occur in Eq. (4.20) if the product |n · Ω| ≈ 0. So one has to control the magnitude of the cosine of the angle near π/2. (4) The statistical estimator or the flux at a point estimator: This method allows for estimating the flux at a point in space. The detector is extremely powerful for deep penetration problems, e.g., radiation shielding calculations of thick shields, for particle streaming through beam holes, etc. (cf. Chapter 7 on page 233). The statistical estimator is composed of two contributions, the first one is the uncollided response representing the source particles and the produced particles, and the second one is the collided response representing all scattered particles leaving a collision site. The uncollided response is given as (r , E)uncollided =

N exp(− i=1

· |r i − r |) wti , 4π|r i − r |2 M total (E)

(4.21)


where • wti is the weight of the source or the produced particle, which is one in analog Monte Carlo, • r i are the coordinates of the source or the produced particles, • r are the coordinates of the detector, • (E) is the total cross section in cm−1 of the matter of interest, total

• N is the sum over all source and produced particles of energy E about E, • M is the total number of source particles. The collided response is given as (r , E)collided =

N g(E → E, → ) · exp(− |r i − r |2

i=1

total (E)

· |r i − r |) wti , M (4.22)

where • g(E → E, → ) is the probability that a particle of energy E and direction will scatter to energy E and direction , which is toward the detector located at position r , • N is the total number of particle collisions, • wti weight of the particle producing the collisions • M is the total number of source particles. • r i are the coordinates of the source or the produced particles, • r are the coordinates of the detector, • (E) is the total cross section in cm−1 of the matter of interest. total

(5) The nuclear response function: The estimated particle flux spectra and particle currents may be used to calculate or to analyze further important nuclear quantities which are the reaction rates or the nuclear response in matter applying the particle fluxes in phase space together with a macroscopic response function (r , E)j,k , which is a function of the spatial coordinates and the energy E. The index j stands for the particle under consideration and the index k considers the material of interest, respectively. The reaction rate Ri can be given by

Ri =

i+1 (r , E)j,k · (r , E, Ω, t)dE, i

(4.23)

i

where (r , E, Ω, t) is the angular particle flux distribution and (r , E)j,k is the macroscopic cross section of a certain nuclear reaction to estimate, e.g., radioactivity, energy deposition, material damage, dose, and other nuclear-induced properties of materials. Databases of nuclear response functions could be retrieved by using the following databases, e.g., the National Nuclear Data Center (NNDC),

199

200


Brookhaven, USA [464], the Nuclear Energy Agency (NEA) Nuclear Data Bank, Paris, France [465], the International Atomic Energy Agency (IAEA) Nucleus-Data Center, Vienna, Austria, [466], and the Radiation Safety Information Computational Center (RSICC-ORNL), Oak Ridge, USA, [467]. 4.4 Range Straggling

Because of fluctuations in energy loss (cf. Section 1.3.6.1 on page 33) a beam of charged particles will have at the end of range a distribution of ranges in the medium. In some applications in spallation research involving low-energy charged particle sources in which the fraction of the beam particles that slow down and stop from ionization losses rather than undergo nuclear collision is significant, it is necessary to take into account the statistical fluctuations in energy loss at the end of range. This also yields for detectors, e.g., for high-energy physics calorimeters segmented of heavy-metal absorber and scintillator plates. These fluctuations result in a variable range distribution at the end of range called range straggling. Calculations have shown that these range distributions for protons in various metals are nearly Gaussian [468]. The probability of energy loss – the straggled range distribution – is taken to be Gaussian about the mean range with a variance given, e.g., by Ref. [469]

σR2i (E

E

→ E ) = 4π · e · 4

Zi2

·N· E

(1 − 1/2β 2 ) · K(E) (Si (E))−3 dE, (1 − β 2 ) · (1 + (2me /mi ) · γ )

(4.24) where σR2i (E → E ) is the variance of the range distribution for a charged particle of type i slowing down from a kinetic energy E to E , • e is the electron charge, • Zi is the charge of the particle slowing down, • N is the electron density of the stopping medium, – the number of electrons per unit volume in the medium4) , • K(E) is the binding correction factor which takes into account the effects of binding on the atomic electrons at low energies of the incident particle. The expression of K(E), Eq. (4.25), has been derived by Livingston and Bethe [95]. • β 2 is given by [(E + 1)2 − 1]/(E + 1)2 , • E is given by E/mi · c2 , • me is the electron mass, • γ is given by (1 − β 2 )−1/2 , • Si is the stopping power of the medium i (cf. Section 1.3.6.1 on page 33). 4) N = NAA·Z·ρ , with NA the Avogadro number, ρ the density of the medium, and Z, A the atomic and the mass numbers of the medium,

respectively. For example, 26 56 Fe is N ≈ 2, 28 × 1024 electrons cm−3 .

4.5 The Elastic Scattering of Protons and Neutrons

The factor K(E) is given by K(E) =

kn I n Z n

2mv 2 Zeff + · ln , 2 Z mv In n

(4.25)

where Zn is the number of electrons in the nth shell of the atoms of the stopping material, In is the corresponding effective excitation potential for the nth shell, kn is a constant taken as 4/3 for all shells [95], the sum over n extends over all shells for which In < 2mv 2 , and Zeff is the effective number of electrons participate in the stopping process at the incident velocity v considered, e.g., Zeff is the sum of the Zn for the shells for which In < 2mv 2 . Typical values of K(E) are, e.g., for Al at Eproton = 5 MeV K(E) = 1.27, at Eproton = 10 MeV K(E) = 1.19, Eproton = 100 MeV K(E) = 1.04, and for Pb Eproton = 5 MeV K(E) = 1.31, Eproton = 10 MeV K(E) = 1.22, Eproton = 100 MeV K(E) = 1.16, Eproton = 200 MeV K(E) = 1.11. K(E) decreases rapidly toward 1 with increasing kinetic energy of the incident particle. The ranges Ri of a particle i are chosen from a probability distribution P(Ri ) P(Ri )dRi =

σR2i

(R − R )2 1 i 1+1 · exp − dRi , √ 2 · σR2i · 2π

(4.26)

where Ri is the mean range of the particle under consideration.


Nuclear elastic scattering at high energies (≥20 MeV) is a mechanism which is often neglected in high-energy radiation transport calculations of neutrons and protons because the energy losses and angular deflections from such interactions are small. However, there are several situations related to accelerator and spallation neutron source shielding, visible energy in high-energy physics calorimeters, heat deposition in material, etc., where elastic scattering affects in the propagation of cascades in matter. For example, small angular changes can affect shield attenuation for very deep penetrations [380]. An example of the elastic scattering cross sections for protons is already given for the materials C, Al, Fe, and Pb on page 50 in Figure 1.31. The data are taken from the NASA compilation [116] and from an evaluation given by [88]. These cross sections are available only for incident proton energies ≥ 100 MeV and are compiled up to energies E ≤ 22.5 GeV. The lower limit for protons is not an important restriction because for most applications protons in this energy range predominately slow down and stop due to ionization energy losses rather than undergo nuclear collisions. The proton elastic cross sections values are assumed essential the same as for neutrons. Neutron-induced elastic cross sections in the energy range from 15 to 100 MeV may be found in the re-evaluated HILO transport cross-section

201


101

5 Total neutron elastic cross section [b]

202

2 Pb 100

5

Fe

Al 2 C

10−1

101

2

5

5 103 2 102 2 Neutron energy [MeV]

5

104

Fig. 4.5 Energy dependence of the total neutron elastic cross section for C, Al, Fe, and Pb in the energy range from 15.0 MeV to 10.0 GeV as used in the elastic scattering model of the HERMES system [88].

library [470] developed originally at ORNL, and also in the NASA compilation of Wilson et al. [116] for higher energies up to 22.5 GeV. The energy dependence of the neutron elastic cross sections as a function of the target material is illustrated in Figure 4.5. A double logarithmic interpolation of the database in the HERMES system [88] is used for both energy and target atomic mass number interpolation. Figure 4.6 illustrates the target atomic mass number interpolation and extrapolation for several energies. Several approaches may be considered for obtaining the angular distributions for the elastically scattered neutrons and protons either to use an approximate optical model, e.g., Marmier and Sheldon [471] or a semiempirical formula based on measurements at very high energies as used in GEANT4 [100, 424]. A rather simple form for the elastic scattering differential cross section is predicted by dσ/d ≈ R4 /λ2

J · [2 · R · sin (/2)/λ] 2 1 , 2 · R · sin (/2)/λ

where • c.m. is the center-of-mass scattering angle • J1 is the Bessel function of first kind, first order

(4.27)


5

Total neutron elastic cross section [b]

2 100 5 50MeV 2 100MeV

10−1 5

400MeV 2

1100MeV

10−2 100

2

5

2 5 101 102 Atomic mass number

2

Fig. 4.6 Target mass dependence of neutron elastic cross sections for several energies. The points are used in the database of the HERMES system [88] for the elastic scattering model, and the lines show the interpolation and extrapolation used within the model for the target mass range.

• • • •

R = r0 · A1/3 + λ and r0 = 1.4 fm λ = h/p is the de Broglie wavelength h is the Planck constant p = 1/c · [E · (E + 2m)]1/2 .

The approximate optical model angular distribution of Eq. (4.27) may be used for both neutron and proton scattering5) . To determine the neutron or proton scattering angle, a random selection is first made of the variable x from a probability distribution function P(x) ∝ [J1 (x)/x]2 , where a series expansion is used for evaluating the Bessel function J1 J1 (x)/x ∼ 1/2 − x2 /16 + x4 /3.84 × 102 − x6 /1.8432 × 104 ,

(4.28)

where the range of x is from 0 to the first zero crossing of J1 , which is xmax = 0.610 · 2π. 5) Details are given in the HERMES program report [88] on pages 99–116.

203


104

103

102 ds/dΩ [mb/sr]

204

101

100

10−1

10−2 0.0

5.0 10.0 15.0 20.0 Proton scattering angle Θ [degree]

25.0

Fig. 4.7 Proton differential elastic scattering cross section on 12 C at 1 GeV as a function of the scattering angle c.m. in the center of mass system as used in GEANT4 (solid line) [424]. The experimental data are measured at Saclay, France [476].

The random number selections are made from a cumulative histogram of P(x) from x = 0 to xmax . The center-of-mass scattering angle c.m. estimated by using the center-of-mass cosine for a nucleon of energy E scattering from a nucleus of atomic mass A is then 1 . cos c.m. = 1 − x2 · −3 1/3 (2 · [7.093 × 10 · A · E(E + 1.878 × 103 ) + 1]2 ) (4.29) The GEANT4 [424] simulation transport system uses a different approach compared to that given in HERMES [88] to estimate hadron–nucleus elastic scattering cross sections. The Glauber model [472] is used as an alternative method of calculating differential cross sections for elastic and quasielastic hadron–nucleus scattering at high and intermediate energies. Scattering of protons of an energy of 1 GeV on nuclei, e.g., 9 Be, 11 B, 12 C, 16 O, 28 Si, 40 Ca, 58 Ni, 90 Zr, and 208 Pb are reviewed in Ref. [473] and also given for 4 He in Refs. [474, 475]. The examples in Figures 4.7 and 4.8 show a comparison of phenomenological cross sections with experiments for protons at 1 GeV energy elastically scattered on 12 C and 208 Pb. The experimental data were obtained at Gatschina [473], Russia, and at CEA-Saclay [476], France.

4.6 The Treatment of Pion Transport in Matter

106 105

ds/dΩ [mb/sr]

104 103 102 101 100 10−1

0

2

4 6 8 10 12 Proton scattering angle Θ [degree]

14

Fig. 4.8 Proton differential elastic scattering cross section on 208 Pb at 1 GeV as a function of the scattering angle c.m. in the center of mass system as used in GEANT4 (solid line) [424]. The experimental data are measured at Saclay, France [476].

4.6 The Treatment of Pion Transport in Matter

Because of the short mean lifetime of pions, the pion transport in matter and their decay has to be taken into account during their transport through matter. Neutral pion decay Neutral pions π 0 s have a very short half-life or mean lifetime of about 0.87 × 10−16 s and are assumed to decay at their collision site where they are created. Within the intra/internuclear cascade usually the source distribution of the π 0 ’s is provided. Therefore, the photons from the π 0 -decay can be obtained either from electromagnetic shower models or codes, e.g., as EGS [477], or in special auxiliary codes taking into account the γ γ - and the Dalitz-decay channels. Charged pion decay For the charged pions ‘‘in flight’’, e.g., before coming to rest due to ionization loss, both decay and nuclear collisions have to be taken into account. π ± -particles which comes to rest may be assumed to decay or undergo nuclear capture. This may depend on the material or the matter in which their transport is considered. Pion decay in-flight is usually determined by defining a ‘‘macroscopic decay cross section,’’ which is in analog to the macroscopic cross section used for nuclear collisions. The capture or decay products have to be treated or transported using other models especially the production of neutrinos. The properties of pions and their decay products are already described in Eqs. (1.1) on

205

206

4 The Particle Transport in Matter Tab. 4.1

Decay properties of pions and muons.

Particle

Mean lifetime τ0 (s)

Mean path length c · τ0 (m)

π+ π− µ±

2.6 × 10−8 2.6 × 10−8 2.2 × 10−6

7.8 7.8 ∼ 660.0

π0

0.87 × 10−16

∼ 10−8

Decay model

π + → µ+ + νµ π − → µ− + ν µ µ+ → e+ + νe + ν µ µ− → e− + ν e + νµ γγ (98.799 %) γ e+ e− (1.198 %)

page 7 and in Table 1.11 on page 29. Important for the decay in-flight is the mean lifetime or mean decay time in the rest frame and the mean free path for decay which are given in Table 4.1. The macroscopic decay cross section is given as D (E) = 1/λdecay with the decay mean path λdecay = v · τ = v · γ · τ0 , where τ = γ · τ0 , and γ = [1 − (v/c)2 ]−1/2 = [1 − (β)2 ]−1/2 . According to Table 4.1 the mean free path for decay can be written as • for pions: Lπdecay = 7.8[(E/139.6 + 1)2 − 1]1/2 (m) • and for muons: µ Ldecay = 660[(E/105.7 + 1)2 − 1]1/2 (m).

207

5 Particle Transport Simulation Code Systems 5.1 Introduction

This chapter summarizes the currently state-of-the-art particle transport simulation code systems used in spallation research and in particle transport through matter. With the advent of high-intensity spallation neutron sources with a proton beam power in the MW range, the SINQ – Switzerland, [478], the SNS – USA [479], the JPARC – Japan, [145], with research projects as the long pulsed European Spallation Neuron Source (ESS) [480], accelerator-driven systems (ADS) for nuclear waste transmutation and energy production [481–483], with the application of radioactive beams, with the detectors and experiments at medium- and high-energy accelerators, and with cosmic and space applications detailed particle production and transport models realized in computer code systems have to be used to demonstrate feasibility and utilization, to optimize the design configurations, and to support the engineering layout. One main goal of the simulation methods is the determination of the particle fluxes for different applications in a three-dimensional complex multimaterial geometry. Particle fluxes and energy deposition influence the engineering design criteria on cooling of targets, windows, and containments. Radiation damage produced by different mechanisms as displacements per atom (dpas), gas production, and nuclide transmutation worsens the mechanical properties and limits the lifetime of structure materials and components. Activation and radiation lead to global hazards rating the different components of the accelerator and the experiments. Shielding in spallation-induced reactions is different compared to nuclear reactor systems and fusion devices due to the high-energy neutron component which influences the safety protection and the environment. Finally the uncertainties of the complex simulation methods have to be assessed based on validation with experiments. The comparison to experiments serves to get confidence in the complex simulations of the physics processes on a level of a broad range of coincident observables, putting strong constraints to the quality and accuracy of modeling (cf. Part 2 on page 277). In Section 5.2, the most useful and recent particle transport simulation code systems and event generator codes for spallation research and applications are summarized. Handbook of Spallation Research: Theory, Experiments and Applications. Detlef Filges and Frank Goldenbaum Copyright  2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40714-9

208

5 Particle Transport Simulation Code Systems

5.2 Particle Transport Code Systems and Event Generators

Several particle transport code systems with high-energy capabilities were developed in parallel in the late 1960s and early 1970s. All employ three-dimensional Monte Carlo methods. The most general are: • the HETC (High Energy Transport Code) of Armstrong and Chandler [192], which is a descendant of the NMTC code (Nuclear Meson Transport Code) [484], developed in the late 1960s for space radiation transport. • the FLUKA/KASPRO system of Ranft and Routti [485], • the CASIM code of Van Ginneken [486], and • the DUBNA code of Barashenkov et al. [487]. The ideas and models realized in these early particle transport codes were further developed and have been partly a basis of all modern simulation systems for particle transport through matter. Today computer simulation opens up new potentials of describing and studying physical and technical issues. In certain circumstances, computer simulation is the only way to understand the complexity of physical phenomena. The classic categories – theory and experiment – are completed or assisted by a third category, the computer simulation. The method of this third category is a realistic simulation by Monte Carlo and the instrument used is the super computer. In many respects, the computer simulation of particle transport in matter looks like an experiment. Not to misunderstood, computer simulation is not to substitute the experiments, but it extends the field of science and enables experiments in a hypothetical world. Because vector and parallel computer systems have tremendous capabilities today, it is possible to simulate and to study a large number of cases and parameters in a very short time. Many complex systems that could not be verified by classical methods are now accessible and can be studied in great detail. Therefore, utilization of supercomputers in science and technology is an important tool and an advantage in parallel to traditional experimental and theoretical research work. The particular challenge requested to the particle-transport codes in spallation research is due to the description of hadronic and electromagnetic phenomena over ten orders of magnitude ranging from the incident particle energies of some GeV down to the energy of the subthermal neutrons in meV energy range. There is one caveat concerning the neutron reaction mechanisms with matter in the low-energy region. The complex features of neutron cross sections in the low-energy region cannot be calculated from first principles using the properties of the nucleus. Hence data must be determined empirically as a function of energy for each nuclide and for each reaction. In general, these data cannot be interpolated over large energy intervals, because of the irregular resonance structure, although Breit–Wigner resonance parameter or other semiempirical relations often allow a characterization of the cross sections in terms of few empirical parameters per resonance. Cross sections of neutron–nucleus reactions as well as their energy and angular distributions of the resulting secondary particles for hundreds of isotopes


over an energy range from 10−5 eV to 150 MeV and neutron scattering kernels with energies in the meV energy range have been evaluated and have been stored on nuclear data files. These data evaluations are restricted to at best to energies of about up to 150 MeV. Nuclear data for all kinds of applications may be retrieved through the international nuclear data centers links such as ANL, LANL, LBNL, LLNL, ORNL, and TUNL (USA), IAEA (Austria), NEA (France), and JAEA-Tokai-mura (Japan). The main evaluator and distributor is the ‘‘National Nuclear Data Center’’ (NNDC), Brookhaven, USA [464], which is responsible for the evaluation of the ENDF library (Evaluated Nuclear reaction Data File) [488] and for the ENSDF library (Evaluated Nuclear Structure Data File) [317]. A useful index of the available nuclear data libraries which can be retrieved through the nuclear data services of the IAEA is given in Ref. [489]. Where multiple channels on particle production via spallation reactions are opened, e.g., above energies of 150 MeV, differential cross sections at higher energies (cf. Eq. (5.1)) need to be described as a function of the energy and the direction of secondary particles of type j produced when a hadron of type hi and energy Ei interacts with a target nucleus of Atarget . dσ/d Ej , j |hi , Ei , Atarget .

(5.1)

This information is not known accurately from measurements over the wide parameter range of interest for practical spallation-research applications, nor is it likely to be because of the large amount of data that would be required. For example, for a target calculation of a spallation neutron source one would need correlated energy and angular distributions for about five emitted particles types (p, n, π ± , π 0 ) produced by four incident particles (p, n, π ± ), at energy points over a range from about 20 MeV to 3 GeV, for over, e.g., ten target nuclei. Furthermore, to predict the residual nuclei mass and charge distributions for induced radioactivity values, it would require additional cross sections for the multiplicities of heavier particles produced (d, t, α, etc.), correlated with the nucleon and pion production. Thus, the state-of-the-art approach is to use theoretical nuclear model codes for generating the spallation reactions with model validation as possible for specific sets of parameters. Furthermore, with such models incorporated in particle transport codes to treat the subsequent collisions generated by spallation reactions, and using Monte Carlo techniques, explicit handling of the vast amount of data can be avoided in spallation reaction calculations. The available particle transport codes and the event generators and models used for spallation research are summarized in the following section. 5.2.1 Particle Transport Systems and Event Generators

The following tables summarize the most advanced Monte Carlo particle transport code systems for beam material interaction studies. Table 5.1

209

210

5 Particle Transport Simulation Code Systems A general overview of the most advanced Monte Carlo particle transport code systems.

Tab. 5.1

MCNPX

PHITS

FLUKA

GEANT4

MARS

Version Institution

2.6 LANL

2.09 RIST GSI

2006.3 CERN INFN

15 FNAL

Cost Manual pages Language Parallel processing

Free 470 Fortran 90/C Yes

Free 180 Fortran 77 Yes

Free 390 Fortran 77 Yes

4.9.1 CERN INFN KEK/SLAC Free 280 C++ Yes

Free 150 Fortran 95/C Yes

Website or contact MCNPX http://mcnpx.lanl.gov/ PHITS http://rcwww.kek.jp/research/shield/phits.html FLUKA http://fluka.org GEANT4 http://geant4.web.cern.ch/geant4/ MARS http://www-ap.fnal.gov/MARS/

gives an overview of the frequently used systems considered in spallation research – mainly MCNPX and PHITS – and in high-energy physics detector simulation – mainly FLUKA and GEANT4. Other more special purpose code systems are summarized in Table 5.3. In Tables 5.2 and 5.4, the implemented physics models and other processes are given which are already described in detail in Chapters 2, 3, and 4. The different tallies and analysis strategies of the particle histories are described in different manuals of the code systems referred to in Tables 5.1 and 5.3 and references therein. Section 5.2.2 gives a short overview about the three-dimensional geometry systems implemented in Monte Carlo particle transport codes. For more details on geometry applications, the interesting reader is referred to different manuals of the code systems summarized in Table 5.1. 5.2.2 The Three-Dimensional Geometry Systems of Particle Transport Code Systems

All general purpose particle transport codes make use of geometry routines and systems defining complex material geometries easy of access by the user. The user can apply either a standard configured geometry package or self-programmed special geometry. Monte Carlo programs typically divide a problem into a number of elementary regions or cells. Each of such regions has a specified composition of isotopes – representing the material in question – and has surfaces that are determined by first- or second-order algebraic equations in a Cartesian coordinate

5.2 Particle Transport Code Systems and Event Generators Physics models, processes, and nuclear data libraries for spallation reactions implemented in particle transport codes.

Tab. 5.2

MCNPX

PHITS

FLUKA

GEANT4

MARS

Particles Energy loss Scattering Straggling Cherenkov

34 Bethe-Bloch Rossi Vavilov No

38 id. Moliere Vavilov No

68 id. Moliere custom Yes

68 id. Lewis Urban Yes

41 id. Moliere custom No

Low-energy neutrons

Cont. ENDF

Cont. ENDF

72 multigroup

Cont. ENDF

Cont. ENDF

Low-energy protons Used models e.g.,

Cont. ENDF Models

Models models

Models Models

Models Models

Models Models

Bertini ISABEL INCL/CEM LAQGSM FLUKA89

Bertini GEMJAM JAM/JQMD >3GeV

PEANUT DPMJET Glauber neutrinos

Bertini INCL ABLA GEM GHEISHA

CEM LAQGSM DPMJET

n s/γ s

n s

βs / γ s

α s/β s/γ s

γ s

Yes Yes Yes

No No Yes

No No Yes

No No Yes

No No Yes

Other features Delayed decay of Eigenvalue Burn-up Fields E, B

Tab. 5.3

Special purpose transport code systems.

Name

Purpose of the system

Main authors

References

CALOR

Calorimeter design/spallation sources

Gabriel et al.

[195, 490]

EA-MC

ADS and energy amplifying

Kadi et al.

[491–493]

HERMES

Spallation sources/calorimeter design

Cloth et al.

[88]

LCS

Spallation sources/general purpose

Prael et al.

[304]

SHIELD

General purpose/spallation

Sobolevsky et al.

[270]

TIERCE

General purpose/spallation

Bersillon et al.

[196]

system. Third-order surfaces, e.g., toroidal-shaped surfaces are also frequently applied. Examples of three-dimensional multipurpose geometry packages are the BOX-concept of the code KENO [503], the Combinatorial Geometry (CG) concept and its further development used in the codes, e.g., MORSE [504], HERMES [88], EGS4 [477] or FLUKA [505] (Table 5.5). The CG-package describes complex threedimensional configurations by the use of unions, differences, and intersections

211

212

5 Particle Transport Simulation Code Systems Tab. 5.4

A summary of important particle event generators for spallation reactions.

INC (intranuclear-cascade) ≤3 GeV

QMD (quantum-molecular-dynamic)

Bertini CEM INCL ISABEL

JQMDa QMD-SDM QMD, BUU SMM

Refs. [183, 184] Ref. [184] Refs. [189, 495, 496] Ref. [497]

GEM GEMINI JULIAN PACE

Refs. [255, 308] Refs. [229, 256] Refs. [499, 500] Refs. [499, 500]

Refs. [12, 77, 153, 158] Refs. [41, 173–175] Refs. [168–170, 172, 210–212] Refs. [162, 164, 167, 236]

Evaporation and fission ABLA/ABRABLA Refs. [221, 222, 351] ALICE/ASH Ref. [498] EVAP-versions Refs. [88, 299, 301, 304] ORNL-fission Ref. [314] RAL-fission Refs. [311–313] Intranuclear-cascade + evaporation BRIC Refs. [260, 261] DISCA Refs. [501, 502] MICRES Refs. [275–277]

a The JQMD and QMD-SDM are applications for spallation reaction calculations implemented in PHITS [494]. Some remarks about QMD and BUU models are already given in Section 2.7 on page 127.

Tab. 5.5

Examples of three-dimensional geometry packages. MCNPX

PHITS

FLUKA

GEANT4

MARS

System

MCNP-based

MCNP-based MORSE-CG

CG

MCNP-based fixed shapes

Viewer and debugger

Built-in

Built-in

Built-in

Solids CSG, boolean, etc. Built-in

Built-in

(OR and AND boolean algebraic equations) of simple geometric bodies. Further developments of the CG-concept are the MARS-Multiple Array System [506] to describe lattices and the CCG (Complex Combinatorial Geometry) [507] where hierarchical structures – lattice inside a lattice – can be designed. HERMES in addition considers inside the CG-package a three-step search algorithm to search for redundant surfaces to save CPU time, which is important for layer geometries often used for spallation target systems or for calorimeter detectors. The three-dimensional geometry scheme developed for MCNP [508] and MCNPX [194, 463, 509] considers a concept based on a combination of regions built from simple surfaces. The concept also allows operators as unions, differences, and intersections, etc. Surfaces of second- and third-order for toroidal-shaped surfaces in three dimensions are used. For complex detector geometry descriptions for medium- and high-energy physics experiments the GEANT4 [284, 424] system


make use of a geometry package built by single three-dimensional bodies without intersections. The GEANT4 geometry system is very flexible by using an objectorientated approach and allowing an easy interfacing of self-programmed geometry modules. For more specific details about the different geometry packages, the interested reader is referred to the user manuals of the program systems summarized in Table 5.1 and the literature therein.

213

215

6 Materials Damage by High-Energy Neutrons and Charged Particles 6.1 Introduction

Radiation effects induced by neutrons were extensively studied in fission reactors. Their radiation environment differs, however, in several important features from that of accelerators, spallation neutron sources, accelerator-driven systems (ADS) for transmutation and energy production, and galactic and cosmic radiation fields in space 1) , e.g., by • the presence of high-energy (several hundreds of MeV) charged particles from the accelerators, usually protons and light heavy recoil products; • a much harder neutron spectrum induced by spallation reactions than a typical fission spectrum (cf. Figure 6.1); • spallation reactions induced by high-energy particles produce a large number of impurities in materials in the form of spallation, fission, and transmutation products which alter their physical and chemical properties. A detailed review of these radiation effects can be found in Refs. of Ullmaier et al. [511–513] and in a recently published report by Broeders and Konobeyev [514]. The accelerator structures and target systems with their associated components such as containment vessel, windows etc. are subject to radiation damage in variousdegrees. Figure 6.1 clearly shows that a neutron energy spectrum for a 0.8-GeV proton beam on a stopping length solid tungsten target has on average a much larger kinetic energy as compared to a fission neutron spectrum. There are several phenomena for damage production, which may change the materials of interest: • atomic displacements induced by recoiling nuclei leading to microstructural defects; • production of hydrogen and helium isotopes in p, n, and d reactions; • formation of heavier chemical impurities caused by spallation-, fission-, and transmutation products. 1) Galactic and cosmic particles produce in spacecrafts single event effects (SEE), classified in three categories: SEU (Single Event Upset),

SEL (Single Event Latch-up), and SEB (Single Event Burnup), NASA Ref. [510].


6 Materials Damage by High-Energy Neutrons and Charged Particles

Energy [MeV] 4

Nucleon emission probability density

216

100

2

10

10−2

100

10

10−4

Spallation protons 10−1

Fission neutrons

10−2

Spallation neutrons

10−3

10−4 −6

−4

−2

0

2

4

6

8

Lethargy Fig. 6.1 The probability of observing a nucleon with energy E in a W target bombarded by 0.8 GeV protons. The corresponding quantity for a typical LWR is shown for comparison. The curves are normalized to unity (after Ref. [515]).

These processes affect many properties of solids. In a nuclear environment the dimensional and mechanical stability of materials is of utmost importance. The irradiation-induced changes in the mechanical properties depend on materials parameters and irradiation parameters in a complex way. General statements can only be made for the temperature ranges where different effects occur, if the temperatures T are normalized to the melting point Tm of the materials (Figure 6.2). It should be mentioned that the evaluations shown in Figure 6.2 are mainly based on extrapolations from fission and fusion research experience since high-dose experimental data on the behavior of materials under conditions of radiation fields of high-intensity spallation neutron sources are not available.

6.2 Displacement of Lattice Atoms

One important type of radiation damage to materials stems from the displacement of lattice atoms resulting from the collision of the projectile particle upon the target atom or from the recoil energy that the atoms of the materials receive when bombarded by nuclear particles – charged particles and neutrons – produced from spallation reactions. Figure 6.3 illustrates the primary knock-on atoms (PKAs) displacing secondary knock-on atoms. The PKAs slow down over distances much larger than a typical lattice distance and can displace more than one atom. The secondary recoils in turn displace other atoms, producing a displacement


Hardening Irradiation creep (+Growth*) Void swelling Helium embrittl.

Hydr. embrittl.

0

0.4

0.2

0.6

0.7

1

T Tm AL

−200

200

0

400

600°C Steel

0

400

600

1200°C W ≈ Ta**

0

800

* In noncubic metals

1600

2400

3200°C

** In hydride-forming metals

Fig. 6.2 Approximate ranges of normalized temperatures T/Tm where the main radiation damage effects occur. In the lower part of the figure the actual temperature scales for three typical materials are given. The cross-hatched areas indicate the ranges of operating temperatures of these materials (after Ullmaier and Carsughi [512]).

cascade2) [513]. This is a hierarchy of secondary, tertiary, etc. displacements, from which, after some atomic rearrangements, a stable defect pattern evolves. The whole process occurs in the order of picoseconds (ps) and cannot be investigated experimentally with present techniques. However, in the last years computer simulation using molecular dynamics methods have clarified many details of the defect production mechanisms by atomic displacements cascades. The displaced atoms finally end up as interstitials and leave vacancies at their original location. The first stage of the damage starts when a PKA is produced. The whole process is finished when the primary and other knock-on atoms have slowed down to energies below the so-called displacement threshold energy, typically in the range between 20 and 90 eV for most metals (cf. Table 6.1). It should be noted here that Seitz [516] introduced the idea of an ionization limit, below which energy losses by electron excitation may be neglected. Kinchin and Pease [517] involved the ionization limit model for estimates of the defect number produced in metals (cf. Eq. (6.6)). 2) Note: A displacement cascade is an interaction of particles – charged particles or neutrons – with the atoms of a solid, whereas

an intranuclear cascade is an interaction of charged particles or neutrons with the nucleons of a nucleus.

217

218


Incident p / n

Incident p/n

Interstitial atom

Vacancy

Transmuted atom

Helium atom

Fig. 6.3 The interaction of high-energy particles with nuclei of a solid leads to primary and secondary knock-on atoms resulting in interstitials, vacancies, and to transmutation products, e.g., helium. Values for the threshold energies for displacements recommended by the ASTM/E 521-89 [538].

Tab. 6.1

Element

Be, C

Na, Mg Al, Si

K, Ca, Ti, V Cr, Mn, Fe, Co Ni, Zr, Nb

Cu

Mo

Ag

Ta, W

Au

Tthreshold (eV)

31

25

40

30

60

40

90

30

Pb

25

An example of the result of a 10 keV atomic displacement cascade simulated by molecular dynamics simulation (MDS) is illustrated in Figure 6.4. One sees in the center of Figure 6.4, a highly disordered liquid-like region surrounded by the well-preserved lattice structure. The transfer of the recoil energy to the lattice atoms leads to the production of point defects, either as single Frenkel pairs or defect clusters. A measure of the irradiation load to a material is the total dpa number (displacements per atom). The displacement rate is given by

5 nm

y


x 5 nm Fig. 6.4 Example of an atomic configuration at 1 ps after a 10-keV PKA event in Au. The atom positions within a cross-sectional slab of three atomic planes near the center of a 10-keV atomic cascade are projected

dpa =

Emax

σdisp (E) · (E) · dE,

on a {100} plane (MDS) by Gahly and Averback. Note that the computer printout of the picture has slightly different scales in x and y directions of the atomic positions (figure adapted from Ref. [513]).

(6.1)

0

where σdisp (E) is the energy-dependent damage cross section in b, often referred to displacement cross section, (E) is the irradiation particle flux in particles · cm−2 s−1 . The details are given by Eqs. (6.3), (6.7), and (6.9), respectively. The two basic measures of material damage produced by charged particles and neutrons are damage energy and atomic displacements. To estimate damage energy the main formalisms needed are those that describe the energy loss ‘‘partitioning’’ between electron and atomic collisions , e.g., the energy dissipated by atomic collisions – producing PKAs, and atom displacements – versus that part of the charged particle energy which is deposited by interactions with the atomic electrons, which is ionization and excitation. The latter effects do not produce displacement damage in metals. The other key relation needed is the total number of displacements created by further secondary displacement collisions from the slowing down of the PKA (cf. Eq. (6.1) and Figure 6.3). The systematic aspects of atomic scattering theory to study radiation damage in metals has been extended primarily by Lindhard and coworkers. Therefore to determine damage energy and total displacements the theory of Lindhard et al. [518–520] and the semiempirical formula of Robinson [521–523] may be used. These relations are rather ‘‘standard’’ in that they have been often used in predicting materials damage from fission and fusion neutron spectra (e.g., Refs. [524–526]) also taking into account the damage by fast neutrons. The physics of the radiation-induced degradation properties of spallation appears to be similar to those of reactor or fusion neutron irradiation. The exposure for materials in spallation sources, high-intensity accelerators or ADS, however, are of a severity in

219

220


terms of neutron and proton fluxes and energies that has no parallel in previous experience (cf. Figures 6.1 and 6.13). In the following, the relations of the parameters determining the radiation damage in matter by spallation reactions are described. 6.2.1 Damage Energy and Displacements

To estimate the radiation damage dose rate in terms of dpa/s, the total displacement cross sections given by Eq. (6.2) σdtotal (E) = σ elastic d (E) + σ nonelastic d (E),

(6.2)

has to be known. The term σ elastic d (E) is the elastic damage cross section which may be evaluated at energies above 20 MeV with various forms of optical models (Refs. [527]), whereas σ nonelastic d is the nonelastic damage cross section which may be evaluated at energies above 20 MeV with the event generators summarized in Table 5.4 on page 212. Below an energy of 20 MeV displacement cross sections especially for neutrons could be retrieved from the evaluated nuclear reaction data file (ENDF) [488] library. Above energies of 1 or 20 MeV displacement cross sections, hydrogen, and helium production cross sections for incident protons and neutrons on target materials from aluminum to uranium are evaluated and published by Lu and Wechsler [528] and by Lu et al. [529], by Korovin et al. [530, 531] and by Konobeyev et al. [532, 533]. Recently, Broeders and Konobeyev et al. [501, 527, 534] published improved displacement cross sections and helium production cross sections for the materials iron, copper, tantalum, and tungsten irradiated with intermediate and high-energy protons and neutrons. Furthermore, Konobeyev et al. [527] performed calculations to compare the different event generators and models given in Table 5.4. To estimate the damage cross sections the NRT formalism of Norgett, Robinson, and Torrens [535] and Robinson [523] is employed as a standard to determine that fraction of the energy of the PKA which will produce damage, e.g., further nuclear displacements. The displacement cross sections can be evaluated from the following expression: σdamage (Eparticle ) =

· i

Timax

Tthreshold

dσ (Eparticle , Ztarget , Atarget , Zi , Ai , Ti ) dTi

× ν(Ti , Ztarget , Atarget , Zi , Ai ) · dTi ,

(6.3)

where • σdamage (Eparticle ) is the damage energy cross section as a function of the incident particle energy Eparticle in (b keV), dσ (Eparticle ,Ztarget ,Atarget ,Zi ,Ai ,Ti ) • is the heavy ion primary recoil spectrum for the ith dTi reaction type as a function of the recoil energy Ti and of the incident particle energy Eparticle ,


• Ztarget , Atarget is the atomic number and mass number of the target, and Zi , Ai are the numbers for the recoil atoms, respectively. • ν(Ti , Ztarget , Atarget , Zi , Ai ) is the number of Frenkel pairs produced by the PKA with the kinetic energy Ti , where Timax is the maximum kinetic energy of the PKA produced in ith reactions. The function ν(Ti ) is the Robinson cascade damage function, here known as the NRT standard. It is that fraction of the recoil energy Ti , which will produce further nuclear displacements. • i is summation over all recoil atoms produced in the irradiation, • Tthreshold is the effective threshold displacement energy. The number of Frenkel pairs or the number of defects ν(Ti , Ztarget , Atarget , Zi , Ai ) produced in irradiated material for spallation reactions using the NRT formalism of Refs. [523, 535] is determined in the following way: Let Ti , Ai , and Zi be the recoil energy, atomic mass number, and charge number of the ion i, respectively, created as a secondary charged particle from a spallation collision. The target material is described by its average mass and charge numbers Atarget =

j

nj · Aj /

nj

and Ztarget =

j

j

nj · Z j /

nj ,

(6.4)

j

where nj is the atom density for the jth element. Based on the Kinchin and Pease formula modified by Norgett et al. [521, 523] and using the Lindhard slowing-down theory [518, 519], the number of defects produced in irradiated material is calculated ν(Ti , Ztarget , Atarget , Zi , Ai ) = NNRT ,

(6.5)

where NNRT is the number of defects calculated by NNRT =

0.8 · Tdamage . 2 · Tthreshold

(6.6)

The constant 0.8 in the formula is the displacement efficiency given independent of the PKA energy, the target material, or its temperature. The value is intended to compensate for forward scattering in the displacement cascade of the atoms of the lattice. Tdamage is the ‘‘damage energy’’ transferred to the lattice atoms reduced by the losses for electronic stopping in the atom displacement cascade and is given by Norgett, Robinson, and Torrens [535]. Tdamage =

Ti , 1 + k · g()

(6.7)

221


where the parameter k, g(), , and α are as follows: 2/3

k=

[0.0793 · Zi 2/3

[(Zi

1/2

· Ztarget · (Ai + Atarget )3/2 ] 2/3

3/2

+ Ztarget )3/4 · Ai

1/2

· Atarget ]

,

g() = + 0.40244 · 3/4 + 3.4008 · 1/6 , = α · Ti × 106 , α=

(0.8853 · Atarget ) 2/3

[27.2 · Zi · Ztarget · (Zi

2/3

+ Ztarget )1/2 · (Ai + Atarget )]

.

(6.8)

The factor α has the unit (eV−1 ), and Ti and Tdamage are given in units of (MeV). The threshold energy Tthreshold is the minimum energy above the PKA that must be transferred to a lattice atom to produce a stable defect. This threshold energy can depend on numerous factors, e.g., such as the directional dependence due to the lattice structure of the target material, material impurities, and the temperature of the material. This is discussed in detail by Jung in Ref. [536]. Therefore, Tthreshold usually represents an ‘‘effective’’ threshold. Values for Tthreshold based on the ASTM standard are given for different materials in Table 6.1. Considerable variations of these values for Tthreshold are found in the literature (e.g., [537]). As an example, Figure 6.5 shows the displacement damage efficiency, Tdamage /T, as a function of the primary recoil energy for several elements calculated from Eqs. (6.7) and (6.8). It shows the effect of electronic losses on the energy available for atomic displacements. Numerous relations have been proposed for relating damage energy to the total number of displacements or defects created and estimating damage cross sections for structure materials in reactor, fusion, and spallation neutron source technology.

Damage efficiency –Tdamage / T

222

1.0 0.8

U Al

0.6 0.4

Au Nb

C Be

0.2 0.0 101

Cu

102

103 104 Recoil energy [eV]

105

Fig. 6.5 Displacement damage efficiency Tdamage /T as a function of the transferred recoil energy T (after Robinson [521]).

106


The following relations in Eqs. (6.9) have been recommended by several evaluation groups [501, 523, 525, 527, 531, 535, 539–542] to calculate the number of produced defects: ν(Ti , Ztarget , Atarget , Zi , Ai ) = η · NNRT , with the relations Tdamage < Tthreshold : NNRT = 0 Tthreshold ≤ Tdamage ≤ 2 · Tdamage : NNRT = 1 0.8 · Tdamage 2 · Tthreshold 0.8 Ti =η· · 2 · Tthreshold 1 + k · g()

Tdamage > 2 · Tthreshold : NNRT = η ·

(6.9)

where • NNRT , Tdam , and Ethreshold are as defined before, • η is the defect production deficiency additional introduced by Caturla et al. [541] and studied by Broeders and Konobeyev [542]. The introduction of the defect production deficiency η in Eq. (6.6) is based on recent investigations by Stoller et al. [540] and by Caturla et al. [541] who have shown that the NRT model (cf. Eq. (6.6)) overestimates the number of displacements. This has been concluded from the analysis of the discrepancy between experimental measurements and calculations using MD (molecular dynamic) models. The most probable difference is related to the excitation energy distributions and recoil kinetic energies of charged residuals computed by intranuclear cascade models. The authors therefore introduced in addition a defect production efficiency η (cf. Eq. (6.9)) evaluated by MD simulations. It was observed that for most metals the value of η decreases as the energy increases, and reaches an asymptotic value of η ∼ 0.2 for recoil energies Ti ≥ 25 keV, e.g., for recoil energies relevant for spallation applications (cf. Figure 6.6). Simulations by other authors have shown that the efficiency η converges to a mean value ofapproximately 0.2–0.4. The relation between the displacement cross section and the damage energy cross section is given by Eq. (6.9). σdisplacement = η ·

0.8 · σdamage energy (b), 2 · Tthreshold

(6.10)

where σdamage energy and Tthreshold in units (b keV) and (keV). In summary, it can be concluded that only a part of the total recoil energy T, deposited in the material by an irradiation particle leads to displacement damage effects, because: (1) The fraction of T going into electronic excitation and ionization does cause displacements (cf. displacement efficiency in Figure 6.5),

223


1.0

0.8 Efficiency h

224

0.6 Tungsten Copper 0.4

0.2 0

20

40 60 Energy [keV]

80

100

Fig. 6.6 Atomic displacement cascade efficiency η for copper and tungsten as a function of the recoil energy. Data are adopted from Caturla et al. [541].

(2) and spontaneous recombination processes in displacement cascades lead to a further reduction of the number of stable defects (cf. cascade efficiency η in Figure 6.6). (1) and (2) operate even at low temperatures where the defects are ‘‘frozen in.’’ At elevated temperatures, thermal migration of the defects enables additional recombinations which further reduce the number of defects causing macroscopic radiation damage effects (cf. Figure 6.2).

6.3 Hydrogen and Helium Production

Hydrogen and its isotopes are soluble in many metals. The effect of hydrogen in metals is well documented in literature related to nuclear technology. More important than hydrogen production is the formation of helium. It is known that especially He produced, e.g., by (p,α)- and (n,α)-type reactions can lead to drastic property changes of the materials even at concentrations in the parts-per-million range. Helium atoms are insoluble in most, if not all, metals. The helium atoms tend to migrate and form bubbles that embrittle the irradiated metal. This effect of helium in metals has been extensively studied. A useful database of measured data is given in [543] recently. The various spallation reactions producing H and He gas atoms within the materials alter the material properties. Thus the atomic parts per million (appm) of H and He are measures of radiation damage via transmutation. It is conventional to relate the H and He production and total displacements in dpa in terms of appm/dpa (cf. Table 6.2 on page 225).

6.3 Hydrogen and Helium Production Comparison of numerical values of evaluated displacement, damage energy, and helium production cross sections for aluminum, iron, and tungsten from incident protons with energies of 0.8, 1.0, and 2.0 GeV of various authorsa .

Tab. 6.2

Incident protons Proton energy (GeV)

Displacement cross section σ disp (b)

Damage energy cross section σ dam (b×keV)

Helium production cross section σ He (b)

[529]

[529]

appm/dpab

Aluminum Refs.

[529]

0.8 1.0 2.0

768 764 708

– – –

53 52 49

– – –

0.43 0.44 0.45

– – –

559 575 635

– – –

151 190 359

194 220 339

76 97 181

115 126 –

Iron Refs.

[529]

[527]

[529]

[527]

[529]

[534]

0.8 1.0 2.0

3298 3131 2878

2610 2610 2380

330 313 287

261 261 238

0.469 0.596 1.035

0.507 0.580 0.807

Tungsten Refs.

[529]

[546]

[529]

[546]

[529]

[534]

0.8 1.0 2.0

9277 10544 13582

6960 7730 –

2087 2372 3056

1566 1739 –

0.706 1.025 2.459

0.805 0.978 1.680

a

It is conventional to relate the He production and total displacements in terms of appm/dpa. b The appm/dpa ratio is determined by the formula (σ /σ 6 He disp ) × 10 .

Investigations on helium production in spallation reactions induced by intermediate and high-energy protons and neutrons are important to adjust the property changes of materials designed for high-intensity spallation sources, ADS applications, and other emerging nuclear systems. There is a significant spread of experimental He cross sections and deficiencies in the model calculations (Refs. [501, 514, 528, 534]). Several experiments are undertaken to measure He

225

Helium production cross section [mb]

6 Materials Damage by High-Energy Neutrons and Charged Particles 104

181

Ta(p,x)He

103

102 Enke (99) Hilscher (01) Systematics ALICE /ASH CASCADE / INPE

101

100 101


104

Fig. 6.7 Comparison of measured and evaluated He production cross sections of 181 Ta. (Refs. shown in the figure are: Enke (99) [114], Hilscher (01) [544], Systematics (Ref. [514] on page 105), ALICE/ASH [498], and CASCADE/INPE [78, 340, 532]) (after Broeders and Konobeyev [514]).

production cross sections induced by protons in the spallation energy range between 0.1 GeV and 2.5 GeV in the recent years. These experiments are described in Part 2. Broeders and Konobeyev [514] have recently compiled He production data for incident protons on nat Fe, 197 Au, 181 Ta, and nat W targets. A comparison of measured and model predictions in Figures 6.7 and 6.8 tantalum and tungsten shows a remarkably good agreement in applying the CASCADE/INPE model [514].


226

104

W(p,x)He

103 Green (88) Enke (99) Hilscher (01) Systematics JENDL-HE ALICE /ASH CASCADE / INPE

102

101

100 101


104

Fig. 6.8 Comparison of measured and evaluated He production cross sections of nat W. (Refs. in the figure the same as in Figure 6.7 and Green (88) [537], JENDL-HE [545]) (after Broeders and Konobeyev [514]).

6.4 Cross Section Examples

It should be mentioned that the radiation damage induced by spallation reactions is particularly characterized by a large He/dpa ratio in appm/dpa (cf. last two columns in Table 6.2) compared to about 0.5 appm/dpa observed in fission reactors.


Based on the above-discussed formalisms, proton and neutron-induced displacement cross sections, hydrogen, and helium production cross section were evaluated by several authors [528, 529, 531, 533, 534, 546] applying the INCE-NRT model and the binary collision approximation model with MD methods (the -BCA-MD) model. Although it is essential to evaluate the radiation damage for spallation sources, for ADS projects, and for space applications, the simulation of radiation damage parameters as displacements, gas production, and transmutation species are often inconsistent and differs due to the various models and codes, which are used for their evaluation. As indicated by Konobeyev et al. [527] there are remarkable differences within a factor of 2 between different models – BCA-MD and NRT – in estimating displacement cross sections, whereas the helium production estimation is much close together. Therefore, it is important to compare the calculated parameters with experimental results and reconsider the currently used methods for calculating spallation damage [527, 528]. A database in ENDF format for neutron-induced displacement cross sections, as well as hydrogen and helium production cross sections from aluminum to bismuth from about 10 MeV up to 1 GeV incident neutron energies is published by Korovin et al. [531]. In addition, improved proton-induced displacement cross sections and helium production cross sections using BCA-MD models are published recently by Konobeyev et al. [527] and Broeders et al. [501, 546] considering iron, tantalum and tungsten, respectively. Lu, Wechsler and Dai [528, 529] have evaluated a radiation database – the NCSU radiation damage database – for 23 target elements irradiated by protons and neutrons for an energy range for incident protons from 1 MeV up to 3.2 GeV and an energy range of incident neutrons from 1.0 × 10−10 MeV up to 3.2 GeV. The contents of the NCSU database considers damage energy cross sections, displacement cross sections, hydrogen-, helium production cross sections, and transmutation cross sections. The data are derived employing different intranuclear-cascade-evaporation and cascade-exciton models (cf. Table 5.4), a multistage-preequilibrium (MPM) model [194], and evaluations taken from ENDF libraries [488]. For low-energy incident protons, e.g., ≤100 MeV damage cross sections calculated with the SRIM model [547] of Biersack and Ziegler et al. [548–550] are employed. The SRIM model uses a more realistic approach for the interatomic potential including Coulomb screening [518, 520], quantum mechanical treatment of ion–atom collisions, and an exchange and correlation interactions between overlapping electron shells. But intranuclear cascades and spallation is not taken into account.

227


Figures 6.9–6.12 illustrate neutron- and proton-induced damage energy cross sections in b keV of Al, Fe, Cu, Nb, W, Au, and Pb, whereas Figure 6.13 shows the helium production cross section from incident protons for Al, Fe, Cu, Nb, W, Au, and Pb target materials. 3 ×103 3

10 Damage energy cross section [b keV]

Al Tthr 27 eV Fe Tthr 27 eV

2

10

Cu Tthr 30 eV Nb Tthr 60 eV

1

10

0

10

10−1 −2

10

−3

10

−11

10

−10

10

−9

10

−8

10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

10

1

10

2

10

3

10 5×103

Incident neutron energy [MeV]

Fig. 6.9 Examples of damage energy cross sections in b keV as a function of the incident neutron energy up to 3.2 GeV for thin target materials of Al, Fe, Cu, and Nb with threshold energies Tthr of 27 eV, 27 eV, 30 eV, and 60 eV, respectively. The data are taken from the NCSU radiation database [528, 529].

2 ×104 Damage energy cross section [b keV]

228

Al Tthr 27 eV Fe Tthr 27 eV Cu Tthr 30 eV Nb Tthr 60 eV

104

103

102

101 100


Fig. 6.10 Examples of damage energy cross sections in b keV as a function of the incident proton energy up to 3.2 GeV for thin target materials of Al, Fe, Cu, and Nb with threshold energies Tthr of 27 eV, 27 eV, 30 eV, and 60 eV, respectively. The data are taken from the NCSU radiation database [528, 529].

5 ×103


Damage energy cross section [b keV]

5×104 10

4

10

3

W Tthr 90 eV Au Tthr 30 eV Pb Tthr 25 eV

102 10

1

10

0

10

−1

10−2 10−3 10−4 10−5 10−11 10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 102 103 5×103 Incident neutron energy [MeV]

Fig. 6.11 Examples of damage energy cross sections in b keV as a function of the incident neutron energy up to 3.2 GeV for thin target materials of W, Au, and Pb with threshold energies Tthr of 90, 30, and 25 eV, respectively. The data are taken from the NCSU radiation database [528, 529].

Damage energy cross section [b keV]

5×104 W Tthr 90 eV Au Tthr 30 eV Pb Tthr 25 eV 104

103

100

101

102

103

5 × 103

Incident proton energy [MeV] Fig. 6.12 Examples of damage energy cross sections in b keV as a function of the incident proton energy up to 3.2 GeV for thin target materials of W, Au, and Pb with threshold energies Tthr of 90, 30, and 25 eV, respectively. Data are from the NCSU radiation database [528, 529].

Table 6.2 shows examples of estimated displacement cross sections, hydrogen, and helium production cross sections from different references for some materials in the spallation energy regime for incident protons. The numbers in Table 6.2

229


101 Helium production cross section [b]

230

100 10−1 10−2 10−3 Al Fe Cu Nb W Au Pb

10−4 10−5 10−6 10−7 10−8 10−9 100

101

102

103

5×103

Incident proton energy [MeV] Fig. 6.13 Examples of helium production cross sections in b as a function of the incident proton energy up to 3.2 GeV for thin target materials of Al, Fe, Cu, Nb, W, Au, and Pb. The data are taken from the NCSU radiation database [528, 529].

referring to [529] are calculated with the NRT model whereas the numbers [527, 534, 546] are evaluated using BCA and MD models. There are, at least, several origins of the difference in estimating damage energy, or displacement, or gas production cross section, e.g., hydrogen and helium: • Differences occur by using different models for the defect production. The results obtained using BCA-MD (or MD) are usually different from the displacement cross sections calculated by using INCE codes applying the NRT model. The differences are some times large especially comparing the number of defects. • Displacement cross sections calculated for incident neutrons and protons below the primary energy 100 MeV can be noticeably different (e.g., different INCE models, BCA or MD, etc.). One main reason is the difference in the contribution of the elastic scattering of protons and neutrons in the total displacement cross section. In case of protons there are contributions from the screened Coulomb [518, 520] and nuclear scattering, and their interference. Also above about 100 MeV the INCE, the BCA-MD, and QMD models may use different physical assumptions that yield to a different cross section estimation especially for calculating the number of defects. • The use of a different effective threshold displacement energy (Tthreshold ) in the NRT formula (6.9) produces also different results in the damage cross section calculation. But this could be corrected by a renormalization to a comparable threshold displacement energy using the formulas (6.9) and (6.10).

6.5 Radiation Damage Effects of High-Intensity Proton Beams in the GeV Range

6.5 Radiation Damage Effects of High-Intensity Proton Beams in the GeV Range

The following example shows a dpa calculation for a high-power neutron spallation source of 5 MW beam power, with a proton current of 5 mA, with a proton beam energy of 1.0 GeV for one year of operation: • the integrated proton flux should be about · t = 2.0 × 1022 protons/cm2 , • the displacement cross section for an iron window is assumed to be about σdisplacement = 3000 b at 1.0 GeV (cf. Table 6.2), • this yields using the formula (6.1) a number of displacements per year: dpa = 3000 b × 2.0 × 1022 cm2 = 60 Figure 6.14 shows estimated dpa numbers, hydrogen (cH ) and helium (cHe ) concentrations as a function of the irradiation time t in the center of a stainless steel proton beam entry window of the spallation source ESS [551] with an incident proton beam energy of 1.3 GeV and a beam power of 5 MW compared to a first wall of a fusion reactor (CTR) with 1 MW/m2 . The helium concentration cHe increases with time as shown by the dash-dotted lines. The values for the hydrogen concentration cH are somewhat fictitious since hydrogen is highly mobile and most of the produced amount will diffuse out of the material. Compared to fast fission systems the dpa rates are similar to the fusion Time 1 day

1 week

1 month

1 year

102

101 100

dpa (ESS)

10−1

dpa (CTR)

100

H (ESS)

10−1

10−2 He (ESS)

10−2 10−3 104

10−3

H (CTR) He (CTR)

105

106 Time [s]

107

10−4 108

Fig. 6.14 Comparison of the number of displacements per atom (dpa) hydrogen concentration cH and helium concentration cHe as a function of the irradiation time t in the center of a stainless steel proton beam entry window of the spallation source ESS [551], incident proton beam energy 1.3 GeV and beam power 5 MW, and in the first wall of a fusion reactor (CTR) with 1 MW/m2 .

CHe,CH [atomic fraction]

Displacement [dpa]

101

231

232


case while the concentration of hydrogen and helium are negligibly small. At operating temperatures of target components of high-intensity spallation sources most of the irradiation-induced defects are mobile and can react with each other and/or with defects already present in the unirradiated material [512]. It is important to note that only a small fraction of the initially produced defects considered by the displacement number dpa contributes to the finally observed property changes of the irradiated material, since the pertinent fraction not only depends on the material, temperature and recoil energy, but also on some other properties illustrated in Figure 6.2. This means the overall ‘‘damage’’ cannot be characterized by a single value. Nevertheless, the dpa number is a useful measure in correlating results determined by different particles and fluxes in an irradiation environment. The temperature during irradiation plays an important role for several reasons: (a) the amount of spontaneous recombination, (b) and the mobility of certain species in the lattice, which is strongly temperature dependent. At material temperatures T ≥ 0.55 · Tm , the melting temperature, the He embrittlement often dominates the mechanical behavior of materials as tensile, creep, and fatigue. In general, displacement effects could diminish with increasing temperature of the material, whereas helium effects gain importance at high temperatures. Experimental investigations on helium production in irradiated materials and material issues on spallation neutron sources and ADS are presented in Parts 2 and 3.

233

7 Shielding Issues 7.1 Introduction

The shielding problem of high-current particle accelerators, high-intensity spallation neutron sources, and ADS systems in the spallation energy regime is unique in terms of the general requirements because of a relatively high-energy and intense particle source, large shielding dimensions, and geometric-material complexities. The solution is usually called the deep penetration problem applied to particle radiation transport. Monte Carlo methods cannot generally be used to determine the radiation paths of particles through a geometrically very complicated shield, although the computer power is available nowadays. Some a priori knowledge of the shield design must be available before beginning a detailed calculation. Therefore, different methods are applied which are semiempirical estimation based on the Moyer model, discrete ordinates, Monte Carlo coupling with discrete ordinates, and Monte Carlo methods using variance and appropriate biasing techniques to direct particles along the ‘‘important’’ paths [552–561]. The general shield design concerns include • bulk shielding for the target stations, • effects of beam holes, collimators, shutters, and other openings inside the bulk shield, • neutron beam line shielding and collimators external to the bulk shield, • accelerator shielding, shielding due to beam losses during acceleration as losses at normal operation, accidental losses, and losses under shutdown conditions of the accelerator, beam dump systems, • groundshine and skyshine phenomena, • energy deposition, heating, material damage, and activation. With the advent of high-energy accelerators, e.g., the Cosmotron, a 3-GeV proton synchrotron [562] at BNL, USA, first operated in 1952 and the Bevatron, a 6.3-GeV synchrotron [563], at Lawrence Berkeley Laboratory (LBL), USA, which first achieved full energy at an intensity of about 1010 protons per pulse in 1954, leads to the development of Moyer model [564, 565]. Moyer developed a phenomenological model capable of estimating the additional shielding required as part of the Bevatron improvement program during 1962–1963. At the time Moyer invented Handbook of Spallation Research: Theory, Experiments and Applications. Detlef Filges and Frank Goldenbaum Copyright  2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40714-9

234

7 Shielding Issues

his model ‘‘the design of shielding for high-energy accelerators was thought to be more an art than a science’’ [566]. The interesting reader is referred to the book of Patterson and Thomas [102] on the history of accelerator shielding and their radiation protection and the recently published article of Thomas [567]. In 1973 these authors postulated that it might eventually be possible to simulate the precise details on shielding and radiation levels via computer calculations, but they argued that such calculations, however, are difficult, expensive, and perhaps several years in future ([102] page 381). Various computer programs to simulate the development of high-energy particle transport in matter are now available (cf. Chapter 5). The explosive development of computer capabilities, the operating experience gained with high-energy proton accelerators and with small- and medium-spallation neutron sources lead to a detailed knowledge on the physics and technology of all aspects of shielding needed for accelerators, high intensity neutron spallation sources and irradiation facilities in the last 10–15 years. Details could be find in proceedings of numerous conferences and workshops, e.g., the International Collaboration of Advanced Neutron Sources, ICANS-I-XVIII, the NEA SATIF 1-7 and SARE 1-6 workshops, or the International Conferences on Radiation Shielding, ICRS 1-11. A collection of computer codes and data sets for accelerator shielding calculations and analysis are collected at the ‘‘Radiation Safety Information Computational Center’’ (RSICC) [467] and the ‘‘NEA Data Bank’’ (NEA DB) [465]. A comprehensive report about radiation physics and protection around accelerators was recently published by Cossart [568]. In the following, the basic assumptions and methods, used nowadays, for the shield design of accelerators and spallation sources are summarized.

7.2 The Attenuation Length and the Moyer Model

The Moyer model [564, 565] in its early form provides a formalism to estimate the high-energy neutron fluxes and the associated biological dose rates at the surface of transverse very thick shields of accelerators with the general assumption that at this local position, the internuclear cascade is well developed and in equilibrium. Although high-energy neutrons are not the only particles propagating the hadronic cascade in matter Moyer assumed that it was sufficient to treat all particles in the cascade as ‘‘neutron like.’’ The fact that those neutrons with energies ≤150 MeV contribute much larger to the dose equivalent as those neutrons with energies >150 MeV which propagate the cascade has led to some confusion in understanding the Moyer model and the early literature ([569] on page 14). The main factors governing the required bulk shield thickness are the attenuation of the high-energy particles and the material density. This is because the particle flux or dose attenuation approximately depends on these factors, and only linearly on the source strength and the dose rate criterion (cf. Eq. (7.1)). This means that the particle flux spectrum at large distances – several mean free paths – from the


Target

Particle beam θ

R D

d1 d2

d

r

d3 d4

Shielding material Fig. 7.1

Secondary particle surface leakage

Coordinates P(r,θ)

Shielding geometry model for the generalized Moyer model.

source is approximately represented by (x, E) ∝ S · B(E) ·

exp(−x · ρ/λatt ) , x2

(7.1)

where • (x, E) = particle flux spectrum at position x, S = source term, • B(E) = build-up factor, x = distance to source in cm, ρ = shielding material density in g/cm3 • λatt = attenuation length in g/cm2 or in cm. 7.2.1 Accelerator Shielding and the Generalized Moyer Model

A generalized formulation for the flux or the radiation level [568, 570] outside a shield is given by Eq. (7.2) considering a neutron source produced by protons interacting on a thick target, e.g., a magnet structure in an accelerator tunnel, as illustrated in Figure 7.1. Assuming that neutrons are the only secondary particles produced, then the number of neutrons dNneutrons /dE at the point P(r, θ ) outside a shield of material thickness d, which are emitted into a given element of solid angle d at angle dθ in an energy interval E + dE relative to a beam by a number of Nproton protons is given by −d · csc(θ ) d2 Y dNneutrons = Nproton · B(E) · d · exp , dE dEd λatt (E) where • B(E) is the buildup factor, • λatt (E) is the effective energy and material-dependent attenuation length,

(7.2)

235

7 Shielding Issues

• d2 Y/dEd the yield of neutrons per solid angle per unit energy interval at energy E, • the exponential function accounts for the attenuation of the radiation field by a shielding thickness d, at an angle θ . In Figure 7.1 the distance of the target interaction point to the coordinate P(r, θ ), where the neutron flux emerges the shield surface, is determined by D = (R + d) · 1/ sin(θ ) = (R + d) · csc(θ ) = r · csc(θ ). Including the factor 1/D2 in Eq. (7.2), the total fluence (E) is then given at point P(r, θ ) by (E) =

Nproton · D2

Emax

B(E) ·

Emin

−d · csc(θ ) d2 Y · exp dE. dEd λatt (E)

(7.3)

In the original formulation, Moyer approximated Eq. (7.3) by a single energy group because of the characteristic functional dependence of the neutron attenuation length shown in Figure 7.2. As seen in Figure 7.2, the neutron attenuation lengths above 150 MeV are approximately independent of the energy, but diminish rapidly with energies below 100 MeV. This consideration is not totally right because the lower energy particles do have a small but finite attenuation length (cf. Figure 7.2). However, for thick shields and for deep penetrations of the particle cascade, the total neutron flux density and as a consequence also the dose equivalent or dose rate are proportional to the high-energy hadron flux density. Because the low-energy particle components are produced by the interactions of the high-energy particles inside the shield, their intensity decreases through the shield in an exponential manner. The general conclusion from the Moyer model, therefore, is that the dose equivalent at any Attenuation length λatt [g·cm−2]

236

150 High-energy limit 100

Concrete ρ = 2.4 g·cm−3

500

0

1

10

100

Neutron energy [MeV] Fig. 7.2 The variation of the attenuation length for monoenergetic neutrons in concrete as a function of neutron energy. Full circles are data from Ref. [571] and open circles are data from Ref. [572]. The dashed line shows the high-energy limit with an attenuation length λatt = 120/2.4 cm = 48.75 cm (after Thomas and Stevenson [570]).

1000


point outside an accelerator shield is mainly governed by a simple line-of-sight propagation of the cascade-generating particles at the first interaction with a target and a kind of multiplication factor accounting for the particle build up. The cascadegenerating particles have an attenuation length that is approximately independent of the energy (cf. Figure 7.2). Therefore for practical accelerator shield calculations (7.3) can be simplified after Moyer in the following way: (1) λatt (E) = constant = λ for E ≥150 MeV and λatt (E) = 0 for E 150 MeV) −d · csc(θ ) Emax Nproton d2 Y dE. exp B(E) · = 2 D λatt dEd E150 MeV

(7.4)

(2) The neutrons emitted at angle θ can be written as a simple function f (θ ) multiplied by a multiplicity factor m(Emax ) that depends only on the incident proton energy [570], thus −d · csc(θ ) Nproton · m(Emax ) · f (θ ) exp D2 λatt −d · csc θ Nproton = · g(Emax , θ ), exp (7.5) 2 D λatt

(Eneutron > 150 MeV) =

where g(Emax , θ ) is an angular distribution function that is constant for a given value of Emax and for a particular target material. (3) The dose rate or dose equivalent per fluence, usually written as H, for neutrons is not strongly dependent near the energy E ≈ 150 MeV (cf. Figure 7.3). Therefore, the dose equivalent outside the shield due to neutrons with about 150 MeV can be written as H150 ≈ C150 · (Eneutron > 150) MeV,

(7.6)

where C150 is the value of the fluence-to-dose conversion factor at 150 MeV. The total dose equivalent, Htotal , is then given by Htotal = k · H150 , where k ≥ 1.

(7.7)

The relation of Eq. (7.7) assumes that the low-energy neutrons are in equilibrium with the neutrons having energies E > 150 MeV after several mean paths in the shield. This is valid for a neutron spectrum in a shield more than a few mean free paths thick. Thus, with the above assumption the dose rate or the dose equivalent Htotal is given by Htotal =

−d · csc(θ ) k · C150 · Nproton · g(Emax , θ ) . exp (R + d)2 · csc2 (θ ) λatt

(7.8)

237

7 Shielding Issues

104 Effictive dose conversion factor [pSv cm2]

238

Protons Pions+

3

10

Muons 102

101

100

Photons

Neutrons

10−10 10−8

10−6

10−4

10−2

100

102

104

Particle energy [GeV] Fig. 7.3 Fluence-to-effective dose conversion factors for AP irradiation as a function of the energy of neutrons, protons, photons, and muons [19].

To calculate the dose rate for personal protection purposes the fluence-to-effective dose factors for the various particles must be known. These conversion coefficients are given in Figure 7.3 for anterior–posterior (AP) irradiation1) and various particles [19, 573]. Other evaluations of conversion factors can be found in Refs. [574– 576] and in periodically published reports of the International Commission on Radiological Protection (ICRP). For example, the effective dose rate from an AP irradiation in a field with neutrons of an energy of 1 MeV with a fluence of 1 neutron cm−2 is about 300 pSv. Using a multilayer shield system shown schematically in Figure 7.1, the ratio d/λatt in formula (7.8) can be replaced by η=

n di , λi

(7.9)

i=1

where the sum is over n shielding layers with thicknesses di and attenuation lengths λi . 7.2.2 The Moyer Model Parameters

The Moyer model parameters are usually determined by an analysis of measurements related to practical shielding configurations at existing accelerators. Basically the function f (θ ) (cf. Eq. (7.5)) must be known. Stevenson et al. [569] and Thomas and Thomas [577] determined from experimental data over a wide energy range 1) AP is a standard irradiation geometry of the human body in radiology for calculations of fluence-to-effective dose conversion

coefficients. AP means the direction of the irradiation – front to back – of the human body.

7.2 The Attenuation Length and the Moyer Model Recommended values of H0 (Eproton ) and Eproton is the proton energy in GeV.

Tab. 7.1

H0 (Eproton )(Sv m2 )

Reference

((2.84 ± 0.14) × 10−13 ) · E 0.80±0.10 ((2.00 ± 0.14) × 10−14 ) · E 0.50±0.10 ((6.90 ± 0.10) × 10−15 ) · E 0.80±0.10 ((7.00 ± 0.14) × 10−15 ) · E 0.80±0.10 ((1.30 ± 0.14) × 10−14 ) · E 0.80±0.10 ((1.30 ± 0.14) × 10−14 ) · E 0.80±0.10 ((1.50 ± 0.14) × 10−14 ) · E 0.80±0.10

[568] [578] [578, 579] [579] [579] [579] [579]

that the function f (θ ) is well described by f (θ ) = exp(−β · θ ),

(7.10)

where θ is given in radians, β is in radians−1 , and β −1 ≈ 2.3 rad−1 for (Eneutron > 150 MeV) and for incident proton energies in the GeV range (1 rad ≈ 57.2958%). This leads to a parameterized expression to determine the dose equivalent H at the surface of a thick shield (cf. Figure 7.1) H=

H0 (Eproton ) · exp(−β · θ ) · exp(−η · csc(θ )) , (r · csc(θ ))2

(7.11)

with r = R + ni=1 di . The expression H0 (Eproton ) · exp(−β · θ ) is determined from target yield data and measurements at accelerators. A best fit for the values of H0 (Eproton ) is by the power law: H0 (Eproton ) = k · E n .

(7.12)

Several values of H0 (Eproton ), k, and n of Eq. (7.12) are given in the literature and summarized in Table 7.1. The Moyer model of Eq. (7.11) is applicable for relatively ‘‘thick’’ targets for 0.80±0.10 Sv m2 as given in Table 7.1. It a H0 (Eproton ) = ((2.84 ± 0.14) × 10−13 ) · Eproton should be mentioned that the Moyer model was further developed by several authors to use this model for shielding estimates around accelerators and spallation neutron sources [553, 559, 580] to overcome the restrictions , e.g., in geometry, energy dependence of the attenuation lengths, variation of H0 (Eproton,θ ) with d/λatt , etc. This could be considered to evaluate certain Moyer parameters with the help of transport codes and Monte Carlo methods (cf. Section 7.3 on page 242). Without going into detail, the Moyer model can be extended to beam losses described by a line source [568]. Using a formulation known as the Tesch approximation [581], this leads to the following equation estimating the dose equivalent: H=

H0 (Eproton ) · 0.065 · exp(−1.09 · η), r

(7.13)

239

240

7 Shielding Issues Earth shield 5.0 m 1.0 m

Accelerator structure

Accelerator tunnel

Concrete shield 2.5 m

Fig. 7.4

Shielding example geometry model for the generalized Moyer model.

where the dose equivalent H is measured in Sv per interacting proton per unit length. The other parameters are the same as before. It should be noted here that the Moyer principle was further developed by Armstrong et al. [553] concerning a semiempirical model code for shielding layout and assessment (CASL, Computer Aided Shield Layout) with a rather general source and 3D material and geometry capabilities. While this model uses approximately, semiempirical physics treatment based on internuclear-cascadeevaporation models and parameters depending upon source energy and shielding thickness, the model provides sufficient accuracy for many practical problems and parameter variation for shielding issues. Applications will be discussed in Part 3. In the following example results are given for transverse shielding estimation for a high-energy linear accelerator applying Eqs. (7.11) and (7.13). Accelerator shielding example A typical shielding geometry arrangement for a high-energy linear accelerator is illustrated in Figure 7.4 with a tunnel built of concrete for the accelerator components and surrounded by an earth shield (concrete thickness d1 = 1.0 m, a tunnel radius tr = 2.5 m, and a thickness of the soil of the top shield of d2 = 5.0 m). (1) LINAC parameters: – proton energy Eproton = 1.5 GeV, – average beam power Pbeam = 5 MW and an average proton current Iproton = 3.33 mA, – number2) of protons = 2.16 × 1016 protons/s. (2) Assumed proton beam losses: – point losses about (10−6 ) = 2.16 × 1010 protons/s= 7.78 × 1013 protons/h, – continuous beam loss during operation assumed 1 Wm−1 or 4.3 × 109 protons s−1 m−1 = 1.55 × 1013 protons h−1 m−1 . 2) 1 mA = 6.24 × 1015 protons/s.


(3) Assumed shield material parameters: – concrete density = 2.4 g/cm3 , assumed attenuation length λconcrete = 0.50 m, – soil density = 1.8g/cm3 , assumed attenuation length λsoil = 0.70 m. To calculate the dose equivalent H with Eqs. (7.11) and (7.13) the following parameters are used: H0 (Eproton ) = 2.84 × 10−13 × E 0.80 Sv m2 ; β ≈ 2.3 rad−1 ; θ = 90◦ = π/2 = 1.57 rad; csc(θ ) = 1.12; r = 2.5 + 1.0 + 5.0 m = 8.5 m; η = d1 /λ1 + d2 /λ2 = 1.0/0.50 + 5.0/0.70 = 9.14. The dose equivalent results for point losses Hpoint and for continuous losses Hcontinuous are then given by Hpoint = 4.21 × 10−21 · 7.78 × 1013 = 0.33 µSv/h , Hcontinnuous = 1.41 × 10−19 · 1.55 × 1013 = 2.18 µSv/h.

(7.14)

7.2.3 Attenuation Lengths

Small uncertainties in the attenuation length λatt may lead to unacceptable errors concerning the shield thickness and therefore the radiation protection goals of a facility by using Eq. (7.1). The following example demonstrates this fact: • A high-intensity spallation neutron source may need a total attenuation provided by the bulk shield of about ∼1012 . • Assuming an iron shield of a thickness of 500 cm with an attenuation coefficient λatt = 20 cm = 156 g/cm2 (shielding material iron with a density ρ = 7.8 g/cm2 ), yields a material attenuation factor alone of about e−25 ≈ 1.4 × 10−11 , or 25 mean free paths of shielding material. • A 10% uncertainty in λatt then corresponds to about e−2.5 , or more than one order of magnitude error in the prediction of particle fluxes or dose rates. One might expect that since λatt is so fundamental to shield design for highenergy radiations, accurate values for common shielding materials would be available from experiments and accelerator facility designs. This is not the case in general. For example, previous measurements of attenuation lengths for iron range from about 120 to 180 g/cm2 . There is also a wide range of measured λatt values reported for concrete ≈ 110–172 g/cm2 . A summary of measured values for λatt and descriptions for most of the experiments are given in the book by Patterson and Thomas [102], and more recently published measurements are summarized by Nakamura [582]. In Table 7.2 some estimates of attenuation lengths are given. There are several reasons for these considerable variations of the measured values for λatt . One reason is that simple measurements for the decrease in radiation intensity with shield thickness may involve some unaccounted values for geometric attenuation as well as material attenuation, e.g., an invalid assumption of a point source attenuation, whereas the actual beam distribution incident on the shield may constitute an extended source. This leads, of course, to a misinterpretation of

241

242

7 Shielding Issues Approximate nominal values for the attenuation length λatt for some shielding materials for accelerators and spallation sources sources. Tab. 7.2

Material

Density (g/cm3 )

Al Fe

2.7 7.8

Steel

∼7.8

Pb

11.3

W U Normal concrete

19.3 18.95 2.4

Heavy concrete

4.0–4.3

Soil Air

1.8 1.2 × 10−3

Attenuation length λatt (g/cm2 ) ± 1σ 135 ± 3 (theory [583]) 119 ±10 − 179 ± 12 ([102, 584]) 164 ±2 (theory ([583]) 161.1 ± 2.1 (exp. ([582, 585]) 150, 3 ± 5.8 (theory ([582, 585]) 200 264 ± 3 (theory([583]) 192 258 ± 3 (theory [583]) 108 ± 20–172 ([102, 584]) 125.4 ± 5.1 (exp [582, 585]) 116.7 ± 3.4 (theo [582, 585]) 133 ± 3 (theory [583]) 130–169 ([102] p. 379) 117 ([102], p. 413) (KENS [586])

cm 50.0 15.2–22.9 21.0 20.7 19.2 17.7 23.3 9.9 13.6 45–72 52.2 48.6 55.4 32.5 (30.2) −39.3 (42.3) 65 8.5 × 104

the proper value of λatt , which is used to represent only material attenuation. Also, λatt has a spectral dependence or a certain energy dependence below the high-energy limit, where the attenuation length is constant. This is shown in Figure 7.2 for the variation of the neutron attenuation length for monoenergetic neutrons in concrete as a function of the neutron energy [570]. The value of λatt measured along the beam axis of an accelerator is different from off-axis measurements of differences in particle spectra, e.g., measurements in 90◦ direction. It may be also different because of the anisotropy of the source term and the spectral differences of the leakage distributions of the targets.

7.3 Advanced Shielding Methods for Spallation Sources

Although the Moyer model has been widely used for accelerator shielding estimates, for high-intensity spallation neutron sources and ADS more sophisticated calculational methods have to be applied, which have been developed with the rapid evaluation in computer technology in the recent years. Table 7.3 shows the relative merits of the basic methods and models. The scheme of performance in Figure 7.5 illustrates the possibilities by using full Monte Carlo or Monte Carlo with discrete ordinates coupling for advanced

7.3 Advanced Shielding Methods for Spallation Sources Tab. 7.3

Relative merits of computational methods on shielding of spallation sources.

Method

Advantage

Disadvantage

Monte Carlo

Three-dimensional, transport of all particle types, generation of high-energy cross sections

Stochastic and time consuming

Monte Carlo coupling with SN -transport

Three-dimensional and deterministic, well suited for deep penetration transport

high-energy cross sections needed, coupling-procedures

1–3 dimensional SN -transport

Deterministic, well suited for deep penetration transport

High energy, cross sections needed, only neutron, gamma transport

Semiempirical formalisms

Easy to use, overview estimates

Attenuation concept, in general only with simple material constants

shielding calculations. Monte Carlo with 1D discrete ordinates3) coupling was first considered by Armstrong et al. [552, 553] and further developed and applied for shielding calculations at the 2.5 GeV proton cooler synchrotron COSY at Jülich, Germany, [555]. Systematic investigations were performed with the Monte Carlo/1D discrete ordinates coupling approach during the study phase of the high intensity European Spallation Neutron Source (ESS) project [551, 556, 558, 559]. The Monte Carlo with discrete ordinates coupling approach was then further extended developing connection methods for the two- and three-dimensional DORT [587] and TORT [588] discrete ordinates codes by several authors [589–591]. A new system named DOORS, 1, 2, and 3D discrete ordinates system for deep penetration neutron and gamma transport [592] is also a useful tool to couple Monte Carlo simulation tools with discrete ordinates. A comprehensive summary of computer codes and data sets for accelerator shielding analysis is given at the SATIF-3 and SATIF-5 workshops by Kirk et al. [593] and Satori et al. [594]. It is important to note that for applying the discrete ordinates transport methods high-energy neutron/gamma transport cross section data are needed. For this purpose, two different cross-section libraries are available for such calculations, the LAHI / LAHIMACK library [338, 555, 556, 595, 596] and the HILO2k library [470, 590, 597, 598]. 3) The discrete ordinates method is the dominant numerical solution to the integro-differential form of the transport

equation, often retained in a shorthand designation SN of historic reasons.

243

244

7 Shielding Issues

3D geometry proton beam parameter

Monte Carlo recommended system MCNPX, PHITS, CALOR, HERMES, FLUKA

3D shielding calculations using biasing techniques

3D geometry calculations angle dependet neutron fluxes/currents at coupling surfaces inside shield

Results dose equivalent energy deposition radioactivity

Surface flux in legendre expansion for discrete ordinates -SN - calculations

n,γ transport cross section libraries HILO/LAHI 10−4 eV − 2.5 GeV

1D/2D/3D discrete ordinates neutron/gamma transport discrete ordinates codes ANISN, DORT, TORT

1,2,3D shielding calculations using discrete ordinates

Results dose equivalent energy deposition radioactivity

Fig. 7.5 Scheme of performance for shielding calculations of accelerators and target stations of spallation sources using Monte Carlo and Monte Carlo discrete ordinates coupling.

High-energy transport library descriptions (1) LAHI / LAHIMACK is a coupled multigroup library of 62 neutron and 21 photon group cross sections and response functions for neutron energies up to 2800 MeV. – The original LAHI library developed during the German spallation source Jülich SNQ project during the 1980s is a combination of the HILO [408] and a high-energy neutron multigroup library from Ref. [599]. The library was updated up to an energy of 2800 MeV during the COSY accelerator project and the ESS project by [555, 558]. The LAHIMACK version of this library contains, additionally, multigroup response functions and reaction cross sections, secondary particle production yields, kerma factors obtained


by HERMES intranuclear-cascade-evaporation calculations [596] above energies of 20 MeV and below 20 MeV from Ref. [600]. – The library contains neutron cross sections for H-1, C, O, Al, Si, Fe, and Pb in 62 energy groups from 0.1 eV up to 2800 MeV and 21 gamma groups from 0.01 up to 14.0 MeV. The library contains Legendre expansions up to P3 for the scattering cross sections (2) HILO2K is a coupled 83 neutron and 22 photon group cross sections for neutron energies up to 2 GeV. – HILO2k is a high-energy neutron and photon transport cross-section library containing neutron cross sections to 2 GeV and photon cross sections to 20 MeV. It represents the updating and extending of the HILO86 transport cross-section library developed by Alsmiller et al. [597] in the mid-1980s. – HILO2k was developed as a part of the neutronics research associated with the design of the Spallation Neutron Source (SNS) currently under construction at ORNL. The HILO2k multigroup cross sections are in the form needed for the DOORS3.2a [592] (ANISN, DORT, and TORT) discrete ordinates codes. In addition to transport cross sections, the HILO library contains flux-to-dose conversion factors, kerma factors, dpa, hydrogen, and helium gas production response functions. – The library contains transport cross sections for 32 nuclides commonly found in the target, reflector, and shielding materials used at spallation neutron source facilities. The high-energy portion (E > 20 MeV) of HILO2k is based on neutron elastic and nonelastic interaction and production data generated using the nuclear collision models in MCNPX [194]. The low-energy portion was derived from ENDF data for all nuclides except Hg. The high-energy neutron interaction and production cross sections were also normalized to the nonelastic cross sections for those nuclides treated in the recently evaluated Los Alamos National Laboratory (LANL) 150-MeV library. Data are provided for H-1, H-2, He, Be, B-10, B-11, C, N, O, Na, Mg, Al, Si, S, K, Ca, Cr, Mn, Fe, Ni, Cu, Zr, Nb, Cd, Ba, Gd, Ta, W, Hg, Pb, U-235, and U-238. – Two versions of the HILO2k library are distributed. In the standard version, all high-energy particle interaction and production data were taken into account to obtain the group total cross section (interaction cross section) and the group-to-group Legendre scattering coefficients (scattering cross sections). In the modified version, neutrons that exited collisions almost straight ahead with more than 95% of the incident neutron energy were treated as uncollided neutrons. Both the libraries contain 105 total energy groups (42 high-energy neutron, 41 low-energy neutron, and 22 photon groups). The modified version of the new HILO library contains P9 Legendre expansions for the scattering cross sections, whereas the standard version contains P10 expansions.

245

246

7 Shielding Issues

7.3.1 Monte Carlo Discrete Ordinates Coupling

As discussed earlier, the particles which cause the radiation problems around accelerators and high intensity spallation neutron sources are the deep penetrating neutrons with energies above 100 MeV up to the energy of the proton beam of the accelerator. These neutrons determine the dose rate at each point inside and outside the shield because they produce via the cascade propagation a neutron spectrum with energies down to thermal energies. This spectrum is far inside the shield more or less an equilibrium spectrum if the shielding material is not changed. The advantage of the Monte Carlo discrete ordinates coupling for shielding calculations is as follows: in the vicinity of the neutron spallation target one uses the Monte Carlo method to treat all important particle types in that area and to calculate the spatial and angular distribution of generated high-energy neutrons at the outer edge of that region. This distribution is handed over to a deterministic transport code, which calculates the attenuation of the high-energy neutron flux inside the shield far away from the target. The incoming proton beam and all particles produced in the spallation process and in the target vicinity are treated exactly by Monte Carlo allowing a proper design of the geometries in three dimensions. Therefore, one gets the correct fast neutron flux distributions inside and at the edge of the target region or the accelerator structure, respectively. The coupling procedure works in the following manner: The particle transport via Monte Carlo has to be used until a depth sufficiently large inside the shield, where the neutrons are the dominant high-energy particles rather than protons, but not deeper than necessary to satisfy the criterion so that the statistics from the Monte Carlo are as good as possible. The Monte Carlo calculated neutron current across this boundary then constitutes the discrete ordinates angular neutron for further calculations. All neutrons from the Monte Carlo calculations crossing this coupling plane in the ‘‘positive’’ direction – larger depth inside the shield – for the first time then constitute a surface source. This is illustrated in Figure 7.6. The experience has shown that a sufficient depth for the coupling plane inside the shield is about several mean free paths or several attenuation length. To analyze the results from the Monte Carlo estimations at the ‘‘coupling plane’’ the boundary source for the 1D discrete ordinates code ANISN is considered here as an example in the following way: • Let MC i,g denote the Monte Carlo calculated neutron flux at the coupling plane for the ith angular interval and gth energy interval, which can be computed as S Wi,g,k /(ωk · N0 ), (7.15) MC i,g = k S is the statistical weight4) of the kth neutron in the Monte Carlo where Wi,g,k calculation which crosses the scoring plane – the coupling surface – in the

4) cf. Section 4.3.1 on page 195.


Monte Carlo

Discrete ordinates Neutron current J (E,µi)

Incident protons

x Z

Coupling surface

End of Monte Carlo history

Fig. 7.6 Schematic of the contribution of a particle to the surface source at its first cross-over point on the coupling surface between Monte Carlo and discrete ordinates.

direction ωk > 0 within the ith angular group and with an energy of the gth group. The direction cosine ωk = k · z, where z is a unit vector along the +z axis (taken to be the beam direction). The index k refers to individual contributions, and the summation over k is for all contributions made for N0 incident beam protons. The contribution to is zero if ωk ≤ 0. • In discrete ordinate calculations, e.g., 1D ANISN, the angular flux is assumed to be directed along a finite number of ‘‘discrete directions,’’ denoted by the cosine µi between the neutron direction and z with a weighting factor wi associated with each direction, e.g., the scalar flux is given by i i,g · wi . Thus ANISN expects as a source from the Monte Carlo calculations the angular flux per unit weight, Si,g = MC i,g /wi .

(7.16)

Note that the source flux is not divided by the energy interval, only the energy group index g, is needed (g = 1 corresponding to the highest energy group in the group cross section library). Thus, the energy boundaries for the Monte Carlo flux tabulation correspond to the energy intervals of the used cross section group structure used for the discrete ordinates calculations. What remains is to define the angular boundaries around the discrete directions for tabulating the Monte Carlo results. Values for µi and wi are defined from the quadrature set used. Normally, classical quadrature sets with µi representing the zeros of Legendre polynomials and wi the weighting factors for Gaussian integration are used, as can be found in numerical analysis handbooks. An example (to only three significant digits) for an SN = S8 quadrature is given in Table 7.4. (By ANISN convention, µ1 = −1 and w1 = 0.) The weights can be envisioned as fractional solid angles ( i wi = 1). (a) µi = zeros of Legendre polynomials, the discrete direction cosines used in discrete ordinates calculation. (b) θi = cos−1 ·µi in degrees

247

248

7 Shielding Issues Example (for S8 quadrature) of defining angular boundaries for neutron source calculations.

Tab. 7.4

Discrete directions and weights (a) µi (= cos θi ) −1.000 −0.960 −0.797 −0.526 −0.183 +0.183 +0.526 +0.797 +0.960

(b) θi (◦ )

(c) wi (weight factor)

180 164 143 122 100 79.5 58.5 37.5 16.5

0.000 0.051 0.111 0.157 0.181 0.181 0.157 0.111 0.051

Source boundaries (d) µi

−1.000 −0.899 −0.662 −0.355 0.000 0.355 0.665 0.899 1.000

+1 (c) wi = weight factors for Gaussian integration −1 (µ)dµ ≈ ni=1 wi · (µi ) (d) µj = boundaries for calculating angular flux source distributions in Monte Carlo calculations = µi−1 + 2 · wi , µi = −1. Apparently, there is no unique method for defining the values µi for the angular source boundaries about the µi directions. One way is to set the interval width about each µi as µi = 2 · wi , then µi = µi−1 + 2 · wi , with µi = −1. The resulting va1ues for the S8 quadrature (given in Table 7.4 are numerically very close to simply taking the midpoints of the µi interva1s, which suggests that the Monte Carlo directions could approximately be scored according to the closest µi . The procedure for coupling with the 2D and 3D discrete ordinates is similar to that described above for the 1D case except that a (R, Z) or a (R, θ ) dependence in the two-dimensional case of the flux and in the three-dimensional case a (X, Y, Z) or (R, θ , Z) must be obtained. Some solutions are given in Refs. [589, 591, 592]. For deep penetration problems, asymmetric quadrature sets are usually required, where more discrete angles are used in forward direction than in backward direction. During the German spallation neutron source, Jülich SNQ project during the 1980s, detailed studies were investigated to show the feasibility of the anticipated nuclear parameters of a high-power neutron spallation target station of 5 MW proton beam power and an incident proton energy of 1.1 GeV [601]. Deep penetration bulk shield calculations utilizing Monte Carlo discrete ordinates coupling with the high-energy neutron/gamma LAHI library [338, 595] mentioned above where done the first time. One concern was that whether the bulk shield thickness of the neutron spallation target station would be the same in the forward direction as in the backward direction for the 1.1 GeV incident proton beam on a water cooled lead target system. The inner part of the shield was composed of cast iron while


Dose equivalent [µSv/h] per 5 mA

1011 1010

Incident proton energy 1.1 GeV on target

107

Forward

Backward

104

Concrete 101

Iron

Concrete Iron

5 µSv/h 10−2 0.5

1

2

3 4 Shield radius [m]

5

5.5

6

6.5

Fig. 7.7 Comparison of the dose equivalent per hour in forward and backward directions in an iron–concrete bulk shield for a proton beam energy of 1.1 GeV and beam power of 5 mA [601].

the outer part was composed of concrete [601]. Figure 7.7 shows a comparison of the space-dependent dose equivalent in µ Sv/h per 5 mA beam power in forward and backward direction to the incident proton beam. It was found by the Monte Carlo 1D discrete ordinates coupling that in forward direction 5.5 m of cast iron and 1 m of concrete and in backward direction 4 m of cast iron and 1 m of concrete are sufficient to reduce the dose rates of less then 5 µ Sv/h outside the shield5) . The difference in the dose rate distributions is the fact, that the energy spectrum in the forward direction is much harder than in the backward direction and the angular distribution is strongly peaked in the forward direction. An example of this anisotropy is shown in Figure 7.8. 7.3.2 Monte Carlo Techniques and Deep Penetration

The Monte Carlo method is a very useful tool for solving a large class of radiation transport problems of complicated three-dimensional geometry and material systems. However, the Monte Carlo method in all its forms involves some sort of random process. Most of the Monte Carlo systems summarized in Table 5.1 are using models in analog fashion. The analog process of ‘‘pure’’ Monte Carlo methods for deep penetration calculations is inefficient because only macroscopic quantities are important. These quantities are often a result from contributions of statically rare events. In thick shields, needed for radiation protection of high-energy 5) It should be noted here that the dose criterion given here is only an example. It depends of the rules of the licensing procedures in

different countries and the radiation protection legislation. Nowadays a much lower value, e.g., 0.5 µ Sv/h, will be assumed.

249

7 Shielding Issues

10° 10

30°

1 dΦ(E)/dΩ [neutrons cm−2 sr−1 p−1]

250

60° 8

1

7 0.1

6

5

4

3

2 90°

0.1

120°

1

150° Beam

1 all neutrons 2> 1 MeV 3> 6 MeV 4> 20 MeV 5> 50 MeV 6> 100 MeV 7> 250 MeV 8> 500 MeV

Fig. 7.8 The measured angular distribution of neutrons in different energy groups for a 20-cm diameter lead target bombarded by protons of an energy of 2 GeV (after Bauer [57]).

and high-intensity spallation sources, particles have to travel a large number of mean free paths to go through, whereas the average number of collisions that particles undergo during their lifetime until their energy has fallen below a certain level is considerably smaller. Neutral particles, e.g., neutrons, however, can travel long distances between collisions, of course, with a low probability for larger path lengths. Thus a few particles can penetrate a thick shield. The calculation of this small fraction is the deep penetration problem. Besides the bulk shielding problems there exist other issues at high-intensity spallation neutron sources e.g., the streaming of neutrons through gaps and ducts in a shield, and the neutron beam line shielding, the instrument shielding, shutters, and collimators which are of high importance. The neutron albedo and multiple scattering of neutrons within the neutron beam line or in the ducts are very important. High-energy neutrons entering a duct in a neutron shield become thermalized and have been able to scatter many times, allowing to stream along


the duct, even with several bends. Also the steel walls of a pipe for cooling water inside a shield can act as a duct for steaming of high-energy neutrons. Unlike low-energy Monte Carlo codes as MORSE [506] or MCNP inside the MCNPX code [509], where a large experience on particle transport biasing exists in reactor and fusion technology shielding [602, 603], this is in general not the case for high-energy intensity spallation sources. Only recently Monte Carlo methods were applied for high-intensity source shielding of spallation sources, e.g., in [560, 561, 604–609] and ADS assemblies [610]. In the following a short overview is given on the available biasing methods in high-energy particle transport system (cf. Table 5.1 on page 210). Details about the main mathematical and statistical considerations can be found in the textbooks by Lux and Koblinger [149] and by Lewis and Miller, Jr. [150]. (1) Russian Roulette and surface splitting: The simplest biasing method is a very well-known Russian Roulette and splitting procedure. The method is based on a region ‘‘importance’’ concept, where a numerical value to each geometry region is assigned, which may be dependent on the particle under consideration. During the transport of the particles through the geometry the number of the selected particle types crossing a given region boundary is decreased or increased by a factor equal to the ratio of the importance on either side of the boundary. Here the relative importance of different regions not their absolute values are considered. In simple deep-penetration calculations with one main attenuation path this biasing procedure is easy to use and very effective [197]. (2) Weight window: The weight window biasing reduces fluctuations of the particle weight within predefined limits. It is also based on Russian Roulette and splitting of the particle weight. It refers to the absolute particle weight and is not linked with geometrical boundaries. It can be finely tuned by region, particle and energy and increases the effectiveness of all other biasing techniques. (3) Leading particle biasing: Leading particle biasing is important for electrons, positrons, and photons. It is used to avoid the geometrical increase with energy of the number of particles in electromagnetic showers. (4) Nonanalog absorption: This biasing method is used mainly for low-energy neutrons. It is the choice between analog absorption with the actual physical probability and a systematic survival with a reduced weight. (5) Biased decay and interaction length: This biasing method is useful to decrease, e.g., the inelastic interaction length in a given material by a particle dependent factor which is useful to increase

251

7 Shielding Issues

the probability of interaction in a low density material. The average decay length of unstable particles can be shortened without stopping the parent particles. (6) Biased down-scattering: This biasing method is used to bias the group-to-group transfer probabilities used in the transport of low-energy neutrons. The method accelerates or slows down the moderation process. The method needs some experience, but can save considerable computer time in systems where the moderation process plays an important role. An impressive example of a deep penetration Monte Carlo calculation for thick bulk shield using the weight window biasing method was given by Koprivnikar and Schachinger [560] recently. The geometry model of the used MCNPX Monte Carlo system [509] was adopted from the JESSICA cold moderator experiment [551, 611], a 1:1 mockup of the target–moderator–reflector system similar to the high-intensity European Neutron Spallation Source (ESS) [551] of 5 MW proton beam power equal to 3.75 mA proton current or 2.34 × 1016 protons/s, and an incident proton energy of 1.334 GeV on a mercury target. The very detailed inner part of the ESS target station is surrounded by a cylindrical bulk shield of iron and concrete, whereas at the outer edge of this inner part at

Target - moderator reflector system x

Iron shield

60°

Incident proton beam

Sampling surface

50

z

∅1

252

Shielding is not to scale

Target

Reflector

Concrete

Fig. 7.9 Horizontal cut through the used Monte Carlo 3D geometry of the target station of an inner diameter of 3 m showing the sampling surface for the incident neutron source incident in a cone into the bulk shield. The target station is surrounded by an iron shield of a thicknesses between 475 cm in forward, 425 cm in backward direction and in addition by a 50-cm thick concrete layer (after [560]).


a radius of 150 cm an energy and angle-dependent sampling surface source was determined for the deep penetration calculations (cf. Figure 7.9). A combination of two biasing techniques, available in the MCNPX system, was used. One is the energy-dependent weight window technique, and the other one is the source biasing. The weight window is a space-energy-dependent splitting and Russian Roulette method in which each space-energy phase-space cell defines a window of a certain acceptable particle weight. If this weight is below a specified weight, then Russian Roulette is played and the particle is eventually killed from the history. If the weight of the particle is above the specified weight, then splitting occurs. All split particles are again within the weight window. For this kind of biasing technique the shield system geometry has been divided into sections. Therefore, the shielding part of the geometry of Figure 7.9 was divided into concentric spherical shells of an incremental radius of 25 cm. The neutron equivalent dose attenuations in (µ Sv/h) as a function of the depth inside the shield are given in Figure 7.10 for 0◦ , 45◦ , and 90◦ directions to the incident proton beam. Assuming a 5 µ Sv/h design criterion for the edge of the shield at 0◦ this yields 475 cm of iron plus 50 cm of concrete, at 45◦ 460 cm of iron plus 50 cm concrete, and at 90◦ about 425 cm of iron plus 50 cm of concrete, respectively. From these calculations the authors [560] determined from the dose distributions, given in Figure 7.10, with the help of an exponential function, neutron attenuation lengths in the energy range between 10 MeV and 1.334 GeV for iron as 22 ± 1 cm and for concrete as 38 ± 2 cm. 1010

Neutron dose equivalent [µSv/h]

Concrete layer 45° Concrete layer 90° Concrete layer 0°

105

100

10−3 100

Total neutron dose rate, 0° direction Total neutron dose rate, 90° direction Neutron dose rate due to neutrrons >10 MeV, 90° direction Total neutron dose rate, 45° direction Neutron dose rate due to neutrrons >10 MeV, 45° direction

150

200 250 300 350 400 450 Distance from inner iron shield surface [cm]

Fig. 7.10 The attenuation of the neutron dose equivalent is plotted in (µ Sv/h) in different directions 0◦ , 45◦ , and 90◦ to the incident proton beam through the bulk shield assuming a thickness of iron to be 475 cm and that of concrete of 50 cm (after [560]).

500

550

253

254

7 Shielding Issues

7.4 Sky- and Groundshine Phenomena

Increased neutron and photon radiation fields have been observed in the vicinity of accelerator and reactor facilities [102, 612], which may produce a radiation hazard for the laboratory staff and the surrounding population. The spatial distributions of these external dose levels are usually affected by the sky- and groundshine effects, due to backscattering of the radiation from the atmosphere, of the secondary radiation from the ground, and of neighboring buildings. Figure 7.11 illustrates these phenomena. The interesting point is that dose rates caused by sky- and groundshine effects are usually measured not directly adjacent to, but at higher distances from a facility. Another problem is caused by the possible contamination of the surrounding air and ground water by the induced radioactivity. These effects can occur at facilities with not adequate roof shielding or by the leakage of radiation underneath the high-density shielding. Its solution is to improve the roof shielding or reduce the scattered radiation by extending the high density shielding into the lighter density foundation of the ground. An overview about evaluation methods of neutron and photon skyshine is given by Hayashi [613] recently. Rindi and Thomas [612] have summarized experimental and theoretical studies from high-energy particle accelerators and suggested a simple formula (cf. Eq. (7.17)) to estimate the neutron fluence at distances up to about 1000 m from the source. (x) =

a·Q · exp(−r/λatt ), 4π · r 2

(7.17) Skyshine

Shielding

Target Direct Ground shine

Ground Fig. 7.11

Sketch of the sky- and groundshine phenomena around a spallation facility.

7.4 Sky- and Groundshine Phenomena 1000

latt in air [m]

800 600 400 200 0 0 10

101 102 103 104 105 Upper energy of neutron 1/E spectrum [MeV]

Fig. 7.12 The attenuation length λatt in air as a function of the neutron energy assuming a 1/E neutron leakage spectrum (after [615]).

where (x)= neutron fluence at a distance r from the source, Q = neutron source strength, a = 2.8 an empirical buildup factor, and the neutron attenuation factor in air is assumed as λatt > 830 m for neutrons with energies E > 150 MeV. Alsmiller et al. [614] calculated ‘‘importance function” for neutron skyshine up to 400 MeV by 2D discrete ordinates adjoint calculation. Stevenson and Thomas [615] extended Eq. (7.17) of Rindi and Thomas using Alsmiller’s response functions. Under the assumption that the leakage neutron spectrum of the facility under consideration is 1/E spectrum, Stevenson and Thomas estimated the attenuation length λatt in air as a function of upper neutron energy Ecut . This functional dependence is given in Figure 7.12. The extended formula is given by Eq. (7.18) and is only valid for distances r > 100 m. H(r) ≈

(1.5 ∼ 3.0) × 10−15 · exp(−r/λatt ), r2

(7.18)

where H(r) is the equivalent dose rate at a distance r from the leakage source in (Sv/per source neutron), r is the distance from the source, and λatt the attenuation length in (m) for high-energy neutrons. To estimate the total dose equivalent H(r) the number of leakage neutrons must be known. Besides these simple methods, nowadays neutron- and photon skyshine distributions are precisely calculated by using Monte Carlo, by multidimensional Sn transport or by Monte Carlo methods coupled with discrete ordinates including the facility and building structure directly, or concatenating leakage calculations from the facility building and air transmission calculations (cf. Figure 7.5). Coupled Monte Carlo discrete ordinates calculations on sky- and groundshine are investigated for the high power neutron spallation target station of 5 MW proton beam power and an incident proton energy of 1.1 GeV of the German SNQ project E[601]. Based on the same geometry used for the attenuation calculations

255

7 Shielding Issues 102 5 2 1

10 5 2 100 5 2

101

Vertical

Horizontal

Relative contribution to dose equivalent

Average neutron energy [MeV]

256

−1

Low-energy neutrons

10−1

Gamms

10−2 −3

10

10−4 10−5

High-energy neutrons

10−6

10 (a)

100

101 2 5 102 2 5 103 2 5 104 Horizontal/verticale distance [m]

101 2 (b)

5 102 2 5 103 2 5 104 Horizontal distance [m]

Fig. 7.13 (a) Average energy of skyshine neutrons versus distance from the targetstation. (b) Relative contribution to the dose equivalent of high-energy neutrons (> 20 MeV), low-energy neutrons (< 20 MeV), and gammas versus the horizontal distances from the target station.

given in Figure 7.7 and using the neutron–gamma transport cross section library LAHI [338, 595, 596] a coupled neutron–photon leakage radiation transport is possible. Zazula et al. [616] have shown for this high-current spallation source facility some interesting features resulting from sky- and groundshine calculations. Up to a distance of 100 m from the station, it is not possible to describe the decrease of the radiation levels proportional to 1/4πr 2 or by an attenuation proportional to exp(−x/λatt ). Numerical fits of the results with semiempirical formulas developed for the description of the skyshine effect were not performed here, since such expressions do not have a universal character and are strongly dependent on the source energy and the geometry details of the facilities considered. The differences of the attenuation lengths are caused by different energy ranges of the secondary particles at different distances. To explain this behavior, the relative contributions to the dose equivalent from different kinds of secondary radiation are compared in Figure 7.13(b). The neutron spectrum distribution in air is equal to the distribution of neutrons at the surface of the shielding, and thus, the low-energy neutrons dominate. Farther away these slow neutrons are absorbed, and the fast neutrons, penetrating the air to large distances (hundreds of meters) in air, are not slowed down very much. Thus, at a distance of about 1000 m and more, the high-energy neutron and gamma components contribute more to the dose. This hardening of the skyshine radiation spectrum was also described by Rindi and Thomas [612]. This effect is clearly shown in Figure 7.13(a), for both, the horizontal as well for the vertical average neutron energy. Another issue is the photon sky- and groundshine problem, where also many theoretical work and measurements including benchmark experiments were done by many investigators. Examples of shielding experiments will be given in Part 2 on page 481. Problems on safety and radiation protection are discussed in Part 3 on page 554. Properties of shielding materials are given in the appendix on page 689.

257

8 The Basic Parameters of Spallation Neutron Sources 8.1 Introduction

The emphasis of this chapter is focused on the basic matters related to the parameter scheme of spallation neutron sources due to their estimated use. As already discussed in Section 1.3 on page 12, besides fission or fusion, spallation is an efficient process for producing neutrons and other particles in large numbers. However, in contrast to fission spallation is an endothermal process and can therefore, a priori not be used for energy production. Compared to fission, spallation has its own technology issues, which need a careful selection of a variety of physical variables. The spallation process is responsible for some technical constraints concerning the neutron producing target as, e.g., the selection of target and structure materials, particle type and kinetic energy of the incident beam, moderators and reflectors, target heating, induced radioactivity, etc. The worldwide operating, under construction and planned facilities of highintensity spallation neutron sources, including the realized accelerators and target systems, are described in Chapters 16,17 on pages 497, 505.

8.2 Parameter Regime for Spallation Neutron Sources 8.2.1 The Particle Type

In considering the merits of protons versus heavier ions as projectiles for spallation sources, a primary consideration is the desire to transfer the beam energy into secondary particle production rather than into deposition power or heating through ionization. The rate of ionization energy loss, or stopping power Si (Ai , Zi , Ei ) for an ion of mass Ai , charge Zi , and kinetic energy Ei can be written in terms of the proton stopping power, Sproton , evaluated at the energy per nucleon, Ei /Ai , of the ion Si (Ai , Zi , Ei ) = Zi2 · Sproton ·

Ei . Ai

(8.1)


258

8 The Basic Parameters of Spallation Neutron Sources

Also, the ion range Ri (Ai , Zi , Ei ) can be scaled from the proton range Rproton evaluated at the energy per nucleon of the ion Ri (Ai , Zi , Ei ) =

Ai Ei · Rproton · . 2 Ai Zi

(8.2)

Proton range curves for iron, tungsten, and lead spallation target materials are shown in Figure 8.2. The nuclear collision cross section is approximately constant with energy and increases slowly with increasing ion mass, 1/2

σin ∝ (Ai

1/3

+ Atarget )2 ,

(8.3)

where Atarget is the mass number of the target nucleus. Thus, to maintain an ion range sufficiently long to allow nuclear collisions to occur requires increasing kinetic energies with increasing ion masses. As an example, consider a lead target with protons and oxygen ions as projectiles. For protons, the nuclear collision mean free path is λ ≈ 200 g/cm2 , and if one specifies a range so that essentially all of the beam protons can interact, say 3 · λ, then the corresponding beam energy needed is ≥1 GeV. For oxygen ions, λ ≈ 135 g/cm2 , and a 3 · λ range requires the beam energy to be about 30 GeV. Once a nuclear collision occurs, the neutron multiplicity will, of course, be higher for higher mass ions, but it is clear that much higher beam energies are involved with higher mass ions, with higher costs and technology requirements for acceleration. Measurements of neutron yields Yneutron per charge number Zi for a lead target bombarded with protons, deuterons, 3 He, 4 He, and 12 C are published by Vassil’kov [617] and shown in Figure 8.1 as a function of the incident energy per ion charge number Ei /Zi . The deuteron is somewhat of a special case in considering ions heavier than protons because: • The deuteron and proton have the same charge, so the Z2 factor in ionization loss is not relevant. • The nucleons of the deuterons are weakly bound and easily ‘‘stripped’’ in nuclear collisions, resulting in a source of neutrons and protons, each with about one-half the deuteron energy, at rather shallow target depths. For 1 GeV beams and a lead target Barashenkov et al. [618] calculated about 15% greater neutron yield for deuterons over protons with essentially no difference in heat deposition. With deuterons a slightly higher neutron yield is produced. This small difference in the neutron yield between incident protons and deuterons is also shown in Figure 8.1. Bauer et al. [619] reported measurements of neutron leakage fluxes from hydrogenous moderators of a realistic target–moderator–reflector configuration as the function of protons and deuterons impinging on lead and uranium targets. The measured neutron fluxes represent not the total fast neutron yield given by Barashenkov et al. [618] and therefore contain some physical parameters as a function of the target–moderator geometry, penetration depth of the primary protons and deuterons, and the coupling efficiency for fast neutron

8.2 Parameter Regime for Spallation Neutron Sources

150 Deuteron

100 Yi /Zi

Proton 3He 12C

50

0

4He

0

1

2

3

4 5 Ei /Zi [GeV]

6

7

8

Fig. 8.1 Measured neutron production yield per incident particle charge, Yi /Zi , of a cylindrical Pb target (diameter = 20 cm, length = 60 cm) as a function of the incident particle energy per ion charge Zi , Ei/Zi , of protons, deuterons, 3 He, 4 He, and 12 C (after Vassil’kov [617]).

from the target into the moderator. A comparison of the flux measurements of Bauer et al. shows a gain for incident deuterons to incident protons on lead at beam energies of 0.4 GeV of about 10% and at an incident beam energy of 0.75 GeV a gain of 23%. For the uranium target experiment, the gain at these incident energies is about 13% and 16%, with a larger value of about 24% at an incident energy of 0.6 GeV, respectively. Theoretical evaluations of Ridikas and Mittig [620] done in the context of ADS studies have shown that there is some advantage to use deuterons instead of protons in ADS devices depending on the choice of the neutron producing target system. They found bombarding beryllium spallation targets, placed inside the same fissionable fuel assembly, with deuterons of an energy at 0.6 GeV, an average energy gain, e.g., the ratio of the energy produced by fissions in the device to the energy delivered by the beam – Efission /Ebeam – of about 28 compared to incident protons of the same energy of about 16 for beryllium. In addition, this study considers for incident deuterons that an energy amplifier could therefore be operated with deuteron beam energies of the order of 0.4–0.8 GeV, e.g., at lower beam energies than suggested for a proton beam, and give a comparable energy gain factor. 8.2.2 The Kinetic Energy

The rationale for the minimum kinetic energy of proton beams appropriate for spallation sources is basically the same as discussed above for particle masses,

259


104

Proton range [g /cm2]

260

103

102

26

Fe, r = 7.87 g/cm2 2

74

W, r = 19.30 g/cm

82

2

Pb, r = 11.35 g/cm

101

100 102

103

104

Proton energy [MeV] Fig. 8.2

Proton energy range curves for target materials Fe, W, and Pb.

i.e., the proton range, a monotonically increasing function of energy, should be sufficiently large compared to the nuclear collision mean free path, which is constant with energy above about 100 MeV. The range dependence on energy is shown in Figure 8.2 for iron, tungsten, and lead. The nuclear mean free path for nonelastic collisions of incident protons for E ≥ 100 MeV and A > 1 and the probability of nuclear collisions Pnuclear-collsion before the proton reaches the end of range are given as λ = 33 · A1/3 (g/cm2 ) Pnuclear-collision = 1 − exp(R/λ).

(8.4)

Figure 8.3 shows for a lead target, the proton range divided by the proton nuclear collision length λnuclear-collision-PB ≈ 192 g/cm2 ≈ 16.9 cm and the probability of nuclear collisions Pnuclear-collsion before the proton reaches the end of range as a function of the proton incident energy on the target. Thus, the probability of obtaining nuclear collisions to produce neutrons is not substantial unless the beam energy is several hundred MeV, or more. This is approximately independent of the target material. For the maximum appropriate neutron production yield, there is no advantage for energies much greater than about 1–2 GeV because, • as mentioned above, essentially all of the beam protons interact if the energy is about 1–2 GeV, • pion production becomes increasingly significant for beams much above 1–2 GeV, and the energy is partly lost for neutron production, in particular the π ◦ decay in 2γ s is not be efficiently converted to neutron production (cf. via the electromagnetic drain already discussed in Section 1.3.8 on page 57),


Proton range / l

101

Proton range / l

101 Lead target

100

100 Probability P

10−1

10−1

10−2 101

102

103

104

Pnuclear collision = 1-exp(-range / l)

102

102

10−2 5 × 104

Incident proton energy [MeV]

Fig. 8.3 Values for the probability of nuclear collisions for a Pb target before the proton reaches the end of its range.

• the spatial dependence of neutron production becomes somewhat less concentrated for higher energies, which may reduce the maximum density of neutrons attainable, • and higher beam energies require larger accelerator systems. Within the above-defined general energy range of about 0.5 up to about 1–2 GeV, the neutron production is directly proportional to the beam energy, which is illustrated in Figure 8.4. The energy dependence of the yield data in Figure 8.4 can be described by a simple empirical relation (see also Section 1.3 on page 12) x Mneutron /proton = M0 + M1 · Eproton ,

(8.5)

where Mneutron /proton is the multiplicity per incident proton with an energy Eproton in GeV. The numerical values for the parameters in Eq. (8.5) are given in Table 8.1. Carpenter et al. [625] have discussed ‘‘The 10,000,000,000-Volt question.’’ What is the advantage to use higher proton beam energies for pulsed spallation neutron sources, e.g., in an energy range of about 10 GeV? Some of the conclusions are as follows: • losses of the neutron yield at higher incident proton beam energies have to be compensated with some higher beam power, • higher incident proton beam energy reduces radiation damage problems in accelerator and target beam windows, • potentially lower beam losses and operating costs of the accelerator, • different shielding problems, e.g., considering neutrons with higher energies and gamma rays produced by electromagnetic cascades, • detailed engineering studies are needed to reach a certain level of confidence.

261

262


200

Mean neutron multiplicity per incident proton

(1)

(2)

150

(3)

100

Hilscher et al. Vassil′kov et al. Arai et al.

50

Protons on cylindrical Pb targets

0

0

5

10

Incident proton energy [GeV] Fig. 8.4 Mean neutron multiplicity per incident proton of Pb targets as a function of the incident proton energy. Data from Hilscher et al. [621] and Filges et al. [622] are for a cylindrical target with diameter 15 cm and length 35 cm, data from Vassil’kov and Yurevich [617] refer to

moderator measurements with a target of diameter 20 cm and length 60 cm, and the data of Arai et al. [623] are for a target of diameter of 20 cm and length 60 cm, respectively. The solid, dotted, and dashed lines correspond to a parameterization (1), (2), (3) given in Table 8.1.

Parameters for Eq. (8.5) to estimate the average neutron yield as a function of the incident proton energy on thick targets, where A is the atomic mass of the target material [621].

Tab. 8.1

Line

(1) (2) (3)

Target material A Pb Pb

Dimension ∅, length (cm)

Ep (GeV)

10, 100 20, 60 20, 60

≤ 1.5 ≤ 3.7 ≤ 8.1

M0

M1

x

−(A + 20) · 0.012 −4.8 ± 1.0 −8.2 ± 1.6

(A + 20) · 0.1 28.6 ± 2.5 29.3 ± 1.3

1.00 0.85 0.75

Reference

[59] [624] [617]

8.2.3 The Target Material

Desirable ‘‘physics properties’’ of target materials for spallation sources are high neutron production, low heating, and low induced radioactivity; a summary of these features are discussed below. There are, of course, other important factors, not discussed here, which enter into target design considerations, such as thermal properties, e.g., conductivity, melting point, mechanisms for heat removal, material damage under irradiation and thermal cycling (affecting target life time), safety, economics, etc. Thus, the overall problem of target design involves tradeoffs of various materials properties, and final selection depends on the magnitude of the


particular proton beam intensities and neutron output goals being considered. The issues and solutions on materials for spallation sources are discussed in more detail in Part 3. 8.2.4 The Neutron Production

One factor in considering appropriate target materials for spallation sources is the total neutron production in phase space produced per beam proton. The determination of this neutron yield, and its dependence upon target material mass number and proton beam energy, has been the objective of several experimental and theoretical studies. It should be noted that values for neutron yields appearing in the literature are not always reported on the same basis. They may refer to neutron absorptions in a moderator surrounding the target, to moderator plus target absorptions, or neutrons escaping from a ‘‘bare’’ target without surrounding the moderator. The numerical difference due to these different definitions can be appreciable, making the interpretation of comparisons between experiments difficult in some cases. Experimentally, the first study of neutron yields for spallation neutron sources was carried out by Fraser et al. [59] at the Brookhaven Cosmotron accelerator in connection with the proposed Canadian ING concept [58] (cf. Section 1.3 on page 1.3) and by Bercovitch et al. [626]. Recent measurement were extensively undertaken together with the research and design of highintensity spallation neutron source projects SNQ [142], SINQ [478, 627], ESS [551], J-PARC [628, 629], LANSCE [630], and in relation with international research on ADS [133, 543]. All these measurements exploit either the neutron time-of-flight (TOF), the moderation and activation, or the threshold method. The various experimental results on thick target measurements can be condensed to a quantity which expresses the ‘‘economy’’ of neutron production, i.e., the number of neutrons produced per incident proton and per unit of beam energy. This number is displayed as a function of the incident proton energy Eincident in Figure 8.5. The presentation could be a useful guideline on neutron generation in the context of new concepts of high flux neutron sources. The neutron number increases sharply with increasing incident proton energy or decreasing electronic loss of the proton in the material and culminates at about 0.8–1.0 GeV when the minimum of ionization is approached. Tungsten gives a 10% higher yield than lead for the same target size, length of 35 cm and diameter 15 cm. Data on mercury give, not shown in the figure, similar results compared to lead. As for example, HERMES simulations reproduce the measured yield for lead and tungsten reasonably well. A calculation for a larger lead cylinder with length of 60 cm and diameter 20 cm is also indicated in order to show the possible gain of some 20% with a larger target and because this has been the standard target in previous investigations [623, 624]. Data from Ref. [624] lie close to the latter simulation. Beyond the maximum near an incident energy of 1 GeV the yield diminishes very slowly, which is corroborated by a recent Mn -bath experiment [623] with a Pb target

263


25

Number of neutrons [np−1 GeV−1]

264

20

15

10 NESSI (Pb, l = 35 cm, d = 15 cm) NESSI (W, l = 35 cm, d = 15 cm) PS208 (Pb, l = 35 cm, d = 15 cm) ORION (Pb, l = 25 cm, d = 12 cm) HERMES (Pb, l = 35 cm, d = 15 cm) HERMES (W, l = 35 cm, d = 15 cm) HERMES (Pb, l = 60 cm, d = 20 cm)

5

0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Incident proton energy [GeV] Fig. 8.5 The average number of neutrons produced in Pb and W cylindrical target per incident proton and per energy unit (1 GeV) as a function of the incident proton energy. Data from NESSI, PS208 [631], and ORION [632] are shown. Results of simulations with

the HERMES code [88, 622] for different targets (Pb: diameter = 15 cm, length = 35 cm, Pb: diameter = 20 cm, length = 60 cm, W: diameter = 15 cm, length = 35 cm)-, are indicated by solid, dashed, and dotted lines.

at incident proton energies at 12 GeV, because at energies well above 1 GeV pion production becomes a dominating process [633] as already discussed. 8.2.4.1 Spatial Leakage Distribution of Neutrons and Target Shape The spatial distribution and intensity of the leakage neutron current from a thick target depend not only on the target radius or thickness but also on the target shape and of course, on the target material. Only an optimal leakage neutron spectrum in space and energy ensures a maximum neutron flux utilization for the spallation neutron source. The leakage current distributions are important for an optimum location of neutron moderators in wing position above and below the target (cf. Section 8.3). Figures 8.6, 8.7, and 8.8 [551] compare calculated axial distributions of leakage neutrons from cylindrical and flat targets of Hg and PbBi bombarded by 1.5 GeV [634, 635] and 1.334 GeV protons [636], respectively (Figure 8.8). The comparison of the neutron leakage flux distributions of Figure 8.7 shows different target geometries and beam shapes where neutron moderators may be located most efficiently. The flat targets give higher leakage neutron fluxes than the cylindrical ones, whereas the rectangular Hg target with a width of 14 cm and a height or thickness of 9.88 cm gives the highest leakage neutron flux. Figure 8.8 shows a comparison of calculated neutron leakage distributions of reflected ESS target geometries of mercury, and water-cooled tungsten and tantalum


Cylindrical targets (Ø [cm], L [cm]) (16.23, 60), (12.36, 60)

Rectangular targets (W × H × L cm3) (14 × 9.88 × 60), (24 × 6.44 × 60)

Proton beam (uniform distribution)

Proton beam shape Cylindrical (Ø [cm])

13.23, 9.36 10 × 6.88, 20 × 3.44

Rectangular (W × H cm)2

Fig. 8.6 Target geometries and beam shapes for neutron leakage distribution calculations. The calculations are given in Figure 8.7.

0.04 Neutron leakage current [neutrons cm−2 protons−1]

14W × 9.88H cm2 (Hg) 24W × 6.44H cm2 (Hg)

0.03

14w × 9.88H cm2 (Pb–Bi) 24W × 6.44H cm2 (Pb–Bi) f 12.36 cm (Hg) f 12.36 cm (Pb–Bi)

0.02

0.01

0

0

10

20

30

40

50

Distance from target surface [cm]

Fig. 8.7 Comparison of calculated axial neutron leakage distributions in (neutrons/cm2 /proton) of bare Hg and PbBi targets bombarded by a 1.5 GeV proton beam with a uniform proton current distribution [634].

(with a 10% average water content) [551, 636] bombarded by a proton beam of 1.334 GeV energy, 5 MW beam power and assuming an elliptical beam profile of 6 × 20 cm2 with a parabolic beam density distribution. The highest neutron current at the target surface is produced by the mercury target and is more than 30% higher compared to the tantalum target . Especially along the target surface, the current gradient in the case of mercury is lower than that of the tungsten target. This means that downstream moderators receive an even higher percentage of neutrons in case of mercury.

265


Total neutron leakage current J [neutrons cm−2 s−1]

266

6× 1014 Target materials Hg W Ta

4× 1014

2× 1014

0

0

20

40

60

Target depth [cm] Fig. 8.8 Comparison of calculated neutron leakage from a lead reflected mercury target and water cooled tantalum and tungsten targets [551, 636].

8.2.5 The Target Heating

The energy carried by the incident beam is dissipated in a spallation source target system in the following way: • target material heating, which is rather concentrated spatially and due to a variety of different deposition mechanisms occurring during the hadronic cascade, • production and emission of high energy particles, mainly neutron, escaping from the target, which deposit energy over a relatively large volume about the whole target station complex, • the binding and mass energies associated with secondary particle production. The amount of energy or heating power in the spallation target depends upon the target dimensions, but since the size of targets for spallation sources are designed to contain most of the cascade collisions approximate comparisons of various heating estimates can be made somewhat independent of the exact dimensions. In Figure 8.9, a comparison of target heating estimates for various spallation target concepts is given in terms of F, the fraction of the proton beam energy which goes into target heating. All of these results are from calculations except for the measurements for U-238 at 300 and 500 MeV and for tungsten at 500 MeV made at the Argonne ZING-P’ facility [637]. Most of the data points are from the calculations of Coleman and Alsmiller [638, 639] and the AECL calculations were made in connection with the Canadian Intense Neutron Generator (ING) study [58], both made for 10.2 cm diameter × 60 cm long geometry. The data point designated ISIS is an estimate by Rutherford Laboratory for a 10 × 10 × 30 cm3 target [640]; those labeled SNQ were computed in the German spallation neutron source study [601] for a ‘‘target wheel’’ geometry with a wheel diameter of 254.8 cm and thickness 10 cm. Results for other spallation source projects are as follows: the Argonne IPNS [641], the Rutherford and Appleton Laboratory ISIS [642, 643],

Fdeposition-heat (energy depositon in spallation targets) (beam energy)


2.2

= Aecl report = Coleman

ZING-P′ (MEAS.)

2.0

SNQ (U) IPNS

1.6

Depleted uranium targets

ISIS (U) ISIS (U)

1.2

ZING-P′(W) (MEAS.) SINQ (Pb-Bi) SNQ (Pb) TRIUMF (Pb-Bi) KENS (W) ¤ SNS (Hg)

0.8 0.4

Lead targets, except as noted

ESS (Hg)

JSNS (Hg)

0 0

500

1000

1500

2000

2500

3000

Incident proton beam energy [MeV]

Fig. 8.9 Fractional heat deposition as a function of the incident beam energy of different existing and projected spallation sources.

the Canadian TRIUMF facility [59], the Japanese KENS source at KEK [644], and the SINQ source [627, 645]. As is evident from Figure 8.9, the estimates for nonfissionable targets are rather consistent. The heating fraction estimates for depleted uranium, which are higher than for lead by factors of about 2 over the energy range from 500 to 1500 MeV because of the energy deposition from fission fragments, show much larger variation. The energy deposition for natural uranium targets is not as simply characterized because the contribution from fissions induced by low-energy neutrons becomes very important, making heating estimates more sensitive to target dimensions and to the presence of moderating materials around the target. Heating for natural uranium targets can be much higher than for depleted uranium (e.g., a fraction of about 6 for a very large natural uranium target and 1 GeV proton beam [268]). The maximum volumetric deposition or peak energy deposition during the pulse train inside the target of a pulsed spallation source depends upon the details of the hadron cascade, the incident beam profile, the pulse frequency and the pulse length. Results for the spatial distributions of energy deposition inside different targets of spallation sources are given in Part 3. 8.2.6 The Induced Radioactivity

Since the binding energy for ejecting a nucleon from the nucleus is about 10 MeV, and since spallation neutron sources need beam energies of about 1–2 GeV and heavy target nuclei with an atomic mass Atarget of about 200 or more, the mass numbers of residual nuclei from spallation collisions can have a wide range of values, extending to about 50 to 100 mass units below the target mass Atarget . These residual spallation nuclei are generally radioactive and neutron deficient relative to the valley of stability. Furthermore, for fissionable target materials, there are additional contributions to the radioactivity from high energy fission reactions,

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8 The Basic Parameters of Spallation Neutron Sources Induced radioactivity at saturation for an incident proton beam of an energy of 1.1 GeV and an average beam current of 5 mA.

Tab. 8.2

Radioisotope production yield per incident beam proton

Induced radioactivity at saturation Bq

Target material Pb–H2 O–Al 3.0

Pb–H2 O–Al 93.6 × 1012

Depleted U–H2 O–Al 15.4

Depleted U–H2 O–Al 960 × 1012

induced by nucleons ≥15 MeV and pions, and from fission induced by low-energy neutrons ≤15 MeV. Also, there can be significant contributions from other lowenergy neutron reactions, such as (n,γ ), (n,2n), etc., for certain target material compositions. One measure of the radioactivity produced by spallation neutron sources is the ‘‘saturation activity,’’ which corresponds to the total decay rate after a very long, e.g., infinite, irradiation at the time-irradiation stops, e.g., for a zero ‘‘cooling’’ time. For these conditions, the rate of decay is proportional to the production rate. Using the conversion factors 6.2 × 1018 protons/s per ampere and 1.0 decay/s per Becquerel1) , the saturation activity, in Peta Becquerel per milliampere of beam current, can be written as Asaturation = 6.24 · G · N ∗ (PBq/mA),

(8.6)

where N ∗ is the number of radio nuclides produced in the total volume, e.g., the target volume, per beam proton. Nominal values for N ∗ for spallation neutron source beam energies are about 3 for nonfissionable target materials and about 5 for fissionable target materials, e.g., depleted uranium (Armstrong et al. [646]). The factor G takes into account the contribution from the decay of daughter or subsequent products. The factor G ≈ 1 for nonfissionable materials, and about 2–3 or more for uranium. Thus, the general magnitude of the induced radioactivity for spallation neutron targets is Asaturation ≈ 18.7 (PBq/mA), nonfissionable targets Asaturation ≈ 62.4 (PBq/mA), fissionable targets.

(8.7)

As an example in Table 8.2 calculated production rates of radioisotopes by an intranuclear cascade model for a water-cooled lead and a water-cooled depleted uranium target [601] are given with the estimated corresponding ‘‘saturation radioactivity’’ using Eq. (8.6). There are several approaches that have been developed and used for estimating the radioactive species produced in spallation targets: 1) 1 Bq = 1 disintegration rate per second = 27 × 10−9 Ci = 27 pCi.

8.3 The Spallation Neutron Source Facility

• the use of measurements, • empirical formula, for ‘‘thin’’ target cross sections, e.g., partial microscopic cross sections for particular spallation product nuclei, and then the integration of these energy-dependent cross sections over thick target particle spectra calculated by a hadronic cascade code, • the use of Monte Carlo methods and the intranuclear cascade model to directly obtain the isotopes produced during the hadronic cascade and depletion codes with evaluated decay libraries. The semiempirical formula for spallation product isotopes, based on sparse experimental data and parametric fits, such as the Rudstam formula [42] or extensions thereof [45, 47], however, can be very inaccurate by an order of magnitude or more. Therefore, Monte Carlo methods are likely to be more accurate coupled with depletion codes and considering evaluated data libraries. Details of these methods [647–651] are given with applications on high-intensity spallation neutron producing targets in Part 3.


A spallation neutron source is usually built of two different systems, a particle accelerator which accelerates charged particles, mainly protons, of energies from about 0.4 GeV up to several GeV and a target station where the charged particle beam interacts with a heavy metal target and produces via the spallation process (cf. Section 1.3.4 on page 21) mainly neutrons and other particles, protons, pions, etc. All these particles are the source for further applications. Without going into detail, Figure 8.10 illustrates possible arrangements for accelerator–target–station systems to produce high-intensity neutron beams. Linear accelerators are an attractive choice to produce the required high-intensity proton currents. They operate most efficiently with duty cycle factors of a few percent in long pulse or continuous wave (cw) mode. Also the development of rapid cycling synchrotrons have enabled their use as high current accelerators. For short pulse spallation neutron sources either a full energy linear accelerator feds an accumulator or a storage ring or a rapid cycle synchrotron is used for proton beam extraction with a cycling frequency between 10 and 60 Hz and pulse duration in the range of ns–µ s. A spallation neutron source fed directly from a pulsed linear accelerator with pulse length of several milliseconds is referred to as a long pulse source. Continuous spallation sources are fed from accelerators with no-macro time structure such as a cyclotron or a cw linear accelerator. The only existing source of this type is the 1 MW SINQ spallation source at PSI, Switzerland [627]. Continuous spallation sources are favorable to accelerator-driven systems (ADSs) as accelerator transmutation of radioactive waste (ATW)s and energy amplifying systems (EASs). The main characteristics of existing spallation sources are discussed in Chapter 17 on page 505 and summarized in Table 17.17 on page 559.

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270


Compressor ring

Linear accelerator

Ion source (a)

Short-pulse target station

Long-pulse target station

Neutron beam lines Ion source

Injector

(b)

Rapid cycling synchrotron

Short-pulse target station

Fig. 8.10 Principle of accelerator–target–station arrangements of spallation neutron sources. Reflector: - Material (Be, C, Pb) - Cooling (H2O, D2O) - Openings, beam holes - Dimension

Target: - Material (Hg, W, Ta, Pb, U) - Cooling (H2O, D2O) - Dimension, geometry

Proton beam: - Energy - Beam profile

Moderators: - Material (H2O, H2, methane, etc.) - Temperature (300 K, 100 K, 20 K) - Premoderator, coupling / decoupling, poisoning

- Position on target, wing, flux-trap, composite - Dimension

Fig. 8.11 The schematics of a TMR arrangement and the main parameters determining the optimal magnitude of the neutron flux.

Figure 8.11 illustrates the principal arrangement of a target-moderator-reflector system (TMR) to produce thermal or subthermal neutron beams via moderation of fast neutrons generated by a high-intensity proton beam of GeV energy impinging on a heavy metal target. The TMR systems is embedded in a thick bulk shield with openings for neutron beam lines guiding the neutrons to the experiments (cf. Figure 8.10). The different parameters which determine the optimal neutron utilization of the TMR system are also depicted in the figure. It is obvious that the construction and the optimization of a TMR system and in general of a spallation

8.3 The Spallation Neutron Source Facility Wing geometry Moderator Target Proton Beam Neutron current to expreiments Slab geometry

Flux-trap geometry

Slab configuration

Proton beam

Fig. 8.12

Slab target configuration with two wing moderators

Basic target–moderator configurations.

neutron source is a very complex process concerning the possible parameter space given in Figure 8.11. As shown in Figure 8.12, there are about three basic target moderator configurations to obtain thermal or cold neutrons from a moderator, the so-called wing-, slab-, or flux-trap configuration [652]. The wing geometry configuration together with a slab target is widely used at the existing or projected high-intensity spallation neutron sources, SNS, USA, J-PARC, Japan, and ESS, Europe (cf. Table 17.17 on page 559 and more in detail in Part 3). It should be noted, that moderators in a slab geometry configuration provide about two times higher neutron fluxes [635]. No experiment views the target directly, therefore a background contamination in the low-energy neutron beam by high energy neutrons from the target may be minimized. In combination with a slab target system, as shown in Figures 8.6 and 8.12, additional moderators with lower intensity could be positioned [551].

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272


Three classes of spallation sources can be distinguished depending of the accelerator parameters as the driving source, the layout of the target station and the considered applications: continuous, long-pulse, and short-pulse spallation sources [57].

8.3.1 Continuous Spallation Neutron Sources

The goal of these sources is to produce a high average neutron flux. Therefore, an accelerator with no macro time structure of the proton beam is used. The use of low-absorption material around and in the TMR system is crucial to produce a long life time of the moderated neutrons and a high thermal neutron flux. Such a system is comparable to a high-flux reactor core-reflector arrangement as ILL [653]. Typically, a 2-m diameter D2 O system acting as moderator and reflector surrounds a target system of low-absorbing target material, e.g., PbBi. The only existing continuous spallation neutron source of this type is the SINQ source [627]. Some issues of continuous spallation sources are also relevant in ADS. Details are given in Part 3.

8.3.2 Short-Pulse Spallation Neutron Sources

Their characteristic feature is that they generate fast neutron pulses. These sources favor neutron application techniques with TOF experiments and are working in pulsed mode with a high neutron peak flux and a short pulse duration at an appropriate repetition rate (cf. Table 17.17 on page 559). Figure 8.13 shows as an example a typical pulse shape of a short neutron pulse with a FWHM of about 100 µ s. The optimization criterion of this type of neutron source is a short pulse duration without disproportionate losses in the peak flux intensity. Examples of these sources are the spallation neutron sources IPNS, ISIS, MLNS, SNS, JPSNS-JPARC. Details are given in Part 3.

8.3.3 Long-Pulse Spallation Neutron Sources

The pulse width of the produced neutron pulse of these sources is essentially determined by the proton pulse duration of the accelerator. Figure 8.13 shows as an example a typical pulse shape of a long neutron pulse with a duration of about 2 ms. High-intensity time averaged neutron flux is an important optimization criterion and a flat top pulse shape as shown in Figure 8.13. Therefore, it is important to find a compromise between high-peak flux and neutron lifetime in the moderator.

Neutron current density [n/cm2 sr Angström]


5× 1013 Peak intensity

4× 1013 Short pulse Exponential pulse decay with decay constant t

3× 1013

2× 1013

Long pulse

1× 1013

0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Time t [ms] Fig. 8.13 Examples of typical pulse shapes of short- and long-pulse spallation neutron target stations.

8.3.4 Scattering of Neutrons by Matter

The pulsed spallation sources make use of the pulse state of the produced neutrons using TOF methods (cf. Figure 8.14). Their merits can be shortly characterized as follows: • A range of wavelengths and corresponding energies of the neutrons can be used at the same time. • The instantaneous luminosity is large enough for signals to be detected with high precision. • It is easy to perform neutron scattering experiments under special condition, e.g., such as pulsed magnetic fields and high pressures. Figure 8.14 illustrates the principle of a neutron experiment applying the time-of flight method. A pulse of protons is injected into a spallation target every 50 ms, shown by the lines of the upper part of Figure 8.14. Accordingly, the neutrons are also produced every 50 ms. The neutrons produced by the spallation process have a broad range of energies. It is possible to determine their energies and wavelengths by using the pulsed nature of the source. The neutrons with energies E1 , E2 , and E3 in Figure 8.14 leave the source at the same time but with different velocities. The neutron with energy E1 is the first one to reach the detector because it has a higher velocity than the neutrons with energies E2 or E3 . Using the distance to the target and the time the neutron reaches the detector, it is possible to determine the velocity, and hence the wavelength and energy of the neutron. The method uses the simple equations as shown in the insert of Figure 8.14.

273


20 Hz 50 ms

Assuming thermal neutrons at T = 293 K:

Proton pulse injection

V = L/t = 2200 m/s E = 1/2 mv2 Energy: = 25.3 meV Wavelength: l = h/mv = 1.798 Å Velocity:

Distance L from the target [m]

274

E1 > E2 > E3

E1

E2

E3

0.0

50.0 Time t [ms]

Fig. 8.14

The neutron time-of-flight method.

The relation between energies and wavelengths of neutrons makes it possible to obtain microscopic information about matter. This depends on the sample which will be analyzed by the low-energy neutrons in the meV energy range. Basically from elastic or inelastic scattering processes of neutrons with matter, information is obtained to determine the relation between the wavelength and energy of scattered neutrons emitted from the sample. As shown from the insert of Figure 8.14, the neutron wave properties obey some relations of quantum mechanics as already given in Section 1.3.5.4 on page 30 which are shortly repeated here. 2π λneutron 1 2 energy: E = mneutron · v 2 = k2 = · ω 2 2mneutron 2π · h neutron wavelength: λneutron = = mneutron · v mneutron · v 3950 ˚ (A), or ≈ v momentum: mneutron · v = p = · k =

(8.8)

where k is the wave vector, mneutron is the neutron mass given in 939.56 MeV/c2 , = h/2π = 6.58212 × 10−22 (MeV s), and λneutron in A˚ is the wavelength of the neutron given by its momentum. With the formulas given in Eq. (8.8) the wavelength of the neutron can be written in a abbreviation form of Figure 8.15 considers the basic ingredients of a neutron scattering experiment [654, 655]. A


Neutron scattering experiment Detector bank Pulsed source from moderator

ki ,

ki ,

Ef

L2 Scattered neutrons

Ei L1 Sample

With the scattering triangle kf

Scattering direction Q,w

Beam direction

2q ki

Fig. 8.15 The principle of a neutron scattering experiment with the use of low-energy neutrons to analyze the structure of a matter.

neutron scattering event can be described by six independent parameters for the initial k initial (kx , ky , kz ) and final neutrons k final (kx , ky , kz ) with the energy E as a function of k . The function Q is known as the scattering vector depicted in the scattering triangle given in Figure 8.15. It represents the momentum transferred to the sample with the momentum conservation Q = k i − k f . Thus, a neutron spectrometer must be able to determine the k i , k f . A neutron scattering experiment measures the number of scattered neutrons as a function of (Q , ω). The parameter space is 4D. The intensity or number of scattered neutrons is proportional to the scattering function S(Q , ω) which depends only on the sample and not on the neutron spectrometer. The scattering function S(Q , ω) gives information about the structure of the sample, whereas the variable ω gives information about the dynamics or motion as elastic, quasielastic or inelastic depending of the momentum Q transferred to the sample by the interacting neutrons. Figure 8.15 illustrates the principle of a neutron scattering experiment. In both elastic and inelastic scattering events, the neutron is scattered through the angle 2θ , Figure 8.15. For elastic scattering, the scattering vector Q can be calculated by the following simple trigonometry relation:

sin θ =

Q/2 k

Q = 2k · sin θ =

4π · sin θ . λ

(8.9)

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276


kf kf

Q

Q

kf

Q

ki ki

ki ki = kf

ki > kf

ki > kf

w=0

w>0

w>0

Elastic scattering

Inelastic scattering (w can vary independly of Q)

Fig. 8.16 The scattering triangles showing the relation between elastic and inelastic scattering.

Comparing elastic versus inelastic scattering the following relations between the wave vector components ki , kf , the momentum Q, and ω are shown by the scattering triangles of Figure 8.16. The energy conservation is given by the following relation: Eneutron = −Esample Esample = Einitial − Efinal =

2 · (k2initial − k2final ). 2m

(8.10)

From the scattering triangles in Figure 8.16 it is shown that in an elastic scattering event in which the neutron is deflected, the neutron does not lose or gain energy (so that ki = kf ) and in inelastic scattering events in which the neutron either loses energy (so that ki > kf ) or gains energy (so that ki < kf ) during the interaction, respectively. The issues on moderator neutron performance, moderator technology, research with neutrons and examples on neutron instrumentation for spallation sources are described in the experimental Part 2 and in the application Part 3.

Part II Experiments

279

9 Why Spallation Physics Experiments? 9.1 Introduction

Contributions in the field of experimental spallation research have been extended and accumulated in the literature over the past 40 years in a vast but scattered array of publications in journals, conference proceedings, in reports and monographs. The following part overviews the most important experiments in spallation research, which were the driving source for the theoretical understanding of the physics of spallation phenomena and have supported applications and developments of spallation neutron sources, of accelerator technology, of accelerator-driven sources as ATWs or energy amplifiers, of detectors for medium- and high-energy particle physics, of space science and astrophysics, etc. The overview given here can only be a selection of some topic experiments, which may be partial to the authors. In the following sections, some arguments are given for the considerable needs to investigate spallation physics by experiments.

9.2 Application-Driven Motivation

Spallation neutron sources for condensed matter research, nuclear transmutation or new concepts for energy production consist basically of a high-energy and highintensity proton accelerator and of target systems or complex target stations for the neutron generation. The spallation neutron sources exploit the thermal excitation of the heavy target nuclei by GeV protons and their subsequent de-excitation by evaporation of neutrons mainly with energies of a few MeV. Typically only 20% of the incident kinetic energy is dissipated in the first reaction into intrinsic excitation giving rise to the emission of about 15 neutrons (for 2 GeV protons impinging on a Pb target, as for example). The larger fraction of the energy, instead, is carried off during the initial intranuclear cascade (INC) by a few energetic (hundreds of MeV) particles, mainly nucleons. These cascade and pre-equilibrium particles in turn initiate further nuclear reactions in the extended target, thereby increasing the total number of neutrons created per incident proton to about 40. Handbook of Spallation Research: Theory, Experiments and Applications. Detlef Filges and Frank Goldenbaum Copyright  2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40714-9

280

9 Why Spallation Physics Experiments?

The best proton energy for the accelerator is still subject of some consideration. From a material’s point of view (e.g., radiation damage) it seems advantageous to use higher energies (some GeV) with the benefit of lower currents, which could also be reasonable from the physics point of view, if the neutron production was indeed to grow linearly with incident proton energy. The latter assumption is not unconfined true for energies far beyond 1 GeV as already discussed in Section 8.2.4 on page 263. One goal of the experiments is to increase the reliability of the models in the GeV energy range where only few and moreover quite diverging data exist. The economic design and operating of new high-current proton accelerators and spallation neutron sources is strongly dependent on the accurate knowledge of neutron, proton, pion, and radioisotope production data, since these reaction products are on the one hand important for the optimal layout of all the systems and on the other hand cause the main technological problems related to shielding, radiation damage, safety, and maintenance aspects. More specifically, it is the aim to accomplish spallation physics experiments for various proton energies in the range from 0.2 up to several GeV to investigate, e.g., • the double differential production cross sections d2 σ/dEd for neutrons, for light particles (p, d, t, and He isotopes), and residual spallation products and their radioactive species of target, window, and structural materials (cf. Section 10.2 on page 288), • the multiplicity distributions of n, p, d, t, and He from the nuclear-evaporationcascade production (cf. Section 10.3.2.2 on page 342), • the neutron multiplication in thick targets, e.g., neutron leakage multiplicity distributions and total inelastic cross sections for ‘‘thick’’ targets of various dimensions, where the incident energy is converted as efficiently as possible into neutron production yield (cf. Section 11 on page 379), • the heat deposition and induced radioactivity in targets and structural materials (cf. Chapter 11 on page 379), • the features and the neutron behavior of ambient temperature water moderators and advanced cold moderators (cf. Chapter 13 on page 431), • the efficiency of shielding in particular for high-energetic neutrons (cf. Chapter 15 on page 481). The experimental efforts contribute to the increasing confidence and in fixing the parameters in the nuclear models currently available. Recently at an international conference on nuclear data for science and technology, an overview about the progress in nuclear data accelerator applications and accelerator-based facility design was given by Goldenbaum [656] and Takada [657]. 9.3 Space Science and Astrophysics-Driven Motivation

Spallation reactions are also important from an astrophysical point of view (cf. already discussed in Section 1.2 on page 5). A large variety of stable and radioactive

9.4 Nuclear Physics Driven Motivation

nuclides is produced by the interaction of cosmic rays and solar particles with extraterrestical matter. To interpret these cosmogenic nuclides in lunar samples, in meteorites, in cosmic dust, and in dense matter, a detailed knowledge of the cross sections of the respective nuclear reactions is necessary. The basic experiments are the measurement of integral excitation functions and depth-dependent radionuclide profiles induced by protons and alpha particles in matter. Experiments can be performed at proton accelerators considering proton energies similar to the energies of the cosmic or solar ray spectrum incident on targets of matter made of elemental compositions identical to extraterrestical matter. Also the measured induced radioactivity in various spacecraft components provides information on the radiation exposure and about single-event-effects (SEE) concerning materials damage in space electronics. The simulation of shielding experiments concerning cosmic ray energies and particle spectra at proton accelerators can be used as part of the safety considerations in the design and operation of long-term space missions, e.g., manned missions to Mars. Experiments in the field of cosmogenic nuclide production and space science are discussed in Chapter 23 on page 653. Spallation reactions and experiments are in addition also important from another astrophysical point of view. Due to their low binding energies Li, Be, and B nuclei are highly unstable especially at temperatures and pressures encountered during stellar nucleosynthesis. The experimental determination of the spallation cross sections for C, N, and O targets will provide a valuable data set to improve the understanding of the anomalous abundance of light elements in the cosmic rays and astrophysical questions of nucleosynthesis of the light nuclei in general. As for example the abundance of Li, Be, and B in cosmic rays is only several times smaller than the abundance of neighboring nuclei C, N, and O, whereas this abundance is 5–6 orders of magnitude smaller when observed in the solar system (the ‘‘LiBeB puzzle’’) (cf. Section 1.2 on page 5). However, to account for all details of the relative abundances of all isotopes on the ground of astrophysics models it is required to thoroughly test their predictions. Therefore, not only the values of the total production cross sections, relevant in the context of the abundance problem have to be determined experimentally, but also the energy distributions should be known precisely enough to allow checking of the applied models. The experiments described in Chapter 23 on page 653 are an approach to fill this serious lack of such data in the intermediate energy range.


Properties of hot nuclear matter can not be described by elementary nucleon– nucleon scattering, because even knowing the hadronic interactions the solution of the many-particle system would cause serious problems for heavy nuclei. Instead generally matter is described by macroscopic observables like temperature, density, and pressure. For infinite nuclear matter, the relation between thermodynamic observables pressure, energy, density ρ, and temperature T is given by the equation

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282


of state [190]. Except of the saturation density ρ0 and the energy at this density the form of the equation is not well understood. When expressed in form of a caloric equation of state the energy EB per Baryon is written as EB = E(ρ, T) = ET (ρ, T) + EC (ρ, T = 0) + E0 (ρ = ρ0 , T = T0 ), A

(9.1)

with the thermal energy ET and the compression energy EC taking the dependency on ρ and T into account and E0 reflecting the energy in the ground state. The relation between the equation of state and the mean nuclear potential U is expressed by the kinetic energy density tv and the density of the total energy hv : EB /A = hv /ρ ∂ (hv − tv ). The equation of state for nuclear matter gives, e.g., insights and U = ∂ρ in phase transitions. At low density (ρ < ρ0 ) and high T due to the superposition of nuclear and Coulomb forces, nuclei can coexist in liquid and gaseous phase analogous to the Van-der-Waals gas. For extremely high T and ρ a quark–gluon plasma is expected. In this new phase of nuclear matter the quarks are supposed to be (quasi-)free. With the energies typically discussed in spallation reactions in this book (few GeV), neither T nor ρ of such high-density phases is approached and the nuclear matter in the form of an ensemble of A nucleons is neither infinitely expanded nor to be reduced to nucleon–nucleon interactions. However, also on the way to this high-density phase numerous interesting phenomena exist. The nuclear physics aspect concerns the decay modes (as being discussed in Chapter 3) of very highly excited nuclei and has been intensively investigated during the last decade mostly with heavy-ion accelerators – with moderate success, however. At high excitation energy one expects more diverse decay modes than evaporation and fission (Section 3.7) to become accessible to the nucleus: multibody fragmentation (Section 3.10) with the emission of many intermediate-mass fragments (IMF), and, when the excitation exceeds notably the binding energy of the nucleus, also even more violent vaporization (Section 3.10) into single nucleons and light nuclei not heavier than alpha particles (complete disintegration of the nuclei into light fragments). In the past heavy-ion collisions in the energy range of up to several 100 MeV/A have often been investigated [658–660] and new decay phenomena or novel and relatively scarce modes have indeed been observed. These exotic modes might be due to the unique feature of nuclear matter – namely the superposition of short range nuclear forces and extremely long ranging Coulomb forces. However at the same time these modes represent the superposition of statistically and dynamically driven fragmentations. Also the exact definition of the decaying source from the correct theoretical description point of view is rather ambitious. These phenomena raise a variety of open questions: Is the multifragmentation driven by thermal excitation of the nuclei, by repelling Coulomb forces or by deformations and high spins? The interpretation of these sequential or possibly simultaneous decay modes requires a clear distinction of statistical and dynamical fragmentation. The driving forces for these fragmentations still remain obscure,


because too many dynamic distortions are inevitably introduced by the heavyion reaction together with the thermal excitation, like large angular and linear momenta, density compression or the formation of peculiar unstable noncompact shapes. Reactions with protons, instead, are much less likely to induce collective excitations in the target nucleus and may thus allow to come closer to what is called statistical fragmentation: the decay of a compound nucleus stocked with purely thermal and equilibrated excitation. The knowledge of these purely statistical decay mechanisms is a prerequisite or key to the understanding of the more complicated effect of collective excitations on the fragmentation. Fundamental properties of hot nuclear matter like heat capacity, specific heat, viscosity and phase transitions are by far not thoroughly explored. Since in this kind of reactions a minimum of compression to the nucleus is induced, the experimental investigation of Eq. (9.1) is enabled for high T at densities ρ ≈ ρ0 . In order to reduce the influence of the entrance channel on the decay modes also the nuclear excitation following annihilation of energetic antiprotons has been investigated as will be illustrated in Chapter 12 on page 425. Detailed discussions on antiproton-induced reactions can also be found in Refs. [379, 661–670]. In short, antiprotons annihilate on a single nucleon at the surface of, or even inside the nucleus, thereby producing a pion cloud containing an average of about five particles. Due to the high center-of-mass velocity (βc.m. ≈ 0.6 in GeV antiprotoninduced reactions) of this cloud it is focused forward into the nucleus. Since the pion momenta are comparable to the Fermi momentum of the nucleons in the nucleus, the pions heat the nucleus in a soft radiation-like way [671], even softer and more efficient than in proton- or other light-ion-induced spallation reactions, which have also been exploited recently for this purpose [114, 219, 622, 672–675]. Due to the small radius of interaction volume of 1.8 fm and a coherence length of cτ ∼1.5 fm in elementary NN annihilation reactions extremely high local energy densities are obtained. For antiproton-induced reactions, INC calculations have been found to provide a reasonable description of the underlying mechanism. They predict that the spin remains low (below maximum 25) and that shape distortion and density compression are negligible [676], in contrast to what is expected in heavy-ion reactions. The reaction time for the achievement of equilibrium conditions is only about 30 fm/c or 10−22 s [169], which is much shorter in general than the dynamical period in heavy-ion reactions [677]. This is all the more important at high temperature (T ≈ 6 MeV) when the characteristic evaporation time reduces to t < 10−22 s, implying little cooling of the compound nucleus during heating. Summarized light-particle-induced reactions – and therefore also protons generally employed in spallation reactions – represent the softest way of producing hot nuclei with the advantage of: • small compression and shape distortions; • small transfer of linear and angular momenta; • good definition of mass and charge of the decaying compound nucleus;

283

284


• fast excitation (much faster than nonrelativistic heavy- ion collisions. Therefore, only little cooling during the formation of the hot nuclei). • existence of reliable reaction models. From previous experiments at Saturne/Saclay [678] (cf. Section 10.2.2.1) and at LEAR1) /CERN2) [379, 663] it is known on the other hand that light projectiles are less effective in energy dissipation than heavy ions and that therefore even with, e.g., the maximum energy at COSY, 2.5 GeV, only about 1000 MeV of excitation with reasonable cross section in heavy target nuclei and somewhat less excitation (but higher temperatures) in lighter nuclei (cf. Chapter 10) is reached. Thus fission and evaporation in heavy fissile nuclei and the new fragmentation phenomena, or rather their onset, in lighter nuclei can be explored. Even the conventional decay modes, evaporation and fission, studied as function of excitation energy become new territory above 200 or 300 MeV of excitation: First, the occurrence of fission up to about 1000 MeV of excitation indicates, by its slow collective nature, that the nucleus has survived this tremendous excitation as a self-bound system which moreover has reached thermal equilibrium [379, 663, 665]. Beyond that, from the competition with the faster and well-known particle evaporation a time scale can be established for fission (and this separately for the motion from equilibrium deformation to the saddle point and from the saddle to scission) [62, 381, 403, 665]. This fission time is related to one of the basic properties of nuclear matter, the nuclear viscosity or dissipation (cf. Section 3.7). The multifragmentation or vaporization phenomena can be observed in lighter nuclei like Cu or Ag [670]. Here it seems important not only to observe the phenomena and deduce a probability or cross section as a function of E ∗ for them, as has been mostly done in the past, but also to specify the phenomena more closely. In this respect, a particular advantage of the NESSI-charged particle detection system described in Chapter 10 can be exploited, namely that it registers not only light particles and lighter fragments (IMFs), but also all heavier residues from each reaction. This allows to built a complete mass- and linear-momentum balance for each reaction and thereby to test the completeness of the multifragmentation process. The need for complete measurements of IMFs at these energies is emphasized by the lack of data measured at incident proton energies below 1 GeV. Drawing conclusion on features of nuclear matter from experimental observations is possible only by an intense comparison between the experiments and the theoretical descriptions. Only using theoretical models assumptions concerning nuclear matter and their influence on observables can be tested. Furthermore, models provide to some extent an insight into the dynamics of processes which is generally scarcely or not possible with the ‘‘static’’ observables accessible in the experiments. Therefore, the primary intention of the experimental investigation in nuclear physics is to provide exclusive data rather than only to improve the database of 1) LEAR – Low- Energy-Antiproton-Ring.

2) CERN – Centre Européen pour la Recherche Nucléaire, European Laboratory for Particle Physics.


inclusive cross sections. With these exclusive data it is possible for instance not only to reconstruct for each initiated reaction the distribution of the deposited thermal excitation energy [663], but also to investigate pre-equilibrium emission for peripheral and central collisions. This in turn allows to test a variety of critical model parameters (cf. Chapter 3) which, e.g., determine the equilibration point after the fast intranuclear cascade stage and define the transition from the INC model to an evaporation model for the statistical decay of the equilibrated nucleus (cf. Section 2.3.1.7).

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10 Proton-Nucleus-Induced Secondary Particle Production – The ‘‘Thin’’ Target Experiments 10.1 Introduction

With the advent of the first spallation neutron sources KENS at KEK, Japan [644, 679], at IPNS at ANL, USA [641], at LAMPF-WNR at LANL, USA [680], and the high-intensity spallation source project SNQ Jülich/Karlsruhe, Germany [142] however, much more strong constraints to the quality and accuracy on spallation reaction data, especially on differential cross-section data, are required to consider the performance of the design parameters. In the context of new sources with high beam intensity, which are built, constructed and planned e.g. SINQ, Switzerland, [627], SNS, USA [681], J-PARC, Japan [629], ESS, Europe [480, 551], numerous projects for accelerator-driven systems (ADS) [483, 682], and high-power targets, e.g., EURISOL [683] more accurate data and a more detailed description of the underlying particle reaction processes are needed [656]. Recently a new neutron facility – the n TOF facility – became available at CERN, Switzerland, well suited for high-quality measurements for capture and fission data for fertile and fissile isotopes for ADS applications. The n TOF facility produces intense neutron beams with a wide energy range of 1 eV to 250 MeV by spallation of a 20-GeV proton beam from the CERN-PS accelerator on a lead target [684–686]. ‘‘Thin’’ target cross-section measurements reduce the uncertainties on the nuclear and other parameters upon which the design and operation of high-intensity spallation sources are based. This demand led to a worldwide effort to extend the experimental database on spallation-induced reactions, e.g., on differential cross sections and multiplicities of neutrons, protons, pions, light charged particles, and residuals nuclei measured over a broad range of target masses, emission angles, and secondary energies well suited to provide a better understanding of the spallation process. Experiments of this kind are called ‘‘code validation experiments’’ using so-called thin targets where the energy loss of the incident proton beam is very small compared to the proton beam energy. The ratio should be of the order Eloss /Eproton ≤ 5%. The produced secondary particles inside the target generally do not undergo further collisions inside the target. In contrast in ‘‘thick’’ targets the energy loss of the incident beam could be large and the produced secondary particles undergo further collisions. These collisions create additional secondary particles and result in an internuclear cascade (INC) (cf. see also Section 1.3.4 on Handbook of Spallation Research: Theory, Experiments and Applications. Detlef Filges and Frank Goldenbaum Copyright  2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40714-9

288

10 Proton-Nucleus-Induced Secondary Particle Production

page 21). The experiments presented in the following section are considered to be a representative selection of ‘‘thin’’ target measurements, although they reflect only a small selection of the numerous investigated ones. Thick target measurements will be discussed in Chapter 11.

10.2 Neutron, Pion, and Proton Double Differential Measurements and Experiments

Above the pion production threshold the majority of existing differential data are charged particle data, which experimentally are easier to detect than neutrons. The detection of neutrons with kinetic energies above several 100 MeV is not trivial caused by the low efficiencies and low time resolutions of suitable detector systems. Until 1980, secondary double-differential neutron measurements at incident proton energies above 100 MeV were rather scarce and cover only a few points in the double-differential scale [125, 126, 687–690]. More systematic studies and measurements were started as part of a study project for the SNQ spallation source [142] by Cierjacks et al. [691] at the PSI cyclotron in 1981. The publication and the controversial discussion concerning the high-energy resolution of these measurements (cf. Refs. [127, 128, 692] led then to a collaborative effort between LANL, Los Alamos, USA, and the research center Jülich, Germany, to measure systematically double-differential neutron-production cross sections of various targets between 0.1 and 0.8 GeV at the Los Alamos LAMPF-WNR facility during 1984–1990. A second research program concerning neutron double-differential production at incident beam energies higher than 0.8 GeV was started at the SATURNE accelerator in France, mainly as a collaboration between Bruyères-le Châtel, SATURNE, Saclay (DAPHNIA), France, and the Uppsala University, Sweden in 1994 [693]. Incident protons and deuterons on different thin target in an energy range between 0.8 and 1.6 GeV were used. The main aim of these investigations was to measure neutron-production cross sections for ADS applications. In addition, during 1995–1998 at the 12 GeV proton synchrotron of KEK, Japan, double-differential neutron-production measurements at incident proton energies from 0.8 GeV up to 3.0 GeV were measured for various targets for applications concerning the design and development of the 3 GeV spallation neutron source at J-PARC in Japan [265, 694] currently being commissioned. There are only few experiments where pion double-differential cross sections in proton–nucleus collisions were measured, which are useful to validate spallation reaction models. This early measurements were originally accomplished to provide sufficient data to allow the optimal design of secondary pion and muon beams at three proton beam ‘‘pion factories’’ as LAMPF, USA, TRIUMPF, Canada, and SIN (now PSI), Switzerland, during 1970–1980. Measured charged pions produced by incident proton energies of 0.585 GeV and 0.730 GeV on various thin targets were reported by Crawford et al. [695] and by Cochran et al. [124].

10.2 Neutron, Pion, and Proton Double Differential Measurements and Experiments Start time t0

Stop time tL

Neutron kinetic energy En

L t = tL - t0 = time to traverse distance L

Fig. 10.1 The kinetic energy Eneutron of the neutron is measured by the stopping t to traverse the flight path of length L.

10.2.1 The Double-Differential Neutron-Production Measurements at the LAMPF-WNR Facility

During the years 1984–1990, a physics program was implemented to measure double-differential neutron-production cross sections at the Los Alamos LAMPFWNR time-of-flight (TOF) facility [680] of various targets up to incident proton energies of 800 MeV. The principle of high-energy TOF measurements of neutrons will be described and the application of this method to measure high-energy neutron-production cross sections will be demonstrated. The principle of the TOF method for relativistic velocities β = v/c is given in Figure 10.1. The kinetic energy of the neutron is measured by stopping the time t to traverse the flight path of lengths L. The kinetic energy Eneutron is then derived using relativistic kinematics by the formula (10.1) Eneutron = m ·

= 939.6 ·

1

1 − β2

−1 1

1 − 11.13[L2 /t2 ]

−1 ,

(10.1)

where Eneutron is given in (MeV), L in (m), the time t in (ns), and m = 939.6 MeV/c2 the mass of the neutron. The relativistic expression of Eq. (10.1) is required above approximately 10 MeV as Figure 10.2 demonstrates. Also from Figure 10.2 it follows that there is certainly a limitation to the TOF method at high energies. For neutrons this is in the several GeV kinetic energy region, where the speed of a neutron is almost same as the speed of light. As an example, typical flight times – as inverse velocities – are 102.286 ns/m at 0.5 MeV, 23.045 ns/m at 10 MeV, and 3.964 ns/m at 800 MeV. The time scale in the experiments is calibrated against the arrival of the prompt gamma flash produced by the incident protons from the target. The inverse speed of light is 3.336 ns/m. For neutrons, the start time has to be taken from the accelerator pulse arrival because neutrons have to be detected by nuclear recoils produced in the

289


104 Non relativistic

102 101 100 10−1

100 10 1 7.8

1

23.0

10 Inverse speed [ns/m]

72.4

103 Inverse speed of light

Kinetic energy [MeV]

290

100

Fig. 10.2 The kinetic energy of a neutron versus its inverse speed. The inverse speed of light is 3.336 ns/m.

detector and, therefore, change their energy by the method of detection. Generally organic scintillators are used as stop detectors with hydrogen-to-carbon ratios of approximately one. In these detectors, neutrons are detected by proton recoils, predominantly at the lower energies, or by inelastic reactions with carbon nuclei. The used detectors are given in Tables 10.3 and 10.4 on pages 297, respectively. Because the slowest neutrons from one initiating process could be mistakenly identified as fast neutrons in the following initiating process, the time spacing between two successive beam bursts has to be long enough to eliminate such misidentifications. For organic scintillation detectors, the practical lower limit of detection is approximately 0.5 MeV which is just above a typical photomultiplier equivalent noise. From this the micropulse spacing required follows as 102.286 ns/m times the length L of the flight path, i.e., approximately 1 µs per 10 m. Employing the experimental uncertainties in total time and flight path uncertainty, the fractional energy resolution of the TOF method can be estimated by the following approximate formula [127]: Eneutron = γ · (γ + 1) · [(L/L)2 + (t/t)2 ]1/2 , Eneutron

(10.2)

where γ = 1/(1 − (v/c)2 )1/2 is the Lorentz contraction factor, and L and t are the flight path length and the neutron flight time, respectively. At nonrelativistic energies γ ∼ 1, the energy resolution is two times the time resolution for (L = 0) and at high relativistic particle energies, the resolution is quadratic in energy. 10.2.1.1 The Time-of-Flight Experiment at the LAMPF-WNR Facility Looking at the accelerator-specific needs for high-energy neutron TOF experiments, the Los Alamos WNR facility is unique in the world because all requirements as the beam intensity and beam time structure are met. An artist’s view of the WNR facility is shown in Figure 10.3. Proton pulses are accelerated in the LAMPF linear accelerator up to an energy of 0.8 GeV. A certain part of the beam is diverted by a


Los Alamos National Laboratory WNR Facility 4 3

6 2 1

5

1. High-current target area 2. Low-current target area 3. Proton storage ring 4. Control/data center 5. Neutron time-of-flight path 6. Proton beam from LAMPF Fig. 10.3

5

7

An artist’s view of the WNR facility (Courtesy LANL).

set of kicker magnets down to the WNR area and handed over to the WNR control system (cf. Figure 10.3 label 6). The beam is then transported to the low current target area (label 2 in Figure 10.3) for the high-energy neutron TOF experiments. The characteristics of the proton beam for TOF experiments at the WNR Figure 10.4 shows, in general, the beam structures of the LAMPF accelerator and the WNR macropulses and micropulses. The LAMPF accelerator delivers of macropulses 800 µs in length at rates up to 60 Hz and consists a series of micropulses separated by 5 ns [696]. Each macropulse contains up to ∼ 1.6 × 105 micropulses. These macropulses are repeated at a rate of 120 Hz. Each micropulse has a full width at half maximum (FWHM) of about 80 ps at LAMPF accelerator but is broadened to approximately 180–200 ps at the WNR facility. For the neutron TOF experiments, a low-energy chopper has to be used to select micropulses separated by 12 µs, which prevents frame overlap at the longest flight path for neutrons above a 250 keV minimum energy. By using this fast chopper located in the injector area of LAMPF accelerator individual micropulses can be selected out of the LAMPF accelerator macropulse for individual applications. Separation between micropulses can be as short as 1 µs over the duration of the LAMPF accelerator macropulse. The chopped macropulses can be repeated at a rate up to 12 Hz or 40 Hz recurrence suitable for neutron energies of 0.256 or 0.8 GeV. The maximum average repetition rate is approximately 8.4 × 103 micropulses per second with 2–3×108 protons per micropulse, corresponding to approximately 40–250 nA. The TOF experimental area Figure 10.3 illustrates the details of the WNR TOF experimental area and the arrangement of different components. The target area is a 12-m diameter dome-shaped room (label 2 in Figure 10.3), which provides a

291

292


80ps

5ns 800 µs

LAMPF macropulses – 120 Hz

81 3 ms

Time [ms]

WNR macropulses – 12 Hz

83 1 3 ms Time [ms]

~200ps

≥1µs

800 µs

WNR micropulses structures Fig. 10.4 The beam structure of the LAMPF accelerator and the beam pulse structure used for the TOF experiments.

low-scatter environment with a minimum distance of approximately 5.5 m from the central target location to the nearest shield wall. Shielding is provided by a 14-m thick compacted tuff wall surrounding the experimental area. The beam stop is embedded into the tuff wall. Six flight paths could be used and penetrate the shield wall to allow angular distributions to be measured at 7.5◦ , 15◦ , 30◦ , 60◦ , 120◦ , and 150◦ . The detector stations are setup outside allowing flight path lengths of at least 60 m. Full details of the experimental setup are shown in Figure 10.5. A detailed description is as follows. The proton beam is transported to the center of the experimental room where the target is located. On its way, the arrival is signaled in a capacitive time pickoff. The


150°

Proton beam

Target

60°

A B

A-E:Filters NeutronC collimator

Time pickoff Secondary Electron monitor Charged particle Collimator Charged particle Sweep dipoles 1kG

120°

D

15°

Compacted tuff

Beam stop

7.5

E

30°

Detector station with veto and neutron detector

Fig. 10.5 The experimental layout of the high-energy neutron TOF experiments at Los Alamos (after Howe [690]).

total charge applied of the proton beam to the target is measured using a secondary emission monitor (SEM). The principle plan view of the arrangement of the neutron TOF lines is shown in Figure 10.6. For most of the experiments the beam lines with 30◦ , 60◦ , 120◦ , and 150◦ are used, whereas Stamer and Scobel et al. [131] experienced measurements in addition with the flight path under 7.5◦ with incident proton energies of 0.256 and 0.8 GeV. The cylindrical tubes of the flight paths have roughly a diameter of 75 cm to minimize the number of in-scattered neutrons. Their clearance assured that all neutrons produced by the target would be seen by the neutron detectors.

293

294


7.5° detector station 50 m 30° detector station 30 m Collimation

120° detector station 60 m Collimation

60° detector station 60 m Target

Protons from LAMPF 150° detector station 60 m Fig. 10.6 The principle plan view of the arrangement of the neutron TOF lines with their distance to the target. The locations of the target in the center and the collimators are also indicated.

The secondary emission monitor SEM (cf. Figure 10.5) is calibrated by measuring its rate relative to the production of 27 Al(p,3pn)24 Na, 27 Al(p,x)22 Na, and 27 Al(p,x)7 Be in irradiating Al foils located at the target position. The dominant uncertainty in this calibration is about 5% cross sections uncertainty for the production of Na. The total uncertainty in the number of protons normalization turns out to be less than 7%, whereas the relative error of the calibration is about 1%, consistent with the TOF system stability [129] and the statistics of counting the calibration foils. Behind the target a collimation system defines the neutron beam in a quasi open geometry. In this system two sets of collimators are separated by a few meters. The first collimator is the charged particle collimator of 50 cm thickness, 1.3 m away from the target. Its aperture is wider than the following neutron collimator, so its surface is not seen by the neutron detector. The charged particle collimator serves to produce a given charged particle beam spot diameter between the poles of the following of the charged particle sweep or clearing magnet which has approximately a 0.1 T magnetic field strength of length 1 m (SmCo permanent magnets). The following neutron collimator is 2.4 m thick. Its aperture defines the neutron beam in such a way that the neutron beam spot at the detector station is larger than the detector, but smaller than the beam pipe diameter to prevent small-angle scattering of neutrons into the detector by the material surrounding the pipe. The design of the shielding between detector and target consists of a 0.5-m concrete wall of the target room and a 14-m compacted tuff wall as indicated in Figure 10.5. Table 10.1 summarizes the parameters for the flight-path collimators and used filter systems used for the beam lines of 30◦ , 60◦ , 120◦ , and 150◦ . Uranium filters were inserted into the flight paths to scatter energetic charged particles out of the beam by successive small-angle scattering. Neutrons interacting

10.2 Neutron, Pion, and Proton Double Differential Measurements and Experiments Parameters of flight paths, collimators, beam diameter at detector, and uranium filter [130].

Tab. 10.1

Scattering angle Flight path length (m) Target to collimator distance (m) Collimator aperture (cm) Beam diameter at detector (cm) Beam/detector solid-angle ratio Iron collimator thickness (cm) Uranium filter (nuclei per mb)

30◦ 28.71

60◦ 58.46

120◦ 23.19

150◦ 30.22

4.80 8.52

4.80 6.95

12.00 7.95

4.80 9.64

25.66

27.86

10.63

33.79

40.06

23.61

6.87

69.49

209.60

248.90

335.28

188.00

138.42

87.69

87.40

1.52

Tab. 10.2 Examples of experimental typical uncertainties for the TOF experiment to measure double-differential production cross sections for 597 MeV protons as given by Amian et al. [130].

Correction Time-dependent background Target-produced background Shadowbar-background Air transmission Uranium transmission < 20 MeV Uranium transmission > 20 MeV Detector efficiency Live time Charge normalization

Magnitude

Uncertainty (%)

< 0.01 < 0.05 < 0.08 > 0.45

042 0.03–0.20 > 071 1.0

5 < 20 5–20 99.75%) Be, BeO, B, BN, C, Al, Fe, Cd, W, Pb, depleted 238 U (> 99.75%

30, 60, 120, 150

7.5, 30, 60, 120, 150

30, 60, 120, 150

Measurements of Stamer and Scobel [131] 256, 800 (thin targets)

7 Li, 27 Al, nat Zr, nat Pb

7.5, 30, 60, 120, 150

a

All target materials selected are in their natural isotopic abundances, if not otherwise indicated. For oxygen and nitrogen targets of BeO and BN were used, again with their natural abundances. The contributions of the Be and the B spectrum were subtracted.

10.2.1.2 The Experimental Results of Double-Differential Neutron-Production Cross-Section Measurements The measurements and also validations with intranuclear-cascade-evaporation models have been published in journals during 1989–1993 [129–131, 696– 698] and conference proceedings [704–708]. Table 10.5 summarizes the studied target materials selected for incident proton beam energy and scattering angle. For the beam energies of 597 and 800 MeV, all targets caused energy loss of less than 3 MeV to the incident protons (cf. Amian et al. [129, 130] Table 10.6). The specifications of the thin targets studied by Stamer et al. [131] at proton beam energies of 256 and 800 MeV are also shown in Table 10.6. For the experiments of incident beam energies of 113 and 256 MeV, the target characteristics are given in Table 11.12 and in Refs. [696–698]. Because these targets are stopping-length or near stopping-length targets the neutron-production spectra are related to thick target experiments and will be discussed in Chapter 11 on page 379. The ‘‘thin’’ target cross sections presented here are retrieved from NNDC experimental nuclear reaction database (EXFOR /CSISRS) [709]. The general behavior of proton-induced double-differential neutron-production cross sections is already

299

300

10 Proton-Nucleus-Induced Secondary Particle Production Tab. 10.6 Thin target materials and thicknesses used at the 597 and 800 MeV proton beam experiments.

597 and 800 MeV incident protons (Amian et al. [129, 130]) Target material thickness (g cm−2 )

Be 1.18

BeO 0.85

B 1.00

BN 0.86

C 0.56

Al 1.29

Fe 1.56

Pb 0.89

238 U 0.78

256 and 800 MeV incident protons (Stamer, Scobel et al. [131]) Target material thickness (mg cm−2 )

7

27 nat Li Al Zr nat Pb Thin metallic foils 70–150

discussed in Section 1.3.7.6 on page 54. Some examples of the measurements of Amian et al. [129, 130] at the LANL-WNR facility for incident proton energies of 597 and 800 MeV and C, Fe, Pb, and U targets are displaced in a double logarithmic scale in Figures 10.9 and 10.10. The total uncertainties of the experimental results are determined by ±1σ = 10–15% and thus are of the order of the data point sizes. The cross-section curves are seem to be very much alike. The absolute double-differential cross sections are seen to increase with increasing target mass number and incident proton energy, e.g., from C to U and from 597 to 800 MeV, for all neutron emission angles. Furthermore, the fraction of cascade neutrons in the total emission spectra increases with decreasing mass number for all laboratory angles. In Table 10.7, a subset of the results of Amian et al. [130] is compared with the measurements of Cierjacks et al. [127] for an incident proton energy of 597 and 585 MeV, scattering angles of 30◦ and 150◦ , and targets of uranium, lead, iron, aluminum, and carbon. Figures 10.11 and 10.12 compare cross-section measurements of Amian et al. [130] and of Cierjacks et al. [127] of uranium and lead at a scattering angle of 30◦ . In the figures the vertical scale uses an energy multiplied cross section, which is equivalent to the cross section per logarithmic energy unit d ln(E). With such a representation discrepancies and agreements between measurements are better illustrated as the commonly used representation in a double logarithmic scale using units of (b sr−1 MeV−1 ]. The error bars depicted in Figures 10.11 and 10.12 on the Amian data are mainly uncertainties due the counting statistics, time-independent and time-dependent backgrounds, whereas the error bars on the Cierjacks data are in general 11% uncertainties referred in [127]. The examples in Figures 10.11 and 10.12 show only an agreement for the evaporation region, whereas at high energies some of the discrepancy might be a result of the short flight paths of the experiments of Cierjacks et al. [127]. These discrepancies are also indicated in Table 10.7.

10.2 Neutron, Pion, and Proton Double Differential Measurements and Experiments 100

100 Eproton = 597 MeV

10−2 10−3 10−4 10−5 10−6 10−7

C - 30° C - 60°

10−8

C - 120° 10−9

Eproton = 597 MeV

10−1 d2s / dΩdE cross section [b sr−1 MeV−1]

d2s/ dΩdE cross section [b sr−1 MeV−1]

10−1

10−2 10−3 10−4 10−5 10−6 10−7

27

Al - 30°

27

10−8

Al - 60°

27

Al - 120°

10−9

C - 150°

10−10

27

Al - 150°

10−10 10−1

100

101

102

10−1

103

Neutron energy [MeV]

Eproton = 597 MeV

10−2 10−3 10−4 10−5 10−6 Fe - 30° 10−7

Fe - 60° Fe - 120°

10−8

Fe - 150°

Eproton = 597 MeV

10−1 d2s / dΩdE cross section [b sr−1 MeV−1]

d2s / dΩdE cross section [b sr−1 MeV−1]

103

100

10−1

10−2 10−3 10−4 10−5 10−6

Pb - 30°

10−7

Pb - 60° Pb - 120°

10−8

Pb - 150°

−9

−9

10

10−10

−10

10

10−1

100

101

102

10−1

103

Reaction proton (597 MeV) + Fe target

100

101

102

103


Neutron energy [MeV] (c)

102

Reaction proton (597 MeV) + Al target

(b)

100

10

101


Reaction proton (597 MeV) + C target

(a)

100

(d)

Reaction proton (597 MeV) + Pb target

Fig. 10.9 Proton-induced experimental double-differential neutron-production cross sections at scattering angles of 30◦ , 60◦ , 120◦ , and 150◦ – measurements of Amian et al. [129, 130]. Each successive curve, starting from the smallest angle 30◦ , is scaled by a multiplication factor of 10−1 , e.g., 60◦ × 10−1 , 120◦ × 10−2 , and 150◦ × 10−3 .

301

10 Proton-Nucleus-Induced Secondary Particle Production 100


10−2 10−3 10−4 10−5 10−6 U - 30° 10−7

U - 60° U - 120°

10−8 10

Eproton = 800 MeV

10−1 d2s/ dΩdE cross section [b sr−1 MeV−1]

d2s/ dΩdE cross section [b sr−1 MeV−1]

10−1

U - 150°

−9

10−2 10−3 10−4 10−5 10−6 Fe - 30° 10−7

Fe - 60° Fe - 120°

10−8

Fe - 150°

−9

10

−10

10−10

10

−1

10

0

10

1

10

2

10

10−1

3

10


102

103


10−1


10−2 10−3 10−4 10−5 10−6

Pb - 30°

10−7

Pb - 60° Pb - 120°

10−8

Eproton = 800 MeV

10−1

Pb - 150°

−9

10−2 10−3 10−4 10−5 10−6

238

U - 30°

238

10−7

U - 60°

238

U - 120°

10−8

238

U - 150°

−9

10

−10

10−10

10

−1

10

0

10

10

1

2

10

10−1

3

10


101

Reaction proton (800 MeV) + Fe target

(b)

100

10

100


Reaction proton (597 MeV) + U target

(a)


302

Reaction proton (800 MeV) + Pb target

100

101

102

103

Neutron energy [MeV] (d)

Reaction proton (800 MeV) + U target

Fig. 10.10 Proton-induced experimental double-differential neutron-production cross sections at scattering angles of 30◦ ,60◦ , 120◦ , and 150◦ – measurements of Amian et al. [129, 130]. Each successive curve, starting from the smallest angle 30◦ , is scaled by a multiplication factor of 10−1 , e.g., 60◦ × 10−1 , 120◦ × 10−2 , and 150◦ × 10−3 .

10.2 Neutron, Pion, and Proton Double Differential Measurements and Experiments Energy-integrated neutron-production cross section of thin targets – U, Pb, Fe, Al, C – for an incident proton energy of 597 MeV evaluated by Amian et al. [130] compared with data evaluated by Cierjacks et al. [127] with an incident proton energy of 585 MeV.

Tab. 10.7

Experiment Experiment 597 MeV protons 585 MeV protons Amian et al. [130] Cierjacks et al. [127] target

dσ/d (mb sr−1 ) 0.9–20 MeV

dσ/d (mb sr−1 ) 20–597 MeV

dσ/d (mb sr−1 ) 0.9–20 MeV

dσ/d (mb sr−1 ) 20–585 MeV

Scattering angle 30◦ Uranium Lead Iron Aluminium Carbon

2428.0 ± 64.1 1866.0 ± 71.1 283.9 ± 7.01 129.5 ± 2.4 51.9 ± 3.71

396.0 ± 1 373.0 ± 1 131.3 ± 0.1 81.2 ± 0.1 39.0 ± 0.1

3129.0 ± 255.2 2226.0 ± 178.4 537.5 ± 44.1 198.0 ± 22.2 92.7 ± 7.7

861.0 ± 96 693.0 ± 77 278.2 ± 31.1 146.6 ± 16.4 52.3 ± 5.8

Scattering angle 150◦ Uranium Lead Iron Aluminium Carbon

1420.0 ± 54 1110.0 ± 47 64.0 ± 2.31 27.2 ± 0.41 5.5 ± 1.3

13.7 ± 0.2 14.6 ± 0.3 2.9 ± 0.1 2.0 ± 0.1 0.93 ± 0.1

1685.3 ± 173.7 1147.0 ± 115.8 116.9 ± 10.7 39.7 ± 3.3 11.0 ± 0.93

64.4 ± 7.2 53.5 ± 6.0 10.8 ± 1.2 7.0 ± 0.78 1.83 ± 0.20

10.2.2 The Double-Differential Neutron-Production Measurements at the SATURNE Facility

The double-differential neutron-production measurements at the SATURNE accelerator, France, during 1994–1997, extend the energy of the incident protons up to an energy of 1.6 GeV and made it possible to measure double-differential neutron-production cross sections in an angular range from 0◦ up to 160◦ . A detailed description of the measurements and the different experimental apparatus is given by Leray et al. [132] and in Refs. [220, 710–713]. In the course of the experimental program several different techniques were developed to measure the double-differential neutron production spectra. The slow extraction of the proton beam of the SATURNE accelerator did not allow a TOF method using a highfrequency pulse from the accelerator as used at the Los Alamos WNR experiment. Therefore, two independent methods were used to measure the neutron spectra in the low-energy range 3 ≤ E ≤ 400 MeV (cf. Figure 10.13) and in the high-energy range 200 ≤ E ≤ 1600 MeV (cf. Figure 10.14). Both methods overlap in the energy range of about 200–400 MeV thus providing also a check on their consistency.

303


Energy x d2s / dΩdE cross section [MeV b sr−1 MeV−1]

1.2 238U (p,xn), 30° Cierjacks et al.

1.0

238U (p,xn), 30° Amian et al. Ep = 585/597 MeV

0.8 0.6 0.4 0.2 0.0 10−1

100


103

Fig. 10.11 Comparison of measured double-differential cross section of Amian et al. [129] and Cierjacks et al. [127] at a scattering angle of 30◦ for the reactions proton (597 MeV) + 238 U and proton (585 MeV) + 238 U. The vertical scale uses an energy multiplied cross section, which is equivalent to the cross section per logarithmic energy unit d ln(E). 1.2 Energy × d2 s/dΩdE cross section [MeV b sr−1 MeV−1]

304

Pb - 300 Cierjacks et al.

1.0

Pb - 300 Amian et al. Eproton = 585/597 MeV

0.8 0.6 0.4 0.2 0.0 10−1

100

101

102

103


Fig. 10.12 The same as Figure 10.11 but for the reactions proton (597 MeV) + Pb and proton (597 MeV) + Pb.

The SATURNE accelerator was a synchrotron located in the CEA Research Centre in Saclay, France. SATURNE could accelerate protons up to 2.95 GeV with a maximum intensity of 8 × 1011 particles per second and heavy ions up to krypton with 1.15 GeV/u with a charge-to-mass ratio of 0.5 [693]. The synchrotron delivers pulses during the beam spill with a pulse length of about 0.5 s with a repetition rate between 1 and 0.25 Hz.

10.2 Neutron, Pion, and Proton Double Differential Measurements and Experiments Charged particles

Collimator Anticoincidence

Magnet

S3 Neutrons

Protons Targets

Scintillator for neutrons

S2

S1

Shadow bars Stop

Start

Time-of-flight 13.4 m

Fig. 10.13 The principle of the TOF arrangement at SATURNE. As an example, an arrangement in the zero degree direction is displayed here [711]. Spectrometer

Collimator Magnet

C2

T2

C3

C1

Proton Neutron Targets

T1 M1 Shadow bars 4.5 m

M3 Hydrogen

Anticoincidence 8.0 m

2.0 m 1.8 m

2.7 m 1.6 m

Fig. 10.14 The principle setup of the spectrometer for high-energy neutron spectra measurements for energies ≥ 200 MeV. As an example, an arrangement in the zero degree direction is shown here.

10.2.2.1 The Experimental Apparatus at the SATURNE Accelerator The low-energy method The principle arrangement for the low-energy measurements for 3 ≤ E ≤ 400 MeV is described by Borne et al. [711] and is shown in Figure 10.13. In the figure the TOF method is shown for a 0◦ measurement, which is adopted with some minor modifications for other angles as shown in Figure 10.15. The TOF is measured between the incident proton beam of an intensity of 106 protons per second tagged by a 1-mm thick plastic scintillator (S1) and a cylindrical neutron liquid scintillator detector (S2) called the Demon detector (cf. Figure 10.15). Beam protons that do not interact with the target and other charged particles emitted in the forward direction are swept out by a magnet about 50 cm downstream of the target. The ‘‘Demon’’ detectors (cf. Figure 10.15) are surrounded by a paraffin shielding loaded with borax and lithium to reduce the background of

305

306

10 Proton-Nucleus-Induced Secondary Particle Production M3 ‘Demon’detectors liquid scintillator

Spectrometer Vé

‘DENSE’ low-energy detectors liquid scintillator

s

M2 Protons

0

115

1000

1350 1150

850

Vé 0

s nu

Rotation 00 – 850

550

70

1450 160

nu

400

0

Liquid hydrogen target

250 100

Neutron beam

00

Concrete

Proton Beam

Target area Beam stop Stop time-of-flight Magnet area scintillator M1

Fig. 10.15 The experimental area at the SATURNE accelerator with the spectrometer and the TOF arrangement to measure angle-dependent neutron-production spectra (after Borne et al [711], Martinez et al. [712], and Leray et al. [132]).

ambient gamma rays. In addition a 3 mm thick plastic scintillator of NE102 (S3) is placed in front of each neutron detector to tag events induced by charged particles. A 1-m shadow bar placed between the sweep magnet and the collimator is used to measure effectively the background produced by the collimator. Measurements with the shadow bar in place found only a small difference in the spectra between target ‘‘in’’ and target ‘‘out’’ of about ≤ 1%. Therefore, the shadow bar is not used for all target-energy combinations. To investigate in addition neutron spectra measurements at lower energies between 2 and 14 MeV, smaller detectors, called ‘‘Dense,’’ are also used as indicated in Figure 10.15. The flight-path length of 13.4 m depicted in Figure 10.13 for the TOF method is reduced to 8 m utilizing the TOF method for the apparatus as shown in Figure 10.15. The main characteristics of the used neutron detectors are summarized in Table 10.8. The high-energy method The principle arrangement for the high-energy measurements is a proton recoil spectrometer for 200 ≤ E ≤ 1600 MeV described Tab. 10.8

The detector parameter of the neutron detectors DENSE and DEMON.

Characteristics Liquid scintillator Diameter Length Photomultiplier Detector threshold

DENSE detector NE-213 127 mm 51 mm 9390 KB 1.0 MeV

DEMON detector NE-213 160 mm 200 XP 4512 1.9 MeV


by Martinez et al. [712] and is illustrated in Figure 10.14 for the 0◦ direction. Figure 10.15 shows the utilization of the spectrometer for angle-dependent measurements. The method is based on the detection of recoil protons produced by the scattering of high-energy neutrons via (n,p) reactions in a liquid hydrogen converter target which is based on an experimental technique published by Bonner et al. [689]. The neutrons emitted from the target are collimated through an 8-m thick concrete shield to an angular aperture of ±25◦ . Behind the collimator the neutrons interact with a liquid hydrogen converter target with a thickness of 1 g/cm2 and a diameter of 14 cm. The spectrometer system (cf. Figure 10.14) contains several components, e.g., a dipole magnet and three double-plan (X–Y) wire chambers indicated in the figure by C1, C2, and C3, to reconstruct the deflected trajectories of the protons. The spectrometer magnet has a field strength of 0.5 T on a 160 × 160 cm2 aperture with a length of 140 cm. The wire chamber C1 covers a 20 × 20 cm2 surface, whereas the chamber C2 covers a surface of 80 × 40 cm2 , and chamber C3 a surface of 100 × 80 cm2 . The wires have a spacing of 1.27 mm for C1 and 2 mm for C2 and C3. The overall XY-resolution gives a 1-mm accuracy on the position measurement for the proton trajectories. From the reconstruction of the trajectories, it is possible to determine the momentum of the protons, which are emitted within the spectrometer acceptance. The wire chambers C1 and C2 measure the emission angle of the protons. The neutron momentum is deduced assuming that the proton is produced by an elastic scattering event in the hydrogen converter target. In the hydrogen converter target several other types of charged particles are produced: np → np, np → npπ ◦ , np → ppπ − , np → dπ ◦ , np → 2nπ + . The charged particles with different masses are identified using the biparametric representation of the TOF of the T1 scintillator-wall versus the momentum measured with the wire chambers and the spectrometer M3 [132]. The plastic scintillator T1 behind the hydrogen converter and a 200 × 200 cm2 plastic scintillator hodoscope T2 of NE102 behind the spectrometer serve in coincidence to provide the trigger for the data acquisition. Both scintillation detectors are also used to measure the TOF of particles crossing the spectrometer identifying the protons from other charged particles produced in the hydrogen converter target. The hodoscope T2 is made of 20 horizontal slats with photomultipliers on each side. Figure 10.15 illustrates the final construction of the experiment, which allows 12 neutron beam lines to measure at neutron scattering angles from 0◦ to 160◦ by steps of 15◦ . In the final setup of the experiment a second magnet M2 is positioned before the hydrogen converter target, which deviates the charged particles created in the target in the horizontal plane. As indicated in Figure 10.15 the whole spectrometer could rotate from 0◦ to 85◦ . This is sufficient, because at larger scattering angles only some few neutrons with energies higher than 400 MeV are produced. At these scattering angles, the spectra could be measured with the TOF method applying the DEMON and DENSE detector systems. The beam intensity is limited to about 1010 protons/s with an average number of 400 triggers/s by dead times of 20% for reading out the wire chamber information.

307

10 Proton-Nucleus-Induced Secondary Particle Production 40.0 NE-213 detector efficiency of the DENSE detector

Neutron efficiency [%]

308

(threshold 1.0 MeV)

30.0

20.0

Experimental datas

10.0

Measured efficiency Corrected efficiency

0.0

1

10 Neutron energy [MeV]

Fig. 10.16 The efficiency of the neutron detector ‘‘DENSE.’’ The triangle symbols are the measured values, whereas the dashed line is a fit of these values. The solid line is the final parameterization used in the data analysis (after Leray et al. [132]).

The number of protons was monitored by two scintillator telescopes viewing a 50-µm thick Mylar foil placed upstream in the beam. The absolute calibration of the telescopes is obtained by irradiated a thin disk of carbon by counting the beta decay of 11 C. The production cross section 12 C(p,11 C) is well known. An estimation of the systematic error of the beam monitoring results in the energy range between 0.8 and 1.6 GeV of about 5.8% [132]. Also the standard beam monitoring with Al foils was used. The detectors, efficiency, energy resolution, and systematic errors in the TOF method Experiments and calculations were performed to estimate the detector efficiency over the whole energy range. Several experiments are investigated with quasi monoenergetic neutron beams to determine the neutron detectors’ efficiencies as a function of the incident neutron energy. Details are described by Leray et al. [132] and in references therein. The characteristics of the DENSE and the DEMON detectors are summarized in Table 10.8. The estimated efficiency functions of the ‘‘DENSE’’ and ‘‘DEMON’’ detectors are adopted from [132] and shown in Figures 10.16 and 10.17. The efficiency measurements were performed for the following energy ranges: • from 2 up to 17 MeV at the Bruyères-le-Chatel Van de Graaff accelerators, France [711] using quasi monoenergetic neutron-production reactions as 7 Li(p,n)7 Be, 3 H(p,n)3 He, 3 H(d,n)4 He, and 2 H(d,n)3 He, • from 30 ≤ E ≤ 100 MeV at the TSL Uppsala accelerator, Sweden [714], • from 100 to 180 MeV using the 7 Li(p,n)7 Be reaction, • from 150 up to 800 MeV at SATURNE using the (d+Be)-breakup reaction to produce quasi monoenergetic neutrons.

10.2 Neutron, Pion, and Proton Double Differential Measurements and Experiments 70 NE-213 detector efficiency of the DEMON detector (threshold =1.9 MeV)

Detector efficiency [%]

60 50 40

Bruyeres-le-Chatel Saturne Uppsala O5S Simulation Modified KSU simulation Efficiency as measured Corrected efficiency

30 10 20 0 1


1000

Energy resolution [%]

Fig. 10.17 The efficiency of the neutron detector ‘‘DEMON.’’ The black symbols are the measurements at Bruyères-le-Chatel, TSL Uppsala, and SATURNE. The gray symbols are simulations based on the codes O5S [715] and the code KSU a modified version of Cecil et al. [703]. The solid line is the final parameterization used in the data analysis (after Leray et al. [132]). 20 18 16 14 12 10 8 6 4 2 0

Total resolution Time component Geometrical component

1

10

102

103


Fig. 10.18 The different contributions to the energy resolution E/E as a function of the neutron energy for the DEMON detector. Estimated for a TOF path length of 8.5 m. The time resolution uncertainty is 1.5 ns and the uncertainty in the flight-path length is about 6 cm (after Leray et al. [132]).

The ‘‘DENSE’’ detectors are only used in the energy range of 2 to 14 MeV. The efficiency calculations for the ‘‘DEMON’’ detector with the KSU code of Cecil et al. [703] are in good agreement with the measurements, which was also found for the efficiency calibration of the LANL-WNR detectors investigated by Amian et al. [129] (see Section 10.8 on page 298). The neutron energy resolution E/E depends on a time and a geometrical component as already discussed in Section 10.2.1 and given by Eq. (10.2) on page

309

310

10 Proton-Nucleus-Induced Secondary Particle Production Tab. 10.9 The estimation of the main systematic errors as a function of the incident proton beam energy for neutron measurements above 0.4 GeV (data are based on Ref. [132]).

Incident beam energy (GeV) Error of

0.8 and 1.2

1.6

Beam intensity Spectrometer Response function Unfolding procedure and analysis

≤5.8% ≤4.0%

≤5.8% ≤11.5%

≤5.8%

≤8.6%

Total uncertainty

≤9.1%

≤15.5%

290. Figure 10.18 shows the energy resolution as a function of the neutron energy for the applied TOF method of the SATURNE experiment. Above 400 MeV (see Figure 10.18) the TOF method does not allow a better resolution than 12%. This was the reason to develop and use a complementary experimental setup for neutron measurement above an energy of 400 MeV, as described in Figures 10.14 and 10.15. Table 10.9 summarizes an estimation of the systematic errors as a function of the incident proton beam energy determined for neutron measurements above 400 MeV applying the spectrometer experimental setup. 10.2.2.2 The Experimental Results of Double-Differential Neutron-Production Cross Section Measurements The measurements at SATURNE and also validations with intranuclear-cascadeevaporation models have been published in journals during 1997–2006 [132, 172, 220, 711, 712] and conference proceedings, reports, and PhD theses [543, 693, 710, 713, 716]. Table 10.10 summarizes the studied target materials selected by incident proton beam energy and scattering angle. The ‘‘thin’’ target cross sections presented are already retrieved again from NNDC experimental nuclear reaction database (EXFOR /CSISRS) [709]. Summary of proton energies, scattering angles, and used targets (the targets are all of natural abundance (data are based on Refs. [132, 220]).

Tab. 10.10

Incident proton energies (GeV)

0.8, 1.2, 1.6

Scattering angles (◦ )

0, 10, 25, 40, 55, 85, 100, 115, 130, 145, 160

Target material Target thickness (g cm−2 ) Target diameter (cm)

Al 8.1 3.0

Fe 23.6 3.0

Zr 19.4 3.0

W 19.3 3.0

Pb 22.7 3.0

Th 39.8 3.0

10.2 Neutron, Pion, and Proton Double Differential Measurements and Experiments 103 Eproton = 800 MeV

d 2 s/dΩdE cross section [mb sr −1 MeV−1]

102 101 100 10−1

10

0

LANL - Pb - 30 Saclay - Pb - 550

−3

×10−2

LANL - Pb - 600

10−4 10−5

×10−1

Saclay - Pb - 250

10−2

LANL - Pb - 1500 Saclay - Pb - 1600

10−1

100


103

Fig. 10.19 Comparison of SATURNE data of Ledoux et al. [220] with measurements at LANL-WNR of Amian et al. [129] of the reaction proton (800 MeV) + Pb. As indicated in the figure, the different curves are scaled successively by a factor of 10−1 . The symbols have a different size to distinguish between the measurements.

Some representative examples of double-differential neutron-production measurements at the SATURNE facility for incident proton energies of 0.8 GeV, 1.2 GeV, and 1.6 GeV for C, Al, Fe, Pb, and Th targets and scattering angles of 0◦ ,10◦ , 25◦ , 85◦ , and 160◦ are displaced in a double logarithmic scale in Figures 10.20, and 10.21 [132, 220]. The total uncertainties of the experimental results are determined by ±1σ = 10–15% and thus are of the order of the data symbol sizes. In Figure 10.19, a subset of the results at SATURNE of Ledoux et al. [220] is compared with measurements at LANL-WNR of Amian et al. [129] for an incident proton energy of 800 MeV and scattering angles of (25◦ , 30◦ ), and (55◦ , 60◦ ), and (150◦ ,160◦ ), and for the reaction proton (800 MeV) + Pb). The data comparison is made for the closest possible angles. It is seen that the SATURNE data fully agree with the LANL-WNR data. For other targets, e.g., Fe, there are some minor differences below neutron energy ≤ 4 MeV. These differences are probably a consequence of the target-enhanced thickness. The employed targets at the SATURNE experiments are not really ‘‘thin’’ compared with the targets used at the LANL-WNR experiment (cf. Tables 10.10 and 10.6). As pointed out by Leray et al. [132] the thickness of the used targets at the SATURNE experiments may produce some distortions in the measured spectra induced by secondary reactions of the particles from the spallation process. The result is a slight shift and a broadening of the quasielastic and inelastic peak at very forward scattering angles and the increase of the number of low-energy emitted neutrons. These peaks vanishes at higher emission angles about ≥ 30◦ as seen in the figures. The LAN-WNR experiments did not show this structure because it was not possible to measure at scattering angles ≤ 30◦ . In contrast to

311

312


SATURNE measurements, the LAN-WNR experiments (cf., for e.g., Figure 10.19) with their lower detection threshold of about 0.525 MeV illustrate much more pronounced broad evaporation region in the neutron-production spectrum which strongly dominates for heavy mass targets, being less pronounced at low masses where the nucleon–nucleon mechanisms dominate. As mentioned earlier, the cross-section curves are seem to be very much alike. The absolute double-differential cross sections increase with increasing target mass number and incident proton energy, e.g., from Al to Th and from 0.8 to 1.6 GeV, for all neutron emission angles. Furthermore, the fraction of cascade neutrons in the total emission spectra increases with decreasing mass number for all laboratory angles. 10.2.3 The Double-Differential Neutron-Production Measurements at the KEK Facility

The TOF method developed at π2 beam line of the 12 GeV proton synchrotron at the National Laboratory for High Energy Physics, KEK, Japan, is to use a very weak secondary proton beam up to 3.0 GeV, which leads to a short flight length of 1 m. The experiments were started with some test experiments by Nakamoto et al. [694] at incident proton energies of 0.8 and 1.5 GeV during 1995 and finished in 1997 by Ishibashi et al. [265]. 10.2.3.1 The Time-of-Flight Method with a Short Flight Path The double-differential neutron-production spectra are measured by the TOF method with detectors of liquid NE-213. The length of flight paths for such experiments has usually been taken to be several 10 m up to 60 m as used at the LAMPF-WNR and the detectors have been heavily been shielded by materials of iron, concrete, etc., against background radiation. However, the proton beam used for the KEK experiments is generated as a secondary particle beam by an internal target, which is placed in the synchrotron accelerator ring. The beam intensity for the experiment was very weak at a level of about 105 protons per 2.5 s. Because of this very low-intensity pulsed proton beam the protons could individually be counted [265]. The experimental arrangement is illustrated in Figure 10.22. As shown in Figure 10.22, a TOF method is used with a pair of Pilot-U scintillators located at a flight distance of 20 m for the separation of incident protons and pions, because of the nature of the used secondary beam. The protons in the beam are contaminated by pions of the same momentum. A time resolution of about 0.25 ns could be achieved to resolve the pions from the incident protons. As indicated in Figure 10.22, NE-102A plastic scintillators were used to define the incident beam of protons on the targets. The coincidence of the scintillators signals of P1 and P2 determines the number of protons. The weakness of the intensity of the incident proton beam comprises not only a short flight paths but also thicker targets (cf. Table 10.12) to maintain the measurement efficiency to a reasonable degree.

10.2 Neutron, Pion, and Proton Double Differential Measurements and Experiments 103

102

Eproton = 0.8 GeV

1

Fe

10

102 d2 s/dΩE cross section [mb sr −1 MeV−1]

d2 s/dΩE cross section [mb sr −1 MeV−1]

103

100 −1

10

10−2 −3

10

10−4 −5

10

00

−6

10

10−7

100

10−8

25 85

10−9

0

0

Eproton = 0.8 GeV Pb

101 100 10−1 10−2 10−3 10−4 10−5 10−6

00

10−7

100 250

10−8

850

10−9

1600

10−10

1600

10−10 100

101

102

103

100

Neutron energy [MeV] Reaction proton (0.8 GeV) + Fe target

(a)



102

Eproton = 1.2 GeV

1

Al

100 −1

10−2 10

−3

10

−4

10−5 10−6

00

10−7

10

0

10−8

25

0

850

10−9 10

−10

10

100 10−1 10−2 10−3 10−4 10−5 10−6

00

10−7

100 250

10−8

10 101


1

850

10−9

1600

100

102

103

1600

−10

100


103

103

102

10

102

Reaction proton (0.8 GeV) + Pb target

(b)

103

10

101


Reaction proton (1.2 GeV) + Al target

101

102


Fig. 10.20 Proton-induced experimental double-differential neutron-production cross sections at scattering angles of 0◦ , 10◦ , 25◦ , 85◦ , and 160◦ –measurements of Ledoux et al. [220] and Leray et al. [132]. Each successive curve, starting from the smallest angle 0◦ , is scaled by a multiplication factor of 10−1 , e.g., 10◦ × 10−1 , 25◦ × 10−2 , 85◦ × 10−3 , and 160◦ × 10−4 .


103

313

314


103

103 Eproton = 1.2 GeV

101



102

Fe

100 10−1 10

−2

10−3 10

−4

10−5 00

10−6

10

10−7

0

250

10−8

85

10−9

0

160

10−10

0

Eproton = 1.2 GeV Th

101 100 10−1 10−2 10−3 10−4 10−5 10−6

00

10−7

100 250

10−8

850

10−9

1600

10−10 100

101

102

103

100


(a)

103


Eproton = 1.6 GeV


102

Reaction proton (1.2 GeV) + Th target

(b)

103 Fe

101 10

101


0

10−1 10−2 10−3 10−4 10−5 10−6

0

0

0

10

10−7

250

10−8

850

10−9

1600

10−10

102

Eproton = 1.6 GeV

101

Pb

100 10−1 10−2 10−3 10−4 10−5 10−6

00

10−7

100 250

10−8

850

10−9

1600

10−10 100

101

102

103

100


(c) Fig. 10.21

The same as Figure 10.20.

101

102



103

10.2 Neutron, Pion, and Proton Double Differential Measurements and Experiments NE102A (veto)

Ø 12.7 cm × 12.7 cm NE213

90°

60°

120°

30°

150° NE102A (beam definition) S01

Proton beam

S02

20m

15°

Target area

P02

Pilot U (p -p discrimination and particle counting)

1.5m

1m

P01

P1 P2

0.6m

0.9m

Proton beam line 15° 30°

150° 120° NE102A (veto)

90°

60° Ø 5.08 cm × 5.08 cm NE213

Fig. 10.22 The arrangement of the KEK time-of-flight experiment (after Ishibashi et al. [265]).

Test experiments referred by Nakamoto et al. [694] were mainly investigated to check the signal-to-background ratio. For this purpose, the NE-213 detectors were shielded with iron blocks with a thickness between 0.5 and 1 m. Measurements with and without shielding of the detectors showed a relatively good agreement at neutron energies above several MeV and a large disagreement below 4 MeV caused by the generation of secondary neutrons produced by high-energy particles in the iron shield surrounding the detectors. Therefore, the KEK TOF experiment was operated in a bare detector condition. Another issue was the background scattering of particles from the concrete floor because of the low beam line height of 1.7 m. Particle transport calculations for estimating the floor particle scattering effects into the detectors and shadow bare experiments showed that the effect of floor scattered neutron is clearly negligible in the energy range up to several 100 MeV [694]. As shown in Figure 10.22 two different sizes of the NE-213 neutron detectors (∅5 × 5 , ∅2 × 2 ) were used simultaneously for the scattering angles of in 150◦ , 300◦ , 600◦ , 900◦ , 1200◦ , and 1500◦ direction. In front of each neutron detector, a NE-102A plastic scintillator is mounted and works as veto detector in anticoincidence to eliminate charged particle background events. The larger detectors are used with flight-path lengths of 1–1.5 m measuring the high-energy neutrons and the smaller ones with 0.6–0.9 m measuring low-energy neutrons, respectively. Table 10.11 summarizes the detector data of the NE-213 detectors. The neutron detector efficiencies were determined by simulations with the standard code of Cecil et al. [703] for neutron energies up to several 100 MeV (cf. see also Sections 10.2.1.1 on page 296 and 10.2.2.1 on page 305) and the SCINFUL code of Dickens [717, 718] for neutron energies below 80 MeV. The results of the simulation with the Cecil code are adjusted together with the SCINFUL results at

315

10 Proton-Nucleus-Induced Secondary Particle Production Tab. 10.11

Summary of the detector data of Refs. [265, 694].

Flight path Flight path length (m) Detector material (liquid) dimension Diameter × thickness (cm) Time resolution (ns) Energy resolution ± 1σ (%)

15◦ , 30◦ , 60◦ , 90◦ , 120◦ , 150◦ 1.5–1.0 NE-213

15◦ , 30◦ , 60◦ , 90◦ , 120◦ , 150◦ 0.9–0.6 NE-213

12.7 × 12.7(5 × 5 ) 0.5–1.0 5.7 for 10 MeV neutrons 14.0 for 300 MeV neutrons

5.08 × 5.08(2 × 2 )

60 Am

Cs

Co

AmBe

Am-bias by SCINFUL for NE213 (5.08 cm) Cs-bias by SCINFUL for NE213 (12.7 cm) Co-bias SCINFUL for NE213 (12.7 cm) AmBe-bias by CECIL (corr.) for NE213 (12.7 cm)

50 Detector efficiency [%]

316

40 30 20 10 0 10−1

100


103

Fig. 10.23 Determined neutron detector efficiencies for liquid scintillators of NE-213. The Cecil code simulations are normalized to the 80 MeV simulations determined by the SCINFUL code. The energy ranges for different biases are indicated in the upper part of the figure [265].

an energy of 80 MeV. The results are shown in Figure 10.23 for four bias levels considering Am, Cs, Co, and AmBe-biases1) . The 241 AmBe-bias is used for the smaller NE-213 detector of 5.08 × 5.08 cm2 whereas the other biases are used for the larger detectors of 12.7 × 12.7 cm2 . 10.2.3.2 The Experimental Results of Double-Differential Neutron-Production Cross Section Measurements Double-differential neutron-production cross section were measured for C, Al, Fe, In, and Pb targets at incident proton beam energies of 0.8, 1.5, and 3.0 GeV. The data presented in the following examples are retrieved from the EXFOR database [709]. Table 10.12 shows the target characteristics and the incident proton beam energy 1) The light output in terms of the scintillator’s response to electrons of energy in MeV is expressed in the electron equivalent MeVee.

10.2 Neutron, Pion, and Proton Double Differential Measurements and Experiments Tab. 10.12

Summary of the target characteristics (data are taken from [265]).

Target material

C

Al

Fe

Fe (1.5 GeV)

In

Pb (plate)

Diameter (cm) 5.0 Thickness (g cm−2 ) 17.5 Density (g cm−3 ) 1.75

5.0 4.9 10.7 23.6 2.69 7.87

4.9 15.74 7.87

4.9 10 × 10 × 1.2 cm3 17.76 13.6 7.31 11.34

Proton energy loss (MeV) : At energy 0.8 GeV At energy 1.5 GeV At energy 3.0 GeV

37 33 33

20 18 18

40 – 36

– 24 –

26 23 23

18 16 16

Mean excitation energya (eV)

73.8

160

278

278

483

819

a The mean excitation energy is the mean excitation and ionization potential in the proton energy loss process in the targets.

Time-of-flight energy resolution [%]

100

80

60

40

20

0 100

Fig. 10.24


103

The resulting energy resolution of the KEK TOF experiment.

loss because the targets are not really ‘‘thin.’’ All targets have a cylindrical shape except the Pb target. The energy resolution of the TOF experiment is important for cross-section measurements. The energy resolution of the KEK experiment is derived from the time resolution and from the neutron detection efficiency. The resulting energy resolution is shown in Figure 10.24. The uncertainty of the neutron detection efficiency was determined to be about 10% in the energy region below 80 MeV, whereas above 80 MeV it was estimated to be about 15% [694]. The time resolution in conventional TOF experiments, e.g., the LANL-WNR experiment, is mainly determined by the pulse width of the incident particle beam usually about several nano seconds. Because in the KEK experiments the protons are counted event by event, the uncertainty of the TOF measurements

317


102

103 Eproton = 0.8 GeV Fe

101 100 10−1 10−2 10−3 10−4 10−5 10−6 10−7 10−8 10

Ishibashi-15° Ishibashi - 30° Amian - 30° Ishibashi - 60° Ishibashi - 120° Ishibashi - 150° Amian - 150°

−9

10−1

100 101 102 Neutron energy [MeV]

103

(a) Reaction proton (0.8 GeV) + Fe target

Fig. 10.25 Proton 0.8 GeV induced experimental double-differential neutronproduction cross sections on Fe and Pb targets at scattering angles of 15◦ ,30◦ , 60◦ , and 150◦ – measurements of Ishibashi et al. [265] in comparison with measurement of

d2s/dΩdE cross section [mb sr−1 MeV−1]

103 d2s/dΩdE cross section [mb sr−1 MeV−1]

318

102


101 100 10−1 10−2 10−3 10−4 10−5 10−6 10−7 10−8 10−9 10−1

Ishibashi-15° Ishibashi - 30° Amian - 30° Ishibashi - 60° Ishibashi - 120° Ishibashi - 150° Amian - 150°

100 101 102 Neutron energy [MeV]

103

(b) Reaction proton (0.8 GeV) + Pb target

Amian et al. [129] of angles at 30◦ and 150◦ . Each successive curve, starting from the smallest angle 15◦ , is scaled by a multiplication factor of 10−1 , e.g., 30◦ × 10−1 , 60◦ × 10−2 , 120◦ × 10−3 , and 150◦ × 10−4 .

is mainly determined by the time resolution of the neutron detectors. The time resolution of the detectors is estimated from the FWHM of the gamma ray peak flash and is obtained to be 0.5–1.0 ns [265]. Then the time resolution have to be converted into the energy resolution considering the uncertainty of the flight path’s length. It is seen from Figure 10.25 that there is a remarkable good agreement between the data of Ishibashi and Amian although the targets studied at the KEK facility are not really ‘‘thin’’ compared to the target characteristics given in Tables 10.5 and 10.12 on page 299 and 317, respectively. The good agreement is also given at other measured scattering angles not shown in Figure 10.25. To visualize the comparison the plotted symbols have different sizes. Figure 10.26 shows examples of the double-differential neutron production for reactions proton (3.0 GeV) + Fe and Pb targets for scattering angles of 15◦ , 30◦ , 60◦ , 90◦ , and 150◦ at the KEK facility of Ishibashi et al. [265]. The energy and angle dependence of the neutron-production cross sections show the expected trend. 10.2.4 The Double Differential Pion Production Measurements

Double-differential pion production measurements of proton–nucleus (p,A)-reactions are very scarce. The production of pions is a major field of intermediate energy physics to validate the secondary particle production channels.


102

103 Eproton = 3.0 GeV

30° 60°

101 10

90° 150°

0

10−1 10−2 10−3 10−4 10−5 10−6 100

Eproton = 3.0 GeV

15°

Fe

101 102 103 5×103 Neutron energy [MeV]

(a) Reaction proton (3.0 GeV) + Fe target



103

102

15°

101

60°

10

30° 90° 150°

0

10−1 10−2 10−3 10−4 10−5

Pb

10−6 100

101 102 103 5×103 Neutron energy [MeV]

(b) Reaction proton (3.0 GeV) + Pb target

Fig. 10.26 Proton 3.0 GeV induced experimental doubledifferential neutron-production cross sections on Fe and Pb targets at scattering angles of 15◦ ,30◦ , 60◦ , 90◦ , and 150◦ – measurements of Ishibashi et al. [265]. Each successive curve, starting from the smallest angle 15◦ , is scaled by a multiplication factor of 10−1 , e.g., 30◦ × 10−1 , 60◦ × 10−2 , 90◦ × 10−3 , and 150◦ × 10−4 .

Only few experiments of double-differential pion production cross-section measurements are reported for incident protons [124, 695, 719–721] and incident neutrons [722] in the GeV energy range. The most comprehensive data sets are published by Cochran et al. [124] for incident protons of 730 MeV, and by Crawford et al. [695, 719] for incident protons of 585 MeV for a large variety of scattering angles and target materials. More recently, the HARP collaboration [723] measured double-differential pion production spectra for incident protons in the energy range between 3 and 12 GeV/c for Be, C, Al, Cu, Sn, Ta, and Pb targets. As already discussed in Sections 1.3.7.6 on page 54 and 1.3.8 on page 57, highpower spallation facilities may provide intense neutrino beams. The main scientific and technical issues investigating an intense neutrino source involve the construction of very intense proton beams, targeting effective capture of produced particles and an accurate knowledge of the pion production mechanisms in proton-induced spallation reactions (cf. e.g., the NuFact-07 conference [724] and references therein). 10.2.4.1 The Double Differential Pion Production Measurements at the Berkley Cyclotron with Incident Protons of 730 MeV The general setup of the pion spectrometer at the Berkley 184-inch cyclotron investigated by Cochran et al. [124] is shown in Figure 10.27.

319

320


Rotation for angles 15, 20, 30, 45, 60, 75, and 90 degree, and 90, 105, 120, 135, and 150 degree

Beam from 184’’ cyclotron Spectrometer

Premagnet collimator Steering magnet

Q 1 Q2

Circe Beam monitors 4.72m

2.65m

Diana

Sic T1 target

T2 target

5.74m

Fig. 10.27 Sketch of the pion spectrometer at the Berkley 184-inch cyclotron using 730 MeV protons of Cochran et al. [124].

The experiment used a low-intensity extracted proton beam of an intensity of about 5–20 nA passing through an adjustable magnet collimator, a steering magnet, and a quadrupole doublet Q1 , Q2 and through a proton beam pipe in the concrete shield into the experimental area. The quadrupole doublet CIRCE is used to focus the beam at the target on position T1 , and the quadrupole doublet DIANA is considered to focus the beam when the target is used at position T2 . Experience has shown that the setting of the premagnet collimator could be used for both the forward- and the backward-angle measurements. As depicted in Figure 10.27 there are a large number beam channels from the target positions of T1 to define scattering angles at 15◦ , 20◦ , 30◦ , 45◦ , 60◦ , 75◦ , and 90◦ with respect to the incident beam, and scattering angles of the target position T2 of 90◦ , 105◦ , 120◦ , 135◦ , and 150◦ . The spectrometer could be moved radially to cover the forward and backward angles. The flight path channel openings are 10 cm horizontally by 12.5 cm vertically and with inserted lead collimators 5 × 7.5 cm2 . The collimators were used to test and estimate the scattering of particles from the channel walls into the spectrometer. The spectrometer was constructed by modifying an existing ‘‘C’’ magnet of the Berkley cyclotron which had a 33 cm × 61 cm pole face and a 15 cm gap. The central field was chosen to be 1.1 T. Further information of the spectrometer itself with the used counters of Ne-102 each 3.8 cm wide and 3 mm thick to assure that a charged particle had entered the spectrometer within its acceptance and had traversed the magnet field, the electronics, the particle identification procedure, and the off-line data reduction is given in great detail in the publication of Cochran et al. [124].

10.2 Neutron, Pion, and Proton Double Differential Measurements and Experiments Used targets and target densities given by [124] (the ‘‘thin’’ targets were either 7.5 cm or 10 cm in diameter).

Tab. 10.13

Target material liquid H2 CD2 Be C Al Ti a

Density (g cm−2 )

Target material

densitya 1.13 0.90 1.10 0.97 0.76

Cu Ag Ta Pb Th

Density (g cm−2 ) 0.97 1.08 1.28 1.90 1.01

The density for liquid H2 is given in 1.59 × 1023 protons cm−2 .

Examples of the double-differential pion production measurements Doubledifferential pion production cross sections d2 σ/ddE by 730 MeV protons at 11 scattering laboratory angles in the interval 15◦ –150◦ and at 12 energies in the interval 25–550 MeV were measured. The used targets are summarized in Table 10.13. The systematic errors in the beam current measurements affecting all cross sections are estimated to be about 10%. The above-discussed flight-path channel in-scattering into the spectrometer gives an average uncertainty of about 10% and is included in the cross-section errors. The statistical uncertainty is typically of the order of 3%. The produced positrons and electrons arise from π and µ decay in flight and from π ◦ –γ decay and subsequent pair production in the region of the target system. It was shown that the (e− /π − ) ratio is in general equal to the (e+ /π + ) and is an order-of-magnitude indication for the background for the experiment [124]. Examples of ratios for e− /π − are selected for the angles 15◦ , 90◦ , and 150◦ and shown in Figure 10.28. Figures 10.29, 10.30, and 10.31 show some examples of double-differential production cross sections of positive- and negative-charged pions at different angles by protons of 730 MeV incident on hydrogen, carbon, aluminum, copper, and lead targets. Note that compared to the production cross sections of neutrons typically in the order of mb, the production cross sections of pions in proton–nucleus (pA) reactions are only in the order of several µ barns. The cross-section production data for π + data of the reaction 1 H(p, π + )X (cf. Figure 10.29) show a very strong forward peaking at all energies. The maximum cross section is seen at 15◦ and about 300 MeV. This behavior is also expected in isobar models, where the (1236) formation and decay dominates the production process. The production cross section of the reaction 1 H(p,π − )X is only a few + − percent of that of the positive pions also shown by the ratio of πtotal /πtotal in − Table 10.16. The production of π from hydrogen comes from the reaction p + p → π + + π − + p + p. The examples of Figures 10.30 and 10.31 show that positive pion production is predominantly at forward angles, and the spectrum peaks at about 250–300 MeV. The shape of the spectra and the angular distributions are essentially the same from the lighter elements to the heavier ones for π + or π − , respectively.

321

10 Proton-Nucleus-Induced Secondary Particle Production 100 15°

Ratio (electron /p−)

90° 150°

10−1

Target Cu ratio: e− / p−

10−2

10−3 2×102

102

3×102

4×102 5×102 6×102

Electron energy [MeV]

Fig. 10.28 Ratios of the number of electrons to the number of negative pions – (e− /π − ) at the same energy. The plotted data are from Ref. [124]. 25 Cross section d2s/dΩdE [µb sr−1 MeV−1]

322

20

15°

20°

30°

90°

120°

150°

Proton 730 MeV 1 H(p, p+)X

15

10

5

0 −1 101

5×101 102 Pion energy [MeV]

Fig. 10.29 π + production spectra from liquid hydrogen for the reaction 1 H(p,π + )X, Eproton = 730 MeV. The pion cross section are plotted in a log-linear scale to show the peaking at the forward angles of 15◦ , 20◦ , and 30◦ . The data are taken from Ref. [124].

5×102

103


102 Cross section d2s/dΩdE [µb sr−1 MeV−1]

Cross section d2s/dΩdE [µb sr−1 MeV−1]

102

101

100

10−1

15° 30° 120° 150°

C-target p− 10−2 101

101

100 15° 30° 120° 150°

10−1

Al-target p− 10−2

102

101

6×102

102

Pion energy [MeV] (a) Reaction C (p, π−)X, Eproton = 730 MeV

(b) Reaction Al (p, π−)X, Eproton = 730 MeV 102 Cross section d2s/dΩdE [µb sr−1 MeV−1]


102

101

100

10−1

15° 30° 120° 150°

Al-target p+ 10−2 101

6×102

Pion energy [MeV]

101

100 15° 30° 120° 150°

10−1

Cu-target p+ 10−2

102

6×102

101

Pion energy [MeV] (c) Reaction Al (p, π+)X, Eproton = 730 MeV

102 Pion energy [MeV]

6×102

(d) Reaction Cu (p, π+)X, Eproton = 730 MeV

Fig. 10.30 Proton-induced double-differential π ± production cross sections for C, Al, and Cu targets at scattering angles of 15◦ , 30◦ , 120◦ , and 150◦ at an incident proton energy of 730 MeV. Data are from Ref. [124].

323

324


102 Cross section d2s/dΩdE [µb sr−1 MeV−1]


102

101

100 15° 30° 120° 150°

10−1

Pb-target p+ 10−2

101

100 15° 30° 120° 150°

10−1

Pb-target p− 10−2

101

102

2

6×10

101

Pion energy [MeV] (a) Reaction Pb (p, π+)X, Eproton = 730 MeV Fig. 10.31

102 Pion energy [MeV]

6×102

(b) Reaction Pb (p, π−)X, Eproton = 730 MeV

The same as Figure 10.30 but for lead targets.

Figure 10.32 shows a comparison of the measured double-differential π + production cross section for incident protons on aluminum with several intranuclear-cascade models, e.g., the Bertini model, the NUCRIN model, and the MICRES model described in Sections 2.2 on page 64 and 2.6.3 on page 123. The Bertini and the MICRES model predictions are in good agreement with the measurements, which is also shown for other target materials in [275]. On page 327, the experimental data of Cochran et al. are compared with the data of Crawford et al. which are described in Section 10.2.4.2. For both the experiments, the total cross sections for the production of charged pions by protons with energies of 585 and 730 MeV are summarized in Tables 10.15 and 10.16. The π + cross sections are normalized to Z1/3 and the π − cross sections to N 2/3 . The Z1/3 law for the π + suggests that the measured π + are produced peripherally. They escape the nucleus with relatively little scattering or energy loss. Therefore, the π + production spectrum peaks at forward angles. The formation of the π − production indicates a more complicated mechanism which is a function of π ◦ formation and the charge-exchange reaction π ◦ + n → π − +p as an important source of π − production. The charge-exchange scattering lowers the energy of the produced π − , broadens the angular distribution , decreases the π + /π − ratio, and makes the total cross section proportional to the nuclear area times N/A, or about N 2/3 . The functional dependence of the total cross sections of π + and π − on Z and N is shown in Figures 10.36 and 10.37 on pages 330 and 331, respectively.


15°

102

101

×10 30°

100

d2a/dΩdE [mb sr−1 GeV−1]

×102 10−1 45°

×103 10−2

60°

10−3 ×104 90°

10−4

×105

10−5

120° 10−6 0

0.1

0.2

0.3

0.4

Pion (p+) energy [GeV] Fig. 10.32 Double-differential π + production spectra for incident protons on aluminum at 730 MeV (p+Al → π + + X) at scattering angles of 15◦ , 30◦ , 45◦ , 60◦ , 90◦ , and 120◦ . Each successive curve, starting from the smallest angle 15◦ , is scaled by

a multiplication factor of 10, e.g., 30◦ × 10, 45◦ × 102 , 60◦ × 103 , 90◦ × 104 , and 120◦ × 105 . Circles are experimental data of Cochran et al. [124], full line Bertini model code, dotted line MICRES model code, and dashed line NUCRIN model code.

10.2.4.2 The Double Differential Pion Production Measurements at the Cyclotron of the Paul Scherer Institut (PSI) The double-differential production spectra of charged pions in proton–nucleus (p,A) collisions were measured from various nuclei at the Paul Scherer Institut (PSI, formerly named as SIN) by Crawford et al. [695, 719] during 1980. The experiment was performed at a proton beam energy of 585 MeV and covered a similar data set as that of Cochran et al. [124]. Pions were measured at energies above 24 MeV [695] and at low energies from 6 to 36 MeV [719]. In Figure 10.33,

325


Scintillation counter P1 P2

Central trajectory incident protons

CH3 target

Superconducting solenoid

Ce of ntra sp l tr ec aje tro ct me ory Sp ter ec tur trom nt ab eter le

Magnet

xr Pion production target

Shielding

326

yr q

Scintillation counter p1

Lead Collimators

p2

1m

p3 Scintillation counter

Fig. 10.33 The experimental arrangement to measure the production of charged pions from various nuclei at the PSI accelerator at an incident proton energy of 585 MeV by Crawford et al. [695, 719].

the general setup of the pion spectrometer is shown. The unpolarized extracted proton beam for the experiment was scattered in an 8-mm thick Be target through an angle of 8◦ and was transverse polarized of about 41.65 ± 0.40%. The transverse polarization of the beam could be rotated through 180◦ using the 1-m long solenoid magnet. The intensity of the proton beam was 5 × 109 protons per second during the experiment. The beam energy of 585 MeV was calibrated with an absolute uncertainty of 1 σ = ± 2 MeV and an energy band Eproton /Eproton of 0.4% FWHM. The beam was monitored with a multiwire proportional chamber near the pion production targets. The beam geometry at the pion production targets was 14 mm horizontally and 9 mm vertically FWHM. The superconducting solenoid magnet (Figure 10.33) has a longitudinal magnetic field of 4.5 T along the beam direction. The beam was monitored by the scintillator hodoscopes P1 and P2 scattered at 30◦ from a 5-mm thick CH2 target. For the calibration of the intensity the beam has to be reduced during the experiment with a factor of about 103 –104 without changing the beam dimensions. The charged particles produced from the target were momentum analyzed in 1 m spectrometer magnet. The TOF through the spectrometer was determined using the scintillator counters π1 , π2 , and π3 indicated in Figure 10.33. The TOF measurement between the detectors π1 and π2 allowed a clear separation of electrons and pions emitted from the targets. The spectrometer could be rotated through the center of the target system to cover backward scattering angles up to 135◦ . The acceptance momentum of the spectrometer is p/p0 = 18.8% (FHWM),

10.3 Proton Cooler Synchrotron COSY at Jülich Tab. 10.14

Pion scattering angles and measured pion energies.

Scattering angle θπ (degree)

Energy Eπ (MeV)

23 45 60, 90, 135

24, 35, 88, 151, 254 24, 35, 88, 151, 192 24, 35, 88, 151

where p0 is the momentum of a particle traveling along the central trajectory of the spectrometer. Examples of the double-differential pion production measurements Pions were measured at the following scattering angles θπ = 23◦ , 45◦ , 60◦ , 90◦ , and 135◦ in the laboratory system with respect to the direction of incident proton beam. Table 10.14 summarizes the studied pion energies and the covered scattering angles. The following pion production targets were used: • CH2 , C6 D12 , Be, Al, Al2 O3 , Ni, Cu, Mo, and Pb. • The CH2 , C6 D12 , and Al2 O3 targets are used to determine the pion production cross sections for the proton, the deuteron, and the oxygen. • The targets were plates with a surface of 5 × 7 cm2 , a thickness of about 1 g/cm2 and were mounted on a remote-controlled target ladder with an empty position to measure the background.

Figures 10.34 and 10.35 show proton-induced π ± production spectra at scattering angles of 23◦ , 60◦ , and 135◦ for C, Al, Cu, and Pb targets. Comparison of total π + and π − cross sections Tables 10.15 and 10.16 summarize the total cross sections determined from of the measured cross-section distributions + − /πtotal π + and π − using the formula σtotal ≡ (d2 σ/ddE)ddE. The ratios of πtotal change from 8.4 of deuterium to 2 of lead for the PSI data of Crawford et al. and rapidly from 45 of hydrogen to 1.9 of thorium for the data of Cochran et al. As mentioned earlier, the functional dependence of the total cross sections of π + and π − on Z and N is shown in Figures 10.36 and 10.37.

10.3 The ‘‘Thin’’ Target Particle Production Measurements at the 2.5 GeV Proton Cooler Synchrotron COSY at Jülich

In this section, the emphasis will be on the description of nuclear data taken by the NESSI (NEutron Scintillator and SIlicium detector) and the PISA (Proton Induced SpAllation) experiment installed at the Cooler Synchrotron COSY in Jülich, Germany. To this, first the accelerator COSY used by both experiments is briefly introduced. The typical energy range of incident protons is 150 MeV–2.5 GeV with luminosities for the internal experiment PISA up to 6 × 1034 cm−2 s−1 . The

327


70

70 Scattering angle 23°

60 50 40 30 20 Pb target Cu Al C

10



Scattering angle 23°

1

60 50 40 Pb target Cu Al C

30 20 10 1

50

50

100 150 200 250 300

Pion p+ energy [MeV]

100 150 200 250 300

Pion p− energy [MeV]

(a) Reaction C, Al, Cu, Pb(p, p+)X, qp = 23°

(b) Reaction C, Al, Cu, Pb(p, p−)X, qp = 23°

70

70 Scattering angle 60°

60

Pb target Cu Al C

50 40 30 20 10


Scattering angle 60° Cross section d2s/dΩdE [µb sr−1 MeV−1]

328

60

Pb target Cu Al C

50 40 30 20 10 1

1 50

100

150

200

50

Pion p+ energy [MeV] (c) Reaction C, Al, Cu, Pb(p, p+)X, qp = 60°

100

150

200

Pion p− energy [MeV] (d) Reaction C, Al, Cu, Pb(p, p−)X, qp = 60°

Fig. 10.34 Proton-induced double-differential π ± production cross sections for C, Al, Cu, and Pb targets at scattering angles of 23◦ and 60◦ at an incident proton energy of 585 MeV. Data are from Ref. [695].

10.3 Proton Cooler Synchrotron COSY at Jülich 100

Scattering angle 1350

90 80 Pb target Cu Al

70 60

C

50 40 30 20 10

Cross section d2 sdΩE [µb sr −1 MeV−1]

Cross section d 2 s dΩE [µb sr −1 MeV−1]

100

0

Scattering angle 1350

90 80 Pb target Cu Al

70 60

C

50 40 30 20 10 0

1

50

100

150

200

1

Pion p+ energy [MeV] (a)

Reaction C, Al, Cu, Pb (p,p+)X, qp = 135°

50

100

150

200

Pion p− energy [MeV] Reaction C, Al, Cu, Pb (p,p−)X, qp = 135°

(b)

Fig. 10.35 Same as Figure 10.34 pion production cross sections for π + and π − at a scattering angle of 135◦ . Data are from Ref. [695]. Total cross sections for π + and π − , the ra+ − /πtotal , and the normalization σtot (π + )/Z 1/3 and tios of πtotal − 2/3 σtot (π )/N , incident proton energy is 585 MeV, and crosssection data are from Ref. [695]. Tab. 10.15

Target/element

σtot (π + ) (mb ± 1σ )

σtot (π − ) (mb ± 1σ )

Ratio + − πtot /πtot

H D Be C O Al Ni Cu Mo Pb

9.7 ± 1.2 8.7 ± 1.1 22.7 ± 2.8 28.5 ± 3.5 34.6 ± 4.2 43.8 ± 5.4 62.5 ± 7.7 60.0 ± 7.4 69.6 ± 8.5 86.0 ± 11.0

– 1.03 ± 0.13 4.75 ± 0.59 4.72 ± 0.58 6.40 ± 0.79 9.80 ± 1.2 14.70 ± 1.8 18.00 ± 2.2 24.50 ± 3.0 41.50 ± 5.1

− 8.4 4.78 6.04 5.4 4.47 4.25 3.33 2.84 2.07

σtot (π + )/Z 1/3 (mb ± 1σ )

σtot (π − )/N2/3 (mb ± 1σ )

9.7 ± 1.2 8.7 ± 1.1 14.3 ± 1.8 15.7 ± 1.9 17.3 ± 2.1 18.6 ± 2.3 20.6 ± 2.5 19.5 ± 2.4 20.1 ± 2.4 19.8 ± 2.5

1.03 ± 0.13 1.62 ± 0.20 1.43 ± 0.18 1.61 ± 0.20 1.68 ± 0.20 1.50 ± 0.18 1.70 ± 0.21 1.72 ± 0.21 1.66 ± 0.20

experimental setups, detection efficiencies, and particularities of the experiments at NESSI and PISA will be discussed and a selection of experimental and theoretical results are shown and compared. The experiments at COSY Jülich (and LEAR, PS/CERN) were carried out within an international collaboration between the FZ-Jülich, the Hahn-Meitner-Institut Berlin, the research center CERN (Geneva),

329

10 Proton-Nucleus-Induced Secondary Particle Production Total cross sections for π + and π − , the + − /πtotal , and the normalization σtot (π + )/Z 1/3 and ratios of πtotal σtot (π − )/N2/3 , incident proton energy is 730 MeV, and crosssection data are from Ref. [124]. Tab. 10.16

Target/element

H D Be C Al Ti Cu Ag Ta Pb Th

σtot (π + ) (mb ± 1σ )

σtot (π − ) (mb ± 1σ )

Ratio + − πtot /πtot

σtot (π + )/Z 1/3 (mb ± 1σ )

σtot (π − )/N2/3 (mb ± 1σ )

13.50 ± 0.73 11.42 ± 0.55 27.30 ± 1.40 35.00 ± 1.80 53.10 ± 2.90 67.00 ± 3.60 77.30 ± 4.30 91.60 ± 5.10 101.00 ± 5.60 104.20 ± 5.80 107.90 ± 5.90

0.03 ± 0.01 1.12 ± 0.06 6.49 ± 0.37 6.64 ± 0.41 13.17 ± 0.90 21.20 ± 1.60 25.20 ± 2.00 35.00 ± 3.00 51.40 ± 4.70 53.70 ± 4.90 60.40 ± 5.50

45.0 10.2 4.3 5.3 4.0 3.2 3.1 2.6 2.0 1.95 1.9

13.50 ± 0.73 11.42 ± 0.55 17.19 ± 0.9 19.28 ± 0.97 22.55 ± 1.24 23.93 ± 1.29 25.12 ± 1.39 25.37 ± 1.41 24.2 ± 1.34 23.99 ± 1.32 24.08 ± 1.32

0.03 ± 0.01 1.12 ± 0.06 2.21 ± 0.13 2.15 ± 0.13 2.26 ± 0.15 2.42 ± 0.18 2.38 ± 0.19 2.26 ± 0.19 2.27 ± 0.21 2.15 ± 0.22 2.22 ± 0.20

30

25 stotal (p+) / Z1/3 [mb]

330

20

15 Data cochran et al. Data crawford et al. 10

5 0

20

40

60

80

100

Nuclear charge Z

Fig. 10.36 Total pion production for π + from various thin targets divided by Z 1/3 as a function of the nuclear charge Z. The plotted data are given in Tables 10.15 and 10.16, where the data of Cochran et al. are for incident protons of 730 MeV, and the data of Crawford et al. are for proton energies of 585 MeV.

GANIL (Caen), INR (Moscow), Rossendorf (Dresden), INP (Orsay), the universities of Rochester and Warsaw and the TU-Munich. The data compiled by the NESSI and PISA collaborations at COSY are of particular interest also for the EU-FP6 integrated project EUROTRANS – (NUDATRA Nuclear data for transmutation of nuclear waste). The goal of that domain is to improve nuclear data in evaluated

10.3 Proton Cooler Synchrotron COSY at Jülich

stotal(π−) / N2 /3 [mb]

3

2

1 Data cochran et al. Data crawford et al. 0

0

20

40

60

80

100

120

140

160

Nuclear neutron number N

Fig. 10.37 Total pion production for π − from various thin targets divided by N2/3 as a function of the nuclear neutron number N. The plotted data are given in Tables 10.15 and 10.16, where the data of Cochran et al. are for incident protons of 730 MeV, and the data of Crawford et al. are for proton energies of 585 MeV.

files and models, which involves sensitivity analysis and validation of simulation tools, low- and intermediate-energy nuclear data measurements, nuclear data libraries evaluation at low and medium energies, and high-energy experiments and modeling. NUDATRA aims at the investigation of: • pA (spallation) reactions in the GeV regime, • data measured from exclusive experiments for testing, validating and developing theoretical models, • double-differential cross sections (DDXS) dσ/dEd of light charged particles (LCP = p,d,t,3 He,4 He, . . .) and intermediate mass fragments (IMFs, Z ≤ 16) in spallation and fragmentation p-induced reactions (0.1–2.5 GeV, C to Au), • reaction mechanism of pN reactions in terms of time scales, simultaneous or sequential emission of IMFs, origin of pre-equilibrium and evaporation processes. 10.3.1 The COoler SYnchrotron COSY

COSY is the abbreviation for ‘‘COoler SYnchrotron,’’ which means as much as accelerator for ‘‘cold’’ particles. However in COSY not only protons can be accelerated up to 96% of the speed of light, but also can stored for ultra slow or fast extraction. The COSY ring as shown in Figure 10.38 consists a 180-m vacuum tube. Protons in the momentum range between 600 and 3400 MeV/c (corresponding to 175 and 2600 MeV) are accelerated and stored. Protons with the desired energy are available for experiments with the circulating beam (”internal experiments”) as

331


Cavity e-Cooler

ANKE

Cosy-11 Fast quadrupoles Stochastic cooling HE polarimeter EDDA

PISA

TOF

NESSI

NESSI

JESSICA

Exchange

332

TOF BIG KARL

GEM, MOMO

LE polarimeter

10 m Cyclotron

Fig. 10.38 Cooler synchrotron COSY in Julich ¨ with the internal and external experimental beam areas. The locations of NESSI and PISA experiment are indicated.

well as for experiments with the extracted beam (”external experiments”). Detailed reports on the performance and perspectives of the Cooler Synchrotron COSY are given in Refs. [725–727]. Twenty-four electromagnets deflect the protons rotating in the COSY around 15◦ each so that a course of approximately 360◦ results. Quadrupoles exert a force on charged particles, which is attractive in one plane and repulsive in the plane perpendicular. These focusing and defocusing planes ensure the protons being held together to a bundle during acceleration.


For the injection process the debit orbit of the particles rotating in the ring is shifted for a short time. The debit orbit is the position intended for the proton beam for a stable closed course in the ring. The particles are shot in on this disturbed course, and during the injection the circulating proton beam is reset on the original debit course. The disturbance of the debit course is accomplished with the use of so-called bump magnets. The entire injection process takes about 0.01 s. By interaction with electron beams (electron-cooling), the proton beam stored in COSY shrinks on the smallest possible expansion. During the ‘‘stochastic cooling’’ the debit course of the circulating proton jet is measured in pick ups and corrected with the help of the kickers. The position of the jet in COSY is measured by 29 position monitors with an accuracy of approximately ∼ 1 mm. This position measurement takes place without contact, i.e., without direct effect on the jet. For the cold moderator experiment JESSICA [551, 611] discussed in Section 13.3.3.2 on page 454, the accelerator COSY was operating in a specific mode: In contrast to the slow extraction needed generally for experiments at COSY or the relatively long spills applied for internal experiments, the single turn extraction requested by JESSICA meets some challenges for the COSY crew. Fast beam extraction in the proton energy range of 0.8–2.5 GeV is accomplished by the use of a kicker magnet generally employed for beam diagnostic measurements. The cycle starts in the same manner as for resonant beam extraction for external experiments. A closed orbit bump in the horizontal plane is located near the electrostatic septum. The proton beam is bunched with a bunch length in the flat top of about 200–500 ns. The beam intensity peaks at about 1010 p in the flat top. By means of the kicker magnet the beam bunch is short time (0.75–2 µs in width, rise- and fall-time ≤ 1µ s) deflected. The kicker excitation is synchronized with respect to the COSY rf-signal and can be adjusted in time by a programmable delay, so a unique deflection of the total bunch can be performed (bunch synchronous excitation). Only horizontal beam deflection is possible with the kicker magnet installed at a repetition rate of 1 Hz. The minimal COSY cycle time varied from 5 ms in case of low energy to 2 s in the case of highest energy. In order to extract the whole beam stored in COSY in one single turn, electron cooling is indispensable. The relatively long cooling time needed results in a repetition rate of the extracted proton beam of ≈0.03 Hz. The reliability of the kicker magnet has been shown to be rather high. Details are given in Ref. [728]. 10.3.2 The NESSI Experiment

All previous data on production yields have been obtained by measuring the integral neutron-induced activation in a moderator bath surrounding the target, by integrating double-differential neutron yields (d2 σ/dEd), or by neutron-induced activation in small samples positioned on the target surface. For a summary of such measurements see Ref. [621]. Instead, in the NESSI experiment all neutrons produced in each individual shower induced by an incident energetic hadron are counted. In order to demonstrate the inherent capabilities of the new approach, the

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characteristics of the employed detection method will be described briefly in the following. On one hand emphasis is put on a comparison of model predictions with the data obtained with thick targets (cf. Section 11.2.2), where incident energetic protons in the energy range up to 2.5 GeV at COSY (5 GeV/c hadrons at PS/CERN and up to 1.2 GeV p at LEAR/CERN, (cf. Section 12)) impinging on massive targets give rise to cascades of nuclear reactions within the bulk target material. In these kinds of setups, charged reaction products are stopped within the target volume and only neutrons are detected. Specifically, of great importance are neutron-production cross sections for various incident hadron energies, various target materials, and different target geometries. On the other hand following the motivation of the fundamental physics aspect, results of production cross sections, particle spectra, angular distributions, etc., for neutrons and charged particles taken at thin target measurements have been compared with the corresponding simulations in order to decouple the primordial spallation reaction from the subsequent INC. The NESSI and former PS208 collaborations have enriched the available information by the event-wise measurement of the number of neutrons (called hereafter neutron multiplicity Mn ) [115, 379, 622, 631, 632, 663, 664, 673, 729, 730] using a high-efficient 4π sr gadolinium loaded scintillator detector [631, 731, 732] and provided a heavyset matrix of benchmark data enabling a validation and possibly an improvement in high-energy transport codes. As mentioned in contrast to previous measurements the event-wise character of the experiment allowed to gain access even to the distributions dMn /dN rather than average values only, thus imposing additional constraints for theoretical models [114, 228, 622, 664, 673, 674]. The first experiment that measured neutron multiplicity distributions for 0.475 and 2 GeV proton bombarding thin targets was carried out at the SATURNE accelerator at Saclay by Pienkowski et al. [219]. The NESSI experimental investigations comprised both, the so-called thick-target scenario, as discussed in Section 11.2.2, where only neutrons can be observed in the NESSI experiment and multiple-nuclear reactions per source particle2) might take place as well as experiments using thin targets (only one nuclear interaction in the target) as discussed in the current section. While the first aspect is triggered by the application driven motivation, the second issue reflects the fundamental physics aspect. The latter enables a code validation not only for neutrons, but also for charged particles or even for correlations between them [114, 228, 664, 673, 674]. Insights into the transport process including all cascade particles in thick targets can be gained only by disentangling intra- and internuclear cascades. In addition to the above-mentioned neutron measurements also light charged particles have been measured with π, p, p,3 He and 4 He projectiles in the energy region of 1–14 GeV [663, 675, 733, 734]. However, most of these studies measured energetic cascade and not evaporative light charged particles as p and He, which are needed to get full information on the reaction process. 2) Not only the intranuclear cascade, but also the INC contributes to the production of neutrons following interactions induced by secondary

particles and therefore resulting in a ‘‘multiplication’’ of neutrons.

10.3 Proton Cooler Synchrotron COSY at Jülich BNB+BSiB S3,S5-S8

Beamdump

S10

S1 p

S11-14 Targets 1110 cm

605 cm

574 cm

Fig. 10.39

Schematic setup for thin target measurements at COSY accelerator.

Fig. 10.40

Picture of the NESSI neutron ball detector in the COSY experimental area.

10.3.2.1 Experimental Setup of NESSI The setup for the NESSI experiment at the COSY facility is schematically shown in Figure 10.39. Figure 10.40 shows a picture of the NESSI-BNB (Berlin Neutron Ball) as mounted in the COSY experimental area. Visible is the 1.4 m diameter and 1.5 m3 scintillator sphere with the photomultiplier tubes viewing directly into the liquid scintillator. The support structure allows for lifting the upper hemisphere in order to have access to the inner scattering chamber and the 4π Si-ball for charged particle detection. As indicated in Figure 10.39 the incoming protons were tagged by a thin, 0.3 mm, plastic scintillator S1 mounted 11 m upstream from the center of Berlin Neutron Ball (BNB) [631, 731, 732]. Their rate was adjusted to give a similar reaction rate independent of target thickness, i.e., to some 104 pps for thin targets and down to some 200 pps for the 35 cm targets. The detector S1 served as a beam counter for absolute normalization of measured cross sections and provided the time reference for BNB. The incident protons were also monitored by a set of scintillator detectors S3 –S14 partly serving as veto counters and tagging those protons entering

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‘‘off-axis.’’ As mentioned previously, thin-target experiments are aimed essentially at studying the physics of intranuclear cascades, with reaction products having negligible chance for secondary interactions with the target matter. Both neutrons and charged reaction products are detected using two concentric 4π sr detector devices, the BNB and the Berlin Silicon Ball (BSiB) [114, 664]. In this setup, the target is placed in the common operational center of the BNB and BSiB. The latter detector is mounted inside the BNB reaction chamber as schematically indicated in Figure 10.41. In the following, a brief description of the 4π sr detectors, their efficiency and the necessary corrections on the data is presented. For more details confer to Refs. [114, 115, 228, 379, 631, 663, 664, 673, 729, 730, 735]. The most recent and comprehensive technical compilations on the two 4π detectors BNB and BSiB are published in Refs. [228, 674, 735, 736]. The electronics and data acquisition are described in [379]. The 4π sr neutron detector The BNB [631, 731, 732] is a spherical vessel – sketched in Figure 10.41 – with an outer diameter of 140 cm and an active volume of 1.5 m3 , filled with a gadolinium-loaded organic scintillator NE343 (1,2,4-trimethylbenzol = C9 H12 ). It contains a central reaction chamber of 40 cm diameter connected to a high-vacuum beam pipe. The active detector volume is viewed by 24 fast photomultipliers mounted on the outer shell of the BNB. The technical data of the 4π sr BNB are summarized in Table 10.17. The operation of the BNB is based on the detection of gadolinium γ -rays from the capture of neutrons thermalized within the scintillator liquid as shown in Figure 10.41(b). The thermalization of the reaction neutrons is a relatively fast process, occurring on a 0.1 µs time scale. It is accompanied by a light flash generated mostly by the interaction of the recoiling nuclei (mostly hydrogen, but also carbon and oxygen) with the scintillator. This flash, combined with the light produced in the interaction of reaction γ -rays and charged reaction products with the scintillator, gives rise to a ‘‘prompt’’ signal – one of the observables in the Tab. 10.17

Technical data of the 4π sr BNB.

Manufacturer Volume Diameter of reaction chamber Scintillator liquid Gadolinium Gd σ capturea for 155 Gd and 157 Gd Number of photomultipliers Energy resolution Time resolutionb Lower trigger threshold a b

Hahn-Meitner-Institut Berlin 1.5 m3 140 cm 40 cm NE343 (1,2,4-trimethylbenzol) C9 H12 0.4% (wt%) 6.1 × 104 and 25.4 × 104 b 24 no ≤ 3 ns 2 MeVee (electron equivalent)

Capture cross section for thermal neutrons Relative to start-detector


Photomultipliers

Reaction chamber BSIB Beam dump

Target

P

BNB

Liquid scintillator Target support (a)

50 cm g

Scattering chamber

p

Light flash

p beam PM

e Target n

Gd p

g e g

Scintillator 10 MeV (b)

10 keV

30 cm, 100 ns

e

0.025 eV

Gd(n,3g)

2 cm, 10µs

Compton eff,Light

Fig. 10.41 (a) Schematic drawing of the NESSI detector system BNB (Berlin Neutron Ball) and the BSiB (Berlin Silicon Ball) in the reaction chamber. (b) The principle of neutron detection in the BNB in three steps: (i) slowing down/thermalization, (ii) storage, (iii) capture, counting.

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NESSI experiments. This signal is an approximate measure of the total kinetic energy of the detected neutrons. A prompt light flash indicates an energy deposition in the detector by any reaction product. As it is detected with virtually unit efficiency, it can be used to measure the total reaction cross section, including reactions without neutron emission. Experimentally, one recognizes prompt signals based on their coincidence with valid ‘‘start’’ signals, which are generated by projectiles traversing the thin scintillation detector S1 placed at the entrance to the BNB reaction chamber. The reaction probability PReac for thin targets is obtained by comparing the number of prompt signals with the number of incident particles. The fast thermalization process is followed by a slow diffusion of the neutrons through the scintillator, before they are eventually captured by the gadolinium nuclei present in the scintillator. There is a statistically distributed time lapse for a thermalized neutron to ‘‘find’’ such a gadolinium nucleus and be captured, which occurs on a µs scale. The most abundant isotopes 155 Gd with 14.7% and 157 Gd with 15.7% have capture a cross sections for thermal neutrons of (6.1 ± 0.1) × 104 and (25 ± 0.2) × 104 b, respectively. The subsequent capture of γ -ray cascade, with a total energy of approximately 8 MeV, produces a delayed light pulse. Due to the statistical nature of the thermalization and diffusion process, individual neutrons entering the detector volume at the same time instance, are captured at different times, spread over several tens of µs. It is this spread in capture times that allows one to count one-by-one the individual light pulses produced in different capture events and thus, ideally the number of neutrons Mn that have entered the detector volume3) . Mn , the neutron multiplicity, is the essential observable of the BNB. As summarized – in terms of time the neutron detection principle in the scintillator volume is a three step process: (1) thermalization of the neutrons, (2) storage, (3) capture on Gd-nuclei and counting the number of light flashes. Efficiency of the NESSI–BNB detector In applications of the BNB, neutron capture γ -rays are counted within a 45 µs counting gate following each reaction event. Hence, as a neutron multiplicity counter, the BNB is a slow device, prone to event pile-up in high-intensity experiments. It is also important to note that not all neutrons are thermalized within the active volume of the detector. Some, especially high-energy neutrons, escape this volume without being captured. Such neutrons are not counted, leading to an overall capture efficiency smaller than unity. In the NESSI experiments, the BNB counts mostly low-energy evaporation neutrons, for which the detection efficiency is typically εn 82%. In contrast, for pre-equilibrium and INC cascade neutrons of higher energy (30–50 MeV), the detection efficiency is of the order of 20–35%. The theoretical neutron detection efficiency ε of the BNB as a function of neutron n is shown in Figure 10.42. This efficiency was calculated using a kinetic energy Ekin Monte Carlo simulation code [737], assuming a light detection threshold of 2 MeVee 3) The principle of neutron detection dates back to the investigations of Frederic Reines. In

1995, he got the Noble Prize for the recovery of neutrinos.


Efficiency [e]

0.9

0.6

0.3

0.0

0

30

60

90

Neutron energy [MeV] Fig. 10.42 Detection efficiency ε of the BNB as a function n , as calculated with the DENIS of neutron kinetic energy Ekin code [737]. A parameterization of this curve is given in Ref. [115] and Eq. (10.3).

(MeV electron equivalent4) ). In the simulation calculations, the latter threshold was matched to the experimental one, reproducing correctly the measured efficiency (82.6%) for 2.16 MeV fission neutrons emitted from a 252 Cf-source. The filter applied for the neutrons, i.e., the efficiency for detecting a neutron in the 4π neutron ball NESSI is parameterized by the following polynomial function which could easily be implemented in any code: 2 3 + 0.000370167Ekin (Ekin ) = 0.820652 + 0.00689154Ekin − 0.00423934Ekin 3 4 5 +0.000370167Ekin − 1.80244 × 10−5 Ekin + 5.2007 × 10−7 Ekin 6 7 −8.97261 × 10−9 Ekin + 9.06944 × 10−11 Ekin 8 9 −4.95096 × 10−13 Ekin + 1.12679 × 10−15 Ekin .

(10.3)

This parameterization is published in Ref. [115] and shown in graphical presentation in Figure 10.42. In the NESSI experiments, the observed neutron multiplicities are averaged over neutron energy spectra and weighted with the respective detection efficiencies. n Since the information on kinetic energies of individual neutrons, Ekin is experimentally not available, the simulation calculations employ the neutron energy as calculated within the model. Additionally, the neutron detection efficiency is n calibrated only at low neutron kinetic energies Ekin . At higher energies, the neutron detection efficiency is extrapolated based on Monte Carlo calculations. The neutron multiplicity distributions measured with the above setup contained random and target frame related background. The magnitude of the target frame background was determined in separate measurements made without target and subtracted from the measured ‘‘raw’’ multiplicities. The random background was measured online using a second, 45 µs long counting gate pulse, started at 4) The total reaction cross section measured using the prompt response of the BNB with this

threshold corresponds to an inelasticity of at least 2 MeV.

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10 Proton-Nucleus-Induced Secondary Particle Production Tab. 10.18

Technical data of the 4π sr silicon ball. Individual silicon detectors Eurisys Meßtechnika IPH750-500 HMI C surface depletion layer Ceramics 763 mm2 500 µm 14285 cm ∼100 V ∼3.2 kV/cm < 100 keV < 250 ps

Manufacturer Type Detector-type ‘‘Backing’’ Active area Total thickness (depletion zone) Spec. resistance Applied voltage Maximum field strength depl. zone Energy res. (5.5 MeV α-source) Time res. (5.5 MeV α-source)

4π sr silicon ball BSiB Granularity Shape Acceptance Radius Weight a

162 detectors, self-supporting 12 penta-, 90 (ir)regular hexagons 91% of 4π sr 10 cm 600 g

Eurisys Meßtechnik, F-67383 Lingolsheim, France.

400 µs after the primary gate pulse. Subsequently, the experimental multiplicity distributions were corrected for this background by deconvolution techniques [379]. All experimental neutron multiplicity distributions shown in the following [115] have also been corrected for the detector dead time of 35 ns and for multiple scattering, but not for the detection efficiency. The latter correction was included in simulation calculations in comparisons to experimental data. The 4π sr silicon detector In addition to the neutrons in the thin-target experiments charged reaction products were detected. Light charged particles (LCP: H and He isotopes), intermediate mass (IMF), and fission fragments (FF) were detected and identified by the Berlin Silicon Ball (BSiB) inside the BNB (see Table 10.18). The BSiB [114, 228, 664, 735] is composed of 158 independent, 500 µm thick silicon detectors approximating a 20-cm diameter sphere and covering a solid angle of about 90% of 4π sr. Charged particles (CP: H + He + IMF + FF) were identified by means of TOF versus energy E correlations with a mass resolution of ±3 units for A = 20 and ±15 units for A = 100. Six of the Si-ball detectors at angles between 30◦ and 150◦ are replaced by E − E telescopes. They consist of two fully depleted E silicon detectors (80 and 1000 µm thick) backed by a 7-cm thick CsI scintillator with photodiode read out. These telescopes allow a fully isotopic separation up to about A = 20 and an extension of the covered range of kinetic energy spectra above the BSiB


thresholds (cf. following paragraph). For thick-target measurements BSiB can be taken out and replaced by thick targets of up to 40 cm length and 15 cm in diameter. Efficiency of the Si detectors Due to the absorption or specific energy loss of LCPs in the target material being evident in particular close to 90◦ , the overall detection efficiency of the BSiB for LCPs, calculated with Monte Carlo simulations [663] is about 79–84%, depending on the atomic number Z of the particle. This already takes into account the active area (94%) of the Si detectors and 11 detectors missing for beam in/out, target in/out, TV-camera, some defect detectors and the six detectors replaced by telescopes. The lower energy threshold of the 500 µm thick Si detectors for all charged particles is 2.2 MeV. Protons and α-particles with energies larger than 8.2 and 32.2 MeV, respectively, are not stopped in the 500 µm silicon detectors. Consequently, the lower detection threshold represents at the same time an upper limit for detecting highly energetic p, d, and t of more than 26, 49, and 76 MeV kinetic energy, respectively. For the same reason minimum ionizing particles, π and K fall below detection threshold. For Z ≥ 2 particles on the other hand practically no such upper energy limit exists. The telescopes used at NESSI allowed for a mass resolution of A ≤ 0.2 amu for E/A > 3.5 MeV/A and A ≤ 0.4 amu for E/A < 3.5 MeV/A [735]. The targets The thin targets nuclei ranging from 12 C up to 238 U have typical thicknesses of the order of some 100 µg/cm2 to g/cm2 . The targets were mounted on a 14 cm long Al flag-pole with 0.5 mm × 5 mm profile and positioned perpendicular to the beam axis. Thinner targets (≤ 1 mg/cm2 ) were generally used to measure proton-induced fission while thicker targets were employed for the measurement of neutron-production and total reaction or inelastic cross sections. The target thicknesses were measured by weighing the energy loss measurements of αparticles from ThC. For thick targets as discussed in Section 11.2.2 three nuclides, Hg, Pb, and W have been chosen which are representative of the target, structure and core materials of the ADS. All pieces were made from chemically pure (≥ 99.98%) material of Pb, W, and Hg, the latter being encapsulated into 1 mm thick stainless steel containers. 10.3.2.2 Experimental Results of NESSI An example of the capability of the NESSI experiment is presented in Figure 10.43. It shows the eventwise correlation of neutron versus charged particle multiplicities measured with the two almost 4π devices. With the simultaneous measurement of mainly evaporative-like neutrons and charged particles (CP) detailed exclusive information is obtained on an event-by event base. The multiplicity correlations are shown in Figure 10.43 representatively for 2.5 GeV proton-induced reactions on target nuclei ranging from 13 Al to 238 U. The target thicknesses between 0.1 and 1.0 g/cm2 correspond to reaction probabilities of 1–5 × 10−3 . The best description of the coincidence data for multiplicities LCPs vs. the multiplicity Mn can be found in Ref. [233].

341

10 Proton-Nucleus-Induced Secondary Particle Production 25 20

E∗

13Al

26Fe

28Ni

29Cu

40Zr

41Nb

47Ag

67Ho

15 10 Charged particle multiplicity

342

5 25 20 15 10 5 25 74W

20

79Au

82Pb

10−2

92U

15

10−3

10

10−4

5 0

25

0

25 0 25 Neutron multiplicity

0

25

10−5

Fig. 10.43 Correlation of measured (NESSI) light charged particle(LCP) vs. neutron multiplicity for 2.5 GeV protoninduced spallation reactions on various targets from Al to U. The color scale gives the relative yield for each target per multiplicity bin; the thermal excitation is following the indicated arrow in the upper left corner of the figure.

As a general tendency, an increase of both, Mn and MCP , with increasing A and Z of the target is observed. This is mainly due to the fact that larger target nuclei incorporate more energy from the INC and have lower particle separation energies than do lighter ones. Also, at lower excitations for heavy nuclei (Au, Pb, U relative to Al, Fe,...) emission of neutrons is strongly favored over that of LCPs and at higher excitations, when comparing Au to U, one observes again a shift of the measured distributions toward larger neutron multiplicity as a result of a further reduction of neutron separation energies. The total number of evaporative particles is strongly correlated with E ∗ indicated in Figure 10.43 by the arrow for Al. The multiplicities are employed to deduce excitation energy distributions dσ/dE ∗ as described in detail in Ref. [379, 663, 664]. The procedure allows to analyze observables as a function of E ∗ . In figure 7 of Ref. [233], a comparison of such a correlation between experiment and filtered (cf. Eq. (10.3)) INC+GEMINI calculation has been given. The particular advantage of the presentation in Figure 7 of Ref. [233] is that the correlation is shown both for GEMINI (evaporation only) as well as for the full calculation (INC+ GEMINI). The z-scale in that figure presents the production cross section expressed in mb per Mn and MLCP units as a function of LCP and neutron multiplicity. Particle multiplicities-neutron production Of significant interest for a wide range of applications and fundamental research, in particular at the crux of spallation neutron sources, transmutation of nuclear waste in ADS, and shielding issues


100

13Al

47Ag

26Fe

67Ho

28Ni

74W

29Cu

79Au

40Zr

82Pb

41Nb

92U

50 0 100 50 0

ds /dMn (mb)

100 50 0 100 50 0 100 50 0 100 50 0

0

5

10 0 10 Neutron Multiplicity Mn

20

30

Fig. 10.44 Measured (symbols) and calculated (histograms) neutron multiplicity distributions of NESSI for 1.2 GeV p+Al,...,U. Note different neutron multiplicities Mn s scales for the left and the right panels.

are also neutron-production double-differential cross sections in GeV protoninduced spallation reactions. In Figure 10.44 neutron multiplicity distributions for 1.2 GeV p+Al,...,U are compared with calculations following the INCL2.0/GEMINI model. These distributions are projections onto the Mn axis of two dimensional presentations like the one given in Figure 10.43. Although generally described satisfying by, e.g., INCE codes, neutrons are more difficult to detect than protons or LCPs. Experimental double-differential neutron-production spectra represent a valuable observable also for validating new model developments or improvements [117, 171, 172, 261]. It is also interesting to look at neutron multiplicities Mn

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as global properties of neutron spectra which are not easily revealed by their inspection. An extensive overview on the observable Mn for thin targets is compiled in Refs. [115, 172, 622, 631]. In order to compare the calculated with the measured distributions the calculations are shown before (dashed histogram) and after (solid, shaded histogram) folding with the neutron energy dependent detection efficiency ε of the BNB (cf. Section 10.3.2.1). Note the different Mn scales for the left and right panels. When taking ε into account, the INCL2.0/GEMINI calculations agree well with the measured distributions. For heavier targets and low Mn there exists a discrepancy between experiment and calculations. A similar discrepancy was reported [730] and ascribed to the sharp cut-off modeling of the nuclear density distribution in INCL2.0. Light charged particle and IMF production The emission of composite particles such as 2,3 H, 3,4,6 He, and Li has long been recognized as an important decay channel in spallation reactions. The pioneering experiments of Poskanzer et al. [738] have shown that the emission of these composite particles could not be accounted for by a single evaporation mechanism. Indeed, the emission is far from being isotropic in the emitter frame and the energy spectra exhibit a high-energy tail in excess of the usual evaporative component. Further experiments [739] have confirmed these findings at different bombarding energies and for several target nuclei. It was also shown that the neighboring isotopes 3 He and 4 He have very different behaviors with a strong and weak component of nonevaporative particles, respectively. However, in all these studies the underlying reaction mechanisms could not be investigated thoroughly due to the lack of additional experimental information. Taking advantage of the very exclusive data brought by the NESSI detector arrays a detailed study of composite particle emission was conducted. As an example of LCP production, double-differential energy spectra of 1,2,3 H and 3,4 He ejectiles following 1.2 GeV p-induced reactions on Ta target as measured by the NESSI collaboration at COSY-Juelich are shown for different angles with respect to the incident proton in Figure 10.45. The experimental data clearly feature two components: an evaporation component dominant for all angles at low-kinetic energies and a high-energy component; The higher the kinetic energy of the ejectiles the smaller their emission angle on average in respect to the incident proton beam. Here [228] for the theoretical description the INCL2.0 [170] intranuclear cascade code is coupled to the evaporation code GEMINI [229]. Only for protons both the components can be well described. Due to the lack of composite particle emission in the early phase of the reaction in the INCL2.0 model, the high-energy tails of the spectra for d, t, and 3,4 He are not described by the calculations. The shape of the calculated evaporation component (shaded histogram in Figure 10.45) however is also well reflected for composite particles. The calculations confronted with the measured total He (3 He+4 He) production cross sections as shown in Figure 10.46 generally tend to underestimate the experimental data because of 5–20% pre-equilibrium emission (not taken into


75°

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Energy [MeV] Fig. 10.45 Energy spectra of 1,2,3 H and 3,4 He for 1.2 GeV p+Ta. Dots: experimental data, shaded histogram: calculated evaporation spectra, dashed histogram: pre-equilibrium protons as calculated by INCL2.0 (after Herbach et al. [228]).

account in INCL+GEMINI). Although it is obvious that there is some discrepancies between different sets of data, Figure 10.46 clearly exhibits the need to solve deficiencies or discrepancies of different models. A comprehensive set of published data and NESSI results for He production cross sections is also compiled in the EU-Hindas Report [543]. Previous measurements essentially exploited mass spectrometry methods [740, 742] for gas extracted from irradiated samples and only a few measurements are based on methods with E − E telescopes for isotope, mass, and energy identification [414, 738]. As for example for p+Fe, the measured helium production cross sections in the NESSI experiment (440 ± 44 mb) are about a factor of two

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INCL2 + GEMINI LAHET(RAL)

2 Total helium production cross section s He [b]

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LAHET(ORNL) HERMES 1.5

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0.5 Herbach2005 Michel1995 0

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Enke1999 Leya2005 80

100

Fig. 10.46 Total He (3 He+4 He) production cross sections for 1.2 GeV p+X (NESSI). Symbols: experimental data: Herbach2005 [228], Enke1999 [114], Michel1995 [740], Leya2005 [741]; Lines: calculation by INCL2.0+GEMINI, LAHET and new version of INCL (cf. Section 5.2.1 on page 209) (after Herbach et al. [228]).

smaller than the other experiments (792 ± 55 mb) [740]. This corresponds to a discrepancy of a factor of 1.8 or about 6 standard deviations. On the other hand, the present results for Fe and Ni agree quite well with the older measurements of Goebel et al. [742]. A large spread of experimental He cross sections found in literature is also evident when comparing model calculations. The cross sections as calculated with the LAHET (with RAL fission model) and the HERMES(with RAL fission model) are somewhat smaller than the NESSI data in case of light targets (Fe,Ni,...) while for heavy targets LAHET (with RAL fission model) and HERMES predict larger cross sections and LAHET(with ORNL fission model) smaller ones (cf. Section 5.2.1 on page 209). The calculations with the INCL2.0/GEMINI model combination are in good agreement with the data throughout all nuclei and incident energies [664]. Recent results of measured mass distributions in the reverse kinematic reaction of 800 MeV/nucleon Au+p are also well described [743] by the INCL code coupled, however, to a different evaporation/fission code [221, 222]. As pointed out above the discrepancies between the data and the HERMES and LAHET calculations are associated with the finding that these codes overestimate


Proton (1.2 GeV) + Au

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Fig. 10.47 Angular distributions dσ/d of 1,2,3 H and 3,4 He for 1.2 GeV p+Au. Symbols: experimental data, lines calculation by INCL2.0/GEMINI (after Herbach et al. [228]).

E ∗ after the prompt INC and with different handling of the Coulomb barriers in the employed evaporation codes. For 1.2 GeV p+Au in Figure 10.47(a), the angular distribution of disentangled evaporation and pre-equilibrium components are shown in Figure 10.47(b) (for details see [228]). For all particle species the evaporation exhibits an isotopic behavior, while more directly emitted particles show larger forward/backward asymmetry. The strong declines of the angular distributions for the pre-equilibrium particles are almost identical for all H and He isotopes. Also the differential cross section for particles with equal Z, but different A does not vary substantially. However, as expected, lower kinetic particles (typically from evaporation process) show an almost isotropic angular distribution, and the slopes are similar for the various isotopes (cf. Figure 10.47(a)). The absolute differential cross sections, however, differ considerably since they are governed by the binding energies of the composite particles. The production cross sections of 6,7,8,9 Li and 7,9,10 Be isotopes for 1.2 GeV protons on different targets (C to Au) as well as the total production cross sections of Li and Be are shown in Figure 10.48. Note that for pre-equilibrium protons the angular dependence is well described in the INCL2.0 model. It would be certainly worth to compare the current experimental data [228] with, e.g., the latest version of INCL4.3 [117] including a coalescence

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Fig. 10.48 Production cross section of (a) and 7,9,10 Be isotopes (b) for 1.2 GeV p+X. Bullets •: NESSI [228], triangles: Dostrovsky [744], stars: Michel [740] data, lines: calculation by INCL2.0/GEMINI.

formalism allowing the cluster emission of composite nucleons (d, t, 3,4 He) in the early phase of the reaction. The production of all individual isotopes does not strongly depend on Z. When looking more carefully at the energy spectra of the IMFs (shown in Figure 10.49), ones more as expected the combination of INCL2.0/GEMINI fails to describe the high-energy tails of the energy spectra. Nevertheless in Figure 10.48 the calculated angle and energy-integrated production cross sections agree generally rather well with the NESSI [228, 740, 744] data, because the pre-equilibrium component may amount to the total cross section only on the percent level. The lines representing the model prediction are reflecting the ejectiles coming from evaporation model only, i.e., GEMINI. The experimental data on 7 Be and 10 Be ejectiles measured for low Z-targets by mass spectrometry [740] coincide with the systematics of the NESSI experiment [228]. In a similar presentation one observes the multiplicity/production cross sections of the neutron rich 6 He isotope strongly increasing with atomic number Z of the bombarded target (cf. Figure 10.50) – a very similar behavior to the one which is observed for the ‘‘neutron rich’’ triton. In contrast to the 3,4 He isotopes, for 6 He the INCL2.0/GEMINI calculations overestimate the experimental results of Herbach et al. [228] by approximately 30%. Please note that in the NESSI data for 1.2 GeV proton-induced reactions mentioned above [228] all cross sections have been corrected not only for double hits, geometrical efficiencies, and background as measured with empty frames and dead times, but also for the contribution below the lower energy threshold (Table 2, page 434 of [228]) of the Si detectors. Indeed this had been done by extrapolating


Production cross section ds/dE [mb MeV−1]

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Fig. 10.49 Energy spectra of Li and Be for 1.2 GeV p+Ti, Ag and W; bullets •: [228], histogram: calculation by INCL2.0/GEMINI.

6He

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Fig. 10.50 Multiplicity of 6 He isotopes as a function of the target charge number Z, bullets •: [228], line: INCL2.0/GEMINI.

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to zero energy for the cross-section contribution. The hydrogen and He production cross sections – including errors – are given for Al, Ti, Fe, Ni, Cu, Zr, Ag, Ho, Ta, W, Au, Pb, Th, and U in Table 3 of Ref. [228]. Thresholds are specified in Table 3. Tables 4 and 5 give the cross sections for the individual p,d,t and 3 He, 4 He and 6 He isotopes, respectively. Table 6 provides the very same targets the production cross sections for 6 Li, 7 Li, 8 Li,9 Li, 7 Be, 9 Be, and 10 Be. Timescale of fission in GeV proton-induced reactions on Au, Bi, and U Integrated fission cross sections have been measured for a large variety of projectiles on thorium and uranium. Simbel et al. [745] studied proton-induced fission on Th and U in the energy range 0.1–30 GeV . These data exhibit an almost constant cross section up to 1 GeV and a decline beyond. This decline is interpreted by Simbel as the onset of multifragmentation [746]. Also for d-, α-induced fission [747] and heavy ion-induced fission of U and Th [748] the fission cross section decreases with incident energy of the projectiles, respectively. This tendency is corroborated by Ar+Th measurements [749], which show complementary an increase of (heavy residual) HR cross sections with increasing incident energy. Ledoux [710] measured the relative fission probability of 475 MeV –p and 2 GeV – 3 He-induced reactions on U as a function of evaporated neutrons. The experimental challenge is to measure fission probabilities of hot nuclei at low spin up to highest temperatures as a function of E ∗ , also including nuclei with higher fission barriers. For this purpose Hilscher et al. [219, 621] proposed to study fission induced by high energetic light particles using the 4π detector array NESSI. As discussed in Section 9.4 light-ion-induced particles provide an ideal tool for the investigation of fission and other decay channels almost free of inducing collective excitations in the target nucleus. In this section, the fission probability Pf (E ∗ ) of highly excited target like nuclei produced in reactions of 2.5 GeV protons on Au, Bi, and U is discussed as a function of excitation energy E ∗ , whereby E ∗ is deduced event-wise from the multiplicity of evaporated light particles[379, 663, 664, 750]. The overall time elapsed from the equilibrium deformation up to the scission (10−20 s) is known to be quite long compared to other processes like the emission of neutrons [751]. Although the entire fission process is slow, the decision to fission can be fast. For the three targets, the dependence of the inelastic cross section, σinel , and the fission cross section, σf , as a function of E ∗ [735, 736] was studied. Their ratio, the fission probability Pf (E ∗ ) = σf (E ∗ )/σinel (E ∗ ), provides the best possible evidence for the presence of dissipative or transient effects in fission and is shown in Figure 10.51. Fission fragments have been separated from heavy residues and intermediate mass fragments by constraints as discussed in more detail in Ref. [750]. The total fission cross sections deduced the amount to σf = 200 ± 60, 320 ± 50, and 1350 ± 120 mb for Au, Bi, and U, respectively, in agreement with Ref. [753]. Despite the large differences in fissility and fission barriers, Bf ≈ 5, 12, and 21 MeV of the initial nuclei U, Bi, and Au, respectively, at the highest E ∗ of 1000 MeV, Pf amounts to 30% for all three target nuclei. Intranuclear-cascade plus statistical model calculations satisfactorily reproduce the observed evolution of Pf


Fission probability Pfission

1

Exp. Simulation U Bi Au Au, td

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0.6

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400 600 800 Excitation energy E* [MeV]

Fig. 10.51 Comparison of the experimental (symbols) and the simulated (histograms) fission probability as a function of E∗ . The simulation for p+Au with af /an = 1.05 and τd = 2 × 10−21 s according to Ref. [752] is shown by the chain dotted histogram (after Jahnke et al. [750]).

with E ∗ – provided that no extra transient delay is introduced. A dynamical fission hindrance or transient time which suppresses fission at higher E ∗ or reduces σf , might be compensated by a larger value of the level density at the saddle point. As for example the chain dotted line for Au target nuclei in Figure 10.51 shows that for τd = 2 × 10−21 s and af /an = 1.05, the calculation fails to describe the actual Au fission data. The differential observation of dPf (E ∗ )/dE ∗ allows an independent determination of af /an and τd . Concluding, fission is decided upon very fast and early in the long deexcitation chain toward scission, e.g., at high excitations. Consequently the large amount of remaining excitation must be carried off, either during the descent from saddle to scission or after scission (cf. Section 3.7.1 and the schematic illustration in Figure 3.9(a)). Indeed according to the calculation at E ∗ = 800 MeV, only between 11%(Au) and 23%(U) of all light particles are emitted before reaching the saddle, or expressed in terms of pre- and postsaddle LCP LCP = 1.9, 1.9, 3.3, and Mpostsaddle = 11.0, 10.4, 7.6 for Au, Bi multiplicities, Mpresaddle and U, respectively. Angular correlations between α-particles and fission fragments (FFs) and the measurement of pre- and postscission α-multiplicities allow insights into the fission dynamics. In the NESSI experiment, post-scission LCP emission can be deduced from the angular correlation with the fission axis. The energy spectra of α-particles for three ranges of emission angles relative to the motion direction of the light fission fragment is shown in Figure 10.52. The spectra are plotted in the center-of-mass system of the FFs from the reaction 2.5 GeV p+Au at E ∗ = 600–900 MeV. The different contributions from the compound nucleus prior to scission (uncorrelated to the scission axis), and from the two FFs which vary in shape and intensity with θα-LF are indicated. The sum of all components (including the neck emission at 70◦ to 100◦ ) shown in

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Fig. 10.52 α-particle energy spectra for three angular domains relative to the motion of the light fission fragment: θα -LF as indicated in the figure; dots: experimental data, lines: contribution from compound nucleus prior to scission (thick continuous lines), from the light (dashed lines) and

10

20

30

40

from the heavy (dotted lines) fission fragments. At 70◦ to 100◦ , a further component for neck emission is added (thin line). The total calculated spectrum is shown by the histogram. Example: 2.5 GeV p+Au at E∗ = 600–900 MeV (after Jahnke et al. [750]).

Figure 10.52 as histograms provides a good approximation at least to the lower energy (Eα ≤ 35 MeV) part, e.g., to the evaporation part of the spectra. The highenergy tails in the measured spectra, instead, originate partly from pre-equilibrium processes and are, since not taken into account in the simulation, deviating slightly from the experimental data. For further details, the interested reader is referred to [750] and references given therein. Vaporization and multifragmentation Besides the evaporation and fission, the event group with IMFs as the heaviest detected fragments (cf. definition of IMFs in Section 3.10) deserves particular attention in the context of multifragmentation (MF). The question is whether these events stem from true MFs, i.e., the complete fragmentation solely into IMFs and light particles, or are only the remnants from events where heavier masses have eluded detection. Such processes can be reliably identified only by using highly efficient detector arrays like NESSI or the one which is reported by the INDRA collaboration [754–756]. The INDRA collaboration observed the onset of vaporization in the Ar+Ni reaction at an available energy of 12 MeV/nucleon for the excitation of the composite system with an overall probability of 5 × 10−6 . In the following the observations made for 1.2 GeV p-induced reactions on Cu (1.1 mg/cm2 ) and Ag (1.5 mg/cm2 ) targets are discussed as a function of E ∗ . It has been demonstrated in Refs. [663–667, 669] that the maximum thermal excitation energy in heavy nuclei (Ho-U) is about 4–5 MeV/nucleon. The conclusion drawn for such energies was essentially, that the nucleus either survives as a self-bound entity (evaporation residue), or undergoes fission. However for lighter nuclei such as Cu, excitation energies larger than the total binding energy of the system are expected and thus vaporization might be reached [362, 439, 445]. The reconstructed experimental excitation energy distributions dσ/dE ∗ for Cu and Ag following p-induced reactions are shown in Figure 10.53 as solid dots.


1.22 GeV p + Cu

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Fig. 10.53 Reconstructed experimental excitation energy distributions dσ/dE∗ ; (•) for p (1.2 GeV) + Cu (a) and Ag (b). Lines correspond to INC-calculations [676, 757]. The bottom scale shows E∗ divided by the average mass of the hot nucleus Ahot = AINC at equilibrium, where AINC is taken from INC model simulation.

The experimental results are in good agreement with the predictions on INC model calculations. Both distributions extend beyond the total binding energy (≈ 8 MeV/nucleon) and, hence, processes such as MF and vaporization are energetically possible in both reactions. For both systems studied, a continuous decrease of the heaviest detected fragment is observed with increasing E ∗ . The ‘‘canonical’’ A = 4 limit is reached only for the p + Cu system, at E ∗ of about 350 MeV or 7.5 MeV/nucleon, taking into account the loss of mass in the INC stage of the process. For the p + Ag reaction, however, the heaviest detected fragment is always larger than about 15 mass units. For the p + Cu reaction about 300 good candidates for vaporization events were detected corresponding to a cross section of about 3 ± 1 mb or 0.3% of the total reaction cross section. The angular distribution of evaporated-like LCPs of this event class is almost isotropic, similar to the distribution of particles from other decay modes, indicating the vaporization of a thermalized source. The total probability for vaporization in the current experiment is found to be a factor of 10 larger than the value reported at similar E ∗ by the INDRA collaboration [754–756]. This discrepancy might be associated with the faster heating mediated by p compared to heavy-ion reactions. The ratio of vaporization cross section relative to the reaction cross section is presented for the p + Cu system in Figure 10.54 as a function of E ∗ . Even for the highest E ∗ (600 MeV or 15 MeV/nucleon) this ratio does not exceed 15%. The dashed line illustrates the same observable predicted by the GEMINI code. As already mentioned note that GEMINI predictions should be taken with some care because the fact that the emission times are large compared to the energy

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Fig. 10.54 Top panel: ratio of vaporization cross section, σv , to the reaction cross section σR for 1.2 GeV p+Cu. The dashed line presents the INC-GEMINI simulation. Lower panels: mean multiplicities of neutrons, LCPs (Z = 1, 2) and IMFs as a

function of E∗ for Cu (left panel) and Ag (right panel). The open (filled) symbols are for all (vaporization) events. The dashed lines illustrate the mean IMF multiplicity for INC plus GEMINI calculations (after Pienkowski et al. [670]).

relaxation times (as is implicitly assumed in sequential-evaporation models) might not be fulfilled at E ∗ larger than about 4–5 MeV/nucleon. However, the excitation energy threshold for statistical decay depends mainly on the mass excess balance and Coulomb repulsion energy, so obviously GEMINI model (cf. Section 3.6 on page 159) is at least able to estimate correctly the observed threshold energy for the p + Cu reaction. For the p + Ag reaction the INC+GEMINI calculations predict the onset of vaporization at excitation energies about 9–10 MeV/nucleon, close to the maximum excitation energies observed in this reaction. This finding corroborates the observed absence of vaporization events among the collected data for the p + Ag system. For both reactions, p + Cu and Ag, the lower panels of Figure 10.54 present the mean multiplicities of neutrons, LCPs (Z = 1, 2) and IMFs (4 ≤ AIMF ≤ 25) as functions of E ∗ . The open and solid symbols in the figure represent all events and vaporization events, respectively. Near the vaporization threshold the ratio MZ=2 /MZ=1 is approximately one and is considerably larger than for other decay modes at the same E ∗ . Beyond the vaporization threshold MZ=2 is decreasing with E ∗ , while Mn is increasing. This finding was also observed by the INDRA collaboration for the Ar+Ni system at 12 MeV/nucleon [754–756]. Close to the


vaporization threshold the INC+GEMINI code also predicts MZ=2 /MZ=1 ≈ 1. The average IMF multiplicity MIMF saturates with increasing E ∗ around MIMF ≈ 1 and MIMF ≈ 2 for the p + Cu and p + Ag systems, respectively. No sudden onset of MF can be observed in the data and the trends in Figure 10.54 do not indicate MF as a conceptually distinct process. Instead the observed multifragment events may reflect statistical fluctuations in the decay modes. This conclusion is justified all the more when looking at the good agreement of multifragment events of the sequential-emission model GEMINI with the experimental data. When defining MF as a process ending with three or more IMFs in the exit channel, one can obtain quantitative characteristics for the so-defined phenomenon. The relative abundances of events with three or more IMFs is discussed in Refs. [664, 670] as a function of E ∗ . A saturation of the probability at about 5% is observed for the p + Cu system, while for the p + Ag system a monotonic increase up to 20% at highest E ∗ is deduced. In agreement with the predictions in Ref. [362], for both systems studied the threshold excitation for such MF is around E ∗ ≈ 4 MeV/nucleon. This agreement, however, does not provide sufficient arguments for a conclusion that the IMF production proceeds in the studied systems according to the general scenario regarded in Ref. [362]. Rather, it is an indication that no exotic scenarios are needed in this case to explain the experimental results. For ion-induced reactions, the phenomenon of MF [658, 675, 734] and the extraction of the time scales [441] have intensively been studied in the past. For p-induced reactions, MF has been measured for the first time. The cross section for the MF process as defined is estimated at 20 ± 5 mb and 30 ± 7 mb for the p-induced reactions on Cu and Ag targets, respectively. The latter value for the p + Ag system is very close to the 35 mb reported [758] for MIMF ≥ 3 events in the 3 He+Ag reaction at 3.6 and 4.8 GeV. In any case, the mean IMF multiplicities of the analysis on NESSI data and the values quoted by Kwiatkowski et al. [675] for 4.8 GeV 3 He+nat Ag and 197 Au reactions are substantially smaller than the published data of Lips et al. [734] for relativistic α-particle-induced reactions. The discrepancies may be based on the larger angular momenta involved in ion-induced reaction mechanisms. 10.3.3 The PISA Experiment

The international collaboration PISA (Proton Induced SpAllation) [656, 664, 759–762] aims at the measurement of total and double-differential cross sections for products of spallation reactions on a wide range of target nuclei (C–U), induced by protons of energies between 100 and 2500 MeV. These cross sections are important for testing physical models of the interaction of protons with nuclei what is of crucial importance for planning and constructing any spallation neutron source. The most restrictive tests of the models are provided by data from exclusive experiments. Therefore, coincidence measurements (high-energy protons with other charged

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particles) are performed besides the inclusive experiments. Additionally experiments with light targets up to Fe will provide the data, which are very important for understanding the anomalous abundance of light elements in the cosmic rays. The mass dependence of the cross sections for full range of targets from C to Au should shed light on the competition of various mechanisms of interaction of protons with nuclei. The PISA project is aiming at a quite similar physics program as NESSI, however with a completely different setup and at an internal beam location. At the internal beam of COSY the investigation of the reactions induced by protons on thin targets (50–200 µg/cm2 ) enables to get the cross sections without the absorption and energy loss involved with the propagation of reaction products in the material of the target. The multiple circulation of the beam in the COSY ring is used to compensate for the small reaction rate of beam protons with the thin targets. The advantages of internal experiments are therefore higher statistics and more precise information on the very tails of double-differential energy spectra, which are important particularly for rare decay channels and low-production cross sections of IMFs. PISA is supported by the European Commission through European Community-Research Infrastructure Activity under FP6 ‘‘Structuring the European Research Area’’ program (CARE-BENE, contract number RII3-CT-2003-506395 and Hadron Physics, contract number RII3-CT-2004-506078) as well as the FP6 IPEUROTRANS FI6W-CT-2004-516520. 10.3.3.1 Experimental Setup of PISA The set up of the PISA apparatus is shown in Figure 10.55. Each of the eight detection arms (as shown in detail in Figure 10.55) mounted at the scattering chamber of the PISA experiment consists of two multichannel plates (MCPs) working as ‘‘start’’ and ‘‘stop’’ detectors for the TOF measurement, a Bragg curve detector (BCD) [759, 760] followed by three silicon detectors of 100, 300 and 4900 µm thickness for particle identification using E-E techniques and kinetic energy measurement of intermediate-mass spallation products, and a set of double layer scintillation detectors – fast and slow (phoswich detectors) in order to identify light charged evaporation and spallation products like p, d, t, and He. It is shown that the TOF plus BCDs provide identification of light heavy ions with mass up to 20–30 and kinetic energy starting from less than 1 MeV/amu. The Bragg curve detector After first successful attempts to use ‘‘Bragg curve spectroscopy’’ to identify highly ionizing particles [763, 764] several detectors exploiting characteristic features of the Bragg curve have been built and used for various applications. The appearance of BCDs has allowed to detect fragments with high precision over a broad range of nuclear charges with low registration thresholds. This has also been demonstrated already in former experiments on fragment production cross sections in carbon at GeV proton beams [765, 766]. The design of the BCD is presented in Ref. [767]. The design features of the BCD are very similar to those of Ref. [768] and references therein. Advantages (as e.g., resistivity to radiation damage and insensitivity to minimum ionizing particles)

10.3 Proton Cooler Synchrotron COSY at Jülich PISA Setup – october 2002 100°

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3

DN 25 ISO–KF DN 25 ISO–KF Electrical Gas Feedthrough out

Turbo pump Membrane pump

(b)

Fig. 10.55 (a) The scattering chamber of the PISA experiment as of October 2002 with two full detector arms mounted at 15◦ and 120◦ equipped with the Bragg curve-, the channel plate-, and the phoswhich-detectors. (b) It shows one of the detection arms in more detail.

9

Si telescope

Membrabe pump

MCP – multichannel plate IMV – intermediate volume BCD – bragg curve detector

357

358


Incident particle

Cathode

800 700

Field shaping rings

Gas input

Amplitude [mV]

600

Gas output

500 400 300 DE

BP

200 100 0

E, R −100

Anode

Frish grid

(a) Bragg curve detector Fig. 10.56 Bragg curve detector (a) as used for spallation studies at the COSY internal proton beam. A detailed description is given in the text. A typical output signal from BCD is shown in (b). The maximum of the curve gives so-called Bragg peak (BP) proportional to the charge of particle. The integral over

0

20

40 60 80 100 120 140 160 Time [samples of FADC]

(b) Typical output signal from BCD the whole length of a signal describes the kinetic energy E of the particle. Other parameters like the range of the particle in the detector medium, R, or the partial energy losses DE can also be extracted from the shape of the curve.

of BCDs compared to alternative detectors (gas-semiconductor ionization chamber, solid-state detectors, CsI(Tl)-crystal scintillators, etc.) are outlined in Ref. [768]. The BCD as shown in Figure 10.56(a) is in principle an ionization chamber with a gas volume of 22 cm in length and 5 cm in diameter. It is sealed off at the entrance by a 3 µm thick carbon-coated mylar foil supported by a wire mesh, which will be operated at ground potential and at the rear end by an anode as printed board. The mesh supporting the entrance window is electrically connected to the cathode in order to avoid undesirable charge collection near the window. The Frish grid, which defines the ionization sampling section in a 2-cm distance from the anode, is made of 20 µm gold plated tungsten wires with 1 mm spacing. The voltage of +1800 V between the Frish grid and the entrance window is divided by a resistor chain, which is connected to nine field-shaping rings in order to maintain a homogeneous electric field over the active detector volume. All internal parts are

10.3 Proton Cooler Synchrotron COSY at Jülich Tab. 10.19

Main parameters of the Bragg curve detector.

Active length Frish grid to anode gap width Cathode voltage Frish grid voltage Anode voltage Number of guard rings Cathode Anode Type of gas Pressure

200 mm 19 mm 0 V grounded 2400 V 2900 V 19 Mylar foil (3.5 µm) Mylar foil (1.5 µm) Isobutane (99.9% purity) 300 mbar

fixed to an isolating skeleton made of plexiglass. The particles enter through the cathode and leave an ionization track parallel to the electric field. For charged, nonrelativistic particles the Bethe–Bloch formula for its specific energy losses in a given medium can be simplified to: −dE/dx ∝ cZ2 /E, where Z and E are the atomic number and kinetic energy of the detected particle, respectively, and c contains all relevant constants together with the quantities characterizing the detector medium. Since the energy loss per single collision is small, dE/dx increases slowly along the particle path. Only when the remaining energy is small, dE/dx increases rapidly forming the so-called Bragg peak (BP). The electrons along the track drift through the grid and are viewed as an anode current. The output signal from the anode as a function of time is proportional to the energy-loss distribution of the detected particle along its path through the detector. The atomic number of the incident detected particle is therefore related to the maximum pulse height, which corresponds to the Bragg peak and the total kinetic energy of the particle is obtained from the integration over the total output signal. The detector is filled with isobutane5) and operated at a pressure of about 300 mb. Practically, for a given gas pressure the voltages for anode and the Frish-grid shall be increased as long as the amplitude of the output signal is saturated and the signal length reaches the minimum. It indicates that the recombination of the electrons traveling through the detector is minimal, all of them are passing the Frish grid and are collected at the anode. The principle of Bragg curve spectroscopy is given in Figure 10.56 where a typical output signal of a BCD is presented. The main parameters of the BCD as being used at PISA are summarized in Table 10.19. In addition to the usual charge identification by Bragg spectroscopy in the PISA experiment an isotope separation for almost all detected particles was achieved. The basic experimental information concerning the BCD was received from a VME-flash ADC module (CAEN Mod. V729A, 40 MHz, 12 bit) allowing to perform data processing of about 1000 sample Bragg curves per second. In order to test 5) Since isobutane is characterized by 30% lower effective ionization potential compared to argon or the so-called P10 mixture (90% of

argon, 10% methane), the number of primary released electrons is increased.

359

360

10 Proton-Nucleus-Induced Secondary Particle Production Tab. 10.20 Lower (Emin ) and upper (Emax ) detection threshold for various isotopes in the PISA experiment.

Particle Emin (MeV) Emax (MeV)

p

d

t

2.0 160

2.6 215

3.0 250

3

He

2.0 580

4

He

2.5 650

6

Li

4.0

7

Be

4.5

10

B

9.0

11

C

11

14

N

14

16

O

16

the reliability of the flash-ADC, the Bragg signals were also digitized with two conventional standard NIM ADC modules. Data were taken with both methods and showed agreement within less than 1%. If the particle was stopped in the active chamber volume, apart from the measured TOF, the following values were calculated from the pulse shape: the integral of the specific ionization over the track (total kinetic energy E of the particle), the maximum of the Bragg peak from the maximum of the specific ionization of the ion (BP proportional to Z), the duration R (corresponding to the range in the BCD gas volume) and a partial integral from the specific ionization at the beginning of the track (∝ E/dx ). Isotope identification has been performed by using the correlations between the parameters R, E, E and TOF. The BCD is capable of measuring isotopic distributions of fragments ranging from Z = 2 to Si down to emission energies as low as 1 MeV/nucleon. The phoswich detectors While the energy and charge of the heavier spallation products (Z ≥ 3) will be determined using BCDs [767] combined with channel plate TOF detectors [664], light spallation products (Z = 1, 2) will be measured by employing phoswich detectors [767] placed behind the BCDs. PISA is using the phoswich scintillation detectors of conical-hexagonal shape, produced by BICRON Corporation. The face of the detector is a 1 mm thick with a slow 940 ns decay time CaF2(Eu) scintillator, acting as an energy-loss (E) detector, and a 313-mm thick fast scintillator BC-412 with a 3.3 ns decay time, acting as energy (E) detector. Particle identification is possible via E − E technique for H, and He isotopes. The front cross section of the phoswich detector is a hexagon of 25.2 mm diameter. In these phoswich detectors 10-stage Hamamatsu HTV 2060 photomultiplier tubes are used. Due to energy losses of particles in the ‘‘thin’’ slow scintillator the energy range of correctly detected light particles is 15–150 MeV/nucleon. In summary the very low-energy thresholds and the upper detection limits for different ejectiles as measured with the PISA detectors are given in Table 10.20. Isotope separation of the ejectiles was done by combining the information from multichannel plates via TOF, silicon detector telescopes and Bragg curve spectroscopy (energy deposited inside BCDs) allowing the separation of the following isotopes: 6 Li, 7 Li, 8 Li– 7 Be, 9 Be, 10 Be– 10 B, 11 B– 11 C, 12 C, 13 C, 14 C and 13 N, 14 N [760]. In [761], the double-differential cross sections (d2 σ/ddE) were measured for the first time with good statistics for isotopically identified intermediate mass fragments


produced by interaction of 2.5 GeV protons with a gold target. For that measurement the following individual isotopes of the elements from hydrogen to boron were resolved: 1,2,3 H, 3,4,6 He, 6,7,8,9 Li, 7,9,10 Be, 10,11,12 B, whereas for heavier ejectiles from carbon to aluminum only elemental identification was done. Measurements of these double-differential cross sections for nine angles of 12◦ , 15◦ , 20◦ , 35◦ , 50◦ , 65◦ , 80◦ , 100◦ , and 120◦ and the investigation of the interaction of medium and high-energy protons with atomic nuclei are important for providing benchmark data in the GeV incident proton energy range, understanding the complex reaction mechanism itself and testing the reliability of physical models describing the fast intranuclear cascade INC-phase as well as the subsequent statistical decay from an equilibrated or thermalized hot nucleus. The GeV proton-induced reactions on nuclei compiled by the PISA collaboration representatively discussed further are of particular interest for developing new models for the description of highly energetic composite particles [117]. 10.3.3.2 Experimental Results of PISA As mentioned the experiments at the internal beam of COSY allow to perform the investigation of the reactions induced by protons on thin targets (of the order of 50–200 µg/cm2 ); thus they enable to get the cross sections without uncertainties (e.g., absorption and energy loss) involved by propagation of reaction products in the material of the target. The multiple circulation of the beam in the COSY ring is used to compensate for the small reaction rate of beam protons with the thin target and to allow for the measurement with optimal counting rates of the order of 1000–2000 s−1 for a total intensity of about 1010 protons in the ring. The constant reaction rate is achieved by a negative back coupling between the counting rate and degree of overlapping of the proton beam with the surface of the target via controlled shifting of the beam in respect to the axis of the COSY beam line. Thus such an internal beam experiment offers a unique possibility to measure efficiently and precisely the cross sections on thin targets. As discussed in detail in Ref. [761], the energy spectra for all nuclear fragments, determined at several scattering angles representatively shown for 4 He and 7 Li fragments in 2.5 GeV proton-induced reactions on Au in Figure 10.57, appear to be of Maxwellian shape with an exponential high-energy tail. The low-energy part of the distribution is almost independent of the angle, but the slope of high-energy tail of the spectrum increases monotonically with the angle.The contribution strongly varying with angle is present at higher energy in all experimental spectra. The slope of this anisotropic energy contribution increases with the angle, which can be interpreted as an effect of fast motion of an emitting source in the forward direction. The shape of the angle-independent part of spectra can be reproduced by the twostage model of the reaction, e.g., an intranuclear cascade of the nucleon–nucleon and meson–nucleon collisions followed by statistical emission from an equilibrated residual nucleus. However, the absolute magnitude of the spectra predicted by two-stage model, using Boltzmann–Uehling–Uhlenbeck model for the intranuclear cascade and the generalized evaporation model (GEM) of [254] for statistical emission of fragments,

361


100

100 Au (p, 4He)

10

Au (p, 7Li) 10−1

1000

1000

1 10−2 0.1 10−3

0.01

100

100 d2 s/dΩdE [mb sr −1 MeV−1]

362

650

10

650

10−1

1 10−2 0.1 10−3

0.01

100

100 160

10

160

10−1

1 10−2 0.1 10−3

0.01 0.001 0 (a)

60

120

180

240 0 30 Energy [MeV] (b)

Fig. 10.57 Energy spectra of 4 He (a) and 7 Li particles (b) for 2.5 GeV p+Au. The bullets • [233] show the NESSI results and the ◦ [761] show the results of the PISA experiment for the corresponding emission angles.

60

90

120

150

10−4 180

The lines show the prediction by evaporation of 4 He and 7 Li evaluated by means of GEM program [254] from excited residual nuclei of the first stage of the reaction with properties extracted from BUU calculations.

is in agreement with the experimental data only for the light charged particles the H and He ions. Furthermore, the theoretical cross sections underestimate significantly the yield of heavier fragments at high kinetic energies for all ejectiles. This indicates that another mechanism plays an important role besides the standard two-stage mechanism. The nice agreement shown in Figure 10.57 of the NESSI data [233] confronted with the PISA data [761] on an absolute normalization scale(!)

10.3 Proton Cooler Synchrotron COSY at Jülich 10

1 3H

Cross section d2s/dEdΩ [mb MeV−1 sr−1]

1 0.1

0.01

0.01

0.001

100

100 2H

10

4He

10

1

1

0.1

0.1

100

10

10

6He

0.1

1H

3He

1

1

0.1

0.1

0.01

0.01

0.001 0 (a)

50

100 150 200 250 300

0

30

60

90

120

150

Particle energy [MeV] (b)

Fig. 10.58 Energy spectra of 1,2,3 H (a) and 3,4,6 He particles (b) for 2.5 GeV p+Au. The ◦ are results of PISA experiment [761] for the corresponding emission angles. Dashed lines show the prediction of evaporation of H and He isotopes evaluated by means of GEM

model [254] from excited residual nuclei of the first stage of the reaction with properties extracted from BUU calculations. The full lines correspond to phenomenological model of two emitting sources as described in [761] (after Bubak et al. [761]).

clearly give confidence to the completely independently analyzed experimental data. In Figure 10.58, the open circles represent typical spectra of protons, deuterons and tritons measured in the PISA experiment [761] by telescopes consisted of silicon semiconductor detectors and 7.5 cm thick scintillating detector of CsI placed at scattering angle of 65◦ with respect to the proton beam. The dashed lines show evaporation contribution evaluated by means of the BUU and GEM evaporation model, whereas the full lines correspond to phenomenological model of two emitting sources described in detail in [761]. Note the change of the scale for the triton spectrum. The right panel of Figure 10.58 shows typical energy spectra of helium ions 3,4,6 He measured by the telescopes of PISA consisted of silicon semiconductor detectors placed at scattering angle of 35◦ with respect to the proton beam – open circles. Note different vertical scales for each spectrum. The lines have the same meaning as for the H isotopes in Figure 10.58(a). Figure 10.59 shows the typical spectra of lithium ions 6,7,8,9 Li and 7,9.10 Be particles and Figure 10.59 for 2.5 GeV p+Au measured in the PISA experiment. The lines have the same meaning

363

10 Proton-Nucleus-Induced Secondary Particle Production 0.1

0.1

0.01

9

Li

10

Be

1E-3

0.01

1E-4 1E-3

0.1 8

1E-3

Li

ds/dΩdE [mb/sr*Mev]


0.01

1E-4 1 7

0.1

Li

0.01 1E-3

0.1 9

0.01

Be

1E-3 0.1 7

1

Be

0.01

0.1

6

Li

0.01

1E-3

1E-3 1E-4

1E-4 0

50

100

150

200

250

0

30

60

E [MeV]

90

120 150 180

E [MeV]

Energy spectra of 6,7,8,9Li

(a)

Energy spectra of 7,9,10Be

(b) 0.1

0.1 B

0.01

O

0.01

12

1E-3 1E-3

1E-4

0.1 11

B

0.01 1E-3



364

0.1 10

0.01

N

1E-3 1E-4 1 C

0.1

B

0.01

0.1

0.01 1E-3

1E-3 1E-4

1E-4 0

30

60

90

120 150 180

0

30

E [MeV] (c)

Energy spectra of 10,11,12B

60

90

120 150 180

E [MeV] (d)

Energy spectra of C,O,N

Fig. 10.59 Energy spectra of 6,7,8,9 Li, 7,9.10 Be, 10,11,12 B, and C, O, N particles for 2.5 GeV p+Au. The ◦ are measured results of PISA [761] for 35◦ with respect to proton beam. The dashed and full lines have the same meaning as in Figure 10.58.


4He

d2s/ dEdΩ [mb sr−1 MeV−1]

101

7Li

10−1

100

10−2

10−1

10−3

9Be

10−1

11B

10−2

10−2 10−3

10−3

0

50

100 0 50 Energy [Mev]

100 150

Fig. 10.60 Typical spectra of 4 He, 7 Li, 9 Be, and 11 B ejectiles measured at 35◦ with respect to incident beam for three proton energies of protons, 1.2 GeV (open circles), 1.9 GeV (full squares), and 2.5 GeV(open triangles) incident on Au targets. The cross sections at 2.5 GeV proton beam energy were published in Ref. [761] and the data at 1.2 and 1.9 GeV are taken from Ref. [762].

as those of Figure 10.58. Finally, Figure 10.59 exhibits the spectra of boron ions 10,11,12 B and spectra of carbon, nitrogen, and oxygen ions and Figure 10.59 without isotopic separation measured in the PISA experiment [761] for 2.5 GeV p+Au at 35◦ with respect to the incident proton beam. Double-differential cross sections d2σ/ddE as a function of scattering angle and energy of ejectiles were also measured as a function of incident kinetic energy of the protons. This allows to study the incident particle energy dependence of the light charged particles production. A detailed analysis of the energy dependence of intermediate mass fragments will be published in a forthcoming thesis by PiskorIgnatowicz [769]. Typical spectra of isotopically identified ejectiles obtained in the PISA experiment as a function of bombarding energy are shown in Figure 10.60. As can be seen in the figure, the shape of spectra does not vary significantly with beam energy. The main effect present for all products is monotonic increase of the absolute value of the cross sections with beam energy. Furthermore, all the spectra for all bombarding energies are bell shaped with two components as described earlier. The low-energy component of the Gaussian shape is attributed to evaporation from an equilibrated, excited nucleus, and the high-energy exponential component my be interpreted as a contribution of a nonequilibrium mechanism. The data for the LCPs represented in Figure 10.60 for alpha-particles have similar character and energy dependence as those for IMFs; however, the nonequilibrium component is more pronounced. The final PISA double-differential cross section

365

10 Proton-Nucleus-Induced Secondary Particle Production 175 MeV p + Ni 100 deg

175 MeV p + Ni

101

0.1 20 deg × 1

10−1 20 deg × 0.5 10−2

0.01 ds/dΩdE [mb/sr∗MeV]

100 ds/dΩdE [mb/sr MeV]

366

100 deg × 0.25

10−3

2H

1E-3 H×1

3

1E-4

3He

1E-5

6Li

× 0.1 7Li

1E-6 1E-7

7

10−4 0 (a)

40

80 120 E [Mev]

0

160

Energy spectra for protons

×1

(b)

20

× 0.1

4He

× 0.01

80

100

× 0.01

Be × 0.001 40 60 E [MeV]

Energy spectra for d, t, 3He, 4He, 6,7Li, 7Be

Fig. 10.61 Kinetic energy spectra for protons at 20◦ , 65◦ , and 100◦ following 175 MeV p+Ni. Symbols are data of the PISA experiment, and lines are from Ref. [770]. The PISA data of d, t, 3 He, 4 He, 6,7 Li, 7 Be spectra for 175 MeV p+Ni and ejectiles measured at 100◦ with respect to incident beam.

experimental data files including ejectiles up to Z = 13 for 1.2 and 2.5 GeV p+Au will be made accessible in electronic form (see also Section 10.3.3.3). A detailed analysis and interpretation of the 2.5 GeV p+Au data can be found in Ref. [761]. The 1.2, 1.9, and 2.5 GeV p+Au data and explanations are of Ref. [762]. The PISA experimental data for 175 MeV proton-induced reactions on Ni target are shown in Figure 10.61. Absolute normalization of the data was done by comparing double-differential cross section of protons measured by PISA experiment with the cross sections published by Foertsch at al. [770]. In this paper the energy spectra of protons were determined for many angles, i.e., from 15◦ to 70◦ in 5◦ steps and for 80◦ , 90◦ , 100◦ , and 120◦ , thus it was possible to compare the PISA data at several angles of 15◦ , 20◦ , 65◦ , and 100◦ with those of Ref. [770]. It was found that the shapes of all compared spectra agree very well and, moreover, the ratio of the data of PISA to those of [770] is the same for all angles. Therefore, the normalized cross sections for all products observed in the 175 MeV p+Ni experiment rely on the absolute normalization of proton data found in [770]. The normalization factor was found using the method of weighted least squares for each angle separately and then mean value of four angles were used as normalization factor. The statistical error of normalization factor was found to be close to 3%. Additionally, systematic


error of data of Foertsch (as authors claimed it is close to 10%) should be taken into consideration. The second method of normalization for the 175 MeV p+Ni reaction was performed independently. It consisted in comparing the experimental total cross section for emission of 7 Be with results of parameterization of 7 Be data published in the literature [760]. This method is less accurate as comparing proton spectra with results of [770] because of two reasons: • statistics of 7 Be spectra in reaction Ni(p,7 Be) is much poorer than statistics of proton spectra, • the experimental total cross sections should be derived from the spectra by integration over angle and energy, which is biased with virtually large error because the experimental spectra are not measured at very low 7 Be energies, where the cross sections are relatively large. Nevertheless, both methods gave normalization factors, which agreed with accuracy of about 20% what are quite good results taking into account the fact of a 10% error of normalization, quoted in [770], inaccuracy of total 7 Be cross sections extracted from the spectra, and the error of parameterization of Bubak et al. [760]. The good agreement gives additional confidence in the absolute normalization. As a function of incident proton beam energy, the He-production cross sections on Fe measured by the NESSI group [228], Hannover group [740], SPALADIN group [771], and PISA group [769] are compiled in Figure 10.62. The latest data points of SPALADIN at 1 GeV and PISA at 175 MeV are also included. The SPALADIN result obtained in inverse kinematics of Fe on protons at GSI shows a value slightly above the NESSI data, but is definitely still smaller than the


103

102

Hannover-group NESSI-group SPALADIN-group PISA-group (proton + Ni) INCL4-ABLA INCL4-GEMINI INCL-CLUSTER-GEMINI

101 0

500


Fig. 10.62 Production cross section of 3,4,6 He isotopes as a function of incident proton beam energy. The symbols: NESSI [228], Hannover [740], SPALADIN [771], PISA [769] data, curves: calculation by INCL4.3+ GEMINI/ABLA.

2500

3000

367

368


systematics of Michel et al. [740]. The data shown here for PISA are for Ni reaction, but a comparison should be legitimate, because Fe and Ni are very close in terms of atomic number. Note that for the PISA data [769] the cross sections for the individual of 3,4,6 He isotopes are given at 175 MeV. The Monte Carlo calculation getting closest to the available experimental He data is the INCL4-Clus-GEMINI version (dashed line in Figure 10.62), which accounts by using a coalescence approach for cluster (here composite He particles) emitted in the first fast phase of the reaction. The two solid lines in Figure 10.62 take into account only the He particles being emitted during the slow evaporation phase and therefore as expected the abundance of production cross sections is underestimated in INCL4/ABLA or INCL4/GEMINI. In conclusion the two experiments, NESSI and PISA, have been consulted to validate models with regard to reaction cross sections or reaction probabilities, neutron-, and charged particle production cross sections and angular-, and energy distributions for GeV proton-induced reactions on various thin targets. PISA experiment e.g., has shown to be able to measure the products of pA collisions with Z-identification up to at least Z = 16 and isotope identification to masses up to 13–14% with a particularly low-energy threshold of less than 1 MeV/A. In brief summary the two experiments PISA and NESSI contribute the following information: • NESSI: – 4π detection system for LCPs and neutrons, and neutron multiplicity spectra, – double-differential cross sections for p,d,t, 3,4,6 He, 6,7,8,9 Li, – targets of Al to U with emphasis on Au, – incident proton beam energies 0.8, 1.2, and 2.5 GeV – Refs. [228, 233, 622, 656, 664, 735, 736, 750] • PISA: – no neutron detection, but due to internal beam operation high statistics, very good Z and A identification of reaction products, – double-differential cross sections for p, d, t, 3,4,6 He, 6,7,8,9 Li, 7,9,10 Be, 9,10,11,12 B, 11,12,13,14 C, N, O,... – targets of Al, Ni, Nb, Ag, and Au – incident proton beam energies of 175 MeV on Ni, and 1.2, 1.9, 2.5 GeV on all targets. – Refs. [664, 760–762] The completely independently performed and analyzed experiments NESSI/PISA are in good agreement. A set of experimental benchmark data is provided for the development and test of models capable of describing among other features the emission of the high-energy component of composite particles When however comparing the NESSI and PISA data to the measurements performed by mass spectrometry [740, 741], the measured He production cross sections do not coincide in particular for light targets. The discrepancies between the two experimental methods for light targets are not yet understood. The huge amount of data collected for proton-induced reactions here and elsewhere by

10.4 Production of Residual Nuclides at Various Proton Energies

Michel et al. [740, 741] will be valuable for the identification of deficiencies of existing INC/evaporation codes. 10.3.3.3 Data Library of H and He in Proton-Induced Reactions At the beginning of 2001 the creation of a database [772] for hydrogen and helium production cross sections in a wide energy range up to several GeV on thin targets has been initiated in the framework of the HINDAS project (High and Intermediate Energy Nuclear Data for Accelerator-Driven Systems). The motivation was essentially driven by the lack of cross sections for production of the lightest isotopes. The database is a compilation of experimental cross sections for proton-induced isotope production at energies from a few MeV to 10 GeV. There are also some data for energies up to 30 GeV. Presently, for proton-induced reactions this compilation contains about 15000 data points for 38 targets and 50 elements. All data are derived from available literature and private communications. Each record of the database contains the following information: atomic mass A and atomic mass number Z of the target, incident energy of the projectile in (MeV), type (A, Z) of ejectile, total production cross section in (mb), error of the production cross section in (mb), angle, references, and certain comments. The whole database was written in ‘‘Excel format,’’ in ASCII format, and is also available for users through the Internet (cf. Ref. [773]).


The production of residual nuclei and the formation of integral reaction cross sections for nuclide production by medium- and high-energy protons are basic data for a large number of applications in science and technology [774]. These applications range from the production of cosmogenic nuclides in extraterrestrial matter by galactic cosmic and solar ray protons over space and environmental sciences, accelerators and high-intensity spallation neutron sources (activation of components, radiation protection, and safety), medicine (radionuclide production and radiation therapy), and at least to the technology of accelerator-based nuclear waste transmutation and energy amplifying (cf. Chapters 20 on page 593 and 21 on page 613). In the recent years, the main goal of the experiments in this field was to characterize the reaction mechanisms of proton-induced collisions in matter [775, 776] and to provide a database for applications in astro and cosmic ray physics [34–38], whereas nowadays experiments are considering target materials in applications for high-intensity accelerator technology, for MW spallation neutron sources, and for ADS research [133, 543]. During the recent years, several comprehensive experimental projects were undertaken to complete the existing database. An extensive database is now available on excitation functions published by Michel et al. [340, 740], Schiekel et al. [777], Gloris et al. [778], Leya et al. [741, 779], Ammon et al. [780], and

369

370


Titarenko et al. [781] in the energy range from 0.2 to 2.6 GeV. Other extensive experimental programs aimed to measure a complete isotopic production cross sections and mass yields of spallation residues, e.g., isotopic and mass yield distributions, fission and evaporation residues, by inverse kinetic methods and proton irradiated thin targets up to proton energies of 2.6 GeV. The data are published by Enquist et al. [68, 782], Ta¨ıeb et al. [783], Bernas et al. [784, 785], Ricciardi et al. [350], Villagrasa-Conton [786], and by Titarenko et al. [787, 788]. Because of the large amount of existing data only some examples for structure and important target materials will be demonstrated in the following subsections. 10.4.1 Excitation Functions and Production Cross Sections

The term excitation function is used in nuclear physics to describe a graphical plot of the yield of a radionuclide as a function of the bombarding projectile energy or the calculated excitation energy of the compound nucleus. The characteristic shape of the excitation function can often be related to the nuclear reaction formalism. The special features as the effective energy threshold, the slope of the function, and if there is a maximum at a certain energy are all important and significant to compare different nuclides and their similarities and differences between their reaction mechanisms. For spallation reactions of high-energy protons with matter one may distinguish, e.g., between the processes as spallation, fission, and fragmentation which are characterized by their typical excitation functions [775, 789]. A nuclear reaction should be described then by a complete study of the exit channels (e.g., 1n, 2n, 3n, etc.). The experiments to measure excitation functions are usually performed in the so-called stacked-foil technique [778], where the foils are arranged in small stacks with thin Al foils in front of the stack to determine via the monitor reaction 27 Al(p, 3p3n)22 Na the proton flux. Al foils inside the stack are used to avoid crosscontamination and recoil effects. A sketch of a typical foil stack is depicted in Figure 10.63. For the experiments exclusively high purity materials must be used. Typical weights of individual targets with diameters of 15.0 mm are 55 or 125 mg/cm2 for lead foils (with a purity of 99.99%+) and for Al foils 10 or 33 mg/cm2 (99.999% purity) [778]. In the experiments performed by Titarenko et al. [781], thin 208,207,206,nat Pb and 209 Bi were used having 10.5 mm diameter with 127–358 mg/cm2 together with Al foil monitors with the same diameter and a thickness of 127–254 mg/cm2 . The proton fluences sufficient for excitation function measurements are usually in the order between 1013 and 1016 per cm2 . The most comprehensive and complete database of excitation functions for many different applications were generated by Michel, Gloris and co-workers at incident proton energy ranges up to 2.6 GeV. Most of these experimental data can be retrieved from the EXFOR database [709] a joint effort of the IAEA-NDS, Vienna and The NNDC, Brookhaven. This database contains more than 15000 individual


Saturne II Saclay

TSL Uppsala

15 mm

15 mm

Bi Al

Pb

Au W Ta Bi Pb Au W Ta

2–5cm

10 mm

5 – 10cm

Al

Al

Proton beam Fig. 10.63 A schematic view of foil stacks used at irradiation experiments at the SATURNEII, Saclay and at the TSL, Uppsala, accelerators for lead targets (after Gloris et al. [778]).

cross sections with more than 550 reactions. Exclusively for lead 2000 individual cross sections with about 127 reactions are available. The used target materials cover almost the whole range of elements ranging from carbon up to lead [340, 740]. Various excitation functions especially for lead as target material are published by Gloris et al. [778] and by Titarenko et al. [781]. From the measurements, the phenomenology of the excitation functions exhibits several reaction modes like multifragmentation, shallow spallation products, ‘‘deep’’ spallation products, and different types of fission reactions. Comparison of the measured data with intranuclear-cascade-evaporation models and codes demonstrates the weakness of the predictive power of the models although the models were permanently improved during the last years [778, 780]. As for example the models cannot accomplish the precise distribution in A and Z of the ejectiles. The ‘‘near’’ target products, the shallow spallation products, and the fission products are predicted mostly fairly well for heavy targets, whereas the quality of the prediction of the ‘‘deep’’ spallation products, mass products far away from the target mass, decreases. Fragmentation products are much underestimated by all the theoretical models [781]. Figures 10.64(a)–(d) illustrate examples of measured excitation functions for Al and Fe targets, usually used as structure materials, and for W and Pb targets as examples for materials for spallation neutron production. The data are based on retrieved data files from EXFOR [709]. The experiments were performed at many different accelerator facilities, e.g., at SATURNEII, Saclay (France), which was shut down in 1997, at TSL, Uppsala (Sweden), at PSI, Villigen (Switzerland), at LAMPF, Los Alamos (USA), CERN, Geneve (Switzerland), and at ITEP, Moscow (Russia). Although the presented excitation functions in Figure 10.64 are only a small sample of the large amount of measured data, it is already clearly seen that proton-induced excitation functions may have a complicated structure and are functions of a multiparameter space in energy, matter, and produced individual

371

10 Proton-Nucleus-Induced Secondary Particle Production 102

5×102 102

101

101

Cross section [mb]

Cross section [mb]

Al-target

100

10−1

W-target

100

10−1 13 2 27Al(p,x)4He

−2

10

- (ley98, mic97)

13 11 27Al(p,x)22 Na - (glo01) 4 13 Al(p,x) 27 7Be - (mic95, mic97)

10−3 10

10−2 300


11 (a) Al(p,x)23He, - 22 Na, - 47Be, (data from mic95, glo01, mic97, ley98 [340, 740, 778, 790]

- (mic02)

500 1000 2000 3000 Incident proton energy [MeV]

3×103 103 2He 4

Pb-target

102

10−1

- (mic02)

74 W(p,x)27 Co 60 0

2He 4

102 25Mn 54

4Be 7

27 Co 58

Cross section [mb]

100

- (mic02)

27

Fe-target

101

74 W(p,x)63 Eu 149 0 74 W(p,x)63 145Eu 0

63 63 (b) W(p,x)149 Eu, - 145 Eu, - 60 Co, (data from mic02 [791]

103

Cross section [mb]

372

82 Pb 204

101 80Hg 193

100 80 Ag 110

10−1

52Te 121

10−2 54Xe 134

10−2

10−3 5 10 100 1000 3000 Incident proton energy [MeV]

25 (c) Reactions Fe(p,x)42He, - 47Be, 54 Mn, 27Co - 58

10


47 Ag, - 52 Te, - 54 (d) Reactions Pb(p,x)24He, - 110 121 134 80 Hg, - 82 Pb Xe, - 193 204

Fig. 10.64 Excitation functions of residual isotope production of protons on Al, Fe, W, and Pb targets. The crosssection data of the lower panels are from: (c) for ( 24 He) 27 from [780], for ( 47 Be) from [791], and for (25 54 Mn, 58 Co) from [340, 740, 777], and (d) from ( 24 He, 54 134 Xe) from [741], for 52 80 82 ( 47 110 Ag, 121 Te, 193 Hg) from [778], and for (204 Pb) from [781].


nuclei. It seems not very surprising that many of the theoretical models based on statistical methods fail to predict residual mass production with a required accuracy needed to estimate, e.g., induced radioactivity or isotope production for spallation relevant technologies. Therefore, experiments will remain indispensable for certain applications in spallation research in future. 10.4.2 Isotopic and Mass Distributions of Residual Nuclides

As described in Section 10.4.1 conventional experiments on residual-nuclide production in proton- and neutron-induced reactions are performed by bombarding various target materials with protons or neutrons of the energy of interest and by analyzing the produced species after irradiation, e.g., by their radioactive decay or by off-line and on-line mass spectrometry [340, 740, 778]. These methods can only give a limited insight into the reaction mechanism, because short-lived products, which form the dominant production in most cases, cannot be observed due to the time delay between the irradiation and the measurement. Information on the reaction kinematics is also not easily accessible. In addition, stable nuclides could only be detected with much effort, e.g., by off-line mass spectrometry. As documented in a comprehensive intercomparison [340] the experimental information was insufficient to develop reliable models. In the course of the concerted action ‘‘Lead for ATD’’ [133] a new experimental method has been developed at GSI based on the bombardment of an hydrogen target with heavy projectiles mainly with Fe-, Pb-, and U-ions. These experiments are known as measurements of spallation reactions in reverse or inverse kinematics methods. The spallation reaction products are identified in-flight by mass and atomic number in a high-resolution spectrometer and with information on the reaction kinematics. Most of the results have been published in Refs. [68, 783–786, 792] or may be retrieved from the EXFOR database [709]. 10.4.2.1 Inverse Kinematics Measurements at GSI The employed experimental apparatus at the fragment separator (FRS) [793] at GSI, Darmstadt, Germany, is illustrated in Figure 10.65. A primary beam of, e.g., from 1 A GeV 56 Fe, 208 Pb, 238 U ions incident on a liquid H2 is delivered by the heavyion synchrotron SIS at GSI. The synchrotron produces, e.g., a 500 A MeV 208 Pb pulsed beam with a pulse duration of 4 s and a repetition time of 8 s. The proton target was composed of 87.3 mg/cm2 liquid hydrogen and enclosed between thin titanium foils of total thickness of 36 mg/cm2 . The heavy residues produced in the target were all strongly forward focused due to the inverse kinematics. They were identified using the Fragment Separator and the associated detector equipment shown in the figure. The FRS is a two-stage magnetic spectrometer (cf. Figure 10.65) with a dispersive intermediate image plane (S2) and an achromatic final image plane (S4) with a momentum acceptance of 3% and an angular acceptance of about 15 mrad around the beam axis. Two position-sensitive plastic scintillators placed at S2

373

374


Beam monitor

Dispersive mid/plane Scintillators ToF (BQ)1 L = 36 m

Ionization chamber music ∆E

Incident beam of 56 Fe, 208Pb, 238U Liquid H2 target

(BQ)2 S4 S2 Degrader

Fig. 10.65 Schematic representation of the fragment separator FRS at GSI with the detector equipment (after VillagrasaCanton et al. [786]).

and S4 provided the magnetic rigidity (B ) and TOF measurements, respectively, which allowed determining the mass-over-charge ratio of the particles. For an unambiguous isotopic identification of the reaction products, the analysis was restricted to ions, which passed both stages of the fragment separator fully stripped. To identify all residues in the whole nuclear-charge range up to the projectile, it was necessary to use two independent methods in the analysis. The nuclear charges of the lighter elements, mainly produced by fission, were deduced from the energy loss in an ionization chamber (MUSIC) with a resolution Z/Z = 170 obtained for the primary beam. Combining this information with the mass-overcharge ratio, a complete isotopic identification was performed. A mass resolution of A/A = 480 was achieved. Since part of the heavier reaction products was not completely stripped, the MUSIC signal was not sufficient for an unambiguous Z identification. Degrader thicknesses of about 5 g/cm2 of aluminum were used. The nuclear charge of the products was deduced from the reduction in magnetic rigidity by the slowing down in the energy degrader. The velocity of the identified residue was determined at S2 from the B value and transformed into the frame of the beam in the middle of the target, taking into account the appropriate energy loss. More than 100 different values of the magnetic fields were used in steps of about 2% in order to cover all the produced residues and construct the full-velocity distribution of each residue in one projectile–target combination. The reconstruction of the full-velocity distribution allows to measure reaction products formed in fragmentation and fission reactions due to their different kinematic properties. A detailed description of the experiment and the analysis procedure can be find in [543, 784, 786]. The measured cross sections The production of residual nuclides was investigated for several systems which are particularly relevant for spallation sources and for ADS. The production rates were measured for the reactions 56 Fe,208 Pb,238 U+1 H. Some measurements were also performed with deuterium as the target material [543]. A large amount of cross sections could be measured using different incident

10.4 Production of Residual Nuclides at Various Proton Energies 208 Pb

82

+ 1H (1 A GeV) 126

Z 50

28

30 mb 10 mb 3 mb 1 mb 100 µb 10 µb 1 µb

82

20 8 2

28 20

50 N

Fig. 10.66 Residual nuclide cross sections for the reaction 208 Pb + p on a chart of the nuclides. Primordial nuclei are marked by open squares, the outer line gives the range of known nuclides, and the shell closures are indicated by double lines (by Leray et al. in [543]).

ion beam energies, e.g., for the reaction 56 Fe+p these were 300, 500, 750, 1000, and 1500 A MeV [786, 795], for 208 Pb+p these were 500, and 1000 A MeV [68, 794], and for 238 U+p for 1000 A MeV [350, 783–785], respectively. In average more than 1000 nuclides could be measured and identified per ion and ion energy. These data can be visualized on a chart of the nuclides given as an example for the 208 Pb +1 H (1 A GeV) case in Figure 10.66. The different regions on the chart of the nuclides, produced by spallation-evaporation and by spallation-fission reactions, respectively, can clearly be distinguished. In Figure 10.67, two experimental samples of results using the inverse kinematics technique are given. In Figure 10.67(a), isotopic cross sections of spallation residues are given for the reaction 56 Fe + p at 0.3 A GeV [786], and in Figure 10.67(b) for the reaction 208 Pb + p at 0.5 A GeV [794]. To visualize in the color plots the dependence of the A, Z-distribution as a function of the production cross section low-incident ion energies 0.3 and 0.5 A GeV are chosen. The isotopic cross sections of 56 Fe are shown for elements from Z = 27 down to Z = 10 with uncertainties around 10% for most of the fragments. For the lightest ones, the uncertainty could reach 20% due to the larger applied transmission correction of the target system of residual nuclei. The evaporation residues produced in spallation by the reaction 2086 Pb + p at 0.5 A GeV are depicted for elements from of Z = 69 up to Z = 83 in Figure 10.67(b). Figure 10.68 illustrates the element distribution of all reaction products from the reaction 208 Pb (1 A GeV) + 1 H. The total cross sections from spallation–evaporation and spallation–fission as a function of the charge number from Z = 22 to 82 are given, whereas Figure 10.69 shows the mass distribution of all reactions products, respectively. The total cross sections from spallation-evaporation

375


101

40

100

30 10−1 20 56

Fe + p at 0.3 A GeV

10−2

10

0

0

5

10

15

Cross section [mb]

Atomic mass A

50

20

25

10−3

Charge Z

(a)

210

208

Pb + p at 0.5 A GeV

101

100

190

180 10−1

Cross section [mb]

200

Atomic mass A

376

170 10−2 160 (b)

68 70 72 74 76 78 80 82 84 Charge Z

Fig. 10.67 Isotopic cross section in A, Z-projection of spallation residues production for the reactions 56 Fe + p at 0.3 A GeV (a), and 208 Pb + p at 0.5 A GeV (b). Data are from [786, 794] and from the EXFOR database [709].

and spallation-fission are given as a function of the atomic mass number A = 44–207. A large amount of individual nuclide production cross sections and velocity distributions in the reactions of Fe, Pb, and U ions with hydrogen and also with deuterium have been studied in the inverse-kinetic experiments at GSI, covering most elements between oxygen and uranium. The reaction products were fully identified in atomic number Z and mass number A using the magnetic


Total prodcution cross section [mb]

5 × 102 10

Total element production cross section of the reaction 208 Pb (1 A) + 1H

2

101

100

10−1 10

20

30

40 50 60 70 Charge number Z

80

90

100

Fig. 10.68 Spallation evaporation and fission isotope production for the reaction 208 Pb(1AGeV) + 1 H as a function of the element number Z. Data are from Enquist at al. [68].

Total production cross section [mb]

102

101

100

10

Total mass production cross section of the reaction 208 Pb (1 A GeV) + 1H

−1

10−2 1

50

100

150

200

250

Atomic mass number A Fig. 10.69 Spallation evaporation and fission isotope production for the reaction 208 Pb (1 A GeV) + 1 H as a function of the atomic mass number A (lower panel). Data are from Enquist at al. [68].

spectrometer FRS. With the measured production cross sections combined with the known decay properties it is possible to estimate the short- and long-term radioactive inventories induced in high-power spallation targets.

377

379

11 Proton-Matter-Induced Secondary Particle Production – The ‘‘Thick’’ Target Experiments 11.1 Introduction

While the ‘‘thin’’ target experiments discussed in Chapter 10 are basically useful to validate and scrutinize the nuclear models in spallation physics, ‘‘thick’’ target experiments are mainly performed as integral experiments. These integral experiments employ extended targets with dimensions large enough to generate an internuclear cascade (INC) inside the material and aim at the study particle leakage distributions, particle yields including secondary particle production, energy deposition, and induced radioactivity. Experimentally, the first studies of neutron yields for spallation neutron targets were carried out by Bercovitch et al. [626] using cosmic ray radiation as a proton source between 250 and 900 MeV and by Fraser et al. [59] at the Brookhaven Cosmotron accelerator in connection with the proposed Canadian ING concept [58] (cf. Section 1.3 on page 12). The Cosmotron measurements were performed as ‘‘thick’’ target experiments measuring the total neutron production yield, e.g., the number of neutrons per incident beam proton emitted from the target surface. Measurements were made for Be, Sn, Pb, and Udepleted at proton beam energies between 500 MeV and 1500 MeV. The neutron yield was determined by measuring the activation of Au foils placed at various positions within a large H2 O moderator surrounding the targets, spatially integrated to obtain the total thermal neutron captures in the moderator, and then applying a ‘‘correction” to account for neutron absorptions inside the targets. The Cosmotron data, although taken some 30 years ago, represented the most comprehensive set of measured data available in terms of beam energies and target materials covered for many years. Recently these measurements were re-evaluated by Zucker et al. [796, 797] and compared with new measurements at the alternating gradient synchrotron (AGS) of the Brookhaven National Laboratory (BNL), USA. A joint Canadian–US experimental program, called FERFICON [385, 798–802], Fertile-to-Fissile Material Conversion was initiated during the 1980s. These measurements are being made at the TRIUMPF facility, Canada, with 480 MeV protons and at the LAMPF accelerator, USA, with 800 MeV protons. These


380

11 Proton-Matter-Induced Secondary Particle Production – The ‘‘Thick’’ Target Experiments

extended target measurements have been studied total neutron yields, angular dependent neutron production cross sections, and fertile-to-fission conversion yields inside thorium and depleted uranium targets. Yield measurements are also reported by Lone et al. [803], Becker et al. [804], and Vassil’kov and Yurevich [617] for various targets. In addition, high-energy neutron leakage spectra emitted from thick targets were measured using the time-of-flight (TOF) method at the 184inch cyclotron at Berkley, USA, by Madey and Waterman [125] with 740 MeV protons during the 1970s and by Meier et al. [696, 698] by the TOF method at the LAMPF-WNR facility, USA, applying 116 and 256 MeV protons during the 1980s. With the advent of high-intensity spallation sources and with ADS projects (see comments in Section 10.1 on page 287), more detailed studies on ‘‘thick’’ targets were investigated using a variety of target materials, target geometries and proton beams up to 2.5 GeV. A group of LANL, USA, considered thick target studies at the LAMPF-WNR facility on neutron production devoted to acceleratordriven targets by Morgan et al. [557], referred to as ‘‘Sunnyside’’ experiments. During the 1990s at the SATURNE accelerator measurements of differential neutron production cross sections and multiplicity distributions of thick targets using the experimental apparatus as described in Figure 10.15 on page 306 are given by Leray et al. [543] and by Fréhaut [805] up to proton energies of 1.5 GeV. More details are given in [806, 807]. At the 12 GeV proton synchrotron at KEK, Japan, neutron leakage spectra of thick lead targets up to proton energies of 1.5 GeV were also studied by Meigo et al. [808]. More systematically the NESSI1) collaboration measured neutron multiplicity distributions and energy deposition of thick targets from tungsten, mercury, and lead with various geometries up to incident proton energies of 2.5 GeV (e.g., Hilscher et al. [631], Letourneau et al. [115], Filges et al. [622, 809]). These neutron multiplicity experiments were performed using a 4π liquid scintillator detector at the cooler synchrotron (COSY) at the Forschungszentrum Jülich, Germany. The ASTE2) collaboration investigated measurements of spatial neutron leakage fluxes and energy deposition of a thick mercury target with incident proton beam energies between 1.6 and 24 GeV at the Alternating Gradient Synchrotron (AGS) at the Brookhaven National Laboratory, USA [810–812]. Experiments on waste transmutation, e.g., the CERN-TOF [685] investigations should also be mentioned in this context. The experiments discussed in this chapter are certainly only a selection of the various efforts done worldwide, and the scope of this book cannot be a detailed comprehensive description of all measurements. In the following in order to get an idea of the ‘‘thick target’’ experiments a representative selection of investigations at various laboratories is presented.

1) NEutron Scintillator and SIlicon Detector. 2) AGS Spallation Target Experiment.

11.2 Proton-Induced Thick Target Experiments in the Energy Range 0.1–2.5 GeV

11.2 Proton-Induced Thick Target Experiments in the Energy Range 0.1–2.5 GeV 11.2.1 Experiments to Measure the Neutron Yield of ‘‘Thick’’ Targets

As already discussed in Section 8.2.4 on page 263, the total neutron production or the neutron yield is an integral parameter which may be used to characterize appropriate target materials for spallation sources. Over the years many experiments were performed applying e.g simple water-bath experiments by Fraser et al. [59], Russell et al. [799], Fraser, Garvey and Milton [798, 813], and by van der Meer et al. [814]. Instead of applying H2 O as neutron moderator, West and Wood [815] used polythene blocks [815] to measure neutron yields. The ‘‘Sunnyside’’ experiment collaboration of LANL (USA), the Uppsala university (Sweden), the university of Michigan (USA), and the CEA (France) used a method named manganesebath technique, a concentration of 1% MnSO4 in H2 O [557]. The most advanced experimental determination of neutron yields and neutron leakage spectra emitted from the target surface of thick targets are the event-by-event measurements of the neutron multiplicity with 4π detector systems by Hilscher et al. [631] and Letourneau et al. [115] and the TOF methods considered by Meier et al. [696, 698], Meigo et al. [808], and by Leray et al. [543]. A compilation of available thick target (n/p)-ratios is summarized in Section 11.2.2.2 on page 402. 11.2.1.1 The Brookhaven Cosmotron Experiments The first experiments investigated to measure neutron yields from thick targets are the experiments by Fraser et al. [59, 813] were carried out at the external proton beam of the Brookhaven Cosmotron accelerator with targets of Be, Sn, Pb, and depleted U bombarded by protons having energies from 0.5 to 2.0 GeV. These measurements were considered to provide information for the first high-intensity spallation source project, the intense neutron generator (ING) by Bartholomew et al. [58]. Six different targets were originally studied: cylindrical targets of Sn, Pb, and depleted U of diameter 10.16 cm and length 60.96 cm, one Pb target with diameter 20.32 cm and length 60.96 cm, and a rectangular Be target square-shaped 10.16 × 10.16 cm2 with length 91.44 cm. The targets were inserted in a large cylindrical vessel with a diameter of 182.88 cm and height 182.88 cm filled with H2 O. A sketch of the experimental apparatus, the so-called water-bath experiment is illustrated in Figure 11.1. Figure 11.1 also shows the lucide frame for about 55 detector foils to measure the absolute neutron density distributions inside the H2 O moderator, which surrounds the target. For most of the experiments foils of Cu with a thickness of 0.03 and 0.0175 mm, Au foils with thickness of 0.05 mm, and also Mn foils with a 10% Ni content with a thickness of 0.0175 mm were used. The foils are Al and Cd covered and were 1 cm2 area. The values obtained by the foil activation method are believed to have an accuracy of about ±5% [58]. A series of experiments with Pb targets

381

382

11 Proton-Matter-Induced Secondary Particle Production – The ‘‘Thick’’ Target Experiments Lucite frame with positions for detector foils

12C monitor foil

Proton beam from BNL cosmotron

Vessel filled with H2O

Targets of Be, Sn, Pb, U

1.83 m

Fig. 11.1 The BNL Cosmotron experiment with H2 O vessel, target assembly, and Lucite frame to position the foil and vanadium β-current detectors (after [813]).

of a diameter 10.16 cm were done using measurements with vanadium β-current detectors. The beam current was determined by the 12 C(p,pn)11 C reaction [59]. The logarithms of the measured fluxes were fitted by least squares to polynomials. Integrating the resulting distributions gives the total number of neutrons captured per second in the H2 O volume. This capture rate in a water-bath experiment is then the number of neutrons per incident proton including a correction for the neutron absorption in the target. The method to evaluate thermal neutron fluxes via reaction rate measurements is well known in classical reactor physics. Details are described in standard textbooks, e.g., [63]. The values of neutron capture rates or the neutron production per incident proton for the Cosmotron experiments are shown in Figure 11.2. A summary of experiments of neutron yield production at thick targets is given in Section 11.2.2.2 on page 402 with a summary comparing different thick target experimental results for thick lead targets. The data were used to estimate the neutron source strength of the Canadian ING project [58], e.g., by bombardment of a 20.4 cm diameter × 61 cm long target of Pb by 65 mA of 1 GeV, one yields about 20 × 65 × 6.28 × 1015 ∼ 8.2 × 1018 neutrons s−1 . 11.2.1.2 The Fertile-to-Fissile Conversion Experiments The Fertile-to-Fissile Conversion (FERFICON) experiments were a collaborative effort between the Chalk River Nuclear Laboratory, Canada, and the Los Alamos National Laboratory, USA, either to measure the total neutron yield from stopping


Number of neutron per incident proton

60

50

- U - 10.2 × 61 cm2 - Pb - 20.4 × 61 cm2 - Pb - 10.2 × 61 cm2

40

- Sn - 10.2 × 61 cm2 - Be - 10.2 × 61 cm2

30

20

10

0 0.25

0.5

1.0 Incident proton energy [GeV]

1.5

Fig. 11.2 The measured neutron production yield as a function of the incident proton energy for various thick targets. The lines are a linear fit of the data. Data are the BNL Cosmotron neutron capture rates measured in a water bath from [58].

length targets and the fertile-to-fissile conversion inside stopping length targets at the LAMPF-WNR in Los Alamos by incident protons of 800 MeV and at the TRIUMF cyclotron in Vancouver by incident protons of 350 and 480 MeV [385, 798–801, 813]. Besides the measurement of the neutron yield per incident proton, the main aim of the FERFICON program was to investigate the production of fissile material for use in power reactors by accelerator breeder systems capable of extending the nuclear resources for an expanding nuclear industry [813]. Several target dimensions and materials were used. Examples are given in Table 11.1 and in more detail in [799, 801, 813]. Neutron yields were measured for single solid targets of W, Pb, Th, and depleted U and for targets composed of Th √ and U rods, where the effective diameter D is calculated by D = d n, with d is the rod diameter and n is the number of rods. The fertile-to-fission conversion yields were measured inside the Th and the depleted U targets inserting Th and U foils into the targets (cf. Figure 11.3) and determining the 233 U and the 239 Pu content by measuring the 233 Pa and the 239 Np formation inside the foils. The thick targets used for the FERFICON experiment at the TRIUMF facility at an incident proton energy of 0.48 GeV have slightly different dimensions [813] compared to the targets used at the LAMPF-WNR experiment. • Pb target: diameter = 10.16 cm, length = 30.48, • Th target, 19 rod array: diameter = 18.272 cm, length = 30.48 with a rod diameter d = 4.192 cm [813], • U target, 37 rod array: diameter = 19.782 cm, length = 30.48 with a rod diameter d = 3.252 cm [813].

383

384

11 Proton-Matter-Induced Secondary Particle Production – The ‘‘Thick’’ Target Experiments Tab. 11.1 Target characteristics of FERFICON targets used at the LAMPF-WNR facility with incident proton energy of 0.8 GeV as given in [799, 801].

Target material

Density (g/cm3 )

Diameter (cm)

Length (cm)

Wa Pb Ub Th Ud

18.26 11.31 18.4 11.38 19.04

4.45 9.85 10.01 18.28c 19.704e

24.13 40.45 40.45 36.31 30.46

U enrichment (wt%) – – 0.19802 – 0.25051

a

With a 5-cm deep reentry hole, diameter 2.54–1.42 cm. Depleted U with 0.198% 235 U. c 19 rod array, rod diameter = 4.194 cm. d Depleted U with 0.251% 235 U. e 37 rod array, rod diameter = 3.239 cm. b

Thorium target 19 rods ∅rod = 3.2393 cm

Central rod of the array

Uranium target 37 rods ∅rod = 4.194 cm

Segmented rods to insert foils

Th or foils inserted in rod Proton beam Th or U rod segmented Fig. 11.3 Illustration showing the 19-rod thorium and the 37-rod depleted uranium targets, the foil location in the arrays, and their position in a rod (after Gilmore et al. [801]).

The FERFICON experiment at the TRIUMF accelerator with 0.48 GeV incident protons were basically performed with the same method used for the Cosmotron experiments. The proton beam intensity was monitored by the 27 Al(p,3pn)24 Na reaction by determining the 24 Na activity produced in the Al foils. The axial and radial neutron flux distributions were measured using bare and cadmium covered Au foils. The array of the foil distribution was somewhat larger than that in the Cosmotron experiments [813].


The FERFICON experiments at the LAMPF-WNR facility with 0.8 GeV incident protons were also performed using the water-bath method by placing the targets inside a cylindrical vessel with a diameter of 2 m and height 2 m, which was located inside the beam channel of the WNR [800]. The proton beam current was estimated by a three-foil aluminum packet counting the 0.025 cm thick foils with a Ge–Li detector system to determine the number of 27 Al(p, x)7 Be, 27 Al(p, x)22 Na, and 27 Al(p, 3pn)24 Na reactions. For a typical experimental run [801] the measured number of beam protons for the three reactions are ranging from 4.25 × 1015 to 1.91 × 1016 total protons for the depleted uranium target and the thorium target. The most useful quantity is the number of neutrons from the target per incident beam proton – the n/p-ratio, and not the total neutron captures in the water bath per proton. Therefore it should be noted how the water-bath experiment affects the total neutron yield. (1) High-energy neutrons above a certain energy, above several 100 MeV, are lost from the water bath. (2) Spallation reactions with the oxygen nuclei by high-energy neutrons and protons emitted from the target into the water bath may produce a secondary neutron source. (3) The neutrons may be reflected back into the target and will be captured. The target can work as a neutron sink. (4) The neutrons that are reflected into the target, if not captured, may produce secondary neutrons via (n, f ) and (n, xn) reactions. The points (1) and (2) may compensate each other and are not very dependent on the target material and the target dimensions. The effects described by points (3) and (4) may compensate, but are also dependent on the target material and the target dimensions. This is especially important for target materials with high absorption and fission cross sections in the thermal, epithermal, and the MeV energy region. For lead the water-bath effects may be minimal. For other target materials as mercury, tantalum, tungsten, thorium and uranium the measured neutron captures in H2 O has to be corrected by simulations to convert the experimental data to neutron yield data. Table 11.2 summarizes some measured H2 O neutron capture rates per beam proton of the TRIUMF at 0.48 GeV proton energy and LANL–WNR water-bath experiments at 0.8 GeV proton energy. More recently measured neutron yields, multiplicities and multiplicity distributions of extended thick targets with a 4π neutron detector by the NESSI collaboration avoid sophisticated corrections by simulations. These measurements are described in Section 11.2.2 on page 390. For both FERFICON targets, the 19-rods thorium and the 37-depleted uranium array target, spatial – radial and axial – distributions of the fertile-to-fissile conversion were measured by bombarded the targets with 0.8 GeV protons. The target rods were loaded with the Th and U foils as indicated in Figure 11.3. The spatial distributions were integrated to determine the total conversion and the total fissions. The amount of 239 Pu and 233 U are determined by measuring the production yield of 239 Np and 233 Pa, respectively. The analysis procedure

385

386

11 Proton-Matter-Induced Secondary Particle Production – The ‘‘Thick’’ Target Experiments Measured H2 O neutron captures per proton for FERFICON experiments (details of the rod array targets are shown in Figure 11.3 and partly given in Table 11.1).

Tab. 11.2

TRIUMF FERFICON results at 0.48 GeV incident proton energy. Data are from [813]. Target material

Pb Pb, 7-array Th Th, 19-array Ua U, 37-array

Target density (g/cm3 )

Diameter (cm)

11.34

10.16 10.15 4.192 18.272 3.252 19.782

11.38 18.94

Target dimensions rod diameter (cm)

3.836 4.192 3.252

Length (cm)

30.48 30.48 30.48 30.48 30.48 30.48

Measured H2 O neutron capture rate per proton

8.3 ± 0.5 8.0 ± 0.4 8.1 ± 0.6 9.6 ± 0.7 9.6 ± 0.7 17.1 ± 1.0

LANL-WNR FERFICON resultsb at 0.8 GeV incident proton energy. Data are from [799, 816]. W Pb Th, 19-array U U, 37-array

18.26 11.31 11.38 18.4 19.04

4.45 9.85 18.28 10.1 19.7

4.194 3.239

24.13 40.65 40.65 40.65 40.65

10.3 (11.3)c 13.1 (13.1) 17.2 (16.5) 25.8 (24.4) 29.3 (26.5)

a

Depleted U with 0.22% 235 U. A detailed analysis of the measured captures per proton indicated an overall uncertainty ±1σ of 6.8% for the solid uranium target and 5% for the remaining targets [816]. c Values in brackets calculated with LAHET–LCS Monte Carlo system [110, 193]. b

Tab. 11.3 Number of fissions and measured production yields of 239 Np, and 237 U of the conversion of 238 U produced by the bombardment by 0.8 GeV protons [801].

Quantity

U-target experiment

Calculation

Number of fissions (fissions/proton) 239 Np production (atoms/proton) 237 U production (atoms/proton)

5.90 ± 2.5 × 10−1 3.81 ± 8.0 × 10−2 0.95 ± 2.0 × 10−2

6.14 ± 3.7 × 10−1 3.875 ± 3.1 × 10−1 –

and further experimental details are described in [385, 799–801]. Tables 11.3 and 11.4 summarize the measured fission and conversion rates together with corresponding calculational results which are useful to estimate the production of 239 Pu and 233 U (see Eq. (11.2)). The calculations were done by using LAHET–LCS particle transport system with the Rutherford fission model [110, 193] (cf. Section 3.7 on page 165), including a

11.2 Proton-Induced Thick Target Experiments in the Energy Range 0.1–2.5 GeV Number of fissions and measured production yields of 233 Pa, 95 Nb, 205 Bi and 227 Th of the conversion of 232 Th produced by the bombardment by 0.8 GeV protons [801].

Tab. 11.4

Quantity

Th target experiment

Calculation

Number of fissions [fissions/proton] 233 Pa production (atoms/proton) 95 Nb production (atoms/proton) 205 Bi production (atoms/proton) 227 Th production (atoms/proton)

1.56 ± 2.5 × 10−1 1.251 ± 3.0 × 10−2 0.068 ± 3.0 × 10−3 0.0056 ± 3.0 × 10−4 0.046 ± 3.0 × 10−3

1.543 ± 6 × 10−3 1.267 ± 7 × 10−3 0.0553 ± 9 × 10−4 0.0077 ± 4 × 10−4 0.0221 ± 7 × 10−4

Production yield [atoms g−1 proton−1]

10−4 239Np,

U-target

237U, U-target 233Pa, Th-target 227Th,

Th-target

10−5

10−6

10−7

−5

0

5

10

15

20

25

30

35

40

Target depth [cm] Fig. 11.4 Axial distributions of 239 Np and 237 U in the depleted uranium target, and 233 Pa and 227 Th in the thorium targets, respectively (data are from Gilmore et al. [801]).

precise three-dimensional geometry description of the experiment. The measured axial conversion distributions are shown in Figure 11.4. For the analysis of the measurements the monitor foils and the dissolved uranium and thorium foils were counted with a Ge(Li) detector system. The axial distributions of the measured reaction products were used to determine the total production per incident beam proton inside the targets. The total number for each reaction is given by Eq. (11.1)

reactiontotal (i) =

M · L · numberproton

L Ni (z)dz, 0

(11.1)

387

388


with M is the mass of the target in gram, L the length of the target in cm, numberproton is the number of incident protons, Ni number of atoms produced per gram of target material, and z is position of foils into the target. The amount of the produced 239 Pu and 233 U is given by the decay of the produced 239 Np and 233 Pa β− 26.967 d β− 233 91 Pa −→ 2.355 d

239 93 Np

−→

239 94 Pu 233 92 U.

(11.2)

11.2.1.3 The PSI Thick Target Lead/Bismuth Experiments Spallation neutron production yields have been measured for protons of 420 and 590 MeV on thick Pb/Bi targets at the cyclotron of the Paul Scherrer Institut (PSI) Switzerland. The measurements of the thick target yields were investigated by using the water-bath method by means of thin 25 µm Au – with and without Cd cover – and 100 µm Mn activation foils placed at certain axial and radial positions within the water bath [814] (cf. see also the Sections 11.2.1.1 and 11.2.1.2 about the Cosmotron and FERFICON experiments). A schematic presentation of the experimental apparatus is shown in Figure 11.5. The water bath is a plexiglass vessel with the dimensions 1000 × 750 × 900 mm3 . A water tight beam line is inserted into the water bath to house the target. The Pb/Bi target consists of Pb and Bi disks each 10 mm thick and 100 mm in diameter. For different incident proton energies 300, 420, and 590 MeV up to 30 disks were used corresponding to the proton ranges (Table 11.5). As indicted in Figure 11.5, a number of plexiglass guide tubes is inserted into the water bath for positioning the Au and Mn foils at well-defined axial – in Z – and radial locations. The following 1000 mm Water bath with plexiglass vessel

Pb disk

Proton beam

Bi disk

100 mm

Al foil

900 mm

Z

Plexiglass guide tubes for foil measurements in Z and radial positions

Fig. 11.5 Sketch – not to scale – of the water-bath experiment to measure H2 O capture rates of Pb/Bi targets at the PSI cyclotron at incident proton energies of 300, 420, and 590 MeV. The different target lengths are given in Table 11.5.

11.2 Proton-Induced Thick Target Experiments in the Energy Range 0.1–2.5 GeV Tab. 11.5

Characteristics of the Pb/Bi targets for the PSI thick target experiments.

Incident proton energy (MeV) 300 420 590

Proton range (cm)

Number of target disks

∼9.5 ∼16.0 ∼27.5

16 18 30

Target lengths (cm) 16 18 30

positions were possible: In axial direction at Z = 25, 55, 85, 125, 155, 175, 205, 235, 265, and 295 mm and in radial direction, measured from the target axis, at R = 76, 126, 176, and 226 mm. The total number of protons was determined by the well-established method irradiating a 0.28-mm thick Al foil placed in front of the target with the same diameter of 100 mm as the target. The activation cross section induced by the beam protons is given by 27 Al(p,3n3p)22 Na. The following cross sections for the reaction 27 Al(p,3n3p)22 Na at different incident beam energies were adopted [814] σ = 14.6 ± 0.6 mb at 300 MeV, σ = 14.8 ± 0.6 mb at 420 MeV, σ = 14.9 ± 0.6 mb at 590 MeV.

(11.3)

The total number of protons for the higher bombarding energies could be determined as numberproton = (6–10 ± 0.3) × 1013 at 300 MeV, numberproton = (3.8 ± 0.2) × 1013 at 420 MeV, numberproton = (7.1 ± 0.3) × 1013 at 590 MeV.

(11.4)

The novel approach analyzing the water-bath experiment at PSI consists in scaling Monte Carlo calculated (n/p)calc -ratios by means of the average ratio measured to Monte Carlo calculated 198 Au foil activities [814]. The total number of neutrons emitted per incident proton usually called the neutron yield per incident proton n/p is determined from the spatially measured distribution of the 198 Au-specific activity. Conventionally the spatial neutron fluence distribution is experimentally integrated over 4π to determine with some corrections the total measured thick target yield (n/p)meas . The procedure adopted by van der Meer et al. [814] circumvents potentially the inaccurate 4π integration of an extended source in the water bath sampled over a relatively small set of measured spatial points via activation foils to estimate the integral quantity n/p. The following steps are considered by the PSI experiment: • The 198 Au measured specific activities are normalized per incident beam proton Eqs. (11.3), (11.4), and (11.6) to receive a measured spatially sampled distribution of measured specific activities per incident beam proton.

389

390


• The Monte Carlo particle transport system MCNPX (cf. Section 5.2.1 on page 209) is then used to calculate the 198 Au-specific activities per incident proton A(ri , Zi )calc and the neutron yield (n/p)calc . • The measured 198 Au-specific foil activities A(ri , Zi )meas per incident proton at each location (ri , Zi ) are divided by the calculated ones. Thus giving a normalized spatial dependent activity Ameas/calc (ri , Zi ). • A weighted average of the values Ameas/calc (ri , Zi ) is then taken over the spatial points (ri , Zi ) on the spatially sampled distribution. • The final experimental (n/p)meas ratio is determined by the following equation (n/p)meas = (n/p)calc · Ameas/calc (ri , Zi ).

(11.5)

It should be noted [814] that the above-described procedure is insensitive to the value of (n/p)calc and to its error, since the product of the two right terms in Eq. (11.5) effectively cancels out the dependence of (n/p)calc . This is only true if the MCNPX Monte Carlo estimation reproduces well the shape and the magnitude of the neutron flux distribution. Also the procedure is free from systematic errors only if the Monte Carlo simulation reliable predicts the principal nuclear reactions, primarily, (1) spectral and angular distributions of the fast neutrons produced and emitted in the spallation process that occur inside the thick target, and (2) the moderating process by such neutrons in the H2 O surrounding the target. A detailed analysis by van der Meer et al. [814] concluded that the conditions to estimate the value of (n/p)meas adopting Eq. (11.5) are indeed fulfilled for both incident proton energies of the PSI experiment, 420 and 590 MeV. The 300 MeV incident proton data were not assigned to an accurate n/pmeas value because the number of protons in the irradiation was determined imperfect. The resulting values (n/p)meas of the PSI experiment following the procedure given by van der Meer et al. [814] are determined including a detailed error estimation as (n/p)meas = 6.0 ± 0.3 at 420 MeV, target length = 18 cm, (n/p)meas = 9.6 ± 0.4 at 590 MeV, target length = 30 cm,

(11.6)

where the uncertainty is given as ±1σ . A comparison with n/p ratios and neutron multiplicity data published in the literature for thick Pb targets showed that the PSI data fall well within the range of former experimental systematics. 11.2.2 Neutron Multiplicities Measured with a 4π Detector at the COSY Accelerator at Jülich

The aim of the thick target neutron multiplicity experiments of the NESSI collaboration at the cooler synchrotron (COSY) at Jülich, Germany, was to enrich the previously obtained data with the use of primary proton beams with energies


Beam pipe with vacuum Proton beam from COSY

S3,S5-S8

Ne343 NESSI liquid scintillator 4Π detector gadolinium loaded S10

To beam dump

S1 S11-S14

Air 60 cm

Thick target 153 cm

480 cm

713 cm

Fig. 11.6 The setup of the NESSI experiment for neutron multiplicity measurements on thick targets.

of 0.4 up to 2.5 GeV bracketing more closely the energy envisioned for future spallation sources and ADS projects. Three materials were studied – W, Pb, and Hg – as thick targets, whereas Hg is the best choice at present for the commissioned high-intensity MW spallation neutron sources SNS, USA, JPARC, Japan, and the future ESS. A large number of cylindrical target sizes have been explored, both in length and diameter. In thick targets the development of the INC depends strongly upon the size of the target and this justifies many different geometries to be explored. Altogether about 300 measurements have been performed, for thick targets providing a wealth of experimental data to be confronted with high-energy transport code systems, which are used as standard tools in ADS and spallation source design [622]. The experimental equipment, a 4π detector, namely a large spherical vessel (BNB3) ), filled with a liquid scintillator loaded with gadolinium, which is described in detail already in Section 10.3.2.1 on page 10.3.2.1, is set up for the thick target measurements without the inner silicon ball detector. The available space is used for the mounting of cylindrical thick targets with diameters and lengths up to 15 and 35 cm. Figure 11.6 illustrates the experimental arrangement installed at an external proton beam line of the COSY accelerator at Jülich showing the neutron ball with auxiliary plastic scintillators to control the experiment. The neutrons produced in the target and emitted from the target surface are slowed down in the liquid scintillator by elastic scattering with the H and C atoms and are eventually captured at different instants with high cross section by the Gd nuclei when reaching thermal energy. The detector is thus used, first, as a nuclear reaction trigger and then as a neutron multiplicity meter. The detector response consisting of γ -rays resulting from the neutron capture in Gd is delayed by about 1 to 50 µs (about 16 µs on the average) with respect to the spallation reaction. This is mainly due to the low Gd concentration imposing an appreciable diffusion time for the thermal neutron before being captured by a Gd nucleus. As a consequence, the neutron signals have been selected within a time gate opened 0.7 µs after the 3) Berlin Neutron Ball (BNB).

391

11 Proton-Matter-Induced Secondary Particle Production – The ‘‘Thick’’ Target Experiments p + Pb 0.8 GeV

60

2 cm Neutron multiplicity (not corrected for efficiency)

392

10 cm

35 cm

2 cm

10 cm

35 cm

2.5 GeV 2 cm

10 cm

35 cm

40 20 0 1.2 GeV

60 40 20 0 60 40 20 0

0

500

0

500

0

500

Light yield [MeVee]

Fig. 11.7 Neutron multiplicity Mn versus the prompt light Esum of the NESSI detector in MeVee (the electron equivalent) for proton-induced reactions on Pb targets of thicknesses of 2, 10, and 35 cm and a diameter of 15 cm bombarded with energies

of 0.8, 1.2, and 2.5 MeV. The color code is normalized to the same intensity for all distributions (in au) in order to directly compare the yield (after [115, 664]).

reaction takes place and closed after 45 µs when the probability to capture a neutron is reduced to 2%. As described in Section 10.3.2.1, the prompt light signal Esum preceding the delayed neutron capture arises from the sum of kinetic energies of all kind of reaction products entering the neutron ball detector, in addition to the neutrons charged particles and γ quanta contribute. Since the threshold for detecting the prompt light flash is as low as 2 MeVee, the electron equivalent, a nuclear reaction is characterized by the occurrence of a prompt light signal larger than this threshold, even if there are no neutrons released in the reaction at all. The complex measured correlation between the neutron multiplicity Mn and the prompt light signal Esum in the NESSI detector is shown for 0.8, 1.2, and, 2.5 GeV for proton-induced reactions on Pb targets of different thicknesses or length in Figure 11.7. For a fixed target thickness and incident proton energy the larger the Esum will be, the smaller the measured multiplicity Mn results. This is due to the high-energy particles generally producing more light in the scintillator than evaporative neutrons or γ -particles. The less energy these ‘‘fast’’ particles deposit in the target, the less effective the nuclei are heated during the intra- and internuclear cascade and the less neutrons


are finally evaporated. On the expense of the prompt light signal for fixed incident energy of the proton Mn increases with increasing target thickness. Consequently for thick targets the major contribution of the prompt light signal rises from evaporative neutrons and γ -particles. Thereby quenching factors for low-energy neutrons scale the light signal down. In other words the originally available energy is converted quite efficiently into the production of neutrons in thick targets of several nuclear interaction lengths. The scintillator light is registered by 24 fast photo tubes distributed evenly at the surface of the vessel. In contrast to measurements implying liquid scintillator detectors for TOF information, the present measurements have been performed without low-energy threshold, since the neutron needs to be thermalized down to 0.025 eV in order to be captured. The efficiency of the BNB detector, already discussed in Section 10.3.2.1 on page 10.3.2.1, is about 55% for neutron energies at 20 MeV and decreases to about 15% at an energy of 100 MeV (cf. Figure 10.42 on page 10.42). For thick targets, an overall efficiency can be taken into account for each experimental condition. The overall detection efficiency increases with the target lengths or thicknesses and material density as depicted in Figure 11.8. The efficiency is excellent in all experimental cases. Therefore, the 4π detector NESSI is a unique system for these thick target experiments. The detection efficiency increases with increasing target thickness, since the leaking neutrons out of the target surface become less and less energetic due to

Overall neutron detection efficiency [%]

0.78

0.76

0.74

0.72

0.4 Gev 1.2 Gev 2.5 Gev

0.7 Pb 0.68

0

20

Hg 0

20

W 0

20

Target length [cm] Fig. 11.8 The simulated overall neutron detection efficiency of the NESSI detector for thick W, Hg, and W targets of 15 cm diameter bombarded by proton energies of 0.4, 1.2, and 2.5 GeV (after Letourneau et al. [115]). Not the offset of the ordinate.

393

394

11 Proton-Matter-Induced Secondary Particle Production – The ‘‘Thick’’ Target Experiments Tab. 11.6

Combinations of target materials, target sizes explored at various energies.

Energy (GeV) Target W

Hg

Pb

0.4

0.8

Diameter (cm)

1.2

1.8

2.5

Thickness (cm)

2 8 12 15

1–34.75 34.75 10–34.75

0.112 0.5–34.75 34.75 2–34.75

0.112 5–34.75 5–34.75 1–34.75

0.112 0.5–34.49 20–34.49 1–34.49

0.112 0.5–34.75 2–34.75 1–34.75

2 15

33.7

2–33.7

0.5 2–30.45

0.5 2–33.7

0.5 5–33.7

2 8 12 15

0.5 1–35 35 5–35

0.2 1–40 1–35 1–35

0.2 35 35 1–35

0.2 0.5–40 15–39 1–35

0.2 2–35 35 2–35

their slowing down within the target. For the thick measurements the experiment has been run with the target in the air (cf. Figure 11.6). The beam pipe was closed by a thin capton foil located about 60 cm downstream from S1 scintillator. The thick targets For thick targets, three materials, mercury as target material for high-intensity spallation neutron sources, lead, and tungsten have been chosen which are representative of the target and core materials of high intensity ADS. Systematic studies of a large variety of different cylindrical geometries and incident energies as summarized in Table 11.6 have been studied. All target pieces were made from chemically pure (≥ 99.98%) material of W, W, Hg, and Pb. Figure 11.9 illustrates how the different target pieces are assembled. The large possible variation of target geometries and materials make the thick target experiment NESSI to an important benchmark for obtaining key parameters necessary to design and construct neutron-producing targets via spallation. The mercury was filled in cylindrical stainless steel containers with a wall thickness of 1 mm, a diameter of 15 cm and different lengths. As indicated in Figure 11.9, the tungsten target was constructed as a ‘‘LEGO’’ design of different hollow and full cylinders. 11.2.2.1 Thick Target Results of the NESSI Experiment at the COSY Accelerator The study on thick target measurements considers data on reaction probabilities Preac , hadronic interaction lengths, average neutron multiplicities Mn with reference either to the number of neutrons generated per reaction or per incident proton Mn /p, and in particular neutron multiplicity distributions dPreac /dMn as obtained with different diameters and lengths of W, Hg, and Pb targets bombarded with 0.4, 0.8, 1.2, 1.8, and 2.5 GeV protons. Due to the multitude of measurements and particle transport simulations investigated by the NESSI only a representative


Tungsten targets diameters = 150, 120, 80 mm

150 mm

120 mm 80 mm

150 mm

10 mm

20 50 mm mm

100 mm

Lead targets diameters = 150, 120, 80 mm

120 mm 80 mm

Mercury targets diameters = 150, 120 mm Fig. 11.9 The dimensions – not to scale – of target geometries for mercury, tungsten, and lead target studied at the NESSI experiment.

selection is given. The interesting reader should refer the secondary literature published recently by the NESSI collaboration [114, 115, 133, 379, 543, 621, 622, 631, 664, 735, 736, 817]. Reaction cross sections and hadronic interaction lengths The key observables of thick target measurements with the 4π neutron ball detector are the neutron multiplicity Mn , measured event-by-event, and the reaction probability Preac . The reaction probability Preac is deduced from the ratio of the number of triggered events (cf. Section 10.3.2.1) divided by the number of incident protons counted by detector S1 (cf. Figure 11.6 on page 391). Moreover, since the incident protons were individually counted and the detector has a very low detection threshold of about 2 MeVee, the neutron multiplicity could be related to both, to the reaction events and to the incident proton events. Figure 11.10 illustrates the dependence of the proton survival probability (1 − Preac ) on target thickness. The data are for 1.2 and 2.5 GeV incident protons and for W, Hg, and Pb target cylinders of 15 cm diameter and a W target of a diameter 8.0 cm. As seen in Figure 11.10 as a function of the target thickness, the experimental data are well fitted by a constant logarithmic representation of the survival probability for all three target materials at 2.5 GeV bombarding proton energy. This indicates a constant reaction cross section. The constant logarithmic behavior is also seen at 1.2 GeV for the Pb target but for the Hg and the W targets with a diameter of 15 cm there is a deviation from the linearity. This effect is caused by the strong absorption of particles, mainly by incident protons, loosing its energy via electronic stopping before undergoing a nuclear reaction. Therefore, the survival probability

395


1.0

Survival probability (1-Preac)

396

1.2 GeV p on Pb 2.5 GeV p on Pb

0.1

1.2 GeV p on Hg 2.5 GeV p on Hg 1.2 GeV p on W 1.2 GeV p on W, Ø = 8 cm 2.5 GeV p on W 0

5

10

15 20 25 Target length [cm]

Fig. 11.10 Survival probabilities (1 − Preac ) of incident protons as a function of cylindrical target lengths for 1.2 and 2.5 GeV protons on Pb, Hg, and W targets of 15 cm diameter and on a W target of 8 cm diameter with 1.2 GeV incident protons. The

30

35

40

straight lines give the best fit of the data for all targets, except the W targets at 1.2 GeV incident protons and of diameters of 8.0 and 15.0 cm. Data are from Letourneau et al. [115].

looks ‘‘artificially’’ too high. This absorption decreases the light yield of the detector and therefore the multiplicity. The effect is partly recovered by reducing the radius thickness as seen in Figure 11.10 for the W target with a radius of 8 cm. The survival probability follows quite accurately the exponential law 1 − Preac = exp (−L/Lint ),

(11.7)

where L is the target length and Lint the interaction length. Measured and calculated interaction lengths are given in Table 11.7. The calculated hadronic interaction lengths resulting from the Monte Carlo calculations agree very well with published experimental values [115]. As a consequence of the constancy of the nucleon–nucleus cross section above some 100 MeV, the gradient of the exponentials does not depend on the incident kinetic energy of the projectile. The difference in the gradient for the three different materials originates mostly from their difference in density and much less from different reaction cross sections.

11.2 Proton-Induced Thick Target Experiments in the Energy Range 0.1–2.5 GeV Comparison of experimental and calculated interaction lengths and reaction cross sections for W, Hg, and Pb targets (data are from [115, 817]).

Tab. 11.7

Material

L int (cm)

W Hg Pb

Experiment 10.84±0.2 15.06±0.4 18.00±0.3

σ reac (b) Calculation 10.05 13.91 17.66

Experiment 1.46±0.03 1.64±0.05 1.69±0.03

Calculation 1.62 1.71 1.73

For a 35-cm length target, the term (1 − Preac ) approaches the 10% level in Pb and 1% level in W, while the range due to electronic interaction is as large as 170 and 97 cm for 2.5 GeV protons in Pb and W, respectively. The reaction probability Preac is used to determine the reaction cross sections σreac σreac =

ln(1 − Preac ) · A (LAv · ρ · d)

(11.8)

with, A, ρ, d and L being the mass, density and thickness of the target and LAv the Avogadro number, respectively. The experimental reaction cross sections deduced from the measured Preac are also given in Table 11.7 of the three target materials W, Hg, and Pb. These values are slightly smaller than the results of the Monte Carlo calculation with the HERMES systems calculations. An agreement of the same order of magnitude is found when the MCNPX code system is used instead of the HERMES code, demonstrating that the description of the observable σreac is well predicted by theoretical models. This analysis is reflected also for the examples in Table 11.8 for different target lengths or thicknesses and the incident proton energies of 1.2 and 2.5 GeV. Neutron multiplicities and neutron multiplicity distributions Here the term ‘‘neutron multiplicity’’ comprises all neutrons originating from primary and succeeding secondary reactions within the target material. In the thick target measurements only neutron leakages can be observed. The neutron yield is not accessible in any experiment since it reflects the neutron production at the point of origin when the neutrons are created, whereas the leakage spectrum can be measured as a source of neutrons leaking from the target surface after they have left the target material. The originality of the NESSI experiment compared with the previous ones lies in the possibility to measure neutron multiplicity distributions and not only their average values. There are essentially two observables measured at thick targets by the NESSI detector [621, 631]: (1) the reaction probability Preac , and (2) the neutron multiplicity distribution dPreac /dMn . In order to compare the measured observables of the NESSI experiment with previous experiments the mean multiplicity Mn must be derived from the

397

11 Proton-Matter-Induced Secondary Particle Production – The ‘‘Thick’’ Target Experiments Pb Average neutron multiplicity per incident proton

398

Hg

W

50 20 10 5

1

2.5 GeV 1.8 GeV 1.2 GeV 0.8 GeV 0.4 GeV

0

20

0

20

0

20

Target lenght [cm]

Fig. 11.11 The average neutron multiplicity per incident proton Mn /proton as a function of the target lengths and thicknesses. Targets are thick W, Pb, and W targets with a diameter of 15 cm bombarded by protons of energies of 0.4, 0.8, 1.2, 1.8, and 2.5 GeV. The measurements are corrected to background and detection efficiency (after Letourneau et al. [115]).

multiplicity distribution dPreac /dMn . From the reaction probability Preac and mean multiplicity Mn , the mean number of neutrons per incident protons is estimated as Mn /p = Mn · Preac .

(11.9)

The integral value Mn /p is of particular importance for the design of spallation sources whereas the experimentally measured neutron multiplicity distribution dPreac /dMn is one of the most sensitive test of intra- and internuclear cascade evaporation models and particle transport codes. In Figure 11.11 experimental results of Mn /p are plotted vs. the lengths of the lead, mercury, and tungsten cylindrical targets, and for the five incident proton energies of 0.4, 0.8, 1.2 1.8, and 2.5 GeV, as obtained in the NESSI experiments. As expected, for every target, the mean multiplicity increases with increasing target length, albeit in a nonlinear fashion. It should be noted that the measured Mn /p is corrected by the efficiency of the detector. A comparison of average multiplicities with theoretical predictions using standard Monte Carlo models HERMES and MCNPX (cf. Section 5.1 on page 207) is given with complete experimental and model average neutron multiplicities Mn and reaction probabilities Preac for different energies and target materials in [622, 664, 817] and it is partly summarized in Table 11.8. This table includes the simulated mean neutron multiplicities Mn , corrected for the NESSI detector neutron detection efficiency and Monte Carlo calculated MnC before taking the detector efficiency into account. All calculations account for the fact that neutrons are slowed down in the target material. Both fission and elastic scattering were included. All other options have been chosen according to the standard set of parameters used in spallation intra–inter nuclear cascade evaporation models. The agreement between calculation and experiment for the second moment of the

Average neutron multiplicity per reaction (<Mn>)


45

Pb-target d = 15cm

Hg-target d = 15 cm

W-target d = 15 cm

40 35 30 25 20 15 1.2 GeV NESSI 1.2 GeV HERMES 1.8 GeV NESSI 1.8 GeV HERMES 2.5 GeV NESSI 2.5 GeV HERMES

10 5 0

0

20

0 20 0 Target length [cm]

20

Fig. 11.12 Average multiplicities per reaction Mn/Preac as a function of the target length of Pb, Hg, and W targets of diameter = 15 cm at incident proton energies 1.2, 1.8, and 2.5 GeV compared with efficiency corrected Monte Carlo model predictions.

distributions within a few percent gives additional confidence to the spallation physics models under consideration, at least as far as the description of neutrons is concerned. Figure 11.12 shows average neutron multiplicities per hadronic interaction Mn /Preac produced as a function of target thickness (diameter 15 cm) for 1.2, 1.8, and 2.5 GeV protons on Hg, Pb, and W targets compared with Monte Carlo model predictions. As seen in Figure 11.12, the model predictions with HERMES code agree very well with the experimental observations, over a wide range of target geometries and target materials. The observed increase in the neutron multiplicity with increasing target length is due to an increase in the reaction probability, Preac and, to a lesser extent, to an increase in secondary reactions with the target length. Some of these data are also given in Table 11.8. For the full set of data the reader may refer to [115]. Hg targets: As far as average values and Hg targets (Table 11.8, Figure 11.12) are concerned, for all Monte Carlo codes considered here, one observes good agreement with the experimental results. Discrepancies between model calculations and experimental data are generally less than 5% for both, Preac and Mn , and a broad range of energies. The maximum discrepancy is 7.4% in the case of the 5 cm Hg target bombarded with 2.5 GeV protons. Pb targets: For the Pb targets, the deviation of theoretical predictions with respect to experimental data decrease with increasing target thickness, while with increasing incident energy divergences increase. The maximum discrepancies are found for thick target bombarded with 2.5 GeV. Note that the divergences for even higher incident proton energies (4.15 GeV) [730] still increase.

399

400


Tab. 11.8 Examples of average neutron multiplicities Mn and reaction probabilities Preac for cylindrical tungsten, mercury, lead targets of 15 cm diameter bombarded with protons of 1.2 and 2.5 GeV energies.a

Target length (cm)

Experiment Mn

Preac

Monte Carlo calculation HERMES C Mn Mn

Preac

MCNPX Mn

MMC n

Preac

Incident proton energy 1.2 GeV – tungsten target with diameter = 15 cm 15 35

21.6 ± 3.2 22.6 ± 3.0

0.729 0.902

20.9 ± 3.2 21.6 ± 3.1

26.9 ± 3.6 27.6 ± 3.4

0.784 0.971

22.5 ± 3.3 23.4 ± 3.1

28.9 ± 3.7 29.8 ± 3.5

0.781 0.964

40.7 ± 4.4 47.3 ± 4.0

52.7 ± 5.0 60.6 ± 4.5

0.781 0.973

21.2 ± 3.3 22.5 ± 3.2

27.7 ± 3.6 29.1 ± 3.6

0.660 0.885

35.0 ± 4.3 40.1 ± 4.1

45.7 ± 4.8 51.6 ± 4.6

0.655 0.884

20.5 ± 3.2 22.1 ± 3.2

27.1 ± 3.6 28.8 ± 3.5

0.567 0.859

33.9 ± 4.2 40.4 ± 4.1

44.7 ± 4.8 52.2 ± 4.6

0.575 0.861

Incident proton energy 2.5 GeV – tungsten target with diameter 15 cm 15 35

36.6 ± 4.1 41.6 ± 3.8

0.758 0.952

36.7 ± 4.2 42.3 ± 3.9

47.6 ± 4.8 54.4 ± 4.4

0.782 0.973

Incident proton energy 1.2 GeV – mercury target with diameter = 15 cm 15 30

20.5 ± 3.1 21.9 ± 3.1

0.645 0.847

19.9 ± 3.2 21.2 ± 3.1

26.1 ± 3.6 27.5 ± 3.5

0.660 0.889

Incident proton energy 2.5 GeV – mercury target with diameter = 15 cm 15 30

33.7 ± 4.0 38.7 ± 3.9

0.647 0.866

33.2 ± 4.2 38.5 ± 4.1

43.9 ± 4.8 50.3 ± 4.6

0.663 0.887

Incident proton energy 1.2 GeV – lead target with diameter = 15 cm 15 35

20.2 ± 3.2 22.2 ± 3.1

0.571 0.848

19.6 ± 3.2 21.4 ± 3.1

25.9 ± 3.6 27.9 ± 3.5

0.579 0.867

Incident proton energy 2.5 GeV – lead target with diameter = 15 cm 15 35

32.3 ± 4.0 38.4 ± 3.9

0.577 0.848

31.7 ± 4.1 37.7 ± 4.0

42.5 ± 4.7 49.7 ± 4.6

0.580 0.865

a The values of M C are Monte Carlo calculated average neutron multiplicities before having taken detector efficiency n into account. The square root of the RMS (root mean square) is also given for all measured and calculated average multiplicities. The experimental data are from [115] and the calculational values are from [622, 664, 817].

W targets: Observations similar to those for mercury and lead were made for the tungsten targets. At 1.2 GeV, agreement with experimental data is very good for the HERMES calculations, while it is still quite satisfactory for the MCNPX calculations. However, for higher incident energies, the data clearly favor HERMES over other Monte Carlo predictions. As mentioned earlier, the originality of the NESSI experiments lies in the possibility to measure neutron multiplicity distributions and not only their average values. In the following some examples of such neutron multiplicity distributions

11.2 Proton-Induced Thick Target Experiments in the Energy Range 0.1–2.5 GeV 1.2 GeV protons on Hg 0.04

5 cm

15 cm

30 cm

0.035

dPreac / dMn

0.03 0.025 0.02 0.015 0.01 0.005 0

0

20

40

0

20

40

0

25

50

Neutron multiplicity Mn

Fig. 11.13 Neutron multiplicity distributions dPreac /dMn as a function of the neutron multiplicity Mn of Hg targets with diameters = 15 cm and lengths = 5, 15, and 30 cm bombarded with protons of 1.2 GeV. The black symbols are the measurements and the thin lines are HERMES computer simulations. 2.5 GeV protons on Hg 0.03 2 cm

15 cm

35 cm

0.025

dPreac /dMn

0.02 0.015 0.01 0.005 0

0

20

40

0 50 0 Neutron multiplicity Mn

50

Fig. 11.14 The same as in Figure 11.13 but with W targets with diameters of 15 cm and lengths 2, 15, and 35 cm bombarded with protons of 2.5 GeV.

given as dPreac /dMn as a function of the neutron multiplicity Mn for W, Hg, and Pb targets are illustrated and compared with Monte Carlo predictions. As seen from Figures 11.13–11.15, the general shapes of the experimental distributions are well represented by simulation calculations obtained with the HERMES Monte Carlo. More details of comparisons between experimental and

401


(a)

1.2 GeV protons on Pb and W targets 0.04 2 cm

15 cm

35 cm

0.035 Pb-target

0.03 0.025 0.02 0.015 0.01 dPreac / dMn

402

0.005 0 0.04

2 cm

35 cm

15 cm

0.035 W-target

0.03 0.025 0.02 0.015 0.01 0.005 0 (b)

0

20

40

0

20

40

0

25

50

Neutron multiplicity Mn

Fig. 11.15 The same as in Figure 11.13 but with Pb targets (a), and W targets (b) with diameters of 15 cm and lengths 2, 15, and 35 cm bombarded with protons of 1.2 GeV.

theoretical evaluated multiplicity distributions of thick targets are given and discussed in [622, 664]. The discrepancies seem to become slightly larger for higher energies [730]. Note in the figures that the enhancement at Mn = 0 includes reactions where a nuclear reaction takes place, however the nucleus de-excited without the emission of a neutron. 11.2.2.2 A Summary of Neutron Yield Experimental Data Tables 11.9 and 11.10 summarize most of the experiments on neutron production measurements on thick targets by incident protons with respect to the time of their publication. Given are the incident proton beam energies, the investigated target materials and their dimensions, the measured quantities, and methods. The different methods for the neutron measurements are also indicated. It is seen from Tables 11.9 and 11.10 that most of the thick target measurements are related to lead targets investigating average multiplicities per incident beam proton. Some of these experiments consider only one target geometry, one or

11.2 Proton-Induced Thick Target Experiments in the Energy Range 0.1–2.5 GeV Summary(I) of published thick target experiments with respect to the time of publication, incident proton energy, target material, and target dimensions.a

Tab. 11.9

Time of publ.

Incident proton energies (GeV)

Targets and dimensions ∅, L (cm)

Measured quantity

Method

1965

0.47/0.54/ 0.72/0.96/ 1.47

Be 10.2 × 10.2 × 91.6 Sn, Pb, U ∅ = 10.2, L = 61 Pb,∅ = 20.4

Mn /p

MOD Au-foils

[59]

1968

0.4/0.5/0.66

Pb,∅ = 10 − 26 L = 55

Mn /p

Fission chambers

[818]

1971

various 0.316 − 0.998

Pb,∅ = 10 L = 60

Mn /p

MOD BF3

[815]

1973

0.73

238 U

d2 σ/dEd

TOF NE202/228

[125]

∅ = 15, L = 60

References

1979

0.26

Fe, 238 U

Mn /p

TOF LQ − SCIN

[804]

1983

0.1

Pb, 7 Li,∅ = 6.2/5.7 L = 1.6/17.4

Mn /p

MOD, Au-foils

[819]

1983

0.25

Pb,∅ = 5 L = 10

Mn /p

235 U,238 U SDTR

[820]

1987

0.1

Fe, Cu, Th,∅ = 6 L = 1.6

MOD Au-foils

[803]

1989

0.113

Be, C, Al, Fe, ∅ = 3.65 L = 5.70/5.83/ 4.03/1.57 238 U,∅ = 4, L = 3

TOF BC418

[697]

d2 σ/dEd

a The measured quantities are given (1) d2 σ/dEd (the angular distribution of the leakage neutron spectra), (2) Mn /p (mean number of leakage neutrons per incident beam proton, and the employed methods are (1) TOF), MOD (moderation in H2 O and polyethylene). The neutron distributions and spectra are measured with Au foils, BF3 counters, plastic and liquid (LQ-SCIN) scintillators, and solid track (SDTR) detectors.

two incident proton energies and are using the water-bath method. In contrary the recent investigations on thick targets of the NESSI collaboration cover a large range of incident proton energies from 0.4 to 2.5 GeV, a large variation in target dimensions on diameter and length, and studied important target materials as W,

403

404

11 Proton-Matter-Induced Secondary Particle Production – The ‘‘Thick’’ Target Experiments Summary(II) the same as in Table 11.9 but recently some additional methods are employed with the n-ball experiment to measure neutron multiplicity distributions dPreac /dMn are indicted (see Table 11.8) (also threshold detectors and 3 H counters are used).

Tab. 11.10

Time of Incident published proton energies (GeV)

Targets and dimensions ∅, L (cm)

Measured quantity

Method

References

1990

0.256

C, Al, Fe, ∅ = 8 L =(C)17.45/30 (Al)12.15/20 (Fe)4.7/8 238 U,∅ = 4 L = 3.5

d2 σ/dEd

TOF BC418

[696, 698]

1991

0.99–8.1

Pb ∅ = 20, L = 60

Mn /p

MOD BF3

[617]

TOF,NE213

[808, 821]

MOD He

[796, 797]

3 He,

1995

0.5/1.5

Pb,15 × 15 × 20 cm3

d2 σ/dEd

1996

0.8/1.0/1.2/1.4

Pb, W, ∅ = 10.2 L = 61/40

Mn /p

3

1997

0.895/1.21

W ∅ = 20, L = 60

Leakage spectra

Threshold detectors

[822]

1998

0.197

Pb,∅ = 12 L = 2 − 25

dPreac /dMn

n-ball,NE343 +0.3%wt Gd

[632]

1998

1–5

Pb,U,∅ = 15/8 L = 5/35/20/40

dPreac /dMn


[631]

1999

0.5/1.5

Pb,15 × 15 × 60 cm3

d2 σ/dEd

TOF,NE213

[808]

2000

0.4/0.8/1.2 1.8/2.5

W, Hg, Pb

dPreac /dMn


[115]

2001

0.8/1.2/1.6

Pb,∅ = 20 L = 105

d2 σ/dEd

TOF NE213

[133]

2004

0.42/0.59

Pb, Bi,∅ = 10 L = 16/18/30

Mn /p

MOD Au-foils

[814]

Hg, and Pb used at the current spallation neutron sources and foreseen for ADS applications. As already mentioned the measured neutron multiplicity distributions are a sensitive test for particle transport simulations. Some experiments consider neutron leakage spectra measured either as d2 σ/dEd or as surface reaction rates with

11.2 Proton-Induced Thick Target Experiments in the Energy Range 0.1–2.5 GeV A compilation of neutron multiplicity data per incident proton of thick Pb targets.a

Tab. 11.11

Incident proton energy (GeV)

Pb-target diameter (cm)

Length (cm)

0.47/0.72/0.96/1.47

10.2

61

0.47/0.72/0.96/1.47

20.4

61

0.4//0.5/0.66 0.99/1.47/2.0/2.56/ 3.17/3.65/4.45

16 20

0.316/0.66/0.997 0.1 0.25 0.8/1.0/1.2/1.4 0.197 1.22/2.2/3.17/4.15 0.4/0.8/1.2/1.8/2.5 0.42/0.59

10 6.2 5 10.2 12 15 15 10

Mn /p

References

7.96/11.76/ 16.64/26.40 8.67/13.94/ 20.29/31.47

[59]

55 60

6.0/8.2/12.5 21.3/31.4/40.2/51.3/ 61.5/68.1/81.1

[818] [617]

60 1.6 10 61 25 35 35 30

3.13/10.61/17.1 0.343 2.64 13.6/17.38/22.31/26.21 1.44 26.7/42.2/53.77/63.85 6.34/17.22/24.97/35.1/43.44 6.0/9.6

[815] [819] [820] [797] [632] [631] [115] [814]

a

The first column gives the incident proton energies, the second and third column the target dimensions, the fourth the multiplicities related to the energies in the first column, and the last column gives the references for the data plotted in Figures 11.16 and 11.17.

threshold detectors positioned on the targets. Table 11.11 summarizes measured Mn /p values of thick Pb targets for incident protons with energies from 0.1 to 5.0 GeV where a large variety of measured data exists. The different experimental Mn /p values are plotted as a function of the incident proton energy in Figures 11.16 and 11.17 for incident proton energy ranges 0.1 GeV ≤ Eproton (incident) ≤ 5.0 GeV and 0.1 GeV ≤ Eproton (incident) ≤ 2.5 GeV, respectively. The data of the NESSI collaboration [115, 631] and the data of Vassil’kov et al. [617] are fitted by a second order polynomial function. As shown in Figure 11.16 up to an incident proton energy of about 2.0–2.5 GeV the neutron multiplicity Mn /p is almost linearly dependent on the incident proton energy. Above a certain energy, other particle production channels as, e.g., π ± , π 0 , and kaon production will be opened, thus the incident proton energy is dissipated. The difference between the NESSI data [115, 631] and the data of Vassil’kov et al. [617] is related to larger target dimensions, diameter of 20 cm and length of 60 cm, compared to the NESSI experiment with a diameter of 15 cm and a length of 35 cm. The neutron multiplicity is still increasing with diameter at higher energies, whereas at lower energies the effect of the target geometry is much smaller as seen in Figure 11.17.

405

Average neutron multiplicity per proton Mn /p

11 Proton-Matter-Induced Secondary Particle Production – The ‘‘Thick’’ Target Experiments 100 Yn = 0.99401+21.52163 Ep-0.80341 Ep2

90 80

Yn = 1.13423+23.70098 Ep-1.97546 Ep2

70 60 50

d=10.2, I = 61cm (Fa65) d=20.2, I = 61cm (Fa65) d=16.0, I = 55cm (Vas68) d= 20.0, I= 60cm (Vas91) d= 10.0, I= 60cm (Wes71) d= 6.2, I= 1.6cm (Lon83) d= 5.0, I= 10cm (Rya83) d= 10.2, I=61cm (Zuc97, Zuc98) d= 12, I= 25cm (Lot98) d=15.0, I =35cm (Hil98, Let00) d =10.0, I =30cm (Mee04)

40 30 20 10 0 −10 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Incident proton energy [GeV]

Fig. 11.16 Measured neutron multiplicities Mn /p of Pb targets in the energy range 0.1 GeV ≤ Eproton (incident) ≤ 5.0 GeV. Complied data are given in Table 11.11 with the references indicated in the figure.

Average neutron multiplicity per proton Mn / p

406

50 40 30 d =10.2, I =61cm (Fra65) d= 20.2, I =61cm (Fra65) d= 16, I =55cm (Vas68) d =20, I =60cm (Vas91) d =10, I= 60cm (Wes71) d= 6.2, I= 1.6cm (Lon83) d =5, I =10cm (Rya83) d =10.2, I =61cm (Zuc97, Zuc98) d = 12, I= 25cm (Lot98) d =15, I =35cm (Hil98, Let00) d= 10, I = 30cm (Mee04)

20 10 0 −10 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Incident proton energy [GeV]

Fig. 11.17 Same as in Figure 11.16 but Mn /p of Pb targets in the energy range 0.1 GeV ≤ Eproton (incident) ≤ 2.5 GeV. Complied data are given in Table 11.11 with the references indicated in the figure.

The difference of the values Mn /p at the same incident proton energies are mainly caused by different target dimensions or by the absolute accuracy, which for the water experiments is believed to be accurate within ±5% to ±10%, whereas the NESSI data are estimated to be accurate within ≤ ±5%.


11.2.3 Neutron Leakage Spectra Distributions of Thick Targets

As already mentioned, the TOF method was considered for thick target experiments to measure the neutron leakage yield spectra at the LANL-WNR, Los Alamos, USA, on stopping-length and near-stopping-length target materials with incident proton beams of 113 and 256 MeV energy by Meier et al. [696–698]. At the synchrotron at KEK, Japan, thick Pb target leakage neutron spectra were measured by Shin et al. [823] and Meigo et al. [808] applying 0.5 and 1.5 GeV proton beams. At the SATURNE accelerator, France, experimental investigations on emitted neutron spectra distributions from thick cylindrical Fe and Pb targets with various lengths and diameters were undertaken with 0.8, 1.2, and 1.6 GeV proton energies by Leray et al. [133, 543]. 11.2.3.1 The LANL-WNR Thick Target Experiments Stopping-length and near-stopping-length target materials (9 Be, C, O, 27Al, Fe, Pb, 238 U) are used at 113 and 256 MeV proton beam experiments by Meier et al. [696–698, 824] to measure the absolute differential neutron yield leakage spectra emitted at scattering angles of 7.5◦ , 30◦ , 60◦ , 120◦ , and 150◦ . The LANLWNR facility and its specific properties of the TOF experiment were already discussed in detail in Section 10.2.1 on page 289. The experiments were conducted

Absolute neutron yield [n sr−1 MeV−1]

101 100

26

10−1

Fe(p,xn) - 256 MeV stopping target

10−2 10−3 10−4 10−5 10−6

7.5°

10−7

30° × 10−1 60° × 10−2

10−8

150° × 10−3

−9

10

10−1

100

101 Neutron energy [MeV]

Fig. 11.18 Experimental 256 MeV proton-induced absolute differential neutron yields for the reaction 26 Fe(p, xn) for a stopping-length Fe target. Each successive curve, starting from the smallest angle 7.5◦ , is scaled by a multiplication factor of 10−1 , e.g., 30◦ × 10−1 , 60◦ × 10−2 , and 150◦ × 10−3 .

102

5×102

407


101 Absolute neutron yield [n sr−1 MeV−1]

408

100

238U(p,xn)

- 256 MeV stopping target

10−1 10−2 10−3 10−4 10−5 10−6

7.5° 30° × 10−1

−7

10

60° × 10−2

10−8

150° × 10−3

10−9 10−1

100

101

102

5×102

Neutron energy [MeV] Fig. 11.19 Same as in Figure 11.18 but for the reaction 238 U(p, xn) and a stopping-length U target.

with incident proton beam energies of 116 and 256 MeV. The target characteristics of stopping-length and near-stopping length targets are summarized in Table 11.12. In Figures 11.18 and 11.19 the absolute differential neutron yields for the scattering angles 7.5◦ , 30◦ , 60◦ , and 150◦ for an incident proton energy 256 MeV on stopping-lengths Fe and U targets are plotted [696, 698]. The target dimensions are given in Table 11.12. Both measurements show at an emission angle of 7.5◦ and a neutron energy of about 200 MeV the quasielastic peak in the neutron yield production, which vanishes at higher emission angles. This effect is also observed in the ‘‘thin’’ target data. All the data are retrieved from NNDC experimental nuclear reaction database (EXFOR/CSISRS) [709]. 11.2.3.2 The KEK Time-of-Flight Thick Pb Target Experiments The TOF experiment at the KEK facility and its specific properties were already discussed in Section 10.2.3 on page 312. The intensity of the proton beam was very weak, ≤ 105 particles per pulse therefore the incident protons on the target could be counted one-by-one. The size of the beam was 2.0 cm in the vertical plane and 1.6 cm in the horizontal one in FWHM, respectively [823]. The used Pb target in natural abundance was of rectangular shape 15 × 15 × 20 cm3 . Protons of an energy of 0.5 GeV were stopped completely, whereas protons of 1.5 GeV caused an average energy loss of about 0.26 GeV. The detector efficiencies were estimated as described in Section 10.2.3 on page 312 and more in detail in [808]. The results of the experiment together with Monte Carlo particle production and particle transport simulations are illustrated in Figures 11.20 and 11.21. The estimated uncertainties of the neutron yield are energy dependent and are given

11.2 Proton-Induced Thick Target Experiments in the Energy Range 0.1–2.5 GeV Stopping-length and near stopping-length target materials used at the 113 and 256 MeV proton beam experiments by Meier et al. [696–698, 824].

Tab. 11.12

Stopping length targets at 113 MeV incident protons Element Be C Al Fe 238 U

Radius (cm)

Length (cm)

3.65 3.65 3.65 3.65 4.00

5.70 5.83 4.03 1.57 3.00

Target mass (g) 434.2 394.8 455.2 517.1 2819.9

Stopping length targets at 256 MeV incident protons Be C Al Fe 238 U

6.60 8.00 8.00 8.00 4.00

27.86 30.00 20.00 8.00 5.00

7048 9928 10918 12644 4770

Near stopping length targets at 256 MeV incident protons Be C Al Fe 238 U

6.60 8.00 8.00 8.00 4.00

17.86 17.45 12.15 4.70 3.00

4517 5775 6632 7429 2863

for neutrons with energies ≤ 20 MeV of about ±1σ = 9–12% and increases up to ±1σ = 19–25% for neutron energies of about 200 MeV [808]. As seen from the figures the measured data were compared with Monte Carlo results using the NMTC/JAERI-MCNP-4A system [808]. In general, the agreement is fairly well in the low-energy regime ≤ 10 MeV whereas in the energy range between 20 and 80 MeV the calculated values are about 50% lower. Also at backward angles, the simulations underestimate the measured results. 11.2.4 The LANL SUNNYSIDE Experiments

The aim of the LANL SUNNYSIDE experiment [557, 825] was to measure the neutron production for prototyping targets for accelerator driven systems ADS and specifically for APT1) applications for incident protons in the energy range from 0.4 to 2.0 GeV. Most of the experiments were investigated at the SATURNE accelerator at CEA-Saclay, France, at incident proton energies of 0.4, 1.6, and 2.0 GeV. Some

409


102 101 100 Neutron yield [n sr −1 MeV −1 p−1]

410

NMTC/JAERI-MCNP-4A Cugnon NN NMTC/JAERI-MCNP-4A Free NN

15° ×102 30°× 10 60°

10−1 90°× 0.1

10−2 10−3 10−4

120° −2 ×10 150° −3 ×10

10−5 10−6 10−7 10−8 100

Proton energy 0.5 GeV

101

102

103

Neutron energy [MeV] Fig. 11.20 Neutron yield spectra for 0.5 GeV protons incident on a thick Pb target measured at scattering angles of 15◦ , 30◦ , 60◦ , 90◦ , 120◦ , and 150◦ . The symbols represent the different angle-dependent measured yield spectra. The dashed and solid lines are results of Monte Carlo simulations with NMTC/JAERI coupled with MCNP-4A (after Shin et al. [823]).

of the experiments were carried out at the experimental LANL-WNR area using a 0.8-GeV proton beam already described in Section 10.2.1.1 on page 290. A sketch of the experimental setup is given in Figure 11.22. A target and a lead blanket assembly of 60 cm outer diameter, a thickness of 16.51 cm, and a length of 200 cm is embedded into a water moderator vessel of 2.5 m in diameter and 3.0 m in length. The neutron multiplicity per incident beam proton Mn /p is determined considering the manganese water-bath technique. The neutrons leaking out of the target are captured in the manganese producing 56 Mn, which has a half-life of 2.6 h and emits gamma rays of 0.846 MeV energy. The upper vessel section contained only water whereas the lower two sections contain a 1–2% MnSO4 solution varying during the experiment. The total volume of H2 O in the upper vessel was 5.047 m3 and the volume of the MnSO4 solution in the two lower vessels was 9.245 m3 . Since most of the 56 Mn is produced near the target a mixing of the manganese solution is accomplished by pumping system before extracting a sample to determine the total 56 Mn production. Details are given in [386].


102 15°×102

101

30°×10

Neutron yield [n sr−1 MeV−1 p−1]

100 10−1

60° 90°×0.1

10−2

120° ×10−2

10−3

150° ×10−3

10−4 10−5 10−6 NMTC/JAERI-MCNP-4A Cugnon NN NMTC/JAERI-MCNP-4A free NN Proton energy 1.5 GeV

10−7 10−8 100 Fig. 11.21


103

Same as in Figure 11.20 but for an incident proton energy of 1.5 GeV.

Four types of targets were chosen which are relevant for various ADS systems. • cylindrical Pb target with diameter of 25 cm and length 120 cm, stopping length for 0.8 GeV protons is about 40 cm, • cylindrical target of 7 Li diameter of 25 cm and length of 170 cm made of seven steel container sections containing the 7 Li metal, the length of the target is only sufficient to stop protons with an energy of 0.4 GeV, • cylindrical target as a mixture of Th in molten salts with Li and florine – ThO2 /Li2 CO3 /teflon, the average density of this target is to low to stop 0.8 GeV protons therefore a teflon plug was added downstream of the target, • cylindrical W target diameter of 15.24 cm and length 81.27 cm, the target was backed by a Pb beam stop with diameter of 25 cm and length of 37.38 cm. The results of the experiment are the total numbers of 56 Mn atoms and the total number of protons, which were determined by the Al foil activation method and by an integrating current transformer measuring the total proton beam charge [557]. A comparison of the beam charge measurements with the transformer and the Al foil activation method agreed in average to 2%. The number of 56 Mn per beam proton is the measured quantity, which must be compared with spallation reaction transport simulations to calculate at least the neutron yield or multiplicity Mn /p.

411

412


Moderator vessel: Ø = 250 cm, length = 300 cm volume: about 14 m3 circulation 12× 103 cm3/s

0.4-2.0 GeV

Lead blanket Ø = 60 cm, length = 200 cm Foil insertion pipe Lead target Ø= 25, length =120 cm Polyethylene plug

Proton beam

1–2% MnSO4 solution Legend Lead Polyethylene H2O/1–2% MnSO4

Fig. 11.22 A schematic sketch SUNNYSIDE experiment with the target holder, the lead blanket, and the water vessel containing the MnSO4 solution (after Morgan et al. [386, 557]).

As discussed in [825], the total systematic uncertainties in the measurements were generally in the range of 3–4%. Some systematic uncertainties are caused from the calibration of the counters to measure the 56 Mn and the determination of the number of beam protons considering the reactions 27 Al(p,3pn)24 Na and Tab. 11.13 Experimentally measured 56 Mn-production rates per proton at different incident beam energies for a Pb and a W target and comparison with MCNPX Monte Carlo calculations given as the ratio exp/calc.

Target

Proton energy (GeV)

MnSO4 (wt%)

Solid Pb ∅ = 25 cm

0.4 0.8 0.8 0.8a 1.6 2.0

0.833 0.833 0.994 1.647 1.647 0.833

0.050 0.171 0.205 0.335 0.855 0.576

3.8 3.3 4.3 3.7 3.1 3.3

0.96 0.97 0.97 0.97 1.01 1.05

Solid W ∅ = 15 cm

0.4 0.8 1.6 2.0

1.647 1.647 1.647 1.647

0.059 0.202 0.519 0.669

3.9 2.8 2.7 3.2

0.89 0.92 0.97 0.98

a

See Morgan et al. [557].

Exp. 56 Mn/proton

Uncertainty (%)

Exp./calc.

average

11.2 Proton-Induced Thick Target Experiments in the Energy Range 0.1–2.5 GeV 27 Al(p,x)22 Na by the Al foil activation, and the total volume of water in the system, the MnSO4 concentration. In Table 11.13 some results for solid Pb and W targets of the SUNNYSIDE experiment are summarized and compared with the calculated values using the MCNPX particle transport system [194]. The data are from Morgan et al. [825]. The examples of the comparison between the experimental and the calculated results, given as the ratio exp/calc, for thick Pb and W targets surrounded by a 60 cm thick Pb blanket (cf. Figure 11.22) show a good agreement. The MCNPX calculations are used with the default parameters, the BERTINI–ISABEL model with pre-equilibrium [825].

11.2.5 Energy Deposition Experiments with Thick Mercury Targets

As already mentioned the spallation neutron sources (SNS) [681] at Oak Ridge, USA, and J-PARC [629] at KEK, Japan, are operated at a proton beam power of about 1 MW, whereas future sources are envisioned to be operated with 5 MW proton beam power and beyond. Liquid mercury has been selected as first priority material for the high power spallation targets originally proposed for the European Spallation Neutron Source (ESS) [551]. Operated in a 1 µs pulsed mode up to 100 kJ per pulse will be deposited into the target systems of such spallation sources. The container material for the liquid mercury will be subject of high radiation damage, high thermal mechanical load and corrosion by the liquid metal. Computer calculations show that because of the high thermal mechanical load, tensile stresses may occur in the container. Therefore, it is essential to understand the energy deposition mechanisms of high-intensity proton pulses incident on thick targets. The experimental collaborations of the NESSI [809] and ASTE [812]2) performed therefore several experiments to measure the thick target energy deposition during the proton pulse train on mercury targets. The NESSI collaboration applied the proton beam of the COSY accelerator at Jülich with 0.8 and 1.2 GeV energy protons for mercury and lead targets of 15 cm diameter and 35 cm length, whereas the ASTE collaboration used the proton beam of the AGS at the BNL, Brookhaven, to measure the energy deposition and the temperature rise in a thick mercury target of diameter 20 cm and length 130 cm during a proton pulse train of 24 GeV incident energy and a deposition power of 30 kJ per pulse. Due to the extremely short proton pulse of 1 µs applied at high-intensity spallation neutron sources, a pressure wave might be induced in the liquid metal. That means the dynamic behavior of the liquid metal and the impulse transfer to the target container must be additionally considered for the design of high-power targets. The alternating stress and pressure make high demands against the fatigue strength of the container material. Since the spatial distribution and intensity of the energy deposition are initial values for the thermodynamic calculations, it has also influences on (1) the design of the target container, like mechanical 1) APT, Accelerator Production of Tritium.

413

414


stability, flow dynamics of the liquid metal and optimization of the cooling, (2) the request at structure materials, like a behavior in relation to strong temperature gradients, fatigue strength, corrosion resistance, (3) and the life time of the target. A reliable prediction of the distribution and intensity of the energy deposition in the target presuppose an accurate simulation of particle production, particle transport and energy deposition mechanisms. Since the rise in temperature produced in the target is caused mainly as a consequence of ionization processes of charged particles the power density in the target can be determined via the measurement of these processes. 11.2.5.1 The Energy Deposition Thick Mercury Target Experiment at COSY Jülich For the proton energies and the intensities available at the COSY accelerator the method of thermoluminescence detectors (TLD)s seems to be suitable for the measurement of the energy deposition in a spallation target. This method is well known from dose measurement in radiation protection and have been successfully employed in Jülich to measure dose distributions and even the Braggpeak in high resolution in phantom irradiation experiments in the incident proton energy range up to 0.3 GeV [826]. Measurements were investigated with incident proton beam energies of 0.8 and 1.2 GeV on lead and mercury targets of 15 cm diameter and 35-cm length which were equipped with 200 TLD detectors. For the measurements thulium doped calcium fluoride thermoluminescence detectors were utilized. For the measurements detectors from Harshaw were used with a chip size of 0.3 × 0.3 × 0.1 mm3 . The dose response of the used TLD is linear in proton – He2+ , Ne10+ – and in π − -fields up to a linear energy transfer (yD ) smaller than 30 keV or 70 keV µm−1 [827]. The detectors are not sensitive to neutrons. More details of the analysis procedure and the Monte Carlo simulations is given in Ref. [828] Figure 11.23 shows the setup of the mercury target and the position of the detectors. The Hg target with a diameter of 15 cm and a length of 35 cm was assembled in stainless container segments of different thicknesses filled with Hg (cf. Section 11.6 on page 11.6). The detectors were positioned between the target segments by a positioning disk of 1 mm thick stainless steel. The first disk was equipped with 73, all following with 33 TLD detectors. TLD detectors and the stopping power Charged particles lose their energy during the passage through matt mainly by the inelastic scattering and by ionizing. The Bethe–Bloch formula (cf. Section 1.3.6.1 on page 33) describes the mean energy loss dE/dx of a charged particle by ionization. To draw conclusions from a dose measured by the TLD material to the energy deposition in a spallation target, the different stopping power values of a charged particle related to the material must be known. The ratio of the stopping power Starget in the target material to the stopping power Sdetector in the detector material converts the dose in TLD CaF2 to the dose deposited in the target material. Figure 11.24 shows the stopping power for protons in the target material Hg and in the TLD detector material as a function

11.2 Proton-Induced Thick Target Experiments in the Energy Range 0.1–2.5 GeV TLD detectors Ø 6 mm

Detector discs

15 cm

Ø 5 mm

35.7 cm

Fig. 11.23 The geometry of the mercury target assembly. Indicated are the stainless steel disks equipped with TLD detectors.

CaF2 Stopping power S [MeV cm2 g−1]

102

Hg S(Hg) / S(CaF2)

101

100

10−1

10−2 1

10 102 Proton energy [MeV]

103

Fig. 11.24 Proton energy dependent stopping power S in the TLD of CaF2 , the Hg target material, and their stopping power ratio.

of the proton energy. The ratio of the stopping power in the target material Hg and in the TLD CaF2 approaches in the area relevant for the measurements above 20 MeV kinetic energy of the particles against a constant value. Due to this behavior of the ratio, the measured dose in CaF2 can be transferred to the target material [828].

415


0.6

0.6

xsr = −0.1 cm sx = 0.56 cm

0.5

Energy deposition [J kg−1]


416

0.4 0.3 0.2 0.1 0

−5

0 2.5 −2.5 x – axis [cm]

5

ysr = −0.03 cm sy = 1.16 cm

0.5 0.4 0.3 0.2 0.1 0

−5

0 2.5 −2.5 y – axis [cm]

5

Fig. 11.25 Gaussian-fitted measured distributions of the horizontal and vertical energy deposition in front of the Hg irradiated with 1.2 GeV protons. σx and σy are the standard deviations of the Gaussian fit and the values xsr and ysr indicate the shift of the maximum in the x- or y-direction.

Proton beam intensity distribution The proton beam intensity distribution was measured with a TLD disk in front of the mercury target at an incident proton energy of 1.2 GeV (cf. Figure 11.23) and is used for the Monte Carlo simulations of the energy deposition inside the thick Hg and Pb targets. The TLD disk in front of the target was equipped with 33 TLD chip detectors along the vertical, y-axis, the horizontal, x-axis, and the 45◦ -axis. The discrete measured values on the x- and y-axis were fitted with a Gaussian function to obtain the intensity distribution. The distributions are given in Figure 11.25, a standard deviation of ± 1σ in the x-direction σx =0.56 cm, and in the y-direction σy =1.16 cm for the y-axis (cf. Figure 11.25) has been found. The zero point of the coordinate system corresponds to the center of the target. It was further assumed that a superposition of the two gaussian distributions in the x- and y-direction represent the distribution of intensity in the xy-plane in the form

f(x,y) = af(x) f(y) +∞ +∞ and f(x,y) dxdy = 1.

(11.10)

−∞ −∞

Figure 11.26 shows the comparison of the measured values to the 45◦ -axis and the corresponding Gaussian fit. The deviation of the measured points x = −0.8, y = 0.8 from the assumed Gaussian fit is shown in Figure 11.26 for example. Related to the measured value the assumed function f(x,y=x) (dashed line) indicates an error of 63%. The deviation of the measured values in relation to the assumed function f(x,y=x) is caused mainly by a shift of the maximum by 0.4 cm. This deviation is situated however in the context of the measurement inaccuracy, which results from the position of the TLDs

11.2 Proton-Induced Thick Target Experiments in the Energy Range 0.1–2.5 GeV 0.6

0.5 0.4

(−0.8; 0.8)

0.3

FHg

0.2 0.1 0

(a)

+x −y

−2

0

2

[cm]



0.6

0.5 0.4 0.3 0.2 0.1 0

−x +y

(b)

−x −y

−2

0 [cm]

2

+x +y

Fig. 11.26 Comparison of the measured values (indicated as triangles) on the 45◦ -axis with the Gaussian fit for Hg target with 1.2 GeV protons. For further explanations, see the text further. Comparison of measurements (exp.) and Monte Carlo simulations (calc.) of the energy deposition along the z-axis of Hg and Pb targets with diameters of 15 cm and lengths of 35 cm.

Tab. 11.14

Incident proton energy 1.2 GeV Position on z-axis (cm)

0.05a 0.05b 1.50 3.70 9.15 14.35 19.55 24.75 a b

exp.

Calc. (J cm−3 p−1 ) × 10−12

Hg 0.436 0.751 0.940 0.732 0.512 0.268 0.146 0.070

Pb 1.29 1.86 2.00 2.06 1.72 1.00 0.49 −

Hg 0.438 0.754 0.857 0.784 0.522 0.368 0.224 0.114

Calc./Exp.

Pb 1.24 1.79 1.91 1.78 1.54 1.03 0.67 −

Hg 1.00 1.00 0.91 1.07 1.02 1.37 1.53 1.64

Pb 0.96 0.96 0.96 0.87 0.90 1.02 1.36 −

Normalized to stainless steel. Normalized to Hg/Pb material.

in the measurement. Taking this into account the above Gaussian approximation of the beam intensity distribution is used for the Monte Carlo simulations of the energy deposition. The results are compared with the measurements and given in Table 11.14. In additional irradiation experiments using a Pb target a higher spatial resolution of the distribution of intensity of the proton beam could be obtained, because of a new position disk. The distribution of intensity of the proton beam could be mapped completely by the TLDs, which cover an area of 2.8 × 2.8 cm2 . The measured and

417

12 10 8 6 4 2 0

−2

−1

0

1

x – axis [cm]

2

Energy deposition in arbitary units


Energy deposition in arbitary units

418

12 10 8 6 4 2 0

−2

−1

0

1

2

y – axis [cm]

Fig. 11.27 Measured (triangle) and simulated (lines) intensity distribution of the proton beam for the irradiation of the Pb target with 1.2 GeV protons.

Monte Carlo simulated response is shown in Figure 11.27. The triangles indicate the measured distribution of intensity and the solid lines of the simulated ones. Results and conclusions The axial energy deposition can be described by the following features: • The energy deposition in the target window, which produces thermal load must be considered for the design of the containment. • Value and position of the peak-energy deposition and the shape of energy deposition along the z-axis in the target determine the pressure gradient in a liquid target material.

Figure 11.28 shows the measured and simulated distribution of the energy deposition along the z-axis of the Hg and Pb target irradiated with 1.2 GeV protons. The triangles indicate the measured values and the circles the Monte Carlo simulated results. The energy deposition at the position z = 0 indicates the energy deposition produced in the stainless window of the target. All other values are the deposition results inside the target material. For the Monte Carlo simulations the contributions of protons, deuterons, tritons, helium, pions, muons, and recoil products from the intra–inter nuclear cascade evaporation process to the energy deposition were taken into account. The agreement between measurements and simulations is fairly well. Underestimated is the maximum of the energy deposition in the target by the simulations, whereas with increasing depth the energy deposition is slightly overestimated. The measured values, the results of the simulation calculations, and the ratio of the calculated results to measured ones are summarized in Table 11.14. The energy deposition calculations were performed by the HERMES code system [88]. The absolute agreement between simulation and experimental results is fairly well. For the Hg target the peak deposition on the z-axis is slightly underestimated by the simulation whereas the tails of the distribution along the z-axis are slightly


Energy deposition [j cm−3 p−1 × 10−12]

1

Experiment MC calculation

0.8 0.6 0.4 0.2 0

0

2.5

5

7.5

10 12.5 15 Target depth [cm]

17.5

20

22.5

25

2.5 Energy deposition [j cm−3 p−1 × 10−12]

Experiment 2

MC calculation

1.5 1 0.5 0

0

2

4

6

8 10 12 Target depth [cm]

14

16

18

Fig. 11.28 Distribution of the energy deposition in J cm−3 per incident proton along the z-axis for the irradiation of an Hg target (a) and a Pb target (b) with 1.2 GeV protons.

overestimated. The accuracy between simulation and experiment is in the order of 5–10%. It was concluded from the results of the validation of the experiments that the predictive power simulating energy deposition in mercury and lead spallation targets is sufficient and accurate to supply reliable parameters for the engineering layout. 11.2.5.2 The Thick Mercury Target Experiment ASTE at the AGS Accelerator at BNL The purpose of the ASTE collaboration was to perform experiments to verify experimentally a number of predictions from theoretical calculations on the neutron and thermodynamical behavior of spallation targets with mercury designed for pulsed operation in the megawatt power regime [812, 829]. The experiments started in 1997 with a first test run with a bare cylindrical (diameter 20 cm, length 130 cm) mercury target and were continued with reflected target systems (cf. Section 13.3.3.3 on page 466 and Figure 13.29 on page 466). The main goals of the experiment were to verify experimentally a number of theoretical predictions: • energy deposition distribution at GeV incident proton beam energy measured as a temperature jump during the pulse tain [810, 830]; • distribution of neutron leakage reaction rates;

419

420


Bars for activation foils

Connection for thermocouples Fig. 11.29 The ASTE stainless steel target container filled with mercury of diameter of 20 cm and length of 130 cm. Also shown are the bars for the activation foil measurements and the connection lines to the thermocouples inside the mercury.

• pressure waves in mercury and stress wave monitoring of the target container [829, 831]; • spallation product measurements, reaction rate distributions in reflected target systems and neutron spectra time-of-flight measurements [812]. The ASTE target, a 20 cm diameter and 130 cm long cylindrical stainless steel container with a hemispherical front filled with mercury, is shown in Figure 11.29. Of major interest of the experiment was the estimation of the energy deposition distribution in the target at short pulsed incident proton beams, which was at the ASTE experiment accomplished by measuring the temperature jump. The temperature jump measurements were carried out by mounting thermocouples at axial positions inside the Hg target medium. Figure 11.30 shows a sketch with the positions of the 32 thermocouples inside the mercury placed in the lower half midplane of the target cylinder. The 1.5 mm thick encapsulated Chromel-Alumel thermocouples (type K) [830] were thinned down to a thickness of 0.5 mm over the last 15 mm in the radial direction in order to improve their time response. The distributions of the thermocouples in radial direction were chosen on the basis of an assumed parabolic radial power distribution. The noise of the system was about 0.5 K dependent from the cable lengths to data acquisition system which was about 50 m. The beam intensity and its profile was measured by an Al foil, 20 × 20 cm2 , 25 µm thick, by the Japanese group of the ASTE-collaboration. The radioactivity of the proton-induced production of 22 Na and 7 Be was determined by placing the Al foil on an image plate and integrating the measured radioactivity over pixel sizes of 2 × 2 mm2 . The resulting beam profile is shown in Figure 11.31. The beam center is not located in the target coordinate center. The distribution is slightly elliptical. The measured profile in the xy-direction was used to compare the experimental results with Monte Carlo calculations given in Figures 11.32 and 11.33.


1300 mm 980 680 480 380

2. 3. 4. 5. 6.

1.

Proton beam Fig. 11.30

7. 12 . 8. 13. 9. 10. 14. 11.

15. 20. 16. 17. 21. 18. 19. 22.

23 . 24. 25. 26. 27.

28 .

31.

32.

200

20 40 60 80

280 230 180 130 80 30

29. 30.

The positions of the 32 thermocouples inside the mercury of the Aste.

24 GeV proton beam profile on the ASTE target Beam intensity

100

3.00 × 10−4 1.82 ×10−4

x [mm]

50

1.14 ×10−4 6.70 ×10−5

0

3.41 ×10−5 −50

4.20 ×10−6 0.0

−100 −100

−50

0 y [mm]

50

100

Fig. 11.31 Measured beam intensity and density profile of the proton beam at an energy of 24 GeV incident on the ASTE mercury target.

Measurements were performed at three different proton energies: 1.5, 7, and 24 GeV. The 1.5 and 7 GeV incident proton pulses incident on the target delivered not enough energy to produce a measurable temperature jump of more than 0.1 K at any thermocouple position in the target. Only a pulse with all protons of the AGS in two bunches, 2 × 4.0 × 1012 protons of 24 GeV in less than 30 ms introduced a maximum temperature jump of 3.4 K in the target (cf. Figure 11.32). A comparison of the axial distribution of a measured and a simulated temperature jump is given in Figure 11.32. The calculations were performed using the measured incident proton beam density profile. The maximum measured temperature jump of 3.4 K shown in Figure 11.32 corresponds to peak power density of about 6.4 J cm−3 using the values of 0.139 J

421


Temperature jump [K]

4.0 ASTE experiment HERMES calculation 0 (inclusive Π -decay deposition)

3.0 2.0 1.0 0.0 −10.0

10.0

30.0 50.0 70.0 Target depth [cm]

90.0

110.0

Fig. 11.32 Measured and Monte Carlo simulated temperature jump produced inside the mercury target during a 25 kJ incident proton pulse of 24 GeV energy. The simulation is predicted by the HERMES Monte Carlo code [88].

1015 Energy integrated neutron surface flux [n cm−2 s−1]

422

Monte carlo simulations

1014

1013

Indium reaction rates of ln(n,n′)115mln

115

1.5 GeV 7.0 GeV 24.0 GeV

1012 0.0

25.0

50.0 75.0 Target length [cm]

100.0

125.0

Fig. 11.33 Comparison of simulated and experimental results of the neutron leakage flux with incident protons of 1.5, 7, 24 GeV measured with indium threshold foil detectors on the surface of the ASTE mercury target. The leakage fluxes are normalized at an incident proton beam power of 5 MW.

g−1 K−1 for the specific heat capacity and 13.52 g cm3 for the density of the mercury. A numerical integration over the measured axial and transversal temperature profiles over the whole target was used to determine the total power deposition per proton in the target during the pulse train [830]. These values are summarized in Table 11.15. A series of experiments on the bare ASTE mercury target were investigated at incident proton of 1.5, 1.6, 7, 12, and 24 GeV on spatial and spectral distributions of the fast neutron leakage using various activation detectors with different threshold energies, e.g., activation foils of In, Nb, Al, Bi, Co, Tm, Y, Ni, and Ti. Details are

11.2 Proton-Induced Thick Target Experiments in the Energy Range 0.1–2.5 GeV Total energy deposition per pulse, total energy deposition per proton, and percentage of energy deposition of the beam power in the ASTE mercury for incident proton of an energy of 24 GeV (data are from Ref. [830]).

Tab. 11.15

Radial extrapolation

Linear Parabolic

Total energy deposited per proton pulse (kJ)

Total energy deposited per proton (GeV)

16.7 18.7

13 14.6

Deposited energy in percent of beam power 54.2 61.0

given by Takada et al. [832]. The 115 In(n, n )115m reaction was selected as the baseline reaction detector because of its low-energy threshold energy of 0.34 MeV compared to the other detectors. Therefore, the reaction rate distributions of 115 In(n, n )115m were measured, most extensively of all detector foils, along the beam axis on the cylindrical surface of the target. The comparison of measured and simulated results gives an agreement of about 12% for the FWHM of the measured and calculated distributions. For all incident proton beam energies there is a tendency of the estimated neutron leakage after the peak position to underestimate by about 3–10% of the measured distribution.

423

425

12 Neutron Production by Proton, Antiproton, Deuteron, Pion, and Kaon Projectiles As discussed for the economy of neutron production in Sections 8.2.2, 8.2.4 and shown in Figure 8.5, there is not much advantage for incident proton beam energies beyond 1–2 GeV. Since at energies well above 1 GeV pion production becomes a dominating process [633] the question arises how efficient pions are for neutron production. Is there a considerable fraction of the initial available energy lost via decay channels like π 0 → 2γ or π + → µ+ + νµ ? To what extent is this loss compensated for by lower electronic energy loss of charged pions prior to nuclear reactions? Pions approach their minimum ionization power at considerably lower energies (0.3 GeV) than protons (2 GeV). To study these questions the comparison of proton- and antiproton-induced neutron production in thick targets is an ideal method, since in the case of p the neutron production is mediated essentially via pions originating from p-nucleon annihilation production on average 5 pions (3π ± + 2π 0 ). That is why in the following more generally hadron (π, K, p, p, d)-induced neutron production in thick Pb targets will be compared up to 5 GeV/c [664, 730]. It might be advantageous for the operation of the accelerator and target assembly of spallation neutron sources to employ at a given beam power considerably higher beam energies than 1 GeV. But since at increasing energy meson production (mainly πs) is increasing, an increasing part of neutron production is due to secondary-meson-induced reactions. How efficient are these pions in producing neutrons? In order to answer this question hadron-induced neutron production as a function of particle energy has been investigated by the NESSI collaboration [664, 730]. In particular neutron production for proton-, antiproton-, and pioninduced reactions has been compared. In the latter reaction neutron production is predominantly induced by π ± and π 0 . No data exist at all for pion-induced reactions which are of importance to account for the yield of secondary reactions, with the pions being produced in a primary (anti-)proton nucleus interaction. In addition, in the framework of the nuclear physics driven motivation described in Section 9.4 especially antiproton-induced reactions in flight and at rest are capable of exciting nuclear matter without disturbing dynamical effects generally induced in heavy ion reactions. Complementary to heavy ion reactions, antiproton-induced reactions represent an alternative method for producing highly excited nuclear matter and enable probing the nuclear equation of state at high temperatures. The Handbook of Spallation Research: Theory, Experiments and Applications. Detlef Filges and Frank Goldenbaum Copyright  2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40714-9

426

12 Neutron Production by Proton, Antiproton, Deuteron, Pion, and Kaon Projectiles

rather high energy transfer is due to the phase-time-structure of the elementary NN-annihilation with a radius of interaction of ≈ 1.8 fm and a coherence length of cτ ∼ 1.5 fm. In case of antiproton annihilation at rest, the keV-antiprotons are captured by the nucleus like ‘‘heavy electrons.’’ They cascade down to lower energy levels emitting Auger-electrons and annihilate with a nucleon of the nucleus at the periphery of the nucleus. On the average 5 pions, as well as strange particles K, , , are produced by partly heating the nucleus in a radiation-like way [671]. Much more thermal excitation energy can be transferred to the nucleus using energetic GeV-antiprotons, because the annihilation takes place much closer to or even inside the nucleus. Increasing the p-energy does not a priori implicit high E ∗ , because at the same time the average kinetic energy of annihilation pions is also increased and the pion absorption via the -resonance becomes less effective. INC model calculations [676] show the spin in p reactions to remain small (≤ 25) and the equilibration time (30 fm/c or 10−22 s) [169] to be much smaller than the dynamical periods in heavy ion reactions [677]. The latter effect is of extremely high importance when for high temperatures (T ≈ 6 MeV) the characteristic evaporation times are less than 10−22 s. The physical picture of annihilation in flight and at rest for pA reactions is drawn in Refs. [379, 663, 664, 673]. The secondary beam facility at the Proton Synchrotron (PS) at CERN, Geneva, enabled the measurement of Mn simultaneously for p, p, d, K, and π ± at the same incident momentum and charge. Details on this experiment can be found in Refs. [219, 631, 664, 730]. For protons, antiprotons, positive and negative pions Table 12.1 gives the first moment Mn of dσ/dNexp as well as the most probable neutron number Mnmax and its standard deviation σ as derived from a Gaussian fit to the multiplicity distribution dσ/dNexp at the position of the maximum. From the measured reaction probability Preac and Mn the mean number of neutrons per incident proton Nn /p = Mn × Preac was deduced. For protons, the NESSI collaboration observed with increasing target thickness an increase of both the mean neutron multiplicity Mn as well as the most probable neutron multiplicity Mnmax with the relation Mn ≤ Mnmax . For the thickest targets of 35 cm Pb Mn has almost attained the value of Mnmax (Table 12.1) simply due to the fact that the intensity of low neutron multiplicity events has become very small due to secondary reactions. For 35 cm Pb as well as 40 cm U, the intensity of the measured distribution which cannot be described by a Gaussian is smaller than 5% while for thin targets it amounts to 20–30%. For target thicknesses higher than 35 for Pb or 40 cm for U any further increase in the neutron yield per incident proton Nn /p is essentially due only to further increase of the reaction probability. The observed steep increase in the number of neutrons per incident proton Nn /p for target thicknesses up to about 10 cm, instead, is due to the combined increase in the reaction probability Preac and the mean neutron multiplicity Mn with target thickness. Contrary to other methods these two quantities are measured independently in the NESSI experiment and not as a product. The observed larger neutron multiplicity for U compared to Pb can be ascribed in the case of thin targets (cf. Table 12.1) to a higher probability for energy absorption in the bigger


427

target nucleus, to the lower neutron binding energies of U spallation products, and eventually to one or more extra neutrons from fission of the residual nucleus (ν = 1.92 for spontaneous fission of 238 U). In a thick target of U fission can be induced by many secondary particles and become a dominating process. It multiplies the number of neutrons to the extent that the neutron yield in a 40-cmlong U-target is nearly doubled as compared to Pb: MnU /MnPb = 1.5 and 1.7 at Ep = 1.22 and 4.15 GeV, respectively. The mean neutron multiplicities Mn for π ± , K+ , p, p, and d+ on a thick Pb-target (35 cm length, 15 cm diameter) is presented in Figure 12.1 at incident momenta 2, 3, 4, and 5 GeV/c, respectively. These momenta are corresponding to somewhat higher kinetic energies for πs than the respective proton energies. Since for such a thick target Preac 1, the given values can be considered also as mean numbers of neutrons per incident particle.

Tab. 12.1 Results for thin and thick Pb and depleted U targets: the mean neutron multiplicity Mn , the number of neutrons per incident particle Nn /p, and Nn /π ± ,the most probable neutron number Mmax n , and its standard deviation σ , the latter two are obtained from a Gaussian fit to the distributions of Mn .a

Incident particles protons/antiprotons Moment Target l inc. part. (cm) (GeV/c)

d (cm)

1.94 1.94 1.94 −1.94 2.00 3.00 4.00 −4.00 5.00 −5.00 1.94 1.94 4.00 5.00 5.00

15 18.0 15 21.3 15 26.0 15 − 15 26.7 15 42.2 15 53.9 15 − 15 63.8 15 − 5 × 5 23.3 8 38.0 8(15) 90.1 5 × 5 35.6 8(15) 112.2

a

Pb Pb Pb Pb Pb Pb Pb Pb Pb Pb U U U U U

0.2 5.0 35 35 35 35 35 35 35 35 0.3 40 40 0.9 40

Mmax n

σ

Mn

7.4 14.5 10.7 19.4 11.5 25.4 − 52.5 13.0 26.7 16.5 41.1 20.4 51.7 − 72.2 24.5 60.8 − 79.5 8.8 19.9 19.4 38.8 28.6 87.6 21.8 28.5 30.5 106.0

Nn /p

Pions π ± Mmax n

0.16 − 4.8 − 20.5 − 35.3 − 22.6 35.5 34.9 48.0 44.0 57.5 65.6 − 51.0 66.0 71.2 − 0.53 − 35.3 − 84.1 99.3 2.56 36.7 101.0 115.3

σ

Mn

− − − − − − − − 16.5 32.5 20.1 46.2 22.9 55.1 − 60.5 27.8 63.3 − 68.6 − − − − 33.1 93.5 24.5 30.5 36.1 107.5

Nn /π ±

− − − − 20.5 34.6 38.7 47.0 50.1 − − − 69.3 1.74 72.4

These values have been multiplied by the factor 1/0.85 in order to account for an assumed mean detection efficiency of 85%. Errors are in the order of 1–5%. Negative momenta in the first column of the table correspond to antiprotons or negative pions (π − ), l and d indicate the target length and diameter, respectively.

12 Neutron Production by Proton, Antiproton, Deuteron, Pion, and Kaon Projectiles p+ p+ thin p+ K+ d+ p− p−

80 70 Mean neutron multiplicity 〈Mn 〉

428

60 50 40 30 20 10 0

0

1

2

3

4

5

6

Incident (available) energy [GeV]

Fig. 12.1 Average neutron multiplicity Mn for incident protons p+ , antiprotons p− , pions π ± , kaons K+ , and deuterons d+ as a function of incident kinetic energy on 35cm-long, 15-cm-diameter Pb target. Mn has been corrected for a mean efficiency of

εn = 0.85 (cf. Eq. (10.3) and Figure 10.42). For p− the small filled squares indicate Mn also at the available incident energy, see text. The filled and open triangles display Mn for thin Pb targets. The curves are and Mmax n to guide the eye.

If Mn induced by p and p is compared at the same incident energy the neutron numbers in case of p are up to a factor of two higher. If however the p-nucleon annihilation energy of 2mp · c2 is taken into account, the ‘‘available energy’’ results p to Einc + 2mp · c2 = 1.22 + 1.88 = 3.1 GeV. Thus, for instance, the comparison with 1.22 GeV p should be rather made at an incident proton energy of 3.1 GeV. Mn is indeed very similar for p and p-induced reactions when compared at the same available incident energy. Proton and π ± -induced reactions should also be compared at the same incident kinetic energy. The multiplicities are somewhat smaller for pions than for protons if compared at the same incident energy1) (cf. Figure 12.1). In the case of π − one might argue that the capture of a π − in a nucleus at the end of the INC converts the rest energy of the pion (138 MeV) to nuclear excitation [775]. This would favor a comparison at an available energy of π− Einc + mπ c2 . In any case the value Mn has been measured at the same momenta and consequently the best approach to the same incident kinetic energy is the comparison p π of Einc = 4.15 GeV and Einc = 3.86 GeV as shown in Figure 12.1. However, the neutron multiplicity distributions and the corresponding average values Mn , as shown here, are relatively independent of the primary hadron species. For π ± and p the total reaction cross section is about 13% smaller [583] and 20% larger [662, 663] 1) While they are slightly larger when compared at the same momenta


than that for protons, resulting in correspondingly lower and higher neutron production per unit length along the trajectory of the beam particles in the target, which is a measure of the produced neutron density. Since the reaction cross section is also somewhat larger for deuterons than for protons this finding implies for the neutron yield per incident deuteron Nn /d = Preac × Mn even larger values, which qualitatively agree with the findings of Vassilkov et al. [617]. However, the uncertainty for deuterons in the present experiment is considerably larger since the secondary beam of the CERN-PS contains much less deuterons than protons (2 and 8) × 10−3 at 5 and 3 GeV/c, respectively. In summary, Mn of the considered hadrons are very similar within 10% if compared at the same incident available energy in the energy range of 1–6 GeV [662, 663, 730]. These findings indicate that neutron production mediated by mesons, which is increasing for proton-induced reactions with bombarding energy, is similarly efficient as spallation reactions without mesons being involved.

429

431

13 Experiments to Study the Performance of Spallation Neutron Sources 13.1 Introduction

To study the performance of spallation sources in terms of neutron flux and neutron spectra output, complex target–moderator–reflector (TMR) experiments were investigated since 25 years. The most common use of spallation produced neutrons is to provide a ‘‘thermal,’’ a ‘‘cold’’ or ‘‘hot’’ neutron source for condensed matter studies, but as already discussed, spallation neutrons are also considered in tritium breeding, as neutron sources for material damage studies, in fertile to fissile breeding and for transmutation of radioactive waste from nuclear power plants. Despite different goals, the same physical processes are involved in all of them and basically they only differ in the selection of materials and geometric configuration that enhances the planned use and the relative importance of the nuclear physics processes. Compared to fission neutrons, neutrons produced by the spallation process have a much broader energy range from 2 to 3 MeV up to several GeV, the energy of the incident protons. The first stages of TMR experiments are facility specific, may demonstrate feasibility, and will then be the choice of materials and geometry that enhance the particular nuclear aspects verified by large-scale experiments. Examples are the the German SNQ study experiments [833], the various examples of TMR experiments conducted by groups from Europe, Japan, and USA at accelerators ANL, BNL, Jülich, LANL, LNS, and PSI published, e.g., at conferences of the ‘‘International Collaboration of Advanced Neutron Sources’’ (ICANS-IV to XVII during 1981–2005 [834]).

13.2 The Target–Moderator–Reflector Issue

An optimal neutron producing system for an ideal neutron spallation source for neutron scattering experiments may be described by a statement of Gary I. Russell of LANL, USA, ‘‘A useful neutron is a neutron with the right energy, at the right time, at the right place.’’ Handbook of Spallation Research: Theory, Experiments and Applications. Detlef Filges and Frank Goldenbaum Copyright  2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40714-9

432

13 Experiments to Study the Performance of Spallation Neutron Sources Tab. 13.1

Moderator classification for spallation neutron sources.

Moderator classification

Neutron energy (meV)

Neutron wavelength ˚ (A)

Moderator temperature (K)

Ultra colda Cold Thermal Hot

≈ 2.5 × 10−4 0.1–10 10–100 100–1000

≈ 500 10–3 4–1 1–0.2

5–8 10–120 120–1000 1000–6000

a Data are based on the SUNS ultracold neutron source at PSI, Switzerland.

The purpose of moderators and reflectors surrounding a spallation target is to convert leakage neutrons from the target to low-energy neutrons to an energy spectrum with continuous or pulse characteristics with the possible highest intensity. A comprehensive description and discussion of the various aspects concerning the TMR systems needed for the utilizing spallation neutron sources in neutron scattering research is given by Carpenter [56], Ikeda and Carpenter [835], Bauer [57], Watanabe [635], and in Refs. [834, 836, 837]. Basic examples of target–moderator configurations are already given in Figure 8.12 on page 271. The process of reducing the energy of the neutrons produced in the spallation target to the thermal or subthermal energy region by elastic scattering is referred to as thermalization, slowing down, or moderation. The material used for the purpose of thermalizing or slowing down neutrons is called a moderator. Table 13.1 shows a classification of moderators with a range of neutron parameters, e.g., energy range, wavelengths, and temperatures suitable for spallation neutron sources. An efficient moderator reduces the velocity of neutrons in a small number of collisions, but does not ideally absorb them to any great extent. Slowing down the energy of the neutrons in as few collisions as possible is desirable in order to produce a large amount of thermal neutrons and also to reduce the number of resonance absorptions in the structure materials, which are contained in the TMR unit. The slowing-down process of neutrons in matter is well known from reactor physics and is published in detail in, e.g., the book of Beckurts and Wirtz [63]. An ideal moderating material used for spallation neutron sources should have the following nuclear properties: • large scattering cross section • small absorption cross section • large energy, loss per collision. A convenient measure of energy loss per collision is the logarithmic energy decrement also known as lethargy:


ξ = ln(Eint ) − ln(Efinal ) = 1 − (α/(1 − α)) Eint ξ = ln Efinal ξ = 1 forA = 1 and ≈ 2/(A + 2/3) for A > 1,

(13.1)

with = ln(1/α), α = (A − 1)2 /(A + 1)2 as auxiliary quantities, which are used to describe the details of the scattering process in the laboratory system [63]. The parameters are, ξ = average logarithmic energy decrement, Eint = average initial neutron energy, Efinal = average final neutron energy, and A = the atomic mass number of the moderator atom. The symbol ξ is commonly called the average logarithmic energy decrement because of the fact that a neutron loses, on the average, a fixed fraction of its energy per scattering collision. Since the fraction of energy retained by a neutron in a single elastic collision is a constant for a given material, ξ is also a constant. Because it is a constant for each type of material and does not depend on the initial neutron energy, ξ is a convenient quantity for assessing the moderating power of a certain material. For hydrogen ξ = 1, and for large atomic masses A, ξ can be very well approximated by ξ ≈ 2/(A + 2/3). Since the quantity ξ is the average logarithmic energy loss per collision, the total number of collisions n necessary to moderate a neutron with an initial energy Eint to a certain energy Ei is given by Eint . n · ξ = ln Ei ln(Eint /Ei ) . (13.2) n= ξ As for example about 22 collisions are needed to slow down a neutron from 20 MeV to a thermal energy of 0.025 eV scattered in a water moderator with a ξ = 0.926 using Eq. (13.2). Sometimes it is convenient to work with an average fractional energy loss per collision as opposed to an average logarithmic fraction. If the initial neutron energy level and the average fractional energy loss per collision are known, the final energy level for a given number of collisions may be calculated using the following formula: Efinal = Eint · (1 − frac)n ,

(13.3)

where Eint is the initial neutron energy, Efinal is the neutron energy after n collisions, n is number of collisions, and frac is the average fractional energy loss per collision. Although the logarithmic energy decrement ξ is a convenient measure of the ability of a material to slow down neutrons, it does not measure all necessary properties of a moderator. The other important parameters of the slowing-down process, the slowing-down power (sdp) and the moderating ratio, are a better measure of the capabilities of a moderating material. The sdp is the product of the logarithmic energy decrement ξ and the macroscopic cross section for scattering s in the

433

434

13 Experiments to Study the Performance of Spallation Neutron Sources Tab. 13.2 Logarithmic energy decrement, slowing-down power, and moderating ratio of common moderator materials (data are from [57, 63]).

Moderator material

Density (g cm−3 )

Logarithmic energy decrement ξ

Slowing down power sdp (cm−1 )

H (liquid) H2 O D2 O Be C

0.07 1.00 1.10 1.85 2.23

1.0 0.926 0.510 0.207 0.158

0.86 1.39 0.179 0.157 0.087

s is the free scattering cross b abs is for thermal neutrons. a

Moderating ratio (mr)

61 66 5406 167 215

Scattering/absorption cross section Σs a Σabs b (cm−1 ) 0.86 1.5 0.35 0.76 0.55

1.4 ×10−2 2.1 ×10−2 3.3 ×10−5 9.4 ×10−4 4.0 ×10−5

section.

material. The most complete measure of the effectiveness of a moderator is the moderating ratio (mr). The moderating ratio is the ratio of the sdp to the macroscopic cross section for absorption a . That means a good moderator should capture only weakly. The higher the moderating ratio, the more effectively the material performs as a moderator. The relations of sdp and mr are given by Eq. (13.4). Table 13.2 shows the slowing-down power and the moderating ratio for different common moderator materials. sdp = ξ · s [cm−1 ] s , mr = ξ · a

(13.4)

with s is the macroscopic scattering cross section and a is the macroscopic absorption cross section in cm−1 . The moderating ratio is usually determined with the average absorption cross section at room temperature. The time dependence of the moderation process can be introduced by the time Ti between the collisions of the neutrons with the moderator atoms which is a function of the mean free path λ of the neutrons in the moderator material given by the inverse macroscopic free scattering cross section of the moderator free in cm−1 . This means the frequency of collisions experienced by a neutron of a certain velocity vi is given by vi free , thus the time between collisions is given as Ti = 1/vi free or Ti · vi = λ ≈constant. This expression means that the total time it takes to slow a neutron down to a final velocity v is inversely proportional to v. Based on superthermal transport theory [838] for slowing-down process of neutrons in a proton gas (hydrogen), Carpenter and Yelon [839] suggested for the time dependence of the slowing-down neutron flux (v, t) after a short source pulse a relation for the product of v · t a functional dependence given by (v, t) ∼ (ξ · free · v · t/γ )2/γ · exp(−ξ · free · v · t),

(13.5)


with γ = 1, for A = 1 and ≈ 4/(3A) for A > 1. The average slowing-down time ts , its standard deviation ts , and the FWHM t1/2 are given by Bauer [57] as 1 v 1 ts = v 1 t1/2 = v ts =

· (1 + 2/γ )γ /(ξ free ), · (1 + 2/γ )1/2 γ /(ξ free ), ·

3 . ξ free

(13.6)

As shown in the appendix on page 687, hydrogen is by far the best moderator. In order to have a high enough density, it must be either in a liquid or super critical state, or a compound with high hydrogen content must be used for which the value v · ts is ≈ 1 cm [57]. In the slowing-down region, the neutron spectrum inside a moderator is approximately proportional to 1/E or more exactly, (1/E)1−α , where E is the neutron energy and α is a number between 0.1 and 0.2 considering the neutron leakage from the moderator during the slowing-down phase [635]. In the thermal equilibrium region, the neutron spectrum represents a Maxwellian distribution due to the balance between the moderator atoms and the neutrons as given in Eq. (13.7). (E)Max =

th · E · exp(−E/(kB · Teff )), (kB · Teff )2

(13.7)

where th is thermal neutron flux, kB = 0.08866165 meV/K , and Teff is the effective moderator temperature. By changing the moderator temperature, Teff , one can shift the thermal equilibrium Maxwellian spectrum to higher or lower energies. This is very nicely demonstrated in Figure 13.1 for H2 O moderators at 20, 70, and 293 K. The example shows the Maxwellian part in the spectrum is shifted to lower energies as the temperature of the moderator decreases. The neutron flux in the slowing-down energy range above 1 eV is independent of the moderator temperature and proportional to 1/E. The deviation from the 1/E behavior below 1 eV for low temperatures is mainly a result of the increase in the scattering cross section and hence a shorter lifetime of the neutrons. The utilization of cryogenic moderators at research neutron spallation is therefore of great importance and are reviewed recently [840]. At energies lower than about 1 eV the time distribution and the pulse shape of a neutron pulse from moderators is determined by the fundamental mode decay time 1/α, which could be evaluated by the time-dependent diffusion theory [63]. The neutron flux I(t) in the moderator after the pulse peak I0 (t) can be written as I(t) = I0 (t) · exp(−α · t) with α = a · v + DB2 − CB4 . . . ,

(13.8)

435

13 Experiments to Study the Performance of Spallation Neutron Sources

Neutrons per incident proton [cm−2 eV−1]

436

10 −6

Ice T-20 K Ice T-70 K Water T-293 K

10 −7

10 −8

10 −9 −4 10

1 10 −3 10 −2 10 −1 Neutron energy [eV]

10

Fig. 13.1 Measured energy spectra from H2 O ice moderator at T = 20 and 70 K and from a H2 O moderator of a temperature at 293 K. The measurements are performed at the JESSICA facility (cf. Section 13.3.3.2[611]).

where a is macroscopic absorption cross section of the moderator material, v the neutron velocity, D is the diffusion coefficient, C the cooling coefficient, and B2 is the geometrical buckling of the moderator for the fundamental mode. Since a · v is constant for a 1/v absorber and the third term in Eq. (13.8) is usually small, the second term in Eq. (13.8) becomes dominant for a small hydrogenous moderator. For a thin moderator the geometrical buckling B2 is mainly determined by the moderator thickness, therefore the decay constant α increases with decreasing thickness of the moderator resulting in a faster decay. The pulse width of the neutron distribution at FWHM, t1/2 , of the exponential decay is given by t1/2 = ln 2/α. For a short incident proton pulse of about several µs the pulse rise-up time is usually much shorter than t1/2 (cf. Figure 13.7). Therefore, the pulse width is mainly determined by t1/2 . But to obtain a very narrow pulse, a very thin moderator is essential which yields to a decrease in the time-integrated neutron flux and also in peak intensity [635]. The other important component, the reflector, of a TMR system should enhance the neutron intensity of the low-energy neutrons by partly reflecting the leakage neutrons from the target, which do not directly enter the moderator. Reflector materials should have a large macroscopic scattering cross section. Common reflector materials are, e.g., D2 O, Be, C, Ni, W, and Pb. Data characterizing reflector materials for spallation sources together with absorption and scattering cross sections are given in tables in the appendix on page 687. For pulsed spallation neutron sources different TMR systems with ambient temperature, cold and ultracold moderators and different reflector materials are used to utilize the different applications. Decoupling and poisoning the reflectors and moderators with neutron absorbing material (B4 C, Cd, Gd, etc.) are applied methods to provide a useful neutron pulse shaping.

13.3 Reflector Experiments with Complex Geometries and Realistic Material Compositions

• applications with high neutron peak flux using high-intensity moderators with a tolerable pulse width, • applications with high neutron peak flux using high-resolution moderators, where absorbing materials as decoupler surrounds the moderator or decouple the reflector to prevent slow neutrons being returned from the reflector, or absorbing material (poisoning) is placed inside the moderator or the premoderator to limit the time slow neutrons spend diffusing out of the moderator (decoupling and poisoning determine the neutron pulse shape), • applications with high neutron average flux using moderators large enough to achieve complete thermalization. Such applications are mostly used by continuous spallation sources. During the 1980s with the startup of several spallation neutron sources, the WNR at LANL, USA, IPNS at ANL, USA, KENS at KEK, Japan, and the construction of the Rutherford source SNS (now ISIS), the planned neutron spallation sources SIN (now SINQ) at PSI, Switzerland, and the high-intensity German SNQ source triggered an intensified research program on TMR and moderator concepts was initiated [834]. This worldwide research is still continued with the emphasis to utilize cold and ultracold moderators to increase the neutron flux at high-intensity MW spallation neutron sources [635, 836, 841, 842]. Some of the experimental investigations on TMR experiments are discussed in Section 13.3.

13.3 Target–Moderator–Reflector Experiments with Complex Geometries and Realistic Material Compositions 13.3.1 The Early Experiments

Thermal neutron flux generation by high-energy protons was experimentally studied during the ING project [58] by Fraser et al. [59]. These test measurements were investigated using the same experiment in measuring H2 O capture rates to determine neutron multiplicities of thick targets discussed in Section 11.2.1.1 on page 381. From these measurements the thermal neutron flux distributions inside a large H2 O water vessel in radial and axial direction were measured estimated for Be, Sn, Pb, and U targets at different incident proton beam energies. The measurements were validated by Coleman and Alsmiller [639] and by Armstrong et al. [338]. With the advent of the ICANS conferences at end of 70s, more realistic TMR experiments were planned and investigated [843]. Measurements are reported for a continuous source system with large H2 O and D2 O moderators, with neutron beam tubes, and using a Pb target with diameter of 13 cm by Thorson [844] at the TRIUMF accelerator. With an incident proton beam of 500 MeV and an average current of 100 µA thermal neutron peak fluxes of about ∼ 5.5 × 1012 cm−2 s−1

437

438


for a H2 O moderator and about ∼ 3.3 × 1012 cm−2 s−1 for a D2 O moderator were achieved. 13.3.2 Neutron Performance Studies at Reflected Target–Moderator Systems 13.3.2.1 Neutron Studies of a Reflected ‘‘T’’-Shape Moderator at LANL-WNR During the years 1980–1982 in LANL, USA, the high current target at the WNR facility was developed. For this purpose, various TMR experiments were performed to measure spectra and neutron beam fluxes of neutrons ≤ 10 eV and at neutron pulse widths of ≤ 0.2 eV. All these measurements were accomplished at the low current time-of-flight (TOF) facility (cf. Section 10.2.1.1 on page 290 and [800, 845, 846]). A sketch of the reflected ‘‘T’’-shape premoderator–moderator system is shown in Figure 13.2. Different targets were used of Pb, W, and U all 24.13 cm long and with a diameter of 4.45 cm. Each of the targets had at the first a 5 cm re-entrant hole with 2.54–1.42 cm diameter. The proton beam had a 1.5-cm diameter beam spot of the targets with an incident energy of 800 MeV. Cadmium with a thickness of 0.076 cm was used as the neutron decoupler of the surrounding Be reflector formed a 46 cm cube (cf. Figure 13.2). The polyethylene thickness of the moderator was measured with thicknesses of 0.64, 1.27, 1.91, and 2.54 cm. This yields an increase Neutron beams from moderator surface

Shadow bar Poison

Decoupler

Reflector

W target Decoupler Proton beam

Neutron beam

Premoderator Moderator

Fig. 13.2 Sketch of the reflected ‘‘T’’-shape premoderator/ moderator experimental model of the LANL-WNR (after [800]).


Neutron flux φ(E) [arbitary units]

104

103

Be reflector Pb/Be reflector CH2 reflector

102

101

100 0.001

0.01

0.1

1.0

10.0

100.0

Neutron energy [eV]

Fig. 13.3 Comparison of measured thermal neutron spectra of the experimental model of Figure 13.2 using Be, polyethylene, and Be/Pb reflectors (after [800]).

in thermal leakage flux from about 0.5 to 1.5 [800]. Beryllium as a moderating reflector was traditionally the standard reflector for spallation neutron sources. Other moderating reflectors, so-called slow reflectors, compound materials with hydrogen as H2 O, D2 O, or instead polyethylene as substitute were investigated and compared with so-called fast reflectors, which could be Pb, iron (steel), Cu, etc. Figure 13.3 shows a measured neutron spectral comparison of a moderator configuration surrounded by decoupled Be, an uncoupled polyethylene, and a decoupled Pb/Be reflector. As shown in Figure 13.3, the Be reflector produces a slowing-down flux about 2.5 times higher than that of a polyethylene reflector. The difference between Be and Pb/Be is only marginal. It has been shown that the time structure of neutron pulses with a Pb reflector may be also marginally superior to other reflector materials [833]. Further studies at the end of the 1980s were continued on the development of the LANSCE system [847] performing measurements of neutron beam fluxes on high-intensity H2 O moderators and high-resolution cold moderators of liquid hydrogen. 13.3.2.2 The SNQ Target–Moderator–Reflector Experiments at the PSI Accelerator (1) Thermal neutron experiments:

During the SNQ study phase from 1980–1985, a large variety of TMR experiments were performed at the accelerator facilities of PSI, Switzerland, CERN, Switzerland, and SATURNE, France, which are mainly investigated by the SNQ teams of the Forschungszentrum Jülich [619, 833, 848–853]. The quantities which were measured were mainly the integrated thermal neutron leakage, the thermal neutron flux distribution, and the time structure of the neutron field for various configurations of target, moderator, and reflector. Among them were a cylindrical Pb–Bi target in a D2 O vessel and a slab target with a hybrid moderator configuration (cf. Figure 13.4) [848]. It should be mentioned that in the SNQ project, the

439


Modenator Reflector

D20-Level Target p Neutrons

p

Al-Table

Neutrons

(a)

Target

D20-Level 85 cm

(b)

170 cm Al-tube 150 f

50

60

10

D20-Level

250 cm

Neutrons

60

110

440

p

Pb-Bi Target 150 f

Grooved moderator

m

0c

17

(c)

170 cm

Fig. 13.4 Schematic illustration of the TMR experiments at PSI, Switzerland with an incident proton beam of 590 MeV. (a) Pb slab target inside D2 O vessel; (b) Pb slab target with hybrid moderator configuration (D2 O below target, and polyethylene (‘‘H2 O’’) moderator with polyethylene reflector above the target), (c) cylindrical Pb–Bi target inside D2 O vessel (after Bauer et al. [848, 850]).

proton beam shall hit the target with pulses of a duration of 0.5 ms once every 10 ms. Therefore, it was essential to study the time structure of the resultant thermal neutron field in different concepts which were considered. The results were used to predict the performance data for a 5-MW spallation neutron source with 500 µs long proton pulses. The targets were mounted inside a vessel of stainless steel 170 × 170 × 250 cm3 filled with D2 O, which could be filled up to a level of 170 cm. Because the SNQ target was designed as a spinning wheel target [142] most of the experiments were investigated with a slab target. The slab target was composed of 27 Pb bricks 50 × 50 × 200 mm3 separated by 6 mm thick polyethylene sheets to simulate the water cooling of the target placed on a Al-table (cf. Figure 13.4). The polyethylene moderator, a so-called grooved type moderator, in the configuration (b)

13.3 Reflector Experiments with Complex Geometries and Realistic Material Compositions Neutron energy [meV] 100 50 20

10 7

5 4

3

Size: height =15 cm Depth =10 cm Width =15 cm Groove depth = 5 cm Groove width =1.6 cm

10000

2

1

GROOVED Neutrons

COUNTER TARGET

Counts

1000

FLAT 15 cm

Neutrons

100 15 cm 5 cm

10

0

COUNTER TARGET

50

100

150

200

250

Channel number [40 µs/ch]

Fig. 13.5 Comparison of TOF spectra measurements from grooved and flat cold (20 K) methane moderators (after Inoue et al. [857] and Ishikawa et al. [858]).

was a 50 mm thick polyethylene slab at the center with 60 mm long and 10 mm thick fins on both sides. This principle of increasing the neutron intensity of a pulsed neutron source was reported by Bauer et al. [848, 849, 854, 855] and by Carpenter et al. [856] for polyethylene moderators and by Inoue et al. [857, 858] for cold 20 K methane moderators. Detailed optimization studies were also investigated by Kiyanagi et al. [859] on grooved thermal moderators for pulsed neutron sources. As seen from Figure 13.5, a grooved moderator produces considerably a higher neutron intensity than a flat moderator, but resulting in broadening of the pulse shape. Therefore, these moderators could only be used effectively in neutron scattering experiments where the resolution is not very sensitive to the pulse width. Experiments show that the intensity gain depends on the reflector material as well as on the coupling among the reflector, the moderator, and the target [854, 859]. It is likely that in small hydrogenous moderators as the used polyethylene moderators in these experiments (cf. Figure 13.4(b)), one decay constant α is usually sufficient to describe the time dependence of the neutron field [848]. This view is supported by the slope of the trailing edge of the neutron pulses measured for the polyethylene moderator as shown in Figure 13.7. It should be mentioned that the decay constants from these slopes are not the true constants due to the contribution of the finite resolution of the proton pulse to the pulse width of the neutron pulse width. A correction may be determined by evaluating the second moment of the proton pulse width distribution. For the proton pulse width distribution of the experiment a value of σproton = 160.3 µs could be determined which yields an

441

13 Experiments to Study the Performance of Spallation Neutron Sources 2500 Pb slab target hybrid moderator Counts/channel [arbitrary units]

442

2000

Measured values Calculated (a = 0.5×10−2 [µs−1)

1500

1000

500

0

0

1

2 Time [ms]

3

4

Fig. 13.6 Time structure of the neutron field as a linear plot in the polyethylene moderator of the hybrid configuration with a Pb slab target. Solid curve is a guide to eye through the measured data given as crosses, the open circles are calculated values with α = 0.5 × 10−2 µs−1 [848]. 2 exponential decay derived by the second moment as σexp = 2 · τ 2 [848]. The ‘‘true’’ decay time τtrue is then determined by

τtrue =

2 τ 2 − σproton /2,

(13.9)

which leads to a ‘‘true’’ decay time of τtrue ≈ 207 µs, which is the case given in Figure 13.6. In Figure 13.6 the open circles represent a calculation performed by Bauer et al. [848] and a decay time of 200 µs using the measured proton pulse. The results agrees fairly well with the measured data. Figure 13.7 shows the measured neutron pulse in linear-log scale with the two decay constants α1 = 0.423 × 10−2 µs−1 and α2 = 0.1205 × 10−3 µs−1 . Although the polyethylene moderator is not really thin, it is clearly seen that α1 dominates the decay process as discussed by Eq. (13.8). In Table 13.3 some ‘‘best’’ values of corresponding equivalent thermal neutron fluxes for Pb flat targets in a large D2 O moderator (cf. Figure 13.4(a)) and for a hybrid moderator (Figure 13.4(b)) are given for an incident proton energy of 590 MeV and an average beam power of 10 mA roughly equivalent to the SNQ values of 1100 MeV protons and 5 mA beam power [848]. The ratio of the peak flux to the corresponding equivalent neutron flux is also given in the table.


Counts/channel [arbitrary units]

104

Pb slab target hybrid moderator 103

a1 = 4.23×10−3 [µs−1]

102 a2 = 1.205×10−4 [µs−1]

101

0

1

2

3

4

5

Time [ms] Fig. 13.7 Same as Figure 13.6 but plotted logarithmic scale. The plot shows that the neutron pulse decay is dominated by one single decay constant [848]. Tab. 13.3 Some ‘‘best’’ values of corresponding equivalent thermal neutron flux for an incident proton energy of 590 MeV and an average beam power of 10 mA neutron leakage of experimental arrangement given in Figure 13.4 (data are from Bauer et al. [848]).

Configuration

Moderator

Pb-flat target (see (a) Figure 13.4) Pb-flat target hybrid version (see (b) Figure 13.4)

D2 O

8.0

D2 O H2 O a

5.6 6.8

a

Neutron equivalent thermal flux Φ(equ) (1014 × cm−2 s−1 )

Peak flux Φ/Φ(equ)

7.9 18.4

Reflected H2 O moderator with grooves (cf. [848] on page 417).

(2) Cold neutron experiments: The experimental setup shown in Figure 13.4(b) was slightly changed, but uses the same large rectangular vessel into which the targets of Pb and U, moderators, and reflectors of D2 O, C plus premoderator, and Pb plus premoderator were installed. The details are given in Figure 13.8. Different cold moderators, grooved and flat moderators of varying size, were used to measure the leakage flux, the neutron life time and neutron spectra distributions with reflectors of heavy water D2 O, graphite

443

444

13 Experiments to Study the Performance of Spallation Neutron Sources C-reflector 3 120×120×80 cm or Pb-reflector 3 70×60×60 cm (L×B×H) H2-moderator

Diaphragm f1 (f50) BF3 - counter Diaphragm f2 (50×25.4)

2350

1550

450

H2O-premoderator (20 mm)

300

90

n 20 Al 5 PE

D2O level

100

Vessel filled with D 2O

850 800

1Cd 10Al 30×30Al

Protons 590 MeV

Fig. 13.8 Schematic sketch of the SNQ experimental setup for cold neutron measurements using a H2 -moderator at the PSI accelerator. Dimensions of the experiment are in [mm]. (In the figure both reflectors – of C and Pb – are indicated, although they are not used in the same experiment.) [851].

C, and lead Pb, respectively. In summary, four pairs of moderator vessels for the cold H2 in normal- and in para-status are investigated (para–ortho ratio of cold H2 see page 450). Each pair consisted of a flat and a grooved vessel with a constant fin width of 1.3 cm for all vessels. Both moderators have the same volume and the same dimensions in direction to the neutron beam. Thus, the moderators differed in an ‘‘effective thickness along the neutron beam direction being equal to 25, 36, 56, and 78 mm. Details of the construction of the cold H2 moderator and its operation are described in [851]. The neutron life time in the cold moderator was measured with a small boron 10 B depleted BF3 counter shielded against thermal neutrons from the reflector by a cadmium cover. The flight path used was a 2.35-m neutron beam channel depicted in Figure 13.8. Tables 13.4 and 13.5 show a selection of the measured results of average-, and peak fluxes, neutron life times, and pulse widths on different moderator sizes and reflector materials. As discussed in Ref. [851], there was apparently no gain factor obtained for cold grooved moderators as it was measured with ambient temperature moderators with grooves (cf. Table 13.3). The determined neutron temperature from the experiments was found to be about 80 K for almost all experimental cases and is therefore much higher than the thermodynamic value. This high neutron temperature is due to a spectral hardening via absorption or by an insufficient moderation. As noted in Ref. [851], an important result is that the combination of Pb reflector with a cold H2 moderator gave highest peak flux and shortest neutron life time.


445

Measured time averaged cold neutron fluxes for cold H2 -moderators and Pb slab targets with different reflector materials.a

Tab. 13.4

Target

Effective moderator thickness (mm)

Φthermal (1014 × cm−2 s−1 )

Time-averaged cold neutron flux

Reflector system

Pb Pb Pb Pb a

D2 O 3.2 3.1 2.7 3.0

25 36 78 96

C + premoderator – 3.0 – 2.5

Pb + premoderator – – 2.2

Given are the measured data from flat vessel moderators. Data are from Bauer et al. [851].

Tab. 13.5 Experimental results for various neutron flux parameters for a liquid grooved H2 -source of a dimension 11 × 11 × 3.5 cm3 as a function of the used reflector material with 238 U flat target (data are from Bauer et al. [851]).

Reflector material and dimensions (cm3 ) D2 O 170×170 × 85 C (cooled) 120 × 120 × 80 Pb 60×60 × 60

Average cold flux Φth (1014 × cm−2 s−1 )

Peak cold flux th Φ

Φ/Φ

Neutron life-time (10−6 s)

(1014 × cm−2 s−1 )

6.6±0.4

79

12

710

620

5.4±0.4

108

20

350

390

4.3±0.3

116

27

230

320

Neutron pulse-width (10−6 s)

13.3.3 Experiments of Short-Pulsed Target–Moderator–Reflector Systems

With the development, the design and the construction of high-power short-pulse spallation neutron sources high-efficiency moderator systems at cryogenic and ambient temperatures became an important area in the last decade. As mentioned earlier, moderators are one of the important devices of a pulsed neutron source, which determine the quality of the experiments performed at these sources. After a long experience on cold and thermal sources at research reactors [653, 860, 861] and at low-intensity spallation sources [641, 643, 679], several groups from Japan, USA, and Europe have performed and initiated research programs on cold and ambient moderator physics [611, 862–869] and theoretical ground-work [870–872]

446


on neutron scattering and other applications of high-intensity pulsed neutron sources [551, 627, 629, 681, 836, 842, 873] during the recent years. Early ideas in this field of research were published by Whittemore and McReynolds [874, 875]. Further basic research studies are undertaken by Carpenter et al. [56, 641] at the IPNS, ANL, USA, by Inoue et al. [876, 877] at the Hokkaido University, Japan, and by Ikeda, Inoue and Watanabe et al. [858, 878, 879] at the KENS facility, Japan. The essentials of these experimental investigations will be described and discussed in the following sections. 13.3.3.1 Target–Moderator–Reflector Experiments at the Hokkaido Electron Linear Accelerator A large variety of TMR experiments was performed at the electron linear accelerator facility at the Hokkaido University, Japan, by Inoue et al. [857, 876, 877, 879–881] and Kiyanagi et al. [859, 865, 866, 868, 882–885]. This electron LINAC facility with an electron energy of 45 MeV (beam power up to 5 KW, electron beam peak current of about 100 mA, pulse rate of 200 Hz, and a duty factor of 0.06%) is mainly used for neutron performance test experiments and detector development. The facility is able to generate about 2.5 × 1012 neutrons s−1 per kW beam power of fast neutrons for this electron energy with a nonfissionable target, e.g., lead [886]. About 1% of this fast neutron source can be converted to thermal or cold neutrons with a proper TMR system. Basic experiments investigated by Inoue et al. [857, 877, 880, 881] were mainly concerned with studies and tests on a variety of cold moderator materials, moderator chambers, and on tests of suitable refrigerators. Among these experiments are the measurements of the cold neutron gain factors [877] of neutrons at 10 A˚ wavelength and the study of different moderator materials of cylindrical moderators with diameter of 20 cm and height 20 cm and flat moderators with width of 25 cm, height 25 cm, and thickness of 5 cm at different temperatures [881]. The moderator materials and moderator temperature used were the following: (1) H2 O at temperatures of 293, 230, 180, 112, 77, 20, and 10 K, (2) H2 at 18 K, (3) CH4 (methane) at 112, 77, 57, 31, and 20 K, (4) C2 H6 (ethane) at 182, 120, 77, 20, and 16 K. From these measurements it was concluded that a moderator with cold methane should be a suitable cold moderator material obtaining a cold neutron gain factor of about 40 at 20 K for long neutron wavelengths. Some of the experiments were performed with BeO, graphite, and paraffin reflectors and a Cd decoupler between moderator and reflector for pulse shaping [881]. Figure 13.9 shows typical TOF neutron spectra from a flat 20 K methane moderator compared with spectra of 300 K ambient- and 22 K ice H2 O moderators. Based on the former experiments of Inoue et al., the group of Kiyangi et al. continued the moderator experiments at the Hokkaido electron linac facility during the 1990s up to 2006. These investigations are mainly performed to optimize the utilization of the planned TMR system of the high-intensity short-pulsed Japanese spallation neutron source, which have been started its operation in 2008 [629]. As mentioned above, these experiments cover a broad range of different moderator configurations and materials. Most of the experiments were performed on different liquid H2 moderator configurations since liquid hydrogen may be from

13.3 Reflector Experiments with Complex Geometries and Realistic Material Compositions Neutron energy [MeV] 500

50 20

10

5

3

2

1.5

3

10

Neutron flux spectrum [arbitrary units]

H2O (300 K) H2O (22 K) CH4 (22 K)

102

5 cm

101

25 cm 25 cm Neutrons 0

4 8 Time-of-flight [ms]

12

Fig. 13.9 Comparison of TOF spectra measurements of a 20 K methane moderator, a 22 K ice moderator , and an ambient temperature H2 O moderator (after Inoue et al. [881]).

certain technology reasons the best choice to utilize high-intensity spallation neutron sources. Also experiments with different moderator materials as methane CH4 , methane hydrate H2 O–CH4 , mesitylene, methanol, ethanol, benzene methanol, and benzine were conducted. Most of these experiments were also validated by particle transport simulations to test the scattering kernels of different moderator materials [865]. Some of the main issues and experimental examples are as follows: • experiments with liquid H2 moderators: coupled, decoupled, with and without premoderators, and poisoned premoderators, [882, 883, 887, 888], • experimental studies of the neutron performance of a hydrogen moderator on para-hydrogen concentrations [889–892], • experiments of tailoring neutron pulse shapes from coupled liquid H2 [885], • experiments on neutron performance with various moderator materials [865, 866, 868], • mockup experiments on flux–trap moderator configurations [884] to enhance the slow neutron intensity. The experimental apparatus The principle experimental arrangement used for the moderator experiments at the Hokkaido electron linac facility is shown in Figure 13.10. The left panel in the figure shows the arrangement for coupled moderator experiments, whereas the right panel shows the principle arrangement for decoupled moderator experiments. The incident electron beam energy was

447

448


Liquid N2

Premoderator (polyethylene 15 mm thick)

Decoupler (Cd or B4C)

Heat exchanger

Neutrons

Moderator 5×12×12 cm3 Reflector (graphite) (a)

Rectangular Pb target 7×8×15 cm3 (b)

Neutron collimator (boric/acid-resin)

Fig. 13.10 The principle experimental TMR assemblies performed for moderator experiments at the Hokkaido electron linac facility. The left panel (A) shows the arrangement for coupled moderator experiments, whereas the right panel (B) shows the principle arrangement for a decoupled moderator experiment (after Kiyanagi [868]).

35 MeV on a lead target. The material composition of the experimental arrangement could easily be changed, e.g., instead of a polyethylene, H2 O or ZrH2 could be used as premoderators in the case of coupled moderator experiments [883]. It was also possible to ‘‘poisoning’’ the premoderator with a 0.5 mm thick Cd sheet [887] for pulse shaping the neutron pulse. The Cd is also used to decouple the moderator from the reflector (cf. Figure 13.10(B)). The reflector was, for all experiments, of graphite with a size of 80 × 80 × 80 cm3 . The viewed surface area of the moderator was 12 × 12 cm2 . The thickness of the moderator could be varied from 2.5 to 5 cm [868, 882]. A mica crystal positioned of about 7 m from the moderator surface was used as an energy analyzer for the pulse shape measurements. For the spectrum measurements 3 He detectors were applied positioned at a distance of 6 m to the moderator surface. The temperatures of the moderators were about 20–30 K [868]. Experimental results (1) Premoderator studies and improved pulse shapes for liquid H2 , CH4 , and mesitylene moderator

Two general methods may be used to improve pulse shapes of cold neutrons from coupled liquid hydrogen moderators with a premoderator. One is to decouple a premoderator from the reflector and the other is to use a poisoned premoderator as given and described in Figure 13.10. Figure 13.11 gives an overview about the pulse shaping of cold neutrons measured on different moderator geometries.


3.94 Å : Coupled (3 cm) : CH4 decoupled : H2 decoupled

Intensity [arbitrary units]

1.0

: Decoupled premoderator (1 cm) : Poisoned premoderator (inner 0.5 cm outer bottom 1.0 cm) 0.5

0.0

0

500 Emission time [µs]

1000

Fig. 13.11 Comparison of cold neutron pulse shapes from five moderator systems: (1) a coupled 5 cm thick liquid H2 moderator with a 3 cm thick premoderator, (2) a decoupled solid CH4 moderator, (3) a decoupled liquid H2 moderator, (4) a a decoupled liquid H2 moderator with a 1 cm thick premoderator, and (5) a coupled liquid H2 moderator with a poisoned premoderator (after Kiyanagi et al. [887]).

Figure 13.11 compares the pulse shapes of cold neutrons at a wavelength of 3.94 A˚ with a premoderator thickness of 3 cm with and without decoupling from the reflector. Liquid H2 and solid methane CH4 at 20 K in the decoupled case were also compared. In the case of a poisoned premoderator the Cd poison sheet of 0.5 mm separates the premoderator of H2 O in an inner part with a thickness of 0.5 cm and an outer part of about 2.5 cm. Comparing the pulse shapes from the various composite moderators with 5 cm thick CH4 at a neutron wavelength of ˚ the solid CH4 is superior to the other systems (cf. Figure 13.11). This about 4 A, advantage vanishes at longer neutron wavelengths where a composite moderator with liquid H2 and poisoned premoderators were close to the results of a decoupled solid CH4 moderator [887]. Measurements on the intensity gain as a function of the premoderator thickness, the moderator thickness, and the moderator material indicate that the spectral intensity in the range from 4 to 10 A˚ does not depend on the moderator thickness, whereas the optimal thickness of the premoderator is about 2 cm in the case of mesitylene as given in Figure 13.12 [868]. This thickness of 2 cm is almost the same as in the case of H2 O and CH4 moderators [868]. Therefore, it may

449


1.0 Reletive intensity

450

Wavelengths from 4 to 10 Å

0.5

Moderator thickness – 2.5 cm Moderator thickness – 4.0 cm

0.0

0

1 2 Premoderator thickness [cm]

3

Fig. 13.12 Premoderator thickness for two mesitylene moderators, 2.5 and 4.0 cm thick, plotted as a function of the relative intensity gain of a neutron wavelength range of 4–10 A˚ (after Kiyanagi [868]).

be considered that the optimal premoderator thickness is not so different for the common materials as H2 O, CH4 , etc. (2) The ortho–para issue of cold liquid H2 moderators Knowing the ortho to para-hydrogen ratio in a liquid hydrogen moderator at a highintensity pulsed spallation neutron source is important when optimizing the design of liquid hydrogen moderators in the energy range below about 50 meV [893]. The equilibrium concentration at T = 20 K of para hydrogen is 99.8%, whereas at T = 300 K the equilibrium is at 25% para-hydrogen. The naturally ortho–para conversion is very slow, which takes about 43000 h from 25% to 99%. Figure 13.13 shows evaluated scattering cross sections for para-hydrogen and a mixture of para and ortho hydrogen [871, 894] which were evaluated for Monte Carlo simulations in MCNPX format. For both there exists an excellent agreement between the calculated and measured cross sections. The sharp drop in the para H2 curve below 50 meV is due to spin coherence and the second drop below 5 meV is due to intermolecular interference [893]. As shown in Figure 13.13 there exists a difference in the para to the mixture of para–ortho cross sections below an energy of some meV. As the para-hydrogen concentration increases with time by natural conversion even without a catalyst [893]. It also increases by neutron- and γ -irradiation. It is therefore essential to understand the influence of the ortho-para ratio on the neutron performance of liquid H2 moderators. More details about scattering models and laws may be found in Refs. [872, 895]. At present liquid hydrogen is the most important cold moderator for the highpower spallation sources J-PARC and SNS. Measurements were performed to

13.3 Reflector Experiments with Complex Geometries and Realistic Material Compositions 100 Para-hydrogen at T = 14 K

10

Total cross section [b per molecule]

IKE SCATHD Seiffert (1970)

1 1.E−05

1.E−04

1.E−03

1.E−02 Energy [eV]

1.E−01

1.E+00

1.E+01

250 Normal hydrogen at T = 14 K 200 IKE SCATHD Seiffert (1970) 150

100

50

0 1.E−04

1.E−03

1.E−02 Energy [eV]

1.E−01

1.E+00

Fig. 13.13 Total neutron scattering cross section for paraH2 (upper panel) and for ‘‘natural’’-H2 , a mixture of orthoand para-H2 (ratio 3:1). Note the different scale at the ordinate in the figure. Data are from Seiffert(1970) [896], calculations are from IKE SCATHD [894]. (after Keinert et al. [894]).

determine the neutronic performance of liquid hydrogen moderators at various para to ortho hydrogen ratios. For this purpose, an ortho–para conversion system was applied using a Fe(OH)3 catalyst to accelerate and control the para-hydrogen concentration [890]. Several experiments were performed at the Hokkaido accelerator and at the Manuel Lujan Neutron Scattering Center at LANL to measure the change of the neutronic performance due to the ortho–para ratio of liquid hydrogen moderators [889–892]. Figure 13.14 shows pulse shapes measured for the coupled and decoupled liquid hydrogen moderators for para concentrations of 35% and 99%

451

13 Experiments to Study the Performance of Spallation Neutron Sources 3×106 16.8 meV 6

2.5×10 Intensity [arbitrary units]

452

Moderator liquid H2 thickness 5 cm

2×106 Coupled para 35 Coupled para 99 Decoupled para 35 Decoupled para 99

1.5×106 1×106 5×105

0

100

200 300 400 Emission time [µs]

500

Fig. 13.14 Comparison of pulse shapes in linear plot as a function of different para-hydrogen ratios of 5 cm thick liquid H2 moderators in a coupled and a decoupled arrangement at a neutron energy of 16.8 meV (after Ooi et al. [891]).

investigated at the Hakkaido electron accelerator. The neutron intensity in arbitrary units is given as a function of the emission time in microsecond. The experimental conditions for the coupled moderator measurements with a 1.5 cm thick polyethylene premoderator are given in Figure 13.10 in the left panel, whereas the decoupled experimental arrangement with decoupler and liner is shown in the right panel of the figure. The para-hydrogen fraction was determined by measuring the change of the thermal conductivity due to the para concentration [891]. It is seen in Figure 13.14 that the pulse peak intensities increase with the para-hydrogen concentration for both moderator systems. In the coupled case the decay of the neutron pulse is very similar, whereas in the decoupled case the neutron pulse decay is much faster for a higher para concentration. This tendency was observed for all neutron energies below 25 meV [891]. The pulse width decreases with increasing para concentration for both moderators. (3) Neutron performance of methane hydrate, mesitylene and other nonexplosive moderator materials. In Figure 13.15, spectral intensities are compared to a mesitylene moderator with those of solid CH4 and liquid cold H2 [868] at temperatures around T = 20–30 K. The moderator of H2 had a para-hydrogen concentration of about 35%. All moderators are measured in the coupled case as indicated in Figure 13.10 in the left panel. The spectra of CH4 and H2 are almost the same [885]. The spectral intensity of the mesitylene moderator is much lower than that for the CH4 and H2 moderators in the cold low-energy region at about ≤ 0.01 eV. The reason of this low spectral intensity is that mesitylene in the cold neutron energy region is effected by


Intensity [arbitrary units]

109

108

107

106

105 0.0001

Mesitylene [coupled] Liquid H2 [coupled] Solid CH4 [coupled] 0.001 0.01 0.1 Neutron energy [MeV]

1

Fig. 13.15 Comparison of measured neutron energy spectra of cold coupled moderators of mesitylene, liquid H2 , and solid CH4 at temperatures around T = 20–30 K. The experimental arrangement is given in the left panel of Figure 13.10 (data are from Kiyanagi et al. [868]).

higher rotational levels of the mesitylene [897, 898]. It should be noted that from the radiation damage point of view, hydrogen and molecules having benzene rings are better moderator materials for high-intensity spallation neutron sources [899]. Mesitylene, methanol, ethanol are such materials. The results shown in Figure 13.15 may indicate that a coupled liquid H2 moderator will be superior to CH4 or mesitylene by using a para-hydrogen moderator with a larger thickness of about ∼ 15 cm [868]. The spectral intensities for other ‘‘nonexplosive’’ moderator materials were measured also to compare them with the nonexplosive mesitylene [868]. The experiments were performed in the coupled case with moderator thicknesses of 4 cm and with a premoderator of 2 cm (cf. Figure 13.10(A)). These moderator materials can be divided into two groups: (1) methanol and mesitylene, and (2) ethanol, benzene, and benzene methanol. The first group, e.g., methanol and mesitylene gives much high intensities for cold neutrons as the moderator materials in the second group indicated in the measurements shown in Figure 13.16. The reason for the higher intensities of methanol and mesitylene are due to the fact that the molecules have methyl groups and the hydrogen atomic number density in these materials is higher than that of ethanol. The spectra intensity of mesitylene and methanol is almost the same, although the intensity of the mesitylene moderator is somewhat higher compared to the methanol moderator above the peak energy (cf. Figure 13.16). From these results it was concluded that the mesitylene moderator material would be best among the nonexplosive materials. The neutron performance of para-rich hydrogen moderators is approaching to that of a methane moderator. Therefore, mesitylene should be used in the case where ‘‘safe’’ materials are required, and

453


1010 Intensity [arbitrary units]

454

109 Mesitylene Methanol Ethanol Benzene methanol Benzene 108

0.001 0.01 Neutron energy [eV]

0.1

Fig. 13.16 Comparison of measured neutron energy spectra for different ‘‘nonexplosive’’ moderator materials (after Kiyanagi et al. [868]).

para-hydrogen moderators should be used at high-intensity spallation neutron facilities where more severe safety regulations are considered [868]. 13.3.3.2 Target–Moderator–Reflector Experiments at the Jülich Proton Synchrotron COSY The JESSICA experiment (Juelich Experimental Spallation Target Set-up In COSY Area) was a full-scale TMR experiment at the COSY cooler synchrotron [725, 726] at the Forschungszentrum Jülich, Germany. The aim of the experiment was to investigate the advanced cold moderators for the upcoming next generation neutron sources like ESS (European Spallation Neutron Source) [551, 873, 900], SNS (Spallation Neutron Source at ORNL, USA) [681], or JSNS (Japanese Spallation Neutron Source) [629] at J-PARC, Japan. The JESSICA assembly JESSICA was an 1:1 mock up of the TMR assembly of a target station of the planned 5 MW source ESS. The data obtained at JESSICA are used to find the best-suited moderator material for the next-generation neutron sources. The main topic of the experimental investigations was the study of the neutron behavior of advanced cold moderators and the validation of Monte Carlo simulation codes as well. Especially, new neutron scattering kernels, which were developed during the course of the JESSICA project by Bernnat et al. [871, 894] could be validated and checked against the measured data. The investigated moderator materials were H2 O at ambient temperature, ice at T = 20 and 70 K, CH4 , methane hydrate, and mesitylene at T = 20 K. The target containing 3500 cm3 of mercury is located in the center of a Pb reflector with a diameter of 1.3 m and height of 1.3 m surrounding the Hg target. The moderators were mounted in the so-called wing geometry. This means two moderators are placed above and below the target to prevent fast neutrons from the target directly leaking out of the system. This

Incident proton beam


Hg target

N eu tro n

be am

Moderator

Reflector of Pb

Fig. 13.17 Schematic drawing of the JESSICA TMR assembly with Hg target, the reflector vessel with the Pb rods, the test moderator, and the neutron beam hole arrangement.

reduces the fast neutron background considerably. Whereas three moderators are filled with water the lower upstream moderator position is used to study various cold moderator materials as shown in Figure 13.17. This moderator can be filled with different moderator materials and could be operated at any temperature between 10 and 300 K. Two different moderator geometries were used: (1) a moderator with a rectangular shape with a height of 12 cm, a width of 15 cm, and a thickness of 5 cm due to the ESS design [551], and (2) a cylindrical moderator with a diameter of 9.5 cm and height of 11 cm with a reentrant hole (diameter = 3.6 cm). Figures 13.18 and 13.19 give a vertical cut through the target–moderator assembly and a photograph of the JESSICA facility installed at an external proton beam area of the COSY accelerator. Because of the low proton beam intensity COSY is particularly suitable for studying the neutron performance of advanced moderators, because radiolysis and activation levels are negligible. Not only can the proton beam energy be tuned from 0.8 to 2.5 GeV, the negligible activation rate permits one to easily modify the geometry, construction details, and the materials involved. Furthermore, the neutron performance can be studied and the gathered data can be linearly scaled to higher beam intensities as they will be available in high-power spallation neutron sources. Figure 13.20 shows the whole experimental setup through a schematic diagram, already depicted Figure 13.19. The neutron flight path as well as the detectors are shielded in order to reduce the background, which is caused by high-energy neutrons leaving the reflector and being reflected at the concrete walls of the experiment hall. These high-energy neutrons are thermalized within a 22.5 cm thick polyethylene layer (42.5 cm in

455


∅ 1300 mm Reflector

Pb

Proton beam 0.8 – 2.5 GeV

1800 mm

1200 mm

Hg target

Beam height over floor

456

Hg storage

Test moderator

Fig. 13.18 Vertical cut through the JESSICA TMR facility with Hg target, moderators and Pb reflector as shown in Figure 13.19.

Neutron detectors

WCM

ICT Reflector JESSICA ess

Graphite crystal

Fig. 13.19

Target

The JESSICA experimental apparatus at the COSY proton synchrotron at Julich. ¨

case of the detector housing) surrounding the neutron flight path (cf. Figures 13.19 and 13.20). The thermalized neutrons are prevented from reaching the detector by being captured inside a boric acid layer with an average thickness of 2 cm. JESSICA was operated with a beam intensity of 4 × 108 up to 4 × 109 protons per pulse incident on the Hg target. The repetition rate was 1/30 Hz with a pulse length of approximately 0.5 µs. The applied fast kicker extraction and the beam instrumentation especially investigated for the JESSICA experiment are given

15

0

cm


Graphite crystal

38

7

cm

Neutron detectors

15 0 cm

Shielding

Moderator

Target Reflector

Start counter

ICT

WCM

Protons

Fig. 13.20 Schematic drawing of the JESSICA experimental setup with the proton beam line, the proton beam monitors ICT and WCM, start counter, neutron detector, crystal analyzer, and a cut through the TMR arrangement (after Nunighoff ¨ et al. [611]).

in [728, 901]. To determine the number of protons per pulse, two proton beam monitors with different working principles are installed in the proton beam line (cf. Figure 13.20). A Wall Current Monitor (WCM) [902] measuring the mirror current in the wall of the beam tube and an Integrating Current Transformer (ICT) [903] measuring the current induced in a coil when the proton beam passes through are used. It has been shown [611] that these monitors agree within about 10% and allow an absolute normalization of the neutron spectra measurements. The number of protons per pulse is indispensable to determine the neutron-to-proton ratio in order to compare the experimental data with Monte Carlo simulations on an absolute scale. The characteristics of the moderators to be investigated are studied by TOF measurements of the neutrons coming out of the moderator surface. Therefore, a 5.37 m long neutron flight path was constructed. At the end a neutron detector is placed to measure the TOF spectra, from which the energy spectra can be deduced. To obtain detailed information of the time structure and wavelength dependency of the neutron pulses, a graphite crystal can be moved into the neutron flight path. Neutrons fulfilling the Bragg condition are reflected by the crystal and can be detected with a second neutron detector viewing the crystal (cf. Figure 13.20). Neutron time-of-flight measurements At the TOF measurements the moderator surface is perpendicular to the neutron flight path. The viewed surface is limited by the cross section of the neutron beam tube with an inner diameter d = 10 cm. The neutron spectra are measured at the end of the neutron flight path by a LiGdBOscintillator detector [611]. This detector has an active area of A = 16.6 cm2 , but to avoid dead time effects the active area was reduced by an aperture to A = 4.5 cm2 of a 1 mm thick cadmium layer and a 1 cm thick B4 C layer. The detector

457

458

13 Experiments to Study the Performance of Spallation Neutron Sources Tab. 13.6 Neutron temperatures compared with moderator temperatures (cf. Figures 13.21 and 13.1 on page 436).

Moderator material Ice Ice Water

Moderator temperature (K) 20 70 300

Neutron temperature (K) 71±4 110± 2 343± 15

efficiency was calculated by Monte Carlo methods for mono-energetic and folding the obtained flux with the neutron absorption cross section for 6 Li [611, 904]. The measurement starts when the proton beam passes through a plastic scintillator, the start counter, in Figure 13.20 in front of the target. In this way the neutron TOF spectrum could be measured for each single proton pulse. The TOFt spectrum of the thermal neutrons were performed by a difference measurement taken with and without a 1 mm thick sheet of Cd in front of the flight path. The difference spectrum is used for the determination of the neutron temperature. An example of ice and H2 O is given in Table 13.6. For measurements within the slowing-down neutron energy region the Cd was replaced by boron carbide (B4 C), because the former is increasingly transparent for neutrons with energies higher than about 0.4 eV. The storage time of the thermalized neutrons within the moderator was determined by measuring the wavelength-dependent TOF spectrum. This is done by placing the above-mentioned graphite mono crystal into the neutron flight path (Figure 13.20), which selects specific wavelengths fulfilling the Bragg condition for the chosen Bragg angle of = 45◦ according to n · λ = 2 · d · sin ,

(13.10)

˚ d is the lattice spacing, is the Bragg angle, and where λ is the wavelength in A, n is the order of the reflection. From the resulting TOF spectrum the pulse width at FWHM and the decay constant is determined assuming an exponential decay of the pulse. With a lattice spacing of d = 3.35 A˚ and a Bragg angle of =45◦ we ˚ Higher are able to analyze the wavelengths λ = 4.74, 2.37, 1.58, 1.19, and 0.95 A. orders can not be resolved and vanish in the high-energy background. Experimental results of JESSICA (1) Time-of-flight neutron spectra of ice and ambient water

In Figure 13.21 the TOF spectra for ice at T = 20 and 70 K are compared to the spectrum of an ambient temperature water moderator. The corresponding energy spectra are already shown in Figure 13.1 on page 436. The spectra have been normalized to the number of incident protons, the active detector area, and per time bin width. Further corrections have been performed for

Neutron TOF spectra [neutrons × 10−10 cm−2 s−1 per proton]


0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

T= 20 K T= 70 K T= 293 K

0

1000 2000 3000 4000 5000 6000 7000 8000 Time [µs]

Fig. 13.21 Neutron TOF spectra of ice moderators at T = 20 and 70 K compared to the spectrum of a H2 O moderator of T = 300 K temperature. The spectra are compared on an absolute scale (after Nunighoff ¨ et al. [611]).

the dead time. As expected, a spectrum shift toward longer flight times – i.e., lower kinetic energy of the neutrons – is observed for lower moderator temperatures. The small peak at about 800 µs is an effect caused by the decreasing absorption cross section of the cadmium layer. In this region, the Cd is partially transparent for thermal and epithermal neutrons but the peak is independent of the moderator material and moderator temperature. Assuming a Maxwellian distribution of the velocity of the neutrons, we are able to determine the neutron temperature. Transforming the time-dependent flux (t) into the velocity-dependent flux (v), the neutron temperature can be determined by the slope of the straight line in a semilogarithmic plot based on Eq.(13.11). (v) mn 2 · 0 ln = ln − · v2 . (13.11) 4 3 v 2 · kB · T vT In Eq. (13.11), v denotes the velocity of the neutrons, mn the neutron mass, kB the Boltzmann constant, vT velocity of the neutron at temperature T, and T the neutron temperature. The neutron temperatures obtained from Eq. (13.11) for two ice moderators and an ambient temperature water moderator are listed in Table 13.6 and compared with the real moderator temperatures. The neutron temperatures are higher than the moderator temperatures. The reason for this discrepancy is that the neutron distribution has a slowing-down source of epithermal neutrons and a loss of 1/v absorption compared to the moderator molecular energy distribution. In general this results in a hardening of the Maxwell–Boltzmann distribution of the neutrons, if the absorption is not too strong and has no resonance structure. (2) Wavelength-dependent TOF neutron spectra of ice and ambient water As described in Section 13.3.3.2, a detailed view on the time behavior of the neutron pulses can be obtained by measuring the wavelength-dependent TOF spectra. In

459

0.16

2.37 Å

1.19 Å

0.18

1.58 Å


Water at T= 293 K Ice at T= 20 K

0.14 0.12 0.95 Å

Neutron TOF spectra [neutrons × 10−11 cm−2 s−1 per proton]

460

0.1 0.08 0.06 0.04 0.02 0

1000

2000

3000

4000

5000

6000

7000

Time [µs] Fig. 13.22 Wavelength-dependent TOF neutron spectra on an ice moderator of a temperature T = 20 K (dotted line) compared with an ambient temperature water moderator at T = 293 K (solid line) (after Nunighoff ¨ et al. [611]).

Figure 13.22, the wavelength-dependent TOF spectra for water at T = 300 K and ice at T = 20 K are shown. From Figure 13.22, a relative shift of the abundance of the spectrum toward lower kinetic energies, e.g., for longer flight times and for lower moderator temperatures can be seen. The neutron intensity of the water moderator at λ = 1.19 A˚ is much higher than for ice, whereas the opposite results for the wavelengths of λ = 2.37 A˚ ˚ At longer wavelengths, the colder moderator shows more intensity. The or 4.74 A. pulse widths in FWHM and decay constants for two moderator temperatures, 20 ˚ and 300 K, have been determined at wavelengths of 1.19, 1.58, 2.36, and 4.74 Aare given together with MCNPX Monte Carlo simulations as shown in Table 13.7. Tab. 13.7 Comparison of measured pulse widths and decay times for several wavelengths (cf. Figure 13.22) for a water moderator at 293 K temperature and an ice moderator at 20 Ka .

˚ Wavelengths (A)

1.19

1.58

2.37

4.74

92.1(73.4±14) 0.014

96.9(87.4±14) 0.015

– –

56(46.2±14) 0.039

101(86.7±14) 0.019

286(280±60) –

293 K water Pulse width FWHM in µs 58.7(59.2±14) 0.016 Decay constant in µs−1 20 K ice Pulse width FWHM in µs Decay constant in µs−1 a

36(40±14) 0.057

The results of the Monte Carlo simulations are given in parenthesis (data are from Pohl [904]).


Such data are important for the design of new neutron scattering instruments at spallation neutron sources. It can be seen that the cold ice moderator reduces the pulse width by about 39% ˚ and agrees within 4% for 2.37 A˚ compared for wavelengths of 1.19 A˚ and 1.58 A, to an ambient temperature water moderator. Furthermore, the decay constant is higher in the case of an ice moderator than in a water moderator at 293 K resulting in a faster decay of the pulse. This will allow a higher time resolution in a real neutron spallation source. (3) Time-of-flight neutron spectra and time structure measurements of methane hydrate and mesitylene moderators. From measurements reported in [611, 867, 875, 877, 879–881, 887] and discussed in Sections 13.3.3.1 and 13.3.3.2 the energy spectra from ice and solid methane at T = 20 K are well known. The ice moderator under consideration [611] has a maximum of the neutron spectrum at an energy of ≈5–6 meV and solid methane have the highest intensity at ≈1–2 meV [868]. The idea was to combine both materials into a new moderator material which should exhibit the maximum neutron flux of solid methane and ice over a broader energy range. Methanehydrate has been sought to be such moderator material. The structure may be described in the following way: a single methane molecule CH4 is encaged in an ice cage formed by approximately six water molecules (CH4 + 5.75 × H2 O). It is further assumed that the methane molecule inside the ice cage behaves like a single methane molecule and is not influenced by the surrounding ice cage. This assumption was confirmed by various inelastic neutron scattering experiments on methane hydrate [842, 905–909]. Especially, the behavior of the trapped methane molecule was intensively investigated. Because ice is one of the main components of methane hydrate more systematic measurements at different ice moderator temperatures were investigated. Figure 13.23 illustrates these neutron spectra measurements. At a temperature T = 248 K close to the freezing point, the neutron spectrum is very similar to the spectrum of a H2 O moderator at room temperature T = 293 K. For the first time a realistic methane-hydrate moderator was investigated with the JESSICA experiment. The rectangular moderator vessel was filled with methane hydrate. The density was estimated to be in the range from 0.44 to 0.77 g/cm3 . In Figure 13.24, the measured energy spectra of methane and ice are compared with the spectrum of methane hydrate at T = 20 K. It can be seen that methane hydrate does not show the expected superposition of the ice and methane spectra. The position of the maxima are the same for methane and methane hydrate. In the energy range below 7 meV the spectrum of methane hydrate has the same shape like solid methane; however the intensity is reduced by a factor three compared to the solid methane. This is due to the lower hydrogen density in methane hydrate–about 5.98 × 10−22 atoms cm−3 – compared to solid methane with 7.958 × 10−22 atoms cm−3 . For higher neutron energies – between 20 and 100 meV – the methane-hydrate moderator shows a slightly higher neutron flux than the methane moderator. However, the flux of the ice moderator is superior

461


10–7 Neutron spectrum [n cm–2 eV–1 per proton]

462

T=20K T=70K T=165K T=248K T=293K

10–8

10–9

10–10

10–11

10–4

10–3

10–2

10–1

1

10

102

Neutron energy [eV] Fig. 13.23 Comparison of measured neutron spectra of ice moderators as a function of the moderator temperature. A water moderator at T = 293 K is shown for comparison also (after Nunighoff ¨ et al. [867]).

to the methane-hydrate moderator as well as the methane moderator in the energy range between 5 and 100 meV. This already indicates that the energy spectrum of a methane-hydrate moderator is not a superposition of the energy spectra of each single material. Especially in the energy region, where ice should be dominant, the intensity is reduced. The intensity of the cold neutrons is reduced by a factor of three comparing a methane-hydrate moderator to a solid methane moderator. Mesitylene is also a potential moderator material as discussed in Section 13.3.3.1 on page 448 and shown by the measurements of Kiyanagi et al. [868]. Compared to methane the low damage effects from induced radiation make mesitylene a potential candidate as a moderator material in high-intense radiation fields of high-power spallation sources [899]. Especially, the rotational modes in the low-frequency range make mesitylene a promising material for cold moderators. Detailed investigations considered that mesitylene has three different crystallographic phases the so-called phase I, phase II, and phase III [910]. The existence of a specific phase or their mixtures depends strongly on the actual freezing conditions. At the JESSICA facility mesitylene cold neutron spectra were measured and compared with spectra of solid methane, liquid hydrogen, and ice at a temperature of T = 20 K. In contrast to other measurements all these spectra could be compared on an absolute level – normalized per incident beam proton – and were measured in a neutron field similar to a neutron spallation source in the GeV energy range. With a new S(α, β) scattering law data file for mesitylene, it is possible to compare the measurements with MCNPX simulations [869].


Neutron TOF spectra [neutrons cm–2 s–1 per proton]

10–5

10–6

CH4 Methane hydrate Ice

10–8

10–10 –4 10

10–2 100 Neutron energy [eV]

101

Fig. 13.24 Comparison of measured neutron spectra of ice, solid methane, and solid methane-hydrate moderators at a moderator temperature of T = 20 K. The measured data are normalized to the number of incident protons and background and detector efficiency corrected (after Nunighoff ¨ et al. [867, 869]).

Figure 13.25 shows a comparison of the measured energy spectrum of mesitylene compared with solid methane, liquid hydrogen, and ice at a temperature of T = 20 K measured with the JESSICA facility. It can be seen that mesitylene shows a very similar spectrum like liquid hydrogen. In the energy range between 20 meV and 1 eV mesitylene is superior to liquid hydrogen. As mentioned above, in the described experiments phase III of mesitylene was used. Maybe an increase in the cold energy range could be achieved by annealing the frozen mesitylene close below the melting point to transform phase III into phase I, as described in [897, 910]. This will open more low-lying energy modes, which could be exited by incoming neutrons and thus resulting in colder energy spectra. A comparison of measured and simulated energy spectra with MCNPX from a mesitylene moderator at T = 20 K using the new developed S(α, β) scattering law data file [870, 911] for mesitylene is shown in Figure 13.26. As seen in Figure 13.26, the agreement between the measured neutron production and the simulated one is very good. For the energy range between 1 and 7 meV the simulation determines a higher neutron flux than measured, while for energies ≥ 7 meV the calculation is below the experimental measurements. The maximum of the neutron spectrum is slightly shifted toward lower energies for the simulated curve. As already discussed, investigations of the time structure of neutron pulses emitted from moderators are of particular concern for the performance of a

463


Neutron TOF spectra [neutrons cm–2 s–1 per proton]

10–5 Ice CH4 Mesitylene H2

10–6

10–7

10–8

10–9 –4 10

10–3

10–2 10–1 100 Neutron energy [eV]

101

Fig. 13.25 Comparison of measured energy spectra for ice, methane, and mesitylene, and liquid H2 normalized to the number of beam protons and corrected to background and detector efficiency. In the experiment, mesitylene in the phase III status was used (after Nunighoff ¨ et al. [869]).

MCNPX phase II MCNPX phase III JESSICA exp.

Neutron spectra [neutrons cm–2 s–1 per proton]

464

10–6

10–7

10–3

10–2

10–1

Neutron energy [eV]

Fig. 13.26 Comparison of measured (phase III) and with MCNPX Monte Carlo simulated cold neutron spectra (phase II and phase III) produced from a mesitylene moderator at T = 20 K (after Nunighoff ¨ et al. [869]).

moderator at a pulsed spallation source. A small pulse width and a fast decay time of the pulse is desired. The measured wavelength-dependent neutron spectra of ice, methane hydrate, mesitylene, and methane were analyzed with a lattice spacing of


Neutron pulse width FWHM [µs]

300

Ice Methane-hydrate Methane Mesitylene 200

100

0.5

0 0.5

1.5

2.5

3.5

4.5

5.0

Neutron wavelength l [Å]

Fig. 13.27 Comparison of neutron pulse widths in FWHM of measured neutron wavelength dependent TOF spectra at JESSICA for ice, methane hydrate, methane, and mesitylene (after Nunighoff ¨ et al. [869]).

3.35 A˚ and a Bragg angle of = 45◦ . Neutrons with wavelengths of λ = 4.74, 2.37, 1.59, 1.19, and 0.95 A˚ could be distinguished. For higher orders of n, even smaller wavelengths cannot be resolved and vanish in the high-energy background. The pulse widths in FWHM and the decay times of the pulses were determined from the measured spectra. Comparisons of ice, methane hydrate, methane, and mesitylene moderators materials of neutron wavelength dependent TOF spectra at JESSICA are shown in Figure 13.27 for the pulse widths and in Figure 13.28 for the decay times. The decay times of the neutron pulses were determined by exponential fitting. 250

Decay time t [µs]

200

Ice Methane-hydrate Methane Mesitylene

150 100 50 0 0.5

1.5

2.5 3.5 Neutron wavelength l [Å]

4.5

5.0

Fig. 13.28 Comparison of the decay time τ determined from measured neutron wavelength dependent TOF spectra at JESSICA of ice, methane hydrate, methane, and mesitylene (after Nunighoff ¨ et al. [869]).

465

466


The ice moderator shows the broadest pulses; the peaks are narrower in case of methane and methane hydrate. It can be concluded that solid methane is a very fast moderator material compared to normal ice. Methane-hydrate lies in between for longer wavelengths and is comparable with methane at shorter wavelengths. From these results, it can be concluded that methane hydrate benefits from the encaged methane molecules as compared to normal ice. The intensity in the lower energy range as well as the time behavior is dominated by the methane molecules of the methane hydrate. 13.3.3.3 Target–Moderator–Reflector Experiments at the Brookhaven Alternating Gradient Synchrotron AGS As mentioned in Section 11.2.5.2 on page 419, the bare thick target experiments of the AGS–ASTE collaboration [812, 829] using the mercury target were continued with a water moderator and a lead reflector [912, 913]. The mercury target (∅ 20 cm × length 130 cm) was surrounded by a lead reflector (1 × 1 × 1 m3 ) and was irradiated by 1.94, 12, and 24 GeV protons. A vertical cut through the TMR ASTE assembly is given in Figure 13.29. The spectral intensities of thermal neutrons from the moderator were measured by the current-mode time-of-flight (CTOF) technique, which is discussed in detail in [914], using enriched 6 Li and 7 Li glass scintillators. With this technique only several incident proton pulses from the accelerator are needed to obtain sufficient statistics for each incident proton beam energy. The results have shown that the neutron spectral intensity per incident beam proton integrated over the Maxwellian energy region of the measured neutron spectrum was almost proportional to the proton energy of the incident beam on the Hg target. The arrangement of different components of the experimental setup is given in Figure 13.30. The ambient water (H2 O) moderator with dimensions of

Hg safety container

Mercury target

1.016 m Lead reflector

Proton beam from BNL-AGS

1.110 m H2O moderator

1.257 m

Fig. 13.29 A schematic view of the ASTE experiment with mercury target, lead reflector and water moderator (after Nakashima et al. [912]).


18 m

Pb reflector Shielding

Hg target

TOF detector

B4C glass scintillator 6Li

Proton beam height H2O moderator

Fe collimator

Fig. 13.30 A perpendicular view of the arrangement of different components of the ASTE experiment as installed at the AGS accelerator proton beam line (after Meigo et al. [913]).

10 × 10 × 5 cm3 was placed at 3 cm below the target container (cf. Figure 13.30). The slow neutrons were extracted from the moderator surface 10 × 10 cm2 and were transported through an iron collimator and an evacuated tube having mylar foil windows with a thickness of 0.23 mm at the entrance and the exit. The neutron detectors of 6 Li and 7 Li glass scintillators with a neutron absorber of B4 C in front were placed at a distance of 18 m from the moderator surface (cf. Figure 13.30) in a shielded detector house. The pulse height distribution of the system was calibrated with an Am–Be source. Neutron detectors in the current-mode time-of-flight technique The applied neutron detectors are an enriched 6 Li glass scintillator with a low γ sensitivity (GS20), and an enriched 7 Li glass scintillator with minimal neutron but high γ sensitivity (GS30). Both detectors have dimensions of 50.8 mm in diameter and 6.4 mm in thickness. The specifications are given in Table 13.8. The scintillation light is produced by 4 He and 3 H nuclei via the reaction 6 Li(n, α)3 H in the 6 Li glass scintillator. To subtract the contribution of γ -rays the enriched 7 Li glass scintillator, GS30, was used. The resulting net signal for neutrons can be obtained by subtracting the GS30 result from the GS20 results. With this technique, common noises, e.g., Tab. 13.8

Specifications of the Li-glass scintillator detectors (data are from [913]).

Parameter Li content wt (%) 6 Li enrichment (%) Density (g cm−3 ) Wavelength (nm) Decay time for γ -rays (ns)

6 Li-enriched

glass (GS20)

7.2 96 2.5 395 100

7 Li

glass, 6 Li depleted (GS30) 8.0 0.1 2.5 395 100

467


from background induced by spikes of the kicker magnets or dark current of the photomultiplier tubes, PMT, could be eliminated [913].

Experimental results Protons with energies of 1.94, 12, and 24 GeV with intensities of about 1012 protons per pulse were injected on the Hg target. The time width of pulses was about ∼100 ns.

Neutron spectrum [n/Lethargy/sr per proton]

10−8

10−9

10−10 1.94 GeV -Z = 11 cm 12 GeV -Z = 16 cm 24 GeV -Z = 20 cm

10−11

10−1 10−2 Neutron energy [eV]

100

10−8 Incident protons 24 GeV Neutron spectrum [n/Lethargy/sr per proton]

468

Exp. Calc.

10−9

10−10

10−2 10−1 Neutron energy [eV]

Fig. 13.31 Normalized thermal neutron spectra measured at the ASTE TOF experiment for beam energies of 1.94, 12, and 24 GeV (upper panel). Thermal neutron spectrum per incident proton with beam energy of 24 GeV compared with Monte Carlo simulations of NMTC-JAM [917] (lower panel) (after Meigo et al. [913]).

100


Several proton beam diagnostics devices were applied: (1) An integrating current transformer, ICT (210-20:1), from Bergoz [903], which could be operated for a pulse width ranging from picoseconds to several microseconds, (2) a foil activation method with Cu foils placed in front of the Hg target vessel, where the protons were determined by the γ -rays of 24 Na produced by the Cu(p,24 Na)-reaction, (3) the beam profile was measured by a segmented parallel-plate ion chamber, and alternatively by an image plate technique with Al activation foils. The image of the image-plate represents the intensity distribution of the radioactivities induced in the Al foil corresponding to the incident proton beam intensity profile. The relative exposure doses of each pixel with a size of 200 µm2 could be analyzed by an imaging plate reader [913]. Measured neutron spectra After subtraction of the signals of the 7 Li glass scintillator from the 6 Li glass scintillator, the thermal TOF neutron spectra measured by the CTOF method was obtained. The detection efficiency of the scintillators were calculated with the MCNP-4A code [915] using an evaluated neutron cross section library [916]. It was found that the detection efficiency of the enriched 6 Li detector was 100% for thermal neutrons. The measured neutron spectra obtained are shown in Figure 13.31. The broad Maxwellian thermal peak distribution is clearly visualized. A comparison [912] of the spectrum measured at 24 GeV with Monte Carlo simulations is given on absolute scale in Figure 13.31 using the NMTC-JAM/PHITS code [917]. The comparison shows an excellent agreement. Thus the NMTC/JAM system for neutron flux calculations could be validated on measurements for a TMR system from a few tens of GeV down to the meV neutron energy region [912].

Maxwellian intensity ×106 [neutrons cm–2 sr–1 W–1]

1.5

1.0

0.5 0

5

10

15

20

Proton energy [GeV] Fig. 13.32 Dependence of the maximum neutron intensity over the Maxwellian peak region normalized to the incident power of protons as a function of the incident proton beam energy (data are from [913]).

25

469

470


By integration of the neutron spectra of Figure 13.31 in the Maxwellian peak region, a value for the neutron intensity as a function of the incident proton beam energy could be determined (cf. Figure 13.32). It can be recognized from the experimental values that the neutron intensity per proton power in Watt decreases as the proton beam energy increases and is almost proportional to the incident proton energies, which is shown in Figure 13.32. Measurements on neutron intensity changes by moving the target along the beam direction within a range of 15 cm resulted only in a difference in thermal peak intensity within 10%. This insensitivity is mainly caused by the broad maximum in the Maxwellian energy range of the thermal neutron flux from the H2 O moderator.

471

14 Experiments on Radiation Damage in a Spallation Environment 14.1 Introduction

Till now there is very scarce experimental information available on the lifetime of structural components in the radiation environment of spallation targets and components. Therefore the lifetime, e.g., for the proton beam window material, the target material if it is solid material (Pb, W, Ta, or U, etc.), the structure material inside the target (Al, Zr, different steel alloys, etc.), the material of the return hull, if a liquid metal target is used, and at last the structure materials of components as moderators, reflectors, neutron beam pipes, etc., is desirably needed. The best way to overcome this lack of knowledge is to investigate irradiation experiments on special material specimens at operating spallation sources [918, 919], at particularly suitable irradiation facilities, or to investigate specimens obtained from spent targets and components of already operating spallation sources, as e.g., LANSCE [630], ISIS [643], and SINQ [627]. Because there are no facilities particularly dedicated for irradiation purposes, generally measured results of a postirradiation examination of spent targets and components are the best way to accumulate experimental data on the irradiation-induced changes of the mechanical properties of these materials. Concerning radiation damage, the most critical region in high-power targets of spallation sources is in the center of the target window which typically accumulates a proton fluence of, e.g., 1.6 × 1026 protons/m2 within one year of full power operation of the ESS [551, 873] with 5 MW beam power. In steels this leads to a displacement dose of about 60 displacements per atom, dpa, and a helium concentration of almost 1 at.%. Based on the experience of fission and fusion materials it is expected that hardening and low-temperature embrittlement is responsible for the main changes in the mechanical properties of the materials due to the considered temperature range of 100–250◦ C. Since 1996, spent target components from LANSCE (Los Alamos Neutron Science Center) and from the ISIS (Spallation Neutron Source at Rutherford and Appleton Laboratory) were became available for postirradiation investigations. The available materials included four different alloys: an austenitic stainless steel (AISI 304L), a nickel-base alloy (IN 718), a martensitic stainless steel (DIN 1.4926), and a refractory metal, tantalum (used as the target material for the ISIS target). The specimen preparation and the measurements were mainly performed in the hot cells of Handbook of Spallation Research: Theory, Experiments and Applications. Detlef Filges and Frank Goldenbaum Copyright  2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40714-9

472

14 Experiments on Radiation Damage in a Spallation Environment

the Forschungszentrum Jülich, Germany. Several mechanical tests, including the Vickers hardness test [920], a three-point bending and tensile tests were conducted. The fracture surfaces were observed by scanning electron microscopy (SEM) after bending and tensile tests. The changes of the microstructures after irradiation were investigated by transmission electron microscopy (TEM). 14.1.1 Irradiation Conditions and Studied Materials

The investigated spent targets and components are a LANSCE Water-Degrader and a LANSCE Beam-Window, a beam window from the Paul-Scherrer-Institut irradiated in LANSCE, and an ISIS tantalum target with structure materials. The ISIS target consisted of a target container with a proton beam window made of AISI 304L stainless steel and 23 D2 O cooled Ta plates as target material. The details of the materials, dimensions and the operation conditions can be found in Refs. [921–924]. The irradiation conditions and other relevant parameters are summarized in Table 14.1, which is adopted from [924]. The material properties of some common target and structure materials are given in the appendix in the tables on page 683. The profiles of the incident proton beam on the spent components were measured by γ -scans in the hot cells at Jülich. It was found that the positron decay of 22 Na, 44 Ti, and 57 Co was mainly caused by the proton beam in the Ni- and Fe-based alloys. Therefore, the integrated positron decay distributions were used. As a result the intensity distribution of the proton beam at all components could be approximated Tab. 14.1

Materials and irradiation conditions (data are given by Chen et al. in [924]).

Materials Protons in (A h) Beam profile σx /σy (mm) Peak fluence (protons/m2 ) Maximum irradiation temperature (◦ C)a Maximum dpa Maximum He (appm) Maximum H (appm)b a

LANSCE water degrader

LANSCE window

PSI window

ISIS target

IN 718 5.29 −

AISI 304L 5.29 22.6/28.8

IN 718 3.41 −

DIN 1.4926 2.83 16.3/23.6

Pure Ta 1.74 17.0/20.7

2.9 × 1025

2.9 × 1025

6.4 × 1025

2.6 × 1025

1.7 × 1025

250

250

400

250

200

8.5 1510 5890

8.5 1680 5270

20 3330 13000

6.8 1510 4720

11 580 −

The maximum irradiation temperature refers to maximum temperature in the beam center at maximum operating power of the spallation source. b The values represent the hydrogen concentration produced in the material, neglecting energetic or diffusion losses, etc.

14.1 Introduction

by a two-dimensional Gaussian shape with variances σx and σy , respectively (cf. Table 14.1). 14.1.2 Experimental Results

Vickers microhardness HV0.2

14.1.2.1 Microhardness and Three-Point Bending Tests The microhardness was investigated applying the Vickers hardness test method [920]. The Vickers number (HV) is calculated by HV = 1.854(F/A), where F is the applied load force in kgf, A is the area of the indentation in mm2 . The applied load is usually specified when HV is cited. It should be noted that the data determined by hardness tests can not be directly used for any engineering purpose. But microhardness tests are easy to perform and provide a quick qualitative information on changes of the mechanical properties. The hardness tests were made on samples which were cut from different components (cf. Table 14.1) to receive specimens with a perfect flat surface for the measurements. To deal with a high-dose gradient, microhardness tests were performed with a load L = 2 N(HV 0.2). The results on each position have been averaged over several indentations along the thickness. In Figure 14.1 the hardness data of the four different materials given in Table 14.1 are shown and compared. 450

550

400

500

350

450 400

300

IN 718

AISI 304L 250

350

(a)

(b)

300 400 250 350 200 300 150 Ta (99.99%)

DIN 1.4926

250 0 (c)

5

10

100

0 (d)

Displacement [dpa] Fig. 14.1 Dependence of Vickers microhardness HV0.2 at room temperature (RT) on displacement dose for (a) AISI 304L, (b) IN 718, (c) DIN 1.4926, and (d) Ta (after Chen et al. [924]).

5

10

473


Attention should be drawn to the IN 718 material, which is the material that, after hardening up to about 1–2 dpa, shows a softening at even higher doses. The hardness level of IN 718 after about 8 dpa is still comparable with that one of DIN 1.4926 (about 400 HV 0.2), but the trend of the two materials is rather different. All the other three materials, i.e., AISI 304L, DIN 1.4926 and Ta, show continuous hardening after irradiation. Both steels reach a level of 400 HV 0.2 at about 7 dpa, while the pure Ta reaches only a hardness of about 250 HV 0.2 at 11 dpa. In addition to the microhardness measurements three-point bending test were investigated on the irradiated materials with samples of 15 × 2 × 3 mm3 in size with an unusual ratio of length to cross section [921]. This technique was preferred to the more standard four-point bending in order to keep the sample as small as possible. The high gradient of the irradiation dose would prevent a constant level of irradiation inside a larger specimen. Figure 14.2 shows a schematic drawing of the three-point bending device used for the bending tests. This system was installed in a standard tensile testing machine equipped with a 5000 N load cell [921]. The tests were performed with a 5 KN load cell and with a crosshead speed of 0.2 mm/min. The experimental setup allowed a maximum deflection of about 2 mm. In order to study possible influences of specimens’ shape and geometry, extensive tests were made on unirradiated specimens before investigating irradiated material. Details are described in Ref. [921]. One of the major problems in these investigations was related to the reference on unirradiated materials. The spent components, from which the specimens were prepared, were operated for several years in spallation environments, but no archive material was left for comparison. Therefore, Inconel 718 was chosen as reference material for studying the size dependence of the bending results. The results of the three-point bending tests are qualitatively similar to the results found in the microhardness tests. The data for all materials are shown in Figure 14.3. The IN 718 specimens did show a trend for softening at an irradiation dose of about 2.1 dpa as already shown in Figure 14.1(b) by microhardness test. In this

.1 m m

Load L

h=2

474

+ I = 10 mm s

Fig. 14.2 A schematic drawing of the three-point bending device with load L and deflection s (after Carsughi et al. [921]).

14.1 Introduction 1000

3500

8.3 4.4

800

0 dpa

3000

2.8 0.3 dpa

7.4

2500

600

2000

400

1500

2.8

5.7

8.5

1000 Load [N]

200 0

500

AISI 304L

0

(a)

(b)

800

1400 1200 1000 800 600 400 200 0 0.0 (c)

IN 718

6 3.4 0.2

11 9 4

600

0 dpa

400

0 dpa

200 DIN 1.4926 0.5

1.0

1.5

2.0

Pure Ta 2.5

0 0.0 (d)

0.5

1.0

1.5

2.0

2.5

Deflection [mm]

Fig. 14.3 Three-point curves of (a) AISI 304L, (b) IN 718, (c) DIN 1.4926, and (d) pure Ta (after Chen et al. [924]).

case a reduction of ductility1) could also be observed. All the other materials, i.e., AISI 304L, DIN 1.4926, and pure Ta, did show a hardening with a reduction of ductility. Among the steels, the austenitic one, AISI 304L, showed a large reduction in ductility, which was clearly visible during the bending test, e.g., the sample irradiated up to about 8.3 dpa broke already after 0.5 mm of deflection. On the opposite, the martensitic steel, DIN 1.4926, irradiated up to about 6.8 dpa did not break within the maximum attainable deflection of about 2 mm. The results for pure Ta were very surprising. A highly brittle material was expected, but the results show instead an irradiation hardening but no reduction in ductility under the given experimental condition. Even the Ta sample irradiated up to 10 dpa did not break during the three-point bending test but showed an extreme ductile behavior and could be bent end to end without failure (cf. Figure 14.4). The Ni-base superalloy, IN 718, showed a trend for softening also, but the values of the failure loads were still much higher than the other materials, although the reduction in ductility was rather strong at 8.5 dpa (Figure 14.3). 14.1.2.2 Tensile Strength, Scanning- and Transmission Electron Microscopy Tensile strength is the stress at which a material breaks or deforms in an unstable manner. The quantitative information of the strength and ductility of irradiated 1) Ductility is a mechanical property used to describe the extent to which materials can be deformed plastically without fracture.

475

476


Ta 10 dpa

2 mm

Fig. 14.4 Optical micrograph of a Ta specimen from the ISIS plate target irradiated to 10 dpa after bending end to end, The given scale in the figure gives a measure of 2 mm (after Chen et al. [923]).

materials will be achieved with tensile tests performed in hot cells. A tensile test is the most fundamental type of mechanical test which can be performed on materials. Tensile tests are relatively simple and fully standardized. Usually a standard test piece or specially formed specimen is gripped at either end by a suitable apparatus in a testing machine, which slowly exerts an axial pull so that the specimen is stretched until it breaks. The test provides information on proof stress, yield point, tensile strength, elongation, and reduction of area. Considering the high spatial gradient of the proton beam distribution in the window and target components, miniature-type tensile samples were machined from the spent components. The specimens used here had a thickness of about 0.5 mm, length of 5 mm, and width in the gauge region of 1.2 mm. All tensile tests were performed at room temperature 25◦ C and 250◦ C in air. For the tests a 2 kN MTS tensile machine was applied which was equipped with a video-extensometer to measure the elongation directly in the gauge area. The strain2) rate was about 10−3 s−1 . Typical stress–strain curves are given in Figures 14.5 and 14.6 for the PSI window material DIN 1.4926 at T = 25◦ C and 250◦ C compared with the material Mod9Cr1Mo at T = 164◦ C from the APT irradiation program [925, 926], and stress–stain curves of the Ta specimens from the ISIS target at temperatures of T = 25◦ and 250◦ C with a strain rate of 10−3 s−1 . The tensile properties of DIN 1.4926 before and after irradiation is depicted in Figures 14.5(a) and (b) for the temperatures 25◦ C and 250◦ C. For comparison a tensile test result is shown in the figure, panel (b), of a quite similar material Mo9Cr1Mo from a previous irradiation program for the Accelerator Production of Tritium (APT) project [926]. From Figure 14.5, it is seen that the strength of the ferritic steels increased with the irradiation dose. All materials begin necking after a relative small strain of about 1 MeV. The results are in good agreement with those given by Drake et al. [1277] for the Martian surface and a cutoff energy of 10 MeV. The depth distribution presented in Figure 23.1 is quite close to an exponential profile. The thermal neutron flux is not greatly affected by the input profile, since the low-energy neutrons have undergone multiple scattering processes, as stated

655

23 Space Missions and Radiation in Space 100 90 Neutron density [10−7 n/cm3]

656

80 70 60 50 40 Via 10B(n,a)7Li reaction: 30

Calculated profile LNPE, Woolum (1975)

20 10 0

100

200

300

400

Depth [g/cm2]

Fig. 23.2 Lunar neutron density profile obtained via the 10 B(n,α)7Li reaction rate. The calculated data are compared to the results from the LNPE (Woolum et al. [1272]). The calculations with the HERMES system [88] include the GCR α-particle contribution and are equivalent to a pure GCR proton flux irradiation of 3.0 protons/(s cm2 ) (after Ref. [1276]).

by Lingenfelter et al. [1278]. Figures 23.2–23.4 show some examples comparing measured and simulated neutron reaction rate depth profiles, e.g., the 10 B(n,α)7 Li reaction rate, the 235 U fission rate, and the 53 Mn production from Fe samples. The measurements were chosen from the LNPE experiments. The 10 B(n,α)7 Li reaction rate (Woolum et al. [1272]) measured in the lunar soil is directly proportional to the neutron density since the reaction rate has a pure 1/v cross section. Therefore, the equilibrium neutron density could be calculated (1) via the 10 B(n,α)7 Li rates by folding the cross section with the neutron flux at each depth and (2) directly from the differential flux profile. Both results are in excellent agreement. Simulations of an alpha-particle irradiation indicate that the alpha-particle contribution of the GCR flux can be accounted for by scaling the proton GCR flux with 3.0 protons/(s cm2 ). The profiles of 235 U fission rates presented in Figure 23.3 are obtained in the same way as described before, by folding the calculated neutron flux with the 235 U fission cross section. Compared to the normalization discussed above, the calculated profile has to be enhanced by a factor of 1.17 to obtain the best fit to the experimental data from Woolum and Burnett [1273]. In order to check the high-energy secondary particle fluxes, the production rate of 53 Mn from Fe was calculated, also including the production from protons. Since the

23.2 Galactic Cosmic Ray (GCR) Induced Reactions in Moon and MARS Soil 4

Fission rate [1/g sec]

3

2

235

U(n,f) rate:

1

Calculated profile LNPE, Woolum (1974) 0 0

100

200

300

400

Depth [g/cm2]

Fig. 23.3 Lunar depth profile of the 235 U fission rate from the LNPE (Woolum and Burnett [1273]) compared to the theoretical profile. The calculated data with the HERMES system [88] had to be enhanced by 17% compared to the standard normalization value of 3.0 protons/(s cm2 ). The calculations include the GCR α-particle contribution (after Ref. [1276]).

high-energy particle fluxes do not change significantly as long as the average atomic mass and charge remain approximately constant, the calculated Apollo-11 neutron fluxes can also be applied for other lunar compositions for the calculation of this reaction rate. The cross-section data were provided by Michel [37] and were calculated with the ALICE LIVERMORE-82 code (Blann and Bisplinghoff [1279]). The theoretical profile can be found in Figure 23.4, together with the experimental data from Imamura et al. [1274, 1275]. A long-term averaged GCR-proton flux of 4.9 protons/(s cm2 ) for Apollo-15 data and 6.0 protons/(s cm2 ) for Apollo-16 data best represents the experimental depth profile, assuming a constant irradiation for several million years (T1/2 = 3.7 × 106 y). A detailed discussion on the production of radionuclides from GCR by neutron and proton fluxes can be found in Michel et al. [37]. 23.2.2 The Mars Observer Orbiter Mission to Measure the Spallation-Induced Gamma Flux Return of GCR on the Martian Surface 23.2.2.1 The Aim of the Mars Observer Mission The Mars Observer mission spacecraft was primarily designed for exploring Mars and the Martian environment. However, in addition to its planetary payload, Mars

657

23 Space Missions and Radiation in Space

103

5 Production rate [dpm/kg Fe]

658

GCR flux = 5.5 protons/cm2 s 2

102 53Mn

5

production from Fe

Calculated profile Apollo 15, Imamura (1973) 2

Apollo 16, Imamura (1974)

101 0

100

200 Depth

300

400

[g/cm2]

Fig. 23.4 Depth profile of the 53 Mn production rate from Fe from lunar samples (Imamura et al. [1274, 1275]) compared to the theoretical profile (calculated with the HERMES system [88]) normalized to 5.5 GCR protons/(s cm2 ). The GCR α-particle contribution is included in the calculations (after Ref. [1276]).

Observer carried some high-energy astrophysics instruments, including a gammaray spectrometer (GRS). Mars Observer was launched on September 25, 1992 from Kennedy Space Center aboard a Titan-III rocket. The spacecraft was lost in the vicinity of Mars after (most probably) an explosion of the fuel and oxidizer elements on August 21, 1993 when the spacecraft began its maneuvering sequence for Martian orbital insertion. While intended as a GRS for mapping the Martian surface composition from orbit, the GRS was also capable of suspending this function in order to make detailed observations of cosmic gamma-ray bursts. The n-type Ge detector for the gamma mapping was 5.5 cm in diameter and length. Passive cooling kept the crystal at less than 100 K. A combined anticoincidence shield and neutron system surrounding the Ge crystal rejected cosmic ray and neutron events when a burst mode had been triggered. Gamma-ray spectra from planetary surfaces provide a tool for the investigation of the chemical composition of the planet’s surface. Galactic cosmic rays (GCR), (cf. Section 1.2 on page 5), penetrate the upper few meters of the soil of planetary surfaces and produce a cascade of hadronic particles and gamma rays. Gamma rays from inelastic neutron scattering or neutron capture are a characteristic fingerprint of the target nucleus, whereas several other gamma-ray sources make up a background spectrum. The gamma-ray spectrum from a planetary surface thus contains information about the elemental composition of the surface material. The

23.2 Galactic Cosmic Ray (GCR) Induced Reactions in Moon and MARS Soil

portion of the gamma-ray flux transferred back through the soil and the atmosphere can be detected from an orbiting satellite. Planetary gamma-ray spectroscopy is an instrument for chemical mapping of planets with a sufficiently thin atmosphere like that of Mars. 23.2.2.2 Application to the Martian Surface To prepare the Mars Observer mission theoretical simulations [27, 1276] and experiments [1280–1282] have to be performed for a variety of expected material compositions in order to extract information about the Martian surface composition from measured gamma-ray spectra [27, 1276]. Calculations were carried out by various authors for the lunar surface, e.g., Armstrong and Alsmiller [1283], Lingenfelter et al. [1278]. Additionally, the lunar photon albedo was calculated by Armstrong [192] using Monte Carlo methods. Reedy et al. [1284] and Reedy [1285] calculated the gamma-ray leakage for a large number of gamma-ray lines using modeled particle fluxes described by Reedy and Arnold [1286]. Experimental lunar gamma-ray spectra had a relatively poor energy resolution since a NaJ(Tl) detector was used. Experimental tests at the SATURNE accelerator with a Ge detector were carried out by Brckner et al. [1287]. Basic studies of gamma-ray emission spectra from the Martian surface were performed by Metzger and Arnold [1288]. The effects of hydrogen in the Martian soil and the macroscopic neutron capture cross section of the Martian gamma-ray spectrum were shown by Lapides [1289]. Evans and Squyres [1290] calculated the gamma-ray-line intensity ratios of selected gamma-ray lines for a variety of expected Martian material compositions and structures such as polar caps. Calculations of Martian neutron leakage spectra for various water concentrations and inhomogeneous material structures were carried out by Drake et al. [1277]. Although extensive calculations on planetary neutron and gamma-ray spectra were carried out in the past, the calculational methods showed some basic disadvantages and gaps. Various Monte Carlo codes for low-energy neutron transport were not coupled properly to the high-energy particle transport codes. Instead, modeled neutron distributions were used as input spectra for the low-energy neutron transport. Important gamma-ray sources produced by the π 0 -decay could not be calculated from first principles. The applicability of spallation reactions of photon evaporation or induced gammaray spectra (cf. page 179) to analyze planetary surface compositions is shown as an example in Figure 23.5 bombarding the moon’s surface assuming a soil composition given in Table 23.1 with an average GCR spectrum (cf. page 10, Figure 1.5). Figure 23.5 shows the total prompt gamma-ray emission spectrum from the moon surface including several resolved gamma lines. The simulation was performed by using the HERMES system [88]. The calculational procedure for the Martian soil was applied to a standard Martian composition [1291] given in Table 23.2 with a water content of 0.9%. Both GCR proton and alpha-particle irradiations were performed. Compared to a simulated proton irradiation, the neutron yield, as well as the low-energy neutron flux per source particle, is increased by a factor of 3.5 for an alpha-particle irradiation.

659


Gamma−ray flux spectrum in arbitray units dΦ(E)/dE [cm−2 s−1 MeV−1]

660

102 5 2 101 5 2

Fe Si Mg

C O

Fe

AI

100 5

Si

Ca AI

Gd −1

10

Cr

5 From O(n,nag)C

10−2 5 10−3 10

20

30

40

50

60

70

80

Gamma−ray energy [MeV ×

10−1]

90

100

Fig. 23.5 The simulation of the prompt photon flux spectrum emitted from the lunar surface bombarded by the cosmic ray flux (GCR). The solid line shows the continuum, which is the sum of the total photon albedo plus the photon contribution produced by the π 0 -decay originating from the intranuclear cascade process. The lines or peaks in the spectrum have a FWHM of about 10 keV (after Ref. [1276]). Tab. 23.2 Elemental composition for the Martian soil (0.9 wt% H2 O and the Martian atmosphere).a

Composition of Martian soil Element wt% Element wt% Element wt%

H 0.10 Si 21.50 Mn 0.34

C 0.60 S 3.00 Fe 13.50

N – Cl 0.70 Co 33 ppm

O 46.60 K 0.12 Ni 52 ppm

Na 0.81 Ca 4.40 Th 0.45 ppm

Mg 3.70 Ti 0.38 U 0.13 ppm

Al 4.10 Cr 0.15

Composition of Martian atmosphere Element C N O wt% 26.52 2.74 70.74 a

The atmosphere of the Martian atmosphere was set to 16 g/cm2 (data are from [1276]).

The low-energy neutron depth profile originating from the GCR alpha-particle contribution was found to be close to that from the GCR proton case. The same results are found for the lunar surface. Therefore, further calculations are

23.2 Galactic Cosmic Ray (GCR) Induced Reactions in Moon and MARS Soil SCR

Observer

&

GCR

Protons & a's Gamma spectrum Prompt g 's, p0-decay, de-ex, of residuals, delayed g 's

Mars atmosphere

Polar ice cap

Mars soil

Fig. 23.6 Schematic sketch of the Martian solar cosmic ray (SCR) and the GCR irradiation environment with the Mars Observer orbiter as used in the simulations.

restricted to a pure GCR proton irradiation. To allow a direct comparison to the lunar calculations, a total 4π GCR flux of 3.0 protons/(s cm2 ) is assumed. The alphaparticle contribution is included in this normalization by the appropriate scaling factor of (0.9 + 0.1 × 3.5)/0.9 = 1.39. The gradient of about 2%/AU of the GCR intensity was neglected for Mars, e.g. [1292]. Figure 23.6 illustrates schematically the Martian irradiation environment with the Mars Observer orbiter applied in the simulations. The geometry used in the Monte Carlo simulations is similar to the geometry applied for the lunar surface. The prompt gamma-ray emission spectrum evaluated for a standard Martian composition is presented in Figure 23.7. The gamma-ray lines shown were obtained by Monte Carlo simulation as applied at the moon soil (cf. Figure 23.5). The statistical errors are about 15%(±1σ ) for strong lines like those from the elements oxygen and iron and are increasing to 40%(±1σ ) for weak lines, e.g., like the capture line from Si. The gamma-ray line fluxes are also determined and summarized in different tables and published in Refs. [27, 1276]. The gamma-ray background contribution from π 0 -decay is shown separately in Figure 23.7. The total gamma-ray flux from π 0 -decay yields 1.95 ± 0.08 photons/(s cm2 ), which is about twice the value given by Armstrong [1293] for the Moon. About one-third of the total gamma-ray continuum up to 10 MeV is produced by π 0 -decay. One of the most important problems is the determination of the water content of the Martian soil. This can be accomplished by two methods: (1) the ratio of the thermal to the epithermal neutron leakage fluxes may be used to determine

661


101 C

5 Diff. gamma-ray flux [1/cm2 sec MeV]

662

Fe

O

2

Fe

100

Cl

Si Si

Mg

Ca

Al H

S

5

Ti Mn

C

2

Al

10−1 5

Cr

0.9 wt% H2O Total photon albedo

2

From p0-decay 10−2 0

2

4 6 Gamma−ray energy [MeV]

8

10

Fig. 23.7 Total prompt gamma-ray emission spectrum from the Martian surface for an average GCR-spectrum including several resolved gamma lines. The dashed curve represents the continuum contribution from the π 0 -decay. The normalization is identical to the lunar case (after Ref. [1276]).

the water content of the Martian soil, and (2) by analyzing the gamma-ray line intensities. Figure 23.8 shows an example to determine the water content by analyzing gamma-ray line intensities (see the following discussion). It could be shown in [27] that the ratio of the inelastic scattering line from Si at 1.779 MeV and the capture line of Si varies significantly for low water concentrations, while the ratio of the capture line from H to a Fe capture line is best suited for the determination of higher water concentrations. A variety of other gamma-ray lines may be used in a similar way. The ratio of the 1.779 MeV inelastic scattering line to the 3.539 MeV capture line from Si is depicted in Figure 23.8 as a function of the water concentration. For the calculation of these data, only the neutron fluxes below 14.9 MeV were taken into account. The intensity of the 4.934 MeV line is about 14% higher than the 3.539 MeV capture line. Although these results are very promising, the interpretation of experimental gamma-ray spectra may be difficult if statistical errors are large. Additionally, the prompt photon contributions to the 1.779 MeV line from neutrons above 14.9 MeV and residual nuclei have to be accounted for. Adding this contribution causes a substantial shift of the line ratios. Furthermore, since prompt and delayed photons cannot be separated experimentally, delayed photons have also to be taken into account. The Si line ratio may thus be suited to show relative changes of the water content, although absolute water concentrations cannot be deduced easily. As can be estimated from Figure 23.8, this method is limited to water concentrations

23.3 The Space Experiment LDEF (Long Duration Exposure Facility)

4 Homogeneous soil + H2O From neutrons 65. This experiment has increased the world sample of ultra heavy cosmic ray measurements by nearly 15 times [1303, 1304]. The radiation exposure experiments including induced radioactivity, absorbed dose, and LET spectra has been extensively modeled [1299, 1300]. With the attitude stability of LDEF, the effects of directional characteristics of the primary environment (trapped protons and heavy ions, cosmic rays, solar energetic particles) (cf. Figure 23.12) could be studied (see [1294, 1295, 1300] and references therein). These measurements lead to the discovery of an ‘‘anomalous’’ quantity of 7 Be [1305]. Other significant results were on radiations emerging from LDEF important on the assessment of risk to crew during long duration spaceflight to determine proton-induced target fragments to the high LET (≥100 keV/µm) region of the LET spectrum [1306]. The radiation quantities and dosimeters applied for the radiation measurements aboard the LDEF satellite are summarized in Table 23.3. Activation results Interesting are the results of the extraction of specific activities from the Al activation tray clamp plates. These were some of the most valuable results from LDEF [1295]. Significant amounts of the two radionuclides, 22 Na and 7 Be (cf. Figures 23.13 and 23.14), were observed, and were found of different

667

668

23 Space Missions and Radiation in Space Tab. 23.3 Radiation measurement methods and dosimeters applied at LDEF (more details are given in [1295] by Harmon et al.)

Radiation quantity

Dosimeter

Total absorbed dose Dose in microscopic volumes LET spectra Neutron fluence and spectra Heavy ion fluence and spectra Proton fluence and spectra Trapped proton directionality

TLDsa microspheres PNTDb Fission foils activation PNTD PNTD/activation foilsc TLDs/activation foils

a

Thermoluminescent dosimeters. Passive nuclear track detectors. c Activation materials include specific metal samples for activation measurements, e.g., Ni, Ta, Va, In, Co, etc. and selected components, e.g., Al, Ti, Pb, steel, and stainless steel trunnions (see also Figure 23.11). b

origins. 22 Na is produced from trapped protons and cosmic ray interactions with higher energy particles in the Al clamps and spacer plates throughout the spacecraft. The distribution of the 22 Na activity on the clamp plates as a function of angle from the leading edge of the spacecraft is shown in Figure 23.13. The 1D approximate calculations were performed using the Watts model [886] simply assuming proton activation along a normal direction to a semiinfinite slab of aluminum. The 3D calculations are more realistic and treated time-dependent proton fluxes, the effects of shielding, and secondary particle production [1297, 1299, 1300]. The models under predict the measured activation by about 30–50%, depending on the strength of the calculated anisotropy. This is also indicted by the model calculations on the dosimetric studies at LDEF [1306]. The regular placement of the clamps over the entire surface of the spacecraft allowed the activation due to the anisotropic proton fluxes to be mapped. The 22 Na activity, with a half-life 2.6 y and a gamma-ray energy of 1.275 MeV from the experiment tray clamp plates allowed the sampling of the flux in a plane aligned with the East, West, North, and South directions. The results indicate a clear peak in the southwest direction for the activation that reflects the maximum in the anisotropic flux of trapped protons [886]. In contrast, there was a major surprise of the LDEF measurements [1305, 1307], comparing the 22 Na distribution with the distribution and the level of the 7 Be activity. A large concentration of 7 Be was discovered on the exposed metallic parts on the leading edge (i.e., the ram direction, the left part of the spacecraft shown in Figure 23.11) of the LDEF as seen in Figure 23.14, while the 22 Na distribution is present in the material from all sides.

23.3 The Space Experiment LDEF (Long Duration Exposure Facility) East 12

North

West

South

East

Measured 22Na activity

10

Approx. calculation, slab geometry, singel altitude = 450 km

Activity [Bq s−1 kg−1]

3D calculation with LDEF mass model, ESA AP8 fluences

8

6

4

2

0

45

90

135

180

225

270

315

360

Angle from LDEF leading direction (east) [degree]

Fig. 23.13 Comparison of measurements and calculations for 22 Na production rates of LDEF Al-clamp-plates as a function of angle from the leading side of the spacecraft. A measured and calculated anisotropy on West versus East is clearly indicated. (The clamp plates are used on outer surface of the LDEF spacecraft to secure experimental trays.) (after Harmon et al. [1295]).

Additional measurements of 7 Be applying a large bank of germanium detectors unevenly distributed around the spacecraft gave similar results [1308]. The conclusion of these exceptional 7 Be isotope distribution could only have been a result of atmospheric deposition combined with adsorption to the metallic surfaces caused by spacecraft motion [1305]. The 7 Be isotope is known primarily as a spallation product of relatively energetic cosmic ray interactions with N and O target nuclei in the atmosphere. Estimations have shown, that the measured concentration of 7 Be per unit mass of air at an altitude of 320 km is three orders of magnitude greater than it would be if it had been produced by cosmic-ray spallation at that altitude. Michel et al. [1309, 1310] have developed various sets of activation cross sections for 22 Na and 7 Be production from aluminum by protons and neutrons in the energy range from about 10 MeV to 1 GeV. The production cross section of 7 Be is in this energy range from about 0.1 to 10 mb very small. One simple explanation is that 7 Be is very fast transported upward from regions of the atmosphere, where its concentration is much higher. This transport has to take place on time scales similar to or shorter than the radioactive half-life = 53.2 d of the 7 Be [1305]. 23.3.3 Hazard Radiations in Space

Hazard sources faced by crews and spacecrafts in LEO or in deep space is the exposure to ionizing radiation in the space environment. The main sources of

669

23 Space Missions and Radiation in Space 160 140 Clamp plate activity per sample in [pCi] or [×3.7 Bq]

670

7Be 22Na (× 5)

120 100 80 60 40 Trailing edge

20 0 −270

−225

−180 7

Leading edge −135

−90

−45

0

45

90

22

Fig. 23.14 A comparison of Be and Na activities for the Al tray clamps taken from all around the LDEF spacecraft. The leading edge is at 0◦ and the trailing edge at 180◦ (after Gregory [1305]).

these radiations are the trapped particles in the Van Allen belts, consisting mainly of protons in the inner belt and electrons in the outer belt, the GCR background, composed of all naturally occurring elements and solar energetic particles, produced by events, such as coronal mass ejections and associated phenomena on the sun. Solar energetic particle events (SPE) are composed of protons and multiple-charged ions, with protons being the main source of concern for human exposures. As these radiations pass through spacecraft and the habitat shielding, their energies and composition are altered by interactions with the shielding. Hence, the radiation fields that actually produce biological damage may be substantially different from the external space environment. The modifications to the radiation fields arise from spallation and atomic processes whereby the charged particles lose energy with orbital electrons and occasionally undergo nuclear collisions. The spallation and fragmentation processes result in the formation of heavy ions that are lighter than the fragmenting nucleus and in the production of large numbers of light ions and neutrons. The potential hazards of space radiation are of concern in several areas: • Understanding the biological effects of space radiation is to be an important research area [1301, 1311]. While in the past, manned missions have been of relatively short duration and the astronaut doses have been relatively small[1312], for new missions such as the international space station (ISS), lunar bases, and Mars visits, the radiation protection to astronauts have to take on increased importance. • As space missions and operations nowadays become more complex and more extensive and higher speed automation requirements increase, more advanced

23.3 The Space Experiment LDEF (Long Duration Exposure Facility)

electronic technologies are needed. Such advanced microelectronics are susceptible to radiation damage, particularly to temporary failure due to single event effects (SEE) and single-event-upsets (SEU) [1313]. Also certain materials and components, such as optical coatings and solar cells, can have their performance degraded by radiation exposure[1314]. As the sensitivity of instrumentation for space observatories has increased, so has the susceptibility to radiation damage increased in many cases. Examples here include star sensors used in spacecraft orientation, CCDs used for visible light detection, and UV detectors [1314]. • Electrons with energies Eelectron > 100 keV can produce a buildup of charge on internal spacecraft structures, spacecraft charging, leading to electrical breakdown and current injection to sensitive electronic components [1315]. • Some satellite experiments include cryogenic systems for cooling instrumentation, and the heating effect of space radiation on these systems is often of concern. • The space radiation environment is a major source of background interference to some types of experiments, and in some cases the induced radioactivity in the spacecraft provides an important background to experiments [1316, 1317]. Astronaut dose limits The biological damage of radiation depends not just on the energy deposition produced but on the density of deposition. The density of energy deposition along a charged particle track is commonly expressed as the Linear Energy Transfer (LET). In radiation protection work, the quality factor (QF) is used to define a ‘‘dose equivalent’’ to relate LET to biologica1 damage. The LET is a physical quantity, representing the energy deposition by ionization processes, and can be calculated for a given ion charge, speed and material composition. Effects of this type are also important in other fields, notably in space physics since a strongly localized LET may induce failures in computer microchips. The QF as a function of LET is set by regulatory commissions and advisory groups based on experiments for various biological systems (ICRU [1318] and NCRP [1319, 1320]). The dose equivalent, H, which is expressed in units of Sieverts (Sv) is defined as H = QF × D with D in (Gy). For example, NASA dose limits for astronauts are given in Table 23.4 set by the National Council on Radiation Protection Measurements (NCRP) [1319]. The limiting criterion is usually the BFO dose, which is the dose at the location of Blood Forming Organs, usually taken to be at a depth of 5 cm in tissue for dose estimates. Dose estimates for manned missions The importance of different space radiation sources (trapped, cosmic, and solar radiation) depends strongly on the mission and associated orbit parameters. Table 23.5 shows dose estimates (taken from various estimates in the literature [1319]) for different types of missions. For missions in LEO, e.g., below about 800 km, most of the exposure is from trapped proton exposure during passage through the South Atlantic region, with contributions from GCR and possible solar proton events (SPE) increasing for increasing orbit inclinations due to the decrease in geomagnetic shielding. The trapped exposure

671

23 Space Missions and Radiation in Space Tab. 23.4 Dose limits set by the USA National Council on Radiation Protection and Measurements (NCRP) for space activities (data based on Ref. [1319]).

Recommended organ dose equivalent limits all ages BFO (Sv) Eye (Sv) Skin (Sv) Career See below 4.0 6.0 Annual 0.50 2.0 3.0 30 days 0.25 1.0 1.5 Career whole-body dose equivalent limit (Sv) for a lifetime excess Risk of fatal cancer of 3% as a function of age at exposurea age

25

Female Male

35

1.0 1.5

55

1.75 2.5

3.0 4.0

Based on 10-y exposure duration. 1.4

.7

1.2

.6 Dose equivalent [Sv/y]

a

Dose equivalent [Sv/y]

672

1 .8 .6

Solar maximum

.4 Solar maximum

0

Aluminum shield

.4 .3 .2 Water shield

.2

(a)

.5

.1

5

10

15

20

25 2

Water thickness [cm or g/cm ]

30

0

10

(b)

20

30

40

50

Shield thickness [g/cm2]

Fig. 23.15 Shielding effectiveness for GCR radiation: (a) dose equivalent in water at solar minimum and solar maximum; (b) dose equivalent behind varying water and aluminum shielding thicknesses.

is strongly dependent on altitude. Outside the earth’s magnetosphere, there is the possibility of very large doses from SPEs, and shielding must be provided. Given that a SPE ‘‘storm shelter’’ is provided for SPEs in the spacecraft design, the main remaining contributor for deep space missions is the GCR. In Figure 23.15 the shielding effectiveness for GCR is illustrated for galactic cosmic rays for hydrogenous material (water) and aluminum shields [1321]. In the trapped electron environment, electrons and secondary photons (bremsstrahlung) are important at small shielding depths. Proton transport is needed in estimating the effects of trapped protons and cosmic ray protons, with the importance of secondary neutron and proton transport. Normal spacecraft

23.3 The Space Experiment LDEF (Long Duration Exposure Facility) Importance of various space radiation environments in terms of biological doses for different space missions (examples from various estimates in the literature, e.g., [1317, 1322]).

Tab. 23.5

Orbit Attitude

Mission Inclination

LEO

Low

LEO

Medium

LEO

High

GEO

0◦

Deep space Deep space

Radiation dose from Trapped protons (Sv/y)

Space Station 450 km, 28.5◦ Skylab 450 km, 50◦ Polar-Platform 450 km, 90◦ Geosynchronous 35790 km 160◦ west long. Lunar base Mars landing

Trapped electrons (Sv/y)

GCRs (Sv/y)

SPEs event (Sv/event)

0.32

∼0

0.03

0.21

∼0

0.06

0.04

0.19

∼0

0.09

0.29

0.47

5.07

0.30 (surface) 0.45 (transit) 0.12 (surface)

3.4

0.04

0.18

0.02

∼0

0.02

∼0

∼0

3.4

shielding, typically about 10–20 g/cm2 on average, is effective in shielding the trapped and SPE radiation, but has limited effectiveness for GCR as shown in Figure 23.15. Heavy ions make an important contribution to biological dose and electronics upset for small shielding thicknesses. Single event effects (SEE) of spacecraft microelectronics exposed to cosmic rays As advanced technology electronic devices have become increasingly smaller, it has become possible that a single charged particle hitting the device can produce sufficient energy deposition and charge release that the logic state of the device is changed. Thus, for example, single-event-upset (SEU) phenomena can change a ‘‘0’’ or ‘‘1’’ bit in an on-board computer, resulting in such effects as a change in the orientation of the spacecraft. A related SEE is latch up, which can cause permanent damage to the device. SEU effects and Multiple Bit Upset (MBU) can be produced both by HZE comic rays, and by cosmic, solar, and trapped protons via the highly ionizing products from proton nuclear interactions. SEE have caused problems on numerous satellites in recent years and costly redesign for spacecraft [1323]. MBU has been observed since 1990s and increase exponentially when the gate length decreases [1324]. One concern is that the dominating sensitivity to SEE arises simply from a cosmic-ray track induced inside the sensitive volume of the device as depicted in Figure 23.16.

673

674

23 Space Missions and Radiation in Space Cosmic-ray track q Surface of silicon chip (greatly enlarged)

≈ 25m Sensitive region Charge cloud

Fig. 23.16 Illustration of SEE phenomena produced by primary and secondary particle collisions inside the sensitive region of a silicon chip.

Various types of SEE have been reported. As mentioned before, the SEU denotes a soft failure which upsets a bit memory from a state to another. This failure is called ‘‘soft’’ because the bit can be restored. In contrast the ‘‘Single Hard Error’’ may cause a permanent device damage, which is then called ‘‘hard’’ [1313]. Some of these problems may be circumvented if the radiation environments are much better understood, which could be done by spallation research on radiation–components interaction. But some problems are only solved by circumvention implementation, e.g., the choice of less sensitive components, the use of redundancy mode, or software protections. Such work was done during the ARIANE 5 design work [1324].

675

Appendix A Values of Fundamental Physical Constants and Relations In this appendix, a selection of the values of fundamental physical constants, units, symbols, conversion factors, and some relations relevant for the spallation physics are summarized in tabular form. For more detailed information, the reader is referred to the standard manifold literature. The 11th General Conference on Weights and Measures (1960) adopted the name Systeme International d’Unites (International System of Units, international abbreviation SI), for the recommended practical system of units of measurement. The same conference laid down rules for the prefixes, the derived units, and other matters. The base units are a choice of seven well-defined units which by convention are regarded as dimensionally independent: the meter, the kilogram, the second, the ampere, the kelvin, the mole, and the candela (cf. Table A.1). Derived units are those formed by combining base units according to the algebraic relations linking the corresponding quantities. The SI unit is not static but evolves to match the world’s increasingly demanding requirements for measurement. The base units are defined as follows: • The meter is the length of the path traveled by light in vacuum during a time interval of 1/299792458 of a second. • The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram. • The second is the duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the 133 Cs atom. Tab. A.1

SI units.

Measure

Unit

Measure

Unit

Meter Kilogram Second Ampere Kelvin Mole Watt

m kg s A K mol W

Newton Farad Joule Coulomb Tesla Hertz Candela

N F J C T Hz cd


676

A Values of Fundamental Physical Constants and Relations Tab. A.2

Some useful conversion factors.

Unit

Conversion unit

1u 1 eV 1J 1 MeV c−1 1C 1T 1 Bq 1 Ci 1 Gy 1 Sv 1 A˚ 1 fm 1b

Tab. A.3

= = = = = = = = = = = = =

1 mu = 12 m(12 C) 1.60217733(49) × 10−19 J 107 erg 5.3443 × 10−17 g cm s−1 10−1 e.m.u. = 2.99792458 × 109 e.s.u. 104 G 1 disintegration s−1 = 2.7 × 10−11 Ci 3.7 × 1010 s−1 1 J/kg = 104 erg/g = 100 rad 100 rem 10−10 m = 10−8 cm 10−15 m = 10−13 cm 10−28 m2 = 10−24 cm2

General constants.

Measure

Symbol/relation

value

Unit

Speed of light in vacuum Planck constant in electron volts, h/e h/2π in electron volts, /e h/2π hc Elementary charge

c h

299792458 ×1010 6.6260755 ×10−34 4.1356692 ×10−15 1.05457266×10−34 6.5821220 ×10−16 197.33 1240 −19 1.602008 √ ×10 1200 MeV fm 9.1093897 ×10−31 2.81794092 ×10−15 1.8835327 ×10−28 105.658389 2.406120 ×10−28 2.4880187 ×10−28 134.9630 139.5673 1.6726231 ×10−27 938.27231 1.6749286 ×10−27 939.56563 6.022045 ×1023 1.30658 ×10−23 8.617385 ×10−5

cm/s Js eV s Js eV s MeV fm/c eV nm C

Electron mass Classical electron radius Muon mass in electron volts, mµ c2 /e Pion mass Pion mass in electron volts, mπ 0 c2 /e in electron volts, mπ ± c2 /e Proton mass in electron volts, mp c2 /e Neutron mass in electron volts, mn c2 /e Avogadro constant Boltzmann constant in electron volts, k/e

hc e e me re mµ π0 π± mπ 0 mπ ± mp mn L k

kg m kg MeV kg kg MeV MeV kg MeV kg MeV mol−1 J K−1 eV K−1

A Values of Fundamental Physical Constants and Relations

• The ampere is that constant current which if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 m apart in vacuum, would produce between these conductors a force equal to 2 × 10−7 N m−1 of length. • The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. • The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kg of 12 C. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles. • The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 × 1012 Hz and that has a radiant intensity in that direction of 1/683 W sr−1 . Other useful relations (a) The de Broglie wavelength, with E = kinetic energy, is given: • λ = /p = (2m0 E + E 2 /c2 )−1/2 = (2m0 E)−1/2 (1 + E/2m0 c2 )−1/2 , √ • for nucleons and E m0 c2 the de Broglie wavelength is λ = (4.55/ E) (fm). (b) The relation between frequency and wavelength: • E λ = = 197.33 (MeV fm) 1/3 1/3 (c) The energy of the Coulomb threshold with R = r0 (A1 + A2 ) (fm): 2 • Vc = Z1 Z2 e /R = 1.44 Z1 Z2 /R (MeV)

677

679

Appendix B Basic Definitions in Nuclear Technology Concerning the Fuel Cycle Some terms, the breeding gain, the conversion ratio, and the reproduction factor, applied in nuclear technology and in the context of transmutation are summarized with examples in Tables B.1, B.2, B.3, and B.4. • η = number of neutrons produced per neutron absorbed in the fuel of a nuclear system. An η > 2 implies breeding is possible. • η = (σfission /σabsorption ) · ν, where ν = number of neutrons produced per fission. Note that a fast spectrum yield higher reproduction factors. • Thus, a fast reactor or ‘‘fast’’ energy amplifier is better suited for breeding fuel. • The highest possible reproduction factor is 239 Pu with an η = 2.7. • Conversion ratio (CR)/breeding gain (BG): fissile nuclei produced CR = fissile = η − 1, where for a CR > 1 the breeding ratio nuclei consumed (BR) is used. • breeding gain (BG) = BR – 1

Tab. B.1

Breeding and conversion of fertile and fissile isotopes.

Fertile isotopes

→

Fissile isotopes

232 Th

→ → →

233 U

238

U 240 Pu

239

Pu

241 Pu

Reproduction factor η for thermal and fast neutrons for different fissionable isotopes.

Tab. B.2

Isotopes

Thermal neutrons

Fast neutrons

233 U

2.30 2.07 2.11

2.45 2.30 2.70

235 U 239 Pu


680

B Basic Definitions in Nuclear Technology Concerning the Fuel Cycle Tab. B.3

Conversion and breeding ratios of different reactor types.

Reactor type

Fuel

Conversion

CR or BR

BWR PWR CANDU HTGR LMFBR

2–4 wt% 235 U 2–4 wt% 235 U nat U 5–90 wt% 235 U 10–20 wt% 239 Pu

238 U

0.60 0.60 0.80 0.80 1.0 to 1.7

Tab. B.4

to 239 Pu to 239 Pu 238 U to 239 Pu 232 Th to 233 U 238 U to 239 Pu 238 U

The status of commercial LMFBRs in 2006.

Reactor

Site

Year

Phenix Superphenix PFBR Monju BN 600 BN 800 Dounray

France France India Japan Russia Russia UK

1974-a 1986–1998 2010 1993–1995 1981-a 2012 1976–1994

a

Power MWe

Type

233 1200 500 246 560 750 250

Pool Pool – Loop Pool – Pool

In operation.

Overview on conversion and breeding ratios for different reactor types and commercial LMFBR reactors1) The most shutdowns or plant closures of fast reactors in the past were the result of sodium leaks in the cooling circuits of LMFBRs. Some shutdowns were caused by financial or political reasons, e.g., the CRBR project (Clinch River Breeder Reactor Plant) in USA or the SNR-300 (Schneller Natriumgekühlter Reaktor) in Germany. With low uranium prices, the costly reprocessing and the fuel cycle, the breeder reactors has had unfavorable economics.

1) Detailed information is given in the ‘‘Fast Reactor Database 2006 Update’’ of the Int. Atomic Energy Agency (IAEA) with a brief history on ‘‘Technology Advances in Fast Reactors and Accelerator Driven Systems,’’ (http://wwwfrdb.iaea.org/auxiliary/history.html).

1.1

232

−4

3.5×10

0.13

2

0.6

1.4

U

0.04

0.2

2.3

0.61

27 d

Pa

(n, 2n) (En: ≥ 6 MeV)

s (bam) Fission

0.31

0.20

22.3 mn

Th

233

233

233

(n, g) capture

U

0.007

0.24

1.3 d

Pa

Th

232

232

t 1/2 b -decay ≤ 10 a (isomer)

s (bam)

Pa

0.0017 25.5 h 0.18

Th 0.04

1.5

0.35

U 0.37

1.55

24.4 mn

Pa

7.2 mn

Th

235

235

235

236

U 0.4

0.08

237

0.35

0.69

1.1

0.3

6.75 d

U

Np

237

238

238

U

0.3

1.8

0.22

0.03

Pu 0.37

1.1

2.12 d

Np

238

239

1.27

Pu 0.31

1.65

2.35 d

Np

0.49

23.5 mn

U

239

239

240

0.4

61.9 mn (7.2 mn) 0.35

Np

14.1 h

U

Pu

240

240

Among the 60% of neutrons not used for fission, 20% are lost and 40% are used to breed 233-U from 232-Th. In this way, new fissile material replaces what is used for fission.

Note the difference between 232-Th and 238-U in terms of TRU access!

U

6.7 h (1.17 mn) 0.3

Pa

0.003 0.25

24.1d

Th

234

234

234

241

241

0.27

2.1

Am 1.2

0.2

14.3 yr

Pu

242

Pu

Am

242

0.47 (0.2)

2.7 (3.1)

0.34

0.24

243

0.24

0.64

1.0

0.17

4.96 h

Pu

Am

243

Fig. B.1 Chart of the actinides: The choice of fuel, Thorium[232 ThO2 (+233 UO2 )]. Thorium is about 5 neutron captures away from the transuranium elements (TRU) one wants to destroy (after Kadi [1325]).

231

231

n + 232Th (1.4 × 1010 a) → 233Th (22.3 m) → 233Pa (27 d) → 233U (1.6×105 a)

B Basic Definitions in Nuclear Technology Concerning the Fuel Cycle 681

683

Appendix C Material Properties of Structure and Target Materials


balanceFe-(50-55Ni)-(17-21Cr)-(4.75-55)Nb -(2.8-3.3)Mo-(0.65-1.15)Ti-(0.2-0.8)Al -(1.0)Co-(0.08)C-(0.35)Mn-(0.35)Si-(0.015)P -(0.015)S-(0.3)Cu balanceFe-(10-14)Ni-(16-18)Cr-(2-3)Mo -(2)Mn-(0.75)Si-(0.1)N-(0.045)P-(0.03)C -(0.03)S balanceAl-(0.4)Si-(0.4)Fe-(0.1)Cu-(0.5)Mn -(2.6-3.6)Mg-(0.3)Cr-(0.2)Zn-(0.15)Ti

INCONEL718b

balanceFe-(11.68)Cr-(1.1)Mo-(0.66)Ni -(0.47)W-(0.63)Mn-(0.29V-(0.45)Si -(0.2)C-(0.03)Nb balanceFe-(8.8)Cr-(0.96)Mo-(0.24)Ni -(< 0.01)W-(0.43)Mn-(0.24)V-(0.32)Si -(0.099)C-(0.3)Nb balanceFe-(12)Cr-(1.8)Si-(0.9)Ni-(0.7)Mo -(0.7)Mn balanceFe-9Cr-1.1W-0.4Mn balanceFe-(7.65)Cr-(2.0)W-(0.2)V-(0.04)Ta -(0.003)B-(0.1)C

2.65

7.9-8.0

8.19

Density (g cm−3 )

15.9 at 273–373 K

∼ 1333

24.0

13.0 at 294-366 K

≥ 1609

863-918

Thermal expansion (µm m−1 K−1 )

Melting point tm (K)

960

500

435

Specific heat (J kg−1 K−1 )

Tantalum is also applied as structure material to clad tungsten targets. For the many details about the properties and applications the comprehensive literature mainly published in handbooks should be considered, e.g. [926, 1153, 1326–1328]. b INCONEL is a trade mark of the Special Metals Corporation group of companies. 316LN/L is the most common stainless steel.

a

Eurofer97 F82H

EP823

T91

HT9

Ferritic–martensitic steels for ADS

AlMg3

316LN/316L

Material composition (% wt)

Properties of common structure materials for spallation neutron source target systems and for proposed ADS applicationsa .

Material

Tab. C.1

684

C Material Properties of Structure and Target Materials

W

82 83

nat Bi

74 73

nat Pb

Solid and liquid target materials:

nat Ta

nat

Solid target materials:

Charge number Z

209

207.2(1)

183.84 180.95

Atomic mass A (g mol−1 )

11.34 (solid) 10.66 (liquid) 9.78 (solid) 10.05 (liquid)

19.25 16.65

(g cm−3 ) at r.t.

Density

544.7

600.61

3695 3290


1837

2022

5828 5731

Boiling point tb (K) at 1 atm

28.9 (solid) 40 (liquid) 13.4 (solid)

4.5 6.3


25.52

26.65

24.27 25.36

Specific capacity (J mol−1 K−1 )

(continued)

0.034

0.17

18.5 22

Thermal cross section (b)

Material properties of target materials for spallation neutron sources and ADS applications. W with about 5 %wt Re and Udep with 7–10% wt Mo were also investigateda .

Tab. C.2

C Material Properties of Structure and Target Materials 685

(continued)

208.2 200.6

(82.5) 80

Isotopic compositions of W, Ta, Bi, Pb, and Hg:

202.6

Atomic mass A (g mol−1 )

(82)

Charge number Z

13.534

10.5 (liquid)

10.6 (liquid)

(g cm−3 ) at r.t.

Density

234.32

398

523


629.88

Boiling point tb (K) at 1 atm

60.4

40 (liquid)


• nat W (180 W(0.12%), 181 W(26.4%), 183 W(14.31%), 184 W(30.64%), 186 W(28.43%)], • nat Ta (180m Ta(0.012%), 181 Ta(99.988%)], • nat Pb (204 Pb(1.4%), 206 Pb(24.1%), 207 Pb(22.1%), 208 Pb(52.4%)], • nat Bi (209 Bi(100%)], • nat Hg (196 Hg(0.15%), 198 Hg(9.97% ),199 Hg(16.87%), 200 Hg(23.1%), 201 Hg(13.18%), 202 Hg(29.86%), 204 Hg(6.87%)]

a

LME (Pb 97.5%, Mg2.5%) PbMg eutectic LBE (Pb 45%, Bi55%) PbBi eutectic nat Hg

Liquid target materials:

Tab. C.3

27.98

31.2

28.4

Specific capacity (J mol−1 K−1 )

389

0.11

0.17

Thermal cross section (b)

686

C Material Properties of Structure and Target Materials

687

Appendix D Moderator and Reflector Materials Tab. D.1 Physical properties of compound moderator and reflector materials. Data based on Bauer [57].

Compound moderator materials Parameter

H2 O

D2 O

CH4

(CH2 )n

NH3

TiH2

A ρ (g cm−3 ) N (1024 cm3 ) σfree (b) σbound (b) σa (b) free =N·σfree (1/cm) ξ κ (2 MeV→1 eV) v · ts (cm)

18 1 0.033 44.8 168.3 0.665 1.5 0.926 15.7 2.15

20 0.001 0.033 10.5 19.5 0.001 0.35 0.510 28.5 14.24

16 0.453 0.017 86.8 333.6 1.334 1.48 0.954 15.2 2.12

14 0.94 0.040 45.7 169.6 0.669 1.85 0.913 15.9 1.77

17 0.682 0.024 71.5 257.6 2.898 1.73 0.879 16.5 1.96

49.92 3.978 0.048 42.43 163.3 6.755 2.04 0.811 17.9 1.81

σfree (b) = free scattering cross section, σbound (b) = bound scattering cross section, σa (b) = thermal absorption cross section, free (1/cm) = N · σfree (1/cm) = macroscopic cross section, ξ = logarithmic energy decrement, κ (2 MeV → 1 eV) = number of collisions for slowing down from an energy E to thermal energy, v · ts (cm) = slowing down length.


H 1 0.07 0.042 2.51 82.02 0.3326 0.86 1.000 14.5 3.47

A ρ (g cm−3 ) N (1024 cm3 ) σfree (b) σbound (b) σa (b) free =N·σfree (1/cm) ξ κ (2 MeV→1 eV) v · ts (cm)

Element

2 0.163 0.049 3.4 7.64 0.0005 0.17 0.725 20.0 21.37

D 9.01 1.85 0.124 6.18 7.63 0.0076 0.76 0.206 70.3 13.58

Be 12.01 2.3 0.115 4.73 5.51 0.0035 0.55 0.158 92.0 24.51

C 14 0.804 0.035 10.03 11.51 1.9 0.35 0.136 106.5 44.48

N 16 1.13 0.043 3.75 4.232 0.0002 0.16 0.120 121.0 108.83

O 55.85 7.9 0.085 11.21 11.62 2.56 0.96 0.035 410.0 59.86

Fe

58.71 8.9 0.091 17.89 18.5 4.49 1.63 0.034 430.8 36.77

Ni

Physical properties of common elemental moderator and reflector materials (data are from Bauer [57]).

Parameter

Tab. D.2

200.6 13.55 0.041 26.53 26.8 372.3 1.08 0.010 1460.1 187.06

Hg

207.19 11.3 0.033 11.01 11.118 0.171 0.36 0.010 1507.9 576.49

Pb

688

D Moderator and Reflector Materials

689

Appendix E Shielding Materials Tab. E.1

Element O Si Al Fe Mn Ti Ca Mg K Na

Soil composition for earth shielding of accelerators (data are from [1329]). Soil composition (wt%) 43.77 28.1 8.24 5.09 0.07±0.06 0.45±0.043 3.65 2.11 2.64 2.84

The density of soil varies between 1.7 and 2.25 g cm−3 depending on the water content and the soil sort.


690

E Shielding Materials

Tab. E.2

Typical compositions of representative concretesa . Concrete partial density (g cm−3 )

Element

Ordinary

Magnetite

Barytes

Magnetite steel

Limonite steel

Serpentine

H O Si Ca C Na Mg Al S K Fe Ti Cr Mn V Ba

0.13 1.165 0.737 0.194 – 0.040 0.006 0.107 0.003 0.045 0.029 – – – – –

0.011 1.168 0.091 0.251 – – 0.033 0.083 0.005 – 1.676 0.192 0.06 0.007 0.011 –

0.012 1.043 0.035 0.168 – – 0.004 0.014 0.361 0.159 – – – – – 1.551

0.011 0.638 0.073 0.258 – – 0.017 0.048 – – 3.512 0.074 – – 0.003 –

0.031 0.708 0.067 0.261 – – 0.007 0.029 – 0.004 3.421 – – – 0.004 –

0.035 1.126 0.460 0.150 0.002 0.09 0.297 0.042 – 0.009 0.068 – 0.002 – – –

Density (g cm−3 )

2.35

3.53

3.35

4.64

4.54

2.1

a Remark: The hydrogen content strongly influences its effectiveness for shielding against neutrons below energies of about 100 MeV (data are extracted from the American National Standard ANSI/ANS-6.4-1985 and from [1330].

Tab. E.3 Comparison of different shielding materials applied for spallation neutron source and accelerator shielding.

Shielding materials

Soil Normal concrete Heavy concrete Steel Cast steel Stainless steel Lead Tungsten

Density (g cm−3 ) 1.8 2.3 4.3 7.85 7.1 7.85 11.3 18.5

Attenuation length λatt [1329] (g cm−2 ) 108 115 130 160 160 160 210 194

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