Nuclear physics. Part A

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Methods of Expe rim en taI Physics VOLUME 5

NUCLEAR PHYSICS PART A

METHODS OF

EXPERIMENTAL PHYSICS: L. Marton, Editor-in-Chief Claire Marton, Assistant Editor

1. Classical Methods, 1959 Edited by lmmanuel Estermann

2. Electronic Methods Edited by E. Bleuler and R. 0. Haxby 3. Molecular Physics Edited by Dudley Williams

4. Atomic and Electron Physics Edited by Vernon W. Hughes and Howard L. Schultz 5. Nuclear Physics (in two parts), 1961 Edited by Luke C. L. Yuan and Chien-Shiung Wu 6. Solid State Physics (in two parts), 1959 Edited by K. Lark-Horovitz and Vivian A. Johnson

Volume 5

Nuclear Physics Edited by

LUKE C. L. YUAN Brookhaven National laborafory Upton, New York

CHIEN-SHIUNG WU Columbia University New York, New York

PART A

1961

ACADEMIC PRESS

@

New York and London

Copyright @ 1961, by

ACADEMIC PRESS INC. ALL RIQHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM BY PHOTOSTATl MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS

ACADEMIC PRESS INC. 111 FIFTH AVENUE NEWYORK3, N. Y.

United Kingdm Edition Published by ACADEMIC PRESS INC. (LONDON)LTD. 17 OLD QUEEN STREET, LONDONS.W. 1

Library of Congress Catalog Card Number 61-17860 PRINTED IN THE UNITED STATES OF AMERICA

CONTRIBUTORS TO VOLUME 5, PART A D. E. ALBURGER, Brookhaven National Laboratory, Upton, N e w York M. BLAU,Institut fur Radiumforschung, Vienna, Austria J. E. BROLLEY, JR.,Los Alamos Scientific Laboratory, Los Alamos, New Mexico B. CORK,Lawrence Radiation Laboratory, University of California, Berkeley, California J. W. M. DUMOND,Department of Physics, California Institute of Technology Pasadena, California

R. D. EVANS,Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts

H. FRAUENFELDER, Department of Physics, University of Illinois, Urbana, Illinois W. B. FRETTER, Department of Physics, University of California, Berkeley, California S. S . FRIEDLAND, Solid Xtate Radiations, fnc., Culver City, California

T. R. GERHOLM, Institute of Physics, University of Uppsala, Uppsala, Sweden W. W. HAVENS,Pupin Physics Laboratory, Columbia University, New York, New York R. HOFSTADTER, Physics Department, Stanford University, Standford, California D. J. HUGHES, Brookhaven National Laboratory, Upton, N e w York* S. J. LINDENBAUM, Brookhaven National Laboratory, Upton, New York G. C. MORRISON, Atomic Energy Establishment, Harwell, Berkshire, England G. D. O'KELLEY,Oak Ridge National Laboratory, Oak Ridge, Tennessee F. REINES,Department of Physics, Case Institute of Technology, Cleveland, Ohio G. T. REYNOLDS, Princeton University, Princeton, New Jersey A. SILVERMAN, Department of Physics, Cornell University, Ithaca

*

Deceased. V

vi

CONTRIBUTORS TO VOLUME

5,

PART A

R. M. STERNHEIMER, Brookhaven National Laboratory, Upton, New York R. W. WILLIAMS,Department of Physics, University of Washington, Seattle, Washington

L. C. L. YUAN,Brookhaven National Laboratory, Upton, New York F. P. ZIEMBA, Solid State Radiations, Inc., Culver City, California

FOREWORD TO VOLUME 5A After a longer delay than originally expected, I a m able to present here the next volume in the series of Methods of Experimental Physics: the first part of the “Nuclear Physics” Methods. Much thought .and work went into this volume and I am in the best position to appreciate all the effort of my fellow editors who devoted so much time in preparing this particular volume. The aims of the publication did not change. The reception of the earlier volumes had proven th a t it was really useful to concentrate on “a concise, well illustrated presentation of the most important methods, or general principles, needed by the experimenter, complete with basic references for further reading. Indication of limitations of both applicability and accuracy is a n important part of the presentation. Information about the interpretation of experiments, about the evaluation of errors, and about the validity of approximations should also be given. The book should not be merely a description of laboratory techniques, nor should i t be a catalog of instruments.” I n these troubled times, when furthering of scientific education is so important, we hope th a t these volumes can be of real help as well to the educator, as to the research worker. At the time of writing my foreword, two other volumes are well under way. The manuscripts of the companion volume of the present one, Volume 5B, are accumulating rapidly and printing should follow this one within a few months. In an even more advanced stage is our Volume 3, “Molecular Physics,” which, under the very valuable leadership of Professor Dudley Williams, promises to become a perfect companion t o the already existing volumes. At the writing of this foreword, Volume 3 is just entering the page-proof stage. I would like to report also on further development. It has been suggested that, in order to enhance the usefulness of this collection to the graduate student, we supplement the planned six volumes with a seventh devoted to nothing but problems. I n discussing this idea with a number of my colleagues I found favorable reaction and a t present I a m investigating how t o organize such a volume. It is a pleasant duty to thank again all those who have devoted SO much time and work to the preparation of this volume. I n the first place come the volume editors whose intelligent and devoted handling of the material is beyond praise. One shouldn’t, however, forget the authors. They have been most accommodating and I join the volume editors in expressing my appreciation. The publishers deserve our gratitude for

vii

viii

FOREWORD TO VOLUME

5A

being very patient and very helpful during this longer delay than we expected. As in the past, Mrs. Claire Marton has been most helpful in handling many of the problems of the editorial office. To all these people go my heartfelt thanks. L. MARTON Washington, D. C. June, 19fl

PREFACE TO VOLUME 5A The field of experimental nuclear physics has in the last two decades, experienced a tremendous growth of activity in all its branches. The difficulty in performing nuclear physics experiments is also greatly multiplied with the increasing complexities of the problems involved. There are, at present, many articles and books which give excellent reviews on basic principles and details of techniques of various detectors, methods and specific topics in nuclear physics. But it is often hard to obtain comprehensive information on the principal methods and their relative merits for the measurement of a specific physical quantity in the field of nuclear physics. This knowledge is especially desirable when one wishes to make a choice among the various methods on the basis of their feasibility, the accuracies attainable, and the limitations in their application under specific conditions. For any specific method of measurement, the comprehensive procedure of converting the experimental data into the desired physical quantities including the necessary corrections involved is often not explicitly mentioned in the literature. It is the intention of the present volume to try to meet some of the requirements mentioned above. All possible methods that deal with the measurement of each particular physical quantity are grouped together so as to achieve a more coherent presentation. Furthermore, the scope of this book is not limited to the usual treatment of low energy nuclear physics only, but it comprises both the high and low energy regions. We hope that this volume will serve as an informative source and as a reference book for physicists in general and, in particular, as an instructive and useful guide for all those who are interested in doing research in this field. Every effort has been made to obtain leading experts in each field to prepare contributions on the specific topics involved so that their intimate knowledge and experience can be shared. Owing to the comprehensive coverage of this book and to the enthusiastic response of a large number of contributors who treated their subject matter so thoroughly, it was found necessary to divide the work into two volumes rather than to publish a single volume a8 originally planned. For this reason and because an unusually large number of contributors have been involved, there has been some unavoidable delay in the completion of this book. ix

X

PREFACE TO VOLUME 5 A

We wish to take this opportunity to express our deepest appreciation and thanks to all the contributors for their understanding and cooperation, to the publisher and to Dr. L. Marton, the Editor-in-Chiefl for their invaluable help and continuous encouragement. CHIEN-SHIUNG Wu Columbia University LUKEC. L. YUAN Brookhaven National Laboratory August 14, 1961

CONTENTS, VOLUME 5, PART A CONTRIBUTORS TO VOLUME 5, PART A. . . . . . . . . . . . . .

v

FOREWORD TO VOLUME 5A . . . . . . . . . . . . . . . . . . vii PREFACE TO VOLUME 5A . . . . . . . . . . . . . . . . . . . CONTRIBUTORS TO VOLUME 5, PARTB. . . . . CONTENTS, VOLUME 5, PART B. . . . . . . .

ix

. . . . . . . . . xv . . . . . . xvii ,

.

1. Fundamental Principles and Methods of Particle Detection 1.1. Interaction of Radiation with Matter . . . . . . . by R. M. STERNHEIMER 1.1.1. Introduction . . . . . . . . . . . . . . . 1.1.2. The Ionization Loss d E / d x of Charged Particles 1.1.3. Range-Energy Relations . . . . . . . . . . 1.1.4. Scattering of Heavy Particles by Atoms . . . 1.1.5. Passage of Electrons through Matter. . . . . 1.1.6. Multiple Scattering of Charged Particles . . . 1.1.7. Penetration of Gamma Rays . . . . . . . .

.

.

. . . . .

. . . .

. . .

. .

1.2. Ionization Chambers. . . . . . . . . . . . . . . . . by ROBERTW. WILLIAMS 1.2.1. General Considerations. . . . . . . . . . . . . 1.2.2. Pulse Formation. . . . . . . . . . . . . . . . 1.2.3. Quantitative Operation and Some Practical Considerations . . . . . . . . . . . . . . . . . 1.2.4. Amount of Ionization Liberated. . . . . . . . . 1.2.5. Noise: Practical Limit of Energy Loss Measurable. 1.2.6. Some Types of Pulse Ionization Chambers . . . . 1.2.7. Current Ionization Chambers and Integrating Chambers. . . . . . . . . . . . . . . . . .

1 1

4 44 55 56 73 76 89 89 95 100 103 105 107 109

1.3. Gas-Filled Counters . . . . . . . . . . . . . . . . . 110 by ROBERTW. WILLIAMS 1.3.1. Gas Multiplication; Proportional Counters . . . . 110 1.3.2. Geiger Counters and Other Breakdown Counters . 118 1.4. Scintillation Counters and Luminescent Chambers . . . 120 by GEORGET. REYNOLDS and F. REINES 1.4.1. Scintillation Counters. . . . . . . . . . . . . . 120 1.4.2. Solid Luminescent Chambers . . . . . . . . . . 159

xi

xii

CONTENTS. VOLUME

5.

PART A

1.5. cerenkov Counters . . . . . . . . . . . . . . . . . and LUKEC. L. YUAN by S. J . LINDENBAUM 1.5.1. Introduction . . . . . . . . . . . . . . . . . 1.5.2. Focusing cerenkov Counters . . . . . . . . . . 1.5.3. Nonfocusing Counters . . . . . . . . . . . . . 1.5.4. Total Shower Absorption Cerenkov Counters for Photons and Electrons . . . . . . . . . . . . . 1.5.5. Other Applications . . . . . . . . . . . . . . .

162 162 168 186 189 191

1.6. Cloud Chambers and Bubble Chambers . . . . . . . . . 194 by W . B. FRETTER 1.6.1. Cloud Chambers . . . . . . . . . . . . . . . . 194 1.6.2. Bubble Chambers . . . . . . . . . . . . . . . 203 1.7. Photographic Emulsions . . . . . . . . . . by M . BLAU 1.7.1. Introduction . . . . . . . . . . . . 1.7.2. Sensitivity of Nuclear Emulsions . . . . 1.7.3. Processing of Nuclear Emulsions . . . . 1.7.4. Optical Equipment and Microscopes . . 1.7.5. Range of Particles in Nuclear Emulsions 1.7.6. Ionization Measurements in Emulsions . 1.7.7. Ionization Parameters . . . . . . . . 1.7.8. Photoelectric Method . . . . . . . . .

. . . . . 208

. . . .

. . . .

. . . .

. . . .

. . . . . . . .

. . . . . . . .

. 208 . 210 . 216 . 224 . 226 . 240 . 245 . 264

1.8. Special Detectors . . . . . . . . . . . . . . . . . . 265 1.8.1. The Semiconductor Detector . . . . . . . . . . 265 by S. S. FRIEDLAND and F. P. ZIEMBA 1.8.2. Spark Chambers . . . . . . . . . . . . . . . . 281 by BRUCECORK

.

2 Methods for the Determination of Fundamental Physical Quantities

2.1. Determination of Charge and Size . . . . . . . 2.1.1. Charge of Atomic Nuclei and Particles . . . 2.1.1.1. Rutherford Scattering . . . . . . . . 2.1.1.2. Characteristic X-ray Spectra . . . . . . by ROBLEY D. EVANS 2.1.1.3. Charge Determination of Particles in graphic Emulsions . . . . . . . . . by M . BLAU

. . . . 289

. . . .

289

. . . . 289

. . . . 293 Photo-

. . . . 298

CONTENTS, VOLUME

5,

PART A

...

XI11

2.1.2. Principal Methods of Measuring Nuclear Size. . . 307 by ROBERTHOFSTADTER 2.2. Determination of Momentum and Energy . . . . . . . 341 2.2.1. Charged Particles . . . . . . . . . . . . . . . 341 2.2.1.1. Measurement of Momentum. Electric and Magnetic Analysis . . . . . . . . . . . . . . . 341 by T. R. GERHOLM 2.2.1.1.4. Measurement of Momentum with Cloud Chambers or Bubble Chambers . . . . . . . . . 375 by W. B. FRETTER 2.2.1.1.5. Momentum Measurement in Nuclear Emulsions 388 by M. BLAU 2.2.1.2. Determination of Energy . . . . . . . . . . . 409 2.2.1.2.1. Energy Measurement with Ionization Chambers 409 by R. W. WILLIAMS 2.2.1.2.2. Scintillation Spectrometry of Charged Particles 41 1 by G. D. O’KELLEY 2.2.1.2.3. Measurement of Range and Energy with Cloud Chambers and Bubble Chambers. . . . . . 436 by W. B. FRETTER 2.2.1.3. Determination of Velocity . . . . . . . . . . 438 2.2.1.3.1. Time-of-Flight Method . . . . . . . . . . . 438 by LUKEC. L. YUANand S. J. LINDENBAUM 2.2.1.3.2. Measurement of Velocity . . . . . . . . . . 444 by W. B. FRETTER 2.2.1.3.3. Measurement of Velocity Using cerenkov Counters. . . . . . . . . . . . . . . . . 454 by LUKEC. L. YUANand S. J. LINDENBAUM 2.2.2. Neutrons. . . . . . . . . . . . . . . . . . . 461 2.2.2.1. Recoil Techniques for the Measurement of Neutron Flux, Energy, Linear and Spin Angular Momentum . . . . . . . . . . . . . . . . 461 E. BROLLEY, JR. by JOHN 2.2.2.2. Time- of-Flight Method . . . . . . . . . . . . 495 by W. W. HAVENS, JR. 2.2.2.3. Crystal Diffraction. . . . . . . . . . . . . . 566 by D. J. HUGHES

xiv

CONTENTS, VOLUME

5,

PART A

2.2.2.4. Determination of Momentum and Energy of Neutrons with He3 Neutron Spectrometer. . . . . 570 by G. C. MORRISON 2.2.3. Gamma-Rays . . . . . . . . . . . . . . . . . 582 2.2.3.1. Internal and External Conversion Lines . . . . 582 by T. R. GERHOLM 2.2.3.2. Determination of Momentum and Energy of Gamma Rays with the Curved Crystal Spectrometer . . . . . . . . . . . . . . . 599 by J. W. M. DUMOND 2.2.3.3. Gamma-Ray Scintillation Spectrometry . . . . . 616 by G. D. O’KELLEY 2.2.3.4. Determination of the Momentum and Energy of Gamma Rays with Pair Spectrometers . . . . . 641 by D. E. ALBURGER 2.2.3.5. Shower Detectors. . . . . . . . . . . . . . . 652 by R. HOFSTADTER 2.2.3.6. Gamma-Ray Telescopes. . . . . . . . . . . . 668 by A. SILVERMAN 2.2.3.7. Measurement of y-Ray Energy by Absorption. . 671 by ROBLEYD. ZVANS 2.2.3.8. Detection and Measurement of Gamma Rays in Photographic Emulsions. . . . . . . . . . . 676 by M. BLAU 2.2.4. Neutrino . . . . . . . . . . . . . . . . . . . 682 2.2.4.1. Neutrino Reactions. . . . . . . . . . . . 682 by F. REINES

AUTHORINDEX . . . . . . . . . . . . . . . . . . . . . . .

699

SUBJECTINDEX . . . . . . . . . . . . . . . . . . . . . . .

718

CONTRIBUTORS TO VOLUME 5, PART B E. AMBLER,Low Temperature Section, National Bureau of Standards, Washington, D.C. F. AJZENBERO-SELOVE, Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania M. BLAU,Institut fur Radiumforschung, Vienna, Austria M. H. BLEWETT, Brookhaven National Laboratory, Upton, New Yorlc 0. CHAMBERLAIN, Department of Physics, University of California, Berkeley, California B. CORK,Lawrence Radiation Laboratory, University of California, Berkeley, California H. DANIEL, M a x Planck Institute for Nuclear Physics, Heidelberg, Germany M. DEUTSCH, Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts H. E. DUCKWORTH, Department of Physics, McMaster University, Hamilton, Ontario, Canada R. D. EVANS,Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts G. FEHER,Department of Solid State Physics, University of California, L a Jotla, California H. FRAUENFELDER, Department of Physics, University of Illinois, Urbana, Illinois W. B. FRETTER, Department of Physics, University of California, Berkeley, California W. GENTNER,M a x Planck Institute for Nuclear Physics, Heidelberg, Germany S. GESCHWIND, Bell Laboratories, Murray Hill, New Jersey J. G. HIRSCHBERG, Project Matterhorn, Princeton University, Princeton, New Jersey J. C. HUBBS,E-H Research Laboratories, Inc., Oakland, California C. D. JEFFRIES, Department of Physics, University of California, Berkeley, California J. K. JEN, Applied Physics Laboratory, The Johns Hopkins University, Silver Spring, Maryland H. W. KOCH,Betatron Laboratories, National Bureau of Standards, Washington, D. C. xv

xvi

CONTRIBUTORS TO VOLUME

5,

PART B

H. KOUTS,Brookhaven National Laboratory, Upton, N e w York D. W. MILLER,Department of Physics, Indiana University, Bloomington, Indiana W. A. NIERENBERG, Department of Physics, University of California, Berkeley, California G. D. O’KELLEY, Oak Ridge National Laboratory, Oak Ridge, Tennessee L. ROSEN,Department of Physics, Los Alamos Laboratory, Los Alamos, New Mexico R. M. STERNHEIMER, Brookhaven National Laboratory, Upton, New York A. N. WAPSTRA,Institute voor Kernphysisch Onderzoek, Amsterdam, Holland W. WHALING, Department of Physics, California Institute of Technology, Pasadena, California

CONTENTS, VOLUME 5, PART B 2.3. Determination of Mass of Nuclei and of Individual Particles 2.3.1. Mass Spectroscopy by H. E. DUCKWORTH 2.3.2. Nuclear Disintegration Energies and Reaction Values by W. WHALING 2.3.3. Microwave Method by S. GESCHWIND 2.3.4. Cloud Chambers by W. B. FRETTER 2.3.5. Photographic Emulsions by M. BLAU 2.4. Determination of Spin, Parity and Nuclear Moments 2.4.1. Spectroscopic Methods 2.4.1.1. Optical and Ultra-Violet Spectroscopy by J. G. HIRSCHBERG 2.4.1.2. Atomic Beam by W. NIERENBERG and J. C. HUBBS 2.4.1.3. MICROWAVE METHOD by J. K. JEN 2.4.1.4. Nuclear Magnetic and Quadrupole Resonances by C. D. JEFFRIES and G. FEHER 2.4.2. Indirect Methods 2.4.2.1. Angular Correlation by H. FRAUENFELDER 2.4.2.2. Conversion Coefficients by A. N. WAPSTRA 2.4.2.3. Oriented Nuclei by E. AMBLER 2.5. Determination of Polarization of Electrons and Circular Polarized Photons by H. FRAUENFELDER

xvii

xviii

CONTENTS, VOLUME

5,

PART B

2.6. Determination of Life-Time 2.6.1. Long Life-Time by W. GENTNER and H. DANIEL 2.6.2. Short Life-Time by M. DEUTSCH 2.7. Determination of Nuclear Reactions 2.7.1.Determination of the Q-Value for Nuclear Reactions 2.7.2.Determination of Nuclear Energy Levels from Reaction Energies by F. AJZENBERG-SELOVE 2.7.3.Total Interaction Cross Sections 2.7.4. Differential Interaction Cross Sections 2.7.5.Elastic Cross Sections 2.7.6.Inelastic Cross"Sections 2.7.7.Nuclear Production Cross Sections by L. ROSENand D. W. MILLER 2.8. Determination of Flux Densities 2.8.1.Flux of Charged Particles by 0. CHAMBERLAIN 2.8.2.Flux of Photons by €1. W. KOCH 3. Sources of Nuclear Particles and Radiations

3.1. Natural Sources-Radioactivity by G. D. O'KELLEY

3.2. Artificial Sources 3.2.1.Low Energy Sources 3.2.1,l.Cascade Transformer 3.2.1.2.Van de Graaff by H. BLEWETT 3.2.1.3.Nuclear Reactor 3.2.1.4.Neutron Sources by H. KOUTS 3.2.2.Medium and High Energy Sources 3.2.2.1. Linear Accelerator 3.2.2.2.Cyclotron 3.2.2.3.Betatron

CONTENTS, VOLUME

5,

PART B

3.2.2.4. Synchrotron 3.2.2.5. Synchrocyclotron 3.2.2.6. Proton Synchrotron 3.2.2.7. Alternate Gradient Synchrotron 3.2.2.8. Fixed Field Alternate Gradient Synchrotron by H. BLEWETT

4. Beam Transport System by R. STERNHEIMER and B. CORK 5. Statistics by R. C’ VANS Appendix 1. System of Units 2. Kinematics by R. STERNHEIMER

AUTHORINDEX

SUBJECTINDEX

xix

This Page Intentionally Left Blank

1. FUNDAMENTAL PRINCIPLES AND METHODS OF PARTICLE DETECTION

1 .l. Interaction of Radiation with Matter*

7

1.1 .l. Introduction In this chapter, we shall discuss the various processes which take place when charged particles and y radiation pass through matter. For any type of charged particle (proton, meson, electron, etc.), there will be a loss of energy as the particle traverses the material, due to the excitation and ionization of the atoms of the medium close to the path of the particle. The loss of energy per cm of path, dE/dx, is generally referred to as the ionization loss. In Section 1.1.2, we give a simplified derivation of the theoretical expression for dE/dx, the well-known Bethe-Bloch formula, including a discussion of the density effect which becomes important at high energies. The ionization loss of a fast charged particle is frequently used as a means of identifying the particle, by observing its track in a cloud chamber, bubbIe chamber, or in photographic emulsion. The ionization loss dE/dx is a function onIy of the veIocity v of the particle (for a given charge), so that a simultaneous measurement of dE/dx and of the momentum p enables one to determine the mass m of the particle. The ionization loss can also be used to determine approximately the energy of the particle, if its identity has been established by other methods. A further important property of the ionization process is that the energy w required to form an ion pair in a gas is approximately independent of the energy and the charge of the incident particle, so that when a particle is stopped in a gas, a measurement of the total number of ion pairs enables one to obtain the energy of the incident particle, provided that the value of w for the stopping gas is known. This property has been widely used in the operation of ionization chambers. 1 In Section 1.1.2, expressions for dE/dx are given for various cases, together with a discussion of the fluctuations of the ionization loss (Landau effect). The recent experiments on the ionization loss of relativistic charged particles will be discussed in some detail. For particles heavier than electrons (e.g., protons, K , T , or p mesons), the ionization loss dE/dx is the most important mechanism of energy loss. As a result, a particle with a given incident kinetic energy T will have a quite well-defined range R, which depends on T, on the mass m and on the

t See also, Vol. 4, B, Parts 6, 7,and 8.

1See also this volume, Chapter

1.2.

* Chapter 1.1 is by R. M. Sternheimer. 1

2

1.

PARTICLE DETECTION

charge z of the particle, as well as on the stopping substance. The relation between R and T is known as the range-energy relation. Tables of the range-energy relation for protons of energies T , = 2 Mev to 100 Bev have been recently calculated by Sternheimer for the following materials : Be, C, All Cu, Pb, and air. These range-energy relations differ from the results of Aron et a1.2 in two respects: (1) the density effect correction is included a t the higher energies ( T , 2 2 Bev); (2) recent values of the mean excitation potential I (which enters into the Bethe-Bloch formula) have been used, which are somewhat higher than the value I = 11.52 ev employed by Aron et al. The tables of the range-energy reIations are given in Section 1.1.3, together with a table of the values of dE/dx which were used in the calculation of R ( T ) . Section 1.1.3 also includes a brief discussion of the range straggling. Section 1.1.4 gives various formulas pertaining to the scattering of heavy particles (heavier than electrons) by atoms. When electrons pass through matter, they lose energy by ionization in the same manner as any charged particle (see Section 1.1.2). However, in addition, a high-energy electron will produce electromagnetic radiation (bremsstrahlung) in the field of the atomic nuclei.* For electrons above the critical energy E , (e.g., 47 Mev for All 6.9 Mev for Pb), the energy loss due to radiation exceeds the ionization loss, and constitutes the predominant mechanism for the slowing down process. The y quanta from the bremsstrahlung can create electron-positron pairs, which in tu r n can produce additional y rays. The resulting electromagnetic cascade is called a shower and has been widely observed in cloud-chamber pictures both with incident electrons and y rays. Th e theoretical expressions for the bremsstrahlung and a discussion of shower production are presented in Section 1.1.5. The multiple scattering of charged particles is considered briefly in Section 1.1.6. The penetration of y rays through matter is characterized by a n absorption coefficient r which determines the exponential attenuation of the y ray beam. The processes which contribute to r are the photoelectric effect, the Compton scattering, and the pair production. A summary of the theoretical expressions for these three processes is given in Section 1.1.7. The discussion of Sections 1.1.4-1.1.7 follows closely the review article

* See also Vol. 4 , A, Section

1.5.2. R. M. Sternheimer, Phys. Rev. 116, 137 (1959). 2 W. A. Aron, B. G. Hoffman, and F. C. Williams, University of California Radiation Laboratory Report UCRL-121 (1 951); Atomic Energy Commission Report AECU-663 1

(1951).

1.1.

INTERACTION O F RADIATION W I T H MATTER

3

by Bethe and Ashkin3 on the “Passage of Radiations through Matter.” I n 1956, in order to solve certain difficulties connected with the decay of the strange particles (particularly the K meson), Lee and Yang4 discussed the consequences of a possible nonconservation of parity in the weak interactions (beta decay, strange particle decay, a- and p-meson decay). They suggested a number of experiments to test this hypothesis. These experimentss7 were performed soon after the publication of their paper, and have shown very clearly that parity is not conserved in the weak (decay) interactions, in contrast to the strong interactions which conserve parity to a high accuracy. An important consequence of parity nonconservation is th at the electrons (or positrons) from the beta decay of unpolarized nuclei should be strongly longitudinally polarized, i.e., the electron spin should be aligned predominantly antiparallel to the electron direction of motion, while for positron decays, the positron spin should be aligned predominantly parallel to the positron direction of motion. The magnitude of the polarization P is predicted to be v/c in each case, where v is the velocity of the particle (electron or positron). Thus for relativistic electrons or positrons, P should be essentially 100%. It should be noted that the prediction that P = v / c follows only from a particularly simple theory of parity nonconservation, namely the two-component theory of the neutrino. I n a separate article,*we have given a discussion of the proposals of Lee and Yang4 concerning parity nonconservation in weak interactions. This article also contains a description of the crucial experiments of Wu et d 6on the beta decay of oriented nuclei (Co60), and of Garwin and co-workersO on the polarization of the p+ from a+ decay, which together with the work of Friedman and Telegdi,7 were the first experiments that demonstrated the violation of parity conservation in weak interactions. We have also summarized*the two-component theory of the neutrino, which was proposed independently by Lee and Yang,g Landau,lo and Salam.” A large number of experiments have been performed to establish the longitudinal polarization of the electrons and positrons from beta decay. 3 H. A. Bethe and J. Ashkin, Passage of radiations through matter. In “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. 1, p. 166. Wiley, New York, 1953. 4 T. D. Lee and C . N. Yang, Phys. Rev. 104, 254 (1956). 6 C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson, Phys. Rev. 106, 1413 (1957). 6 R. L. Garwin, L. M. Lederman, and M. Weinrich, P h p . Rev. 106, 1415 (1957). 7 J. I. Friedman and V. L. Telegdi, Phys. Rev. 106, 1681 (1957). 8 R. M. Sternheimer, Advances in Electronics and Electron Phys. 11, 31 (1959). 9 T. D. Lee and C . N. Yang, Phys. Rev. 106, 1671 (1957). 10 L. D. Landau, Nuclear Phys. 8, 127 (1957). 11 A. Salam, Nuovo n’mento 1 101 6,299 (1957).

4

1. PARTICLE

DETECTION

These investigations involve a variety of methods to determine the longitudinal polarization : scattering of the polarized electrons on nuclei (Mott scattering) ; scattering on polarized electrons (ferromagnetic 3d electrons of iron in a magnetic field), which is often referred to as Mdler scattering; circular polarization of the bremsstrahlung emitted by the polarized electrons; and annihilation of the polarized positrons in various materials. The experiments have in turn led to important developments of the theories presented in Sections 1.1.4 and 1.1.5 on the scattering and interaction of electrons in matter. These new theoretical results, as well as a review of the experiments on the longitudinal polarization, are presented in the latter part of the article on parity nonconservation." 1.1.2. The Ionization Loss dE/dx of Charged Particles 1.1.2.1. The Bethe-Bloch Formula. The theoretical expression for dE/dx is based on the Bethe-Bloch formula, which has been derived from the work of Bohr,12Bethe,13 Bloch,I4 and others. The Bethe-Bloch formula for particles heavier than electrons is given by

dx

-2p-6-

u

]

(1.1.1)

where n = number of electrons per cma in the stopping substance, m = electron mass, p = v/c, where v = velocity of the particle, z = charge of the particle, I = mean excitation potential of the atoms of the substance, Wmsx= maximum energy transfer from the incident particle to the atomic electrons, 6 is the correction for the density effect, which is due to the polarization of the medium, as will be discussed below, and U is a term due to the nonparticipation of the inner shells ( K ,L, .) for very low velocities of the incident particle. This term is generally called the shell correction term, and will be discussed below [see Eq. (1.1.34)l. The maximum energy transfer W,, is given by

..

Wmsx= 2mv2/(1 - p 2 )

( I .1.2)

for energies E (mi2/2m)c2], W,,, increases approximately a s (1 - p2)-1/2. Indeed, Vmax approaches the value E - (mi2/2m)c2,a s is shown by the following general formula due to Bhabha.23

where E is the total energy of the particle (including the rest mass). Thus in the extreme relativistic region, the logarithm of (1.1.1) has a term - ln(1 - p2)3/2,whereas 6 has a term - ln(1 - p2), so that the relativistic rise is only one-third as large as it would be without the density effect. 1.1.2.3. Energy Loss due to eerenkov Radiation. It should be noted that the relativistic rise includes the energy loss due to Cerenkov radiation. The Cerenkov loss is given by the formula of Frank and T t ~ m m . ~ ~ (1.1.16) where the integral extends over the frequencies v for which pn > 1, and where n(v)is the index of refraction of the medium. The Cerenkov loss is 0. Halpern and H. Hall, Phys. Rev. 67, 459 (1940); 73, 477 (1948). G. C . Wick, Ricerca sci. 11, 273 (1940); 12, 858 (1941); Nuovo cimento [9] 1, 302 (1943). 20R. M. Sternheimer, Phys. Rev. 88, 851 (1952); 91, 256 (1953); 93, 351, 1434 (1954). 2 1 R. M. Sternheimer, Phys. Reu. 103, 511 (1956). 22 P. Budini, Nuovo cimento [9] 10, 236 (1953). 23 H. J. Bhabha, Proc. Roy. SOC. A164, 257 (1937). Z4 I. Frank and I. Tamm, Compt. rend. acad. sci. U.R.S.S. 14, 109 (1937). l8

l9

1.1.

INTERACTION OF RADIATION WITH MATTER

9

zero at low energies, and increases to a small saturation value in the region of the relativistic r i ~ e . ~ " The ~ ' magnitude of (1.1.16) is in all cases small compared to the magnitude of the relativistic rise. This result arises from the fact that there is a large number of absorption lines and cont i n ~ a , ~corresponding ~-~~ to excitation of the electrons in the K , L, M . . . shells, except for the very light atoms (H, He) where, however, there is still a wide spectrum of absorption frequencies corresponding to the continuum above the discrete spectrum for excitation from the 1s shell. As a result, the expression for the atomic polarizability C Y ( V ) contains a large number of terms, one term for each absorption frequency (discrete line spectrum or excitation to the continuum). For such a behavior of C Y ( V ) , the index of refraction %(.) is less than 1 over a considerable region of V . As a result the region of integration of (1.1.16) is considerably restricted, and actually the only region which makes an important contribution to (dE/dx)c is the region of v below the first absorption limit, i.e., effectively the optical and near-ultraviolet region, as has been shown by Sterr~heimer.~~ It should also be noted that the width of the spectral lines gives rise to a strong absorption of the cerenkov radiation for values of v close to the frequencies of the atomic transitions, thus resulting in a further reduction of the Cerenkov energy loss. For condensed materials, - (l/p)(dE/dx)c is of the orderz0 of lop3 Mev/g cm-2 and hence completely negligible compared to the total ionization loss (> 1 Mev/g cm-2).31 For gases, the cerenkov loss is somewhat more important,29of the order of 0.1 Mev/g for Hz and He, and 0.01 Mev/g cm-2 for gases with medium and large Z . Even for H,, the cerenkov loss accounts for only -15% of the relativistic rise. Comprehensive treatments of the stopping power problems in dense materials, including the Cerenkov radiation, have been recently given by fan^,^^ Budini and Taff Bra,3 5 and Tidman. 3 4 The Bethe-Bloch formula, Eq. ( l . l . l ) , includes the cerenkov loss. l7 Thus, in order to obtain the energy -(dE/dx)d deposited close to the path of the particle, it is necessary in principle to subtract - (dE/dx)c. A. Bohr, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd. 24, N o . 19 (1948). H. Messel and D. M. Ritson, Phil. Mag. [7] 41, 1129 (1950). 17 M. Schoenberg, Nuovo cimento [9] 8,159 (1951); 9,210,372 (1952); M. Huybrechts and M. Schoenberg, ibid. 9, 764 (1952). 28 P. Rudini, Phys. Rev. 89, 1147 (1953). 99 R. M. Sternheimer, Phys. Rev. 89, 1148 (1953); 91, 256 (1953); 93, 1434 (1954). 80 G. N. Fowler and G. M. D. B. Jones, Proc. Phys. SOC.(London) A66, 597 (1953). 81 See also J. R. Allen, Phys. Reu. 93, 353 (1954). 84 U. Fano, Phys. Reu. 103, 1202 (1956). 88 P. Budini and L. Taffara, Nuouo cimenlo [lo] 4, 23 (1956). *4 D. A. Tidman, Nuclear Phys. 2, 289 (1956); 4, 257 (1957). 26 24

10

1.

PARTICLE DETECTION

We have (1.1.17) However, as pointed out above, (dE/dx) c is generally negligible compared to the relativistic rise of dE/dx, except for gases of low 2 (H2, He). While the energy loss due to Cerenkov radiation is very small compared to the total ionization loss, the Cerenkov effecta6has received an important application in the design of Cerenkov counters," which are based on the property that the Cerenkov radiation is absent unless the velocity of the particle exceeds a critical value vc determined by v,no/c

=

(1.1.18)

1

where no is the index of refraction of the radiating substance (usually a liquid) in the optical region. The Cerenkov counter is used as a velocity selector, and together with a momentum measurement, it enables one to identify charged particles by placing an upper or lower limit on the mass. 1.1.2.4. Evaluation of dE/dx. The M e a n Excitation Potential 1. Equation (1.1.1) can be written in the following form: Pdx:

P2

B

P + In Wmax,Mev + 0.69 + 2 In mic - 2p2 - 6 - U -

(1.1.19) where p is the density of the medium in g/cm3, so that -(l/p)(dE/dz) gives the energy loss in Mev/g cm-2; A and 3 are defined by:

A = 21rnx2e4/(mc2p) 3 = 1n[mc2(1OE ev)/P].

(1.1.19a) (1.1.19b)

I n Eq. (1.1.19), Wmax,Mev is the value of Wmax[Eqs. (1.1.2), (1.1.15)] in MeV. In order to obtain - (l/p)(dE/dz) in Mev/g cm-2, one must take A as: 0.1536(Z/A0)x~,where 2 = atomic number, A0 = atomic weight of stopping substance. The following expression for 6 has been obtained by Sternheimer :20,21 6 = 4.60GX 6 = 4.606X

+ C + U ( X I- X)' +C

(Xo < X < X i ) (1.1.20) ( X > Xl) (1.1.20a)

where X = loglo(p/m,c), X Oand X I are particular values of X which depend on the substance. X Ois the value of X below which 6 is zero; XI is the value of X above which the high-energy expression, Eq. (1.1.20a), applies.

* See also this volume,

Chapter 1.5. A review of the applications of the eerenkov effect has been given by J. V. Jelley, Progr. in Nuclear Phys. S, 84 (1953). as

1.1.

11

INTERACTION O F RADIATION WITH MATTER

In the region X > XI, the ionization loss becomes independent of the excitation potential I , as has been first shown by Fermi.I7 I n Eqs. (1.1.20) and (1.1.20a), a, s, wd C are constants which depend on the substance and on the value chosen for I . C is defined as

C = -2 ln(I/hv,)

-

1.

(1.1.21)

The mean excitation potential I has been the subject of numerous investigations. I n 1933, Bloch14showed that, on the basis of the ThomasFermi model, I should be proportional t o the atomic number: I = kZ,but he could not determine theoretically the value of the proportionality constant lc, which therefore has to be obtained from experiment. An early determination by Bethe36of I for air from the range-energy relation of (Y particles gave I,, = 80.5 ev. I n 1940, Wilson3? obtained a value for aluminuni,IA1= 150 ev. Both of these results giveI/Z 11.5 ev. I n 1951, Bakker and Segr&38 measured dE/dx for 340-Mev protons in a number of materials, and obtained values of I of the order of -92 - 1 0 2 ev for heavy elements. Measurements of the ranges of 340-Mev protons by Mather and Segr&39 led to similar values of I . For Al, Mather and Segr83g = 148 i -3 ev, in good agreement with Wilson’s37earlier result. found IA1 On the other hand, Sachs and Richardson40 from a determination of the absolute stopping power for 18-Mev protons in A1 obtained a substjailtially namely I , = 168 k 3 ev. For heavy elements Sachs higher value for IAl, and Richardson obtained I ,- 1 4 2 - 182 ev, which is also considerably higher than the results of Bakker and Segr&38From measurements of the range in A1 of protons of various energies from 35 to 120 MeV, Bloembergen and van Heerden4l deduced a value I A 1 = 162 & 5 ev. A similar measurement for 18-Mev protons in A1 by Hubbard and M a ~ K e n z i e ~ ~ gave I A I = 170 ev.43 I n 1955, C a l d ~ e 1 1recalculated ~~ the values of I from the data of Sachs and Richardson40 with the inclusion of the low-energy shell corrections [ C , and CL, see Eq. (1.1.34)]. Th e resulting values of I are somewhat smaller than those originally obtained by Sachs and Richardson, but are still considerably above the Bakker-Segr& values. Thus for Al, C a l d ~ e 1 1 ~ ~ found = 163 ev, and for the heavy elements, I,- 132 - 1 4 2 ev. M. S. Livingston and H. A. Bethe, Revs. Modern Phys. 9, 261 (1937).

R. R. Wilson, Phys. Rev. 60, 749 (1941).

** C . J. Bakker and E. SegrB, Phys. Rev. 81,489 (1951). R. L. Mather and E. SegrB, Phys. Rev. 84, 191 (1951). D. C . Sachs and R. J . Richardson, Phys. Rev. 83,834 (1951); 89, 1163 (1953) 41 N. Bloembergen and P. J. van Heerden, Phys. Rev. 83, 561 (1951). 4 * E. L. Hubbard and K. R. MacKenzie, Phys. Rev. 86, 107 (1952). 43 See also D. H. Simmons, Proc. Phys. Sac. (London)A66, 454 (1952). 4 4 D. 0. Caldwell, Phys. Rev. 100, 291 (1955). 40

12

1.

PARTICLE DETECTION

Caldwell also showed that the various experiments are generally consistent with I values which are independent of the velocity of the incident particle. 4 6 This result was important, since it had bee%previously believed that I might be velocity-dependent, in order to reconcile data from different experiments. 4 6 Recently, two accurate determinations of I have been made from measurements of the range and stopping power of low-energy protons ( s 2 0 Mev). From measurements of the range of protons of various energies from 6-18 MeV, Bichsel et aL4’ have obtained the following I values for Be, Al, Cu, Ag, and Au: I B e = 63.4 f 0.5 ev, I A I = 166.5 k 1 ev, Icu = 375.6 k 20 ev, I A g = 585 k 40 ev, and I A u = 1037 k 100 ev. The result for Be confirms an earlier determination by Madsen and V e n k a t e ~ w a r l uwho , ~ ~ obtained IBe= 64 f 5 ev. The large value of I / Z for Be (1/Z i X 16) had been previously predicted by A. B ~ h (who r ~ ob~ tained I = 60 ev) on the basis of polarization effects caused by the presence of the two conduction electrons per atom. For Al,Cu, Ag, and Au, 13 ev, which is somewhat smaller the I values of Bichsel et al. give I / Z than Caldwell’s results,44but is considerably higher than the values of I / Z obtained by Bakker and Segr&.@ The other recent determination of I values has been made by Burkig and M a ~ K e n z i eThese . ~ ~ authors measured the stopping powers relative to aIuminum of a number of metals of 19.8-Mev protons. The resulting values The values of I of I are based on the value I A I = 166.5 ev of Bichsel et of Burkig and MacKenzie for Be, Cu, Ag, Au, and P b are: IBe= 64 ev, Icu = 366 ev, I,, = 587 ev, IAu = 997 ev, and I,, = 1070 ev. These values of I are in good agreement with the results of Bichsel et aL4’ In 1959, Zrelov and stole to^^^" measured the range R in copper of 660-Mev protons from the Dubna synchrocyclotron. These authors have obtained a value R = 257.6 f 1.2 gm/cm2, which leads to a calculated mean excitation potential ICu= 305 f 10 ev ( I / Z = 10.5 k 0.3 ev). This value of Icuis appreciably smaller than the values obtained by Bichsel et al.47and by Burkig and M a c K e n ~ i eat~ ~lower energies (6-20 Mev). Zrelov and stole to^^^^ have also determined the stopping power relative to copper for H, Be, C, Fe, Cd, and W for 635-Mev protons. For H, Be, and

-

See also W. Brandt, Phys. Rev. 104,691 (1956);111,1042(1958);112,1624(1958). M. Scharff, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd. 27, No. 15 (1953). 47 H. Bichsel, R. F. Mozley, and W. A. Aron, Phys. Rev. 106, 1788 (1957). 48 C. B. Madsen and P. Venkateswarlu, Phys. Rev. 74, 648 (1948). 49 V. C.Burkig and K. R. MacKenzie, Phys. Rev. 106,848 (1957). 4OeV. P. Zrelov and G. D. Stoletov, Zhur. Eksptl. i Teoret. Fiz. 86, 658 (1959): [translation: Soviet Phys. J E T P 9,461 (1959)l. 45

teJ. Lindhard and

1.1.


13

C, the resulting values of I / Z are -14-15 ev (IH = 15, I B e = 61 f 6, I0 = 85 f 8 ev). The value of I B e is in good agreement with the results of earlier experiment^.^'-^^ On the other hand, for Cd and W, the values of I / Z are 9.8 and 9.2 ev, respectively, indicating that for heavy elements, the value of I / Z may be appreciably lower than the results (-13 ev) of references 47 and 49. The values of the constants a, s, and C for the density effect correction 6 which are given in reference 20 were based on the Bakker-Segr838values of I , whereas the results for 6 of reference 21 were obtained by means of . ~ ~denote these values of I by I1 and Iz, the higher I values of C a l d ~ e 1 1We respectively. The values of 6 for any intermediate 1 value, I = I O (e.g., that of Bichsel et aZ.47),can be obtained by logarithmic interpolation, as follows. Let 61 and 8 2 denote the values of 6 pertaining to I I and 12, respectively.20.21Then 60 appropriate to I = I0 is given by 60 =

where q is defined by

761

+ (1 -

7162

(1.1.22) (1.1.23)

Of course, Eqs. (1.1.22) and (1.1.23) apply also, within reasonable limits, if I0 is outside the range (IJZ), i.e., for I0 < I1 or I 0 > Iz. When the stopping material is a compound containing several atomic species, the stopping powers of the individual elements are approximately additive (Bragg’s rule). Thus Eq. (1.1.1) still holds for the compound, provided that the mean excitation potential I in this equation is replaced by the following average potential 1: (1.1.24) with (1.1.25) Here n, is the number of atoms of the ith type in the compound, with atomic number Z, and excitation potential Ii; fi of Eq. (1.1.25) is the oscillator strength of the atomic electrons belonging to the ith species. The density effect correction 6 of Eq. (1.1.1) must be replaced by 8 defined bv (1.1.26) i

where 6i is the value of 6 for the ith constituent.

14

1.

PARTICLE DETECTION

In order to obtain - (l/p)(dE/dz) in Mev/g cm-2, the constant A of Eq. (1.1.19a) must be taken as follows:

(1.1.27)

where Ai is the atomic weight of the i t h element. Extensive experiments have been carried out by Thompsons0t o verify the additivity of the stopping powers for a number of organic compounds (containing C, H, N, 0, and Cl). I n these experiments, the 340-Mev protons from the Berkeley cyclotron were slowed down to 200 MeV. It was found that the stopping powers of the constituent elements in the compound are additive to within 1%. Small deviations (less than 1%) from additivity were observed; these were attributed to the influence of the chemical binding. Thompson also obtained values of I for the elements C, H, N, and 0, using liquid targets for H, N, and 0. 1.1.2.5. The Restricted Ionization Loss, - (l/p)(dE/dz)~,. Equation (1.1.1) or (1.1.19) gives the average energy loss of the charged particle. These expressions include the possibility of large energy transfers, up to the maximum value W,,, [Eqs. (1.1.2), (1.1.15)]. I n certain applications, one is, however, interested in the restricted energy loss with energy This is true in particular for transfers less than a certain fixed value WO. the grain count in nuclear emulsion* and for the drop count of tracks in cloud chambers. t For the grain count in emulsion, the relevant quantity is the ionization loss with energy transfers less than W O 5 kev,26because larger energy transfers generally result in the development of neighboring grains not directly in line with the track, so that the grain count along the track is no longer proportional to the complete ionization loss. A similar phenomenon takes place for cloud chamber tracks. For energy transfers which are larger than -1 kev, a cluster of drops is formed, so that the drop count along the track is not proportional to the complete ionization loss. The restricted energy loss, with maximum energy transfer W o is , given by

-

* See also this volume,

Chapter 1.8.

t See also this volume, Chapters 1.6 and

1.7. T. Thompson, University of California Radiation Laboratory Report UCRL-1910 (1952). See also T. Westermark, Phys. Rev. 93, 835 (1954). 60

1.1.

15


or in terms of the constants A and B

P

p)

dx wo

=

$ [B + 0.69 + 2 In P + In -

wO,,,

- /Y

-6-u

mic

1

(1.1.29) where W0,Mevis the value of W Oin MeV. Whereas the average ionization loss - ( l / p ) ( d E / d z ) continues to rise the indefinitely with increasing energy (due to the increase of W,,,), restricted energy loss - ( l / p ) (dE/dz) W , levels off to a constant value at high energies, which is generally referred to as the Fermi plateau. This I I1111111

I11111111

I

11111111

I11111111

I Ill1

72

fi MESON MOMENTUM (IN Mev/c)

FIG.1. The ionization loss of p mesons i n 02 as a function of the p-meson momentum. The solid curve represents the restricted energy loss, - ( l / p ) ( d E / d x ) w ,with W o= 1 kev, as obtained from Eq. (1.1.28). The crosses represent the experimental data of Ghosh et aL61The theoretical curve and the experimental drop count (number of ion pairs per cm) have been normalized a t the minimum of ionization.

result arises from the fact that the logarithmic term in 6 exactly cancels the effect of the (1 - ,@) denominator in Eq. (1.1.28), as has been discussed above. An example of the relativistic increase of - (l/p) (dE/dz)w,, and the Fermi plateau at very high energies is presented in Fig. 1, which shows the restricted energy loss of p mesons in oxygen (at normal pressure) for an assumed value Wo = 1 kev. A measure of the relativistic increase is given by the ratio R = Jplat/Jmin, where JP1,, and Jmi,are the values of - (l/p)(dE/dz)~,in the plateau region and at the minimum of ionization 61 S. K. Ghosh, G. M. D. B. Jones, and J. G. Wilson, Proc. Phys. SOC.(London) A66, 68 (1952); A67, 331 (1954).

16

1.

PARTICLE DETECTION

(v = 0.96c), respectively. As is seen from Fig. 1, for 0 2 , with W O= 1 kev, = 1.08 Mev/g cm-2, so that R = 1.51. we have Jplat= 1.63, Jmin Figure 1 also s h o w the data of Ghosh et aLblobtained from the drop count in an expansion cloud chamber filled with oxygen a t normal pressure (see Section 1.1.2.9). The theoretical curve and the experimental data have been normalized at the minimum of ionization (Jmi.= 44 ion pairs/cm). The equivalent number of ion pairs is indicated on the right-hand scale.

'*4"

- 1.2

z 2

1.11 10'

' " t l l l l l

"111111'

'

1

111111

' ' I J I I I '

lo3 lo" 10' lo6 p MESON MOMENTUM (IN Mev/c)

' " ' 1 ' 1 '

10'

FIG.2. The ionization loss of p mesons in He as a function of the p-meson momentum. The solid curve represents the restricted energy loss, -(l/p)(dE/dx) wO1with W O= 1 kev. The dashed curve shows the energy deposited along the track, -(l/p)(dE/dx)d, after subtraction of the estimated energy escape due to Cerenkov radiation, - (I/p) ( d E / d s ) c [see Eq. (1.1.17)]. The flat part of the curves a t very high momenta ( >lo5 Mev/c) is often referred to as the Fermi plateau.

The theoretical curve is in good agreement with the data, except a t the highest momenta of the experiment ( p , 2 10 Bev/c), where the data give an indication that the relativistic rise may be somewhat smaller than predicted by the theory. However, the uncertainties of the measurements make it impossible to decide at present whether there is a real discrepancy. The Cerenkov loss ( d E / d z ) c may also be partly responsible for the apparent disagreement, since it reduces the energy deposited along the track (dE/dz)d. However, - (l/p)(dE/dx)c is expected to be quite small for oxygen (-0.02 Mev/g ern+ in the region of the Fermi plateau29). Figure 2 shows the relativistic rise in helium at normal pressure. In this = 1.22, Jplst = 1.79 Mev/g so that R = 1.47. We have case, Jmin made an estimate of the cerenkov energy ~ o s sand , ~ ~the dashed curve of

1.1.

17


Fig. 2 showsthe resultant energydepositedalong the track - (l/p) (dE/dz)d. The difference between the solid and the dashed curves represents the cerenkov loss, - (l/p)(dE/dz)~[see Eq. (1.1.17)]. The relativistic rise of - (l/p) (dE/dz)w o in gases has been observed in a large number of experiments. A summary of these experimental investigations is given in Section 1.1.2.9. 1.1.2.6. The Most Probable Ionization Loss Eprob. Fluctuations of the Energy Loss. As has been shown by Williams,62Landau,63and others, the ionization loss in a thin absorber is subject to appreciable fluctuations, because of the statistical nature of the ionization process. The energy loss in a thin absorber has a considerable spread about the most probable value Eprob. This spread is often referred t o as the Landau effect, since Landau was the first to calculate the expected distribution of the energy losses. Further contributions to this problem have been made by S ~ r n o nand ,~~ by Blunck and L e i ~ e g a n gThe . ~ ~ distribution is asymmetric, with a long tail on the side of high-energy losses which is due to the infrequent collisions with very large energy transfers which result in a relatively large angle scattering of the incident particle. The full width of the Landau distribution at half-maximum is of the order of 20% of Eprob for typical cases. From Landau’s theory,63one obtains the following expression for eproh for a thin absorber (of thickness t g/cm2) : Eprob

=

~

mv2p -

- P2

+ 0.37 - 6 - U (1.1.30)

Equation (1.1.30) can be written as follows:

B

P At + 1.06 + 2 In mic - + In

As an example, a thickness of 6.97 g/cm2 of Be gives a most probable loss = 10 Mev for 3.0-Bev protons. The Landau distribution for this case is shown in Fig. 3. This figure shows the rapid rise of the probability P ( E ) on the side of low-energy losses, and the long tail on the side of large energy losses. The maximum energy transfer of a 3-Bev proton to a single electron, Wmx = 17 MeV, is indicated on the abscissa for comparison with eprob

E. J. Williams, Proc. Roy. SOC.A126, 420 (1929). D.Landau, J. Phys. U.S.S.R. 8, 201 (1944). 6 4 K. R. Symon, quoted by B. Rossi, in “High-Energy Particles,” p. 32. PrenticeHall, New York, 1952. 6 s 0. Rlunck and S. Leisegang, Z.Physin‘k 128, 500 (1950). 61

sSL.

18

1.

PARTICLE DETECTION

the values of Epr& and e~~ (average energy loss). can, of course, be ob= eP(e) de. The tained by integrating over the distribution, i.e., values of the loss E for which the distribution has half its maximum value 1 = a 1 - 8 ~€ ~ 0 ~9.13 ~ Mev and E Z = 11.20 MeV. The fractional spread (€2 - t l ) / ~ p r o= h 0.21 is a measure of the width of the distribution. The ratio ( € 2 - eprob)/(Eprob - el) is 1.38, which is a measure of the skewness of the distribution. The average loss in the same thickness of Be is: eAv = 6.97 X 1.593 = 11.10 MeV, where 1.593 Mev/g cm-2 is - ( l / p ) ( d E / d x ) (see Table 111). The difference between and Eprob is also

lo"

-

t

0.35

n

0.30 -

7

-

gr

W

=0.25 -

--zu0.20 -I

-

a

0.15 -

-

i m

0

0.10

-

0.05-

-

0- L

7

ENERGY LOSS 6 (IN M e V )

FIG.3. The Landau distribution of energy losses E for 3-Bev protons traversing a thickness 6.97 gm/cm2 of Be, for which €pr& = 10 Mev, EA" = 11.10 Mev, and W,, = 17 Mev.

an indirect measure of the importance of the infrequent large energy transfers. In similarity to the restricted energy loss, the most probable loss eprob also levels off to a constant value (Fermi plateau) at very high energies. This result is, of course, due to the fact that the close collisions (which would result in an unlimited increase of dE/dx) do not contribute to eprob, but only to the tail of the Landau distribution. A summary of the experimental determination of Eprob in thin absorbers will be given in Section 1.1.2.9. 1.1.2.7. Ionization Loss of Electrons. Equations (1.1.2.8) and (1.1.2.9) for (dE/dz)w, and Eqs. (1.1.30) and (1.1.31) for eprab are valid for any type of charged particle: electron, meson, etc. These expressions do not iiivolve the close collisions which differentiate slightly between

1.1. INTERACTION

O F RADIATION W I T H MATTER

19

electrons and particles heavier than electrons. On the other hand, Eqs. (1.1.1) and (1.1.19) for the average energy loss are applicable only for particles heavier than electrons. These expressions include a term due to close collisions. For electrons, this term is somewhat different, and the average ionization loss is given by:56

where T , is the kinetic energy of the electron. The factor $T, in the logarithm represents the effective maximum energy transfer W,,,. The reason for this result is that the maximum possible energy transfer from the incident electron t o the atomic electron is T,. However, since the two electrons are indistinguishable, one can call the incident electron after the collision that which has the highest energy. Since this energy is 2 $ T e , the effective maximum energy transfer is $T,. Aside from the replacement of W,, by +Te,the square bracket of Eq. (1.1.32) for electrons differs from that for heavy particles [Eq. (1.1.1)] by an amount: A = (17/8) - In 2 = 1.43

(1.1.33)

a t very high energies ( p = 1). In (1.1.33) the term In 2 is due t o the fact atomic electron) that the reduced mass of the system (indident particle is +m for a n incident electron, as compared to = m for a heavy p a r t i ~ l e . ~ ’ The term 17/8 is due t o the difference between the cross sections for close collisions of electrons as compared t o heavy particles. The effect of A on dE/dx is relatively small ( 5 10%) since the value of the square bracket of (1.1.32) is generally -20. 1.1.2.8. The Shell Correction Term U. We shall nbw discuss the correction U for the nonparticipation of the K , L, . . . electrons a t low energies of the incident particle. This correction has been introduced b y Bethe.36 U is given by

+

(1.1.34) where CKand CLare the K and L shell corrections, respectively. CKand C L are negligible at high energies, and become appreciable only when the velocity v of the particlgis decreased to a value of the order of the velocity 66

H. A. Bethe, in “Handbuch der Physik” (H. Geiger and K. Scheel, eds.), Vol. 24,

p. 273. Springer, Berlin, 1933; C. Mgller, Ann. Physik [5] 14, 531 (1932). 67 H. A. Bethe and J. Ashkin, i n “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. 1, p. 253. Wiley, New York, 1953.

20

1.

PARTICLE DETECTION

of the atomic electrons in the K and L shells, respectively. Thus the shell corrections will enter at a somewhat higher energy for heavy elements than for light elements. As an example, for Cu, 2 C ~ / 2is less than 0.05 for proton energies above T , = 65 MeV, corresponding to v, = 0.35~.CLbecomes appreciable only at still lower energies. For Cu, ~ C L / Z < 0.05 for T, > 11 Mev (v, > 0.15~).Detailed calculations of CK and CL have been carried out by Walske.68 The corrections for the M , N , and higher shells of heavy atoms are generally negligible, except at very low energies (T, 5 1 MeV). ,, where the present theory becomes unreliable for other reasons (capture and loss of electrons by the incident particle, see Section 1.1.2.10).

I

I

I

I

I

I

I

I

I

%

sz Q

tE

1.8 1.6

1.4

1.2 1.0’ I02

I

’ 1 1 1 1 1 1 1

lo3 p

’

‘ ‘ * l t 1 l ’

lo4

’

lo5

’

lo6 ’

11111111

lo7

MESON KINETIC ENERGY (IN MeV)

FIG.4. The average ionization loss of p mesons in Be, Al, Cu, Ag, and Au, as a function of the p-meson kinetic energy [Eq.(1.1.19)].

1.1.2.9. Example: - (l/p)(dE/dx) for /I Mesons in Various Materials. Experimental Verification of the Bethe-Bloch Formula at Relativistic Energies. In summary, Eqs. (1.1.1), (1.1.19) and (1.128)-(1.1.32) givethe

expressions for the ionization loss for 4 different possibilities: (1) average energy loss of particles heavier than eIectrons; (2) average energy loss of electrons; (3) restricted energy loss (energy transfers less than a fixed value W O;) (4) most probable loss in a thin absorber. As an example of the behavior of dE/dx as a function of energy, Figs. 4 and 5 show curves of -(l/p)(dE/dx) versus kinetic energy T, for /I mesons in various solids and gases. In calculating the curves of Fig. 4 for the solids, we used for the excitation potential I of each substance the 68

M. C. Walske, Phys. Rev. 88, 1283 (1952);101,940 (1956).

1.1.

INTERACTfON OF RADfATfON WfTH MATTER

21

average of the I values determined by Bichsel et aL4' and by Burkig and M a c K e n ~ i e .The ~ ~ resulting I values are: IBe= 64 ev, Id = 166 ev, Icu= 371 ev, IA== 586 ev, and I , = 1,017 ev. For Fig. 5, we used the I values given in reference 21 for Hz, He, and air: I H 2= 19 ev, I H e= 44 ev16Q and Isir= 94 ev (the last corresponding to I = 132 ev). For Ar and Xe, the values of I / Z were obtained by interpolation from the above I values = 230 ev, and ZXe = 684 ev. for Cu, Ag, and Au. This gives: IAr 4 and 5 show the ionization minimum (for T , 200-300 MeV) Figures and the relativistic rise at higher energies. The value of - (l/p) (dE/dz)a t the minimum decreases with increasing 2, on account of the increase of I

-

p MESON KINETIC ENERGY (IN MeV)

FIG.5. The average ionization loss of p mesons in Hz, He, air, Ar, and Xe, as a function of the p-meson kinetic energy [Eq. (1.1.19)]. The curves for He, air, Ar, and Xe pertain to normal pressure.

in the denominator of the logarithm of Eq. (1.1.1). For H,, three curves are presented corresponding to different pressures. On account of the density effect, the ionization loss decreases with increasing pressure a t very high energies ( T , 2 10 Bev). It may be noted that, in contrast to Figs. 1 and 2 for - ( l / p ) ( d E/d z)w0,which show a plateau at high energies, the curves of Figs. 4 and 5 have an unlimited logarithmic increase, which is, of course, due to the fact that they represent the average energy loss, including all possible energy transfers up to Wmax[cf. discussion following Eq. (1.1.29)]. Figures 4 and 5 also apply for protons, provided that the numbers on the abscissa are multiplied by the factor mp/m,, where m pand m,, are the 69 E. J WilIiams, Proc. Cambridge Phit. Soc. 33, 179 (1937).

22

1.

PARTICLE DETECTION

proton and the p-meson mass, respectively. [ d E / d x for protons of energy T, is equal to dE/dx for p mesons of energy T,(m,/m,).] The Bethe-Bloch formula has been verified in numerous experimental investigations. A summary of the low-energy work on the ionization loss d E / d x and on the range-energy relation is to be found in the articles of Bethe and A ~ h k i n Allison ,~ and Warshaw,60and Taylor.61The experimental studies at relativistic energies are discussed in the review articles of Price62and of Uehling.'j3 Here we shall present a summary of some of the experiments performed at relativistic energies to verify the existence of the relativistic rise and of the Fermi plateau (due to the density effect). An outline of the main features of some of the experimental investigations on the relativistic rise and the density effect is given in Table I. The most extensive experiments on the energy loss in gases a t relativistic energies have been made by means of expansion cloud chambers,* by obtaining the drop count along the tracks of the particles. The momentum is determined by measuring the curvature in the magnetic field of the cloud chamber. One of the earliest experiments of this type is that of Ghosh and co-workerssl who measured the restricted energy loss (W O= 1 kev) of 1.1 mesons in 0 2 a t normal pressure. They observed a rise from 44 drops per cm to 60 drops per cm in going from the minimum ionization at p , = 0.4 Bev/c to the Fermi plateau starting at p , = 20 Bev/c. From their determinations of the drop count at p , = 7, 15, and 30 Bev/c, it is clear that the energy loss - (dE/dx)w-,does not increase indefinitely with increasing p,, but instead levels off to a saturation value. This result provides a direct confirmation of the existence of the density effect in gases a t high energies (see Fig. 1). Aside from the work of Ghosh et u Z . , ~ ~there have been several other cloud-chamber determinations of the relativistic rise, although these have generally somewhat poorer momentum determinations. Carter and W h i t t e m ~ r eperformed ~~ measurements in a helium-filled chamber, both for p mesons and electrons, and obtained evidence that high-energy electrons ionize more heavily than minimum, which would confirm the relativistic rise. These authors also obtained direct evidence for the relativistic rise by comparing the ionization due to p mesons with momenta between 70 and 250 Mev/c (14.7 f 0.35 droplets/mm on the photographic film) with the ionization for a group with p , > 1500 Mev/c

* See also in this volume, Chapter 1.6. S. K. Allison and S. D. Warshaw, Revs. Modern Phys. 26, 779 (1953). A. E. Taylor, Repfs. Progr. in Phys. 16, 49 (1952). 6a B. T. Price, Repis. Progr. in Phys. 18, 52 (1955). 83 E. A. Uehling, Ann. Rev. Nuclear Sci. 4, 315 (1954). G4 R. S. Carter and W. L. Whittemore, Phys. Rev. 87, 494 (1952). 60

1.1. INTERACTION

23

O F RADIATION WITH MATTER

TABLEI. Summary of Some of the Experimental Investigations on the Relativistic Rise and the Density Effect for the Ionization Loss For each particular method of determination, the experiments are listed in chronological order. For additional details and a more complete list of references, see text (Section 1.1.2.9).

Author

Method of determination

Corson and Cloud chamber (drop count) Brodeae (1938)

Sen Gupta67

Cloud chamber

(1940)

Hazen66(1945)

Cloud chamber

Haywardas (1947) Cloud chamber

Carter and Whittemore'4

Cloud chamber

(1952)

Ghosh et aL61

Cloud chamber

(1954)

Kepler et aLa9 (1958)

Cloud chamber

Type of particle, energy range, and material traversed

Results

Electrons (0.3-60 MeV) Observation of miniin cloud chamber mum of ionization filled with Nzat 1.5 (at T , 2 MeV) and atmos pressure. relativistic rise for electrons. Electrons (2-500 MeV) Observation of minimum of ionization and relativistic rise for electrons. Electrons in air. Two Observation of relativistic rise (of -40% energy groups: 1.42.1 Mev and 30-240 between two energy groups) in agreement MeV. with Bethe-Bloch formula. Electrons in He. High-energy electrons ( T , > 100 MeV) have 1.4 times minimum ionization, in agreement with theory. Increase of ionization p mesons in He (at between the two pressure P = 98 cm momentum groups Hg). Two momenin good agreement tum groups: p = 70with theory. 250 Mev/c, and p > 1500 Mev/c. p mesons in 0, (at nor- Observation of relativmal pressure); p = istic increase of ionization and levelling 0.3-30 Bev/c. off to Fermi plateau above 6 Bev/c in reasonable agreement with calculations including the density effect. p mesons with p / p c = Observation of relativistic rise and Fermi 3-80, and electrons plateau: in good with p/pc = 50agreement with 2000, in following

-

24

1.

PARTICLE DETECTION

TABLE I. Summary of Some of the Experimental Investigations on the Relativistic Rise and the Density Effect for the Ionization Loss (Continued) ~~

Author

~



gases: He (1.3 atmos); Ar (0.2 atmos) ; Ar-He mixture (each a t 0.2 atmos); Xe-He mixture (each at 0.1 atmos) p mesons (0.3-70 Bev/c) in mixture of argon (P = 774 mm Hg) and ethylene (P = 46 mm Hg). p mesons in Ne.

Results theory for He; for other cases, calculated rise is somewhat larger than experimental values.

.

Parry et al.1° (1953)

Eyeions et

al.7'

(1955)

Proportional counter


Palmatier, et aZ.I2 (1955)


p

mesons (0.2-15 Bev/c) in argon at pressures from 2 to 40 atmos.

Lanou and Kraybi11728 (1959)


p

mesons (3.3-140 Bev/c) in a mixture of 95% He and 5% CO2 a t a total pressure of 2.7 atmos.

Barbersl (1955)

Ionization chamber

Electrons (1-35 MeV) in Hi, He, and N2 (normal pressure).

Observation of relativistic rise and Fermi plateau in good agreement with theory. Observation of relativistic rise and Fermi plateau in good agreement with theory. Decrease of the relativistic rise with increasing pressure, in good agreement with calculations of Sternheimer20 on the density effect. Observation of relativistic rise in He and saturation of the most probable ionization loss a t p/m,c 200 (Fermi plateau). In this region, the ionization loss is 1.28 & 0.04 times minimum. Observations in good agreement with calculated relativistic rise for Nz, but experimental increase of ionization somewhat smaller than calculated increase of IdE/drl for Hzand

2

1.1. INTERACTION

O F RADIATION WITH MATTER

25

TABLEI. Summary of Some of the Experimental Investigations on the Relativistic Rise and the Density Effect for the Ionization Loss (Continued)

Author

Barbers2 (1956)


Ionization chamber


Electrons (1-35 MeV) in H2 and He a t 1 and 10 atmos pressure.

Results He, possibly clue to production of Cerenkov radiation which does not contribute t o ionization.20 Observations in good agreement with calculated relativistic rise. At 10 atmos, reduction of ionization due to density effect is observed for T, 10 Mev in HZ and for T. >, 18 Mev in He. The reductions at 35 Mev are in reasonable agreement with calculations of Sternheimer.20 Observed relativistic rise between 1.7 Mev (minimum) and 9.0 Mev in reasonable agreement with calculations. Relativistic rise of 1.17 k 0.03 between sea level spectrum and underground spectrum, in good agreement with calculations using density effect correction.

2

Herefords7 (1948)

Low-pressure counter

Shamos and Hudes88 (1951)

Low-pressure counter

McClureS6 (1953)

Pickup and Vo.yvodics9 (1950)

Electrons (0.2-9.0 MeV) in H 2 ( P = 7 cm Hg).

Cosmic-ray p mesons at sea level (average momentum = 3.5 Bev/e) and under 140 feet of rock (average momentum = 48 Bev/c); primary specific ionization in H2 filled counter ( P = 2.0 cm Hg). Observations in good Electrons (0.2-1.6 Low-pressure agreement with MeV) in He, He, Ne, counter Bethe’s theory of and Ar. primary specific ionization. p-decay electrons and First observation of Nuclear relativistic fi mesons -10% relativistic emu1sion rise of grain count in (grain count) in plate exposed to

26

1.

PARTICLE DETECTION

TABLE I. Summary of Some of the Experimental Investigations on the Relativistic Rise and the Density Effect for the Ionization Loss (Continued) ~

Author



Results

emulsion between sea-level cosmic-ray T/m& 3 and spectrum; and highT/mor2 20, in reaenergy electrons, n sonable agreement mesons, and protons with theoretical prein plate exposed t o dictions. cosmic rays at high altitudes. Blob count increases Electrons (5 Mev-5 Nuclear Bev. emulsion -5% between 5 Mev and 15 MeV, then re(blob count) mains constant to 25 Bev; relativistic rise is smaller than value predicted by theory (14%). Grain count increases Nuclear n mesons (200 Mev/c-3 by -8% between Bev/c). emu1sion 500 and 1500 Mev/c, (grain count) in reasonable agreement with calculations of Budini.z* Ratio R of plateau to Electrons (y > lo), Nuclear minimum blob emulsion T mesons (y < loo), count, R = 1.14 k (blob count) and protons (y < 0.03, in good agree10). ment with theoretical value 1.14;slow rise of grain count until saturation is reached for > 100, in good agreement with calculations of Sternheimer.20 Nuclear Electrons from p decay Relativistic increase of emulsion (average energy = 14% (between mini(blob count) mum and plateau 34 MeV) and nionization), and slow mesons (31-230 Mevl. rate of rise, in good agreement with theory. Ratio R of plateau t o Nuclear T mesons (109.1 Mev) minimum grain emulsion from Krz decay, and count, R = 1.133. (blob count) p mesons (152.7

N

Morrishgo (1952)

Daniel et aLS4 (1952)

Stiller and Shapiros2 (1953)

Fleming and Lord93 (1953)

Alexander and JohnstongTa (1957)

1.1.

27


TABLE I. Summary of Some of the Experimental Investigations on the Relativistic Rise and t h e Density Effect for the Ionization Loss (Continued)

Author


Type of particle, energy range, and material traversed MeV) from K,? decay.

JongejansSTb (1960)

Nuclear emuIsion (blob count)

Beam pions (5.2-5.7 Bcv/c) from Berkeley Bevatron; secondary pions produced by beam pions (1.8 < y 5 3.5); electron pairs (65 < y < 1100).

Whittemore and Street104 (1949)

Crystal counter (pulse-height distribution)

Cosmic-ray p mesons in AgCl crystal. Two energy groups: T, = 0.3 Bev (minimum ionization), and T, > 1.6 Bev.

Bowen and Roser1°6 (1952)

Scin tillator (pulse height distribution)

Cosmic-ray p mesons (30 Mev-3 Bev) in anthracene crystal.

Hudson and Hofstadterllo (1952)

Scintillator

Cosmic-ray p mesons in NaI (Tl) crystal ( p r > 225 Mev/c).

Baskin and Winckler (1953)

Scintillator

Cosmic-ray p mesons (80-2200 MeV) in xylene solution (with terphenyl).

O6

Results The authors have obtained an accurate calibration curve for grain count versus p p c for the region 0.5 < p < 0.95. Ratio R of plateau t o minimum grain count, R = 1.129 f 0.010. The relativistic rise of the grain count is slow, with an appreciable increase (-4%) taking place between y = 40 and y 1000 (plateau). Observed relativistic increase between the two energy groups is in agreement with the prediction of the Bethe-Bloch formula, including density effect correction. No detectable relativistic rise of most probable energy loss eprob, in good agreement with theory including the density effect. Observed pulse-height distribution in good agreement with calculations including the density effect. No relativistic rise is observed, in agreement with calculat i o n 9 including the density effect.

-

28

1.

PARTICLE DETECTION

TABLE I. Summary of Some of the Experimental Investigations on the Relativistic Rise and the Density Effect for the Ionization Loss (Continued) ~~~

Author Bowen111 (1954)

Millar et (1958)

uZ.109

Paul and Reich"4 (1950)

Goldwasser, MiIIs, and Hanson 119 (1952)

Goldwasser et diZ1 (1955)


~~


Results

Relativistic increase in Accelerator-produced reasonable agreeT- mesons (60-220 ment with calculaMev), p- mesons tions; rise to plateau (245 Mev), and cosvalue may be somemic-ray p mesons (4 what faster than pregroups with average dicted by theory [obenergies: 0.37, 0.76, served rise: (10.9 i 1.47, and 5.23 Bev); 1.0)% at T, = NaI(T1) crystal. 50m,c*]. Scintillator Cosmic-ray p mesons Observed value of €i,rob (two energies: T p = (0.30 Bev)/e,,,b (2.2 0.30 Bev and 2.2 Bev) = 1.016 2c Bev); liquid scintil0.005, in good agreelation counter filled ment with BetheBloch formula, inwith triethylbenzene (plus tercluding density efphenyl). fect correction. Energy loss in Electrons (2.8 and 4.7 Observed mean energy loss is in better agreethin sample Mev) in samples ment with calculated (-0.3 gm/cma) of Be, C, H20, Fe, and value if density efPb. fect correction is included. Electrons (9.6 and 15.7 Observed most probaEnergy loss ble energy loss (-1 MeV) in thin samples (-1 gm/cm*) of Mev) is in good Be, polystyrene, Al, agreement with Cu, and Au. Landau formula, including correction for density effect. Energy loss Electrons (15.7 MeV) Direct observation of in thin samples of the density effect by comparing energy Teflon and Kel-F, and in the correloss in solid and gasesponding gases (same ous samples of the chemical composisame substance. The tion) : perfluororeduction of the ionization loss in the cyclobutane and chlorotrifluorosolid samples is in good agreement with ethylene. calculations of the density effect.

Scintillator

1.1.


29

TABLEI. Summary of Some of the Experimental Investigations on the Relativistic Rise and the Density Effect for the Ionization Loss (Continued)

Author Hudson121 (1957)

Method of determination Energy loss


Results

Electrons (150 MeV) in Observed energy loss thin targets (-2.5 eprob in good agreement with Landau gm/cma) of Li, Be, C, and Al. formuIa including the density effect correction.

(17.9 f 0.25 droplets/mm). The theoretically predicted value for this high-energy group of p mesons is 18.5 droplets/mm, in good agreement with the experimental result. Hazene6 verified the relativistic rise for electrons in air. Even earlier experiments by Corson and Brodese and b y Sen GuptaG7gave convincing evidence for the relativistic rise b y using cosmic-ray electrons. More recently, Hayward68 showed that high-energy is the minielectrons have a n ionization of 1.4Jni, in helium, where Jmin mum ionization. 9 measured the relativistic rise I n a recent experiment, Kepler et ~ 1 . 6 have of the ionization loss of p mesons and electrons in He, Ar, and Xe, by obtaining the drop count in a n expansion cloud chamber. Measurements were made for He a t -1.3 atmos, for Ar a t -0.2 atmos, for a n Ar-He mixture, each a t -0.2 atmos, and for a Xe-He mixture, each a t -0.1 atmos. I n each case, the cloud chamber contained alcohol and water vapor a t a partial pressure of -5 cm Hg. The p-meson momenta extend from minimum ionization (p/m,,c = 3) to p/m,c = 80. The electron momenta extend from p/mc = 50 to z 2 0 0 0 . Thus the entire region of the relativistic rise is covered in these measurements, including the Fermi plateau, which starts a t p/mc E! 1000. For the helium experiment, the theoryzkz2is in very good agreement with the data, if Williams’ valuesg of the excitation potential I for He is used (IH, = 44 ev;69 I for gas mixture = 49.4 ev69). For the argon and the argon-plus-helium experiments, the calculated rise is larger by a t least one standard deviation than W. E. Haaen, Phys. Rev. 67, 269 (1945). D. R. Corson and R. B. Brode, Phys. Rev. 63, 773 (1938). 67 R. L. Sen Gupta, Nature 146,65 (1940); Proc. NatE. Znst. SCi. fndiu 9,295 (1943). 88 E. Hayward, Phys. Rev. 72, 937 (1947). 60 R. G. Kepler, C. A. d’Andlau, W. B. Fretter, and L. F. Hansen, Nuovo cimento [lo] 7, 71 (1958). See also A. Rousset, A. Lagarrigue, P. Musset, P. Rancon, and X. Sauteron, Nuovo cimento [lo] 14, 365 (1959). 66

68

30

1.

PARTICLE DETECTION

the experimental values. Thus for the Ar-He mixture, a t p/mc % 1700,the is 1.59 f 0.04, whereas the calculated resultz0is observed value of J/Jmin is 1.64. Here J is the observed ionization loss, - (l/p)(dE/dx)w,, and Jmin the value of J at the minimum of ionization. In this experiment, the effective maximum energy transfer W O(determined by the size of a blob = 40 drops) was 700-1000 ev. For the Xe-He mixture, the discrepancy is appreciably larger. For p/mc E 1000, the experimental J/Jmin = 1.58 -t 0.05, whereas the values calculated from the theories of BudiniZ2and SternheimerZ0are 1.78 and 1.75, respectively. On the whole, it appears that the experimental points for 4, AF-He, and Xe-He lie on a curve which increases less rapidly with increasing momentum than the theoretical curve obtained from the expression for - (l/p) (dE/dz)w, [Eq. (1.1.28)]. Kepler et ~ 1 have . given ~ ~ various possible reasons for this discrepancy for the heavy gases, in particular: (1) a variation of the maximum energy transfer W Owith atomic number 2, due to the 2 dependence of the binding energy of the struck electron; (2) the ratio of the energy loss to excitation and to ionization may depend on the velocity v and on 2, in such a manner as to decrease the slope of the curve of - ( l / p ) ( d E / d z ) w , versus p / p c at high momenta beyond the ionization minimum; (3) shielding effects of the inner electron shells in the heavy elements are important at very high energies. However, these shielding effects are taken into account, a t least in first approximation, by the density effect term 6 in Eq. (1.1.28). As a check on the Xe-He experiment, as Kepler et al.69 have determined the number of drops per cm at Jmin 28.9 k 0.6 under well-controlled conditions. This experimental value can be compared with the theoretical predictions: 31.8 0.4 drops/cm for the Bakker-SegrP excitation potentialszoI , and 29.6 ? 0.4drops/cm for the higher C a l d ~ e 1 values 1 ~ ~ ofz1I . It is seen that the experimental value is in good agreement with Caldwell's results, which are also favored by several other ionization loss experiments (see Section 1.1.2.4). Important experiments on the ionization loss a t relativistic energies have been carried out with proportional counters. I n these experiments, the Landau distribution is measured, from which, of course, one obtains the most probable loss Ep& The most accurate determinations are those of Parry et aL7Oon the ionization loss eprobof p mesons in argon, and those of Eyeions et d."who obtained Eprob for p mesons in a neon-filled counter. In both cases, a relativistic rise of -50% was found, and the leveling off to

+

70 J. K. Parry, H. D. Rathgeber, and J. L. Rouse, PTOC. Phys. Soc. (London) A66, 541 (1953). 7l D. A. Eyeions, B. G. Owen, B. T. Price, and J. G. Wilson, PTOC. Phys. Soc. (London) A68, 793 (1955).

1.1.


31

the Fermi plateau was clearly observed. These two experiments used a cosmic-ray magnetic spectrometer, in which the high-energy particles are passed through a strong magnetic field and the resulting deflection is measured in a hodoscope array of Geiger counters placed below the proportional counter. Palmatier and c o - w o r k e r ~have ~ ~ investigated the relativistic rise and the Fermi plateau of Eprob for p mesons in a counter filled with argon a t various pressures up to 40 atmospheres. These authors have directly verified the dependence of Eprob on the pressure, i.e., the increase of the density effect with increasing pressure,20 and the resulting decrease of cprob,plat/Eproh,min, the ratio of the plateau to the minimum value of eprob. The calculated values of €,,rob are in reasonable agreement with these experimental results. Lanou and Kraybil1728have recently carried out a-similar investigation using p mesons of momenta 3.3-140 Bev/c in a proportional counter filled with a mixture of 95% He and 5% COZ at a total pressure of 2.7 atmospheres. These authors have observed the relativistic rise of cprobin helium, and have found that the rise saturates at momenta p/m,c 2 200 due to the density effect. In the region of the Fermi plateau, the most probable ionization loss is 1.28 k 0.04 times the value a t the minimum, in agreement with the calculations of Sternheimer.20,21 Several other experiments with proportional counters demonstrate the relativistic rise, but were not accurate enough to establish the existence of the Fermi plateau. Among these studies, we may mention the experiments of Kupperian and Palmatier,73Becker et aZ.,74 Price et aZ.,76and Eliseiev et aZ.76 Several experimenters have investigated the width A of the Landau distribution as a function of the value of A t / ( p 2 1 ) ,which enters as a parameter in Landau’s theory. West77found that, although the percentage width 100A/Eprob decreases with increasing At/(P21), as required by Landau’s theory, the value of A/cprob is larger than Landau’s result by a factor of -2. This discrepancy for the width of theidistribution can be reE. D . Palmatier, J. T. Meers, and C. M. Askey, Phys. Rev. 97, 486 (1955). R. E. Lanou and H. L. Kraybill, Phys. Rev. 113, 657 (1959). 73 J. E. Kupperian and E. D. Palmatier, Phys. Rev. 91, 1186 (1953). T 4 J. Becker, P. Chanson, E. Nageotte, P. Treille, B. T. Price, and P. Rothwell, Proc. Phys. SOC.(London)A66, 437 (1952). 75 R. T. Price, D. West, J. Becker, P Chanson, E. Nageotte, and P. Treille, Proc. Phys. soc. (London)A66, 167 (1953). 7 6 G. P. Eliseiev, V. K. Kosmachevsky, and V. A. Lubimov, Doklady Akad. Nauk. S.S.S.R. 90, 995 (1953); (English translation: NSF-tr-163, Dept. of Commerce, Washington, D.C.) 77 D. West, Proc. Phys. SOC.(London)A66, 306 (1953). 71

788

32

1.

PARTICLE DETECTION

moved by improvements in the Landau theory which have been discussed by fan^^^ and H i n e ~ . ~ ~ Igo and co-workerssOhave measured the distribution of energy losses of 31.5-Mev protons in a $inch proportional counter filled with an Ar-C02 mixture (96 % Ar, 4 % C02). The pulse-height distribution was in reasonable agreement with the Landau distribution, although slightly wider in the region of the tail for large energy losses. Barbers1.s2has measured the specific ionization of electrons in Hz, He, and Nz in a n ionization chamber. The electrons were obtained from the Stanford linear accelerator and had energies ranging from 1 to 35 MeV. A collimated beam of electrons was sent through an ionization chamber into a Faraday cup, so that the ratio of the collected ionic charge to the charge collected in the Faraday cup is proportional to the specific ionization. In Barber's first experiment,S' H2, He, and Nz at atmospheric pressure were used. Under these conditions, one does not expect any density effect correction, since the density effect sets in above 35 Mev for gases a t normal pressure.20 A t minimum ionization, the number of ion pairs per cm (probable specific ionization) was 7.56 f 0.09,6.15 k 0.08, and 53.2 f 0.7 in H2, He, and N2, respectively (at normal temperature and pressure). These results were compared with the theoretical expression for the ~ , (l.l.28)] with W o= 17.4 kev restricted energy loss, - ( l / p ) ( d E / d ~ ) [Eq. for Hz, 16.4 kev for He, and 70 kev for Nz,as determined from the size of the ionization chamber and the experimental conditions. Barbers1 thus obtained the following values for w, the average energy required to produce an ion pair: 37.8 k 0.7, 44.5 k 0.9, and 34.82",; ev for Hz, He, and N'L,respectively. These results are in reasonable agreement with ,~~ and H ~ r s t , ~ ~ the values of w obtained by Jesse and S a d a u s k i ~Bortner and Bakker and S e g r P (see Section 1.1.2.12). The total number of ion pairs per cm a t the ionization minimum as obtained from the average energy loss, - (l/p)(dE/ds) [Eq. (1.1.1)] without any limitation on the maximum energy transfer, was found to be: 9.19 f 0.18,7.55 k 0.16, and 61.62::; for the three gases. The relativistic increase of the ionization from minimum (at -1.7 MeV) to 35 Mev is 1.17 for H2,1.20 for He, and 1.24 for Nz. For N P , the calculated increase of the ionization loss agrees within 1% with the observed rise, but for Hz and He, the predicted increase is U. Fano, Phys. Rev. 92, 328 (1953). K. C. Hines, Phys. Rev. 97, 1725 (1955). G. J. Igo, D. D. Clark, and R. M. Eisberg, Phys. Rev. 89, 879 (1953). *1 W. C. Barber, Phys. Rev. 97, 1071 (1955). 82 W. C. Barber, Phys. Rev. 103, 1281 (1956). W. P. Jesse and J. Sadauskis, Phy8. Rev. 90, 1120 (1953). R 4 T. E. Bortner and G. S. Hurst, Phys. Rev. 90, 160 (1953). 79

1.1.


33

somewhat higher than the observed value, assuming that the energy loss per ion pair w is independent of the electron energy. In particular, for Hz, the deviation between the calculated and the observed values would correspond to an increase of (3.3 i-0.7)% in w as the electron energy is increased to 35 MeV. It is possible that the lowering of the rate of rise of the ionizatJionis due t o the production of c e r p k o v radiation29which is not reabsorbed t'o form ions in the gas. In Barber's second experiment,82 the specific ionization of electrons in H2 and He was measured at 1 and 10 atmospheres pressure. At 10 atomspheres, a sizable density effect is expected for both gases a t 35 MeV, whereas at 1 atmosphere the density effect correction is negligible. The experimental setup was essentially the same as in the first experiments1 (ionization chamber; Faraday cup). The theory is in good agreement with the experimental results which show that at 10 atmospheres, the specific ionization J [Eq. (1.1.28)] a t first increases above the minimum, from -2 to 10-15 MeV, but levels off above 10 Mev for H2 and 18 Mev for He. For Hz at 10 atmospheres, the value of J ( 3 5 Mev)/Jmi, is 1.12 upon inclusion of 6 (density effect), as compared to the experimental value: J / J m i ,= 1.11. (The calculated value of Jmin is 3.39 Mev/g cm-2.) Without the density effect, the calculated J / J m i ,would be 1.20. Thus the data provide a good confirmation of the existence of the density effect for gases at high pressure. Upon taking into account the experimental uncertainties, BarberB2finds that the ionization loss is decreased by (8 f 1 ) % by the density effect, as compared to the theoretical reductionz0of 6.5 %. Similar agreement is obtained for the measurements in He at 10 atmospheres, where the observed decrease of the ionization J at 35 Mev is (3.5 f 1.3) %, as compared to the calculated valuez0of 3%. Low-pressure Geiger counters have also been used to determine the relativistic rise of the ionization loss in gases. In a low-pressure counter, one attempts to measure the total number of ionizing collisions:

N = /owm"xP(W)dW where P(W) dW is the probability of a collision with energy transfer between Wand W dW. By contrast, a proportional counter measurement gives the total energy deposited: & = J ~ " " " P ( W ) W dW. The fluctuations in & are largely due to the presence of the large energy transfers W of the order of W,., which are weighted by the factor W in the integrand for &. For N , on the other hand, there is no factor Win the integrand, so that the large energy transfers are weighted much less heavily than for &, and the fluctuations in N are correspondingly reduced. The condition under which the Geiger counter measures N rather than & is that the incident

+

34

1.

PARTICLE DETECTION

particles make on the average less than one ionizing collision in traversing the counter. N is proportional to the primary specific ionization J,, of the incident p a r t i ~ l e . ~ ~ Several experiments have been performed using low-pressure Geiger counters. The most extensive recent measurements for electrons in the 0.2-1.6 Mev energy rangepre those of McC1ureS6who obtained the primary specific ionization J,, of electrons in this energy range for Hz, He, Ne, and Ar. For Hz, the results could be fitted to the theoretical curve of J,, versus p/mc obtained by Bethe.86Somewhat earlier, H e r e f ~ r d , using ~’ counter measurements, obtained evidence for the relativistic rise of the ionization loss of electrons by measuring J,, for electrons in hydrogen in the range from 0.2 to 9.0 MeV. Evidence for the relativistic rise for p mesons in hydrogen has been obtained by Shamos and Hudes.88 The relativistic rise of the ionization loss in photographic emulsion has been the subject of numerous investigations. The first definite evidence for a -10% rise in the grain count in emulsion was obtained by Pickup and VoyvodicS9in 1950. The minimum value of the grain count G is obtained for a ratio T/moc2 3 of the kinetic energy T to the rest energy mgc2 of the particle. The rise starts at T/moc2 N 3 and continues until the plateau value G,,,, is reached for T/moc2 10-100. The precise value of T/moc2at which the plateau is reached has been the subject of some controversy, with some experiments favoring a rapid rate of rise of G to GPIsta t T / m d 10, while others give evidence of a more gradual rise for which the plateau is reached only for T/moc2 50-100. In the experiments, following a suggestion of Morrish,Soone obtains generally the blob count rather than the grain count. Here a blob is defined as either a single grain or a group of overlapping grains which cannot be resolved. It was found that the blob count is considerably more independent of the observer than the grain count.91 In the region between the minimum and the plateau, the blob count is proportional to the grain count.92For comparison of the theory with the grain or blob count observations, one must calculate the restricted energy loss [Eq. (1.1.2S)l with maximum energy transfer W O 5 kev. The reason is that for energy transfers W >, 5 kev, the delta-ray will have a large enough range to traverse one or more addi-

-

-

-

-

-

G. W. McClure, Phys. Rev. 90, 796 (1953). H. A. Bethe, in “Handbuch der Physik” (H. Geiger and K. Scheel, eds.), Vol. 24, p. 515. Springer, Berlin, 1933. *’ F.L. Hereford, Phys. Rev. 73, 982 (1947);74, 574 (1948). M. H. Shamos and I. Hudes, Phys. Rev. 84, 1056 (1951). 88 E. Pickup and L. Voyvodic, Phys. Rev. 80, 89 (1950). A. H. Morrish, Phil. Mug. [71 43, 533 (1952). 91 See also L. Jauneau and F. Hug-Bousser, J . phys. radium 13, 465 (1952). 92 B. Stiller and M. M. Shapiro, Phye. Rev. 92, 735 (1953). 86

88

1.1.

35


tional grains not directly in line with the path of the particle. If only the grains along the track are counted, or if a blob count is made, in which all of the overlapping grains due to an energetic 6 ray are counted as a single unit, then the observed count will be essentially unaffected by the presence of the high-energy 6 rays, so that a cutoff at WO 5 kev is indicated, as was first pointed out by Messel and Ritson.26The resulting values of - (l/p)(dE/dx)wo as obtained by Sternheimer20n21 give a relativistic rise of 14% which saturates slowly and does not level off until T/moc2 100. The magnitude of the increase and the gradual character of the relativistic rise are in good agreement with the observations of Stiller and Shapirog2 using cosmic rays, and those of Fleming and Lordg3using cosmic-ray electrons and accelerator-produced 7-mesons. On the other hand, Budini’s calculations22*2s give a more rapid rate of rise, with GpIstbeing reached for T/moc2 between 10 and 40, depending on the specific assumptions made about the widths of the spectroscopic lines of the Ag and Br atoms of the emulsion. Budini’s calculations are in reasonable agreement with the results of Daniel et al. 9 4 which indicate a more rapid rate of rise than those of references 92 and 93. Data on the ionization loss in emulsion have also been obtained by McDiarmidlg6Michaelis and Violet,g6Morrishlg7and others. Alexander and Johnstong7ahave obtained a very accurate calibration curve for the grain density as a function of ppc for ?r and p mesons. In this work, the authors used T and p mesons of constant and precisely known energy from the K,2 and Kp2decays of K partides at rest. The calibration curve extends from p = 0.5 to0.95, corresponding to 1 < g* < 3, where g* is the grain density normalized to the minimum of ionization:

-

-

g* = (dE/dX)w,/[(dE/dX) Wolrnin.

I n the range 1 < g* < 1.6, the accuracy of the calibration curve for g* is estimated to be better than 1%. For the ratio of plateau to minimum grain count, the authors have obtained g,*l, = 1.133. Recently, J ~ n g e j a n s ~ has ’ ~ measured the relativistic rise of the grain density in Ilford G5 emulsion, using pion tracks of energy -5.4 Bev from the Berkeley Bevatron, and secondaries produced by the pions in the emulsion [pions stopping in the emulsion, with y between 1.8 and 2.5; J. R. Fleming and J. J. Lord, Phys. Rev. 92, 511 (1953). R. R. Daniel, J. H. Davies, J. H. Mulvey, and D. H. Perkins, Phil. Mag. [7]43, 753 (1952). 9 5 I. B. McDiarmid, Phys. Rev. 84, 851 (1951). 06 R. P. Michaelis and C. E. Violet, Phys. Rev. 90, 723 (1953). $7 A. H. Morrish, Phys. Rev. 91, 423 (1953). 978 G. Alexander and R. W. H. Johnston, Nuovo cimenfo [lo] 6, 363 (1957). wb B. Jongejans, Nuovo cimenlo [lo] 16, 625 (1960). 98 94

36

1.

PARTICLE DETECTION

other secondary pions; electron pairs with y between 65 and 1100, where y = (T/moc2) I]. The momentum of the tracks was determined from multiple scattering measurements. For the ratio GPlat/Gmin,a value 1.129 k 0.010 was obtained, in good agreement with the results of Stiller and Shapirog2 (1.14 2 0.03), Fleming and Lordg3 (1.14 f O.Ol), and Alexander and Johnston97a (1.133 f 0.008). The rise of the grain density G to the plateau value was found to be slow, with a n appreciable increase 1000. This result is in good agreetaking place between y = 40 and y ment with the calculations of Sternheimer,20*21 and with the experiments of references 92 and 93. A calibration curve for g” versus (y - 1) is given in the paper of J0ngejans.97~This curve was calculated using a value of the excitation potential I = 501 ev for AgBr. In comparing the theory with these data for emulsion, it must be borne in mind that one would expect large fluctuations of the ionization loss in each grain, since the most probable energy loss Eprcb for a minimum ionizing particle in a 0.2 p AgBr grain is only -50 ev. On the other hand, the threshold value 7 for the energy deposit below which the grain is not exposed is of the order of several hundreds of volts. Thus the effect of Landau-type fluctuations on the grain count is expected to be quite important, as was first pointed out by BarkasI9*and subsequently by Brown.99I n view of these difficulties and limitations of the theory, the detailed quantitative agreement which has been obtained for emulsion is surprisingly good. In connection with the emulsion measurements on electron-positron pairs produced by y rays of very high energy ( E , 2 10 Bev), Perkins’** observed that there is a reduction of the ionization loss J below the value of twice minimum ionization (2Jmi,) due to the interference between the electromagnetic fields of the positron and electron. With increasing distance x from the origin of the pair, the ionization J varies form 0 to 2Jni,, as a result of the increase of the distance d between the positron and electron. Several authorslo’ have treated theoretically the problem of the ionization loss of an electron-positron pair, along the lines of Fermi’s cal~ulation’~ of the ionization loss of a single particle. The theory gives the dependence of the ionization J on the distance d. The asymptotic value 2Jmi. is attained when d becomes large compared to c/(2rvP)

+

-

W. H. Barkas, in “Colloque Bur la sensibilite des cristaur et des emulsions photographiques,” Paris, September, 1951; see also W. H. Barkas and D. M. Young, University of California, Radiation Laboratory Report URCL-2579, revised (1954). 99 L. M. Brown, Phys. Rev. 90, 95 (1953). loo D. H. Perkins, Phil. Mag. [7] 46, 1146 (1955). 1olA. E. Cudakov, Izvest. Akad. N a u k S.S.S.R. 19, 650 (1955); I. Mito and H. Ezawa, P r o p . Theoret. Phys. (Kyoto) 18, 437 (1957); G. Yekutieli, Nuovo cimento [lo] 6, 1381 (1957).

1.1.


37

( = 0.51 X lodEcm for emulsion). Upon using the theoretical dependence of J on d, the measurements of J ( x ) enable one to obtain the value of the opening angle of the pair 0 ( = d/x), from which in turn one can estimate the energy E , of the parent y ray.lo2The results obtained by this methodLo3 are in reasonable agreement with the conventional determination of the y-ray energy from the subsequent development of the shower of electrons and y rays (see Section 1.1.5.2). Besides the experiments on photographic emulsion, crystal counters and scintillators* have also been used to observe the relativistic increase of the ionization loss of p mesons in condensed materials. The first of these experiments was carried out by Whittemore and Streeti0*in 1949, using a silver chloride crystal. These authors compared the ionization pulses produced by p mesons of range > 112 cm of P b (T, > 1.6 Bev) with those produced by minimum ionization p mesons (T, = 0.3 Bev) , and found a definite relativistic increase. The results were in agreement with the predictions of the Bethe-Bloch formula including the density effect correction. Experiments with p mesons passing through an anthracene scintillator have been performed by Bowen and Roser,lo6who obtained no detectable relativistic increase of the ionization loss Eprob above the minimum value. This result is in agreement with the theoretical predictions, since the density effect sets in at a relatively low energy (near the ionization minimum) for low atomic number, and is large enough to prevent any rise of €pro,, from occurring. Similar results were obtained by Baskin and W i n ~ k l e r using , ~ ~ ~a ~liquid ~ ~ ~scintillator of low Z (xylene). In these experiments, it is assumed that the light output of the scintillator is proportional to the energy deposited by the incident particle ( p meson). This assumption has been verified by Chou,lo8who showed that the response of most scintillators is nearly linear up to 3Jmin-4Jmi,. Millar, et al. lo9 have exposed a large-area liquid scintillation counter to cosmic-ray p mesons. The counter was filled with triethyl-benzene (plus terphenyl). The most probable loss Eprob and the Landau distribution were obtained both for T, = 0.30 Bev and T , = 2.2 Bev. The value of fprob

* See also in this volume, Chapter 1.4. A. Borsellino, Phys. Rev. 89, 1023 (1953). W. Wolter and M. Miesowice, Nuovo cimento [lo] 4, 648 (1956). lo4 W. L. Whitternore and J. C. Street, Phys. Rev. 76, 1786 (1949). 106T.Bowen and F. X. Roser, Phys. Rev. 86, 992 (1952). 108 R. Baskin and J. R. Winckler, Phys. Rev. 92, 464 (1953). 107 See also A. G. Meshkovskii and V. A. Shebanov, Doklady Akad. Nauk S.S.S.R. 83, 233 (1952). 108 C. N. Chou, Phys. Rev. 87, 903 (1952). 109 C. H. Millar, E. P. Hincks, and G. C. Hanna, Can. J . Phys. 36,54 (1958). 102

108

38

1.

PARTICLE DETECTION

at 0.30 Bev is higher by (1.6 & 0.5) % than the value a t 2.2 Bev, in good agreement with Eq. (1.1.30) including the density effect correction 6. The prediction of the Landau theory for the width of the pulse-height distribution (18 % at half-maximum) is in reasonable agreement with the observed width (20.5% at half-maximum in the central area of the counter) when the width due to the counter resolution function (8%) is taken into account. Hudson and Hofstadterl'O have exposed a thallium-activated sodium iodide crystal [NaI(TI)] t o the cosmic-ray p-meson spectrum and have found that the resulting observed pulse-height distribution is in much better agreement with the theoretical distribution obtained upon inclusion of the density effect correction 6 (as calculated from the paper of Halpern and Hall's) than with a theoretical distribution obtained by setting 6 = 0. In each case, the theoretical curve was obtained by folding the Landau straggling distribution6awith the cosmic-ray p-meson spectrum. In a later investigation, Bowen'" used a NaI(T1) crystal to measure the energy loss of T- and p mesons of selected energies or energy groups. The T- mesons were produced by the Chicago 450-Mev cyclotron and had well-defined energies extending from 61 to 222 MeV. In addition, 245-Mev p- mesons arising from the decay of 227-Mev T- mesons were used. Moreover, four energy groups of the cosmic-ray p-meson spectrum were studied. These groups were separated in energy by using various thicknesses of iron absorber. The average energies of the p mesons in the four groups were: T, = 368,755,1470, and 5230 MeV. The energies of the T- from the cyclotron were: 61, 85, 118, 163, and 222 MeV. At each of these energies, the most probable loss Eprob was obtained from the observed pulse distribution. Bowen thus obtained values of epr& as a function of T/moc2. The theoretical prediction20321 for 6prab versus Tlmoc2 is in reasonable agreement with these data. Thus from the calculations one obtains an 8.2% increase (relative to minimum ionization) at T,, = 50mpc2,and an asymptotic value of the rise (at very high energies) of 11.4%. The experimental value is 10.9 5 1.0% at T, = 50m,c2. This result may indicate that eprob rises to the plateau value somewhat more rapidly than predicted by the theory. It may be noted that the reason why there is a relativistic rise of Eprob for NaI but none for anthracene or xylene is that with increasing 2, the density effect correction 6 sets in a t higher energies, thereby leading to a relativistic rise before the energy loss eprob saturates due to the onset of 6. The density effect has been extensively studied by observing the energy Hudson and R. Hofstadter, Phys. Rev. 88, 589 (1952). T. Bowen, Phys. Rev. 96, 754 (1954).

11oA. l11

1.1.


39

loss of electrons in passing through thin foils. The straggling of the energy loss in thin foils was already clearly demonstrated in 1928 by the work of White and Millington,"2 as well as that of Madgwick.'13 More recently, Paul and Reich114measured the energy loss of 2.8-Mev and 4.7-Mev electrons in foils of Be, C, Fe, and Pb. Chen and Warshaw116showed that eprob for electrons with energies T , < 2 Mev is correctly given by Landau's theory.63 However, from their data they were unable to discriminate between the Landau distributionKaof energy losses, and the (somewhat wider) distribution of Blunck and Leisegang.66On the other hand, in the experiments of Kalil and Birkhoff,'ls an accurate comparison could be made with the Blunck-Leisegang distribution, and it was found that while for the heavy elements (e.g., Pb) this distribution is in essential agreement with the observations, for the light elements (e.g., Be) the predicted width of the distribution at half-maximum is too small by a factor of -1.8. However, the discrepancy for light elements was not observed in a more recent experiment by Hungerford and Birkhoff."' ~ ~ good ~ agreement with the BlunckKageyama et ~ 1 also. obtained Leisegang distribution for foils of Al, Cu, In, and Pb. Goldwasser, Mills, and Hansonllg have measured the energy loss of 15.7-Mev electrons in passing through thin samples of Be, polystyrene, Al, Cu, and Au. With the exception of Au, they found that good agreement for eproo could be obtained by using the asymptotic value of the density effect correction [Eq. (1.1.2Oa)l. For the case of Au, the expression for 6 for intermediate energies, Eq. (1.1.20), must be used, as was pointed out by Warner and Rohrlich.120The energy loss distributions of Goldwasser et aZ.l19 are in essential agreement with those predicted from the Landau theory.6a Goldwasser, Mills, and RobillardlZ1have obtained a direct demonstration of the density effect, by measuring the energy loss of 15.7-Mev electrons in (solid) Teflon and Kel-F, and then in the gases corresponding to Teflon and Kel-F (i.e., gases having the same chemical composition). It was found that the difference between the values of Cprob P. White, and G. Millington, Proc. Roy. SOC.A120, 701 (1928). E. Madgwick, Proc. Cambridge Phil. SOC.23, 970 (1927). 114 W. Paul and H. Reich, 2.Physik 127, 429 (1950). 116 J. J. L. Chen and S. D. Warshaw, Phys. Rev. 84, 355 (1951). 116 F. Kalil and R. D. Birkhoff, Phys. Rev. 91, 505 (1953). 117E. T. Hungerford and R. D. Birkhoff, Phys. Rev. 96, 6 (1954). 118 S. Kageyama, K. Nishimura, and Y. Onai, J . Phys. Sor. Japan 8, 682 (1953); Kageyama, S., and Nishimura, K., J . Phys. SOC.Japan 7 , 292 (1952). 119 E. L. Goldwasser, F. E. Mills, and A. 0. Hanson, Phys. Rev. 88, 1137 (1952). 120 C. Warner and F. Rohrlich, Phys. Rev. 93,406 (1954). lZ1 E.L. Goldwasser, F. E. Mills, and T. R. Robillard, Phys. Rev. 98, 1763 (1955); see also A. M. Hudson, Phys. Rev. 106, 1 (1957). 112

11*

40

1.

PARTICLE DETECTION

in the solid and the gaseous phases is given by the predicted density effect correction 6. The ionization loss is rapidly becoming an important tool in bubble chamber investigations. The bubble count (number of bubbles per cm of path) is a function only of the velocity of the particle and the temperature . the ~ first ~ to~make .a systematic study of of the liquid. Glaser et ~ 1 were the bubble count as a function of the velocity of the particle, by using secondary protons and T+ mesons of momenta between 0.53 and 1.60 Bev/c from the Brookhaven Cosmotron. They found that the bubble density b is approximately proportional to 1/P2. This indicates that the bubble formation is proportional to the number of slow 6 rays (secondary elect,rons). The number of 6 rays per gm/cmZ is given by

where El' is the lower limit and Ez' is the upper limit of the energies of the 6 rays considered in ns; El' and E l are in electron volts. I n similarity to the grain count in emulsion or the drop count for cloud-chamber tracks, E 2 is taken as the energy of a S ray that has a long enough range to extend to a visible distance from the track of the incident (primary) particle. One thus obtains E2' = 50 kev. El' is taken as -3 times the mean excitation potential I of the atoms of the liquid, so that the 6 rays with energy E1' can be treated as free electrons during the collision. Thus El' is a t most a few kev, and therefore 126 is not very sensitive to the precise value of E i , since 1/Ez/ )> ~ Vo) (1.1.37) where v o = e2/h is the velocity of an electron in the first Bohr orbit of hydrogen (of radius aH),and v is the velocity of the incident particle. Equation (1.1.37) holds for heavy eIements, which have several atomic electrons with velocities vel larger than v. For light elements, where this condition is not fulfilled, Brinkman and K r a m e r P have derived the following formula for u c : UC

uc =

(2’~,/5)a,2zz(v,/v)’2

which is expected to hold for ( v / v o )

(1.1.38)

2 10.

124 G. A. Blinov, Iu. S. Krestnikov, and M. F. Lomanov, Zhur. Eksptl. i Teoret. Fiz. 31, 762 (1956); [translation: Soviet Phys. JETP 4 , 661 (1957)l. 126 N. Bohr, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd. 18, No. 8 (1948). 126 H. C. Brinkman and H. A. Kramers, Koninkl. Ned. Akad. Wetenschap., Proc. 33,

973 (1930).

42

1. PARTICLE DETECTION

The theory of the loss of electrons by a charged particle (ion) is less complicated. The cross section for loss al is of the order of r u H z S10-16 cm2 for v V O , i.e., for protons of -25 kev. uz falls off rather slowly with increasing v. The capture and loss cross sections of protons are equal (a, = u l ) for air at -20 kev and for hydrogen a t -50 kev. Above this energy, we have ae < a. For intermediate 2 values, BohrlZ6has obtained the followingestimate of a1: a1 7raH222/3(421). (1.1.39)

-

-

Thus for protons, q / a Cvaries as Z 1 / 8 T 6 for / 2 medium 2. The ratio is nearly independent of 2 and increases rapidly with energy. In agreement with this result, it is found experimentally that at predominates rapidly over ac as the energy is increased above the value (-25 kev) for which ac = a ~ . The processes of capture and loss of electrons are very important for the energy loss and range of fission fragment^.^ 1.1.2.11. The Stopping Power at Very Low Energies. For very low energies, where the velocity of the particle is less than the velocity of the atomic electrons ( T , 5 25 kev for protons), Fermi and Teller127have obtained the following expression for the energy loss: (1.1.40) where Ry is the Rydberg unit, and urn is the maximum velocity of the electrons of the substance if the latter are regarded as constituting a where TL is the number degenerate Fermi gas. Thus urn = (3~~/8?r)'/~(h/m), of electrons per ~ m Equation . ~ (1.1.40) shows that the energy loss increases with increasing v in this region, in contrast to the decrease with increasing v at higher energies ( = l / v 2 ) . Experimentally, good evidence has been obtained for the increase of the stopping power with increasing velocity a t low energies. Warshawl28 has made careful measurements of dE/dx for protons in Be, Al, Cu, Ag, and Au in the energy range from 50 to 400 kev. For all cases, he obtained a maximum of d E / d x in the neighborhood of T, = 100 kev. In the region below the maximum, an extrapolation of Warshaw's results (from -50 to -25 kev) could be well fitted by the Fermi-Teller formula. The maximum of the ionization loss dE/dx, to be denoted by J,, has the value 640 Mev/gmcm-2 for Be, where it occurs a t TP.,= 75 kev. For Al, J, = 440 Mev/gm cm+ and T,,, = 72 kev. For Cu, Ag, and Au, J , = 230, 140, and 100 Mev/gm cm-2, respectively. The corresponding values of T,,, are 140, 160, and 160 kev, respectively. 127

la*

E. Fermi and E. Teller, Phys. Rev. 73, 399 (1947). S. D.Warshaw, Phys. Rev. 76, 1759 (1949).

1.1.


-

43

For somewhat higher energies (T, 400 kev), above T,,,, BohrlZ6has given an approximate theory based on the Thomas-Fermi model of the atom, and has obtained the following expression for the energy loss: dE-- 1 6 ~ n Z l ' ~ h e ~ -dx mu

(1.1.41)

In this region, -dE/dx goes as l / u , instead of the l / u 2 dependence which prevails at somewhat higher energies. Warshaw l Z 8has also obtained reasonable agreement of Eq. (1.1.41) with stopping power data for Cu, Ag, and Au in the range from T, = 350 to 550 kev. For a more detailed discussion of the stopping power measurements at low energies ( T , 2 2 MeV) the reader is referred to the review article of Allison and Warshaw.60 1.1.2.12. The Energy w Required to Produce an Ion Pair in a Gas. When a heavy charged particle passes through a medium, it excites and ionizes the atoms of the material. The ion pairs which are formed by direct action of the particle in the immediate vicinity of its path are called the primary ions. The most energetic of these primary ions, called delta rays, may travel a considerable distance before being themselves stopped by the medium. I n the slowing down process, the delta rays produce additional ions called secondary ions. The sum of the primary and secondary ionization constitutes the total ionization produced by the passing charged particle. It has been found experimentally that the energy w required to produce an ion pair is approximately independent of the energy and charge of the incident particle. Moreover, w does not vary appreciably for different gases, all values being of the order of 25-35 ev/ion pair. Typical values of w, obtained by Jesse and S a d a ~ k i s are ' ~ ~ as follows: 36.3 ev for H2, 42.3 ev for He, 35.0 ev for N2, 26.4 ev for Ar, 34.0 ev for air, and 32.9 ev for COz. Fano130has proposed a theory which explains both the fact that w is independent of the energy and charge of the incident particle, and also the smallness of the variation of w with the atomic number 2. The constancy of w as a function of E has been widely used in ionization chambers for the determination of the energy of particles. Thus if a particle is stopped in the gas of an ionization chamber, its initial energy is proportional to the total number of ions produced, which can be electronically measured by means of a linear amplifier. It is necessary to know the value of w for the gas in the ionization chamber; w can be determined by measuring the number of ions produced by a particle whose lZ9 W. P. Jesse and J. Sadaukis, Phys. Rev. 107, 766 (1957); see also T. E. Bortner and G. S. Hurst, Phys. Rev. 95, 1236 (1954); R. H. Frost and C. E. Nielsen, Phys. Rev. 91, 864 (1953). u0U. Fano, Phys. Rev. 70, 44 (1946); 72, 26 (1947).

44

1.

PARTICLE DETECTION

energy is known by other means (e.g., natural a particle or particle originating from an exothermic nuclear reaction). 1.1.3. Range-Energy Relations The mean range R of a particle of kinetic energy TI is given by (1.1.42) where -(l/p)(dE/dx) is the average energy loss as obtained from Eq. (1.1.1) or (1.1.19). In this section, we shall restrict ourselves to particles heavier than electrons, since the range of high-energy electrons is determined by the bremsstrahlung and shower production rather than the ionization loss, as will be discussed in Section 1.1.5 [see Eqs. (1.1.81) and (1.1.82)]. 1.1.3.1. Summary of Range-Energy Relations. Range energy relations have been obtained by several authors. I n 1937, Livingston and Bethe36 published range-energy relations for protons, deuterons, and a particles in air. In obtaining these results, various experimental data on the ranges of natural a particles were used. The ranges of a particles and protons are related by (1.1.43) R,(T,) = 1.0072 R,(3.971 T,) - 0.20 cm where R,(T,) is the proton range for an energy T p ( E*T,), and R,(T,) is the a-particle range for an energy T,; 1.0072 = za2(mp/ma);3.971 = (m,/m,) [Eq. (1.1.44)], and the constant term131-0.20 cm is due to the capture and loss of electrons at low energies which has a somewhat different effect on a particles and protons. The proton range-energy relation of Livingston and Bethe extends up to 15 MeV. In 1947, Smith1S2obtained range-energy relations for protons up to 10 Bev, both for air and aluminum. For air, Smith used the same value of I as Livingston and Bethe:36 Iair = 80.5 ev; for Al, Wilson’s value3’ l a l = 150 ev was used. Somewhat later, Aron el aL2 calculated proton range-energy relations for a number of metals and gases, up to 10 Bev, using a value of I = 11.52 ev, which was essentially derived from Wilson’s result for Al. The calculations of Aron et al. as well as those of Smith neglect the density effect, which becomes important for proton energies T, above -2 Bev. The tables of Aron et al. have been extended by Rich and made^.'^^ A summary of the range measurements at various energies has been given by Bethe and A s h k h 3 P. M. S. Blackett and L. Lees, Pmc. Roy. SOC.A134, 658 (1932). J. H. Smith, Phys. Rev. 71,32 (1947). l a 3M. Rich and R. Madey, University of California Radiation Laboratory Report UCRL-2301 (1954). l31

182

1.1.


45

There have been several determinations of the range-energy relation for nuclear emulsion. In 1953, V i g n e r ~ n 'obtained ~~ a range-energy relation, based on older data, particularly those of R ~ t b l a t , and ' ~ ~ Cuer and Jung. l3-5 Vigneron's results were later extended by Barkas and Young.137 Calculations of the range-energy relation for high energies, including the density effect correction in dE/dx, have been carried out by Baroni et ~ 1 . Friedlander, l ~ ~ Keefe, and M e n ~ n , ' ~have ~ " made a comparison of the ranges in emulsion and in aluminum for protons of energies 87, 118, and 146 MeV. Recently, Barkas and his c o - w o r k e r ~have ~ ~ ~ made very extensive measurements of the ranges in Ilford G5 emulsion, taking into account the effect of the water content of the emulsion on the rangeenergy relation. The water content determines the density of the emulsion. Barkas140 has calculated a new and very accurate range-energy relation for Ilford G5 emulsion for a "standard density" of 3.815 gm/cm3, and has given the correction which must be applied to ranges measured under nonstandard conditions to obtain the corresponding ranges for the standard density (and hence the energy of the particle). In obtaining the values of d E / d x used in calculating the range [Eq. (1.1.42)], Barkas has included both the shell correction U at low energies and the density effect correction 6 a t high energies. The mean excitation potential I was used as an adjustable parameter, to be determined so as to give the best fit of R ( T ) t o the available range measurements. I n this manner, a value I = 331 f 6 ev was obtained, which gives an average I / Z = 12.1 & 0.2 ev for the elements of emulsion (excluding the hydrogen). This value of I/2 is in good agreement with the recent results of Bichsel el aL4' and of Burkig and M a c K e n ~ i e . ~ ~ 1.1.3.2. Calculations of the Range-Energy Relations of Protons for 6 Substances. As mentioned above, the range-energy tables of Aron et aL2 do not take into account the density effect correction 6. Moreover, these tables were calculated for an excitation potential I = 11.52 ev, which is somewhat lower than the most recent value, I 12.5-132 ev, as obtained

-

l a 4 L.

Vigneron, J. phys. radium 14, 145 (1953) J. Rotblat, Nature 167, 550 (1951). 136 P. Cuer and J. J. Jung, Sci. et ind. phof. 22, 401 (1951). 137 W. H. Barkas and D. M. Young, University of California Radiation Laboratory Report UCRL-2579, revised (1954). 188 G. Baroni, C. Castagnoli, G. Cortini, C. Franzinetti, and A. Manfredini, Report BS-9, Istituto di Fisica dell'Universit8, Rome, 1954. la*s M. W. Friedlander, D. Keefe, and M. G . K. Menon, Nuovo cimenfo [lo] 6, 461 (1957). 189 W. H. Barkas, P. H. Barrett, P. Cuer, H. H. Heckman, F. M. Smith, and H. K. Ticho, Phys. Rev. 102, 583 (1956); Nuovo cimento [lo] 8, 185 (1958). $40 W. H. Barkas, Nuovo cimento [lo] 8, 201 (1958). la6

46

1.

PARTICLE DETECTION

from references 47 and 49. Sternheimer' has carried out calculations to determine new range-energy relations for some of the commonly used materials, using the higher I values and including the density effect correction. Range-energy relations have been obtained for 6 substances : Be, C, Al, Cu, Pb, and air, for proton energies T , from 2 Mev to 100 Bev. The reason for choosing 100 Bev as the upper limit of the tables is th a t 10 Bev, upon this enables one to obtain pmeson ranges u p to T, applying a small correction for the fact th at the maximum energy transfer W,,, becomes slightly dependent on the mass ratio m,/m a t the highest energies considered. This correction will be given below. The ranges obtained in this work' are -1 to -9% higher for T , = 10 Bev than t,hose of Aron et aL2 and of Smith.132 The largest differences occur for Be (9.2%) and C (6.4%). Details of the calculations of the ranges are given in reference 1. The values of - ( l / p ) ( d E / d z ) were calculated from Eq. (1.1.1). The mean excitation potentials I were obtained from the work of C a l d ~ e 1 1 , ~ ~ . ~ ~ following I values were Bichsel et U Z . , ~ ~and Burkig and M a ~ K e n z i eThe used: IBe = 64 ev, Ic = 78 ev, Iair= 94 ev, IAl= 166 ev, Icu = 371 ev, and Ipb= 1070 ev. The density effect correction 6 was evaluated from the calculations of Sternheimer.20*21 The K and L shell corrections C K and CL a t low energies were obtained from Walske's Table I1

-

TABLE 11. Values of the Constants Used t o Obtain the Ionization Loss' Z is the mean excitation potential. A and B are the constants appearing in Eq. (1.1.19). C, a, s, X O ,and XI enter into the expression for the density effect correction 6 [Eqs. (1.1.20)and (1.1.2Oa)l. Material Z (ev) A

Be C A1 Cu

Pb Air

64 78 166 371 1070 94

(s) B

0.0681 0.0768 0.0740 0.0701 0.0608 0.0768

18.64 18.25 16.73 15.13 13.01 17.89

-C

2.83 3.18 4.25 4.71 6.73 10.70

a

S

0.413 0.509 0.110 0.118 0.0542 0.126

2.82 2.67 3.34 3.38 3.52 3.72

xo

XI

-0.10 -0.05 0.05 0.20 0.40 1.87

2 2 3 3 4 4

gives the values of the constants which were used in the calculation of the ionization loss [Eq. (1.1.19)] and the density effect correction 6 [Eqs. (1.1.20) and (1.1.2Oa)l. Table I11 gives the values of - (l/p)(dE/dx) for protons. The resulting range-energy relations are presented in Table IV. Recently Sternheimer140ahas derived an expression for the rangeenergy relation R(T,) for protons as a function of the mean excitation 140s

R. M. Sternheimer, Phys. Rev. 118. 1045 (1960).

1.1. I N T E R A C T I O N

O F RADIATION W I T H MATTER

47

TABLE 111. Values of the Ionization Loss -(l/p)(dE/dz) (in Mev/gm crn-l) for Protons in Be, C, Al, Cu, Pb, and Air1

2 3 4 5 6 7 8 9 10 12 14 16 18 20 22.5 25 27.5 30 35 40 45 50 55 60 65 70 75 80 90 100 110 120 130 140 150 160 180 200 225 250 275 300 325 350 375

131.9 97.45 78.06 65.59 56.69 50.15 45.03 40.99 37.63 32.44 28.62 25.65 23.30 21.38 19.41 17.80 16.47 15.34 13.53 12.15 11.05 10.15 9.412 8.788 8.254 7.791 7.385 7.026 6.424 5.933 5.527 5.187 4.896 4.644 4.424 4.232 3.908 3.647 3.384 3.173 3.000 2.853 2.730 2.625 2.534

140.6 104.4 83.97 70.74 61.29 54.28 48.81 44.47 40.87 35.29 31.17 27.96 25.42 23 34 21.21 19.46 18.01 16.79 14.82 13.32 12.12 11.14 10.33 9.645 9.062 8.556 8.112 7.719 7.061 6.526 6.079 5.706 5.388 5.112 4.872 4.659 4.304 4.016 3.728 3.497 3.307 3.148 3.013 2.896 2.797

A1

cu

Pb

Air

110.8 83.16 67.44 57.19 49.84 44.38 40.09 36.67 33.80 29.35 26.04 23.45 21.39 19.70 17.95 16.52 15.32 14.31 12.67 11.41 10.41 9.584 8.902 8.325 7.831 7.402 7.026 6.693 6.132 5.674 5.292 4.973 4.700 4.464 4.258 4.077 3.768 3.522 3.272 3.072 2.908 2.771 2.655 2.555 2.469

78.93 61.83 51.27 44.08 38.73 34.71 31.50 28.94 26.77 23.38 20.83 18.82 17.22 15.91 14.54 13.42 12.48 11.68 10.38 9.383 8.584 7.925 7.378 6.914 6.514 6.167 5.861 5.590 5.133 4.760 4.449 4.187 3.961 3.767 3.594 3.445 3.192 2.989 2.783 2.616 2.480 2.366 2.268 2.185 2.112

41.14 34.62 29.85 26.36 23.65 21.54 19.81 18.40 17.18 15.23 13.73 12.52 11.54 10.73 9.874 9.163 8.564 8.050 7.203 6.548 6.020 5.581 5.213 4,900 4.629 4.391 4.181 3.996 3.682 3.424 3.209 3.027 2.870 2.734 2.616 2.511 2.333 2.189 2.042 1.924 1.828 1.747 1.678 1.619 1.568

134.0 99.86 80.53 68.00 58.99 52.32 47.11 42.96 39.51 34.15 30.20 27.10 24.66 22.66 20.61 18.93 17.53 16.35 14.44 12.98 11.82 10.87 10.09 9.420 8.852 8.360 7.928 7.546 6.904 6.382 5.950 5.587 5.276 5.007 4.773 4.567 4.221 3.942 3.660 3.434 3.248 3.093 3.961 2.848 2.751

48

1.

PARTICLE DETECTION

TABLE 111. Values of the Ionization Loss - (l/p) ( d E / d x ) (in Mev/gm crn-)) for Protons in Be, C, Al, Cu, Pb, and Air1 (Continued)

400 450 500 550 600 700 800 900 1000 1250 1500 1750 2000 2250 2500 2750 3000 3500 4000 4500 5000 6000 7000 8000 9000 10,000 12,500 15,000 17,500 20,000 22,500 25,000 27,500 30,000 40,000 50,000 60,000 70,000 80,000 90,000 100,000

2.453 2.321 2.215 2.129 2.059 1.950 1.871 1.812 1.767 1.692 1.649 1.623 1.608 1.599 1.595 1.593 1.593 1.597 1.604 1.612 1.621 1.638 1.655 1.670 1.685 1.699 1.728 1.753 1.774 1.792 1.808 1.822 1.835 1.847 1.886 1.915 1.939 1.959 1.976 1,991 2.005

2.709 2.563 2.448 2.355 2.278 2.159 2.074 2.009 1.960 1.879 1.833 1.806 1.791 1.782 1.778 1.777 1.778 1.784 1.793 1.802 1.813 1.834 1.854 1.873 1.890 1.905 1.939 1.968 1.993 2.014 2.033 2.050 2.065 2.077 2.122 2.156 2.183 2.206 2.225 2.242 2.257

2.392 2.268 2.169 2.090 2.022 1.921 13 4 9 1.795 1.754 1.687 1.649 1.629 1.618 1.613 1.611 1.613 1.615 1.624 1.635 1.647 1.659 1.682 1.704 1.724 1.743 1.759 1.796 1.827 1.853 1.876 1.895 1.913 1.929 1.944 1.991 2.027 2.056 2.080 2.100 2.118 2.134

2.049 1.945 1.863 1.795 1.741 1.658 1.598 1.555 1.522 1.471 1.443 1.429 1.422 1.420 1.422 1.425 1.429 1.440 1.452 1.465 1.478 1.502 1.524 1.544 1.562 1.579 1.615 1.645 1.671 1.693 1.712 1.729 1.745 1.759 1.804 1.839 1.866 1.890 1.909 1.926 1.941

1.523 1.448 1.390 1.343 1.305 1.246 1.205 1.175 1.153 1.120 1.104 1.099 1.099 1.102 1.108 1.114 1.121 1.135 1.150 1.164 1.178 1.204 1.227 1.248 1.267 1.284 1.321 1.351 1.377 1.399 1.418 1.436 1.451 1.465 1.511 1.546 1.574 1.597 1.616 1.633 1.648

2.666 2.524 2.413 2.323 2.249 2.136 2.055 1.995 1.950 1.877 1.838 1.819 1.809 1.806 1.808 1.812 1.818 1.834 1.851 1.870 1.889 1.924 1.958 1.989 2.017 2.044 2.102 2.151 2.194 2.232 2.265 2.296 2.323 2.348 2.433 2.499 2.552 2.597 2.631 2.661 2.687

1.1.


49

TULE IV. Range-Energy Relations for Protons in Be, C, Al, Cu, Pb, and Air1 The range R is given in grn cm-z.

TP (MeV) 2 3 4 5 6 7 8 9 10 12 14 16 18 20 22.5 25 27.5 30 35 40 45 50 55 60 65 70 75 80 90 100 110 120 130 140 150 160 180 200 225 250 275 300 325 350 375

Be

C

A1

cu

Pb

Air

0.0091 0.0180 0.0296 0.0436 0.0601 0.0789 0.0999 0.1232 0.1487 0.2061 0.2719 0.3459 0.4278 0.5175 0.6404 0.7750 0.9212 1.079 1.426 1.817 2.249 2.722 3.234 3.784 4.371 4.995 5.655 6.349 7.840 9.461 11.21 13.08 15.06 17.16 19.37 21.68 26.61 31.91 39.03 46.67 54.78 63.33 72.30 81.64 91.34

0.0084 0.0168 0.0275 0,0406 0.0558 0.0732 0.0926 0.1141 0.1376 0.1904 0.2508 0.3187 0.3937 0.4759 0.5884 0.7116 0.8452 0.9891 1.307 1.663 2.057 2.488 2.954 3.456 3.991 4.559 5.160 5.792 7.148 8.623 10.21 11.91 13.72 15.62 17.63 19.73 24.20 29.02 35.49 42.42 49.77 57.53 65.65 74.12 82.91

0.0115 0.0221 0.0355 0.0517 0.0704 0.0917 0.1155 0.1416 0.1700 0.2337 0.3062 0.3872 0.4766 0.5742 0.7073 0.8526 1.010 1.179 1.551 1.967 2.427 2.928 3.469 4.051 4.670 5.327 6.021 6.750 8.313 10.01 11.84 13.79 15.86 18.04 20.34 22.74 27.85 33.34 40.72 48.61 56.98 65.79 75.02 84.62 94.58

0,0190 0.0335 0.0513 0.0724 0.0967 0.1240 0.1542 0.1874 0.2234 0.3035 0.3943 0.4954 0.6066 0.7276 0.8922 1.071 1.265 1.472 1.927 2.434 2.992 3.599 4.253 4.954 5.699 6.488 7.321 8.195 10.06 12.09 14.27 16.58 19.04 21.63 24.35 27.19 33.23 39.71 48.39 57.66 67.49 77.82 88.61 99.85 111.5

0.0410 0.0676 0.0988 0.1345 0.1746 0.2190 0.2674 0.3198 0.3761 0.5000 0.6385 0.7912 0.9576 1.138 1.381 1.644 1.926 2.229 2,885 3.614 4.411 5.275 6.202 7.192 8.243 9.352 10.52 11.74 14.35 17.17 20.19 23.40 26.80 30.37 34.11 38.02 46.29 55.14 66.98 79.61 92.95 107.0 121.6 136.7 152.4

0.0087 0.0175 0.0287 0.0423 0.0581 0.0761 0.0963 0.1185 0.1428 0.1974 0.2598 0.3299 0.4073 0.4920 0.6078 0.7346 0.8720 1.020 1.346 1.712 2.116 2.557 3.035 3.549 4.097 4.678 5.293 5.940 7.327 8.835 10.46 12.20 14.04 15.99 18.03 20.17 24.73 29.64 36.23 43.29 50.79 58.68 66.95 75.56 84.50

50

1.

P AR T I C L E DETECTION

TABLE IV. Range-Energy Relations for Protons in Be, C, Al, Cu, Pb, and Air1 (Continued)

400 450 500 550 600 700 800 900 1000 1250 1500 1750 2000 2250 2500 2750 3000 3500 4000 4500 5000 6000 7000 8000 9000 10,000 12,500 15,000 17,500 20,000 22,500 25 ,000 27 ,500 30,000 40 ,000 50,000 60,000 70,000 80,000 90,000 100,000

101.4 122.3 144.4 167.5 191.3 241.3 293.7 348.1 404.0 548.9 698.8 851.7 1007 1163 1319 1476 1633 1946 2259 2570 2879 3493 4100 4702 5298 5889 7347 8784 10,202 11,604 12,993 14,370 15,737 17,095 22,450 27,711 32 ,899 38,030 43,112 48,152 53 , 158

91.99 111.0 131.0 151.8 173.4 218.6 265.9 314.9 365.3 495.8 630.7 768.2 907.3 1047 1188 1328 1469 1750 2029 2308 2584 3133 3675 4212 4743 5270 6570 7850 9112 10,359 11,595 12,820 14,036 15,243 20,003 24,677 29,286 33,843 38,356 42 ,833 47,278

104.9 126.4 148.9 172.4 196.7 247.6 300.7 355.6 412.0 557.7 707.7 860.4 1014 1169 1324 1479 1634 1943 2250 2555 2857 3456 4046 4629 5206 5777 7183 8563 9922 11,262 12 ,588 13,901 15,202 16 ,494 21,574 26,550 31 ,448 36,284 41,067 45,807 50,509

123.5 148.6 174.9 202.2 230.5 289.5 350.9 414.4 479.4 646.8 818.7 992.9 1168 1344 1520 1696 1871 2220 2566 2908 3248 3919 4580 5232 5876 6512 8077 9610 11,117 12,604 14,072 15,525 16,964 18,391 24,002 29,491 34 ,888 40,214 45,477 50,692 55,863

Pb

Air

168.6 202.3 237.6 274.2 312.0 390.5 472.2 556.3 642.2 862.7 1088 1315 1543 1770 1996 222 1 2445 2888 3326 3758 4185 5024 5847 6655 7450 8234 10,153 12,023 13,856 15,657 17,432 19,184 20,915 22,629 29,344 35,883 42,290 48,596 54,820 60 ,975 67,070

93.73 113.0 133.3 154.4 176.3 222.0 269.8 319.2 370.0 500.9 635.7 772.5 910.3 1049 1187 1325 1463 1737 2008 2277 2543 3067 3583 4089 4589 5081 6287 7462 8612 9742 10,853 11,950 13,032 14,102 18,282 22 ,336 26,295 30,177 34,002 37,781 41,519

1.1.


51

potential I . The expression for R(T,) is obtained by an interpolation of the previously calculated range-energy relations’ for Be, Al, Cu, and P b (see Table IV). The result for R(T,) is accurate to within 1% for values of I between 60 and 1100 ev. The expression for R(T,) can be used to calculate the range-energy relation for any substance, provided a n appropriate value of I is assumed. 1.1.3.3. Range-Energy Relations for Particles other than Protons. The Correction Factor Fi. The range R for any other particle i (heavier than an electron) with energy Tican be obtained from the proton ranges of Table IV by means of the relation

(tl)

R;(Ti) = - - R, zi2

)

3 Ti F i

(mi

( 1.1.44)

where z, is the charge of the particle, mi is its mass, mp = proton mass, and R,[(mp/m,) Ti] is the proton range for the appropriate energy (m,/m,) Ti. I n Eq. (1.1.44), the factor Fi corrects for the slight dependence of the maximum energy transfer W,,, on mi a t very high energies. Thus Wmax for 1.1, T , and K mesons is slightly smaller than for protons with the same value of yi = Ei/mic2, where Ei is the total energy (including rest mass) of the particle. Hence - (l/p) (dE/dz) is decreased and the range Riis slightly increased for mesons (Fi > 1). From Eqs. (1.1.1) and (1.1.15), one finds that the change of - (l/p)(dE/dz) is given by

Values of F, for p mesons are given in Table V. These values were obtained’ by numerical integration of Eq. (1.1.42) with - (l/p) (dE/dz) calculated from the appropriate W,,, for p mesons. Table V shows that the correction for 1.1 mesons is very small ( F , - 1 5 0.01), and that the values of F , are practically independent of 2, being nearly the same for Be and Pb. For 7r and K mesons, the corrections F , and F K are not tabulated, since one will not generally be interested in the ranges of these particles for y i 2 5 , in view of the large probability that they will interact before coming t o the end of the range. Actually for a given yi, the corrections are even smaller than for 1.1 mesons. Thus for Pb, F , = 1.0095 for Y~ = 100, and F K = 1.0017 for Y K = 100. It should be noted that a t very high energies [E >> (mi2/m)c2], spindependent effects on the energy loss in close collisions will be present,141 which are not included in the Bethe-Bloch formula. 1.1.3.4. Range Straggling. When a beam of particles loses energy by 141 See, for example, B. Rossi, York, 1952.

“

High-Energy Particles,” p. 14. Prentice-Hall, New

1.

52

PARTICLE DETECTION

TABLE V. Values of the Factor F, Which Enters into the Expression for the p-Meson Range R,, at Very High Energies [Eq. (1.1.44)11 F,, is given for Be and Pb, as a function of 7, = E,/m,,ea.

4 6

8 10 15 20 25 30 40 50 60 70 80 90 100

1.0013 1.0017 1.0021 1.0025 1.0032 1.0039 1.0047 1.0054 1.0066 1 -0077 1.0088 1.0098 1.0107 1.0116 1.0125

1.0010 1.0014 1.0017 1,0020 1.0027 1.0034 1,0041 1.0047 1.0058 1.0068 1.0079 1.0089 1.0098 1.0107 1.0115

ionization, all of the particles do not come to the end of their range and stop after traversing the same thickness of material. Instead there is a distribution of the ranges due to the statistical nature of the ionization loss process. This distribution is a Gaussian. The probability p(R) dR of a particle of well-defined initial energy T ohaving a range between R and R 4-dR is given by

p(R)dR

= __ 1 cYT1'2

[

exp - ( R

a~0)2] dR

(1.1.46)

where CY'

E

2(R - RQ);,= 2Jp(R)(R- Ro)'dR.

(1.1.47)

In Eqs. (1.1.46) and (1.1.47), Ro is the mean range142obtained above [Eq. (1.1.42)]by integration over the average energy loss - (l/p)(dE/dx). An approximate equation for ( R - Ro);, has been obtained by Bohr.12 For sufficiently large initial velocities v of the particle (2mv2 2 I K , where IK is the ionization potential of the K shell), Bohr's formula gives

loTo ( d E / d ~ )dT -~

( R - Ro);, = 4re4z2NZ

(1.1.48)

where T Ois the initial energy of the particle, N = number of stopping atoms per ema, and Z = atomic number of stopping material. I n practice, it is difficult to obtain directly the value of ( R - RO);, from the observed distribution of ranges. Instead one obtains the number142 See

also H. W. Lewis, Phys. Rev. 86, 20 (1952); U. Fano, ibid. 92, 328 (1953).

1.1.


53

FIG.6. Schematic number-distance curve, showing RQ, Rextrand S (cf. Bethe and Ashkin, reference 3, p. 246, Fig. 15b).

distance curve by plotting the number of particles which survive as a function of the thickness traversed. An example of such a curve is shown in Fig. 6. From the Gaussian of Eq. (1.1.46), one finds that the curve of N has half its maximum value, N = +No, at R = Ro,the mean range of the particles which are assumed t o be initially homogeneous in energy. Moreover, a t R = Ro,the theoretical curve of N versus R has its maximum slope, - (1/ad2).By drawing a tangent to the N versus R curve a t its steepest point ( R = Ro)and obtaining the intersection of the tangent with the R axis (see Fig. 6), one finds the extrapolated range, Re,,,, which is given by (1.1.49) Re,,= = RO TT 1 112a.

+

The difference Rextr- ROis defined3 as the straggling parameter S. Thus

S2 = h a 2 = +(R - Ro);,

(1.1.50)

and the distribution function p(R)dR [see Eq. (1.1.46)] can be written as f o l l o ~ s : ~ 1 (1J.51) p(R)dR = - exp ( R - R d 2 ]dR. 2s

[ (&)

The value of ( R - Ro);, obtained from the measured S by means of Eq. (1.1.50) can be compared with the theoretical expression, Eq. (1.1.48). Good agreement has been obtained in several experiments. As an example, the calculated percentage straggling of protons in 100S/Ro decreases slowly from 2.29 for RO= 5 cm, to 2.13 a t 149 H. A. Bethe and J. Ashkin, in “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. 1, p. 244. Wiley, New York, 1953.

54

1.

PARTICLE DETECTION

10 cm, 1.78 at 100 cm, 1.57 a t lo3 cm, 1.31 a t lo4cm, and 0.97 a t lo5em. Millburn and S ~ h e c t e r 'have ~ ~ found experimentally that SIRo varies very slowly with 2. Thus the value of SIR0 relative t o copper is 0.90 for Be, 0.95 for Al, 1.02 for Ag, and 1.06 for Pb. The range straggling in emulsion has been thoroughly investigated by Barkas et U Z . , ' ~ ~ who used protons and T+, T - , and p+ mesons, These authors have found that the Bohr formula [Eq. (1.1.48)] gives reasonable agreement with their data. They have also discussed various additional straggling effects which are present in range measurements with nuclear emulsion. Values of the range straggling for six substances (Be, C, Al, Cu, Pb, and air) have been recently calculated by Sternheimer.'45a In these calculations, the following expression for ( R - Ro)i,was used:

Equation (1.1.48a) differs from the Bohr formula [Eq. (1.1.48)] by the inclusion of the following factors in the integrand: (1) the factor (1 - &P2)/ (1 - pz), which is a relativistic correction that was first derived b y Lindhard and S ~ h a r f f (2) ; ~ ~the factor [l (2m/mi)y]-' which is derived in reference 145a, and which becomes important only for very high energies (y >, mi/2m); (3) the correction factor K which takes into account the effects of binding on the atomic electrons a t low velocities of the incident particle [v 5 (IK/2m)1'2].The correction K is similar to the binding effect corrections C K and CL which enter into the Bethe-Bloch formula [Eq. (1.1.34)]. K becomes 1 for sufficiently high energies ( T , 2 100 Mev for Al; 400 Mev for Pb). The expression for K has been obtained by Bethe.3v36 / R ~ The percentage range straggling t = 100[(R - R O ) ~ " ~decreases with increasing energy until a minimum is reached for T,/mLc2 2.5, which is in the same region as the minimum of the ionization loss d E / d x . Beyond the minimum, e increases with energy, as a result of the effect of the factor (1 - p2)-' in the integrand of (1.1.48a). We note th a t c as defined above is related to S by: t = 100(2/~)"~S/Ro.As a n example, for p mesons in Cu, e, decreases from 3.94 a t T , = 10 Mev to a minimum of 2.69 a t 280 MeV, and then increases to 3.07 at T, = 1 Bev, 4.07 at 3 Bev, and 5.74 a t 10 Bev. It is that B is almost independent of 2, showing only a small increase in going from Be to Pb (at a constant

+

-

144 G. P. Millburn and L. Schecter, University of California Radiation Laboratory Report UCRL-2234, revised (1953). l P sW. H. Barkas, F. M. Smith, and W. Birnbaum, Phys. Reu. 98, 605 (1955). 145B R. M.Sternheimer, Phye. Rev. 117, 485 (1960).

1.1.


55

energy T). As an example, for 300-Mev protons, Q,(Z)/~,(CU) is 0.885 for Be, 0.895 for C, 0.941 for Al, and 1.113 for Pb.

1.1.4. Scattering of Heavy Particles by Atoms For nonrelativistic velocities, the Rutherford formula for the elastic scattering of heavy particles by nuclei of charge Z e is given b y d@Ll(B.) =

2?re422Z2sin e de 16T2s i n 4 ( p )

(1.1.52)

where d% is the differential cross section, e is the scattering angle, ze is the charge of the particle, and T is its nonrelativistic kinetic energy. Equation (1.1.52) can also be written (1.1.53) where TYevis T in units MeV. For the scattering of identical particles of spin the M ~ t t formula l ~ ~ gives for the cross section:

a%(@)=

2T24e4 cos f? sin T2

e de 1 -~ [sin4

L

-

1

e +X e

1 cos sin2 e cos2 e

+ (protons, electrons),

tz

In tan2 e)].

(1.1.54)

The last term in the square bracket (involving ti) arises from the quantummechanical exchange phenomena which are a consequence of the identity of the incident particle and the scatterer. I n the inelastic collisions of heavy charged particles with atoms, it is of interest t o obtain the energy distribution of the secondary electrons (6 rays). The angle of ejection $ in the laboratory system is related as follows t o the energy W of the secondary electron:

W

=

(1.1.55)

2 mu2 cos2 $

where m = electron mass, v = velocity of incident heavy particle. The maximum value of $ is 90’ in which case W = 0. The cross section for ejection of a 6 ray with energy between W and W dW is

+

(1.1.56) 148 N. F. Mott, Proc. Roy. Soc. Al26, 259 (1930); see also N. F. Mott and H. S. W . Massey, “The Theory of Atomic Collisions,” 2nd ed. Oxford Univ. Press, London and New York, 1949.

56

1.

PARTICLE DETECTION

The cross section for finding a 6 ray between $ and $

+ d$ is (1.1.57)

For relativistic energies, BhabhaZ3has shown that the collision cross section for a particle with spin 0 is given by ( 1.1.58)

where W,,, is the maximum possible energy transfer to the atomic electrons [Eq. (1.1.15)]. Of course, for p+ 0, Eq. (1.1.58) reduces to the nonrelativistic expression (1.1.56). It may also be remarked that for energy transfers W > 137rn~~Z-'/~), = I ' [ 4 ln(183Z-1/3)

where

I' E Z(Z

+ $1

( 1.1.67)

+ {)5.79 X 10-z8 cm2

(1.1.68)

and where { is the correction for the contribution of the atomic elect r o n ~ ; {' ~is~of the order of 1.2-1.4. The distance Xo over which the electron has its energy decreased by a factor e is called the radiation length. Thus XOis defined by

l/Xo = 4NI' ln(183Z-1'3).

( 1.1.69)

For large energies, we have [from Eq. (1.1.67)]: (1.1.70)

where b = 1/[18 1n(183Z-1/3)]is very small compared to unity (b = 0.012 for air, 0.015 for Pb). Table VI gives values of the radiation length XO for various materials. TABLE VI. Values of .the Critical Energy E, and Radiation Length XOfor Various Substances This table is taken from Bethe and Ashkin, reference 3, p. 266. Substance

E , (Mev)

X O(gm/cm2)

340 220 103 87 77 47 34.5 24 21.5 6.9

58 85 42.5 38 34.2 23.9 19.4 13.8 12.8 5.8 36.5 35.9

~~

Hydrogen Helium Carbon Nitrogen Oxygen Aluminum argon Iron Copper Lead Air Water

~

83 93

60

1.

PARTICLE DETECTION

Bethe and Heitler14’ have given the following approximate formula for the critical energy: E, S 1600mc2/Z (1.1.71) from which they have obtained the following expression for the ratio of the radiation loss to the collision loss:

-

(dEo/dz)r,fi - EoZ (dEo/dz),,11 - 1 6 0 0 ~’~ ~ ’

(1.1.72)

It should be noted that Eqs. (1.1.71) and (1.1.72) are very approximate, as can be seen by comparing the values of E, calculated from Eq. (1.1.71) with the actual values given in Table VI. With increasing energy of the electron, the radiation becomes increasingly peaked forward. Aside from a factor which depends slowly on E O and hv, the angular distribution of the radiation du/dQ is determined by:154

(1.1.73) where 0 is the angle of emission of the radiation and B is a constant. Thus the average angle of emission is given by

(1.1.74)

mc2/Eo

which becomes very small with increasing Eo. Recently tfberal1155has investigated the bremsstrahlung produced by fast electrons in single crystals. He has shown that interference phenomena are expected to occur which can enhance the radiation and markedly change the y-ray spectrum. A similar effect for pair production (see Section 1.1.7.3) is also discussed by Uberall. The crystal effect is small at low energies, and sets in for q noh/a, where q is the momentum transfer to the target atoms, a is the lattice constant, and no is of the order of 2 or 3. This condition corresponds to an electron energy To 200 Mev 1 Bev for pair production. The for bremsstrahlung, and y energy hv interference effect is confined to angles 00 of order 0 0 5 (137Z-’I3) X (mca/Eo) between the primary beam and the line of atoms participating in the interference. I n a second paper, tfberal1165has discussed the polarization of bremsstrahlung emitted from a monocrystalline target. The polarization P is all), where ul and ~ 1 are 1 the cross secdefined as: P = (aL - q ) / ( a l tionsjor producing radiation polarized perpendicular and parallel, respec-

-

-

+

lS4 A. Sommerfeld, “Atombau und Spektrallinien,” Vol. 2, p. 551. Vieweg, Braunschweig, Germany, 1939. lKK H. Vberall, Phys. Rev. 103, 1055 (1956); 107,223 (1957).

1.1.


61

tively, to the production plane (formed by the incoming electron momentum PO and the emitted y-ray momentum k). He has shown that, in typical cases, P is increased by a factor of -1.5 above the value obtained when an amorphous target is used. Moreover, there is a net polarization with respect to the plane formed by the incident direction (PO) and the crystal axis. Thus for a Cu crystal, at T = O", with an incident electron energy EO= 600 MeV, for 0 0 = 20 X rad, the polarization POof the entire bremsstrahlung cone is E0.15 for z 3 kv/Eofrom 0 to 0.2. Between 2 = 0.2 and 0.5, PO decreases slowly to 0. Here eo is the angle between the primary direction and the crystal axis. For Oo = 5 X 10-8 rad, Po is 0.31 a t x = 0, and decreases rapidly with increasing x, becoming negative at z = 0.19, with minimum value PO= -0.11 at x = 0.33. Thus by using an appropriate angle 00, it may be possible to obtain partially polarized y radiation of sufficient intensity to perform high-energy polarization experiments. 1.I .5.2. Shower Production. As the eIectron proceeds through the material, it will create a shower, which is produced as follows. The electron loses energy by bremsstrahlung, producing a high-energy y ray. The y ray in turn can produce an electron and positron by pair production (see Section 1.1.7.3below). The pair in turn can radiate energy by bremsstrahlung, thereby producing photons, which can then create more pairs. In this way a cascade of photons and high-energy e+ and e- is produced, which is called a shower. The number of e+ and e- present increases at first with increasing thickness t, then attains a maximum a t a certain thickness t,, and decreases for larger t. I n this connection, as was mentioned above, it is convenient to define a critical energy E, by the condition that for EO = E,, the radiation loss, Eq. (1.1.701, is equal to the energy loss by ionization. Values of E , for various materials are given in Table VI. Figure 8 shows the expected number of electrons n as a function of the thickness t in radiation lengths Xo. I n the figure, loglo n is plotted against t for 4 different values of the total energy E Oof the primary electron, which is given in units of the critical energy E,. These curves were taken from the work of Rossi and Greisen.lS8Figure 8 shows that for a shower withiiincident energy E O = 100E,, n increases from n = 1 at t = 0 to a maximum n, G 10 a t t = t, 4, and thereafter decreases to n = 1 at t g 12, and is negligible for t >, 12. The thickness t, at which the maximum is reached, and the value n, at the maximum both increase with increasing energy EOof the primary electron. Thus for E O = 104E,, we have n, 1000, t, E 9. I n this case, n becomes negligible only for t 2 30. Many authors have treated analytically the complicated mathe166

B. Rossi and K. Greisen, Revs. Modern Phys. 13, 240 (1941).

1.

62

PARTICLE DETECTION

matical problems involved in shower production. A review of these calculations is given in the book by Ro ~ si.'~ ' Wilson168has recently treated the problem of the shower development by a Monte Carlo method, in which a large number of electrons are followed through the material, with a statistical (probability) determination of the bremsstrahlung and pair production processes in each particular case history." Neglecting scattering, by means of an approximate ((

5

4

3 C

2 2

s

I

0

-I

0

t

FIG.8. The number n of electrons in a shower as a function of the thickness traversed of B. Rossi and K . Greisen [Revs. Modern Phys. 13, 240 (1941)l. t in radiation lengths. These curves were taken from the work

theoretical model of shower production, Wilson finds for the mean range r (in units Xo) of an electron of initial energy Eo r

=

ln 2 In

(-E ,EoIn 2 + 1).

(1.1.75)

The distribution of ranges around r is approximately Gaussian, and the root mean square straggling s (in units X , ) is given by (1.1.76) Wilson has shown from his Monte Carlo calculations that for a n incident electron, one is more likely to find 1, 3, 5 . . . than 2, 4,6, . . . electrons and positrons a t a given thickness in the shower, since electrons lS7

B. Rossi, " High-Energy Particles," Chapter 5. Prentice-Hall, New York, 1952. R. R. Wilson, Phys. Rev. 84, 100 (1951); 86, 261 (1952).

1.1.


G3

and positrons are formed in pairs. On the other hand, if the shower is initiated by a y ray, one is more likely to find 2 , 4 , 6 . . , than 1 , 3 , 5 . . . electrons and positrons a t any thickness in the shower, Wilson’s shower curves as obtained b y the Monte Carlo method are more spread out than those of the general (analytic) shower theory, i.e., the shower penetrates to a greater thickness t than according to the general t,heory. One of the reasons for this difference is that Wilson’s calculations take into account the fact that the low-energy y rays have a relatively long mean free path (see Section 1.1.7.4). 1.1.5.3. Production of Secondary Electrons by Electrons. Scattering of Electrons by Electrons and Nuclei. The cross section for ejection of secondary electrons (6 rays) by an electron passing through matter is given by146

d@e(T.W) ., I

=

( + ( T -1W)’ - W ( T 1- W )cos [

re4 - dW wi1

T

ln

(‘+)I} (1.1.77)

where T is the kinetic energy of the primary electron, and W is the energy transfer, i.e., the kinetic energy of the secondary electron. In Eq. (1.1.77), the second and third (cosine) terms in the curly bracket are exchange terms. For small W , these terms become negligible, and the resulting cross section [re4/(TWZ)]dW is the same as that for primary heavy particles (e.g., protons) [see Eq. (1.1.56)]. For relativistic energies of the incident electron, the electron-electron scattering cross section has been obtained by Mdler,L69and is given by

1

+

( T - W ) z+ ( T

+ mcz)2

(1.1.78)

For T < mc2, Eq. (1.1.78) reduces to the nonrelativistic Mott formula, Eq. (1.1.77), in which the cosine factor in the last term is ~ 1unless , W7 is very close to either 0 or T. I n his calculations,169MGller summed over both directions of polarization of the two electrons, so that Eq. (1.1.78) represents the average cross section for unpolarized incident electrons. However, recently in connection with the experiments on parity n o n co n ~e r v a tio n ~in-~beta decay, it has become of interest to evaluate the cross sections for polarized electrons scattered by polarized electrons, in particular for the case that the two spin directions are parallel or antiparallel to each other, and along 159 C. Mdler, Ann. Physik [5] 14, 531 (1932).

64

1.

PARTICLE DETECTION

the relative direction of motion of the electrons. This problem arises because it has been shown from the two-component theory of the neutrino,*ll that the electrons and positrons from beta decay are expected to be longitudinally polarized10.16't-162 (i.e., with average spin direction along the direction of motion). The value of the polarization P is predicted to be: P = T v/c, where v = velocity of p particle, and the minus sign applies to electrons, while the plus sign applies to positrons. Thus high-energy electrons from p decay (v c) are almost 100% polarized, with the spin pointing opposite to the direction of motion. A method of determining the longitudinal polarization consists in scattering the &decay electrons on the electrons in a ferromagnetic sample of material. As is well known, if a strong magnetic field is applied to an iron sample, the 3d electrons of iron will be polarized in the direction of the applied field.* Thus if a magnetic field is applied along a direction parallel or antiparallel t o the direction of the incident p electrons, one can obtain the polarization P of the incident beam from a comparison of the c o u n h g rates of scattered electrons a t a particular angle, for the two field directions. This result arises from the fact that the cross sections for parallel spin directions (bp and for antiparallel spin directions & are appreciably different from each other, for all values of the incident energy, provided that the energy transfer W is sufficiently large (WIT 2 0.2, where T is the kinetic energy of the incident electron). The first calculation of the spin-dependent cross sections r#Jp and (ba was carried out by Bincer,163and we shall here briefly summarize his results. For the differential cross sections in the center-of-mass system of the two electrons (to be abbreviated as c.m. system), one obtains

-

d(bp =

eddn [2 C O S ~t7 2~284sin4 t7

dr#Ja=

e4dQ [I -2 ~ ~ sin4 8 4 t7

--

+p(3

G + C O S ~t7)

COS~

+

B4(1

+

s)]

COS~

(1.1.79)

+

G + B2(2 + 3 C O S ~t7 - C O S ~8)

COS~

+ 84(5 - 4

G + C O S ~t7)]

COS~

(1.1.80)

where 8 is the c.m. scattering angle, B is the c.m. velocity of either electron (in units of c), is the total c.m. energy of either electron, and d o is the element of solid angle in the c.m. system. We have S2 = (y - l ) / ( y l),

+

* See also Vol. 4, A, Chapter

3.5. J. D.Jackson, S. B. Treiman, and H. W. Wyld, Phys. Rev. 106, 517 (1957). Iel L. Wolfenstein and L. A. Page, BUZZ. Am. Phys. Sac. [Z] 2, 190 (1957). 162 R. B. Curtis and R. R. Lewis, Phys. Rev. 107, 543 (1957). A. M. Bincer, Phys. Rev. 107,1434 (1957);see also G.W. Ford and C. J. Mullin, Phys. Rev. 108, 477 (1957). l60

1.1.

65


where y = E/mc2,with E = total laboratory energy of incident electron. Upon defining x = cos' 8, Eqs. (1.1.79) and (1.1.80) can be rewritten as follows : d+P

(1.1.81) @a

( 1.1.82)

We note that cos

e = 1 - 2w

(1.1.83)

where w is the fractional energy transfer, w = W/T. Thus after integrating over the azimuthal angle, one obtains

J dn

=

27r sin ede

=

-2nd(cos #)

=

47r dw.

(1.1.84)

I n view of Eq. (1.1.83), we have: x = (1 - 2 ~ ) Upon ~ . substituting these results in Eqs. (1.1.81) and (1.1.82),one finds for the average differential cross section per unit w (for unpolariaed electrons) :

x [ y y i - 2~

+ 3 ~ -2 2w3 +

w4)

- (27

- i ) ( w - 2w3

+

W4)l.

(1.1.85)

Finally, upon using the relations: T = (y - 1)mc2, and W = wT, one can easilyshow that Eq. (1.1.85) is equivalent to Eq. (1.1.78) ford@(T,W). From Eqs. (1.1.81) and (1,1.82), one obtains

+ +

&, -- ~ ' ( 1 &

+

62 x2) - 2 y ( l - Z) 1 - 5' 87' - 2y(4 - 52 2') 4 - 62 22"

+ +

+

( 1.1.86)

Figure 9 shows the curves of as a function of w, for y = 1,3, and co , as obtained by B i n ~ e r . 'It~ ~ is seen that decreases rapidly with increasing w, independently of y. The minimum value is attained for w = 0.5 (0 = go"), and is given by (Y - 1)' = 4(2y2 - 2y

(1.1.87)

+ 1)

which becomes 0 for y 4 1 (nonrelativistic energies) and

+ for y

4

co

.

66

1.

PARTICLE DETECTION

The dependence of the electron-electron scattering cross section on the relative directions of polarization has been used in a few experiments t o determine the longitudinal polarization of electrons from B decay. 164,18i, The arrangement of the experiment of Frauenfelder et ~ 1 . lis~ shown ~ I .3

.8

.6

0, +o

.4

.2

0 0

.I

.2

.3

A

.5

W

FIG.9. The ratio &/&, for electron-electron scattering, as a function of the relative kinetic energy t,ransfer w.This figure is taken from the work of A. M. Bincer [Phys. Rev. 107, 1434 (1957), Fig. 11, and is reprinted with the permission of the author and the Editor of the Physical Review.

schematically in Fig. 10. The scattered electrons are recorded in coincidence by the counters 61 and Cz.The counting rates are compared for opposite directions of the magnetizing current around the Deltamax scattering foil. The scattering angle 0 is usually so chosen that it corre164 H. Frauenfelder, A. 0. Hanson, N. Levine, A. Rossi, and G. De Pasquali, Phys. Rev. 107, 643 (1957). lo5 N. Benczer-Koller, A. Schwarzschild, J. B. Vise, and C. S. Wu, Phys. Rev. 109,

85 (1958).

1.1.

67


sponds approximately to the maximum relative energy transfer w = 0.5, for which has its smallest value, as discussed above. For w = 0.5, the two final electrons have equal energies in the laboratory system and are emitted symmetrically with respect to the incident direction a t a n angle 8, given by sin2 8, = 2/(7 3). (1.1.88)

+

A description of the experiments on the M3ller scattering of P-decay electrons, as well as a more complete discussion of the theory, can be found in reference 8.

COLLIMATOR

& SOURCE

FIG.10. Schematic view of the experimental arrangement of Frauenfelder et a1.ln4 used to demonstrate the longitudinal polarization of electrons from the fl decay of P32 and Pr144,by means of the Mflller (electron-electron) scattering.

For the scattering of relativistic electrons b y nuclei, McKinley and FeshbachlGB have obtained the following expression :

I

+ ZsP - s i n ( p ) [ l - sin(+0)] . 137

(1.139)

This expression applies provided th at 2/137 is not too large, Le., not for the heaviest nuclei. For P + 1, Eq. (1.1.89) may be rewritten as follows:167 Ze2 cos2(&8) sin4((B8) -{I+--

-)

d @ = ( 2E

] (2ssin

a 2 sin(;O)[l - sin(@)] 137

cos2 (Be)

0 dB)

(1.1.90)

where E is the total laboratory energy of the electron. We note th a t both in (1.1.89) and (1.1.90), 0 is the angle of scattering in the center-of-mass system. W. A. McKinley and H. Feshbach, Phys. Rev. 74, 1759 (1948). R. Hofstadter, Revs. Modern Phys. 28, 214 (1956).

lB8

68

1.

PARTICLE DETECTION

In connection with the longitudinal polarization of the electrons from beta decay,loJao-lezthe Mott scatteringleaof the electrons from a heavy nucleus has also been used to detect the po1arization.t I n this type of experiment, the longitudinal polarization of the electrons is first transformed into a transverse polarization, for instance, by deflecting the electrons through -90" by means of an electrostatic field. A system of crossed electric and magnetic fields may also be used, t o take advantage of the fact that the focusing condition for the electrons can then be made identical with the condition for turning the spin through 90". After the particles have thus acquired a substantial amount of transverse polarization (spin d perpendicular to momentum p), they are scattered through an angle 0 in the plane perpendicular to the (d,p) plane, which we assume to be horizontal, and the up-down asymmetry of the scattered intensity is observed. That is, the intensity of the electrons scattered through an angle 8 in the upward direction is different from the intensity of the electrons scattered through the same angle 8 in the downward direction. As was first shown by Mottlesin 1929, the asymmetry in the scattering of transversely polarized electrons is largest for heavy elements and for large scattering angles (6 90"-150"). For a beam with transverse polarization P , the ratio R of the scattered intensities in both azimuthal directions perpendicular to the (d,p) plane (i.e., upward and downward in the example discussed above) is given by

-

(1.1.91)

where s(0)is a function, first calculated by Mott,le8which depends on the atomic number of the scatterer, the incident electron energy, and the angle of scattering 8. The most complete recent calculation of S(0) has been carried out by Sherman,169who has tabulated s(e)at intervals of 15" for various values of the electron velocity p, for three elements: mercury (2 = 80) ; cadmium (2 = 48);and aluminum (2 = 13). s(0) is given by

- p*)1'2 s(e) = 2PE(1 [F(B)G*(B)+ F*(e)G(0)] sin e(da/dQ)

(1.1.92)

where 5 = 2/(137p), X is the de Broglie wavelength, P(0) and G ( 0 ) are the

t See also Vol. 4, A, Section 3.5.1. 168 N. F. Mott, PTOC. Roy. Soc. Al24,425 (1929);A136,429 (1932);see also the discussion in N. F. Mott and H. S. W. Massey, "The Theory of Atomic Collisions," 2nd ed., pp. 74-85. Oxford Univ. Press, London and New York, 1949. lEg N. Sherman, Phys. Rev. 103, 1601 (1956).

1.1.

69

I N T E R A C T I O N O F RADIATION WITH MATTER

regular and irregular solutions, respectively, of the Schroedinger equation for the electron in the field of the nucleus. These functions are in general complex, and the asterisk denotes the complex conjugate. In Eq. (1.1.92) du/dQ denotes the differential cross section for an unpolarized beam of electrons, which is given by

+ ]GI2 sec2(+0)].

du/dQ = X2[t2(1- pz) ]PIz csc2(+0)

(1.1.93)

has also tabuIn addition to the asymmetry function i3(0), Sherman*eB lated the values of the real and imaginary parts of F and G, as well as the

FIQ.11. The asymmetry factor -S(O) for mercury (2 = 80) as a function of the scattering angle e. The values of S(O) were obtained from the results of Sherman.leg The curves for electron velocities j3 = 0.2, 0.4, 0.6, 0.8, and 0.9 correspond to electron energies T, = 10.5, 46.6, 128, 341,and 661 kev, respectively.

differential cross section du/dQ. As mentioned above, Ii3(0)l is largest for heavy elements and large values of 0. Figure 11 shows the curves of s(0) versus 0 for 2 = 80 and for various velocities p. x(e)is zero for 0 = 0" and 180" for all energies, and a t B = 1for all angles 0. Several experiments have 0.6 (T, 130 kev), been carried out at 0 = 90" using electrons with p for which lS(90")1 has its maximum value. It may be noted that for p = 0.6, IS(0)l increases from 0.271 a t 90" to 0.424 a t 120" and 0.418 at 135'. Nevertheless, it has been found desirable t o work a t -90" because of the rapid decrease of the cross section du/dQ with increasing angle.

-

-

70

1.

PARTICLE DETECTION

FIQ.12. The ratio 9 = (du/dn)/(duz/dn)for mercury (2 = 80) as a function of the scattering angle 0. The solid curves of 7 were obtained from the results of Sherrnan.lBg The dashed curve of 9 for fi = 0.6 was calculated from the formula of McKinley and FeshbachlG6[Eq.(1.1.89)].

Figure 12 shows the values of the ratio q for mercury, as obtained by Sherman,169 where q is defined by (1.1.94) where duR/dQ is the Rutherford cross section, which is given by the factor outside the curly bracket of Eq. (1.1.89): (1.1.95) The dashed curve in Fig. 12 shows the values of q predicted by the formula of McKinley and FeshbachIG6for = 0.6, i.e., the curly bracket of Eq. (1.1.89). It is seen th at the actual values of q differ appreciably from the McKinley-Feshbach result, as would be expected in view of the large 2 value (2/137 = 0.58). Among the earlier determinations of S(O), we may mention the calculations of Mott, 1 6 * Bartlett and Watson,17o Bartlett and Welton,'" and Mohr and T a ~ s i e . " ~ J. H. Bartlett and R. E. Watson, Proc. Am. Acad. A d s Sci. 74, 53 (1940). J. H. Bartlett and T. A. Welton, Phys. Rev. 69, 281 (1941). l72 C. B. 0. Mohr and L. J. Tassie, Proc. Phys. Soc. (London) A67, 711 (1954); C. B. 0.Mohr, Proc. Roy. Soc. A182, 189 (1943). 170

1.1.


71

It should be pointed out that the function S(0) was originally introduced by Mott"j8in connection with the double scattering of a n initially unpolarized beam of electrons. I n this case, the polarization P after a single scattering through an angle O1 is given by S(B1),and the direction of the spin d after the scattering is perpendicular to the plane of the scattering. After a second scattering, the relative intensity of the beam as a function of the angle cp between the first and second planes of scattering is given by I(el,ez,cp) = 1

+ s(e1)s(e2)co~ (o

(1.1.96)

where 0 2 is the angle of the second scattering. Among the more recent double scattering experiments which have attempted to verify the theoretical values of S(0),we may cite the works of Shull et u Z . , ' ~ ~ Ryu et u Z . , ' ~ ~ and Louisell et A review of these investigations has been given by Tolhoek. 176 The Mott scattering has been used in several e x p e r i m e n t ~ ' ~ ~on - ~the 8~ longitudinal polarization of p-decay electrons, and has shown th a t the polarization agrees with the predicted value, P = v/c, within the experimental errors.8 The same conclusion was obtained from the experiments using the MIdller scattering.'64s165 1.1 -5.4. Range of p Rays in Matter. In some cases, a crude value of the energy of a beam of homogeneous p rays is obtained by measuring the socalled practical range R, in some material, such as aluminum. The practical range is obtained by extrapolating the straight-line (maximum slope) part of the graph of transmission versus thickness traversed, and taking into account the background (cf. Fig. 6). For p rays in aluminum, Katz and P e n f ~ l dhave l ~ ~ given the following expressions for R, as a funcC. G.Shull, C. T. Chase, and F. E. Myers, Phys. Re*. 63, 29 (1943). N. Ryu, K. Hashimoto, and I. Nonaka, J . Phys. Soc. Japan 8, 575 (1953). l76 W. H. Louisell, R. W. Pidd, and H. R. Crane, Phys. Rev. 94, 7 (1954). 178 H. A. Tolhoek, Revs. Modern Phys. 28, 277 (1956). 177 H. Frauenfelder, R.Bobone, E. von Goeler, N. Levine, H. R. Lewis, R. N. Peacock, A. Rossi, and G. De Pasquali, Pkys. Rev. 106, 386 (1957). 178 H. De Waard and 0. J. Poppema, Physica 23, 597 (1957). 179 P. E. Cavanagh, J. F. Turner, C. F. Coleman, G. A. Gard, and B. W. Ridley, Phil. Mag. [8]2, 1105 (1957). l80 A. I. Alikhanov, G. P. Eliseiev, V. A. Lubimov, and B. V. Ershler, Nuclear Phys. 6, 588 (1958). 181 A. de-Shalit, S. Kuperman, H. J. Lipkin, and T. Rothem, Phys. Rev. 107, 1459 (1957). 182 H. J. Lipkin, S. Kuperman, T. Rothem, and A. de-Shalit, Phys. Rev. 109, 223 (1958). 183 L. Katz and A. S. Penfold, Revs. Modern Phys. 24, 28 (1952). l73

17*

72

1.

PARTICLE DETECTION

tion of the incident electron energy TO(in MeV) :

R,

=

412Tonmg/cm2

n = 1.265 - 0.0954 In TO

(1.1.97)

for 0.01 d To =< 2.5 MeV, and

R,

=

530To - 106 mg/cm2

(1.1.98)

for 2.5 5 T O6 20 MeV.

FIG.13. The practical range R P of low-energy electrons in aluminum as a function of the electron kinetic energy To.This curve was calculated from the range-energy relation given by L. Katz and A. S. Penfold [Revs. Modern Phys. 24, 28 (1952)]. See also Eqs. (1.1.97) and (1.1.98) of text.

Figure 13 shows a plot of Eqs. (1.1.97) and (1.1.98). This curve is in good agreement with the experimental data on the maximum range of electrons from natural beta emitters. Among the earlier works on the range of low-energy /3 rays, we may mention those of Marshall and Ward, l S 4 FeatherJ1S5Flammersfeld, l 8 6 Bleuler and Zunti, l*7 Glendenin, and Hereford and Swann.lE9 lS4

J. Marshall and A. G. Ward, Cun. J. Research A16, 39 (1937).

N. Feather, Proc. Cambridge Phil. SOC.34, 599 (1938). A. Flammersfeld, Nuturwissenschaften 33, 280 (1946). lg7 E. Bleuler and W. Ziinti, Helv. Phys. Actu 19, 137, 375 (1946); 20, 195 (1947). la*L. E. Glendenin, Nucleonics 2, 12 (1948). lag F. L. Hereford and C. P. Swann, Phys. Rev. 78, 727 (1950). lS6

I**

1.1.

INTERACTION OF RADIATION W I T H MATTER

73

1.1.6. Multiple Scattering of Charged Particles When a charged particle penetrates a thick absorber, it undergoes a large number of small-angle Coulomb scatterings. This process, which is called multiple scattering, was first treated quantitatively by Williams.l90 In addition, the particle may undergo a small number of relatively largeangle scatterings, for which the probability can be directly obtained from the Rutherford scattering formula. We shall here be concerned with the multiple scattering only. The resultant distribution of the space angle 6 between the incoming and outgoing directions of the particle is given by (1.1.99)

where

<e2> is the mean square value of

(2)

<e2>

= 2OI2In - =

8 and is given by1g1

el2 In [ 4 ~ Z ~ / % ~($)2]iVt

(1.1.100)

Here Omin is the minimum angle of scattering in a single encounter, Omin Ei X/a, where X is the de Broglie wavelength of the particle and a is the radius of the atom, a a& 1‘3. In Eq. ( l . l . l O O ) , t is the thickness of material traversed, z is the charge of the particle, m is the electron mass (regardless of the type of scattered particle: proton, meson, etc.), N is the number of atoms of absorber per cm3, and el is that angle for which there is, on the average, only one collision with 8 > 01 throughout the absorber. el2 is given by e12 = 4 T ~ ~1)9e4t/(p42. ( ~ (1.1.101)

-

+

Equation (1.1.100) can be written as follows: <e2>

=

0*1572(2 4- 1)z2t ln[1.13 X 104Z4/3z2A-1tp-2](1.1.102) A (PO)

where pv is in MeV, t is in gm cm-2, and A is the atomic weight in gramsThe expression preceding the logarithm in (1.1.102) is el2. Rossi and GreisenlS8have given a somewhat different formula for < e2> which has been frequently used in experimental applications. This expression is given by (1.1.103) E. J. Williams, Proc. Roy. SOC.A169, 531 (1939);Phys. Rev. 68,292 (1940). H. A. Bethe and J. Ashkin, in “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. I, p. 285. Wiley, New York, 1953. 1**

191

74

1.

PARTICLE DETECTION

where Xo is the radiation length in the material [cf. Eq. (l.l.G9) and Table VI], and E, is a constant energy given by

E,

=

( 4 X~ 137)1%c2

=

21.2 MeV.

( 1.1.104)

As was shown by Bethe and Ashkin,lgl Eq. (1.1.103) applies only for relatively large thicknesses t > to, where t o is given by

to

=

6.7(137/2)2A113gm cm-2.

(1.1.105)

For Pb, to = 110 gm cm-2, while for C, to = 8000 gm cm-2. For small thicknesses, Eq. (1.1.103) overestimates the mean square multiple scattering angle. Thus for 3-Bev protons and samples of thickness t = 10 gm cm-2 of C, Cu, and Pb, < 0 2 > u2 = 0.123", 0.248', 0.388' from Eq. (1.1.102) for C, Cu, and Pb, respectively, whereas the corresponding values from Eq. (1.1.103) are: 1'2 = 0.159", 0.289", and 0.429', respectively. The factor F by which Eq. (1.1.103) differs from (1.1.102) is: F = 1.29, 1.17, and 1.11 for C, Cu, and Pb, respectively (for t = 10 gm cm-2). It is often useful to consider the projected angles 8, and 01/, i.e., the projections of the angle 8 on the zy plane perpendicular to the direction of motion of the particle. The distribution of the 8, values is a Gaussian: (1.1.106) where

< eZ2> is the mean square value of <e,2>

=

;<e2>

8, and is given by

(1.1.107)

with < 8 2 > given by Eq. (1.1.102). Thus the denominator of the exponent is the same ( = <e2>) for both P(8) and P,(Q [Eqs. (1.1.99) and do, for the projected angle 8, (1.1.106)]. Of course, the distribution P1/(02/) has the same form as P,(8,) do,. The distribution of the lateral displacement r of the particles has been determined by Ferrni,Ig2and is given by (1.1.108) where t' = t/X, is the thickness traversed in units of radiation lengths X O . More elaborate theories of the multiple scattering have been developed Snyder and Scott,Ig6 by Goudsinit and S a ~ n d e r s o n , ~ ~ ~ E. Fermi, quoted by B. Rossi and K. Greisen, Revs. Modern Phys. 13,265 (1941). S. Goudsmit and J. L. Saunderson, Phys. Rev. 67, 24 (1940). ls4 G. MoliBre, 2. Naturforsch. Sa, 78 (1948). 195 W. Paul and H. Steinwedel, in "Beta- and Gamma-Ray Spectroscopy" (K. Siegbahn, ed.), p. 1. Interscience, New York, 1955. 196 H. S. Snyder and W. T. Scott, Phys. Rev. 76, 220 (1949). 192

193

1.1.


75

Lewis,lg7and Bethe. l g 8 These theories treat more accurately the transition from the small-angle region of multiple scattering to the large-angle region where single scattering predominates. This transition region is sometimes called the region of plural scattering. I n a recent investigation, Nigam and c o - w ~ r k e r s have ' ~ ~ made a critical study of the MoliBre theorylS4of multiple scattering, and have obtained a consistent treatment of the scattering of a charged particle by the field of an atom, up to the second Born approximation. I n this work, Nigam et al. have used the expression of Dalitz2""for the scattering cross section of a relativistic particle of spin in a screened atomic field, for which the potential is: V = -(Ze2/r)exp(--r), where K is a n arbitrary constant, and r is the distance from the nucleus. It was found that the deviation of the complete expression for the "screening angle " 8, from the value given by the first Born approximation is considerably smaller than was obtained by MoliBre. Moreover, the expression for the distribution function P(e) contains additional terms of order xZ/137, which were not obtained by Moliitre. Nigam et al. have carried out calculations of the distribution function P(0) for the case of 15.6-Mev electrons scattered by Au and Be, in order t o The compare their theory with the experimental results of Hanson el aLZo1 theoretical results are in good agreement with the data. The two cases considered correspond to electrons of average energy 15.7 Mev scattered by a gold foil of thickness t = 37.2 mg/cm2, and 15.2-Mev electrons scattered by a beryllium sample of thickness t = 491.3 mg/cm2. The experimental distributionsof Hanson etal. havea llewidth O,(exp) = 3.78" for Au and 4.33" for Be. Here 0, denotes the angle (measured from the direction of the incident beam) a t which P(0) has fallen off t o l/e of its value a t e = 0". e, is thus given by < 02> l i 2 for a Gaussian distribution [Eq. (1.1.100)]. It may be noted, however, th a t the actual multiple scattering d i s t r i b ~ t i o n ~ ~deviates ~ - ' ~ ~ somewhat from a Gaussian a t all angles. I n particular, at large angles (0 >, 28,), the actual P(0) lies above the Gaussian of Eq. (l.l.lOO), and slowly approaches the single-scattering cross section (which decreases only as m e - * ) . For comparison with the values of e,(exp), the theory of Nigam et a l l g 9gives 0, = 3.80" for Au, and 4.35" for Be, in very good agreement with the data. On the other hand, MoliBre's theorylg4gives e, = 3.83" for Au, and 4.56" for Be. The result

+

H. W. Lewis, Phys. Rev. '78, 526 (1950). 1**H.A. Bethe, Phys. Rev. 89, 1256 (1953). 199 B. P. Nigam, M. K. Sundaresan, and T. Y. Wu, Phys. Rev. 116, 491 (1959). zoo R. H. Dalitz, Proc. Roy. Soe. A206, 509 (1951). 201 A. 0. Hanson, L. H. Lanzl, E. M. Lyman, and M. B. Scott, Phys. Rev. 84, 634 (1951).

76

1. PARTICLE DETECTION

for Be is thus too large by 5%. It may be noted that from the simple expression of Bethe and Ashkin given above [Eq. (1.1.102)], one obtains 0, = 3.94" for Au and 4.33" for Be. The value for Au is too large by 476, while the result for Be agrees exactly with B,(exp). On the other hand, the formula of Rossi and Greisen [Eq. (1.1.103)] would give the values 5.82" for Au and 6.56"for Be, which are both considerably larger than O,(exp). The multiple scattering has been frequently used for a crude measurement of pv for charged particles in nuclear emulsion.202If the track of the particle is subdivided into sections (cells) of length t , the average angle between successive sections is given by (1 .l.109)

where ,6 = v/c, t is measured in microns, pv is in MeV, A~ is in degrees, and K(t,,6) is a slowly varying function of t and p. The theoretical value194*196 of K is between 23 and 24, which is in satisfactory agreement with the experimental results both of the Bristol groupzo3and of C o r ~ o n , * ~ ~ namely K = 25.1 f 0.6 for P = 1 and t = 100. 1.1.7. Penetration of Gamma Rays For y rays passing through matter, there is an exponential attenuation such that the intensity I ( z ) after traversing a thickness 2 is given by

I ( z ) = I(O)exp( - Nuz)

=

I(O)exp( -m)

(1.1.110)

where I(0) is the incident intensity (at 5 = 0), N is the number of atoms of absorber per cm3,u is the total cross section for absorption or scattering of the y rays, and r = Nu(cm-') is the absorption coefficient of the radiation. There are three processes which contribute to 6: (1) the photoelectric effect, which consists of the ionization of atomic electrons by the incident photon. (2) The Compton scattering of the photons by the atomic electrons. I n this process, the atomic electrons can generally be considered as free, and the energy transfer to the electron is a function of the scattering angle O of the ? ray and its initial energy hvo. The energy transfer is determined in a straightforward manner from conservation of momentum and energy. (3) The production of an electron-positron pair in the field of a nucleus. P. H. Fowler, Phi.?. Mag. [7] 41, 169 (1950). Gottstein, M. G. K. Menon, J. H. Mulvey, C. O'Ceallaigh, and 0. Rochat, Phil. Mag. [7] 42, 708 (1951). go4 D. R. Corson, Phys. Rev. 80, 303 (1950); 84, 605 (1951). Zo2

2osK.

1.1.

77


The photoelectric effect predominates at low y energieszo6(hv < 0.05 Mev for All hv < 0.5 Mev for Pb). The Compton effect gives the main contribution at intermediate energies (0.05 < hv < 16 Mev for Al; 0.5 < hv < 4.8 Mev for Pb). The pair production predominates a t high energies (hv > 16 Mev for Al; hv > 4.8Mev for Pb). We shall now consider separately each of these three processes. For a general review of the subject of the interaction of y rays with matter, the reader is referred to the review articles of Bethe and Ashkinj3 Davisson and Evans,206and D a v i s ~ o n . ~ ~ ’ 1.1.7.1. Photoelectric Effect. For energies far above the K absorption edge and in the nonrelativistic range (hv > m d , relativistic effects become important. This problem has been treated by SauterlZ1OHulme,211 and others. The following formula obtained by Hall is valid in the limit hv >> mc2, and includes the effect of the Coulomb field of the nucleus on

-

2 0 6 See, for example, W. Heitler, “The Quantum Theory of Radiation,” 2nd ed., p. 216, Fig. 21. Oxford Univ. Press, London and New York, 1944. 206 C. M. Davisson and R. D. Evans, Revs. Modern Phys. 24, 79 (1952). 207 C. M. Davisson, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 24. Interscience, New York, 1955. 208 H. Hall, Revs. Modern Phys. 8, 358 (1936). a** M. Stobbe, Ann. Physik [5] 7, 661 (1930). Z1O F. Sauter, Ann. Physik [5] 9, 217 (1931); 11, 454 (1931). 211 H. R. Hulme, Proc. Roy. Soc. A133, 381 (1931). P1z H. Hall, Phys. Rev. 46, 620 (1934); 84, 167 (1951).

78

1.

PARTICLE DETECTION

the outgoing electron, which becomes appreciable for heavy elements : @phot.K

3 Z6damc2 2 1374 hv

= - - -exp[ -TCY

+ 2 d ( 1 - In

CY)]

( 1.1.113)

where CY = 2/137. Equation (1.1.113) shows that a p h o t , R decreases quite slowly with increasing v (only as v-l) in the relativistic region, as compared to the k 7 1 2 decrease at nonrelativistic y energies [Eq. (1.1.111)]. Aside from the approximate calculations mentioned above, which are based in part on the Born approximation, and on the use of plane-wave or nonrelativistic wave functions, Hulme et ~ 1 . have ~ ~ 3 carried out exact calculations for the photoelectric effect from the K shell, using the appropriate Dirac wave functions in the field of the nucleus. The calculations of Hulme et al. were carried out for two y-ray energies, hv = 0.354 and 1.13 MeV, and for three values of 2 : 26, 50, and 84. These results have been extensively used to check the validity of various approximation formulas and to obtain smooth curves of versus hv in the intermediate energy region (hv 1 Mev). I n obtaining the photoelectric absorption coefficient, one must include the contribution of the absorption from the L, M , . shells. Latyshev214 has made direct measurements of the photoelectrons ejected from the K and L shells of P b and Ta, for the ThC’ y rays (hv = 2.62 Mev). ,,, have Detailed calculations of the total photoelectric cross section ,,@ been carried out by White1215who used the results of StobbelZo9 Sauterj210 and Hulme et aL213According to White,21bthe ratio [ of the total photoelectric cross section @phot to the K shell contribution aphot.K is -1.15 for heavy elements. White has obtained E for various values of 2, for two y-ray energies: (1) a t the K absorption edge; (2) for mc2/hv = 1.5 (i.e., hv = 0.340 Mev). At the K edge, [ = 1.02 for 2 = 6, 1.11 for 2 = 29, 1.14 for 2 = 50, and 1.167 for 2 = 92. For hv = 0.340 MeV, [ = 1.01 for 2 = 6, 1.07 for 2 = 29, 1.10 for 2 = 50, and 1.138 for 2 = 92. Figure 14 shows the plot of loglo(@ph,t/&,)versus hv for C, All Cu, Sn, and Pb, as obtained from the results of White. For C and All and decreases approxifor the heavier elements a t low photon energies, aphot mately as Y - ” ~ , as expected from Eq. (1.1.111). On the other hand, for Pb is proportional to v-1 [see at high energies, between 5 and 50 MeV, aphot Eq. (1.1.113)]. White’s calculations include the effect of the L and M

-

. .

Z13

H. R. Hulme, J. McDougall, R. A. Buckingham, and R. H. Fowler, Proe. Roy.

Sac. A149, 131 (1935).

G. D. Latyshev, Revs. Modern Phys. 19,132 (1947). G. R. White, Natl. Bur. Standards Rept. 1003 (1952); see also Appendix I by C. M. Davisson, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 857. Interscience, New York, 1955. 214 216

1.1.


79

shells, which becomes the sole contribution to the photoelectric effect a t low energies below the K absorption edge. Thus for Sn, the break in the cm2 curve of @)phot at 29.25 kev is due t o the K edge [@,hot = 1.05 X = 8.58 x on low-frequency side of K edge ( L and M shells only); cm2 on high-frequency side ( K L M shells contribute)]. Similarly, for Pb, the break a t 88.2 kev is due to the K edge, while the discontinuity in the region of 15 kev is due to the L absorption edges (LI, LII, ,5111). Below 13.07 kev (LIrledge), only the M , N , and 0 shells contribute to the photoelectric effect.

+ +

PHOTON ENERGY h v ( I N MeV)

FIG.14. The cross section *,,hot for photoelectric absorption for C, Al, Cu, Sn, and Pb, as a function of the photon energy hv. These curves were obtained from the results of White.216

The theoretical results presented above are in fairly good agreement with various experiments on the photoelectric e f f e ~ t . ~ ~ ~ s ~ ~ ~ , ~ ~ ~ 1.1.7.2. Compton Scattering. The Compton scattering consists of the scattering of y-rays b y atomic electrons which can be considered as free (no atomic binding forces) for sufficiently high y-ray energies. From the laws of conservation of energy and momentum, one finds th a t the frequency v of the scattered quantum is given by v =

yo

1

+ (hvo/mc2)(1 - cos e)

(1.1.114)

where vo is the frequency of the incident quantum, and 0 is the angle of H. A. Bethe and J. Ashkin, in “Experimental Nuclear Physics” (E. Segr6, ed.), Vol. 1, p. 304. Wiley, New York, 1953.

1.

80

PARTICLE DETECTION

FIG.15. Schematic diagram of Compton effect, showing notation used in the text: hvo and hv are the energies of the incident and the scattered quanta, respectively;

T

is the kinetic energy of the recoil electron.

scattering of the y ray. The notation used is shown in Fig. 15. The kinetic energy T of the electron is given by

T

=

h(vo -

V)

=

2 m ~ ~ ( h vCOS' o ) ~cp

(hvo

+ mc2)2- (hvo)2cos2

cp

( 1.1.115)

where (a is the angle between the directions of the outgoing electron and the incident y. The angles cp of the electron and 6 of the y ray are related as follows: tan

cp = Y Os c:;':

0 = (mcT$hv)

cot(&@.

(1.1.116)

The energy of the scattered photon decreases with increasing 6. The minimum value, attained for 6 = 180", is given by (1.1.117) The maximum possible angle of the recoil electron is cp = go", in which case the energy of the electron is T = 0, while the scattered y ray continues with its initial energy (hv = hvo) in the forward direction. The differential cross section for Compton scattering was first obtained by Klein and NishinaZ17in 1929. The Klein-Nishina formula gives

) 817

0. Klein and Y . Nishina, 2.Physik 62, 853 (1929).

(1.1.118)

1.1.

81

INTERACTION OF RADIATION W I T H MATTER

where ro = e2/mc2, k = hv, ko = hvo, and d@c is the cross section for scattering of the y ray through an angle e into the solid angle dQ. Upon substituting Eq. (1.1.114) for k = hv, one obtains

(1.1.119) where y 3 ko/mc2. Equation (1.1.119) gives the cross section as a function of the angle 8. For small values of y, the distribution follows the 1 cos2 0 law characteristic of classical electromagnetic theory. As y increases, the distribution becomes increasingly peaked forward, as is generally the case for any high-energy process. The differential cross section as a function of the energy k is given by

+

d@c =

[

?rro2mc2dk 1 + J2;( kko

- 2(y Y+2 1)

+

(1

+ 2r)k y2ko

1

I k~ r2k

(1.1.120)

with

1

1

+

5 -k 5 1 27 - ko

(1.1.121)

Detailed calculations of various quantities and spectra pertaining to the Compton scattering have been carried out by Figure 16 shows the spectrum of the scattered quanta [Eq. (1.1.120)] for incident y energies hvo = 0.5, 1, 2, and 3 MeV. The minimum value hv,i. [Eqs. (1.1.117), (1.1.121)] increases slowly with increasing hvo. Thus hvmin= 0.169 Mev for hvo = 0.5 MeV, and hvmi, = 0.236 Mev for hvo = 3 MeV. (The asymptotic value in the limit v o 4 co is m c 2 / 2 = 0.255 Mev.) It is seen from Fig. 16 that, as hv is increased above hvmin,d @ ~ / d ( h vfirst ) decreases t o a minimum value, and then increases uniformly up to hv = hvo. The minimum of the cross section becomes increasingly more shallow as the primary energy hvo is increased. The total Compton scattering cross section @C is given by

-

(1.1.122) where 40is the Thomson cross section [Eq. (1.1.112)]. For small y (y

> l), one obtains (1.1.122b) Equations (1.1.122) show that the Compton cross section decreases uniformly with increasing energy of the y quantum. Figure 17 shows a , was taken from Bethe and Ashkin, referplot of % / 4 0 versus h v ~ which ence 3, p. 322. For comparison, we have also shown the photoelectric

PHOTON ENERGY hv (IN MeV)

FIG.16. Spectrum of Compton scattered quanta, as obtained from Eq. (1.1.120), for incident photon energies hvo = 0.5, 1, 2, and 3 MeV.

cross section divided by 2,in the same units 40 [i.e., @phot/z+O]. The reason by 2 is that a p h o t pertains to the photoelectric effect for for dividing aphat the entire atom, so th at @.phot/Z represents the photoeffect per atomic electron and is therefore the quantity t o be compared with 9 c (Compton scattering per electron). We note th at the energy hvo E h ; ~for which +phot/Z = 9~increases with increasing 2. Thus hih = 0.02, 0.05, 0.13, and 0.53 Mev for C, All Cu, and Pb, respectively. As hvo is increased above h h , the photoelectric effect rapidly becomes unimportant compared to the Compton scattering as a source of y-ray attenuation. It should be noted th at the expression for @c [Eq. (1.1.122)] no longer applies for very low photon energies, where the binding of the atomic electrons must be taken into account. I n this case, the incoherent (Compton) scattering will be reduced, both because of the binding of the atomic

1.1.


83

electrons, and because of the effect of the exclusion principle in preventing transitions to occupied atomic levels. On the other hand, there will also be a substantial amount of coherent scattering from the atom as a whole, so that the total scattering cross section will generally be larger than the value Z@cwhich would be calculated from Eq. (1.1.122). For a discussion of these effects, the reader is referred to the review article of Davisson.*07 The Klein-Nishina formula has been tested in various experiments, and has been shown to be in good agreement with the experimental dat~.206,207.216

and the photoelectric cross section per FIG.17. The Compton total cross section electron, iP,hot/Z, for C, Al,Cu, and Pb, as functions of the incident photon energy hvo. The curves of GphOt/Z were obtained from the results of White.216

1.1.7.3. P a i r Production. The theory of the pair production by y rays is closely related to the theory of the bremsstrahlung by a high-energy electron. The general formula obtained by Bethe and Heitler147using the Born approximation is very complicated and will not be given here. However, the formula simplifies considerably if the energies of both positron and electron are not too high so that screening can be neglected, i.e., if ( 1.I . 123)

where E+ and E- are the total energies of the positron and electron, respectively, and k = hv is the energy of the incident photon. If in addition to Eq. (1.1.123) , all energies involved are large compared to mc2,the

1. PARTICLE

84

DETECTION

energy distribution of the positrons (or electrons) is given by Q(E+) dE+

=

45 dE+

ka

kmc2 ( I. 1.124)

where 5 is defined by

5

(Z2/137)ra2

(1.1.124a)

with T O = e2/mc2 (classical radius of the electron = 2.82 X cm). As is also true for the general Bethe-Heitler formula, Eq. (1.1.124) gives a symmetric energy distribution for the positron and electron. Actually for small velocities v+ and v- of the pair, and for large 2, the Coulomb effect (which is neglected in the Born approximation) becomes important and results in a somewhat asymmetric distribution favoring higher energy positrons. For very large energies E+, E-, the screening is complete (4 = 0 ) , and Q(E+)is given by Q(E+) dE+

=

45dE+

[( +

~-

k3

+ Em2+ +E+E-)ln(183Z-1’s) - +E+E-]. (1.1.125)

Figure 18 shows the energy distribution of the pair particles (positrons or electrons) as obtained from the calculations of Bethe and Ashkin (reference 3, p. 328). For hv up to 10mc2,the curves do not include screening and are valid for all elements; for higher photon energies] the calculations of Bethe and Ashkin were done for Pb and include the effect of screening. It is seen that for small values of hv, the energy distributions are generally quite flat between the minimum and maximum values, T+,mi,= 0 and T+,,,, = k - 2mc2, where T+ is the kinetic energy of the positron. For k/mc2 5 30, the distribution has a broad maximum at T+ = +T+,mx= +lc - mc2.For larger k/mc2, Q(E+) has a broad minimum at +T+,m.xand two subsidiary maxima on each side of the minimum, which implies that for large k/mc2, either the positron or the electron tends to carry off most of the energy of the y ray. For any finite k/mc2, = k - 2mc2. the distribution is zero a t the two ends, T+,mi,= 0 and T+,msx Figure 18 shows that the distributions are symmetrical with respect to T+ = $T++,., = +lc - me2. This is a consequence of the use of the Born approximation for Q(E+), which gives identical spectra for the positron and electron. In analogy with the bremsstrahlung in the field of the atomic electrons] which has been discussed above [Eqs. (1.1.62), (1.1.68)], there is also the possibility of pair production in the field of the atomic electrons. Wheeler

1.1.

85


and Lamb14*have shown that, for complete screening, the electronic pair production is a fraction [/Z of the production in the field of the is the same quantity that appears in the nucleus, where [ (-1.2-1.4) formula for the bremsstrahlung [Eq. (l.l.68)l. Thus, for the case of complete screening, the total @(E+)dE+ (including the electronic con[) in the definition of b tribution) is obtained by replacing Z2 by Z(Z [Eq. (1.1.124a)I. For low energies, where screening can be neglected (hv 5 20 Mev), the pair production in the field of the atomic electrons 15* Rohrlich,'j' Nemirovhas been calculated by B o r s e i l i n ~ , 'Votruba, ~~ sky,L62and Watson.153

+

10

a

NE N I Lb : I

- 4

'? -

w"

a

2

'0

0.1

0.2

0.3

0.4 (E;

0.5

0.6

07

Q0

0.9

I

mc2)/~v-2mc2)

FIG.18. Energy distribution of the positron (or electron) in an electron pair as a function of the positron kinetic energy for various energies hv of the incident y ray. These curves were taken from the calculations of Bethe and Ashkin (reference 3, p. 328, Fig. 38). For hv 5 IOrnc~,the curves do not include screening and are valid for all elements. For higher photon energies, the curves pertain to Pb and include the effect of screening.

The angular distribution of the pair electrons becomes increasingly forward with increasing energy of the primary quantum, in analogy with the bremsstrahlung distribution. In particular, for high y energies k, the average angle between the incident y ray and the direction of motion of the electron (or positron) is given by

Emmc2/E

(1.1.126)

where E is the energy of the electron (or positron). Equation (1.1.126) is completely analogous to Eq. (1.1.74) for the bremsstrahlung. The pair production in monocrystalline targets has been discussed by ubera11.166

86

1.

PARTICLE DETECTION

The total cross section for pair production cally for two limiting cases:

aPaiF can be obtained analyti-

' / ~ screening), Eq. (1.1.124) is valid, (1) For me2 1 3 7 m ~ ~ Z - "(complete (1.1.125) gives = $[? ln(183Z-1'3) - A]. (1.1.128) For intermediate values of hv, the total cross section @pair must be obtained by numerical integration. Figure 19 shows the resulting curves of 14

12 10 @'pair

v

@

8 6

4 2 n "I

2

5

10

20

100 200 hv/mc2

500

2000

l0,OOO

FIG.19. The total cross section for pair production Qlpair as a function of the 7-ray energy, for air and Pb, and for the hypothetical case of no screening. These curves were taken from the calculations of Bethe and Ashkin (reference 3, p. 338, Fig. 41).

for the (hypothetical) case of no screening, and for air and Pb (including screening), as obtained by Bethe and Ashkin (reference 3, p. 338). It is seen that for large energies (hv/mc2 2 50), the values of for air and P b fall below the curve for no screening, and slowly approach the asymptotic value [Eq. (1.1.128)] which is 14.1 for air and 11.6 for Pb. As mentioned above, the Bethe-Heitler theory based on the Born approximation cannot be expected to give accurate results for high Z and low energies of the positron or electron, since the wave functions for these particles will then be appreciably distorted by the Coulomb field

1.1.

87


of the nucleus. Jaeger and Hulme220have calculated the pair production cross sections at two photon energies (hv = 3mc2 and 5.2mc2), using the exact Dirac wave functions for the pair particles. At hv = 3mc2, they obtained results for 2 = 50, 65, and 82; at hv = 5.2mc2, the calculations were carried out for 2 = 82 only. For the worst case, 2 = 82, hv = 3mc2, the Born approximation cross section is too low by a factor of 2 (0.34 X cm2, as compared to 0.67 X cm2 from the exact calculation). For the other 2 values, and hv = 3mc2, the Bethe-Heitler result is also too small, but the deviation decreases rapidly with decreasing 2 or increasing hv. Thus for hv = 5.2mc2, 2 = 82, the error is only 16%. Recent experimental results a t high energies (210 Mev)221-227 have shown that the measured pair production cross sections are appreciably lower than the Bethe-Heitler calculated values, and that the deviation is proportional to Z2. Thus Lawson,222from his measurements at 88 MeV, concluded that the ratio of the experimental to the theoretical cross section can be approximately represented by

~

~= 1 -~1.5 x 1~ 0-522. /

a

(1.1.129) ~

In view of these results, Bethe et al.2zshave carried out an accurate calculation of the pair production, in which the Born approximation was not used. They have found that the correction to the Born approximation result of Bethe and Heitler147is a reduction of the cross section proportional to Z2, as is indicated by the experimental data. Upon applying this correction, Bethe et a1.228have obtained excellent agreement with the measurements of LawsonZP2 at 88 MeV, and those of DeWire et aLZz4a t 280 MeV. It may be noted that the correction to the Born approximation is a reduction of the cross section a t high energies, as compared to an increase of the cross section at low energies.220 The correction goes through 6 Mev.226n228 zero at hv 1.1.7.4. Total Absorption Cross Section for y Rays. The total cross section u for the removal of a y-ray photon from the incident beam is given by u z= a p h o t 2a.c @'pair. ( 1.1.130) Z z O J . C. Jaeger and H. R. Hulme, Proc. Roy. SOC.A163, 443 (1936); J. C. Jaeger,

-

+

+

Nature 137, 781 (1936). 2 2 1 C. D. Adams, Phys. Rev. 74, 1707 (1948). a22 J. L.Lawson, Phys. Rev. 76, 433 (1949). 223 R. L. Walker, Phys. Rev. 76, 527 (1949). 224 J. W. DeWire, A. Ashkin, and L. A. Beach, Phys. Rev. 83, 505 (1951). 2zaC. R.Emigh, Phys. Rev. 86, 1028 (1952). z*BE. S. Rosenblum, E. F. Shrader, and R. M. Warner, Phys. Rev. 88, 612 (1952). 227 A. I. Berman, Phys. Rev. 90, 210 (1953). 228 H.A. Bethe and L. C. Maximon, Phys. Rev. 93, 768 (1954); H. Davies, H.A. Bethe, and L. C. Maximon, ibid. 93, 788 (1954).

~

88

1.

PARTICLE DETECTION

The complete absorption coefficient c equals Nu.Figure 20 shows the mass absorption coefficient T/P (in units cmZ/gm) for Al, Cu, and Pb. For Pb, we have presented the separate contributions to r / p due to the photoelectric effect, Compton effect, and pair production (dashed curves). The values of r / p were obtained from the tables of White.216It is seen from Fig. 20 that, as a function of frequency, T has a minimum, which 4 for Pb. occurs at hv ? 20 Mev for Al, .=8 Mev for Cu, and ~ 3 . Mev For Pb, the minimum lies in the region where the Compton effect is predominant. For lower v, the photoelectric effect predominates, and the with increasing v is responsible for the decrease of rapid decrease of apbot ale c

$'

0.16

E

0

-$g

0.14

0.12

I-

0.10

2

'a

0.08

0.06

t 0.04 a 0

2 O.O2 0

0.1

0.5

0.2

I

2

5

10

20

50

100

PHOTON ENERGY h Y (IN MeV)

FIG.20. The mass absorption coefficient T/P for Al, Cu, and Pb, as a function of the photon energy hv. For Pb, the separate contributions of the photoelectric effect, Compton effect, and pair production are shown by the dashed curve8. The curves of r / p shown in this figure were obtained from the tables of White.216

the total absorption coefficient c. At frequencies somewhat above the position of the minimum, the pair production becomes the main effect, and is responsible for the rapid increase of with increasing v, until a t very high energies (hv 5 Bev), T approaches a constant value as a result of the saturation of due to screening. The minimum of r implies that y rays of energies of the order of 5-20 Mev have a relatively long mean free path in matter. The values of ~ / at p the minimum for Al, Cu, and Pb, are as follows, according to the data of White: 0.0217 cm2/gm for Al; 0.0306 cm2/gm for Cu, and 0.041 cm2/gm for Pb. The corresponding values of the maximum mean free path are: Amsx = 46.1 gm/cm2 for A1 (at hv = 20 Mev); ,A, = 32.7 gm/cm2 for Cu (at hv = 8 Mev); X, = 24.4 gm/cm2 for P b (at hv = 3.4 Mev).

-

1.2.

IONIZATION CHAMBERS

89

C ~ l g a t has e~~ carried ~ out measurements of the total y-ray absorption cross section a for y energies of 0.411, 0.664, 1.33, 2.62 Mev using radioactive sources, and 4.47,6.13, 17.6 Mev using y rays from nuclear reactions. The measurements were made for a variety of absorbers (polyethylene, C, Al, Cu, Sn, Pt, Pb, Bi, and U). I n general, the theory is in good agreement with these data, when account is taken of a small correction due to Rayleigh in addition to the three principal effects: the photoelectric effect, Compton scattering, and pair production in the field of the nucleus and the atomic electrons. The present theory is in reasonable agreement with measurements of the absorption coefficient at various energies up to 280 M ~ v . ~ ~ ~ , ~

1.2. Ionization Chambers* 1.2.1. General considerations’ An ionization chamber is a device which measures the amount of ionization created by charged particles passing through a gas. The basic processes of ionization of gases by charged particles have been discussed in Chapter 1.1.7 If an electric field be maintained in the gas by a pair of electrodes, the positive and negative ions will drift apart, inducing charges on the electrodes which can be detected as a voltage pulse. Or if a steady flux of particles enter the chamber one can measure the average current caused by the ionization. The latter application will be specifically considered in Section 1.2.7; the other sections of this chapter, however, will be principally devoted to the ionization chamber as a detector of single particles and therefore as a pulse instrument. 1.2.1.1. Essentials of a Pulse Ionization Chamber. Figure 1 shows schematically the essential parts of this very simple device. One of the electrodes, the “collector” (a misnomer, as we shall see), is designed to have a low capacity both to the other electrode and to ground, so that a very small charge will still give a measurable potential change. The small amount of charge is characteristic: if the particle loses 1 Mev in collisions S. A. Colgate, Phys. Rev. 87, 592 (1952). W. Franz, Z. Physik 98, 314 (1935); P. Debye, Physik. Z. 31, 419 (1930). t See also Vol. 2, Chapter 4.1 and Vol. 4, B, Chapter 7.5 and Section 9.2.3. 1 General references for Sections 1.2 and 1.3 are: D. H. Wilkinson, “Ionization Chambers and Counters.” Cambridge Univ. Press, London and New York, 1950; B. B. Rossi and H. Staub, “Ionization Chambers and Counters.” McGraw-Hill, New York, 1949; S. C. Curran and J. D. Craggs, “Counting Tubes.” Academic Press, New York, 1949; see also Vol. 4, A, Section 2.1.5. **o

*an

-

* Chapter

1.2 is by Robert W. Williams.

90

1.

PARTICLE DETECTION

with the gas, the number N of electrons released will be about 30,000, or coulomb. The passage of a particle creates the ionization, for all practical purposes, instantaneously. The positive ions then drift toward the negative electrode with a velocity of the order of (1cm/sec) X [(760 mm Hg)/p X [ E / ( l volt/cm)] or in a typical case 0.001 cm/psec. The electrons, if they do not suffer attachment and thereby become heavy negative ions, will drift toward the positive electrode with velocities, under comparable conditions, of 1-5 cm/psec. For definiteness assume, as is usually the case, that the collector is the anode. The collector potential is lowered both by the motion of electrons toward it and by the motion of positive ions away from it. It is instructive to calculate explicitly the potential change in a highly idealized case; for

a charge of 5 X

'\ \

I

I

= -k

FIG. 1. The essentials of a pulse ionization chamber (schematic). The dotted line illustrates the path of an ionizing particle whose passage leaves pairs of ions in the gas of the chamber.

example, insulated long cylindrical electrodes with cylindrical sheets of positive and negative charge, Q+ = Q- = Q formed a t radius r1 (Fig. 2). Let the initial potential difference between inner and outer electrodes be Vo(b)- V O ( Uwhere ), V,(r) is the potential at any point in the chamber before the charge sheets have been moved apart, and assume that the negative sheet of charge collapses uniformly toward the central electrode and the positive one expands toward the outer electrode. When the two charge distributions are at r- and r+ respectively we find from elementary calculation that the potential has changed by an amount which is independent of the magnitude of the initial potential, and which can be written (1.2.1)

This result, that the potential change of the electrode is proportional t o the fraction of the total potential drop through which the charge has

1.2.

IONIZATION CHAMBERS

91

moved, is not restricted to this special geometry; it will be discussed in more detail in Section 1.2.2. We use it here to note that the total voltage pulse, if all the ions are ultimately collected, is Q/C as expected, and that the part of this pulse corresponding to electron motion occurs orders of magnitude more rapidly than the part due to ion motion. It is therefore important t o know under what circumstances the electrons will remain free as they drift through the gas, and what mechanisms may prevent complete collection of all ions.

FIG.2. Idealized cylindrical ionization chamber.

1.2.1.2. Behavior of Ions and Electrons in Gases. The positive ions which are formed upon passage of a charged particle through the gas remain nearly in thermal equilibrium with the gas. The presence of the electric field causes them to drift toward the cathode, but the increase in kinetic energy is very small and they rapidly reach a terminal mean velocity for which the energy gained from the electric field is dissipated in molecular collisions. This “drift velocity’’ would be expected to depend on the ratio of field strength to mean free path, and indeed it is observed experimentally t o be, in the ionization-chamber range, a linear function of E / p : 760E (1.2.2) w = Ko-

P,

with E in volts per centimeter and p in mm Hg, K Oranges from about 8 cm/sec for Ne, t o 2.5 cm/sec for Ar, t o 1 cm/sec for Xe.2 Negative ions 2 A. M. Tyndall, “The Mobility of Positive Ions in Gases.” Cambridge Univ. Press, London and New York, 1938.

92

1.

PARTICLE DETECTION

(not electrons) have about the same drift velocities. A simple theory of drift velocity8 yields the expression

_e _X -E muP where is the mean free path and u the ion’s speed, assumed the same for all ions. Since practical limitations (Section 1.2.3) usually restrict the average value of E / p to not more than -1 volt/cm/mm Hg, the minimum time required for ions to cross even a small ionization chamber, say 1 cm, will be -0.2 msec, and in most cases it will be considerably longer. In nearly all cases where ionization chambers are used to count individual particles only the fast portion of the pulse, due to the motion of the electrons, is utilized. The electrons which are released in the initial ionization prove to remain free, in most gases, until they impinge on an electrode (or other surface). The exceptions are electronegative gases which have an appreciable probability to form negative ions by electron attachment. For 0 2 , the most dangerous offender, the probability of attachment, per collision, is 10V to Water vapor has a similarly large attachment probability, and Clz, NH,, N20, HzS, SO?, NO, and HC1 are all known to be bad. In a nonattaching gas-Ar, N2, CH4 are among the commonly used ones-the electrons continue to drift toward the anode, but acquire from the field a kinetic energy many times their thermal energy, because the mechanisms of energy transfer to the gas are relatively inefficient. The denoted by the ratio of mean kinetic energy to thermal energy (#KT), “agitation energy” 7,is typically of the order of 100 in a noble gas, where elastic collision and atomic excitation are the only available energytransfer mechanisms; it is down by an order of magnitude in diatomic gases, and is not much greater than 1 in polyatomic gases. Table I gives some values5 for the drift velocity and agitation energy of electrons in various gases. Argon is a particularly important gas; it is convenient and is widely used. Its first excitation level is very high, 11.5 volts, and in consequence the electron agitation energy is large. A small amount of polyatomic impurity gas lowers the agitation energy greatly; Rossi and Staub’ find w

=

8 13. B. Rossi and H. Staub, “Ionization Chambers and Counters,” p. 6. McGrawHill, New York, 1949. D. H. Wilkinson, “Ionization Chambers and Counters,” p. 41. Cambridge Univ. Press, London and New York, 1950. 6Based on R. H. Healey and J. W. Reed, “The Behavior of Slow Electrons in Gases,” Amalgamated Wireless, Ltd., Sydney, Australia, 1941; and tables in Wilkineon,‘ and Rossi and Staub,s

1.2.

93

IONIZATION CHAMBERS

about a factor of ten for 10% COZ at E / p = 1. This has the effect of increasing the drift velocity, for two reasons: the electron speed is lowered [see Eq. (1.2.2)], and because of the Ramsauer resonance effect on the cross section of noble gas atoms for electrons the mean free path proves to increase, in the region of interest (which is from -10 ev to -1 ev). Table I includes the drift velocity in 5 % and 10% CO,; at E / p = 1 the increase over pure Ar is a factor of ten; there is a comparable increase when the mixture is compared to pure C02, because the mean free path of -1 ev electrons in argon is so great. TABLEI. Drift Velocity w and Ratio of Agitation Energy t o Thermal Energy 9 for Electrons in Various Gases a t Room Temperature; Principally from Healey and Reed5 Values are approximate and in noble gases are strongly impurity sensitive.

E/p

He Ne Ar HI

N2

coo

0.95 Ar 0.05 COz 0 . 0 9 Ar 0 . 1 coz

=

0.2 v/cm/mm Hg

w (cm/piiec)

tl

0.5 0.5 0.3 0.4 0.4 0.1

62 120 2.7 6.5 1.5

11

E / p = 1 v/cm/mm Hg

---

w (cm/Mec) 0.9 1.5 0.5 1 .o 0.8 0.55

3.3

4.3

0.9

5.3

9

53 216 285 9.3 21.5 1.5

R. H. Healey and J. W. Reed, “The Behavior of Slow Electrons in Gases.” Amalgamated Wireless, Ltd., Sydney, Australia, 1941. 0

The greater drift velocity of Ar-C02 mixtures is often of great practical value in obtaining a faster pulse, and these mixtures are widely used. Unpurified tank argon alone will give rise to an electron drift velocity considerably greater than that of Table I, and is satisfactory for many applications where E / p is large (-1) and where accurate pulse height is not essential. In high-pressure cylindrical-geometry chambers, where E / p at the outer electrode is usually quite low, the Ar-C02 mixture may show some attachmentJ6and pure Ar will give better performance. There are two additional complications in the motions of ions or elec6 Under ordinary conditions COO has negligible electron attachment. Experience with cosmic-ray ionization chambers [H. S. Bridge, W. E. Hazen, B. B. Rossi, and R. W. Williams, Phys. Rev. 74, 1083 (1948)Jindicates that a t very low E / p values (-0.01), “pure” Ar has distinctly less attachment than Ar-COZ mixtures.

94

1.

PARTICLE DETECTION

trons in gases which can lead to reduction in pulse height; we now consider the first of these, diffusion. The center of gravity of a group of ions which is liberated at a point in an electric field will have a displacement proportional to the time, while the mean distance of the ions from the center of gravity will increase as the square root of the time. The importance of diffusion can be measured by the ratio of the latter distance to the former one. This ratio can be calculated from kinetic theory; the mean free path cancels out, and one has, for a gas at room temperature.’ diffusion distance drift distance

=

o.18

d+

(1.2.3)

where q is the agitation energy and V the voltage difference between the point of release of the ions and the final position of the center of gravity. For heavy ions 7 = 1 and the effects of diffusion will generally be small. For electrons 7 may be -100 in noble gases, and diffusion may be important; electrons may diffuse back to the cathode, or out beyond the boundaries of the apparatus. The “ C 0 2 effect ” may be utilized to reduce r] and therefore decrease the electron diffusion. The second effect is recombination, the neutralization of positive and negative ions before they are collected. This is a complex subject, considered in detail by Wilkin~on,~ and we shall only summarize the principal results. The recombination coefficient a is defined by writing the rate of disappearance of ions, when n+ positive ions and n- negative ions per cubic centimeter are present, as an+n-,8 with a depending on agitation energy, and, of course, on the nature of the negative ions; a is cm3/sec for heavy negative ions, and cm3/sec for electrons. In air chambers or other chambers where attachment is more or less complete, recombination can be a serious cause of pulse loss, particularly in those current chambers which contain a large density of ionization (Section 1.2.7). Free-electron chambers are better both because of the smaller value of 01 and the shorter time which the electrons spend in the gas. The over-all improvement factor is -lo7; therefore free-electron chambers do not suffer from recombination effects under any circumstances ordinarily realized. Ionization chambers used as monitors of the direct beams of pulsed machines (synchrotrons, etc.) present special problems. They will be considered in Section 2.7.1. 7 D. H. Wilkinson, “Ionization Chambers and Counters,” p. 37. Cambridge Univ. Press, London and New York, 1950. * This assumes that an ion, when created, does not recombine preferentially with the other member of the pair. The assumption is surely correct if the negative ions are electrons, b u t it might be expected to break down in very high-pressure air.

1.2.

95

IONIZATION CHAMBERS

1.2.2. Pulse Formation Quantitative use of the ionization chamber as a particle detector requires consideration of shape and magnitude of the voltage pulse caused by release of ionization. We at first calculate the potential change of the collecting electrode due to the motion of a single charge, assuming that the electrode is essentially insulated (i.e., referring to Fig. 1, that the time constant RC of the grid resistor and the capacity of the collector is large compared to the collection time of an ion). The result which was obtained for a special case in Section 1.2.1 can be obtained for arbitrary two-electrode geometry under the assumption (always true in practice, even in a Geiger counter) that the charge released in the gas is small compared to the charges residing on the electrodes which give rise to the initial potential difference Vb Va. Consider an electron of charge -e at point r l in the gas. The potential at rl will consist of two parts: the potential Vo(rl) due to the charges on the electrodes, which is, by our assumption, essentially independent of any charges which may be present in the gas, and V8(rl), the potential due to other ionization which may be present in the gas. Then the electrostatic energy of the electron consists of two independent parts, the spacecharge energy and the energy in the field of the electrodes, and we may consider the latter ~ e p a r a t e l y The . ~ energy of the system of electrodes, plus the electron, is

-

+BqiVi = i[-eV,(r,)

-k

&ova0

- QoVao].

where V , and v b are the electrode potentials, and the subscript 0 refers to initial values. As the electron drifts from rl to r 2 the work done on it by the field must be extracted from this system; QO and VbO must remain fixed, but Val the collector potential, can change by some amount AV. Thus we have -e[Vo(rz) - Vo(r1)l

=

+[-eVo(rd

+ eV&d +

QO

AVI (1.2.4)

which is the desired result. 9 The space charge has no direct effect on the size of the pulse due to one electron but it may affect the velocity of the electron and therefore the pulse shape, and secondary effects of the electron. In pulse ionization chambers these questions are unimportant because the space charge is so small, but they are crucial in Geiger counter operation (Section 1.3.2).

1.

96

PARTICLE DETECTION

Two examples will serve to illustrate the behavior of pulse ionization chambers utilizing electron collection (the pulse shape of “slow” chambers is not usually interesting). First the parallel-plate chamber of separation d, ion-pair formed at X;O from the positive electrode, at t = 0. AV will have a rapid contribution from the motion of the electron which has drift velocity w-, and a much slower contribution from the positive ion (drift velocity w+). -e A V = - (C =

w-t

+ wft )

-e

(7w+t + 2)

-e C

finally.

--

until w-t until w+t

= =

xo

d - xo

Figure 3 illustrates this pulse using drift velocities characteristic of argon. To take advantage of the fast electron-collection pulse one must eliminate the effects of the slow positive ions by incorporating a low-frequency rejection network in the collector circuit of the chamber or, more commonly, in the amplifier-e.g., a short time-constant T in a resistancecapacitance coupling stage such that RC = r > t- where t- is the collection time of the electrons, but the positive-ion pulse will be reduced by roughly r/t+. For a given form of input pulse the detailed shape of the pulse after passing through the low-frequency rejection network can be obtained by standard transient-response analysis-Wilkinsonl gives several examples. The dotted line in Fig. 3 illustrates the effect of a time-constant r = 5t(10% pulse-height loss). An approximation which is sufficient for some purposes is to assume that the voltage rise due to the electrons is undistorted, but that the voltage then returns t o zero with the time constant r, and with no positive ion contribution. A network which gives a more nearly square-topped pulse is illustrated in Fig. 4;it consists of a shorted delay line in series with a resistance equal to its characteristic impedance. It is less convenient than the RC, and causes a 50% amplitude loss, so that it is usually used only when there is some reason to require a good pulse shape. The pulse-shaping action of this network is illustrated in Fig. 5. The second example of the voltage pulse in an electron-collection chamber is that of cylindrical geometry: a small central electrode, of radius a, is surrounded by a concentric cylinder of radius b. This is a simple, lowcapacity arrangement, and its chief virtue is that the electron-collection

1.2.

IONIZATION CHAMBERS

97

C 3Q.C d C

nv t

, I \'\ \

'.. -

FIQ.3. Idealized voltage pulse from a parallel-plate ionization chamber. An ion pair is released at distance Xa from the negative electrode. The time scale would be approximately right if, for example, d were 2 cm, V were 1000 volts, and P were 1 atmos. The dashed line indicates the effect of a five-microsecond time constant.

0-jD.L.

FIQ.4. Shorted-delay-line pulse-shaping network. Rk is equal to the characteristic impedance of the delay-line.

t

-AV

FIQ.5. Effect of delay-line pulse shaping on a typical pulse from a cylindrical ionization chamber; 270 is the round-trip time of the line, and must be greater than t-, the electron collection time, if t h e pulse is to have a flat top.

98

1.

PARTICLE DETECTION

pulse is reasonably independent of the point at which the electron is released, since most of the potential drop occurs near the central wire, Figure 6 shows the electron pulse in a chamber with b/a = 120, calculated for a single electron [curves (a) and (a’)]; for uniform ionization in the chamber [curves (b) and (b‘)]; and for uniform ionization along a straight line passing through the axis of the chamber [curve (c)]. The pulse shape,

FIG.6. Electron pulse in a cylindrical ionization chamber: 1- is the drift time from outer to inner electrode; curves (a) and (a’) are for a single electron, (b) and (b’) for a uniform distribution, and (c) for a linear distribution.

AV(t)/AVfina~,is calculated from Eq. (1.2.1), with elapsed time related to drift velocity by

t=

J:&

and with two assumptions for w: constant (solid curves) and proportional to (E)”q(dotted curves). Inspection of curve (a) shows that a considerable portion of the chamber volume gives rise to pulses of nearly maximum height. It is easy to show that uniform ionization gives a pulse which is a fraction f of the total-charge-collection pulse, f = b2/(b2 - u2)

This is 0.90 for b/a

=

120.

- 1/[2 ln(b/u)]

1.2.

IONIZATION CHAMBERS

99

The parallel-plate chamber can be modified so that it gives an electron pulse independent of the point of production,1° by adding a third electrode. A grid at a fixed potential, near the collecting electrode, will shield the collecting electrode from the rest of the chamber (Fig. 7) so that only that portion of the electron’s travel which takes place between the grid and the collector will cause a pulse to be induced. The electrostatic situation near the edge of an electrode will in general be complicated-either the field will be quite distorted and therefore the effective volume, and expected pulse shape, somewhat uncertain, or the field shape can be maintained by “guard electrodes”-extra electrodes, Grid, Useful Volume

-

Collector

h

FIG. 7. Schematic diagram of a gridded chamber. Electrons originating in the shaded area will give pulses of nearly uniform height as they pass between the grid and the collector.

held at the average potential of the collecting electrode, which maintain the symmetry of the field beyond the edge of the collecting electrode. The pulse induced by an electron near the edge of the collector is more complicated in this case; detailed calculations in simple cases are given by Rossi and S t a ~ b . ~ In many applications of ionization chambers one is interested in average current or in total amount of charge collected (Section 1.2.7). For these chambers (as well as for “slow” pulse chambers which respond to the motion of ions) the details of the eIectron pulse are unimportant. The guard electrode helps t o define accurately the volume of gas from which ionization is collected. The role of the guard electrode in preventing leakage current is discussed in Section 1.2.7. 10 For the detailed theory of this device, Bee D. H. Witkinson, “Ionization Chambers and Counters,” p. 74. Cambridge Univ. Press, London and New York, 1950.

100

1.

PARTICLE DETECTION

1.2.3. Quantitative Operation and Some Practical Considerations 1.2.3.1. Attachment, Diffusion, Recombination. Those basic phenomena of the passage of electricity through gases which affect the operation of ionization chambers are discussed in Section 1.2.1. Their importance may range from very little in the case of large E / p and nonabsolute pulseheight requirements (e.g., a low-pressure parallel-plate chamber used as a counter) to considerable in the opposite extremes (e.g., a high-pressure cylindrical chamber used for proton recoil pulse-height spectrum work). The maximum value of E / p which can be used is determined by the condition that there should be no gas multiplication (Section 1.3.1) even in the region where E / p is largest. For cylindrical or spherical chambers this means that E / p will necessarily be low in the region near the outer electrode; and in general the difficulties mentioned are associated with low E / p . Even with parallel-plate chambers the inconvenience of working with high voltages will often put a practical limit on the available E / p . Electron attachment can be eliminated by sufficiently rigorous exclusion of electronegative gases, of which O2 and H20 are the most frequent offenders. Noble gases can be purified very effectively by circulation over hot calcium c h i p ~ . ~Purified Jl argon in a clean metal-and-glass chamber with soldered seals has been found to remain free from attachment for years, in an application in which the minimum E / p was about 0.01 v/cm/mm Hg. However, a chamber containing volatile material (e.g., rubber gaskets) must be purified frequently if quantitative performance under low E / p conditions is to be maintained. The Ar-C02 mixture previously referred to can also be purified with hot ~ a l c i u r nand , ~ is free of attachment under most conditions. However, there are indications12 that at very low E / p this mixture, unlike pure argon, shows noticeable attachment. Gases which cannot be purified by such drastic methods, e.g., BF3 (which in pure form does not have serious attachment), must be prepared with great care. Graves and Fromanla describe a suitable technique for preparing BF3for ionization chambers. Recombination can be shown to be negligible14under nearly any circumstances in chambers in which no attachment takes place. This subject will therefore be treated in Section 1.2.7, in connection with current chambers. L. Colli and U. Facchini, Rev. Sci. Instr. 23, 39 (1952). H. S. Bridge, W. E. Hazen, B. B. Rossi, and R. W. Williams, Phys. Rev. 74,1083 (1948). l 3 A. C. Graves and D. K. Froman, “Miscellaneous Physical and Chemical Techniques of the Los Alamos Project,” p. 154. McGraw-Hill, New York, 1952. l4 D. H. Wilkinson, “Ionization Chambers and Counters,” Chapter 111. Cambridge Univ. Press, London and New York, 1950. 11

la

1.2.

101

IONIZATION CHAMBERS

Diffusion effects are important only for electrons under conditions of large agitation energy q [Eq. (1.2.3)].A typical situation in which diffusion may be important occurs when ionization is released adjacent to the wall of the chamber, as when a noncollimated alpha-particle source is incorporated in the negative electrode as a calibrating standard. It can be assumed that electrons which diffuse back t o the negative electrode are lost, and with the help of Eq. (1.2.3) an estimate of the pulse loss from this effect can be made. Addition of a polyatomic gas, with consequent 10 r

0

0.5

1.0

15

2 .o

2.5

E (ev) FIG.8. Attachment probability h upon collision of an electron with an oxygen molecule, as a function of electron energy E. From Wilkinson,6 by permission.

reduction of 7, is of course desirable for applications where diffusion must be minimized. An example of a complete diffusion calculation is worked out by Rossi and Staub.16 1.2.3.2. Checks for Quantitative Operation. Spurious Effects. The most commonly used test for proper ionization-chamber operation is the examination of the pulse height, from some reproducible source, as a function of collecting voltage. If a chamber exhibits a good “plateau”region in which pulse height is independent of voltage-it is usually considered free from the defects we have outlined. However, examination of the dependence of electron attachment coefficient on electron energy, Fig. 8, shows that in oxygen, at least, there is a region in which increasing electron energy causes an increase in attachment coefficient, which might l6 B. B. Rossi and H. Staub, “Ionization Chambers and Counters,” p. 27. McGrawHill, New York, 1949.

102

1.

PARTICLE DETECTION

compensate the decreased number of collisions which electrons suffer at higher energies. Also, the drift velocity (in impure argon, for example) l6 may not increase with E. The existence of a plateau is therefore not a sufficient indication, in a pulse chamber, that all electrons are being collected. A source of alpha particles of known energy can be used to release a known amount of ionization in the chamber (Section 1.2.4), say Qo. The pulse height due to the electron motion should then be

Vo = (fQo/C,)G where f is the (average) fraction of total voltage drop through which the electrons move, C, the collector capacity, and G the amplifier gain. A quantitative check system based on this principle is outlined by Bridge and associates.6 A more elaborate method of checking, which does not depend on knowing the capacity of the chamber, has been used by Hazen and collaborators.I6 They provide a polonium source which remains at the potential of the negative electrode but can be moved toward the collecting electrode, reducing the degree of attachment by reducing the path length through which the electrons move. Constant pulse height as the source is moved in is a reliable check in this case. A still more elaborate method, using a pulsed X-ray source and measuring the fraction of total current carried by electrons, is described by Rossi and Staub.” In the design and construction of pulse ionization chambers a reasonable care must be taken to avoid spurious pulses from high-voltage leakage or breakdown or electrical pickup. Any good insulators can be used (in contrast to the current chambers described in Section 1.2.7, which require very high quality insulators) ; in particular, glass-Kovar seals are very useful. In most applications it is possible to provide a grounded conductor (guard electrode) which separates the high-voltage insulator from the collector insulator, thereby greatly reducing the dificulties caused by leakage across the high-voltage insulator. It is sometimes convenient to have the collector at high voltage, and to connect it to the amplifier through a coupling condenser. The condenser must then be selected very carefully. Ceramic condensers seem to be the most satisfactory. The signal obtained from a pulse chamber is often of the order of a millivolt or less. Obviously the collector and the amplifier input must be completely shielded. Ordinarily a double shield (i.e., a grounded case See F. E. Driggers, Phys. Rev. 87, 1080 (1952). B. B. Rossi and H. Staub, “Ionization Chambers and Counters,” p. 58. McGrawHill, New York, 1949. l8

l7

1.2.

IONIZATION CHAMBERS

103

outside the high-voltage electrode) is necessary; occasionally one must take special precautions-such as connecting all grounds together only a t the first tube of the amplifier-to prevent electrical pickup. The use of a short time-constant in the amplifier eliminates most ac and microphonic difficulties, although in low-level work it is often necessary t o operate the preamplifier filaments on dc.

1.2.4. Amount of Ionization liberated The basic process by which a fast charged particle loses energy in a gas have been discussed in Chapter 1.1,* where it is pointed out th a t the total energy loss of a particle of charge Z and velocity @ is, within certain limitations, just proportional to Z 2 times a function of @.Experimentally it is found that a given energy loss gives rise t o a number of ion pairs that depends on the gas, but is approximately (to at worst 10%) independent of the nature and speed of the particle. One can understand in a qualitative way why this should be so: the primary energy-loss event results either in excitation of the gas molecule, or in ionization. I n the latter case the electron may be ejected with considerable energy, but if so it will itself undergo further excitation or ionization collisions, 80 that the energy ultimately either goes into ionization or into excitation (whence it is dissipated in collisions or escapes as radiation.I8 The partition of energy between ionization and excitation depends mainly on the behavior of rather slow electrons even though the primary particle may be of very high energy. The energy loss corresponding to the formation of one ion pair W proves t o be a few times the ionization potential. T he constancy of W means th at the energy of a particle which stops in an ionization chamber can be measured by measuring the quantity of ionization released, or more generally the energy lost in the chamber by particles passing through is directly proportional to the ionization. This is an important and much-used property of the ionization chamber, and for quantitative work i t is clearly necessary to have accurate empirical data ~ ' provided a large amount on W . The work of Jesse and S a d a u k i ~ ' ~ -has of information on W ;it extends previous work, and where a cross-check

* See also Vol. 4, A, Parts 1 and 4. Xenon and, to a lesser extent, krypton and argon give off a considerable fraction of this energy as "scintillation" light in the visible and ultraviolet region. See R. A. Nobles, Rev. Sci. Znstr. 27, 280 (1956);C. Figgler and C. M. Huddleston, Phys. Rev. 96, 600 (1954);A. Sayres and C. S. Wu, Rev. Sci. Instr. 28, 758 (1957).Such noble gas scintillations are discussed in Chapter 1.4. 18 W. P. Jesse and J. Sadaukis, Phvs. Rev. 97, 1668 (1955);100, 1755 (1955);W.P. Jesse, H. Forstat, and J. Sadaukis, ibid. 77,782 (1950). 20 W.P. Jesse and J. Sadaukis, Phys. Rev. 102, 389 (1956). 21 W.P. Jesse and J. Sadaukis, Phys. Rev. 107, 766 (1957). 16

104

1.

PARTICLE DETECTION

is possible it agrees with other contemporary data; the relative accuracy between different gases is a few tenths of a per cent, and the absolute accuracy about 1%. Their principal results are summarized in Table 11, TABLE 11. Average Energy, in ev, to Make an Ion Pair W for Beta Particles, Po210Alpha Particles, and Low-Energy Alpha Particles, in Pure Gases* Gas He Ne Ar Kr Xe Hz

Air Nz 0 2

COZ CaHi CzHe CHI CzHz

Mean W,g for beta particles

W , for PoZlO alpha particles

W , for alpha particles of 1.2 Mev

42.3 36.6 (26.4) 24.2 22.0 36.3 34.0 35.0 30.9 32.9 26.2 24.8 27.3 25.9

42.7 36.8 26.4 24.1 21.9 36.3 35.5 36.6 32.5 34.5 28.0 26.6 29.2 27.5

42.4 37.4 (26.4) 24.1

37.1 38.1 36.3 29.8 28.5 31 .O 29.0

W. P. Jesse and J. Sadaukis, Phys. Rev. 102, 389 (1956); 107, 766 (1957).

where values of W are listed as the energy loss in electron volts corresponding to one ion pair. The table is actually based on the assumption that in argon W is a true constant for different particles; this assumption is strongly supported in reference 21. The constancy of W in argon for alpha particles of varying velocities has been checkedlS for a range of 1 to 9 MeV. There is evidence from a study of Po210 recoil nuclei20 that these extremely slow and heavy particles have a W ,in argon, about four times that of alpha particles. However, fission fragments already exhibit “normal” behaviorzz-they have a W of 36 ev in air, and presumably would show the standard W in argon. Bakker and SegrtP found W for 340-Mev protons to be 35.3 for Hz and 33.6 Nz,3% lower than the W,gvalues of Table 11. BarberZ4studied the specificionization of high-energy electrons (1 to 34 Mev). By assuming the validity of the theoretical energy-loss expression he found, for elec22 D. H. Wilkinson, “Ionization Chambers and Counters,” p. 21. Cambridge Univ. Press, London and New York, 1950. [See D. West, Can. J. Research A26, 115 (1948); N. D. Lassen, Phys. Rev. 70, 577 (1946).] ** C. J. Bakker and E. Segr6, Phys. Rev. 81, 489 (1951). 24 W. C. Barber, Phys. Rev. 97, 1071 (1955).

1.2.

IONIZATION CHAMBERS

105

trons of -2 MeV, W values of 37.8 0.7 in Hz, 44.5 t- 0.9 in He, and 34.8 5-0.9 in Nz. At higher energies the Nz value remains constant but the Hz value increased by about 3% a t 34 MeV, presumably because Cerenkov radiation begins to carry off some of the energy. The general conclusion to be drawn is that in pure noble gases, and in hydrogen, W is independent of the energy and nature of the particle, to about 1%, over a wide range. I n air and other complex gases one can expect a lessened ionization efficiency a t high specific ionizations, i.e., an increase in W which can be of the order of 10%. However, it has been found’g that in the noble gases of very high ionization potential, notably helium, minute admixtures of impurity reduce the value of W very decidedly: for example, 100 parts per million of Ar in He has a 25% effect. Presumably this is caused by ionizing collisions of metastable He atoms with argon atoms (the first excited state of He is higher than the ionization potential of Ar). Argon should be free of this difficulty, since it has a much lower ionization potential. Statistical fluctuations in the actual number of N of ions formed in a chamber around the average number of fl = A E / W (where energy AE is lost in the chamber) present two quite different problems. If the particle loses all its energy, fluctuations in the rate of energy loss will be correlated, since the total energy is fixed, and the standard deviation in N will be it has been calculated to be about two-thirds of this.25 less than 4%; But if the energy lost in the chamber, AE, is only a small fraction of E , the fluctuations in AE will depend on the number of primary collisions N , and on the energy given to the delta ray in each collision; the standard deviation will be considerably greater than The problem is more characteristic of proportional counters than of ionization chambers, since the former must be used when AE is small; discussion is therefore postponed to Section 1.3.1.

dz.

1.2.5. Noise: Practical Limit of Energy Loss Measurable

The smallest charge which can be detected as a pulse in an ionization chamber is limited by the intrinsic noise of the first stage of the amplifier. Amplifier noise as it affects the sensitivity of ionization chambers is discussed by ElmoreZ* and Gille~pie.~’ * Depending on conditions (fast or slow rise, large or small chamber capacity), different sources of noise may become the predominant source of noise charge. The optimum signal-to-

* See also Vol. 2, Chapter 12.5. U. Fano, Phys. Rev. 72, 26 (1947). W. C. Elmore, Nucleonics 2, (3), 16 (1948). 27 A. B. Gillespie, “Signal, Noise, and Resolution in Nuclear Counter Amplifiers.” Pergamon, New York, 1953. 26

26

106

1.

PARTICLE DETECTION

noise situation proves to be one in which rise time is comparable with pulse duration. Elmore shows that in a typical short-pulse-duration chamber, if C,,, is the chamber capacity in micromicrofarads, R, the “equivalent shot noise resistance” in ohms, and T the pulse duration in microseconds, the most probable noise charge in electron charges is Q/e

-

CPpfdR./7

for the best signal-to-noise ratio. The equivalent resistance R, is defined so that 4kTR, = 2eI,/gm2 where k is the Boltzmann constant, and I , and gm are the plate current and transconductance of the first tube of the amplifier. A typical value for R, is 10000, for 7 , 10 psec, and for C , 30 ppf, so that Q / e 300 ion pairs equivalent noise. This would correspond to about 9000 ev of energy loss in the chamber. Of course fluctuations of three or four times the most probable noise occur frequently, and the minimum charge which can be detected reliably corresponds to about 5 times this, or an energy loss of 50,000 ev. This is somewhat better than is usually achieved in practice, although Wilkinson28cites some experience indicating that it may be attainable. Improvement by further lengthening T is not very effective, even if speed of response can be sacrificed, since grid resistor noise, independent of T , becomes important. For T not restricted, Elmore finds for the optimum case the most probable noise is

-

Q / e = 735[C,,t(R,/R,)11’2 where R, is the effective grid resistance (the grid resistor in parallel with the equivalent noise resistance of grid current). For a high gm tube such as the 6AK5, R, is limited by grid current, so one is limited to a threshold sensitivity which proves to be two or three times better than that of the short-pulse limit. The pulse duration and rise time, for a typical case, might be 50 Msec. The selection of amplifier rise time and clipping time depends not only, or even principally, on the noise problem, but on the particular application in hand-the speed of counting, necessity for accurate timing, shape of ionization chamber pulses, quantitative preservation of pulse height, etc. If accurate reproduction of pulse shape is not important, signal-to-noise and pulse-height reproduction can both be improved by using a rise time and clipping time which are equal, and somewhat longer than the rise time of the slowest chamber pulse. For a detailed discussion of several specific cases, see Wilkinson,’ Chapter 4. x8 D. H. Willcinson, “Ionization Chambers and Counters,” p. 142. Cambridge Univ. Press, London and New York, 1950.

1.2. IONIZATION

CHAMBERS

107

1.2.6. Some Types of Pulse ionization Chambers 1.2.6.1. Alpha-Particle Chambers. The typical energy release of a radioactive or induced alpha emission is several MeV, so it is clear from the foregoing that the ionization chamber is very well suited to the detection of these particles, or measurement of their energy. Typical ranges of these alpha particles in air at one atmosphere are a few centimeters, so that it is easy to contain the entire path of the particle in the chamber. Absolute counting of alpha particles is usually accomplished by placing a thin deposit containing the alpha-active material on the negative electrode of the chamber. Ideally this constitutes a ‘ I 27r” counter (assuming the electrode is plane), so that the fractionf of all disintegrations counted is +. However, the finite thickness t of the deposit will introduce a correction; the fraction which escapes can easily be shown to be $(l - t/2R), where R is the range of the particle in the material; since there will be which is the least energy an escaping alpha can have some energy Emin and still be counted, because a nonlinear discriminator of some sort must be set to reject the unwanted small pulses due to noise, etc., the range is the range of a particle of energy Emin, and the fraction reduced by R(Emin), becomes f = +{ 1 - t/2[R - R(EmiJ]}. A second correction is required to take into account the backscattering of alpha particles which start into the material in a direction away from the gas, but are deflected into the counting volume by multiple scattering. Rossi and S t a ~ give b ~ ~ numerical results of a calculation of this effect for various materials and energies. Typical values would be, for an alpha particle of 3.68 cm range, an increase in f of 8 % for gold or 2 % for aluminum. Measurement of alpha-particle energy by means of ionization chambers is considered in Section 2.2.1.2. 1.2.6.2. Proton Recoil Detectors. Neutrons can be studied by observing the ionization of recoil protons from n - p collisions in the counter gas, if it is hydrogenous, or in a hydrogenous foil or lining of the chamber. A proton recoil at an angle 0 with respect to the direction of a neutron of energy EOwill have an energy of EOcos2 e (nonrelativistic). I n the energy range from a few hundred kilovolts to a few Mev the recoils can be stopped in the gas (a 5-Mev proton has a range of 34 cm in air at 1 atmosphere) and the recoil chamber can be used as a measure of the energy and absolute flux of neutrons; this application is discussed in Section 2.2.2.1. As a method for counting neutrons on a relative basis the recoil ioniza29 B. B. Rossi and H. Staub, ‘‘Ionization Chambers and Counters,” p. 127 McGrawHill, New York, 1949.

108

1.

PARTICLE DETECTION

tion chamber is usually less satisfactory than the proportional counter (Section 1.3.1) because the continuous energy distribution of the protons leads to many undetectable recoils. A large number of specific designs of proton recoil detectors are discussed in references 3 and 4. More recent work is discussed by Johnson and Trai130 and by Berenson and S h ~ r m a n . 3 ~ 1.2.6.3. Boron Trifluoride Chambers and Fission Chambers. Neutron: of all energies can be detected by the energy released in an ionization chamber by a nuclear disintegration; it is sometimes convenient to detect high-energy protons this way also. For neutrons the most widely used reaction is Bl0(n,a) Li7, which releases 2.34 Mev for thermal neutrons, and of course more for fast neutrons. The cross section for this reaction is very large, so that even though natural boron contains only 19% BO ' it leads to relatively high efficiency counters. Among the gaseous compounds of boron BFt is the most stable and satisfactory. It is reasonably free from electron attachment when pure, but the commercial gas often contains impurities which are difficult to remove and which lead to attachment. A method of preparing the pure gas is described by Graves and Froman. l3 Boron enriched in Bl0is available from suppliers of stable isotopes. BO ' has a cross section for slow neutrons which varies as l / v up to energies in the kilovolt range. Thus it lends itself to absolute measurements of neutron density n (rather than flux nv) in the low-energy range, since the disintegration rate is proportional to nvu, and therefore, to n, since u l/v. An ionization chamber filled with BF3provides a very stable method for a relative measurement of neutron flux, since it can be arranged so that the majority of the pulses are the same height (by using cylindrical geometry or a gridded chamber) and the pulse height is essentially independent of the applied voltage. However, the pulses are small and for most applications a proportional counter (Section 1.3.1) is more convenient. For measuring an integrated flux or time-average flux a current chamber (Section 1.2.7) filled with BF3 is very satisfactory. Another useful reaction for neutron measurement is the fission of heavy nuclei. Fission releases nearly 200 MeV; the fragments have a range of about 2 cm in air, and ionize most heavily at the start of their range. Very thin deposits of fissionable material are necessary to obtain the full energy of the fragment, but the energy release is so large that for counting this is not very critical. UZ3& has a large slow-neutron fission cross section, and competes in

-

30 31

C . H. Johnson and C. C. Trail, Rev. Sci. fnstr. 27, 468 (1956). R. E. Berenson and M. B. Shurman, Rev. Sci. Instr. 20, 1 (1958).

1.2.

IONIZATION CHAMBERS

109

efficiency with boron. U233and Pu239 are also slow-neutron fissionable, but have higher alpha activities. However, none of these substances is generally available. At higher neutron energies several heavy elements show convenient fission “thresholds”; some of these are listed in Table 111. Above bismuth TABLE 111. Approximate Thresholds of Various Heavy Substances for Fission by Neutronsn Nucleus

Threshold

NpZ3’

0.4Mev 0.5 Mev 1.1 Mev 1.3 Mev 60 Mev

Pa232 U238

Th234 Bi2o9

*From B. T. Feld, in. “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. 2, p. 347. Wiley, New York, 1953.

all of these elements are alpha-active. Rapid electron collection and a short resolving time are necessary to prevent “pile up”-imitation of a large fission pulse by two or more alpha pulses.

1.2.7. Current Ionization Chambers and Integrating Chambers* Historically the earliest use of ionization chambers was not as singleparticle detectors but as meters for the average rate of ionization occurring in a gas, and such current-or charge-meters still have very wide utility; for example, in detecting and measuring radioactivity, in radiological health measurements, in cosmic-ray intensity studies, and in beammonitoring at particle accelerators. In general the problems of current chambers are quite different from those of pulse chambers and they will not be discussed here in The current t o be measured is usually very small and the use of good insulators is essential. Polystyrene, amber, quartz, and Teflon are satisfactory. Guard electrodes to prevent a direct leakage path between high-voltage electrode and collector (Section 1.2.3.) are essential in this application. Care should be taken th a t ionization cannot collect on an insulating surface. The second principal problem of current chambers is recombination. Since the recombination rate of ions is proportional to the square of the ionization density, recombination affects the linearity as well as the

* See also Vol. 4,A, Section 2.1.5. For more information on current chambers see D. H. Wilkinson, “Ionization Chambers and Counters,” Chapter 5. Cambridge Univ. Press, London and New York, 1950. 32

110

1.

PARTICLE DETECTION

absolute accuracy of ionization chambers. I n many applications reproducibility and accuracy are the essential qualities demanded of the current chamber. A pure, nonattaching gas is essential unless the E / p values are everywhere high. A small amount of attachment, which may be quite acceptable in a pulse chamber, will often lead to recombination and therefore to nonquantitative, nonlinear effects in a current chamber. Recombination is particularly serious in chambers used with pulsed accelerators, since the ion density is high during the pulse even though the average ion current may be low. A well-designed ionization chamber can have a time-constant of weeks, and its accuracy is usually limited by the precision of measurement of current or amount of charge. The transient response of a current chamber is of course quite different depending on whether or not attachment is taking place. I n a free-electron chamber the part of the current carried by electrons will have a response in the microsecond region. Rossi and S t a ~ describe b ~ ~ a chamber which was used to detect changes in gamma-ray flux occurring in less than 1 psec.

1.3. Gas-Filled Counters* 1.3.1. Gas Multiplication; Proportional Counterst

I n the discussion of the behavior of electrons in gases (Section 1.2.1) it has been assumed that the electrons released by the initial ionizing particles do not create further ionization after they have been slowed down to the mean agitation energy. They continually gain energy from the electric field, at, an average rate weE,where w is the drift velocity, but it has been assumed that they lost this energy in elastic collisions or excitation collisions with gas molecules. If the field strength is sufficiently great, however, some electrons will acquire an agitation energy greater than the ionization‘potential of the gas molecules, and new ionization will be formed. If-the average rate of ionizing collisions is a per centimeter, the average number of electrons after I centimeters, per original electron, would be n = eaz. This effect is called gas mu2tipZication.f By its use the total amount of ionization from a given initial act can be increased, thereby increasing the available signal. It is, of course, an undesirable 33 B. B. Rossi and H. Staub, “Ionization Chambers and Counters,” p. 106. McGrawHill, New York, 1949. t See also Vol. 4, A, Section 2.1.4. $ See also Vol. 2, Chapter 4.1.

* Chapter

1.3 is by Robert W. Williams.

1.3. GAS-FILLED

COUNTERS

111

effect in an ionization chamber, where its presence destroys the quantitative relationship between the energy lost in the chamber and the amount of ionization released. A counter which is designed to use gas multiplication as a n amplifying device is called a proportional counter. It will normally have a very smallradius positive electrode, so that the region of high field and therefore of gas multiplication will be confined to a small volume near the electrode. The most frequently used geometry is cylindrical, with concentric electrodes of radii a and b, and with a l; or toluene cadmium propionate 3. toluene samarium or gadolinium propionate [Sm149( n , ~ ) ] .

+

+

}

Generally however, quenching sets in before much of the desired substance is in solution, so that concentrations have until recently been restricted to approximately one percent. Recently studies b y Kallmann" and Swank have shown that intermediate solvents such a s naphthalene and biphenyl can be used to extend the amounts and types of compounds that can be successfully dissolved without quenching. Recent work by Hyman16 has resulted in plastic scintillators containing u p to 5% b y weight of lead, with a response that is about 50% that of a n unloaded plastic scintillator. 1.4.1.8. Noble Element Scintillators. In the past several years efforts to prepare gaseous scintillation counters have resulted in certain successes in the use of the noble gases. These counters have provided properties of interest in speed, large light output, linearity, simplicity and flexibility in Z and density. Work has been done by Northrup and noble^,^^-^* Eggler and H u d d l e ~ t o n , 'and ~ Sayres and Wu120among others. Early results were difficult to correlate until the importance of eliminating impurities was fully realized. Other factors that must be taken into account are the size of the container and t,he effect of wavelength shifters. Consistent and successful gas scintillation counters have been reported by Sayres and Wu20 constructed along the lines shown in Fig. 8. The gases used were helium, argon, krypton, and xenon. Since the gas is H. Kallmann, I R E Trans. on Nuclear Sci. NS-3, No. 4 (1956). J. A. Northrup and R. Nobles, IRE Trans. on Nuclear Sci. NS-3, No. 4 (1956). 19 C. Eggler and C. M. Huddleston, IRE Trans. on Nuclear Sci. NS-3, No. 4 (1956). 10 A. Sayres and C. S. Wu, Rev. Sci. Znstr. 28, 758 (1957); W. R. Bennett, Jr. and C. S. Wu, Bull. Am. Phys. SOC.[2] 2, No. 1 (1957). 17

18

138

1.

PARTICLE DETECTION

susceptible to impurities it is necessary to take special precautions in construction. Teflon is used as gasket material; metal components and gaskets are carefully baked out. I n addition a gas circulating pump and calcium purifier were also incorporated. With proper precautions and initial purification Sayres and Wu found no measurable deterioration in performance in their counters over a period of five days. Most of the light coming from the noble element scintillators lies in the ultraviolet so that wavelength shifters are necessary. Some of the ,Q.

phenyl,

as ut

u234

I' alpha source

1

0

I in.

m

u Scale

FIG.8. A typical gas scintillator. (From Sayres and WulZoRev. Sci. Instr. 28,759 (1957), Fig. 3.)

early work was confused by the quenching effects of certain wavelength shifters. The most successful method for wavelength shifting has been to deposit thin layers of quaterphenyl or diphenylstilbene (30 to 50 pgm/ emz) on the walls of the container and adjoining photomultiplier face. Difficulties in the interpretation of the role of nitrogen as a wavelength shifter have been removed by the work of Bennett and WuZowhich indicates that the observed spectra, rise times, etc., when nitrogen is present are explained on the basis of collision phenomena rather than photoexcitation. Using a chamber of the sort shown in Fig. 8, Sayres and Wu have made systematic observations using helium, argon, krypton, and xenon as the scintillators count Po210a particles. In each case the pressure was adjusted t o be as high or higher than that needed to stop the a particles in the chamber volume. The results of these tests show that pulse heights reach

1.4.

SCINTILLATION

139

COUNTERS

a maximum when the particle is stopped in the chamber, and that the resolution also improves with increasing pressure. Resolutions (in per cent peak width) of 5 % to 10% were obtained. The relative performances as well as the effect of wavelength shifting is shown in Table IV. TABLE IV. Performance of Noble Element Scintillators Glass phototube (6292)

Gas (optimum pressure) Xenon (6 psi) Krypton (8 psi) Argon (10 psi) Helium (45 psi) [Noise/background]

Without quaterphenyl

With quaterphenyl

6 hcCompton mean free path is 17 cm implying absorption lengths a h i t 35 cm. The Comptmi recoil electrons are for these energies readily :~l~sorJwci by the scintillator with a n energy loss I

1

I

I

I

I

4

5

6

7

8

9

I

I

I

1

30

a

e II -

aQ

10

5

0

I

2

3

1 0 1 1

1 2 1 3 1 4

NEUTRON ENERGY ( M E W 1710.

11. Neutron, proton cwlhsion incan free path in toluene.

of about 1.6 Mev/cm so that the energy deposition of a gamma ray is given essentially by the Comptori process. Neutrons give up their energy to the scintillator largely by elastic c~ollisionswith protons. The process of nentroii slowing down arid diffusion has been extensively and it is known that the distance travelled by a fast neutron prior to its thermalieation is of the order of the mean free path for the first collision. Figure 11 gives the mean free path for n,p collisions in toluene. The slowing down process is quite rapid (-2 X

146

1.

PARTICLE DETECTION

sec) so that the sequence of proton recoils involved occurs within the resolving time, 2 X lo-’ sec, of more or less conventional electronics. Under these circumstances the neutron slowing down pulses “pile up” giving one pulse. Because of the nonlinearity of the scintillator response to protons and the various combinations of recoil energy loss possible for a neutron, the sum pulse varies. Consequently, there is no unique scintillation response for a given neutron energy.36The role of the other major scintillator constituent, carbon, in slowing down the neutron is small because of the relatively great mass of the carbon nucleus and also because the neutron collision cross section presented by the scintillator protons is somewhat larger than that of scintillator carbon. Nevertheless, the neutron loses an average of 14% of its energy in each collision with a carbon nucleus and since the carbon recoil nuclei are so inefficiently signaled by the scintillator, this fact introduces additional nonlinearity into the response to neutron energy. The light output of a liquid scintillator (5 gm/liter terphenyl in toluene) for electrons and protons as determined by Harrison87is shown in Fig. 12. As an example of the photon yields to be expected from liquid scintillators we quote the absolute photon yield determined by Post38for terphenyl (8 gm/liter) in toluene, 150 ev/photon. 1.4.1.10.3. DESIGN CONSIDERATIONS. As outlined in Section 1.4.1.10.1, the response of the detector to a primary event is a consequence of several factors : 1. the energy deposited in the scintillator and the fraction of this energy which appears as scintillation light, 2. the transparency of the liquid to its own scintillation; 3. the reflectivity of the container walls and the fraction of the wall area covered by photocathodes; 4. the photoelectric efficiency of the photocathodes and the electrical characteristics of the photomultipliers, photomultiplier ganging circuits, and amplifiers. This lack of uniqueness could be eliminated in principle by the use of a system fast enough to observe the individual proton recoils. Indeed, since one is here concerned with correcting for a nonlinearity, it need not be made with precision. Thus far such corrections have not been attempted although the multiplicity of recoiLs has been observed by D. W. Mueller’s group a t Los Alamos (private communication), and was actually used to discriminate between neutrons and gammas by F. D. Brooks, in “Liquid Scintillation Counting” (C. G. Bell and F. N. Hayes, eds.), pp. 268-269. Pergamon, New York, 1958. F. B. Harrison, Nucleonice 10(16), 40 (1952). 88 R. F. Post, Phys. Rev. 79, 735 (1950), as interpreted by S. C. Curran in hie book entitled “Luminescence and the Scintillation Counter.” Academic Press, New York, 1963.

1.4.

147

SCINTILLATION COUNTERS

The fraction F of scintillation light which eventually reaches a photocathode can be related to the mean transmission over the light path between reflections t of the scintillation light by the liquid, the reflectivity I2

I

I

I

I

I

I

I

I

I

10

_I W

a

0

‘I,s

ELECTRONS

* U a

a

t

m

R

a

-.

6

%

P W

I $

4

_I

2

a

0

1

2

3

4

5

6

7

8

’

9

1

0

PARTICLE ENERGY (MEV)

FIG.12. Scintillation light output for electrons and protons versus particle energy in a 5 gm/liter terphenyl toluene solution (Harrison37).

of the wall r, and the fraction f of the wall surface uniformly covered by photocathodes by the simple formula:* F

=

tf/[l - tr(1 - f)].

(1.4.10)

The mean transmission t is in general a complicated function of the mean free path for scattering A, and absorption A. of the light by the liquid and the detector size and shape. For t,he simple case of a spherical detector and a liquid which has a scattering mean free path much greater than the detector diameter, t is given by the relationship A,

t = - (1 d

- e-”Aa).

(1.4.11)

* A more precise formula would replace t in the numerator by to, where fa is the light transmission, averaged over points uniformly distributed throughout the scintillator, to the wall.

148

1.

PARTICLE DETECTION

As a n example, suppose d = 1 meter, A, = 2 meters, r = 0.9, f = 0.05. Then t = 0.787 and the fraction of scintillation light collected F = 0.121. The enhancement of light collection caused by the reflectivity of the container walls is seen in this example, to be a factor of 2.4. The reflectivity of a given wall coating, scintillator combination can be measured by using a light source centrally placed in a sphere which is so small that t = 1.0. A uniform distribution of holes around the sphere would enable the light to emerge and appropriate filters can be used around the light sensing element, to match scintillation light. Small optical mockups have been used to measure the uniformity of light collection in this limiting, t = 1, case.39 The total mean free path for scattering plus absorption can be measured for the scintillating liquid by means of a Beckman spectrophotometer, provided a standard is available for comparison. As a standard we have used reagent grade toluene measured in a six foot long box with a modulated light source at one end to provide an ac signal, a photoelectric detector a t a variable distance from the source, and light baffles between for collimation. The total mean free At path was measured as the departure from inverse quar re,^" i.e., I(r,At) = Ioe-r/X1/4~r2.

(1.4.12)

With this arrangement we have found reagent grade toluene to have a total mean free path of 5 meters. Thus far no measurements have been made which enable a direct and accurate determination of the absorption mean free path A, for a given scintillator. In principle, given the total mean free path ',A ( = A'; )';A and a light, source which mocks up the scintillation light, an experiment the result of which depends on the ratio X,/At can be done. For example, a centrally placed light source surrounded by a spherical volume of liquid with photoelectric detectors on a spherical black wall represents a calculable system. A drawback is the which is required in order to make a useful rather large radius (-A,) measurement. Because of the complicated dependence of the various light collection factors on the geometry, the eventual optimum design and resultant, characteristics for any given application is probably better determined from studies of complete detectors than from an a priori synthesis. 1.4.1.10.3.1. Scintillator Transparency and ReJlecting Paint. The problem of scintillator transparency4I has been met, by standard chemical

+

H. W. Kruse and F. Reines, unpublished. C. L. Cowan, Jr., F. B. Harrison, A. D. McGuire, and F. Reines (unpublished). 4 * For more details see A. R. Ronzio, C. L. Cowan, Jr., and F. Reines, Rev. Sci. Instr. 29, 146 (1958). 3a *O

1.4.


149

means in the case of triethylbenzene (TEB, C2Hy, p = 0.87 gm/cm3). Reagent grade toluene with a total mean free path for s5attering and absorption of about 5 meters at a wavelength of 4200 A is generally acceptable without further treatment. Purified triethylbenzene has a mean free path of from 5 to 10 meters. Purification of the crude T E B is accomplished by digestion wit,h sodium methylate (about one pound per hundred gallons of TEB) for four hours followed by fractional distillation in which a narrow cut is t,aken at. the constant boiling point range. The liquid is best st,ored in glass or metal cont,ainers painted wit8hEpon base enamel because of it,s tendency, especially in t,he case of toluene, to leach impurities from t,he container walls. Storage under an inert atmosphere such as argon is advisable in order to avoid the lowering of scintillation efficiency due to the act.ion of dissolved oxygen. An essential ingredient in any large scint’illator is the so called wavelength shifter which absorbs the primary scintillation light and re-emits it a t a lower frequency which is: (a) less readily absorbed by the scintillat,ing solution; (b) more easily reflected by available inert,, adhesive container coatings;42(c) more efficient,ly detected by the photomultiplier tubes. A popular shifter is 1’0P0P43 which moves the maximum in the emitted spectrum from the region of 3800 to 4200 A. The 5-meter mean free path quot,ed above is in purified solvent and does not apply to t,he case of an actual scintillator in which e.g. 3 gm/liter of terphenyl and 0.3 gm/liter of POPOP have been added t,o the solvent. For a scintillating solution the total mean free path drops to about one meter, presumably due to fluorescent or reradiative scattering. Judging from the actual performance of large detectors, itJ seems that only a small part, of the mean free path reduction is due to increased absorption by t,he terphenyl and POPOP. 1.4.1.10.3.2. Uniformity of Light Collection. If it is desired to employ the detector in any manner which requires energy resolution then it is clearly necessary t o be able to make the amount of light collected by the photocathodes dependent only on the number of photons emitted. In view of the fact that the optical t#rarismission of the liquids and the reflectivity of the container are imperfect, this aim is accomplished in an approximate sense by surrounding the liquid with a multitude of photomultiplier tubes. The fraction of the liquid surface area covered by photoin the largest detectors (-1.5 meterss) constructed at cathode is Los Alamos. Numbers from 45 2-in. h b e s to 110 5-in. tubes have been

+

4 2 Plasite paint with TiO:! pigment has been found to be suitable coating material. Ohtained from Wisconsin Protective Coat.ing Go., Green Bay, Wisconsin. F. N. Hayes, D. G. Ott, and V. N. Kerr, Nucleonics 14, 42 (1956). See also Table I, Section 1.4.1.3.1 of this volnmc.

1.

150

PARTICLE DETECTION

used to help obtain uniformity of light collection. Figure 13 shows three arrangements designed to this end. All the tubes are usually connected in parallel although it is on occasion useful to divide them into two interleaved banks which are connected in prompt coincidence to disSCINTILLATING

ISOLATION LlQUlO TO /OBTAIN MATCH OPTICAL

.TRANSPARENT WINDOW

x-

x--- - SCINTILLATING LlOUlO

PHOTOMULTIPLIER TUBES DISTRIBUTED AROUND CYLINDRICAL WALL

\

I

X X X

SCINTILLATING LlOUlD

X X

X

X X X

\ r c ) RECTANGULAR

1

X X X

PARALLELOPIPED

FIQ. 13. Photomultiplier tube arrangements designed for uniformity of light collection.

criminate against tube noise. In arrangement (a) light from events which occur over an appreciable distance compared with the photomultiplier tube dimensions is collected in a uniform manner. In (b) the photomultipliers are isolated from the scintillating liquid by a nonscintillating, optically matching so that short-range particles cannot deposit their energy in the near vicinity of a photomultiplier cathode. Arrange-

'* Cerenkov radiation will of course reRult in some light from energetic particles passing through this region.

1.4.


151

ment (c) is a variant of (b) which has been found useful in work where a large, unobstructed area was necessary. The light collection characteristic of a uniformly distributed source was measured in an optical mockupa9 of a rectangular detector 9 X 4+ X 2 ft to have a half-width a t halfmaximum of 7%. The over-all light collection was estimated as %. Given this figure, 1 Mev deposited in the scintillator, a conversion to photons of 1 photon/l50 ev deposited, and 1 photoelectron emitted by the photocathode per 10 incident photons, we conclude that 170 photoelectrons are produced per Mev absorbed by the liquid. This figure implies a statistical uncertainty of k 4 1 7 0 or ?8% due to fluctuations in the number of photoelectrons. The over-all uncertainty for 1 Mev deposited in this example is therefore conservatively taken as k 15%. 1.4.1.10.3.3. Photomultiplier Selection and Circuitry. * Very large detectors built thus far have employed 2-in. RCA and Dumont and 5-in. Dumont (K-l198), photomultiplier tubes. Sixteen-inch tubes which are just becoming available in quantity have a wide variation of response across the photocathode and hence, assuming only a few are used, introduce nonuniformity in the light collection of the system. As mentioned, photomul4iplier tubes have been used in gangs with as many as 110 5-in. tubes connected in parallel. Under these circumstances, uniformity of light collection requires that all the tubes have the same gain. Since tubes differ from each other in gain for a fixed voltage distribution along the dynodes, it is necessary to adjust the voltage in each case so as to equalize the gains of all tubes in the gang. This is done by observing the response to a source such as Csla7 placed in a reproducible fashion near a NaI crystal which is viewed by the photomultiplier tube under adjustment. The tubes are similar enough to require only the selection of an appropriate load resistor. Figure 14 shows a resistance network on one tube. Figures 15 indicate how a number of tubes can be connected in parallel. Also shown in Fig. 15(a) is a simple switch which enables a given tube to be thrown out of the gang. This switch is useful in checking the performance of individual tubes in situ, especially in the location of tubes which have become noisy, In addition to equalizing gains it is also of importance to select tubes of acceptable noise level because despite the large capacitance of the ganging network even one noisy tube can result in variable and unacceptable backgrounds. Occasionally problems arise in which even the best collection of tubes results in unacceptable levels due to tube noise. In such cases it has been found that tube noise can be greatly reduced by the expedient of dividing the tubes into two interleaved bunks axid requiring a coincidence between the two banks. The price one pays for this reduction in background is in a smaller light collec*See also Vol. 2, Section 11.1.3.

152

1. PAltTICLE DETECTION GAIN BALANCE RESISTOR

H.V.

-

2 MEG I

I

I

1

G-

SCREEN (13)

CATHODE (14) SWITCH

FIG. 14. Voltage divider network used on a 5-in. 1)umont K-1198 photoInultiplier tube.

t,iori and hence in decreased energy resolution per bank. I n c,ases where the energy resolution required in making a coiiicidence is not unduly restrictive it is possible t,o add the signal from the two banks through iso1ztt)ion networks (i.e., separate preamplifiers) and regain the resolution lost in t,he division. A problem occasionally eucountered is in the electrical oscillations which sometimes result., in the ganging “yoke.” Parasitic resistors c:m be added t,o suppress such ringing as show11 in Icig. 15(b). The variatiou of

1.4.

I53

SC'INTILLATION ('OUNTElIS

FIG, 15(:1).Ganging yokc for a largc number of 5-in. photomultiplier t rrhes. 93"COAXIAL SIGNAL LEAD ( T O PREAMPLIFIER) COAXIAL HIGH VOLTAGE LEAD

SIGNAL YOKE

FIG.15(b). Schematic drawing of ganging yokc for a large number of 5-in. photoinultiplier tubes.

154

1.

PARTICLE DETECTION

photomultiplier response due to ambient magnetic fields can also be a cause for concern. In general the use of mu metal shields is cumbersome, expensive, and in many cases impractical because of the restriction imposed on light collection. One solution to this problem is to build the 18

17 16

15

t

14

13

z

3 12

>

a a

11

g

10

E U

-

9

W

G

8

0

7

&

z F

2 3

6

8

5

0

20

30

40

50

60

70

80

PULSE HEIGHT (VOLTS)

FIG. 16. Peaked spectrum due t o cosmic rays which pass through the giant slab detector of Fig. 5. The liquid depth, 56 cm, corresponds to a peak energy of -90 MeV. These data were taken under about 200 f t of rock at an altitude of 7300 f t and correspond t o a muon rate through the detector of 13/sec. (Cowan and Reines, unpubished, 1957.)

detector of steel which will shield the tubes, another is to recalibrate the system a t each position of use simply foregoing the loss of resolution due to the magnetic field effects. 1.4.1.10.4. CALIBRATION AND USES. Two general calibration schemes

1.4.

155


are available for use with large detectors: t,he first makes use of cosmic radiation, the second of radioactive sources. Minimum ionizing cosmicray particles, for example, deposit an amount of energy which is proportional to the track length in the scintillator. In consequence a pulse-height spectrum due to cosmic rays penetrating the detector has a peak which may be associated with the mean energy loss of minimum ionizing cosmic rays. A typical “through peak” is shown in Fig. 16. Once the through peak is located, the detector can be replaced by a standardized pulser which is then adjusted to give an output voltage equal to the through B

I

I

I

1

I

I

I

PULSE HEIGHT ( M E W

Fro. 17(a). Mu-meson decay electron spectrum seen with 75-crn cylindrical detector. Data were taken at 7300 ft above sca level and 40,000 counts were recorded in 36 hours. (Reines et al.so)

peak value. Energy gates can then be set using the pulser. An independent energy calibration can also be obtained, along with a check of the system used to study events in delayed coincidence, by employing the phenomenon of muon decay, p*-+ 8’ Yv+. (1.4.13)

+ +

In this instance, a delayed coincidence is required electronically with a fist pulse of, say 20-40 Mev energy followed within 10 psec by a second pulse of energy > 15 MeV. The energy spectrum of these second pulses is due t o the decay electron. Fortunately, backgrounds are small, and despite considerable distortion of the decay electron spectrum due to bremmstrahlung losses and edge effects, the end point (53 MeV) is sharp. Figure 17(a) shows such a decay spectrum measured with the 75-cm cylin-

1

156

1.

I’AHTICLE

DETECTION

drical detector. Figure 17(b) shows the associated time interval or decay time spectrum. These calibration schemes give the energy response for a distributed source: the variation of response across the detector can be investigated by means of aperture detectors used in coincidence to gate the detector under study. The distributed or localized response can also be determined by means of radioactive sources such as the posit,ron I

I

I

I

I 8

I

10

v)

Iz

3

0 0

10

z

IC

I

I

I

2

4

6

I, \ 10

12

MICROSECONDS

FIG.17(b). Half-life measurement of the 8-decay of cosmic fi mesons stopping in the detector used as a check on the time calibration of the apparatus. The entry of the meson yielded a ‘ I first pulse ” and the decay electron the “second pulse.’’ This measurement was made in conjunction with the energy calibration using the decay electron spectrum end-point. (Reines et ~ 1 1 . 3 ~ )

emitter Cu‘j4which can be dissolved as a salt into the scintillator and the resultant spectrum observed. If the Cu64is encapsulated, only the annihilation radiation enters the detector, giving gamma rays of unique energy for measurement. These calibrat,ioii techniques incidentally indicat,e some of the uses which can be made of large volume liquid scintillation detectors. Another degree of freedom which can be incorporated into the detector is a sensitivity to neutrons which are fast or are produced in association with a

1.4.

SCINTILLATION COUNTEItS

157

charged particle. In this case the delayed coinviderice technique mentioned above in connection with muon decay can be employed, the first pulse being due to the associated rharged particle or a recoil proton, the second to the capture of the moderated neutron by the scintillator solution. A good neutron capturer is cadmium which has a large capture cross section (5300 barns at thermal energies for the natural isotopic mixture) and on the average, four capture gamma rays with energy totalling 9 MeV. The use of energetic gamma ray.; helps discriminate

K >

0 20

1.0 c W

a

0.16

zI-

0 12

0 t W

K

t a m 4

0 5 2

a a

0 08

w

a

u

0 04

r a a

0

0

0 CAPTURE TIME t ( p SEC)

Fro. 18. Keutron capture versus time spectrum seen with 75-cm cylindrical detector = 0.003%.These measurements mere made using cosmic-ray neutrons: 9 X lo4 neutrons were rounted in 20 hours. The solid lines represent theoretical values ohtained by means of a Monte Carlo calculation. (Iteines et ~ 1 . ~ ~ ) CY

against background. Cadmium-bearing compounds such as Cd propionate and more recently Cd octoate have been used with some success.41 Figure 18 shows a typicd neutron capture versus time curve for the 75-cm cylindrical detector with a Cd to H at,omic ratio, a = 0.0032. I n this case the neutrons were fast, arising from various cosmic-ray events such as p- cnpt'ure and stars in 90 kgm of P b placed on top of the detector. 4 5 The kinds of electronic circuitry employed with large detectors are indicated schematically in Fig. 19. Shown are the positive high-voltage supply (h.v.) (required t o maintain the photocathode a t ground potential so as to prevent degeneration due to high-voltage gradients at the photocathode), preamplifiers, amplifiers, coincidence circuits, scalers, and 45 In some experiments this cosmic-ray neutron background can be quite troublesome. A partial remedy is to construct the detector of light elements and avoid using heavy elements close to the detector as part of the shielding.

158

1.

.PARTICLE DETECTION

pulse-height analyzer. * Photographic records of oscilloscope traces triggered by appropriate pulses from coincidence circuits are sometimes employed to assist in the identification of the signals and the elimination of noise and background events. As an example of how this circuitry is used, consider the case of muon decay as measured with a large liquid scintillation detector. A pulse corresponding to an entering muon of 20-40 Mev passes through circuit I and registers as a pulse on the scalar. In addition a pulse is sent to coincidence unit I1 making it sensitive for, say, 10 psec. If a pulse in the energy range 15-60 Mev passes through circuit I1 during this 1 0 ~ s e c

‘ I l -

SWITCH

FIG.19. Schematic of electronics associated with a large liquid scintillation detector to measure delayed coincidences, e.g., muon decay.

it registers a delayed coincidence on the scaler and, a t the same time triggers the pulse-height analyzer gating circuit. The second pulse is then analyzed by the pulse-height analyzer. If it is desired to analyze the first pulse, the delay line can be used to store it pending the electronic decision that an appropriate delayed coincidence has occurred. In addition to this sequence, the time interval between the two pulses which comprise the delayed coincidence is measured by a time delay analyzer triggered by the two pulses. The second pulse scaler reads the rate,at which single pulses occur in the energy range 15-60 MeV. The example just given is only meant to be indicative of the kind of use to which such detectors can be put. The reader is referred to the literature for more details. a0--aa-46 I n general, however, it should be noted that the field of large scintillation counters is still a new one and in many *On these circuit elements consult also Vol. 2, Chapters 6.1, 6.2, Sections 7.2.2, 9.1.1, and Chapter 9.6. 4 4 F. Reines, in “Liquid Scintillation Counting” (C. G. Bell and F. N. Hayes, eds.), pp. 246-257. Pergamon, New York, 1958.

1.4.

SCINTILLATION COUNTEItS

159

cases the answers to questions associated with specific uses must be found by t,he experiment,alist.*

1.4.2. Solid Luminescent Chambers Almost as soon as the nature and usefulness of scintillation counters became apparent, the possibility of “seeing” the path of a charged par-

FIG. 20. (a) Sample scintillator filaments. (b) A filament scintillation counter. comparison solid scintillation counter.

(c) A

ticle in the scintillator was discussed. Early reports, later borne out by publication^,^^ of Russian work date back to 1954. This work involved the photography of tracks of protons a t two times minimum ionization in CsI crystals, with the aid of an image intensifier tube.48 I n this arrangement, involving as it does a solid CsI crystal, the depth of focus

* The author acknowledges collaboration with Dr. C. L. Cowan, Jr. on the problems associated with the development of large volume liquid scintillation detectors. 47E.K. Zavoiakii, M. M. Butslov, A. G . Plakhov, and G. E. Smolkin, J . Nuclear Energy 4,340 (1957). B. R. Linden and P. A. Snell, Proc. IRE (Insl. Radio Engrs.) 46, 513 (1957).

problem is severe in the optical link coupling the crystal with the image intensifier. This is particularly true since, as will he discussed later, the light output of a scintillator is very low compared with photographable intensities. Recently a technique has been d c t v e l ~ p e dfor ~ ~the ~ ~ preparation ~ of plastic scintillators in the form of long, thin filaments as shown in Fig. 20. Such filaments are now available commercially. 61 The filaments are

FIG. 21. Crossed filament array with simulated stereo views light piped from rear to front faces.

arranged in rows, stacked alteriiatively a t right angles as shown in Vig. 2 1, to furnish the stereoviewing necessary for three-dimensional reconstruction of the particles’ path. Each of the two orthogonal sets of filaments is viewed separately by image intensifiers. The major advantage of such a system is that since the filaments act as light pipes for the scintillation output, only those filaments traversed by the particle actually put out light, and this light is piped to the end of the respective filaments, so that the optics problems are restricted to a plane source. Thus the coupling to the image tubes can be made directly. Filament arrays with individual diameters from 0.5 to 1.0 mm have been prepared. G. T. Reynolds and P. E. Condon, Rev. Sci. Instr. 28, 1098 (1957). G. T. Reynolds, N?ccleonies 16 (6), 60 (1958). 51 Available froin Pilot Cherniral Corp., Watertown, Connrcticrit, and Xuclear Enterprises, Edinburgh, Srotland. 48

1.4.

SCINTILLATION C O U N T E R S

161

It has been s h o ~ 1 that 1 ~ ~for ~ a 1-mm diameter filament traversed by a minimum ionizing particle, approximately 16,000 photons/cm’ result. This shows that recording on fast film requires an image intensifier with a gain of lo5. Figure 22 shows a track of a minimum ionizing p meson

FIG.2 2 . A minimum ionizing p-meyon track 1 in. long obtained with a scintillation chamber niadc up of &5-1nm diameter fi1ament.s 1 i in. long.

1 in. long obtained in n chamber made up of ~.5-111mdiameter filaments. Six stages of image intensification prereded an image orthicon tube and the track was photographed on the face of 3, kinescope. The Russian reports imply n tube of that gain, and there is every evidence that similar tubes will he available commercially to Western scieiitists i l l t.he near future.

162

1.

PARTICLE DETECTION

A solid scintillator detector composed of plastic scintillator filaments has the advantage of simultaneous fast timing (approximately 3 X sec) and good space resolution, allowing the detection of relatively rare events in fluxes 1000 to 10,000times those possible with bubble chambers. With the image tube? gating techniques available, the scintillation chamber can be triggered after the event, similar to cloud-chamber operations. There are no moving parts; the nuclear composition is simple (carbon and hydrogen) and loading with selected 2 material can be easily accomplished by placing thin sheets between the rows of filaments. Following a suggestion of Kalibjian,62Jones and Per16a have applied the idea of regenerative feedback to the problem of viewing a CsI crystal. Although lack of precise registry of successive images in practice prevents simple application of the regenerative idea, forced registration, or alternatively, alternate cycling of two image tubes in the regenerative chain, offer promising approaches for this general idea. Several commercial laboratories are currently engaged in the development of channeled image intensifiers in which secondary electron cascade (and possible subsequent photon internal regeneration) paths are restricted to small cross-section channels.

1.5. cerenkov Counters* 1.5.1. Introduction

cerenkov counters have recently been playing an increasingly important role in the detection of high-energy particles, especially in experiments performed in particle accelerators in the multi-Bev range. Not only do these Cerenkov detectors prove to be extremely useful in many counter experiments but they can also be employed in conjunction with bubble and scintillation chambers to select and identify high-energy particles as desired. $

t See also Vol. 2, Section 11.2.3. R. Kalibjian, UCRL 4 i 3 2 (1956). L. W. Jones aud M. L. Perl, Rev. Sci. Instr. 29, 441 (1958). $ Regarding the Cerenkov effect see also Vol. 4, A, Section 1.5.3.

6*

65

-

*Chapter 1.5 is by S. J. Lindenbaum and Luke C. 1. Yuan.

1.5. EERENKOV

163

COUNTERS

A cerenkov counter can be constructed from any relatively transparent optical medium which possesses an index of refraction sufficiently greater than 1 in the region of the visible spectrum and its neighborhood. When a charged particle of velocity v(cm/sec) travels in a medium of index of refraction n such that v > (c/n)-i.e., when tJheparticle velocity exceeds the velocity of light in the medium Cerenkov radiation (first observed by Cerenkov) is 'The Cerenkov photons are radiated with uniform probability along the elements of conical surfaces of angle 6 relative to the direction of motion of the particle, where 0 is given by2nS ( I .5.1)

and p = the ratio of the particle velocity to the velocity of light in vacuum. n(v) = the optical index of refraction of the medium at the frequency v of the emitted photon. The instantaneous apex of the cone passes through the position (macroscopic) of the particle. The Cerenkov radiation is polarized such that the electric vector lies in the plane formed by the photon direction and the direction of motion of the particle. The intensity of cerenkov radiation per unit length per unit frequency interval is then given by -d2N =-

dx dv

sin2 e

= 2Tz2 -sin2 0

137c

(1.5.2)

where d 2 N / d x dv is the number of photons emitted per cm of path per unit frequency interval, v is the frequency of emitted photons, e is the electron charge, Z is the ratio of the magnitude of the charge of the moving particle to the electronic charge, c is the velocity of light in vacuum in cm/sec, and h is Planck's constant.

Figure 1 summarizes the relevant features of the Cerenkov radiation. 1P. A. cerenkov, Compt. rend. acad. sci. U.R.S.S. 2, 451 (1954);Phys. Rev. 62, 378 (1937). * I. Frank and I. Tamrn, Compt. rend. aead. sci. U.R.S.S. 14, 109 (1937). 3 G. B. Collins and V. G . Reiling, Phys. Rev. 64,499 (1938). 4 H.0.Wyckoff and J. E. Henderson, Phys. Rat. 64, 1 (1938). 6 J. Marshall, Ann. Rev. N d e a r Sci. 4 , 141 (1954);CERN Sumposium, ffenevo, Proc. 2, 62 (1956).

164

1.

PARTICLE DETECTION

Figure 2 depicts the relationship between index of refraction and velocity for a series of Cereiikov angles. Figure 3 depicts the variation of Cerenkov angle with p for various fixed indices of refraction corresponding to some of the commonly available values. For most practical cases the index of refraction is relatively constant over the visible spectrum which is contained in a frequency

CHARGED

CTnRY

FIG. 1 . Relevant features of Cerenkov radiation. (Instant

1.28 1.26 1.24 1.22 1.20 1.18 1.16 n 1.14 1.12 1.10 1.08 1.06 1.04 1.02 I.00.

I \

\,,

I

=

Instantaneous.)

I

I

,975

1.00

-

BOO

1

,825

,050 .075

,900

.925

,950

B

FIG.2. Index of refraction

a8

a function of velocity

for a series of Cerenkov angles.

interval -3.5 X 10'" cycles/sec. Furthermore, practical Cerenkov counters use photons only in the visible and near ultraviolet regions of the spectrum. For a particle with 2 = 1 traveling in a medium of coilstant, index of refraction, the number of photons in the visible spectrum generated per em of path length ( d N / d z )is found by evaluating (1.5.2) to be

I

x

500 sin2 8

(1.5.3)

1.5.

165

EERENKOV C O U N T E R S

where I = number of photons generated in the visible spectrum per cm of path. Commercial photomultipliers in general use a t present, have equiva lent photocathode efficiencies of -0.05 to 0.10 electron per photon over a frequency interval approximately equal to the visible frequency interval. Therefore if all the generated Cerenkov light is collected without absorption or other loss and a conservative average photocathode efficiency of -0.05 is assumed for the photomultipliers, one obtains as the

50”-

40’-

0 60

070

0 80

0 90

10

B FIG.3. Variation of Cerenkov angle with B for various fixed indices of refraction.

resultant electrical signal, S(photoelectrons/cm), generated at the photomultiplier cathode : (phototdtrons

= 25 sin2 8.

(1.5.4)

The maxipum Cerenkov signal will obviously be obtained for any medium when the particle velocity approaches c, i.e., when 0 3 1. This will be the case, for example, for a relativistic electron 2 10 MeV. If the particle traverses wat>er (n = 1.33) one can evaluate (1.5.4) and one finds S 10 photoelectrons/cm. For Lucite, another commonly 14 photoelect,rons/t:m. For Pb loaded used medium, ( r ~= 1.5) arid 8 16 photoelectrons/cm. glass TL = 1.7 and S On the other hand a minimum ionizing particle in plastic scintillator would lose about, 1.5 Mev/cm and generate -6300 photons/Mev. *

-

-

-

* Other organic phosphors such as anthracene, stilbene, and diphenyl acetylene are either generally equal or superior to plastic scintillator in photon yield. The numerical evaluation is based on data listed in “Handhuch der Physik-Encyclopedia of Physics” (S. Fliigge, ed.), Vol. XLV, p. 145 (Nuclear Instrumcntation 11). Springer, Berlin, 1959.

1.

166

PARTICLE DETECTION

This would yield -475 photoelectronsjcm if all the light were collected. Hence for a relativistic ionizing particle of @ -+ 1 the ratio of scintillator photoelectrons/cerenkov photoelectrons n. 33 for Lucite and = 44 for water. In many practical cases the ionization loss in plastic scintillator would be several times minimum with a corresponding proportional* increase in photoelectrons/cm generated in the scintillator, whereas also in many practical cases @ < 1 which reduces the Cerenkov signal. Hence one can say that generally speaking the ratio of photoelectrons generated by charged particles in plastic scintillator/cm to that generated by Cerenkov radiation in optical media/cm is greater than -30 to 50, i.e., S(scintillator)/S(Cerenkov) > 30 to 50. From the foregoing it is obvious that Cerenkov counters would only be employed when one wishes to make use of the special characteristics of this radiation. The main use of Cerenkov counters is to restrict the velocity range of the particles counted. This can be done in the following ways. 1. Detection of cerenkov light sets a threshold for /3, i.e., @ > l/n. 2. Measurement of the angular range of the Cerenkov cone determines the generating particle velocity to lie in a range < @ < @ 2 where 81 and @ z are determined by the index of refraction of the medium and the details of the measuring system.

One might remark at this point that with the best of modern techniques signals of -5 to 10 photoelectrons on the average can be utilized to count with an efficiency approaching 100%. Average signals of even 2-3 photoelectrons have been used to count with maderate efficiency.6sCHence although the Cerenkov signal levels are much lower than scintillation signal levels it has proven quite feasible to construct many types of useful Cerenkov counters. There are two general categories of Cerenkov counters: focusing or angular selection counters and nonfocusing count,ers. In the focusing type, a system for focusing the photons in the Cerenkov cone on the detector photomultiplier or photomultipliers is included. This type allows a selection of a range of angles O1 < 8 < e2 of the Cerenkov light, The nonfocusing type most commonly used merely att$empts to collect as

* Saturation effectsfor very high specific ionization cases are neglected in the above treatment. S. J. Lindenbaum and A. Pevsner, Rev. Sci. Znstr. 26, 285 (1954). 6a Using modern high gain photomultipliers such as the RCA-6810A, the authors have found it quite easy even without further amplificationto attain efficiencieswhich correspond to counting all cases where one or more photoelectrons are generated i.e. eff. = 1 e-< where 7t is the mean number of photoelectrons generated.

-

1.5.

167

~ E R E N K O V COUNTERS

many cerenkov photons as possible onto the photomultipliers more or less independent of their cone angle. However, there is a special class of nonfocusing counters which select an angular range of cerenkov light by making use of the properties of internal reflection and the appropriate use of absorbing coatings of black paint. Both types and combinations of them have been extensively employed. Probably one of the first working Cerenkov counters was built by D i ~ k e It . ~ mas the focusing counter schematically depicted in Fig. 4. A design proposal for the type of counter Dicke used was previously made by Getting.s Dicke employed the 20-Rlev electrons from a betatron to test his counters. A 20-Mev electron t,raveling parallel to the axis generates Cerenkov radiation (as shown) which is internally reflected

EERENKOV RAY INCIDENT PARTICLE- TRAJECTORY

PHOTO M ULT IPL IE R

FIG.4. Cerenkov counter designed by Dicke.

by t he rod and cone until it leaves the base of the cone and is focused by the lens a s shown on the 1P28 photocathode. A fast particle of different velocity such that 0 differed sufficiently would not be focused a t the photocathode and hence could be discriminated against. Although Dicke probably detected Cereiikov light he was not able at the time to rule out all other possibilities. Jelleyg later achieved success with a nonfocusing water cerenkov counter shown in Fig. 5. The cone of Cerenkov light generated in the water is relected by the silver-coated glass cylindrical container, and the light enters the photocathode of the photomultiplier at the bottom directly, and also from the back side of the photocathode by reflection from the MgO cone. The black paint on the outside of the tube was removed t o allow this. Jelley showed quite clearly that the cosmic-ray counts were due to cerenkov light since he painted black the end wall of the glass container opposite the photomultiplier (i.e., the top end in the diagram) and by a R. H. Dicke, Phys. Rev. 71, 737 (1947). I. A. Getting, Phys. Rev. 71, 123 (1947). J. V. Jelley, PTOC. Phys. SOC.(London) A M , 82 (1951).

1.

I G8

PARTICLE DETE("PI0N

roincidencc method selected part,icles moving downward. He then showetl tha t rotating the counter by 180" so that the photomultiplier was on top, caused the counts in the photomultiplier to be reduced to a small fraction of their former value. This demonstrated conclusively that most of thc counts were due to light which was dirrcted forward, and cerenkov radiation is the only known possibility. GLASS END PLATE WITH BLACK PAPER

DISTILLED WATER

CONTAINER SILVERE D ON THE OUTSIDE

LIGHT TIGHT ENVELOPE

'

LIGHT GATHERING CONE COATED WITH M g 0

PHOTOMULTIPLIER

AMPLIFIER

FIG.5. Nonfocusing water Cerenkov coiinter designed hy Jelley.

Several general reviews6,l0--lla have been written on Cerenkov counters. I n the present article representative types of the most generally useful types of counters will he discussed without necessarily including all reported counters. "

1.5.2. Focusing Cerenkov Counters

Basically a focusing Cerenkov counter consists of three elements: a radiator, a focusing system including in some cases an output coupler, and a photomultiplier or series of photomultipliers. 1.5.2.1. Radiator. This is the opticd medium in xvhich the Cerenkov radiation is generated. The radiator is generally designed so as to main10 Cerenkov and other fast countcr terhniqurs. P E R N Symposium, Geneva, Proc 2, 61-103 (1956). 11 J . V. Jelley, "Cerenkov Radiation." Pergnmon, Kew York, 1958 1 1 D. ~ Blanc, " DPtecteurs de particulrs," pp. 170-187. Masson, Paris, 1950

1.5.

6ERENKOV COUNTERS

169

tail1 a defiiiite relatioil between the cone angle of Cerenkov light a i d the direction of the particle in order to allow a measurement of this angle. A common type of radiat,or consists basically of a cylinder of solid optical medium such as Lucite or glass. The bases and cylindrical surface are optically polished. The charged particles are incident on one base of the cylinder in a direction parallel t o the axis as shown in Fig. 6. If a fast charged particle does not scatter or interact, or appreciably slow down in such a radiator, the unique angle 0 of the cerenkov photons relative to the cone axis is maintained regardless of the number of reflections from the cylindrical surfare.

FIG.6 . Cercmkov rays in a radiator.

This can easily be seen since the photon momentum p originally has a component pll = p cos 8 along the cone axis and a (*omponelitp , = p sin 8 perpendicular to the cone axis. The perpendicular component can be further broken down into pI = ps p. where p+ is tangent to the cylindrical surface a t the point of contact hut perpendicular to the axis, and p. is perpendicular to the surface and the axis. lcor a specular reflection a t the cylindrical surfwe pi1 and ps are obviously unaffected, p, is reversed iii direction from an outward going normal to the surface to an inward going normal to the surfave without change of magnitude. Hence i t is obvious that the angle 8 relative to the axis is maintained and also that the minimum distance of appiwch to the axis for a skew ray is maintained. If @ < I, the critical angle i h exceeded a t the cylindrical surface and all light is internally reflected. When the photon reaches the exit base surface of the cylinder and enters the air, its angle to the axih is changed due to refraction from 0 tn 8 , where

+

( I .5.5)

If sin 8 < 1 the ray is transmitted (at least partially) to the air. However, if sin F) I the ray is interirally reflecte,! and is trapped.

>

170

1.

PARTICLE DETECTION

Since P 5 1 ; it follows that: sin 0 5 -\/nz - 1. Hence, if n 5 -\/2 there is always some transmission out of the cylinder. However, for even common optical media such as glass or Lucite, if ---t 1 entrapmeiit results. Hence in these cases an output coupling section is necessary to allow escape of the light from the radiator. In the Getting-Dicke counter (Fig. 4) the flared cone of the radiator cylinder serves this purpose. Another type of radiator for use with liquid optical media is a cylindrical container with specularly reflecting walls which is filled with a transparent liquid and is fitted with a thin glass window at the exit end to allow escape of the Cerenkov radiation. The principle of operation is similar to the Lucite or glass rod described above, except for the fact that specular reflection a t the boundary replaces the internal reflection. If a thin polished glass cylinder is used as the cylindrical container internal reflection can still be employed. One should remark that for p = 1 the internal reflection will even, in the case of the solid cylindrical radiator (Lucite), be complete only for particles exactly parallel to the axis, and hence a reflecting coating or a slight outward taper toward the front face of the cylinder to insure internal reflection may be in order if the beam divergence is appreciable. In many cases the base of the radiator where the charged particles enter is coated with black paint to absorb Cerenkov radiation of particles proceeding in the wrong direction. Gas radiators have also been employed although not to the extent of liquid or solid radiators. Gas is used for very fast particles where /3 -+ 1 and it is desired to employ a low index of refraction medium to set a high p threshold or to improve the velocity revolution de/d,B. A major advantage in a gas Cerenkov counter is the feasibility of varying the index of refraction n by simply varying the pressure of the gas and, to a lesser extent, by varying the temperature. Thus charged particles of a desired velocity or momentum can be easily selected during the course of an experiment by providing the appropriate pressure of the gas in the counter. 1.5.2.2. Focusing System. The focusing systems generally make use of n series of cylindrically or spherically symmetrical surfaces around the axis parallel to the designed-for direction of motion of the incident particle. Following the work of Jelley, Marshall6employed the focusing counter system shown in Fig. 7. The Lucite radiator is joined to a Lucite heniispherical lens which has a focal length equal to twice its radius of curvature, Therefore there is a sharp focus a t 3 radii for rays froin the radiator coplanar with the axis of the system. A cylindrical mirror constructed of glass tubing which is aluminized on the inside is inserted such that the rays strike its surface at a distance -1.5 radii from t,he radiator end so

171

h

D

172

1.

PARTICLE: D E T E C T I O N

that a sharp focus is made a t the axis of the lens. The hemispherical lens serves as a coupler to efficiently remove the photons from the radiator and avoid the total internal reflection for very fast particles ( p + I ) . For rays skew to the axis the focusing does not work due to conservation of angular momentum, and has been shown5 t o lead to an image diameter; sin e D>_nd--(1.5.6) sin 8’ where D is the diameter of the image and the equality holds under ideal conditions; n is the index of refraction; d is the diameter of the radiator; fl is the C‘erenkov cone angle and 8’ is the half cone angle of the rays which form the image. CYLINDRICAL MIRROR

>

BAFFLE

PHOTOMULTIPLIER

Lll!LL-

FIG.8. Schematic of the most commonly applied t,ype of focusing counter.

At the position of the image a photomultiplier is used a s a detector which converts the incident photon into an electronic pulse. A diaphragm can be used a t the image to limit the acceptance circle. Diaphragms can also be used in other parts of the system. Marshall has described5 variations of his counter in one of which the hemispherical lens is split by a light shield, and two plane mirrors are placed colinear with the axis to form two images on two photomultipliers such that stray eerenkov light? produced in the lens cannot lead to a roincideiice but the desired light from tjhe radiator does. Also a two photomultiplier coincidence eliminates the counts due t o a particle directly striking the photomultiplier and greatly reduces phototube noise counts. A schematic of the most commonly applied type of focusing COUIIter6~12J3 is shown in Fig. 8. It uses the cylindrical radiator and cylindrical mirror but not the hemispherical coupler. The radiator can be a solid polished cylinder of an optical medium such as Lucite, glass or quartz, or a polished glass

’*

S. J. Liudenliaiim and I,. C. I,. Yuan, (’ERN Symposic~m,OPrwiw, Z’io(. 2, (23 (1956). l 3 0 Chamhrrlain and C Wicgand, (‘ERN Syi:iposzum, Geneva, Proc 2, 6 3 (IR5ti)

1.5.


173

cylindrical shell container filled with a liquid which acts as the radiator. One can also use as a radiator a metal cylindrical shell which is filled with liquid or gas and contains a polished aluminum cylindrical inner wall, or a separate polished aluminum cylindrical mirror and an exit window a t one end with plane surfaces perpendicular to the axis to allow the photons to escape. The cerenkov light will escape from the end of the radiator only if eo is small enough. Let us denote the index of refraction of the generating medium by no, that of the glass by n,, and that of the air by n,. Also denote the angle of the Cerenkov light to the axis by 80, OQ, and 8,. Then due to the relations no sin eo = n, sin $# = n, sin 8, and the geometries used, the angle to the axis ea of the cerenkov light escaping into the air is given by sin 8, = 2 sin na

eo c-. no sin eo

and, provided 0, is real, is independent of the index of refraction of the glass exit window. 0, is real provided

Since n, = 1 this reduces to (no2 - l/p2) 5 1 < pno. Thus if Cerenkov light is generated in the radiator these inequalities are always satisfied for no I z/% For no > 4 3 , P must not be too large to satisfy these inequalities. The cerenkov photons of cone angle 8, are then reflected from the cylindrical mirror as shown in Fig. 8. The position of the image along the axis is determined by the Cerenkov cone angle 8. Provided that the diameter of the cylindrical mirror D, is 2 3 times the diameter of the radiator d, the image is not much affected by the optical aberrations of the focusing system. The magnification of the system is approximately one and the image is a circle of diameter equal to the effective radiator diameter. However, it can be shown that the effective angular uncertainty of acceptance of such a system is proportional to d/D,. Hence high-resolution counters require a large D,/d. In many practical cases, two or more photomultipliers, generally off the axis, are employed in coincidence to effectively eliminate counts due to tube noise coincidences, direct excitations of the photomultipliers by a charged particle, and stray light coming through a part of the baffle system due to a particle of wrong velocity proceeding through the radiator in the wrong direction. Such a particle may be due to a scattering or inelastic interaction of the incident particle, or due to background

174

1.

PARTICLE DETECTION

particles. Possible ways of splitting the light are indicated in Figs. 9 and 10. One should remark here that in the case of liquid or gas radiators, cerenkov light will be generated in the solid transparent exit window. u

0 2 4 6 8

INCH

HALF TRANSPARENT MIR

LINED WITH ALUMINIZED POLYSTYRENE

FIG.9. High-velocity resolution gas counter designed by Lindenbaum and Yuan.

/ / /

/

/ /

-

\

\

PHOTOMULTIPLIER

\

/

\ \

'

/

/

PLANE MIRROR

\ \

\

I--.---\'

I

'

I--/

t

----

ZERENKOV RADIATOR

ED - BLACKEN BAFFLE

FIG.10. Velocity selecting counter designed by Chamberlain, SegrB, Wiegand, and Ypsilantes.

For p + 1, if the exit window, as is generally the case, is made of glass, quartz, or other transparent optical media of n > 4 2 , the light generated by a particle parallel to the axis will be trapped in the window and can be absorbed by blackening the outer boundary of the window.

1.5.

EERENKOV

COUNTERS

175

However, a sufficiently slow particle will produce light in the window of the same cone angle as the desired signal and hence will get through the optical system. But the number of photons involved will be smaller than those of the desired signal by approximately the ratio of the window thickness to the radiator length. Hence by designing a large ratio of radiator length to window thickness, they can be discriminated against. Particles proceeding along some directions in the window may also generate an appreciable signal some of which gets through the baffle system. However, in general this can also be discriminated against by pulse height and light splitting with a double coincidence requirement. Furthermore, a directional requirement can be made by requiring one more coincidence after the particle passes through the focusing counter. 1.5.2.3. Resolution. The velocity resolving power of a focusing cerenkov counter can be expressed in terms of the partial derivative ae/ap which from Eq. (1.5.1) is

ae -ap

1 p2n

sin 0'

(1.5.7)

For counters designed for high energy resolution of relativistic particles

p -+1 and n = 1. Hence, since the variation of p with energy is very slow, large values of ae/ap are required to obtain good energy resolution. But a6 ap

N

-. 1

sin 6

(1.5.8)

-

Hence small values of 6 are required for high resolution. However, intensity const sin2 6. Therefore : ae 1 const - W v N W s1n dintensity

(1.5.9)

Hence an index of refraction must be used such that 6 - 0 in order to obtain a good resolution. However, it is also obvious from Eq. (1.5.9) that 6 must be sufficiently larger than zero to allow an adequate number of photons to be generated. Gases under variable pressure and temperature are the practical sources of these low index optical media. Figure 11 shows the index of refraction of one of the commonly used gases as a function of pressure a t several temperatures. If it is desired to cover a wide range of index of refraction from near unity to -1.2 to 1.3, then a gas with a critical temperature near or above room temperature is desirable. This allows liquefaction and the resultant high values of index of refractions to be obtained with moderate pressures, at temperatures which are not excessively high or low.

1.

176 1.20 1.18

PARTICLE DETECTION

-

co2

25.05%

321)80 40°C

1.16

1.14 c

49.71'

4

x

1.1 2

2

1.10

5

1.08

n

1.06 1.04

1.02 1.00 PRESSURE -(ATOM.)

FIG.11. Index of refraction of COZ gas as a function of pressure at several temperatures.

1.5.2.4. Practical limitations. Let us now consider the various practical limits to the resolution of a focusing cerenkov counter. These fall into three categories : (a) the finite width (AO) of the cerenkov angular cone radiated relative to the instantaneous position of the particle; (b) the deviations of the particle trajectory tangent from the optical system axis direction, and various geometrical and optical factors contained in the resolution; (c) characteristics of the incident beam of particles and the effects resulting from their interaction with the cerenkov counter itself. I n category (a) we have the following effects. 1. Difraction-The Cerenkov cone angle has a width A6 due to diffraction which essentially depends on the length of path in the radiator over which the coherence conditions are unchanged. Although it has been ~ h o w n ' ~that - ~ ~ the emission of individual photons do not affect this coherence, large enough Coulomb scattering and nuclear shadow scattering do. Lil43lSand Dedrickle have shown th at the characteristic distance which determines the diffraction width of a Cerenkov cone is much greater than l4 16

l6

Yin Yuan Li, Phys. Rev. 80, 104 (1950). Yin Yuan Li, Phys. Rev. 82, 891 (1952). K. G . Dedrick, Phys. Rev. 87, 891 (1952).

1.5.


177

the mean distance between emission of successive photons but much smaller than the total path length in the radiator material. In practical cases the diffraction width is generally negligible. 2. Dispersion-Since the index of refraction n is a function of frequency Y , it follows from Eq. (1.5.1) that there will be a dispersion width A0, introduced. The dispersion width Ad, can be estimated as follows:

ae

AB, = - A n an

(visible) =

An pn2 sin 0'

(1.5.10)

For a relativistic particle in Lucite A0, = 0.8", in fused quartz A0, = 0.6" in water AO, = 0.5'. It is obvious from Eq. (1.5.10) that At?, can become large a t small angles. In this connection it is interesting to compare the behavior of the ratio of the dispersion width AO, to the velocity resolving power a0/ap as a function of angle 8. Using the foregoing we find

that the above ratio is independent of angle. Hence, the increasing dispersion width a t small angles is accompanied by a proportional increase in velocity resolving power a t small angles. Therefore, it is generally desirable to go to small angles for increased resolution, since beam angular divergence, and Coulomb and shadow scattering widths are more or less independent of angle and generally larger than dispersion widths. In category (b) we have the following effects. 1. Scattering-Even for a particle originally parallel to the optic axis of the system, both Coulomb and nuclear shadow scattering change the direction and position of the trajectory in the counter. Since the Cerenkov photons are radiated at a polar angle 0 to the trajectory direction] the scattering of the trajectory leads to a distribution in 0 when all the photons radiated from the various parts of the trajectory are considered together. Light media of low atomic number such as water or Lucite or especially gas minimize both optical and Coulomb scattering. 2. Optical resolution-Any practical optical system has a characteristic angular resolution due to the finite size of the object, the inherent resolution limits of the optical system and optical aberrations. For the optical system shown in Fig. 8, the angular resolution can be defined as l/A0, where A 0 is the width of the range of cone angles leaving the exit window of the radiator which are transmitted with greater than

178

1.

PARTICLE DETECTION

half the peak intensity by the optical system to the photomultipliers comprising the detectors. D,/d. Hence it is For a fixed angle 8, the optical resolution l / A e obvious that a large enough mirror to radiator diameter ratio is required for high optical resolution. Practical counters used by the authors and others use a D,/d ratio -3 to 10. For these values of D,/d spherical and other optical aberrations of the system have little additional effect on the resolution. In category (c) (beam characteristics and effects of interactions) we have the following effects. 1. Ionization loss in the radiator-The ionization loss leads to a systematic decrease in p as the particle progresses through the radiator. This of course leads to a decreasing C'erenkov cone angle which gives an energy loss width term A8dr,dz to the cone angular spread. This effect is only important for thick radiators and lower energy particles. Marshall6 has shown how the use of tapered outward toward the front radiators can correct for this efiect. Another indirect effect of the ionization loss on the Cerenkov radiation is the production of 6 rays of sufficient velocity to themselves produce Cerenkov radiation. This latter effect, of course, leads to a type of general background light. However, this effect is generally not a serious background limitation in practical Cerenkov counter applications. 2. Nuclear interactionsThe special cases of Coulomb and nuclear shadow scattering which lead to small changes of particle direction have been previously considered. In addition, one can also have inelastic nuclear interactions which change the direction, energy, and type of particle as well as adding new particles. These interactions generally terminate the Cerenkov radiation pattern of the original particle a t the point of interaction, but also in many cases supply new Cerenkov light emitted by the products of the interaction which acts as a background. Actually background-producing particles can enter the counter from any point of its surface and both directly emit C'erenkov light and also indirectly via products of nuclear interactions which they induce. It is again desirable to use light media as radiators to reduce the number of nuclear interactions. 3. Beam characteristicsA practical beam of particles even if momentum analyzed so as to be nearly monochromatic in energy has both an energy spread, and an angular spread. These two effects obviously lead to a spread AObeamin the cerenkov cone radiated which limits the practical velocity resolution of the counter. 4. Magnetic J i e l d s T h e presence of strong stray magnetic fields can cause a curvature of the radiating charged particles path and hence impart a distribution to the Cerenkov cone angle relative to the axis of

-

1.5. EERENKOV

COUNTERS

179

the system. This effect is not important in most practical cases. Stray electric fields can also in principle modify the cerenkov cone angle but the fields usually encountered are too weak. 1.5.2.5. Photomultipliers.* The photomultiplier characteristics most useful for application to Cerenkov counters are the following. 1. End window semitransparent photocathode type of large enough cathode area to efficiently cover the image in a focusing type counter and collect as many photons as possible. For nonfocusing counters the large area end window type are also desirable for highest detection efficiency. 2. A high efficiency for converting photons to photoelectrons over as wide as possible a section of the visible and ultraviolet spectrum as is transmitted by the radiator. A peak of conversion efficiency in the blue or ultraviolet is in general desirable. 3. As high a gain as feasible to reduce the need of electronic amplification of the small Cerenkov pulses. 4. A good signal-to-noise ratio. 5. Preferably a small spread of transit time from photocathode to first dynode structure) and a small time spread in the photomultiplier structure, in order to take advantage of the very short time spread in the cerenkov pulses. The authors) experience has been that the fourteen-stage 56AVP (Philips) and the 6810A and 7264(RCA) represent a reasonable compromise with the above requirements for general purpose use. There are many other phototubes manufactured by Dumont, EMI, RCA and others which are more suitable for particular cases. Some of these are described in the various references given for individual counters. I n particular the 5-in. and 16%. diameter RCA and Dumont phototubes are useful for large counters. 1.5.2.6. Some Practical Focusing Counters. Figures 9 and 10 show various practical focusing counters of the type depicted schematically in Fig. 8. Figure 9 shows a type of high-velocity resolution counter with both a liquid and a gaseous radiator which was first constructed in 1952 by the authors and tested at the Brookhaven Cosmotron.lEs The gas generally employed is C O z which can be varied continuously in index of refraction over the range 1.004 to 1.21 by varying the pressure over the range 0 to 200 atmospheres and the temperature over the range 25" to 50°C (see Fig. 11). *See also Vol. 2, Section 11.1.3. Recent modifications include an anti-coincidence channel to improve background rejection when K mesons are detected in the presence of a large r-meson background and these changes are not shown. S. Ozaki a.nd J. Russel have collaborated with the authors in these modifications. 16*

180

1.

PARTICLE DETECTION

At 25"C, a pressure of approximately 75 atmospheres liquefies the CO, which then has an index of refraction -1.2. Hence when the counter is set for a particular angle of detection, the index of refraction and hence the velocity interval accepted can be changed a t will. For the counter shown, the mean value of P accepted can be varied from = 0.83 to /3 = 1.00 with a velocity resolution AD = ,0.005. Since the cerenkov cone angular range is not changed, the photon intensity and geometrical resolution properties are constant as the index of refraction is changed. Hence also the efficiency of detection is approximately constant. When used a t detection angles of -10" the efficiency is 290%. The liquid radiator can be used with Minnesota Mining & Mfg. Co. fluorochemical 0-75 with a variable temperature to cover the range n = 1.26 to 1.31. Water, sugar water, and then various standard liquids can be used to cover the range n = 1.33 to 1.7. A list of the index of refraction of some solid and liquid substances is shown in Table I. TABLE I. Index of Refraction of Solid and Liquid ~

~~~

Index of refraction Material

nd

Solid (at 18°C) Fused quartz Polymethyl methacrylate (Lucite) Quarts Polystyrene Glass (ordinary Crown) (light fiint) (dense flint)

1.458 1.489 1.493 1.5443 1.592 1.517 1.580 1.655

Liquid (at -20°C) Fluorochemical FC-75 Water Paraldehyde Carbon tetrachloride Toluene Benzene Chlorobenzene Carbon-di-sulfide

1.276 1.333 1.405 1.46 1.497 1.501 1.525 1.630

-

~~~

Reciprocal dispersion V = (na - l ) / ( w - nJ 65 49 70 30 60 42 34

56 49 30 31

The small gaps that exist are easily filled by using the movement of the photographic bellows to change the angle of detection. With a particular liquid the mean value of fi accepted can be varied over a considerable range by changing the distance between the light splitter and the radiator via the bellows. Although the efficiency can in principle change

1.5.

EERENKOV COUNTERS

181

as the angle of detection is changed this effect is small since theefficiency can be made close to 100%. The velocity resolution however does change somewhat in a calculable way with angle. The major use of such a counter is as a mass spectrometer wherein small changes of resolution are not too important. A velocity selecting counter of this general type was employed by Chamberlain et ~ 1 . 'in~ their discovery of the antiproton at Berkeley. Figure 10 shows a diagram of their counter. STAINLESS CYLINDER 4 " 1.D x 8" LONG I" QUARTZ

WINDOW

FRONT SURFACE

TWO ELEMENT LUCITE LENS

RING APERTURE PHOTOMULTIPLIER C7170 (RCA)

FIG.12. Focusing counter using gaseous or liquid fluorochemical 0-75 designed by Baldwin et al.

Using a fused quartz solid radiator and a cylindrical mirror arrangement as shown in Fig. 10 they were able to attain a velocity resolution such that when AD 0.03 the counting rate dropped to -3% of the peak value. This counter was used as an element in a counter telescope to select antiprotons from negative pions in a momentum analyzed beam. A gas focusing counter using gaseous or liquid fluorochemical 0-75 (normally liquid) at elevated temperatures and high pressures has been designed and used by an M.I.T. group, Baldwin et ul.18 A diagram of this counter is shown in Fig. 12. By varying the temperature up to -255°C

-

l7 0. Chamberlain, E. SegrB, C. Wiegand, and T. Ypsilantis, Phys. Rev. 100, 947 (1955). . E. Baldwin, D. Caldwell, S. Hamilton, L. Osborne, and D. Ritson, Scintillation Counter Conference, Washington, D.C., January, 1958; D. Hill, private communication.

182

1.

PARTICLE DETECTION

and the pressure over the range 1-20 atmospheres the index of refraction can be varied from near unity to -1.28. The optical system employed essentially consists of a lens system with Schmitt correction which focuses light of a particular Cerenkov polar cone angle relative to the lens axis into a narrow ring located approximately one focal length behind the lens. For small angles the ring diameter is proportional to the Cerenkov cone angle and hence a small annular disc opening before the photomultiplier restricts the photons accepted to a small Cerenkov cone angular interval.

FIG.13. Focusing liquid counter employing a spherical mirror, designed by Huq and Hutchinson.

One advantage of this system is that for obtaining high-velocity resolution with large diameter radiators the necessity for a large cylindrical mirror can be avoided. However the alignment of the lens system must be rather carefully performed. This counter has been used in the detection of K+ mesons in positive analyzed momentum beams a t the Bevatron. The velocity resolution is such that the counting rate dropped by a factor of 20 when Afl 0.006. Another variation of the ring type of focusing counter employing a spherical mirrorIg instead of a series of spherical lenses is shown in Fig. 13. This counter is particularly suited for liquid radiators. The liquid radiator fills the space between a spherical mirror through which the beam enters and a plane mirror face through which the beam leaves. Both mirrors are front silvered. After reflection from the plane

-

1s

M. Huq and G. W. Hutchinson, Nuclear Instr. and Methods 4, 30 (1959).

1.5. EERENKOV

COUNTERS

183

mirror the Cerenkov cone is focused into a ring defined by an annular stop of inner radius equal to 4.25 cm and external radius of 4.5 cm. The focused ring diameter is an increasing function of Cerenkov angle. The counter described was used in a 900-Mev proton beam and exhibited an over-all angular resolution of for a cone angle -35'. This corresponded to an energy resolution -54%. A counter similar in principle using ethylene gas as the radiator has been designed by von Dardel and his co-workers*Oat CERN t o be used for experiments with the 25-Bev alternate gradient synchrotron beam. Operating a t a gas pressure of up to 70 atmospheres (at room temperature) they were able to separate antiprotons from k- and T- up to 16 Bevlc momentum. A drawing of this counter is shown in Fig. 14a. A counter designed by the authors and their co-workers for particle separation at the Brookhaven 32 Bev Alternate Gradient Synchrotron is shown in Fig. 14b. The major differences from the CERN counter is the use of COS gas and the addition of an anti-coincidence channel which collects the n-meson light when K-meson or anti-proton light is tuned into the signal channel. This technique greatly reduces the background level. One should remark a t this point that cerenkov radiators and optical systems of the focusing type with film or other integrating detectors2I have been employed to measure the velocity distribution of particles in a beam. Since we are concerned here with cerenkov counters we shall not describe these. There is a special class of velocity interval selecting counters which do not use focusing. The lower velocity limit is set by the threshold and the upper velocity limit by the internal reflection and subsequent absorption of light with a cone angle greater than ec where ec is that angle for which total internal reflection occurs. A counter of this type designed by Fitch and Motleyzz is shown in Fig. 15. The velocity range selected is 0.65 < p < 0.78. From the discussion in Section 1.5.2.2 it is clear that n L .\/z is required in order for total internal reflection to occur at the exit face. This is the major limitation of this kind of velocity interval selector. One is restricted to a low threshold velocity and also one has a much poorer resolution than obtainable in the focusing type of counter. However the simplicity of the device is of course an advantage for those cases where it can be used. Another variation of velocity interval selection can be obtained by

*+'

G. von Dardel, private communication (1960). R. L. Mather, Phys. Rev. 84, 181 (1951); J. V. Jelley, AERE NP/R 1770 Atomic Energy Research Establishment, Harwell (1955), unpublished. 22 V. Fitch and R. Motley, Phys. Rev. 101, 496 (1956). 20

21

FIG.14a. Focusing gas counter designed by von Dardel et al.

signal light; (F), special collecting mirror for Cerenkov anti-coincidence light; ( G ) , reflecting mirror for directing light into photomultipliers; (a), RCA type 6810A or Amperex 56UVP photomultiplier; (I), 4 RCA type 6810A or Amperex 56UVP photomultiplier connected in parallel.

186

1.

PARTICLE DETECTION

using black paint or other arrangements to eliminate the largest cerenkov cone angles corresponding to the highest velocities and of course a threshold velocity is set by the index of refraction. A counter of this type has been designed by Hughes, Palevsky, and co-workers23 for use in detecting high-energy neutrons. The counter consists of a high-pressure COZ cylinder which allows a variable index of refraction to set a variable threshold velocity; the cylinder is painted with black paint so that only small angle light will escape. BLACK PAINT

FIQ.15. Velocity interval selecting counter designed by Fitch.

1.5.3. Nonfocusing Counters The counter described in Fig. 5 constructed by Jelley represents one general type of nonfocusing counter, namely what can be. called the “end on type.” This ‘counter is mainly suitable for use as a last element in a telescope, as the thickness of material in the counter and the presence of the phototube in the beam do not make it convenient as in one of several counter elements in a telescope. A relatively thin transmission type nonfocusing cerenkov counter has been designed by Lindenbaum and Pevsner.s In this counter two 23

D. Hughes, H. Palevsky, and co-workers, private communication.

1.5. EERENKOV

COUNTERS

187

5819 RCA photomultiplier tubes faced the sides of an aluminum liquid container of 3 in. X 3 in. cross section and 14 in. thick with 31-mil walls. Liquids of various indices of refraction were used as cerenkov radiators. The ends of the phototubes were immersed directly in the liquids so that the semitransparent photocathode was covered by liquid. O-ring seals around the tubes were used to make a tight seal. Aluminum foil lined the inside of the container except for the ends of the phototubes. This counter was used as an element in a counter telescope in an 87-Mev negative pion beam which had been momentum analyzed. Differential range curves taken in this beam with turpentine (n = 1.475), ethylene glycol (n = 1.427), and water (n = 1.33) all exhibited the usual T - and pmeson peaks appropriately shifted to correspond to a velocity threshold a t that cerenkov angle which provided -2 photoelectrons at the photomultipliers. The only observable background was small and due to accidental coincidences. The absolute counting efficiency obtained was greater than 90% relative to filling the counter with a scintillator. There was no evidence for any background counts due to scintillation or other noncerenkov counts. It has been the general experience with Cerenkov counters that, except for substances that tend to scintillate, the cerenkov effect is large compared to general light background due to other sources even for nonfocusing counters. A number of improved versions of this type of counter using 6810A 14-stage RCA photomultipliers have been designed and employed by the authors at the Brookhaven Cosmotron for the past few years. A typical example is shown in Fig. 16. Counting efficiencies of -95-98 % have been attained for relativistic particles. The counter output was amplified and limited with one distributed 140 mc amplifier with 18 db gain. The output of the amplifier was fed to one grid of a 6BN6 dual grid tube coincidence circuit. Another counter element or series of elements were put in coincidence with the Cerenkov counter via the other grid. The inside of the counter is coated with white reflecting paint to diffuse the light or in some cases coated with aluminum foil. A gas counter of this type designed to use CO:, mainly is shown in Fig. 17. The mirrors increase the efficiency of light collection at very small angles. The pressure can be varied from one to 200 atmospheres. The temperature can be regulated from 0" to 100°C. This allows the index of refraction to be varied continuously from 1.004 to 1.21. The liquid version of this counter (Fig. 16) allows the index of refraction to vary with suitable liquids over the range 1.33 to 1.7. The index of refraction of fluorochemical o-75 can be varied over the range 1.26 to

188

1.

PARTICLE DETECTION

1.32 by varying the temperature. Hence except for some small gaps most of the index of refraction range 1.004 to 1.7 can be attained by these counters. A series of these transmission type counters were employed by the authors24 a t the Brookhaven Cosmotron in an investigation of positive SCALE 1:2

LIT€ WINDOW

DIRECTION OF BEAM

-

-

-

-

-

FIG.17. Nonfocusing gas counter designed by Lindenhaum and Yuan.

K-meson production in positive proton collisions. After momentum selection via magnetic deflection, a velocity interval is selected by requiring a coincidence in a counter with a threshold p1 and a n anticoincidence in a counter with a threshold pz where pz > pl. Hence the velocity range selected is represented by: p1 < p < p2. 2 4 S. J. Lindenbaum and L. C. L. Yuan, Phys. Rev. 106, 1931 (1957).

1.5. EERENKOV

COUNTERS

189

Such a coincidence anticoincidence pair then acts as a mass spectrometer in a momentum analyzed beam, and can be employed t o detect only K + or another mass component in the beam. To insure a high efficiency in the anticoincidence, 2 or 3 counters are used. As a matter of fact the major practical use of the focusing type of Cerenkov counter is also to act as an element which by selecting a velocity interval in a momentum analyzed beam a t a high energy accelerator selects a particular mass of particles. Another major advantage of Cerenkov counters is that they do not detect background due to low velocity particles (i.e., below their threshold). Hence the accidental counting rates are reduced and jamming is avoided. Heiberg and Marshallz6and also Porter26have reported using a fluorescent material additive to a water Cerenkov counter so that the violet and ultraviolet components of the radiation can be transformed into a nondirectional light of more usable wavelength for the photomultiplier. Gains of less than a factor of two have been realized in certain cases with this technique. Atkinson and Perez-Mendez have reportedz7a gas Cerenkov threshold device for discriminating against inelastically scattered pions in a negative pion momentum analyzed beam.

1.5.4. Total Shower Absorption eerenkov Counters for Photons and Electrons*

Kantz and Hofstadter first suggested28 the principle of using a total absorption cerenkov counter to measure the energy of a photon or electron of energy >,100 MeV. The basic idea is that a block of a relatively clear optical medium of short radiation length such as lead loaded glass, with dimensions equal to many radiation lengths is used as a shower producing medium for a photon or electron entering near its center. If the block is large enough, the mean total path length of electrons and photons is approximately linearly related to the energy of the incident photon or electron. Since for electrons of energy 2 several Mev the mean number of Cerenkov photons emitted per unit path length is independent of the energy, the mean number of Cerenkov photons emitted in the counter is also a linear function of the incident energy. If the side walls of the block are reflecting for the shower photons, and the front end is

* Refer to Section 2.2.3.7. 26

E. Heiberg and J. Marshall, Rev. Sci. Znslr. 27, 618 (1956).

** N. Porter, Nuowo cimento [lo] 6, 526

(1957). H. Atkinson and V. Perez-Mendez, Rev. Sci. Znstr. SO, 864 (1959). 28 A. Kantz and R. Hofstadter, Nucleonics 12, (3), 36 (1954); R. Hofstadter, CERN Symposium, Geneva, Proc. 2, 75 (1956). 2’J.

190

1.

PARTICLE DETECTION

optically coupled directly to one or a series of large photocathode photomultipliers, a sizeable fraction of the photons generated will strike the photocathodes. A fraction of the light emitted will be reabsorbed before reaching the photocathodes. Both this reabsorption and the fraction of photons collected on the photocathode will be only slightly dependent on the incident energy over the energy range for which a well-designed counter is useful. Hence the mean number of photoelectrons generated at the photomultiplier photocathodes will be approximately a linear function of the energy of the incident photon or electron. Known energy electron beams can be used to calibrate the counter. Obviously there will be appreciable fluctuations from the mean number of photoelectrons for individual showers caused by the same energy electron. These fluctuations arise from the stochastic nature of the shower, the partial loss of electrons from the counter even in large counters, the fluctuations in Cerenkov photon emission, photon collection, and photoelectron production. The over-a11 effect of the fluctuations can be expressed in terms of the per cent energy resolution. Although different definitions have been employed, a convenient one is the full width at half-maximum of the counter response to monoenergetic electrons divided by the energy. The most commonly used type of total shower absorption Cerenkov counters employ Pb-loaded glasses or heavy crystals which are nearly colorless. A typical glass of this type is manufactured by the Corning Glass Co. It has a radiation length of 1-in., a density of 3.9, an index of refraction of 1.65, and a critical energy of 16 MeV. The critical energy is that energy at which ionization loss equals radiation loss. The Schott glass works in Germany makes two varieties of Pb-loaded glass suitable for total absorption counters. The lighter one, type SF-1, is clear white, contains 62% PbO, has a density of 4.44, and a radiation length of 2.0 cm. Counters of this type have been designed and utilized by Kantz and Hofstadter12*C a s ~ e l s , ~ ~ Brabant et u Z . , ~ Swartz13* ~ Filosofo and Y a m a g a t ~Koller , ~ ~ and S a c h ~and , ~ ~others. Jester30 employed a 12-in. diameter Corning glass cylinder 14 in. long (2 optically coupled 7-in. cylinders). Four 5-in. diameter Dumont 6364 photomultipliers were placed with their cathodes in optical contact with J. M. Camels, CERN Symposium, Geneva, Proc. 2, 74 (1956). M. H. L. Jester, Univ. of California Radiation Laboratory, Report No. 2990 (1957). 91 J. M. Brabant, B. J. Moyer, and R. Wallace, Rev. Sci. Znstr. 28, 421 (1957). C. Swartz, I R E Trans. on Nuclear Sci. NS-8, 65 (1956). 33 I. Filosofo and T. Yamagata, CERN Symposium, Geneva, Proc. 2, 85 (1956). 34 E. L. Koller and A. M. Sachs, Phys. Rev. 116, 760 (1959). 29

30

1.5.


191

one end of the cylinders using Dow-Corning Silicone No. 200 as the optical coupling. He has reported obtaining a linear response from 50 to 200 Mev with a resolution of -45%. An improved version of Jester’s and its linear response until counter was developed by Brabant et -1.5 Bev was demonstrated. A typical counter of this type designed by Swartz a t Brookhaven is shown in Fig. 18. The observed resolution was better than 30% for 400-Mev electrons.

FIG.18. Typical shower detector designed by Swartz.

Another obvious way to obtain a linear relation with electron or photon energy is to use a total shower absorption scintillation counter. Various versions of this type have been reported10*11.31utilizing NaI at low energies ( 5100 Mev), ordinary and heavy liquid scintillators, and combinations of liquid or plastic scintillators sandwiched between Pb plates to reduce the radiation length. In general the cerenkov type appears more likely to provide a compact, high resolution, low background, higher absolute energy accuracy, counter for the several hundred Mev to several Bev region. 1.5.5. Other Applications Several useful applications of cerenkov counters which have not been discussed are the following.

192

1.

PARTICLE DETECTION

1. As a directional device-All focusing counters are highly directional. Furthermore front to back discrimination is easily attained with nonfocusing counters by painting the back end of a cylindrical radiator black. Wincklerss used this method to measure albedo in the atmosphere. Various other methods of obtaining directional sensitivity are also possible. 2. Since the number of Cerenkov photons is proportional to Z2 for a known velocity particle, various charged particles can be separated by pulse height. There is some advantage over scintillators in that the broad Landau ionization loss distribution does not exist. Of course the statistical fluctuations of the number of photons and photoelectrons are larger in the Cerenkov case due to the smaller numbers. Nevertheless one can probably do better in many cases with a cerenkov counter than a scintillator. 3. A s a source of very fast light pulses for very short resolution time of flight work-A particle traveling along the axis of a radiating cylinder like that in Fig. 1 will produce a light pulse a t the front face of the cylinder of width in time equal to:

(-p

At8o,, = 1 v

1

e - 1)

COS~

(1.5.10)

where 1 is the cylinder length in cm, v is the particle velocity in cm/sec, and 0 is the Cerenkov cone angle. The cause of this time width is due to the fact that photons emitted in the interior of the radiator arrive later a t the exit than those emitted at the exit face. This is due to two reasons: (a) the velocity of light is less than the particle velocity; (b) the particle travels along the axis while the photons travel a t angle 8 to the axis. For a relativistic particle p + 1 (1.5.11) For a Lucite radiator of 2411. length At

-

sec.

For a gaseous cerenkov radiator of 6411. length operated a t a 10" Cerenkov angle with p 4 1, At

-

2X

J. R. Winckler, Phys. Rev. 86, 1034 (1952).

sec.

1.5. EERENKOV

193

COUNTERS

An optical system of the type shown in Fig. 8 will in principle approximately preserve the value of At for a photocathode placed in the image plane. In this respect one should note that an additional path length of 1.2 in. second and a n additional path length of in air gives a delay of -1O-lO 0.12 in. gives a delay of -10-l1 second. Hence great care must be taken to maintain a pulse width of to 10-l1. Of course there is no existing production type photomultiplier of end type photocathode of sizeable area which will allow one to transform these short time width light pulses into electrical pulses without appreciable lengthening. The best photomultipliers presently available would lead to a width of one to several millimicroseconds sec) for a n instantaneous light sec pulse. Although developmental types may provide widths of -1O-lO in the near future. Techniques employing rf gating of the first dynode for short intervals to reduce timing errors to -10-10 sec have also been considered. I n this connection one should note that the other general and older method for measuring velocity is by electronically measuring the time of flight between two counters. Presently available photomultipliers and ordinary coincidence and chronotron techniques allow a time of flight measurement of a n accuracy close to 10-lo second, when one demands counting each individual particle with a moderate efficiency. Therefore for a typical relativistic particle ( p 4 1) timed over say 10 f t , which is a typical telescope distance, we would have A@ 0.1. If the distance were increased to -50 f t we would have Ap 0.01. With a 50-ft distance and an accuracy of timing of -1O-lO sec we would have A 0 0.001. Of course short lifetime particles cannot be effectively counted over such large distances. I n order to generate pulses with sharp enough timing to maintain 10-10 sec coincidence resolution, entails a series of severe problems. Also changes of signal path length of -1 in. would bring the counters out of coincidence. Gas cerenkov counters of the focussing type can in th e future be expected with proper design and precautions t o reach velocity resolutions of AD < 0.001. This would allow one, for example, in a 20 Bev/c beam a t the Brookhaven AGS or CERN strong focusing 25-Bev proton accelerators, t o separate antiprotons from all other known particles. * From the foregoing one can conclude th at cerenkov counters appear most promising in providing highest velocity resolution. Furthermore they can be used with existing photomultipliers and electronic techniques

-

* Refer to Section 2.2.1.3.

- -

194

1.

PARTICLE DETECTION

most conveniently, and do not require large time of flight distances. Finally their direct velocity selectivity makes them extremely useful in reducing background and pileup problems.

1.6. Cloud Chambers and Bubble Chambers* Cloud chambers and bubble chambers are used to make visible the paths of high-speed charged particles. In cloud chambers, the track of the particle is formed when a supersaturated vapor condenses preferentially on the ions formed by the charged particle as it passes through a gas. Droplets formed on the ions grow large enough so that, with the proper illumination, they are visible and can be photographed. Bubble chambers operate quite differently. The path of the particle is delineated by the bubbles formed when a charged particle passes through a superheated liquid. Energy deposited along the track by the ionizing particle creates locally heated centers around which bubbles of vapor start to grow. When these bubbles reach a suitable size they, too, may be illuminated and photographed. The most important basic difference to be noticed between cloud chambers and bubble chambers is that the former operate with gases, and the latter with liquids. There are many other differences, aside from technical operating problems, and they will be discussed in a section concerned with the advantages and limitations of each method. We will now treat only the fundamental principles involved in the operation of these devices.

1.6.1. Cloud Chambers The supersaturation of vapor needed to form droplets on ions may be obtained in two ways: (1) by the rapid expansion of a volume of gas containing the vapor (expansion cloud chamber) ; and, (2) by the diffusion of vapor from a warm region where it is not supersaturated to a cold region where it is supersaturated (diffusion cloud chamber). The technical problems of construction and operation of these two types of cloud chambers are quite different. However, once supersaturation is obtained, by whatever method, the formation of the droplets proceeds according to well-known thermodynamic principles. *Chapter 1.6 is by W. B. Fretter.

1.6.

CLOUD CHAMBERS AND BUBBLE CHAMBERS

195

The theory of the formation of liquid droplets from a supersaturated vapor, treated by J. J. Thomson’ in 1888, and developed by various other physicists is summarized by J. G. Wilson in his excellent treatise on cloud chambers,2 and by Das Gupta and Ghosh3 in their review article. More recent developments in cloud-chamber technique are described in a report4 of a conference on cloud chambers. Liquid droplets may form in a supersaturated vapor on nuclei present in the gas or, if the supersaturation is sufficiently high, spontaneously on microscopic fluctuations in density in the vapor. The latter process determines the upper limit of supersaturation desirable to attain in an expansion type cloud chamber, but is not of use in track formation. The nuclei present in cloud chambers, upon which droplets form at lower supersaturations include ions, both those of the track and others formed in the chamber, foreign suspended particles such as dust, chemical compounds which may act as condensation nuclei, and re-evaporation nuclei. The latter are produced by the evaporation of large droplets to a point where further evaporation ceases. Of all these condensation nuclei, the only ones wanted to exist in the chamber a t the time of production of supersaturation and subsequent photography are the first, those ions produced by the passage of the charged particles. All others must be cleared from the chamber in various ways. 1. Unwanted ions are removed by an electrostatic clearing field. 2. Dust particles and re-evaporation nuclei are removed by production of supersaturation successively, in the case of expansion chambers, and continuously, in the case of diffusion chambers, until the nuclei are carried to the bottom of the chamber, where they adhere to the wall. The condensation of vapor on ions involves in the first approximation the dielectric constant of the liquid and the external medium, the surface tension of the liquid, the molecular weight of the vapor, and the degree of supersaturation of the vapor. The theory is given by Wilson2 and only the broad outlines will be given here. The effect of the charge is to modify the surface energy of an incipient droplet in such a way as to permit it to grow by the condensation of molecules out of the vapor. If the supersaturation, that is the ratio of the vapor pressure existing ( p ) to the 1 J. J. Thomson, “Applications of Dynamics in Physics and Chemistry.’’ Macmillan, London, 1888. 2 J. G. Wilson, “The Principles of Cloud-Chamber Technique.” Cambridge Univ. Press, London and New York, 1951. a N. N. Das Gupta and S. K. Ghosh, Revs. Modern Phys. 18, 225 (1946). 4 ‘‘Report of the Conference on Recent Developments in Cloud-Chamber and Associated Techniques, March, 1955’’ (N. Morris and M. J. B. Duff, eds.), University College, London, 1956.

196

1.

PARTICLE DETECTION

equilibrium vapor pressure (PO)at the temperature after the expansion is sufficiently high, charged droplets will grow and continue to grow until other limitations occur. The value of the supersaturation necessary for drop formation on ions depends on the nature of the vapor and the sign of the ion. The latter fact indicates that, in the initial stages of formation of the drop, the polar nature of some of the vapors used plays an important role. For example, water vapor condenses preferentially on negative ions, and higher supersaturation is needed for condensation on positive ions, while for ethyl

EXPANSION RATIO

FIQ. 1. Positive and negative ion thresholds, 70% ethanol and 30% water, in a cloud chamber filled with oxygen. Curve at left is for positive ions; curve at right is for negative ions.

alcohol, the opposite effect occurs, as is shown6 in Fig. 1. The value of the supersaturation required2 varies from p / p o = 4.14 in water to 1.94 in ethyl alcohol, to name two commonly used vapors. Mixtures of alcohol and water are also sometimes used, in which case the value of p / p ~may be as low as 1.62. The rate of growth of drops from a supersaturated vapor determines the length of time required for a drop to reach visible (or photographable) size. Rapid growth makes possible short photographic delay times, thus minimizing distortion effects due to motion of the gas, and it also is desirable in producing large droplets which fall quickly to the bottom of the chamber, leaving no residue of re-evaporation nuclei. The rate of drop growth has been studied by Hazen" and by Barrett and Germain.' C. E. Nielsen, Ph.D. Thesis, University of California, Berkeley, California, 1941. (1942). '0. E. Barrett and L. S. Germain, Reu. Sci. Instr. 18, 84 (1947).

* W. E. Hazen, Rev. Sci. Znstr. 13, 247

1.6.


197

Theoretically, the drop growth is determined by the diffusion of vapor toward the growing drop and by the conduction of heat away from the growing drop. Practically, it has been founda that the heat conductivity of the gas is the predominant factor in ordinary operation of cloud chambers near atmospheric pressure. The rate of growth of droplets in xenon, which has a very low heat conductivity, is very small, but if a n equal pressure of helium, with high heat conductivity, is added to the xenon, the drop growth is nearly as rapid as it is in pure helium. Kepler et al. also found th at the rate of drop growth is not very dependent on the gas pressure in the range 0.2 atmos to 1.4 atmos, indicating th a t diffusion processes are not limiting the growth. In practice, the time required for a drop to reach visible size is between 50 and 250 milliseconds, depending on the observational conditions. The diameter of the drops in a cloud chamber, a t the instant of photography, is of the order of 10W cm. 1.6.1.1. Expansion Chamber. Supersaturation is produced by a rapid, nearly adiabatic expansion of the mixture of gas and vapor. The drop in temperature during such an expansion is given by T1/T2= ( V Z / Z J I ) Y - ~ where y = c,/c,, the ratio of specific heats a t constant pressure and dependconstant volume of the gas mixtures, or Tl/T2 = (pl/pz)(~-l)’u, ing on whether the expansion is volume-defined or pressure-defined. Here T lis the initial (absolute) temperature, T 2 the final temperature, v 1 the initial volume, and v 2 the final volume, with p l and pz the corresponding pressures. The change in temperature is clearly dependent on y, and the large value of y for monatomic gases makes them desirable for use in cloud chambers. Most cloud chambers in current use are volume-defined, generally by mechanical means, but pressure-defined cloud chambers, with only a thin rubber diaphragm separating the pressure vessels, are sometimes used, especially in cloud chambers containing metal pIates. There are .many factors to consider in the design of a n expansion cloud chamber, aside from the purely mechanical ones, and those connected directly with the experimental setup. Some of these factors are discussed briefly a s follows. 1.6.1.1.1. PRESSURE AND TYPE OF GAS. Cloud chambers may be operated a t widely varying pressures, from a fraction of a n atmosphere, to 50 atmospheres. At low pressure, the vapor becomes a n appreciable fraction of the total amount of gas present, changes the value of y, and provides an appreciable amount of ionization. I n the range of pressures from 0.1 atmos to 2 atmos, the operation of a cloud chamber is not 8 R. G. Kepler, C. A. d’Andlau, W. B. Fretter, and L. F. Hansen, Nuovo cimento 1101 7, 71 (1958).

198

1.

PARTICLE DETECTION

difficult. At higher pressures, the chamber becomes more difficult to clear of old droplets, and the time necessary to wait between expansions increases as the operating pressure increases. Scattering of the passing particle by high-density gas reduces the accuracy of curvature measurements if the chamber is placed in a magnetic field. On the other hand, the sensitive time of high-pressure chambers is long compared with chambers operated near atmospheric pressure. A good discussion of the properties of high-pressure chambers has been given by Burh0p.O When the type of gas is not specified by the experiment to be done, noble gases are preferred because of the lower expansion ratio required. Argon is most commonly used, or a mixture of half argon and half helium which gives easily visible tracks and rapidly growing droplets. 1.6.1.1.2. USE WITH MAGNETIC FIELD.Very often information on the momentum of the tracks passing through a cloud chamber is required, and the cloud chamber must be placed in a magnetic field. One factor to be considered here is the design of the chamber and its expansion mechanism to make the best use of the field. See Fig. 2. It is desirable to make the expansion mechanism occupy the least space possible, and often most of it can be placed outside the magnet, with a rod or tube leading through the iron to compress or expand the chamber. The magnetic field should be as uniform as possible to avoid large corrections in the momentum, and accurate fiducial marks should be provided in the chamber to serve as points of reference. Generally the larger the magnetic field, the better, since the spurious curvatures produced by scattering and by motion of the gas are relatively less import,ant if the field is large. Wilson2 summarizes the relative importance of these two types of error for various lengths of track and magnetic field values. The term “maximum detectable momentum” is often used as an index of performance of a chamber. This is the particle momentum for which the true curvature is equal to the uncertainty in curvature, and for a chamber where tracks of length about 40 cm can be measured in a field of lo4 gauss, the maximum detectable momentum under very good conditions may be as high as 50 Bev/c. 1.6.1.1.3. USE WITH PLATES. Since the stopping power of gas is so low, and the probability of nuclear interaction in a typical cloud chamber is very small, it sometimes is desirable to place sheets of heavier material in the chamber, leaving gas spaces between, in which the tracks may be seen. The minimum distance between such plates is about 3 in., and they must be coated with reflecting materials to increase the light scattered by the droplets. Such multiplate chambers have proved valuable in investigations of nuclear reactions, and the short-lived unstable E. H. S. Burhop, Nuowo dmento [9] 11, Suppl. No. 2 (1954).

1.6.

199


particles produced in these reactions. The range of particles can be measured if the particle stops in one of the plates, and scattering in the plates can also be determined. If y rays traverse the chamber, the probability of pair production may be increased if plates are introduced, and

A I

REAR

C

\ i? APPROX

FRONT WINDOW

,//Y,///; .// B

,/.v

//+

6

1

SCALE IN INCHES

2

3

4

FRONT

FIG.2. Cloud chamber designed for use in a magnetic field. The back plate of the cloud chamber moves to produce the expansion. (A) Vertical section parallel to front. (B) Vertical section parallel to side. (C) Horizontal section.

details of nuclear reactions can be observed. Multiplate chambers do not operate well in regions of high background unless sufficient shielding is available. A multiplate chamber has been operated near the Bevatron'O with a strong pulsed electric field between the plates, which reduced the ion background to the point where counter control may be used. 1oR. W. Birge, H. W. J. Courant, R. E. Lanou, and M. N. Whitehead, Univ. of California Radiation Laboratory Report UCRL-3890 (1957).

200

1.

PARTICLE DETECTION

1.6.1.1.4. COUNTERCONTROL.Expansion cloud chambers may be operated a t random, with a repetitive cycle accelerator, or with counter control. Normally counter control is used for cosmic-ray experiments. Here a particle passes through the chamber and associated counters, giving a signal which triggers the expansion of the chambers.* Since the speed of expansion is of the order of 10 milliseconds, counter controlled tracks are broadened by diffusion to widths of about one or two millimeters. If the chamber is triggered before the particles pass through, as with an accelerator, the tracks are much sharper, easier to photograph, and to measure. If unusual events are to be observed, however, it is sometimes advantageous to use counter control even a t an accelerator. CONTROL. No cloud chamber operates con1.6.1.1.5. TEMPERATURE sistently without adequate temperature control, and if accurate momentum measurements are required, extreme precautions must be taken to avoid certain types of temperature gradients. The order of magnitude of temperature control required may be 10.1"C for most applications, and +O.Ol"C for accurate momentum measurements. Normally a temperature gradient of about O.Ol"C/cm is maintained from top to bottom of an expansion cloud chamber to provide stability of the gas. It is also good practice to measure the temperature and the temperature differences around a cloud chamber as routine operating procedure. The speed of expansion of a cloud 1.6.1.1.6. SPEEDOF EXPANSION. chamber is usually an important factor only in the case of counter control, when diffusion of the ions before they are immobilized by the forming drops may make the track too wide for accurate measurement. Cloud chambers can be made to complete their expansion in as little as 0.004 set," but expansion times of 10 to 20 milliseconds are more commonly used. The width of a track is given by2

X

=

4.68(D7)'I2

where X is the width which contains 90% of the drop images, D is the diffusion coefficient in cm2sec-1, and 7 is the expansion time. Tracks about 1 mm wide are obtained with expansion times of 14 milliseconds in air a t NTP. The speed of expansion is increased if the moving parts are of low mass; however, if the speed is too great and gas must move at speeds near the speed of sound, undesirable shock wave effects occur which often can completely spoil the operation of the chamber. In cloud chambers expanded by a moving piston, provision should always be made to catch and damp the motion of the piston at the end of its stroke.

* See also Vol. 2, Part 8. R. V. Adams, C. D. Anderson, P. E. Lloyd, R. R. Rau, and R. C. Saxena, Revs. Modern Phys. 20, 334 (1948). 11

1.6.


20 1

1.6.1.1.7. RECYCLING TIME.A conventional expansion cloud chamber requires a t least one minute to prepare for each expansion. This time is occupied by slow, clearing expansions, waiting for the motion of the gas to cease, and the vapor to diffuse back through the gas. Such chambers cannot operate a t the repetition frequency of a pulsed accelerator, and in this respect are far inferior to bubble chambers, which can recycle every few seconds. Various attempts have been made12 to shorten the resetting time by quick recompression, overcompression, etc., but it seem difficult to operate on much less than a one minute cycle. Under these circumstances the operating characteristics are quite different from those at longer times, and the chamber must be adjusted to take these differences into account. 1.6.1.1.8. PHOTOGRAPHY AND ILLUMINATION. Although the illumination problems of diffusion and expansion cloud chambers have some common features, the illumination of an expansion chamber is sometimes easier because it may be possible to illuminate from the rear and use the large amount of forward-scattered light. Illumination a t right angles to the direction of viewing gives about 100 times less light than illumination from the rear but in many cases, because of mechanical reasons, right angle illumination is necessary. The design of the illumination system depends on the degree of detail required in the tracks. If individual drops must be photographed, the requirements are stringent. Flash-tube light sources are now universally used. A brief discussion of recent techniques of photography and illumination is given in the paper by flretter;I2 and Wilson2 discusses fully the photographic problems involved in drop photography. 1.6.1.2. Diffusion Chambers. Supersaturation in a diffusion cloud chamber is produced by the diffusion of a vapor from a warm region where supersaturation does not exist into a cold region where supersaturation occurs. The diffusion cloud chamber is a continuouslg sensitive instrument. The region of supersaturation is necessarily horizontal because it is maintained by temperature gradients in a gas, where stability in the gravitational field occurs when the thermal gradient is vertical. The thickness of the sensitive region depends on the thermal conditions, but cannot be made more than two or three inches. A review article on diffusion cloud chambers, covering the theory of operation and the techniques of experimental use was published by Snowden,l3 and more recent developments were reviewed by Fretter. l 2 In several ways, the design factors for diffusion cloud chambers are similar to those for expansion cloud chambers. A magnetic field is often I* 18

W. B. Fretter, Ann. Reu. Nuclear Sci. 6, 145 (1955). M . Snowden, Prop. in Nuclear Phys. 3, 1 (1953).

202

1.

PARTICLE DETECTION

required, and the chamber, together with the cooling arrangement must then be designed to fit in a magnet. The illumination and photography present similar problems, but the photographic background in a diffusion chamber may consist of a black-dyed liquid which gives excellent contrast with the brightly illuminated track. It is considerably more difficult to utilize plates of heavy material, as in a multiplate expansion chamber because of thermal problems involved, and although some attempts have been made along these lines, the use of plates in a diffusion chamber has not become an important device. There are, however, certain unique design factors in diffusion cloud chambers. 1.6.1.2.1. PRESSURE AND TYPE OF GAS. Diffusion cloud chambers operate satisfactorily with air and argon at atmospheric pressure. Methyl or ethyl alcohol are commonly used as vapor. Dry ice (solid CO,) usually provides the cooling for the bottom of the chamber, either directly by contact, or by circulation of a liquid such as acetone over dry ice and through cooling tubes on the bottom of the chamber. A low-pressure chamber has been constructed for use with helium by Choyke and Nie1~en.l~ In this chamber the bottom was cooled with liquid air and the chamber operated in the pressure range of 75 cm to 15 cm Hg. The temperature of the top had to be maintained a t less than -20°C to provide mass stability. Such a chamber might be used, for example, for the observation of low-energy electrons whose range would be small in an atmospheric pressure chamber. The most' useful diffusion cloud chamber for high-energy nuclear research has been the high-pressure hydrogen chamber. Shutt16has shown that the light gases such as hydrogen, deuterium, and helium are unsuitable for use in diffusion chambers near atmospheric pressure, but work well a t pressures of the order of 25 atmos. Thus a desirable increase in density is obtained along with proper operation. Such chambers have been widely used in connection with accelerators and until the advent of the hydrogen bubble chamber provided the only means of observing directly interactions of fast particles with protons and deuterons. Although the technical problems of operating a t 25 atmos pressure are substantial, the high-pressure hydrogen diffusion chamber is an important instrument in nuclear physics. 1.6.1.2.2. USE WITH ACCELERATORS. The diffusion cloud chamber, being continuously sensitive, is adaptable to the rapid cycling of pulsed accelerators. For such operation, the recharging of the condenser bank supplying energy to the flash tubes illuminating the chamber must be done at a rapid rate. More basic problems are those of background W. J. Choyke and C. E. Nielsen, Rev. Sci. Insts. 2S, 207 (1952). 1sR.

P. Shutt, Rev. Sci. Instr. 22, 730 (1951).

1.6. CLOUD

CHAMBERS AND BUBBLE CHAMBERS

203

produced in the chamber during the acceleration cycle, and the ion load supportable by the chamber during the pulse. The first of these is handled by proper shielding of the chamber and insertion of the target late in the acceleration cycle. The ion load allowable in the chamber is limited by the diffusion rate of the vapor into the region depleted by formation of tracks during the previous exposure. If the cycling time is not less than five seconds, the chamber will usually recover adequately. 1.6.2. Bubble Chambers

It has long been known that liquids may be heated above the boiling point, without actually boiling. Such superheated liquids are unstable and erupt into boiling after short periods of time. Boiling may start, that is bubbles may form, at surfaces or a t nucleation centers within the liquid. D. A. Glaser was the first to conceive the idea that nucleation centers within the liquid might be created by deposit of energy by passing charged particles, and to see that such a process could be used to detect fast-moving charged particles. The bubble chamber can be thought of as the inverse of a cloud chamber, with a gas bubble forming in a superheated liquid instead of a liquid drop forming in a supersaturated gas. The first bubble chambers were constructed so that the only nucleation centers were provided by the ionizing particle. They were made entirely of glass and were thus limited in size. Later experiments showed that gasketed chambers could operate satisfactorily if the expansion conditions were properly controlled. Development of this technique has been rapid and bubble chambers of large size are in operation or under construction. Many different types of liquids have been used, for example, liquid helium, liquid hydrogen, organic liquids, liquid xenon, and certain other inorganic liquids. Although the general principles of the operation of a bubble cloud chamber are known, there is as yet no satisfactory theory which predicts, for example, the degree of superheat required or the number of bubbles formed as a function of energy loss. It is found experimentally that bubble chambers operate with pressure'8 about two-thirds of the critical pressure and the temperature about two-thirds of the way from the normal boiling temperature to the critical temperature. Some examples of pressure and temperature are given in Table I. The liquid in a bubble chamber is superheated by a sudden reduction of pressure. After the track forms and the photograph is taken, the pressure is increased to the initial value, the bubbles collapse and the chamber is ready for another expansion. The great advantage of the bubble chamber over the expansion cloud chamber is that all this can take place in a few 16

D. A. Glaaer and D. C. Rahm, Phys. Rev. 97, 474 (1955).

204

1.

PARTICLE DETECTION

TABLE I. Operating Conditions of Typical Bubble Chamber Materials For the methyl iodide-propane chamber the ratiation length is 7 cm; for liquid xenon it is 3.1 cm. Operating pressure (psi)

Operating temperature

Density gm/cm3

Hydrogen Heliums Xenonb Propane

70 15 300 315

28°K 4°K - 19°C 58°C

Isopentane Methyl iodide-propane WFs*'

350 450 426

157°C 125°C 149°C

0.07 0.07 2.3 0.4 (0.078 gm/cm* of H) 0.5 1.3 2.42

Substance

~

0

---

~

W. M. Fairbank, E. M. Harth, M. E. Blevins, and G. G. Slaughter, Phys. Rev.

100, 971 (1955). * J. L. Brown, D. A. Glaser, and M. L. Perl, Phys. Rev. 102, 586 (1956).

seconds, making the bubble chamber match a pulsed accelerator in its duty cycle. A chamber described by Glaser and Rahm16 filled with isopentane, became fully sensitive to minimum ionizing particles 3.5 milliseconds after the expansion was initiated, and remained sensitive for about 10 milliseconds. Photographs must be taken within this interval. In liquid hydrogen bubble chambers" the bubbles grow much more slowly, and delay times of the order of 50 milliseconds are required. Although the exact process of nucleation of bubbles is still not under~ ~ of stood, the rate of growth of the bubbles can be e ~ p l a i n e d ' *in~terms the heat flow in the liquid. In liquids of high thermal conductivity the rate of growth of bubbles is expected to be large, and the experimental results give close agreement with the theory. The nucleation process itself has been discussed16,20.21 in connection with measurements on bubble density. Deposit of a substantial amount of energy, such as might occur when a delta ray is made, seems to be necessary. That the bubble density varies with velocity in the same way delta rays do supports this idea. Another theoretical approach to this problem has been made by Askar'ian,22who finds an expression giving the specific number of bubbles. 17 D. Parmentier, Jr. and A. J. Schwemin, Rev. Sci. Znstr. 26, 954 (1955); also D. E. Nagle, R. H. Hildebrand, and R. J. Plano, ibid. 27, 203 (1956). 18 H. K. Forster and N. Zuber, J . Appl. Phys. 26, 474 (1954). 19 M. S. Plesset and S. A. Zwick, J . Appl. Phys. 26, 493 (1954). 20 D. A. Glaser, D. C. Rahm, and C. Dodd, Phys. Rev. 102, 1653 (1956). 11 G. A. Blinov, I. S. Krestnikov, and M. F. Lomanov, Soviet Phys. J E T P 4, 661 (1957). 22 G. A. Askar'ian, Soviet Phys. J E T P 4, 761 (1957).

1.6.


205

1.6.2.1. Bubble Chambers. Design Factors. 1.6.2.1.1. TYPEOF LIQUID. The choice of liquid depends primarily on the nature of the experiment to be done. Liquid hydrogen has the advantage of presenting a purely protonic target to the incident particle, but its low density makes it necessary to construct rather large chambers to have appreciable probability for interaction, and the low temperature involved creates cryogenic problems. A source of liquid hydrogen must be a t hand. Organic liquids such a s pentane or propane present a mixture of nuclei as targets, complicating the analysis of the pictures, but their density is much greater. The density of free protons is about the same in liquid hydrogen as in propane. Organic liquids must be heated above room temperature to produce the required superheat. Neither liquid hydrogen nor organic liquids are very efficient in materializing photons. Liquid xenon, liquid SnC14, and liquid WFs23 may be used as h i g h 4 materials with good detection efficiency for photons, but the cost of sufficient xenon to fill a reasonable size bubble chamber is very high. Another type of bubble chamber24 is th a t in which a gas is dissolved in a liquid under pressure. When the pressure is released the gas forms bubbles. The liquids used are usually organic liquids; thus the matter of choice of liquid is the same a s for a n ordinary organic liquid bubble chamber. 1.6.2.1.2. CONTROL OF TEMPERATURE. In order to ensure reproducible conditions, particularly for bubble counting, the temperature of a bubble chamber must be controlled to about 0.1"C. This implies accurate thermostatting and consideration of temperature gradients. Rapid recycling causes heating of the liquid, and compensation for this must be provided. Generally the production and maintenance of the proper temperature conditions presents a major problem in bubble chamber design. 1.6.2.1.3. MAGNETIC FIELD. It is often desirable to immerse the bubble chamber in a magnetic field so th at measurements of momentum may be made. See Fig. 3. For liquid hydrogen the multiple Coulomb scattering is not important compared to the deflection in the magnetic field a t 10 kilogauss or higher. Scattering is more serious in organic and other heavier liquids. Since single scattering can often be detected by visual inspection of the track with fields of 20 kilogauss measurements of momentum accurate to 10% may be made in bubble chamber filled with organic liquids. I n most cases it is desirable to design chamber and magnet together because of the many interlocking mechanical and thermal problems. 23 J. H. Mullins, E. D. Alyea., L. R. Gallagher, J. K. Chang, and J. M. Teem, Bull. Am. Phys. Soe. 2, 175 (1957). 2 4 P. E. Arzan and A. Gigli, Nuovo czmento [lo] 3, 1171 (1956); 4,953 (1956); see also B. Hahn and J. Fischer, Rev. Sci. Instr. 28, 656 (1957).

206

1.

PARTICLE DETECTION

1.6.2.1.4. ILLUMINATION. The first photographs of tracks in a bubble chamber were taken with bright-field illumination. I n this system a bright diffuse source of light is placed back of the chamber and light is scattered out of the beam by the bubbles. Thus the bubbles appear dark against a bright background. The scattering angle can be quite small, of the order

FIG.3. Large liquid hydrogen bubble chamber and associated magnet.

of one degree. Dark field illumination has also been developed, similar to the " straight-through" illumination in cloud chambers. This may be done by a series of plastic strips placed a t an angle, illuminated by a flash tube. 1.6.2.1.5. PHOTOGRAPHY AND REPROJECTION. Photography of a small bubble chamber presents no serious problems, and reprojection of the photographs is similar to reprojection of cloud-chamber photographs. For a large bubble chamber, however, the optical problems may be formidable.

1.6.


207

The index of refraction of the liquid is not negligible and the glass through which the photograph is taken may be quite thick. Thus the images are displaced, and for a large chamber the displacements may not be linear with distance off axis. The analysis of the pictures is very complicated unless suitable optical elements are introduced. 1.6.2.1.6. SENSITIVETIME, COUNTERCONTROL.Measurements of sensitive time in a pentane chamber have been made by Glaser, who found that the chamber was sensitive for 10 milliseconds. Further experiments on the nucleation centers in this chamber showed that the lifetime of such centers was never more than 1 millisecond and was usually less. Since several milliseconds are required to perform the expansion, counter controlled expansions seem to be impossible, at least with present techniques. Thus for cosmic ray studies where countercontrolled expansion are often required, bubble chambers have not been very useful. Some attemptsz5have been made to recycle bubble chambers a t a very high rate and have therefore a large fraction of the time during which the chamber was sensitive.'Photographs of the chamber would be taken only when an interesting event is detected by a system of counters, analyzed for events occuring during the sensitive time. However, it seems likely that the principal use of bubble chambers will be with accelerators, to which they are ideally adapted. 1.6.2.1.7. SAFETY. The dangers inherent in operation of bubble chambers are so great that all possible precautions against accident must be taken. Hernandez et aLZ6described the safety measures taken in the construction of hydrogen bubble chambers, and tests on explosions of hydrogen gas. The safety precautions to be taken with bubble chambers containing organic liquids must be also carefully considered when the chamber is large, since such liquids are necessarily hot and at high pressure, and an explosion would be disastrous. 26 E. V. Kurnetsov, M. F. Lomanov, G . A. Blinov, and Chuan Chen-Niang, Soviet Phys. JETP 6, 773 (1957). 2 6 H. P. Hernandez, J. W. Mark, and R. D. Watt, Rev. Sci. Instr. 28, 528 (1957).

1.7. Photographic Emulsions* 1.7.1. Introduction The use of in the field of nuclear physics dates as far back as the discovery of radioactivity, the latter being first observed by photographic methods. A new era in the use of photographic emulsions was initiated when Kinoshita8 and Reinganumg were able to identify a trajectories-rows of developed silver grains-marking the passage of an a particle through emulsions. After Rutherford’s discovery of the disintegration of light elements by a particles, there arose a definite need for sensitive tools to detect and measure protons emitted in these disintegrations. Since only a few sensitive instruments were available a t this time, experiments with photographic emulsions were initiated. The trajectories of slow protons were first detected in 1925;‘O in the following years the tracks of faster protons-up to about 50 MeV-were observed, due to the subsequent improvements of the quality of emulsions, processing techniques, and thickness of emulsion layers. The grain density in proton tracks was smaller than in a tracks of equal velocity, and it was soon definitely established that the grain density in tracks is a function of the specific ionization loss which a particle suffers in the penetration of matter. The earliest experiments were concerned with particles emitted in the disintegration of nuclei by a particles of radioactive origin. Attempts were made to determine the yields and angular and energy distribution of disintegration products in these reactions. The low intensity of radiation, available from radioactive sources seriously limited the accuracy of the 1

a

M. M. Shapiro, Revs. Modern Phys. 13, 240 (1941). P. Demers, Can. J . Research A26, 223 (1947). H. Yagoda, “Radioactive Measurements with Nuclear Emulsions.” Wiley, New

York, 1949. J. Rotblat, P r o p . i n Nuclear Phys. 1, 37 (1950). A. Beiser, Revs. Modern Phys. 24, 273 (1952). OL. J. Vigneron, J . phys. radium 14, 121 (1953). Y.Goldschmidt-Clermont, Ann. Rev. Nuclear Sci. 3, 141 (1953). L.Voyvodic, i n “Progress in Cosmic Ray Physics” (J. G. Wilson, ed.), Vol. 11, p. 219. North Holland Publ., Amsterdam, 1953. Ib M. M. Shapiro, in “Handbuch der Physik-Encyclopedia of Physics” (S. Flugge, ed.), Vol. 45, p. 342. Springer, Berlin, 1958. ?OC. F. Powell, P. H. Fowler, and D. H. Perkins, “The Study of Elementary Particles by the Photographic Method,” Pergamon Press, London, 1959. S. Kinoshita, Proc. Roy. SOC.A83, 432 (1910). 9.M. Reinganum, Phys. 2. 12, 1076 (1911). l o M. Blau, J . Phys. 34, 285 (1925).

* Chapter 1.7 is by M. Blau. 208

1.7.

PHOTOGRAPHIC EMULSIONS

209

measurements. Emulsions were exposed to cosmic radiation on high mountains and in balloon flights, leading to the discovery of fast neutrons in cosmic radiation simultaneously with cloud chamber experiments. In these exposures multiple disintegration of emulsion nuclei by cosmic radiation-stars-were observed for the first time. l1 With the availability of collimated proton-deuteron- and a-particle beams from accelerators, it became possible to correlate the residual range of particles in emulsions and the grain density in tracks with mass, charge, and energy of the incident particles; these calibration tracks were then used for the energy determination of particle tracks emitted in di~integrations.l~-'~ The investigations were greatly enhanced by the availability of new emulsion types, containing higher concentration of silver halides, with which much denser and therefore better defined tracks could be obtained. These emulsions were manufactured first by Ilford and later also by Kodak and Eastman-Kodak. However, the photographic method up to 1948 was limited to the detection of particles with velocities p 5 0.4; the then available emulsions were not sensitive enough to record particles of higher velocity and therefore smaller specific ionization. I n 1948 Kodak, Ltd., in England, and soon afterward Eastman-Kodak and Ilford were successful in manufacturing the so-called electron sensitive or minimum ionization emulsions with which all charged particles, regardless of velocity, can be recorded. Another shortcoming of the older emulsion techniques was overcome by a new processing technique16-the so-called temperature methodwith which plates with up to 1 mm emulsion thickness can be developed. Before the invention of this procedure, the maximum thickness which could be developed within a reasonable length of time was 200 microns. The development of plates with emulsions thicker than 1 mm, although possible, is very lengthy; it is also difficult to obtain uniform development and to avoid distortion. However, since many experiments in the highenergy range require thicker emulsion layers, the manufacture of stripped emulsions or pellicles must be considered a very great improvement in emulsion techniques. Tightly compressed stacks of these emulsion sheets are exposed to the radiation and later developed separately. Various marking systems have been devised which make it possible to follow the particle trajectories through adjacent sheets. This increases the measural 1 M. Blau and H. Wambacher, Sitzber. Akad. Wiss. Wien, Math.-naturw. Kl. Abt. Zla 146,623 (1937). l2 W. Heitler, C. F. Powell, and C. E. F. Fertel, Nature 144,283 (1939). I3T. R. Wilkins, J . Appl. Phys. 11, 35 (1940). l 4 J. Chadwick, A. N. May, C. F. Powell, and T. C. Pickavance, Proc. Roy. SOC. A183, 1 (1944). l 6 C. C. Dilworth, C. P. S. Occhialini, and R. M. Payne, Nature 162, 102 (1948).

210

1.

PARTICLE DETECTION

ble path length of high-energy particles and therefore allows a greater number of measurements on a single track to be made. Consequently the statistical error in the measurement is diminished. The greater observable path length is of special importance in the investigation of interaction and decay events because a greater number of events becomes observable. Another milestone in the development of the photographic method is the introduction of a measuring technique with which multiple scattering in particle trajectories can be determined.16~~7 This technique is indispensable for mass measurements of particles in the relativistic energy range; for lower energy particles the results of scattering measurements supply a valuable complement to ionization and range measurements. Perhaps the greatest triumph of the photographic method is the discovery of unstable particles. In 1947 Perkins18 discovered the negative ?r meson and shortly afterward Lattes et ~ 1 . found ~ 9 the positive counterpart. Since then a great number of unstable particles-heavy mesons and hyperons-have been detected and their properties investigated through work in nuclear emulsions. The first heavy meson, unambiguously defined by its decay, was the 7 meson, discovered by Brown et a1.2Qin nuclear emulsions. The contribution of nuclear emulsion work in the field of strange particles could be adequately described only together with the development of particle physics. The recent improvements in mass and energy measurements are primarily the result of these experiments. So far the above discussions have mainly dealt with field of high-energy particles. The method has also been successfully applied in the field of slow neutrons, or photo-disintegrations, and in problems connected with fission. In most of these experiments the emulsions are loaded (impregnated) with small amounts of the element under investigation. There is, furthermore, a large field of application for nuclear emulsions in problems of biochemistry, biophysics, medicine, and mineralogy.

1.7.2. Sensitivity of Nuclear Emulsions The process of latent image formation for particles in nuclear emulsions is essentially the same as the case of light in ordinary emulsions. The fact that one observes rows of silver grains in the former case, and not in light exposures, may be explained by the larger energy of the particles and by the different mechanism of energy dissipation. The general theory of P. H. Fowler, Phil. Mag. 171 41, 169 (1950).

Y.Goldschmidt-Clermont, Nuovo cimento [91 7, 331 (1950). D.H.Perkins, Ndure 169, 126 (1947). l9

C. M. G. Lattes, G . P. S. Occhialini, and C. F. Powell, Nature 160, 486 (1947).

ao

R. H. Brown, U. Camerini, P. H. Fowler, H. Muirhead, C. F. Powell, and D. M.

Ritson, Nature 163,82 (1949).

1.7.


21 1

latent image formation is a solid state problem and will not be discussed in detail here. I n a recent articlez1the latest experimental and theoretical data were reviewed on which the theory of the latent image formation is based. Electrons and positive holes are liberated through irradiation and move independently through the crystal. The authors consider as a first step in the latent image formation the creation of a “pre-image speck,” which is a combination of an electron and a silver ion, absorbed near a dislocation site. The pre-image speck is unstable and decays in a fraction of seconds, if not, another electron and silver ion is deposited at the same dislocation site. The pre-image is now converted into the “ sub-image,” a neutral complex Agz. The sub-image has a longer lifetime and can be developed, but only with strong developers or large induction periods. The next step then is the absorption of another silver ion and the subsequent neutralization by a photoelectron. Experimental investigations make it plausible that the neutral aggregate Ag, can be considered as the origin of the stable latent image. Since silver in contact with silver halide acquires positive charge, it is likely that Ag3 will combine with a silver ion to form a stable tetrahedral combination Ag,. Ag, due to its positive charge, is now easily reduced by the developer. It is believed that in exposures of extreme short duration predominantly sub-images are formed, since recombination phenomena prevent the formation of stable Ag, complexes. Fast particles traverse a silver halide grain in about 10-14 sec, a time interval which is exceedingly short in comparison to the small mobility of ions. Therefore most of the ionization energy of fast particles will be spent in the formation of sub-images, which are less effective than stable latent images in the subsequent developing procedures. That explains why, in spite of the relatively high ionization power of fast charged particles, special types of emulsions are necessary for the detection of particles. Problems connected with the sensitivity of nuclear emulsions, i.e. , the maximum particle energy which can be detected and the maximum grain density in tracks of particles of given properties, have been well studied.1~22~23 A few a ~ t h o r s ~ *attempted ~ 4 , ~ ~ to describe the relation between a grain density and specific ionization loss by semiempirical mathematical expressions. However, in later experiments with particles of higher energy and in emulsions of higher sensitivity it was found that 21

T. W. Mitchell and N. F. Mott, Phil. Mug. [8] 2, 1149 (1957).

** C. M. G. Lattes, P. H. Fowler, and P. Cuer, PTOC.Phys. Soc. (London)69,883 (1947). J. H. Webb, Phgs. Rev. 74, 511 (1948). M. Blau, Phys. Rev. 76, 279 (1948). a5L. van Rossum, J . phys. radium 10, 402 (1949). 25

a4

212

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PARTICLE DETECTION

these relations are valid only within a limited energy range. More detailed information about the phase of the theoretical and experimental investigations prior t o the use of electron sensitive emulsions, may be found in * These articles also contain discussions of the first the literat~re.26~-~~ experiments in electron-sensitive emulsions and the earlier investigations on stopping power and the range energy relation in emulsions. Electron-sensitive emulsions are able to detect all charged particles, no matter what their energy is, thereby opening a completely new area of nuclear physics to emulsion research. It became necessary to re-examine and to revise the methods of earlier investigations in order to adapt the techniques to the new problems. In particular, the study of grain density as a function of specific ionization or energy loss has been resumed during the last years. As a result it has been found necessary to introduce certain changes due t o theoretical considerations and because of practical reasons connected with the new measuring techniques. At very high energies the grain density decreases slowly approximately proportional to the ionization loss until a minimum value is reached a t energies of about three times the rest mass of the particle. The grain density starts then to rise again slowly for still higher energies (relativistic increase)a1up to the so-called plateau value, which is about 10% higher than the minimum grain density. Grain density is not only a function of energy loss, but depends also upon emulsion sensitivity and development conditions. However, it has been found that the ratio g/gminor g/gpl is nearly independent of development and changes in emulsion sensitivity; where g is the grain density in the track element under investigation and gminand gpl are the grain densities a t minimum ionization and plateau value. For slower or multiply charged particles the relationship between specific energy loss and grain density becomes more complicated. With increasing ionization loss the grain density in particle tracks tends to reach a saturation value which is due to the limited number of grains per unit length. The saturation value depends strongly on development (size of grains) and upon the emulsion sensitivity.

* See also Vol. 4, A, Section 2.1.7. J. W. Mitchell, ed., “Fundamental Mechanism of Photographic Sensitivity.” Butterworths, London, 1951. 2 6 P. H. Fowler and D. H. Perkins, in reference 25a, p. 340. R. Morand and L. van Rossum, in reference 25a, p. 317. ** R. W. Berriman, in reference 25a, p. 272. ISL. Vigneron and M. Boggardt, in reference 25a, p. 265. J. Rotblat and C. T. Tay, in reference 25a, p. 331. a1 E. Pickup and L. Voyvodic, Phys. Rev. SO, 98, 251 (1950). Also refer to Section 1.7.6 of this volume. 258

1.7.


213

I n a later chapter we will discuss theoretical and semiempirical equations, governing the correlation between specific energy loss and grain density or related parameters. These relations are extremely important in problems of particle identification. It can be considered as a general statement that quantitative results in nuclear emulsions can be obtained only if appropriate calibration methods are employed. Therefore, it is understandable that, for emulsion experiments, as for any other measuring technique, based on calibration methods, technical details and reproducibility considerations will play an important role. We will return to this topic later; here, only the more technical aspects of emulsion properties, sensitivity requirements, and processing conditions will be treated. The chief purposes of emulsion experiments are: (a) the identification of particles by mass and charge measurements; (b) the determination of particle energies; (c) the investigation of lifetime and decay characteristics of unstable particles; (d) the study of scattering, interaction, and production cross sections; and (e) the detailed study of the nature and energy a s well as angular distribution of the particles emitted in these events. In many problems the emulsion serves only as the detector of particles from an external source, while in others it is utilized as a reaction chamber in which the particles interact with nuclei of the emulsion itself, or with additional nuclei introduced into the emulsion for the specific purpose of the experiment. Such experiments can be carried through successfully only if: (1) the emulsion sensitivity is sufficient for the detection of particles in the energy interval under consideration; (2) the discrimination among trajectories of particles with different properties is satisfactory, i.e., the difference in grain density is appreciable; (3) the emulsion thickness is large enough to observe, on the average, appreciable segments of the trajectory; and (4) the geometrical relations prevailing a t exposure are well reproduced in the developed emulsion. The latter depends on the processing conditions. Item (3) depends upon the size of the emulsion as well as the thickness of the emulsion layer, if glass-backed plates are used, or on the number of sheets within the emulsion stack; (1) and (2) depend mainly on emulsion properties but can be changed slightly through development conditions. The simultaneous attainment of the highest sensitivity and the best discrimination properties is not always possible. I n emulsions of high sensitivity the number of developed grains tends to reach the saturation value for particles of relatively high kinetic energy (about 30 MeV for protons). Thus, a further increase in ionization energy does not essentially change the grain density in the trajectory. Therefore it is very fortunate

214

1.

PARTICLE DETECTION

that various types of emulsions are available, permitting a choice appropriate to the problem at hand. The various types of emulsions and their specific properties have been discussed in great detai1;1-7b,32 in the following discussion the various types of emulsions are merely enumerated. Ilford, Kodak (England), and Eastman Kodak (Rochester) manufacture various types of emulsions of different sensitivity. The most sensitive emulsions, which are selected for work with fast electrons and in the field of high energy are: G6 by Ilford, NTd by Kodak and NTB-, by Eastman-Kodak. The emulsion in widest use is the G-5 emulsion which is available in various sizes and thicknesses either with glass backing or as free sheets called pellicles. The emulsions next in sensitivity are the CZ, NTZa, and NTB emulsions from Ilford, Kodak, and Eastman Kodak respectively in which protons up to 5G70 Mev and electrons up to 30-100 kev can be detected. In El, NTA emulsions, where protons up to l(f20 Mev can he detected, the discrimination between proton and (Y particle tracks is very good. Ilford’s D and Eastman Kodak’s NTC emulsions detect only slow particles and no protons. They are designed for the detection of fission products which can easily be distinguished from particles in these emulsions. Ilford manufactures another type of emulsion, the Go, with a sensitivity between El and CZ emulsions; it can be developed in the same way as G6 emulsions and is very useful if sandwiched between GSpellicles for the detection of multiple charged particles of high energy (e.g., heavy primaries in cosmic radiation). Ilford manufactures G5 emulsions in gel-form, with which one can prepare fresh emulsion layers and avoid background tracks, due to cosmic radiations. This technique is used whenever the intensity of the radiation under investigation is very low as in cosmic-ray experiments at great depths below the earth’s surface or in measurements of the radiation of pure isotopes. It is furthermore useful in geological and biological problems where the emulsion can be directly poured over the substances whose radioactivity is to be measured. Within the last few years, Ilford Ltd., had introduced new types of small grain emulsions, sensitive to particles of minimum ionization, with essentially the same constitution as G-5, which has a mean crystal diameter of 0.27 j~ (microns). The new emulsions, K-5 and L-4, with mean crystal sizes of 0.20 p and 0.15 p diameter respectively, thus exhibit developed grains of smaller diameter than does G-5. The fine grain emulsions prove especially useful in the identification of dense tracks, in the analysis of large stars, and in general provide sharper resolution in the measurement of small distances. However, these emulsions must be 32 C. Waller, J . Phot. S&. 1, 41 (1953).

1.7.


215

processed very soon after exposure, because the fading of the latent image is more rapid in emulsions of smaller crystal size. Among other types are the so-called diluted emulsions which have a smaller silver halide content than other nuclear emulsions. These emulsions are used usually together with ordinary emulsions in experiments in which interactions of particles with light and heavy elements are compared; the former being more frequent in diluted emulsions. A similar purpose is served by plates with alternating emulsion and thin gelatin layers where the trajectories of particles emitted in the interactions with the light elements of the gelatin can be followed into both adjacent emulsion layers. Furthermore, all companies supply emulsions loaded with boron, lithium, and bismuth for experiments in which certain reactions with these elements are studied. Unfortunately, the amount of foreign element which can be introduced is small. Experiments connected with loaded and sandwiched emulsions are described in references 1-7b, mentioned above. Most nuclear emulsions have very similar chemical composition, with the exception of diluted emulsions and Eastman-Kodak N T C emulsions which contain smaller amounts of silver halides. TABLE I. Composition of Dry Ilford Gg Emulsions. Element

Weight in gm/cm3

Silver Bromine Iodine Carbon Hydrogen Oxygen Sulfur Nitrogen

2.025 1.496 0.026 0.30 0.049 0.20 0.011

0.073

a Ilford Nuclear Research Emulsions (Ilford Research Lab., Ilford, London, England, 1949).

The composition of the Gsemulsion is given in Table I. The density of the dry emulsion is 4.18. However, emulsion density changes if the emulsion is brought into surroundings of higher relative humidity; because the changes take place very slowly, an appreciable length of time will elapse before equilibrium is reached. These problems were recently investigated with great care by Barkas and co-workers and are discussed below. An exact knowledge of the emulsion composition, which of course includes the water content, is very important in investigations of the range-energy

216

1.

PARTICLE DETECTION

relation in emulsions and for cross section experiments. Since the increased water content changes the volume of the emulsion layer, the spatial relationship between particle trajectories can only be evaluated if the actual volume a t exposure is known. During processing, the emulsion layers experience several large density changes, the most important occurring during fixing, when most of the original silver halide is dissolved. After drying, the thickness of the emulsion layer is considerably decreased with respect to the original value. The ratio of emulsion thickness before and after processing (provided that the emulsion was mounted on glass during the processing), the socalled shrinkage factor, determines the relationship between geometrical conditions a t exposure and in the processed emulsions. However, inasmuch as the processed emulsion is also hygroscopic, emulsion work should be done in humidity controlled laboratories and the emulsion thickness should be checked frequently through repeated emulsion thickness measurements. The magnitude of the shrinkage factor depends slightly on the processing conditions and is greatly influenced by the concentration of the glycerin, or other plastisizer solutions, which is used in the last bath to which the emulsions are subjected. Details on shrinkage factor measurements may be found in references 1-7b. The large thickness of emulsion layers required the development of new processing methods in order to achieve uniform development throughout the emulsion and to avoid distortion. The latter is very important not only for the true reproduction of the geometrical condition prevailing a t exposure, but even more so on account of multiple scattering measurements which represent one of the most important measuring techniques in nuclear emulsions. Emulsions, if not developed immediately, should be kept in deep and narrow wells both before and after exposure in order to minimize exposure to cosmic radiation. The latent image tends to fade under the influence of humidity and high temperature, necessitating special care in the storage of exposed emulsions. The fading effect is discussed in references 1-7b and is treated exhaustively in a more recent paper b y Demers et 1.7.3. Processing of Nuclear Emulsions

1.7.3.1. Processing Techniques. The development of various types of emulsion of thicknesses u p to 2 0 0 p has been discussed in detail in references, 3, 4, and 5. The larger the emulsion thickness, the more difficult it is to obtain uniform development because of the time needed for thorough penetration of the developer. This difficulty was removed by the so-called temperature 33

P. Demers, T. Lapalme, and T. Thonvenin, Can. J . Phys. 31, 295 (1953).

1.7.


217

development method.34-36The principle of this technique is based on the fact that a developer is chemically inactive at very low temperatures (about 4%). Amidol has been generally adopted as developing agent. The plates are soaked in cold developer until its diffusion in the emulsions is completed. Then the plates are removed from the solution and warmed to the required temperature in stainless steel containers. After the development at the higher temperature is concluded, th e developing action is stopped by a weak solution of acetic acid. The emulsion is then washed and finally readied for the fixing solution. The temperature of the dry-development stage determines the degree of development, i.e., the size of the developed grains and the number of minimum ionizing tracks. The fixing procedures for thick emulsion layers have also been completely changed just as in the development stage. While fixing a t room temperature would be rapid, it has been found to lead to grave distortion in the emulsion; hence the fixing solution is also kept a t very low temperatures. I n order t o minimize distortion, the fixing solution is never changed abruptly during the fixing process which lasts several days, but is slowly and carefully replenished by fresh solution. When the fixing process is concluded, the solution is slowly removed and replenished b y cold water. The plates are then soaked in a plasticizing solution, placed between guard rings (old emulsion sheets), and dried under controlled temperature and humidity conditions. The plasticizer is introduced in order to avoid stripping of the emulsion from the gIass and to restore part of the original thickness; the latter has been considerably decreased by the dissolution of all the silver halide which was not activated during the exposure. The dry emulsions are often covered with a thin plastic coat in order to protect the emulsion and t o avoid excessive fluctuations in the water content of the processed pellicle, even if the ambient relative humidity should change abruptly. Some authors introduce a clearing solution after fixing, especially if plates thicker than 600 p are used. The purpose of all these involved procedures is t o ensure the most homogeneous development possible and to minimize distortion. The latter is accomplished by avoiding abrupt changes in the pH of all solutions brought into contact with the emulsion, and by avoiding sharp temperature changes. To this end, it has been proposed that the temperature of the hot stage be lowered, or that only cold developer be used, consequently increasing the development time. However, i t has been found that such 34 C. C. DiIworth, C. P. S. Occhialini, and L. Vermassen, Bull. centre phys. nucltaire, univ. libre Bruselles No. 13a (1950). 35 A. Bonetti, C. C. Dilworth and C. P. S. Occhialini, R 711L. centye phys. nucldaire, univ. libre Bruxelles No. 13c (1951).

218

1.

PARTICLE DETECTION

emulsions show a smaller ratio of grain density to background density.3e In the processing t e ~ h n i q ~ e sfor ~ ~ thick , ~ emulsions ~ ~ ~ ~ - (Ilford, ~ ~ G-5), Amidol is used as the basic developing agent. However, the Brussels group use boric acid Amidol while the Bristol group uses a combination of Amidol and bisulfite. TABLE 11. Temperature Development for 600 p Plates Used in Belgian Laboratories. Boric Acid Amidol Developera Amidol Sodium sulfite (anhydrous) Potassium bromide (10% solution) Boric acid Distilled water PH a

4 . 5 gm 18 gm 8 cma 35 gm 1000 ml 6.4

From Dilworth, Occhialini, and V e r m a ~ s e n . ~ ~

TABLE 111. Temperature Development for GOO p Plates Used in Belgian Laboratories" Operation

Bath

Preliminary soaking Distilled water Cold stage Boric acid Amidol Warm stage slow Dry (after wiping the heating plate surface with a soft tissue) Development Dry Slow cooling Dry Stop bath Acetic acid, 0.2% Silver deposit cleaning Washing Running water Fixing Hypo 40% (sodium sulfate u p to 10% is added if swelling is excessive) Slow dilution Water (adding sodium sulfate) Glycerin bath Glycerin, 2% Slow drping with guard rings

Temperature

Time

Cooling down to 5°C 5°C 5°C to 28°C

120 min 120 min 5 min

28°C 28°C to 5°C 5°C to 14°C

60 rnin 5 min 120 min

14°C 14°C cooling to 5°C

5°C

5°C to room temperature 20°C

120 min Until clear

100 h r 120 min 7 days

a From Y. Goldschmidt-Clermont, Photographic emulsions. Ann Rev. Nuclear Sci. 3, 149 (1953).

*OA. J. Her2 and M. Edgar, Proc. Phys. SOC.(London) A66, 115 (1953). 87 A. D. Dainton, A. R. Gattiker, and W. 0. Lock. Phil. Mag. 171 42, 396 (1951). 38A.J. Herz, J . Sci. Znstr. 29, 60 (1952). 39 B. Stiller, M. M. Shapiro, and F. W. O'Dell, Rev. Sci. Instr. 26, 340 (1954).

1.7.

219


Tables 11, 111, IV, and V describe processing solutions and processing procedures used in Brussels and in the Naval Research Laboratory in Washington. TABLE IVa. Developer for 600 p Emulsions in Development Procedures by Stiller, Sha.piro, and O'Dell of the Naval Research Laboratory88 Amidol Sodium suIfite (anhydrous) Potassium bromide (10% solution) Boric water Distilled water

4 . 5 gm 18 gm 8 cm 35 gm 1000 ml 6.6

PH TABLE IVb. Fixing Solution (pH 5.3) ~~

Distilled water Sodium thiosulfate cp Sodium bisulfite Ammonium chloride

~

1000 400 7 7

om3 gm gm gm

500 15 5 5

cm2 gm gm gm

TABLE IVc. Clearing Solution (pH 5.2) Distilled water Ammonium acetate Citric acid Thiurea

TABLE V. Processing Procedure

Presoaking in distilled water Penetration of cold developer Warm dry development Dry cooling Acid stop (0.5%) Fixing Clearing solution Washing Plasticizing solution (Flexoglass) Coating and drying

Temperature

Time

Room temp. to 5°C 5°C 18°C 18"C-5"C 5°C 5°C 5°C 5°C 5°C Room temp.

150 rnin 150 rnin 180 min 5 min 150 rnin Clearing time 50% 24 hr 36 hr 1h r 5 days

+

After each exposure a sample emulsion should be carefully examined for the degree of development by grain density measurements on the tracks of fast electrons. The uniformity of development may be investigated by determining the variation of grain density of fast tracks which

220

1.

PARTICLE DETECTION

traverse the thickness of the emulsion, or by measuring the grain density in electron tracks near the top, center, and bottom of the pellicle. Finally, one has t o determine the degree of distortion in the processed emulsion; this may be accomplished through a technique described by Cosyns and Vanderhaeghe.40 First, several steeply dipping tracks are chosen at various positions in the emulsion; the distortion is then determined by measuring the angle, fi - a, between the tangent and the chord of the curved line representing the projection of the distorted track. The quantity

Distorted trock

FIG.1. Perspective drawing of a distorted track inclined to the emulsion surface.

d sin(@- a) as shown in Fig. 1, where d is the chord length of the track, is then the component of the distortion vector perpendicular to the projection of the track on the emulsion surface. Here AC would be the path of the particle if no distortion were present; AB is the chord length of the track; and ARB is the actual distortion path of the particle. The problem of distortion measurements and the various types of distortion configurations are treated in greater detail in various papers on multiple scattering, where methods for distortion diminution in scattering measurements are also described (see Section 1.7.5). I n some laboratories, emulsions are subjected to erradication processes 4 0 M. G. E. Cosyns and C . Vanderhaeghe, Bull. centre phys. nuclbaire, univ. libre Bruxelles No. 16 (1951).

1.7.


221

before expoeure. The latter is based on the destruction of the latent image by water v a p ~ r . ~ ~ , ~ ~ The wide use of large stacks of emulsions necessitates a description of the assembly and processing techniques peculiar to stacks. Pellicles were first used by D e m e r ~ and , ~ ~ the first commercial pellicles were manufactured by Eastman-Kodak-NTB emulsion sheets, 250 p thick. The pellicles most commonly used a t present are Ilford G-5 emulsions which are available in thicknesses ranging from 200 to 1000 p. Inasmuch as the pellicles suffer considerable lateral expansion and contraction during processing, it is necessary to mount them on glass plates prior to development. 43,44 The mounting techniques in various laboratories differ slightly from each other, although the essential point is the uniform wetting of both the emulsion surface and the glass plate in order to ensure freedom from air bubbles. The pellicle and glass are generally soaked in cold water and are then pressed to each other either with a rubber roller or by passing the pellicle and glass plate through a mangle in which the pressure can be adjusted. Again, care must be taken that the pressure is uniform over the entire emulsion sheet. After mounting, the pellicle is developed and fixed in exactly the same way as are ordinary plates. For exposure the emulsion sheets are tightly pressed together by blocks of Bakelite to ensure close contact of emulsion surfaces. Holes are sometimes punched in the pellicles to aid in the packing and subsequent alignment of the stack. Prior to development the plates are sometimes exposed to narrow beams of X-rays passing near the edges of the stack in order to facilitate, after development, the aligning of emulsion sheets relative to each other and thereby aid in tracing tracks from one plate to another. Other more elaborate have also been developed for successful alignment. One method which has gained wide acceptance is the printing of labeled grids on the surface of each pellicle. Before printing each plate is carefully adjusted in a jig with respect to small holes which were previously punched in the stack. Another method is the placement of brass tabs at the corners of the emulsion which then allow each emulsion to be mounted a t the same relative position of the microscope stage. The alignment of the emulsions by this method is generally within 50 microns. M. Wiener and H. Yagoda, Rev. Sci. Znstr. 21, 39 (1950). P. Demers, Can. J. Research A24 628 (1950). 43 B. Stiller, M. M. Shapiro, and F. W. O’Dell, Phys. Rev. 86, 712 (A) (1952). 4 4 C. W. F. Powell, Phil. Mag. [i]44, 219 (1953). 46 R. W. Burge, L. T. Kerth, C. Richman, and D. H. Stork, U. C. R. L. 2690 (1954). 46 G. Goldhaber, S. J. Goldsack, J. E. Lannuti, and H. L. Whetstone, U. C. R. L. 4l

42

2928 (1955). 47

E. Silverstein and W. Slater, J . Sci. Instr. 33, 381 (1953).

1.

222

PARTICLE DETECTION

I n a novel a p p l i c a t i ~ nof~ ~the stripped emulsion technique the two outside emulsion sheets move relative to a fixed stack, providing a time record for particles entering or leaving the stack. This method is especially suited for investigation of the heavy primary component in cosmic radiation. 1.7.3.2. Water Content of Emulsions. The emulsion is composed of a mixture of silver halides, gelatin, and glycerin; the last two constituents are hygroscopic and will normally contain a certain amount of adsorbed wat,er. A precise knowledge of the water content of emulsion is of great importance in cross section and range measurements and for a n exact determination of angular relations in scattering, disintegration, and decay events. The influences of the water content of emulsion on shrinkage factor and range measurements have been treated e ~ t e n s i v e l y . ~ ~ ~ ~ ~ - ~ ~ It is assumed that the absorption of water by gelatin and the consequent swelling of the emulsion occur without strong chemical interaction. On the strength of this assumption, one obtains that the absorption of w gram of water, of density one, by 1 cm3 of emulsion, density p , should result in a final volume, (1 - w) cm3, of material of density d. The latter is then related t o the emulsion densit,y p through the expression d = [ ( p to)/(l w)] gm cm-3. However, detailed experiment^^^-^^ have shown that Av/Aw, the volume change in cm3per mass change in grams, due to absorption or evaporation of water, is a quantity smaller than unity. The deviation of this ratio from unity is particularly evident if the time interval between the accrual or removal of water from the emulsion and the actual measurement is short. After long time intervals-several days -Av/Aw reaches an equilibrium value, which is 0.875, 0.84, and 0.94 for G5 emulsions, according to determinations by Batty, Ilford Lab., and Barkas respectively. This effect, which Barkas attributes to the porosity of the emulsion, requires th at both the mass and volume of the emulsion be obtained when precision measurements of particle ranges are needed. The slow diffusion of water vapor into and out of emulsions was measured carefully by Oliver and B a r k a ~ .Table ~ ~ , ~VI~ gives the loss of

+

+

J. J. Lord and M. Schein, Phys. Rev. 80, 304 (1950). F.K.Goward and T. T. Wilkins, Proc. Phys. Soc. (London) A63,662, 1171 (1950). 6o H. Bradner and A. S. Bishop, Phys. Rev. 77, 462 (1950). 61 J. Rotblat, Nature 166, 387 (1950). 62 J. J. Wilkins, A. E. R. E., Harwell c/r 664 (1951). 6 8 A. J. Oliver, Rev. Sci. In&. 25, 326 (1954). 6 4 W. H.Barkas, Rev. Sci. Instr. 25, 329, (1954). 66 C.J. 'Batty, Nuclear Znstr. 1, 138 (1951). 6 e W. H. Barkas, P. H. Barrett, P. Cuer, H. H. Heckman, F. M. Smith, and H. K . Ticho, Nuovo cimento [lo]8, 185 (1958). 48 49

1.7.


223

weight due to the ambient humidity of a 1000 fi G5 emulsion kept a t 50% r. h. (relative humidity) after unpacking. Only after 30 days in a vacuum do 1 0 0 0 ~plates lose their water content entirely, exhibiting then a density of 4.03 gm/cm3. TABLE VIm Days at 50% r. h.

Weight loss in grn per gm of emulsion 5.02 X 10-3 8.07 8.36 9.07 9.'16 9.22

1 4

5 11

15 18

"From A. J. Oliver, Rev. Sci. Instr. 26, 327 (1954).

Table VII gives the ratio of thicknesses of plates, kept a t various values of relative humidity, to the thickness at 50% r. h.; the thicknesses represent equilibrium values. Thickness rather than volume determinations are permissible if changes in length and width of the emulsion are negligible. TABLEVIIa % r. h.

T/Taor.h.

10 20 50 60 70 81

0.9657-0.0035 0.9720-0.0011 1.ooo 1.0202-0.0016 1.0466-0.004 1.1090-0.0035

5From A. J. Oliver, Rev. Sci. Instr. 26, 326 (1954).

If the physical conditions a t the time of exposure are known precisely, the shrinkage factor of the processed emulsion may be determined by measuring the thicknesses of the processed and unprocessed emulsions. Because this factor depends strongly on the humidity of the surroundings, measurements must be made repeatedly if the work is not done in humidity controlled rooms. Only if the shrinkage factor is known exactly can one relate the measurements in the processed plate to the situation prevailing at the time of exposure. The shrinkage factor also depends strongly on the concentration of plasticizer used in the last step of processing, on the concentration of hardener used in the fixing solution, and according t o Oliver, on the time of fixing and washing.

224

1.

PARTICLE DETECTION

One of the most accurate measurements of the shrinkage factor employs a narrow beam of particles (e.g., a particles from a radioactive source) inclined at a small angle to the emulsion surface. The ratio of the tangent of this angle to the tangent of the angle observed in the processed emulsion gives the shrinkage factor directly. The density of emulsion may be made by comparing residual ranges of particles from accelerators or in decay processes (e.g., p mesons from r - p decay) with the ranges found in standard emulsion.

1.7.4. Optical Equipment and Microscopes The contents of this section do not purport to be a complete description of microscope procedures for nuclear emulsions, but rather to highlight certain aspects which differ from ordinary microscopic work. Because many hours are spent in searching the emulsions and in measuring events, the use of a binocular microscope is necessary for the viewer’s comfort. The total magnification depends on the specific problem at hand, and may vary between 100 and 2000 times. The eyepiece lenses should be of the best possible quality; the magnifications which are commonly used may go from 6 to 20X. Dry objectives, lox, 20X, or 25X can be used when low magnification is desired, while oil immersion objectives, 45-70X and 90-100X, are commonly used for higher magnifications. The aperture of these objectives should be as high as is possible on account of depth measurements and in order to ensure optimum working conditions. Emulsions which are thicker than 400 p (or emulsions whose shrinkage factors are reduced by special processing) require objectives with long working distances, such as those now manufactured by Cooke, Throughton, and Sims, Leitz, and Koristka. The relatively low numerical aperture (n. a.) is a drawback which must be tolerated in order to allow observation of the entire emulsion thickness. Table VIII describes the characteristics of various objectives with long working distance. TABLE VIII

Manufacturer Cooke, Throughton, and Sims Leitz KS Objective Leitr KS Objective Leitr KS Objective Koristka Koristka Koristka

Magnification 45 x 22 x 53 x 100 x 30 X 55 x 100 x

n. a.

0.95 0.65 0.95 0.95

Working distance

1.50mm

1.05

2.30 m m 1.00m m 0.370m m 3.00m m

0.95 1.25

1.35 m m 0.530 mm

1.7.


225

The low magnification oil immersion objectives are useful because they permit better visibility and because their use avoids changing from dry to immersion lenses when switching magnifications. Dry objectives with large working distances such as the Newton and reflecting objectives which are manufactured by Beck (England) have the advantage of enabling one to place one emulsion atop another, and then to trace directly a track which passes from one pellicle to the other; this procedure is often convenient when one is examining a stack of emulsions. However, the magnification is so small that it is difficult to follow minimum tracks; furthermore, the visibility is impaired by the emulsion-glassemulsion sandwich, and the degree of optical alignment which is necessary to permit easy tracing of tracks is quite critical. Measurements of length are performed with an eyepiece micrometer which has been calibrated against a stage micrometer. The other eyepiece may contain a reticle in which a line, or two parallel lines, are engraved; this line is used as a fiducial line in making angular measurements in the plane of the emulsion. The rotational movement needed to superimpose this line on the track under consideration may be determined by a protractor device connected to the eyepiece. Angular measurements to within fractions of a minute may be made with precise eyepiece goniometers. For very accurate length measurements, so-called filar eyepiece micrometers are available; these devices contain one or two hairlines which are moved normal to a calibrated scale by a micrometer screw. Depth measurements are performed with the fine vertical adjustment screw, the emulsion being viewed with an oil immersion objective of highest magnification and n. a., in order to minimize the depth of focus. Another method of depth determination utilizes depth gages for measuring the vertical motion of the objective. Microscopy with nuclear emulsions differs markedly from ordinary applications in its requirements for precise and extended stage movements. The microscopes which are most widely used in emulsion work are the Cooke, Throughton, and Simms, type M4000, and the Leitz Ortholux; both instruments possess sufficiently smooth movements along the two stage axes. The former microscope has the advantage of a micrometer movement, thereby enabling one to read the stage position to within 2 p , while the rigid structure of the Ortholux is a desirable feature. A serious drawback in the use of both microscopes is the fact that neither can accommodate plates which are larger than 3 in. X 4 in. These disadvantages are especially evident in high-energy work, where large emulsion sizes are now widely used. In order to overcome these difficulties, nuclear emulsion workers have themselves designed and built completely new stages, or modified the

226

1.

PARTICLE DETECTION

original stages, to allow them to accommodate the large plate sizes. The simplest kind of modification consists of merely extending the linear dimensions of the stage itself; the chief disadvantages of this method are the fact that the very largest plates cannot be used and that the relatively short movements must be tolerated. A further step towards versatility is the construction of a superstage which is mounted atop the original stage on runners, allowing movement of the plates in addition to that of the stage. Many workers have retained only the frames and the optics of their microscopes, and have constructed entirely new stages, according to their specifications. Others have gone even further and have utilized only the optical components from commercial microscopes and have, in effect, manufactured their own microscopes. Such an instrument has been built by ZornS7from a universal table and drill press stand; it can accommodate the largest emulsionfi used and its movement is adequate for use as a scattering microscope. Another complete microscope has been constructed by Schein at the University of Chicago. In all of these endeavors the requirements of precision and care are understandably high. The above-mentioned microscopes are not always satisfactory for scattering measurements, where the movement must be accurately linear over a range of several inches. One must check each microscope for the linearity of its movement, which is found to vary with the individual instrument. 1.7.5. Range of Particles in Nuclear Emulsions

1.7.5.1. Measurement of the Residual Range of Particles in Nuclear Emulsions. The length of a track in the emulsion is determined by first ) then measuring the projection of the track in the focal plane ( 2 , ~and finding the angle of inclination (dip angle) to this plane. The former measurement is executed with a carefully calibrated eyepiece micrometer, while the latter is found by measuring the z coordinates, of two more grains in a track, on the fine adjustment depth screw of the microscope. If the direction of the trajectory changes, separate determinations of these quantities must be made for each segment of track with a different dip. The actual length of a track is sometimes given by

R

=

(P

+ S2z2)1/2 Z(l + S2tan2 =

+

a)lI2

(1.7.1)

where 2 = AX)^ ( A Y ) ~ .The shrinkage factor S is defined as the ratio of the original emulsion thickness to t o the thickness after development t d , while a is the angle of inclination as measured in the processed emulsion. 0’

G, Zorn, Rev. Sci. Instr. 27, 628 (1955).

1.7.

227


Inasmuch as the track length depends on the magnitude of the shrinkage factor, the latter must be determined with great accuracy. The shrinkage is a function of the water content of both the undeveloped and shrunken emulsions. Furthermore, the slow diffusion of water into and out of the emulsion63requires that before exposure the plates be kept in surroundings of constant humidity for an extended period of time (several days). The humidity content of the developed emulsion should also be kept constant during search and measurement. Methods of shrinkage factor measurements are described by many a~thors.4~~8-6~ Dip angle measurements must be made under the highest magnification (smallest depth of hcus) and with oil immersion objectives. The index of refraction varies slightly with the water content of emulsions; for G5 emulsions, it has the values 1.539, 1.533, and 1.521 for 31, 51, and 75% r. h., respectively. The projected length, dip angle, and shrinkage factor must be determined separately for each emulsion sheet when a track passes through several sheets in a stack. These measurements presuppose a knowledge of the thickness of each plate a t exposure, the value of which may vary as much as 5% throughout the stack. As a consequence of this fluctuation in thickness, when accurate range determinations are required, it is not adequate merely to measure the total thickness of the stack and then to divide by the number of sheets to obtain a mean value. describe the measuring techniques and all the necesBarkas et uZ.66,62a sary precautions in obtaining accurate values of the density and of water content of emulsions. A precise knowledge of the emulsion composition is a prerequisite for range measurements, the latter being meaningful only in a well-defined medium. The residual range Ad, of a particle in a dry emulsion (it may be dried in a vacuum or over Hzso4) of density do is related to its range in an emulsion of density d and water content w through the Eq. (1.7.2) :

-Ad

rd - 1 r(da - d ) Ado rdo - 1 -l- rdo - 1

L‘

’

(1.7.2)

Here, Ado and Ad are the ranges in emulsions of densities do and d, respectively. A, is the range of the particle in water, and T [ = ( A v l A w ) 5 I] is the ratio of the increase of volume to that of weight of an emulsion after L. Vigneron, J . phys. radium 10, 309 (1949). J. RotbIat, Nature 167, 550 (1951). E o V. L. Telegdi and W. Zunti, Helv. Phys. Acta 23, 754 (1950). 61 F. A. Roads, in “Fundamental Mechanism of Photographic Sensitivity” (J. W. Mitchell, ed.), pp. 327-330. Academic Press, New York, 1951. 82 M. Gailloud, C. H. Heanny, and R. Weill, Helv. Phys. Acta 27, 337 (1954). 8% W. H. Barkas, F. M. Smith, and W. Birnbaum, Phys. Rev. 98, 605 (1955). 6*

68

228

1.

PARTICLE DETECTION

the absorption of a certain amount of moisture. In the case of dry emulsions, the authors find r = 0.94 and 4.004.03 gm/cma for the density, while Ilford Laboratories obtain 0.84 and 4.033 gm/cm3 for the same quantities. Another factor which, if not properly taken into account, may lead to inaccuracies in range determination is emulsion distortion. Cosyns and Vanderhaeghe40 were the first workers to treat the distortion problem mathematically. Distortion calculations are based on simple geometrical considerations, which describe the coordinate displacement of an originally straight track, due t o the action of stresses present in the emulsion before solidification or induced d\ring the processing and drying of the emulsion. The maximum displacement A O will occur a t the air surface of the emulsion, since the other surface is firmly attached to the glass and will remain fixed. The displacement A of any other depth h in the emulsion will be a function of the ratio h :t, where t is the thickness of the emulsion sheet. The authors introduced a unit distortion vector, called the “Covan” and defined by C = Ao/t2, where the surface displacement A0 is measured in microns and the emulsion thickness t in mm. The relation of distortion to range measurements is discussed by Barkas et aE.6sand its connection with angular and scattering measurements by La1 et u Z . ~ ~ The apparent range of a particle in distorted portions of the emulsion differs from the value in the undisturbed part. If A0 = Ct2is the maximum displacement occurring on the emulsion surface and if the second derivative of the distortion vector with respect to the depth coordinate in the emulsion is constant (C-shaped curvature, which is the most widely observed), then the change p in the position of a point in the xy plane is given by p = Ao(1 - h2/t2) where h is the z coordinate of the point, measured from the emulsion surface. The range variance arising from small local distortions is often called “microscopic distortion straggling,” which is related to cavities and irregularities in the emulsion that are caused by the fixing process. The variance due to this type of straggling depends on the grain diameter, and lies between 0.02fi and 0.03E where R is the mean particle range. The finite grain size and separation of the grains introduces another uncertainty into range measurements in that the actual range may be larger than the measured range. However, the effect is generally small, except in the case of trajectories of very small residual range. Finally, the observer error must also be taken into account. While this contribution is small for experienced workers, it may be considerable for steeply dipping or strongly scattered tracks. The resultant of all errors and uncertainties responsible for range 68

D. Lal, Y. Pal, and B. Peters, Proc. Indian Acad. Sn’. A38, 398 (1953).

1.7.


229

straggling, which have been mentioned thus far, are smaller than the effect of “Bohr straggling.” The latter is inherent to the process of energy loss and will be discussed below in the section on range-energy theory. 1.7.5.2. Range-Energy Relation in Nuclear Emulsions.* The residual range of an ionizing particle is a function of its velocity, charge and mass.

R

=

F(v,Z,mi).

(1.7.3)

The range-energy relation in emulsions is derived partly from theoretical considerations and partly from actual range measurements on particles whose energies have been accurately determined. Because the stopping power of nuclear emulsions is relatively high (it has more than 1000 times the stopping power of air) even fast particles, available from high-energy machines, can be brought to rest in the emulsion, provided th a t thick layers or stacks of emulsions are employed. This feature is one of the greatest assets of the photographic method in th a t it permits the observation of the entire trajectory of a particle and its investigation b y a variety of experimental methods. Although the relation is derived from protons, the ranges of other singly charged particles with velocities equal to th a t of protons of range R , can be obtained immediately from th e following equation:

where mi and m p are the masses of the particle and proton respectively. This relationship follows from the energy loss equation (1.7.5), which asserts t ha t the energy loss is independent of particle mass. The range of a particle, R(miz), with a velocity equal to that of a proton, but with different mass mi and charge z is given by the expression

(1.7.5) The quantity f(z) represents a range correction due to electron capture, and will be discussed later in connection with the ranges of multiply charged particles. The range-energy relation for e l e c t r ~ n s ~differs ~ - ~ ~somewhat from the case of heavier particles; but will not be discussed here, inasmuch as range

* Refer to Section 1.7.6. H. Ross and B. Zajac, Nature 162, 923 (1946). R. H. Hertz, Phys. Rev. 76, 478 (1949). 6 6 J. Blum, Compt. rend. 228, 918 (1949). B. Gauthe and J. Blum, Compt. rend. 234, 2189 (1952). 68 J. P. Lonchamp and C. GBgauff,J . phys. radium 17, 132 (1956).

64

66

230

1.

PARTICLE DETECTION

measurements of electrons are rarely performed in emulsions. The trajectories of slow electrons are not straight, but, due to scattering phenomena, curved in a complicated way. The rectified length of electron tracks or the number of grains rendered developable in electron tracks of certain energy is of importance in problems of &ray emission and will be discussed in the section on particle charge (2.1.1.3). The residual range of particles of given mass, charge, and energy can be calculated if the emulsion composition and the differential stopping power of the constit,uent emulsion elements are known. This follows from the fact that the stopping power of a homogeneous mixture is equal to the sum of the contributions from each element. I n earlier emulsion experiments the differential stopping power relative to air was used because the a particles from radioactive elements were residual ranges-in air-of well-known quantities. The range-energy relation in emulsion was first calculated by Webbz3 for Eastman-Kodak emulsions. Similar calculations were later performed by Wilkins,62based on the experimental datasg-72 for the computation of the irlntegral stopping power of emulsions. These data were adjusted to provide a smooth curve of stopping power versus particle velocity. Wilkins calculated range-energy curves for protons and a particles with velocities up t o p = 0.31; the proton values, with the exception of the very low energy region, are in excellent agreement with the latest and most carefully determined data of Barkas. 7a Wilkins’ calculations are extremely useful in that they allow evaluation of the residual range of particles in emulsions of various compositions-in th e so-called “loaded emulsions,” and in emulsions which contain a higher percentage of water as a result of humidity conditions. A more direct approach to the calculation of the residual range is the evahation of the integral of the reciprocal of the energy loss per unit length which a n article suffers along its trajectory through matter.

(1.7.6) The energy loss, * d E / d x is a function of the mean excitation potential I, of the elements of the stopping material and of the charge z, and velocity B = v / c of the ionizing particle. The following equation shows th a t the

* See Eq. (1.7.13) Section 1.7.6. M. S. Livingston and H. A. Bethe, Revs. Modem Phys. 9, 245 (1937). J. D. Hirschfelder and J. L. Magee, Phys. Rev. 73, 207 (1948). ’1 R. Warshaw, Phys. Rev. 76, 1759 (1949). 7 1 E. L. Kelly, Phye. Rev. 76, 1006 (1949). 73 W. H. Barkas, Nuovo cimento [lo] 8, 201 (1958). 88

To

1.7.

PHOTOGRAPHIC

EMULSIONS

231

energy loss does not depend on the mass mi of the particle, provided that P / m c < mi/m, where p is its monentum and m the electron mass.

Here N is the number of stopping atoms per unit volume of atomic number Z and Ci is a correction term for non-participating electrons in the i t h shell of the atoms. The Bethe-Bloch equation (1.7.7) is valid for heavy particles with velocities greater than approximately 1.5 X lo9 cm/sec. The factor ZCi becomes negligible if the particle velocity is well above the orbital velocity of K electrons. At still higher velocities a different correction term must be introduced into the Bethe-Bloch equation which represents the reduction of energy loss arising from the polarization of the medium. This polarization effect or the “density effect,”* so called because of its evidence in dense media, was first treated mathematically by Fermi.74 The density correction term (6), which must be added to the energy loss equation, depends both on the particle velocity and the mean excitation potential of the medium. This effect will be discussed in greater detail in the section on grain density and energy loss. The energy loss, and hence the residual rangeof the particle [Ey. (1.7.6)], can be determined in the region of validity of Eq. (1.7.7) if I and Ci are known. However, until recently, because these quantities were not known directly from experiments, appreciable uncertainties existed in the rangeenergy relation. It is generally quite difficult t o perform very accurate ionization potential measurements. The magnitude of ionization potentials I, may be obtained from energy loss and range measurements on particles with known momentum; however, in both cases In I , and not I, enters into the equation. Therefore, extremely accurate measurements are required in order to determine the value of I and to decide about the velocity dependence of the ionization potential. Another difficulty arises from the dependence of dE/dx on the electron density of the medium, and therefore on the density and composition of the emulsion. A number of a ~ t h o r s 7 ~ - have 7 ~ attempted to represent the rangeenergy relation empirically by power law equations which, however, are valid only in a restricted energy region. A semiempirical relation, * Refer to Section 1.7.6. E. Fermi, Phys. Rev. 67, 485 (1940). U. Camerini and C. M. G. Lattes, Ilford technical data (Ilford Research Lab., London, 1948). 76H. Bradner, F. M. Smith, W. H. Barkas, and A. S. Bishop, Phys. Rev. 77, 462 (1950). 77 W. M. Gibson, D. J. Prowse, and J. Rotblat, Nature 173, 1180 (1954). 78 H.Fay, K. Gottstein, and K. Hain, Nuovo cimento [91 11,Suppl. No.2,234 (1954). 74 7b

232

1.

PARTICLE DETECTION

R E 2 = K . Ral * RPb, for high-energy particles has been proposed.79 This equation compares the range in emulsions, RE, with ranges in aluminum and lead, as determined from absorption experiments where K is a slowly varying constant. The first theoretical range-energy curve was calculated by VignerongQ for particles with velocities p 5 0.3 in dry emulsions of density 3.815 gm/cm3, employing a constant value of I = 332 ev. The curve is based on the most reliable experimental data known a t that time, and extends down to proton energies of 0.1 MeV. The low-energy portion of the curve is based on measurements b y Mano.81 Baroni et aLsz have calculated the range-energy relation for energies up to several Bev, taking into account the density effect by utilizing Sternheimer’se3 equation. The emulsion density used in deriving this curve is taken t o be 3.92 gm/cm3. Barkas and Youngs4 have also extended Vigneron’s curve t o higher energies by calculating the ranges from the Bethe-Bloch equation, in which they used an ionization potential of 331 ev and a n electron density corresponding to an emulsion density of 3.815 gm/cm3, the same value used by Vigneron. The calculated values for high energy protons are based on Sternheimer’s work. I n a more recent paper, B a r k a refines ~ ~ ~ the approach to this problem by performing much more detailed calculations. The correction factor C, in the Bethe-Bloch equation is evaluated for the K and L shells of all emulsion nuclei, except hydrogen, by using Walske’sss calculations. Sternheimer’s expression for the density effect correction is used at very high energies; the mean excitation potential, which enters this correction, is determined experimentally. The value of the mean ionization potential can, in principle, be obtained from a single accurate measurement of the range in a n emulsion of known composition (humidity content) and shrinkage factor on a particle of welldefined momentum. The energy of the particle must lie below the value where the density effect becomes noticeable, in order to utilize the simplest form of the Bethe-Bloch equation. Once I has been determined, however, the exact ranges or higher energy particles may be employed for the direct evaluation of the density effect correction.

R. R. Daniel, G. G. George, and B. Peters, Proc. Indian Acud. Sci. A41,45 (1055). 8 o L .Vigneron, J. phys. radium 14, 145 (1953); Compt. rend. 232, 1199 (1951). M.G. Mano, Compt. rend. 197, 1759 (1933); Ann. phys. [ l l ] 1, 407 (1934). 8 2 G. Baroni, C. Castagnoli, G. Cortini, C. Franrinetti, and A. Manfredini, Bureau of Standards, Bull. No. 9, CERN, Geneva (1956). 81 R. M. Sternheimer, Phgs. Rev. 103, 511 (1956). 81 W. H. Barkas and D. M. Young, U. C . R. L. 2579 (1954). 8 b M. C. Walske, Phys. Rev. 88, 1283 (1952), 101, 940 (1956). 70

1.7,


233

A mean value of the excitation potential of 331 zk 6 ev has been established from several very exact range determinations. If one uses the Bloch relations6 I = kz, and substitutes for Z the mean atomic number of the emulsion elements, one obtains the value 12.25 -t 0.22 ev for k. The latt,er value is in satisfactory agreement with the measurement of Bichsel et aZ.,s7 who find that k varies between 12.5 and 13.1 in absorption measurements of low-energy protons in various elements. The above value of k is definitely greater than that of 9.1 which was found from the experiments of Mather and Segr&8s The excellent agreement between calculated and measured values presents a strong argument in favor of adopting a mean excitation potential near 331 ev; furthermore, it seems to strengthen Caldwell’s*9 assumption concerning the constaiicy of I . The calculated values of I have ~ ~ with ~ experimental not only been compared with the work of B a r k a but data from other laboratories as well. The calculated range, (602 2.2)p, of the p meson in the T-P decay is in excellent agreement with the Berkeley results and with measurements on the G stack.g0The ranges of mesons from the decay of K,, mesons agree with Barkas-Birge tables91 to within an experimental error in the determination of the K meson mass. The measurements of Heinzg2on 342 Mev protons are about 1.5% lower, those of Friedlander el aL93on protons of 87, 118, and 146 Mev are lower by about 1%,and of De Carvalho and Friedman94 on 208 Mev protons are in good agreement with Barkas’ values. Figure 2 gives the residual range of protons as a function of kinetic energy in the emulsion; the curves are drawn according to the tabulated range data in Barkas’ paper.73 For energies below 1MeV, the ranges were calculated from the E3’2 relation of Geiger and Bohr. The range-energy relation due to Baroni et aZ.S2deviates from Barkas’ curve only at proton energies greater than 1 Rev, while the curve from Fay et aZ.,78which is based on a power law, exhibits considerable deviation a t proton energies as low as 150 MeV. Figure 3, taken from Barkas’ paper,73gives the percentage increase in F. Bloch, 2. Physik 81, 363 (1933). H. Bichsel, R. F. Morley, and W. A. Aron, Phys. Rev. 106, 1788 (1957). 88 R. L. Mather and E . SegrB, Phys. Rev. 84, 191 (1951). *9 D. 0. Caldwell, Phys. Rev. 100, 291 (1955). G. Stack Collaboration, Nuovo cimento 1101 2, 1063 (1955). $1 R. W. Birge, D. H. Perkins, J. R. Peterson, D. H. Stork, and M. N. Whitehead, Nuovo cimento [lo] 4, 834 (1956). *z 0. Heinr, Phys. Rev. 94, 1726 (1954). *3 M. W. Friedlander, D. Keefe, and M. G. X. Menon,Nuovocimento [lo] 6,461 (1957). *4 H. G. De Carvalho and J. I. Friedman, Rev. Sci. Instr. 26, 261(1955). 86

234

1. PARTICLE

DETECTION

FIG.2. Range-energy relation for protons in emulsions. The energy of protons in Mev is plotted versus the range in cm. (According to tabulated data by Barka~'~.)

-

2 -

I

I

I

I

I

I

I

1

I

FIG.3. The percentage increase of ionization potential causing a 1% increase in emulsion range is plotted versus particle velocity (after Barkas73).

1.7.

PHOTOGRAPHIC

235

EMULSIONS

the mean excitation potential which would cause a one per cent increase in emulsion range, as a function of particle velocity. Similarly, Fig. 4, also from Barkas’ work,66gives the relative decrease in range resulting from a one per cent increase in emulsion density, as a function of P ; these curves are drawn assuming that the ratio of water volume decrease to water I .o

I

0.9

-

0.8

-

0.7

-

0.6

-

0.5

-

0.4

-

0.3

I

I

I

I

I

I

I

I

I I l l

I I I l l

0.01

I

I

I

I

1

I

0.1

I

I

I

I

l

l

I I l l

I

B FIG.4. The percentage range decrease for one per cent increase in emulsion density is plotted versus particle velocity. The curves are calculated for 3 different assumptions about the ratio of the water volume to water weight decrease in emulsions (after Barkas et al.66).

-

weight decrease is A * * equal to unity, B * * equal t o 0.94, and C * equal to 0.84. It may be noted how critically the residual range is affected by density variations, especially a t high particle velocities. Fowler and Scharffg6propose a simple range-energy formula which is believed to be accurate to within 5% for ranges lying between 0.1 and 3000 gm/cm2:

--

E 96

=

+(R)[l . 1R

+ 25 z / R - 21.

(1.7.8)

P. H. Fowler and M. Scharff, cited by Friedlander et al., see reference 93.

236

1.

PARTICLE DETECTION

Here R is measured in gm/cm2 and the factor 9 is close to unity but varies somewhat with emulsion density. Another range-energy relation which is valid within a rather wide energy interval and which was used extensively in the early fifties, may be written in the form:

E

=

aZzn&fl--nRn.

(1.7.9)

1.7.5.3. Range Straggling.* The effect of range straggling on the range-energy relation, due to fluctuations in the rate of energy loss, must be considered if experimental range data is*to be interpreted correctly. Range straggling in homogeneous matter was first studied by B0hr.~6The problem was treated relativistically by Lindhard and Scharff,97 who showed that the exact straggling may be obtained from the equation:

where n is the electron density; E the kinetic energy, dE/dR, the mean the mean value of R2. rate of energy loss, R the mean range, and Lewisgs has applied several corrections to the nonrelativistic form of Bohr’s equation without appreciably altering the magnitude of the Bohr straggling effect in emulsion. Furthermore, the difference between the mean range and the most probable range found by Lewis to be caused by a slight skewness of the range distribution, is of negligible magnitude. Barkas et aLg9 have tabulated the percentage range straggling for protons in emulsions. Inasmuch as both quantities u = Z2ZB/mi1/2and r = Z2E/mi,where u is a measure of the range straggling and r is a measure of the residual range, depend only on the velocity of the particle, the percentage straggling 100a/r is also a function, albeit slowly varying, of the velocity alone. The quantity (100a/r) = (100mi1’2Z~/R),and hence the percentage straggling, does not depend on the charge of the particle, but varies inversely with the square root of its mass, since varies directly with the mass. According to Barkas, the percentage straggling is greater than 2% for very slow protons and about 1% for very fast protons. However, the actual range straggling in emulsion is greater than the Bohr effect alone, as a result of straggling due to distortion and inhomogeneity

0,

* Refer to Section 1.7.5.1. N. Bohr, Phil. Mag. [6] SO, 581 (1915). J. Lindhard and M. Scharff, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd. 27, No. 15 (1953). 96 91

98 99

H. W. Lewis, Phgs. Rev. 86,20 (1952). W.H.Barkas, F. M. Smith, and W. Birnbaurn, Phys. Rev. 98,605 (1955).

1.7. PHOTOGRAPHIC

EMULSIONS

237

of emulsion, reading errors, and end effects, as has been noted in previous sections. 1.7.5.4 Range-Energy Relation for Multiply Charged Particles.* Residual ranges of a particles of various energies were known even before those of protons, because the former were available a t well-defined energies, emitted from radioactive sources. These experiments are described in references 1-7b. The problem of a-particle ranges was also extensively studied by Wilkins,62 Cuer and Lonchamp,'O' and Neuendorffer et c z L 1 0 2 Wilkins determined the ratio of ranges of a particle and proton of equal velocities by means of a comparison with the like ratio in air (Livingston and Bethesg), and by baking into account the stopping powers of both media. He found t h a t the relation obeyed in emulsion by a particles is given by (1.7.11) where the range extension C = 1.5 N for CZ emulsions of density 3.92 gm/cm3. The excess of the range of multiply charged particles over the range of protons of the same velocity is caused by the occasional capture of orbital electrons from atoms of the traversed medium. This effect becomes important, when the ion velocity is equal or smaller than the orbital velocity of electrons. Consequently during a portion of its trajectory through the emulsion the effective charge of the a particle ZHe< 2. Thus the energy loss is not proportional to Z2,but is given by -dE/dR = Zf(P), wheref(P) depends only on the particle velocity. The effect of range extension becomes very noticeable for ions of still higher charge, since the ion velocity a t which electron pickup sets in increases with higher ion charge. It is usually assumed that the cross section for electron capture and electron loss become comparable when the ion velocity approaches the velocity of the most loosely bound electron, and that the capture cross section increases rapidly and hence the effective ion charge decreases when the ion velocity drops below this value. Knipp and TellerlO3 have calculated the ratio ionic t o nuclear charge for slow ions in gases, using range data, available at this time. The authors describe in detail the process of orbital electron capture, basing their calculations on the Thomas-Fermi charge distribution, and calculate the effective charge as a function of the ion velocity. I n these calculations

* Refer to Sections 1.1.3.2and

1.1.3.3.

P. Cuer and J. P. Lonchemp, Compt. rend. 232, 1824 (1951). 102 T. A. Neuendorffer, D. R. Inglis, and S. S. Hanna, Phys. Rev. 82, 79 (1951). 103 J. Knipp and E. Teller, Phys. Rev. 69, 659 (1941). 101

238

1.

PARTICLE DETECTION

enters a parameter, y, assumed to be constant, which has to be determined by comparison with experimentally determined range energy curves. A semitheoretical study of the range-energy relation of heavy ions in emulsions, based on the calculations of Knipp and Teller was made by WilkimS2The range-energy relation for heavy ions is given by

(1.7.1l a ) where the range extension C Z ( @is a function of velocity: C Z @ )is different for ions of different charge. Wilkins determined C Z @ )for various heavy ions in emulsions of density 3.92 gm/cm3 and compared the results with the experimental data of various authors. Since then a great number of range measurements of heavy ions, Li, Be, B, C, and N in emulsions have been made. The heavy ions used for the purpose of analysis originated from disintegrations of light emulsion nuclei or from boron atoms (boron-loaded emulsions), caused by irradiation with particles of well-defined energy. In other experiments carbon or nitrogen ions, recoiling in elastic collisions with monoenergetic beam protons, and finally magnetically analyzed ions, produced in the target of accelerators were used. The latter method was used by Barkaslo4who has measured the ranges of H and He isotopes and of Li8 and Be8-the last two being easily identified by their decay schemes (“hammer” track). The particles were emitted in the bombardment of a thin target by a! particles from the 184-inch cyclotron. The difference in range extension can be measured for particles of equal velocities. Barkas derives from Knipp and Teller’s work a general relation for the quantity BZ which B Z is thus a function is defined by the equation r = (Z2R/mi)- B z ; r of velocity only. He assumes the validity of the relation Bz = aZ3 and determines a from experimental data to a = 1.2 X cm. The range extension CZ is then given by CZ = Bz(mi/Z2). The value of C Z for hydrogen is just equal to BZ = 1.2 X cm and thus evidently a negligible quantity; CZ was determined for a particles Li7, Be9, and C12 ions to Cz = 0.8 p, Cs = 2.4 p, Cq = 4.2 p, and Ca = 8.5 p, where the Cz values of the heavier ions were found with reference to the range-energy relation of a! particles. A very similar procedure was previously used by Lonchamp,105 while in his later paper106range-energy relations of heavier ions are calculated with reference t o protons. There is furthermore an essential difference in the

+

W. H. Barkas, Phys. Rev. 89, 1019 (1953). J. P. Lonchamp, J. phys. radium 14, 89 (1953). lo8J. P. Lonchamp, J. phvs. radium 18,239 (1957). 104

106

1.7.

239


treatment of the relationship of effective charge versus ion velocity. The ratio between effective charge and ion velocity, y, which in the work of Knipp and Teller,’Wilkinsand Barkas was assumed to be constant, is now treated as a variable. This assumption is justified, since according to measurements of Reynolds et aZ.lo7in gases and Reynolds and Zuckerlo8 in emulsions, y decreases with increasing velocity. The experiments were performed with nitrogen ions in the energy interval between 4-28 MeV. The range-energy relation for various heavy ions in CZemulsions were investigated by Lonchamplo6 and the corresponding values for range extension were derived; Lonchamp’s values are in general somewhat lower than Barkas’ and Wilkins’ values, with exception of Cz, which was found to be Cz = 1.6 p instead of 0.8 p , given by Barkas. The later paper of Lonchamplo6is dedicated to the investigation of the range-energy relation of Li7 ions. The ions were emitted from the target of the 184-in. Berkeley cyclotron, and after being deviated by the cyclotron magnet, strike photographic emulsions, situated a t various distances from the target ; each position corresponds to a different radius of curvature and therefore different particle velocity. In a painstaking way the author compares the energy loss in trajectories of protons and Li7 ions, starting in each case from a point of known energy; the values of d E / d X were measured within very small energy intervals. Since the range energy relation for protons is known, one finds the energy loss of Li7 ions according to the relation (1.7.12)

provided that the velocity of protons and Li7 ions is the same. 2, in the above equation is equal to 1, a t least down to energies larger than 0.32 Mev.lo9In this way Lonchamp has determined the effective charge of Li7* ions for various energy values, from which in turn the range extension and thus the complete range-energy relation can be derived. Figure 5 gives the range-energy relation for Li7 ions, drawn according to the tabulated values in Lonchamp’s paper. From the data of Li7 the curves for Li8 and Lie ions can be obtained easily, considering the proportionality between mass and residual range for particles of equal charge and velocity. The author compares the calculated data with experimental values of various authors and especially with the more recent range measurements of Livesy,l10 which are in excellent agreement. Livesy discusses in detail the difficulties confronting any theoretical approach to the problem of H. L. Reynolds, J. W. Scott, and A. Zucker, Phys. Rev. 96, 671 (1954). H. L. Reynolds and A. Zucker, Phys. Rev. 96, 393 (1954). 109 T. Hall, Phys. Rev. 79, 504 (1950). 110 D. L. Livesy, Can. J . Phys. 34, 219 (1956). 107

108

240

1.

PARTICLE DETECTION

effective charge versus particle velocity. He proposes a semiempirical method in which the range-energy relation is approximated by a power law with coefficients determined from experimental data. However up to now the range energy relation in emulsions is exactly known only for He and Li ions, while the question of range extension in heavy ions is still unsettled. It is evident that the knowledge of CZvalues is especially important for slow ions, if energy determination from range measurements is needed. Problems of this kind arise in investigations of I00

I

0.1

FIG.5. Range-energy relation for Li7, Lie, and Lie particles. The energy E is given in Mev and the range in microns.

binding energies and studies of energy levels of disintegrating atoms and calculations of the total energy release in stars caused by the capture and subsequent decay of unstable particles. Fortunately several heavy ion accelerators were recently built and it is to hope that in the near future a great number of experimental data will become available, so that the problem of velocity dependence of effective ion charges can be solved in a general way.

1.7.6. Ionization Measurements in Emulsions The problem of energy loss in emulsions as a function of particle charge and velocity has been discussed in a previous section in connection with the range-energy relation in emulsions. The theory is baaed on the BetheBloch relation and modified for high-energy particles to include the

1.7.

241


relativistic deiisit y effect . Sternheimer 83, l s l l2 performed detailed calculations of the density effect in various materials, including nuclear emulsions. Inasmuch as the theory is fully treated in Section 1.1, here we will discuss only Sternheimer’s equation for fast particles in connection with the corresponding ionization parameters in the emulsion. This relation is :

( ) $, [In 2me2W0 Iz + In

g,=- 1 dE P

r i b wo

=

PZ 1 - 02

~

- 61.

(1.7.13)

In this equation, which is valid for singly charged particles heavier than electrons, the energy loss per gram cm-2 is represented as a function of particle velocity, p, and depends on the value of the mean excitation . latter represents the potential I , the polarization effect 6 , and W OThe maximum energy transfer, which contributes to the local ionization, limited to the silver halide crystals within the path of the particles; g, does not include the energy spent in the production of fast knock-on electrons (6 rays), whose range in silver bromide exceeds the mean grain diameter. The choice of W Ois somewhat arbitrary and estimated values between 2 and 30 kev can be found in various publications; m is the mass of the electron, and A is a constant defined by A = 4?me4/mc2p,where n is the number of electrons per cm3in the substance; thus A = 0.0698 MeV/ gm cm-2 for Ilford emulsions, and A = 0.0671 Mev/gm cm-2 for AgBr. The importance of the mean ionization potential I for energy loss calculations was already mentioned in the section dealing with the rangeenergy relations in emulsions. Sternheimer has chosen C a l d ~ e l l ’ svalue, ~~ I Ei‘ 132. The value of 6 in AgBr is given by 6 = 4.606 loglo(py)

- 5.95 + 0.0235[4 - logl~(pr)]4.03 (1.7.14a)

for all values of loglo(py) 0.30

< logio(P7) < 4

and 6 = 4.606 loglo(py) - 5.95

*

*

10glo(Pr) < 4.

(1.7.14b)

The constants in these equations were calculated by Sternheimer, taking into consideration mean values of excitation potentials of Ag and Br. According to Eqs. (1.7.13), (1.7.14a), and (1.7.14b) the ionization loss decreases slowly for high-energy particles until a minimum is reached a t 0 = 0.95. For still higher velocities, the second term in Eq. (1.7.13) rapidly increases so that a continuous increase in energy loss would be expected if it were not for the 6 term, which a t very high energies increases proportionally to loglo(0y). 111 111

R. M. Sternheimer, Phys. Rev. 88, 851 (1952). R. M. Sternheimer, Phys. Rev. 91, 256 (1953).

242

1.

PARTICLE DETECTION

It has been found that the grain density closely follows, within a wide energy range, the energy loss versus velocity curve. The grain density curve goes through a broad minimum, reaching the lowest point a t y = 4 and then increases slowly for still higher energies. The relativistic increase was first observed by Pickup and V o y ~ o d i c ; ~these ~ 3 authors found that the plateau value, which is reached for y = 50, is about 14% higher than the minimum grain density. Other authors110-118have more recently confirmed the realtivistic rise in grain density. However, there is still disagreement among the various authors about the ratio gPl,t,,u/g,i, as well as the rate of increase of grain density with y beyond the value y = 4, qplateaubeing the maximum value the grain density reaches for values of y > 4. The experimental results of Stiller and Shapiro"' and Fleming and LordlZ1seem to be in good agreement with Sternheimer's calculations. Alexander and Johnston122have determined the rate of plateau to minimum grain density to be 1.133; the error in these measurements is estimated to be less than 1 %. Actually this investigation was not based on grain densities but on blob densities; the relation between these two parameters will be discussed in the next paragraph. The authors, furthermore, determine from the ratio of blob densities Bpiateau/Bmio the parameters I and Wowhere Bplatellu is the maximum blob density beyond y = 4 and I and Wo have the same meaning as in Eq. (1.7.13). This can be done because the energy loss at the plateau value does not depend on Eo/I but only on Wo =

A[ln 2mc2W0 - 21n(hvp)]

(1.7.15)

where v p is the plasma frequency of the mean emulsion nucleus,l13m the mass of the electron, and A the constant defined in Eq. (1.7.13). On the other hand In(2mc2Wo/12)can be calculated from (1.7.16) (1.7.16) E. Pickup and L. Voyvodic, Phys. Rev. 80, 89 (1950). A. H. Morrish, Phil. Mag. 171 43, 555 (1952). us A. H. Morrish, Phys. Rev. 91, 425 (1953). llE M. M. Shapiro and B. Stiller, Phys. Rev. 87, 682 (1952). 117 B. Stiller and M. M. Shapiro, Phys. Rev. 171 92, 735 (1953). 11* R. R. Daniel, J. H. Davies, J. H. Mulvey, and D. H. Perkins, Phil. Mag. [7] 43, 753 (1952). ' 1 9 M. Danysz, W. 0. Lock, and G. Yekutieli, Nature 169, 364 (1952). 120 R. P. Michaelis and C. E . Violet, Phys. Rev. 90, 723 (1953). 121 J. R. Fleming and J. J. Lord, Phys. Rev. 92, 511 (1953). 122 G. Alexander and R. H. W. Johnston, Nuovo n'mento [lo] 6, 363 (1957). 118 114

1.7.


243

where j3 is the particle velocity (determined from the residual range) for which minimum ionization is observed. In these investigations ?r and p mesons from K,, and K,, decays were used. The authors find for W othe value (2.9 & 0.5) X lo4 ev and for the mean excitation potential I = 12.92; the latter value is in good agreement with Caldwell’s value but differs somewhat from the value found by B a r k a ~ ’for ~ particles of lower energy. The above value of W Oseems high, considering that an electron ejected with this energy produces an easily visible &ray track in the emulsion. Before analyzing the theoretical basis of various ionization parameters, it is well to discuss experimental det,ails and methods. The measurement of grain density requires oil immersion objectives of 90-1OOx magnification and eyepieces with carefully calibrated micrometer scales. One counts the number of grains lying within given scale intervals, being careful to choose track segments which lie near the center of the field of view. When the trajectory is inclined to the horizontal plane, the true grain density is found by multiplying the measured value by cos a,where cr is the dip angle of the track in the emulsion before development. It is generally assumed that there exists a simple relationship between the probability of activating an emulsion crystal and the ionization loss of the particle traversing the emulsion. This relationship can be represented, assuming a (Poissonian) distribution law, P = 1 - exp( - p ~=) 1 - exp( - y) (1.7.17) where P = n,/nt is the probability of rendering developable nu crystals out of a total number nt crystals. In Eq. (1.7.17) y is given by y = qv2 where q is a parameter which depends only on development conditions, 2 is the mean path length of the ionizing particle through the crystal, and v is the number of ionization acts per crystal; the latter must be identified with the restricted ionization loss, since, otherwise (including 6 rays), would depend on the velocity of the particle activating the grain. Equation (1.7.17) assumes that all crystals have equal size and that their centers are aligned. If one assumes with Demers2 that the crystals are spheres of equal size, distributed at random about the trajectory of the particle then

] exp (:)- - .

(1.7.18)

Fowler and PerkinsIzapropose a more general approach accounting for fluctuations in crystal sizes; they assume random distribution of crystals 1zSP.

H. Fowler and D. H. Perkins, Phil. Mag. [7] 46, 587 (1955).

244

1.

PARTICLE DETECTION

and a distribution of crystal sizes in which all diameters between 0 and 22 are equally probable. P is then given by 1

+ (I + y)exp(-zy)

].

(1.7.19)

Because of the wide distribution chosen (the distribution of crystal sizes is sharper) Eq. (1.7.18) can accommodate variations in the parameter q, by expressing y = f[(QZ)v].

FIQ.6. A plot of the relations between mean development probability and restricted ionization loss. The curves refer to Eqs. (1.7.17), (1.7.18), and (1.7.19) in the text. The experimental points are: o experiment A; experiment B; 0experimental values which have been corrected for the apparent loss of grains in clusters; X track with dip angle 40", A tracks with dip angle of 30", calculated according to Eq. (1.7.43a).

+

In Fig. 6, the values of defined by Eqs. (1.7.17), (1.7.18), and (1.7.19) are plotted as functions of y. (The experimental points in this figure will be discussed later.) The three curves are nearly identical €or small values of y, but differ considerably for larger values. From grain density measurements in tracks of lightly ionizing particles it is known that values of normalized grain density (g* = g/go) as a function of ( S / g o ) ? can be easily fitted to the initial part of the 3 curves. However, thus far, it has not been possible to decide which one of the 3 curves, or if indeed any one of them, does truly represent g* = f[(g/go),] for g* 2 6. The reason for this failure is due to difficulties of performing grain density measurements in dense tracks.

1.7.


245

Grain counting is a simple procedure in tracks of very fast particles, especially if the development is light. I n this case the grains are well separated from each other and clogging of grains occurs only occasionally. However, in denser tracks cluster formation is quite frequent and it becomes difficult t o resolve the clusters into single grains. The number of grains per cluster is either estimated b y the observer, or determined by the application of length criteria based on the mean grain diameter of developed grains, or finally by investigating the diffraction patterns of grain clusters. However, all these methods are subjective and tedious, leading t o the belief that the grain counting method should be replaced by other more objective procedures.

1.7.7. Ionization Parameters: Blob Density, G a p Density, Mean G a p length, and Total G a p Length

At present the general practice is to measure “blob” density instead of grain density, a blob being a single grain as well as a cluster of grains. An advantage of the blob counting method is that it leads directly to another ionization parameter, the gap ” density, where a gap is the blank space between blobs. However, it is necessary to emphasize that the grain density, or the number of silver crystals per unit length, made developable by the traversing particle, and not the blob density, is directly related t o the ionization loss. The blob density depends upon the spacing of crystals in the emulsion and upon the size and configuration of the developed grains, these being influenced by the processing conditions. As in the case of grain density, one introduces, instead of blob density 3,the normalized value 3*,which refers t o the ratio of actual blob densities t o the respective value at minimum ionization, or more often to blob densities a t the plateau value. It has been found th a t for low ionization densities-singly charged particles of high energy-the normalized blob densities are reasonably independent of development. Therefore, by exposing emulsions to particles in this energy region one can obtain calibration values of B”, which are valid also in emulsions of a different batch or which were processed under slightly different conditions. However, for higher ionization densities the independence of B values of development conditions ceases to be valid, and blob density, therefore, loses its value as an ionization parameter. There is still another reason why blob density measurements are not meaningful in the region of higher ionization density. The blob density increases with increasing ionization loss up to a certain maximum value and then decreases slowly for still higher ionization values; thus in a certain energy interval, blob density is a bivalent function of energy loss.

246

1.

PARTICLE DETECTION

H o d g s ~ nrecommended ~~~ gap length, the total blank space between blobs per unit length, as a useful parameter in ionization measurements in tracks with blob densities larger than 2 times plateau value. Shortly afterward Renardier and AvignonlZ6modified this technique by counting, instead of measuring, the total number of gaps per unit length. Since that time gap measurements in a variety of forms have played a n important role in nuclear emulsion problems. Gap measurements are now generally used instead of, or in combination with, blob density measurements if the ionization exceeds 2 times the minimum value or when light but steeply dipping tracks have to be measured. The following parameters are used in ionization measurements : (a) blob density B , the number of blobs per unit length; (b) H , the number of gaps per unit length; H is, of course, equal to B, the number of blobs per unit length; (c) L H , gap length or the total width of gaps per unit length; (d) A, mean gap length, defined as the mean distance between the inside edges of neighbor grains. The last parameter was proposed by O'Ceallaigh. l Z 6 Ritson127 has devised a very convenient method for gap measurements : The track is aligned in the microscope with one of the two stage movements which in turn is driven by a motor a t a low constant speed. The observer is provided with two counters, driven by a single pulser. One of the counters runs continuously with the stage movement, while the other one is activated only when the observer presses a button which is done whenever the hairline in the eyepiece lies over a region of track unoccupied by grains. The ratio of the two counter readings gives directly the total gap length in the track under investigation or the gap length LHper unit length if counter readings are made at certain predetermined time intervals. Baroni and CastagnolilZ8describe a similar arrangement, using a motor-driven stage moving with constant velocity. The authors add several improvements, the most important being a n RC circuit connected with a counting device which enumerates only those gaps which are longer than a certain predetermined minimum value. Therefore, one can separately record, with this apparatus, the total number of gaps, the number of gaps longer than a minimum value I, the total width of gaps P. E. Hodgson, Phil. Mag. [7] 41, 725 (1950). M. Renardier and Y. Avignon, Compt. rend. 233, 393 (1951). IaSC. O'Ceallaigh, Proc. Intern. Union Pure and Appl. Phys. Conf. on Cosmic Radiation, BagnBres, France, 1953 (unpublished). l27 D. M. Ritson, Phys. Rev. 91, 1572 (1953). l28 G. Baroni and C . Castagnoli, Nuovo cimento [9] 12, Suppl. No. 2, p. 364 (1954). Iz4

'26

1.7.


247

per unit length and the total width of gaps surpassing a minimum length. The introduction of a minimum gap length is connected with the findings of various authors th at the inclusion of gaps below a certain minimum length in gap measurements makes the measurements unnecessarily tedious and dependent on observational errors and optical conditions (resolution). Therefore, only those gaps are included whose widths exceed a certain minimum value 1 which should be large enough for accurate and rapid measurements, but small enough to prevent loss of information, especially in tracks of high ionization density. The choice of 1 depends upon the problem, the length, density, and dip angle of the track. The corresponding parameters referring to data above a minimum value 1 will be denoted in the following expressions: H(1), X(I), and A H @ ) For . tracks inclined t o the emulsion surface the optimum value of gap length 1 is given by 1 = I’ sec a,where (Y is the dip angle and 1‘ the minimum value corresponding t o a track with equal gap density, but negligible dip angle. I n the following paragraphs we will discuss the relationships which exist among blob density, total gap length, mean gap length, and grain density. Grain density, as pointed out earlier, is a direct measure of the energy loss of the traversing particle; it depends, of course, also upon the processing conditions; however, the dependence is of such a nature th a t the normalized value g* becomes independent of development conditions and so represents, for a given emulsion type, the true ionization parameter. The normalized blob density B* is, as stated before, also reasonably independent of development in tracks of low ionization density; this is due t o the fact th at only a small percentage of grains will coalesce to greater complexes. However, for higher ionization densities, the process of cluster formation becomes more and more important and the degree of coalescence depends on the strength of development which determines the final size t o which the crystals grow during processing. The relationships between ionization loss and blob density or any other parameter connected with gap measurement, are much more complicated than in the case of grain density. An analytic expression for the former relationship must be based on a theory of track formation in the emulsion. Various a~thors123~12~-1a7 have worked on this problem and several models describC. O’Ceallaigh, Bureau of Standards Document No. 11, CERN, Geneva (1954). M. G. K. Menon and C. O’Ceallaigh, Proc. Roy. Soc. A221,292 (1954). 131 R. H. W. Johnston and C. O’Ceallaigh, Phil. Mag. [7] 46, 424 (1954). 132 M. Della Corte, M. Ramat, and L. Ronchi, Nuovo cimento [9] 10, 509, 958 (1954). 133 M. Della Corte, Nuovo cimento [9] 12, 28 (1954). 134 M. W. Happ, T. E. Hull, and A. H. Morrish, Can. J . Phys. SO, 669 (1952). 1 3 5 A. J. Hem and G. Davis, Australian J . Phys. 8, 129 (1955). 136 J. M. Blatt, Australian J . Phys. 8, 248 (1955). 137 C. Castagnoli, G. Cortini, and A. Manfredini, Nucwo cimento [lo] 2, 301 (1955). 139 130

248

1.

PARTICLE DETECTION

ing this process were proposed. The essential difference between these models lies in the treatment of spatial distribution of silver crystals. O’Ceallaigh’s work126~12p-1s1is based on the assumption, shared by Happ et al.,la4 that there exists no correlation between the position of silver crystals in the emulsion, while Herz and Davis136visualize the emulsion as a lattice with silver crystals spaced a t regular intervals; other authors use somewhat modified models supporting their assumptions by comparison with experimental data. BlattL3‘jand Fowler and PerkinslZ3discuss in great detail the theory of track formation and the importance of the theoretical assumptions for the practical use of emulsions in ionization measurements. In comparing their theories with experimental data the authors arrive at different conclusions, Blatt supporting a modified constant spacing model, while according to Fowler and Perkins’ findings, agreement between experiment and theory can be obtained with a variable spacing model; the authors introduce certain refinements in the earlier theory which are connected with fluctuations occurring in the distribution of crystal positions, size, and developability. For low ionization densities all theories give satisfactory agreement with experiments, due to the fact that the blob separation in lightly ionized tracks is not greatly influenced by any type of fluctuation, and exceeds greatly such distance8 as required in the constant spacing model. We will discuss here first O’Ceallaigh’s work based on the simplest model. O’Cealla.igh126~12v and Menon and O’Ceallaigh130have shown that the distribution in the length of gaps along tracks is exponential and given by (1.7.20)

where h is the mean gap length. The connection between mean gap length A, grain density g, and total gap length LH,is given by the equation 1

-

9

- a!= x and LH

=

1 - ag

(1.7.21)

where (Y is the mean diameter of the silver halide crystal.’30 During development the grain increases in size and reaches finally a diameter which is about twice the size of the original value. Therefore, the free space between grains becomes smaller, and some free spaces may disappear completely while the grains coalesce and form larger complexes. However, because of the exponential distribution, the homogeneous growth of all grains does not change the mean gap length A; the latter is, therefore, a parameter which is largely independent of the diameter of the developed grains and hence of developing conditions. The model is based

249

1.7. PHOTOGRAPHIC EMULSIONS

on the assumption of a random distribution of crystal centers along the track. The blob density B which is equal to the number of gaps H ( 0 ) can then be written B

=

gexp(--)

(1.7.22)

where K is a constant, depending strongly on developing conditions, must be determined for each stack and it is related to the crystal diameter, a, by the relation K = d - a e, where d is the diameter of the developed grain and e is the smallest distance between grains which can be resolved microscopically. K can be found by the simultaneous measurement of B and X in track segments of constant ionization. The expression exp( - K/X) represents the conversion factor for grains into blobs, which for constant energy loss depends only upon developing conditions. Alexander and JohnstonlZ2develop a relationship which allows the conversion of normalized blob densities from one stack to another (different development); however, this relation is limited to tracks of low densities for reasons previously mentioned. In this case one can replace B* = (B/BO) by

+

B*

= g*

exp[-K(g - go)]

= g*

exp[-K(B

- Bo)]

(1.7.23)

where go and Bo refer to the values of grain and bIob density in the plateau or minimum ionization region. If now K1 and K z are the developing constants for two different stacks, it can be easily seen that the two blob densities B1* and Bz* are related by B1* = Bz* exp([Kz(Bz - B231

- [K1(B1 - BIO)]). (1.7.24)

The procedure of finding the mean gap length can be greatly simplified by using the method proposed by Johnston and O’Cealleaigh.’31 The authors propose to count the number of gaps H(I1) and H(E2),exceeding two predetermined lengths l1 and l z ; these data are related to each other and the mean gap length by (1.7.25) With a set of 4 t o 5 different values of I , the mean gap length X can be quite accurately determined. Such measurements, performed with conventional methods (microscope and eyepiece scale), ar’e lengthy and time consuming. This difficulty can be removed by the use of “gap analyzers” ;

250

1.

PARTICLE DETECTION

recently a number of such devices have been c o n ~ t r u c t e d . ~ ~ *The -'~~ principle of such gap analyzers is a system of counter circuits, which are actuated by the closing of a microswitch; the microswitch is closed by a number of equidistant contacts on a motor-driven disk; if the system is actuated by photoelectric means, then the contacts are replaced by holes in the disk. I n this way the gap lengths are chopped into increments which are determined by the constants of the quantizer and the gaps are measured in subunits of time, in a time dependent circuit, or in subunits of length if the stage and disk are driven by synchro motors. Another method for the measurement of X was proposed by Fowler and perk in^,'^^ in which blob measurements are combined with the measurements of gaps exceeding a predetermined length I

H ( I ) = B exp

(-

$a

(1.7.26)

This method has the advantage, that i t can be performed with conventional methods, fixed eyepiece scale and manually driven stage, especially if a conveniently large value of I is chosen; however, one is likely t o lose information if E is too large since, especially in denser tracks, the number of small gaps is large and thus important for the measurement. The authors have calculated the optimum gap length A, which, of course, depends on the track density. They found a relatively broad optimum region between 1.5X < 1 < 2.5X. A great advantage of mean gap length measurements by (1.7.25) or (1.7.26) is, th at these methods can be easily adapted for dipping tracks. Mean gap length measurements, instead of blob measurements alone, should be made, even in very light tracks, if the ~ - ' ~measured ~ distance I dip angle exceeds 15" (developed e m ~ l s i o n ) . * ~The in (1.7.25) or (1.7.26) is the projected length and has t o be replaced by It which is related t o I and dip angle a b y It =

I sec a

+ d(sec a - 1)

(1.7.27)

where d is grain diameter and d(sec a - 1) accounts for the obscuration due to the apparent increase of blob sizes. The "smooth model" by O'Cealleigh neglects any type of correlation, which may exist among the positions of silver crystals in the emulsion. I n la* J. E. Hopper and M. Scharff, Bureau of Standards Document No. 12, CERN, Geneva (1954). la8 K. Enstein, Electronic Eng. 29, 277 (1957). 140 A. DeMarco, R. Sanna, and G . Tomasini, Nuovo cimento [8]9, 524 (1928). 141 S. C. Bloch, Rev. Sci. Instr. 29, 789 (1958). 144 M. Della Corte, Nuovo cimento [lo] 12, 28 (1959). 148 H. Winzeler, Nuovo cimento [lo] 4, Suppl., p. 259 (1956). 14( R. C. Kumar, Nuovo cimento [lo] 6, 757 (1957).

1.7.

251


fact, such a correlation may be of no consequence in light tracks with widely separated grains; however, it will become important in dense tracks. Fowler and PerkinsIz3 assume that the gaps, found in very dense (saturated) tracks, can be identified with the gelatine gaps existing between neighboring crystals. Furthermore, they assume (based on experiments) that the gap length distribution is exponential, but cuts off sharply a t a distance-lco-equal to the mean crystal diameter. The distribution function of gaps between developed grains is found in the following way. If u is the distance between the grain centers of two grains, bounding a gap of n undeveloped crystals, then the frequency with which a gap can ) p2(1 p ) n is appear will depend on the product a(n)F,(u)du. ~ ( n = the probability that one grain is developed and followed by an undeveloped and finally another developed grain. The function Fn(u) is defined by

-

Wn n!

Fn(u) = - exp(-W)

(1.7.28)

+

where W = (u/ko) - (u 1)2 is the path length through gelatine, expressed in units of lco. The total differential distribution, or the number of gaps per unit length between u and u du, is obtained by summation over all values of n, and is given by

+

The calculation of the integral distribution is quite complicated, because the sum appearing in (1.7.29) cannot be simplified. Numerical evaluation shows for small grain separations a certain degree of roughness in the distribution curve; which strongly depends on the choice of the numerical values for LO and 2 . However, for grain separations (center to center) greater than about 0.8 p and reasonable assumption for the values of Lo and 2, the curves appear smooth. Fowler and Perkins assumed LO = 2 = 0.2 p. I n general it will be possible to describe the total integral gap length by (1.7.30)

which is a n asymptotic approximation of the distribution function for u -+ a.I n this equation X is defined by (1

- $)exp(-

:)

=

1 - F.

(1.7.31)

252

1.

PARTICLE DETECTION

In one assumes that the asymptotic distribution form is also valid for the calculation of blob densities and if u’ is the distance between neighboring crystal centers, then B is given by

B

=

+(u’P) = -exp X

5k03 (- ): [ + 4l l k8o +~m3 -

1

*

*

*

1

(1.7.32)

where the series in the squared parenthesis converges rapidly and reaches a maximum value for ( k o / X ) = 1. Thus far we have neglected to take into account fluctuations in the size of the developed grains in the distribution function. The existence of such fluctuations can be easily observed; Fowler and Perkins showed that the distribution function is Gaussian and determined the standard deviation t o be 0.14 p or 20%, with a mean grain diameter of 0.7 p. They then calculated a correction factor C = [ l f (p2/4X2)] by which both blob and gap densities have t o be multiplied in order to account for fluctuations in the size of the developed grain. It is easily seen that this correction factor is negligible for light tracks. Fowler and perk in^'^^ have plotted 1 / X as a function of residual range for proton and pion tracks in two sets of emulsions which were developed in different ways. The curves for both emulsions coincide over a wide range of energy, down to about 100 Mev (protons) ; but from there on the curves diverge considerably and hence A * in this region cannot be considered as a parameter which is independent of development conditions; A * = X/Xu, where Xo is the mean gap length found in tracks of high-energy electrons; X* was determined from blob density in the region of small energy loss and in the denser region from blob and gap measurements [Eq. (7.1.26)]. The constant spacing model of Herz and Davis (H-D) is based on the following assumptions: the crystals in the emulsion are arranged in a definite lattice, so that each silver crystal can be assumed to lie within a cell of certain small dimensions. Therefore, there exists a strong correlation between crystal positions, which will affect the results in the dense region of the track; however, in light tracks where grain spacing is usually many times the length of the small unit cells, the correlation will cease to be effective and the (H-D) model leads to the same results as the two previously discussed models. For very high densities the gap length will not reduce to zero, but will assume a constant value which is determined by the original lattice spacing and the dimensions of the developed grain. The grain size is assumed to have a constant value (grain size fluctuations are neglected) and the model has been calculated in “linear approximation” (the crystal centers are assumed to be aligned along the path of the particle). The intrinsic difference in the calculations of the

1.7.

PHOTOQRAPHIC EMULSIONS

253

various parameters of the (H-D) model in comparison with other models stems from the assumption of a discrete gap length distribution. I n the following we call b the length of the unit cell containing one crystal and d the diameter of the developed grain; the zero gap length is 1) - d, where r is the largest integer less than then given by Go = b(I’ d/b, and it is the integral part of a factor determining the growth of the crystal during development. The size of the next gap (first-order gap) will 2) - d and the probability of finding a n n t h order gap is be G = b(I’ One can now calculate the integral given by ~ ( n=) P z ( l - P)(r+n-l). gap length distribution and the various track parameters exactly as in the case of other models. Although the problem was calculated rigorously by Blatt,136we indicate here only the results in the simplified form used ’ following equations (1.7.33-1.7.35) refer to by Castagnoli et ~ 1 . l ~The blob density B = H(O), the number of gaps larger than 1 per unit length H(Z), and, the mean gap length X :

+

+

(1.7.33) (1.7.34) where

rl is the integral part of

(d

+ Z)/b and (1.7.35)

The expression for X in (32) becomes identical with O’Cealleigh’s value for P > b. The total gap length LH per unit length is equal t o the product B X X. Blatt has compared the O’Ceallaigh and (H-D) models and various other slightly modified models with experimental data, and has found good model. Castagnoli et al. and agreement with the predictions of the (H-D) model more recently O’Brien, 146 based their investigation of the (H-D) on a large number of experimental data. Castagnoli et al. measured about 100 particle tracks in an energy region from 600 Mev (protons) down to very small energies using the gap counting machine, previously mentioned. Equations (1.7.33)-(1.7.35) show that the various parameters are related to each other through the probability P . I n order t o calculate P one has to know the mean grain diameter d, which can be found by direct measurements and b and r. The unit cell is not strictly defined; it is at least the size of the mean crystal diameter and can probably be identified X where 8 is the projected mean distance between centers with 8 = ko of successive crystals as defined by Fowler and Perkins. 12* Castagnoli 146 B. T.O’Brien, Nuovo cimento [lo] 7, 147 (1958).

+

254

1. PARTICLE

DETECTION

a

et aZ. define b by b = 1/0.455($a), where is the mean crystal diameter and 0.455 the fraction of emulsion volume occupied by silver bromide. Finally r, which by definition is an integer number, can for experimental reasons assume only values between 1 and 3. Although neither b nor I’ are exactly known, one can determine these magnitudes from the following considerations. Only certain pairs of values of b and r are compatible with the observed grain diameter and the crystal diameter known to be between 0.2-0.3 p. The best combination of b and I? can be found by comparing calculated and observed values at the maximum of the B versus P curve, since B,, = ( l / b ) [ l / ( l r)][l- 1/(1 I?)]’ depends only on b and y . The value of r, of course, is development dependent, and must be determined for each set of emulsions. With the values of r and b found at the blob density maximum, B versus X curves were calculated and compared with experimental data. The authors found good agreement between experimental and calculated curves within a wide region of ionization, starting with near minimum densities down to values of about 8 times minimum value. O’Brien145tested the (H-D) model for relativistic heavy primaries in G-5 emulsions which were very lightly developed; he found good agreement for light and medium tracks, but disagreement for very dense tracks (relativistic Fe nuclei). ~ 7 a) plot of the measuring parameters Figure 7 (Castagnoli et ~ 1 . ~ is B, A, LH,and H(Z) per 200 track length versus R / m = r, where m is the electron mass and R the residual range measured in microns; the range of a particle with a mass mi times the electron mass is then given by r X mi. It can be seen that for dense tracks (small values of r ) the slope of the LH curve is steep (contrary to B and X curves) and therefore, in this region, L H is sensitive to changes in ionization loss. Furthermore, it has been found, that LH in the dense region is less subject to personal errors than other parameters, since the occasional omission of small gaps in the determination of the total gap length does not seriously affect the results. In the light region the slope of the Lcr curve decreases and LH ceases to be sensitive to changes in ionization loss; in this region the best parameters are R and A, the latter having the additional advantage of being independent of development conditions. The role of the parameter H(1) (the curve refers to 1 = 0.8 p ) has been discussed before. It must be emphasized that in the (H-D) model the parameter X is not, as in the (F-P) model, derived from the slope of the gap distribution curve. but is found from the ratio L H / B . The experimental values should be identical in both cases if the gap length distribution were truly exponential However, experiments show that this is only approximately true for light tracks, but certainly not for medium and heavy tracks. The discrepancy

+

+

1.7.


255

between the two X values is quite serious, because Fowler and Perkins (and subsequently many other authors) express the ionization loss in tracks by the parameter l/X*. In general, the value 1 / X derived from the slope of the gap distribution curve is larger than the value found by L H / B ,and consequently the values of the probability derived from Eqs. (1.7.31) and (1.7.35) will be quite different when tracks of medium or large densities are considered.

200 100

50

1

1 -

-

I

0/200p

-

20 10

5.0 H(l)/200p

0.5 2.0 1.0

_/_-------

-

k(microns)

0.2

-

The situation is rather complicated for the following reasons. The ionization parameters proposed thus far are applicable only within certain density intervals, and furthermore, their mutual interdependence as well as their relationship to are based on assumptions (models) and emulsion parameters [Eqs. (1.7.31) and (1.7.35)] which cannot be verified in a simple experimental way. While the exact values of these emulsion parameters are of no consequence in light tracks, they are of great importance for tracks of medium and large density. Thus the problem of

256

1.

PARTICLE DETECTION

ionization measurements in the latter regions has not yet been solved satisfactorily, even though this is an important region for the identification of slow singly charged, and multiply charged particles of all velocities. 100

90 00 70 60 50 40

30

-8

20

---

u

I

10

9 8

7 6

5 4

3 2

\ ,

I

0

1

2

3

4

5

6

GAP LENGTH (IN MICRONS)

FIQ.8. Gap length distribution curves (normalized). Experiment A: pion, 1.3Bev/c. Experiment B: proton, R = 9300 p.

The discrepancy between the X values, measured by different methods can be partly understood from observations recently published by Cortini et ~ 1 . The ' ~ ~authors found that the gap length distribution deviates from an exponential curve for small gap lengths, an observation which was confirmed by other authors. Figure 8 gives the semilogarithmic plot for a pion with momentum I46 G.

Cortini, G. Luzzatto, G . Tomasini, and A. Manfredini, Nuovo cimento [lo] 9,

706 (1958).

1.7.


257

1.3 Bev/c and a proton of 9 mm residual range and it can be easily seen that the curves deviate from a straight line for gap lengths 1 5 0.5 p. If h is calculated from the straight line (F-P), the result will be smaller than the value found from h = LH/B and consequently the parameter 1/X in the (F-P) model becomes too large. This effect will be more pronounced, the denser the track, since the fraction of small gaps (which are neglected in the slope measurements) will increase with ionization loss. Cortini et ul.l4*explain their observations as the result of a development effect and propose a modification of Eqs. (1.7.26) to (1.7.36)

[

H ( Z )= B exp -

1 - F(Z)

(1.7.36)

where 6 is a function, which is zero at 1 = 0 and increases to a constant value a t the gap length, where (1.7.36) becomes exponential in 1. If h is defined by Eq. (1.7.36), then evidently Eq. (1.7.31) which relates h and P is no longer valid, but must be replaced by another relation which now contains in addition to ko and x still another parameter 6. The exact value of 6 is difficult to determine, since it involves distances, which are close to the limit of optical resolution. Because of the ambiguities contained in the use of the parameter A, another more empirical approach to the ionization problem has been attempted, based on the use of a parameter which is more directly related to grain density.ld7 The method is based on the blob length distribution; the importance of the latter has already been pointed out by Della Corte132-133 who investigated the differential blob length distribution in tracks of various density. The differential distribution curves consist of a distinct Gaussian part and a tail, which later becomes more important with increasing ionization. The integral blob length distribution curves for tracks of various densities are plotted in Fig. 9. (The curves refer to measurements in two different experiments A and B.) The number of blobs B(Z)as a function of blob length decreases exponentially with increasing blob length, except for the initial part of the curve. The curved part in the integral distribution reflects the Gaussian part of the differential distribution. It is reasonable to assume that all blobs falling within the Gaussian part are single grains, while all blobs outside the Gaussian part are clusters, composed of at least two grains. In the following the density of single grains will be denoted by N , and the cluster density by N,. The dividing point between grains and clusters can be determined from the shape of differential and 147 M. Blau, S. C. Bloch, C. F. Carter, and A. Perlmutter, Rev. Sci. Instr. 31, 289 (1960).

258

1.

PARTICLE DETECTION

integral distribution curve. If S is the abscissa of the dividing point, then all blobs with 1 5 S are considered to be single grains while all blobs with 1 > S are clusters with a mean cluster length of w = A S , if A is the reciprocal slope of the exponential curve. The percentage a of blobs, being single grains ( S 5 I) decreases with increasing ionization loss, while A and therefore, w increases (see Figs. 10a, 10b and 1Oc). Since the

+

FIG.9. Integrated blob length distribution curve; the blob length is given in microns. The curves represent measurements on trajectories of the following particles. Experiment A: A 1 - proton, Reff = 1890 p ; A B - proton, Reff = 1.5 cm; A 3 proton, R,rr = 3.66 cm; A , pion with equivalent proton range, R = 17.7 cm. Experiment B: I?, - antiproton, Ref{= 1.01 cm; Bf - antiproton, R,ff = 1.45 cm; B3 - 680 Mev/c pion. The curves were fitted t o the experimental points by the method of least squares. The errors shown are statistical.

-

-

exponential behavior of the curves is probably related to the randomness of the agglomeration process, one can assume that w is a function of the number of grains contained in a cluster. In order to evaluate the blob length distribution curves certain assumptions have to be made: The first assumption is connected with the number of grains per cluster in minimum tracks and it was assumed that the mean cluster length w o corresponds to just two grains per cluster; w o is smaller than 2d, where d

1.7.


259

is the mean grain diameter found from the Gaussian distribution. I n the minimum case the size of the second grain is defined by d, = wo - d, which is less than d, due to geometrical conditions or reasons connected with development conditions. It is furthermore assumed that the number of grains per cluster, g,, in denser tracks with w > W O is given by ge

= 1

+ wwo --d d

(1.7.37)

*-

This assumption seems to be reasonable, as a first approximation, at least in tracks of medium density. Linearity of the value of g, - 1 as a function of w - d u p to P = 0.6 follows for instance from the (H-D) model, using parameters /3 = 0.35, 'I = 2, and d = 0.7 p . However, it is clear that the linearity relation will break down for large densities for a variety of reasons connected with the spatial distribution of silver crystals as well as with phenomena appearing in the saturation region. Therefore, relation (1.7.37) is no longer valid a t high ionization densities and a certain value of w might correspond to a larger number of grains than expected from the proportionality relation; the number of apparent grains per cluster is too small, because a certain number of grains are not efficiently observed, while others, due to saturation processes, are not efficiently produced. This statement is equivalent to the assumption that the grain diameter in clusters d, decreases with increasing ionization and this effect was considered tentatively by writing d,

= (w, -

(

3

d) exp - -

(1.7.38)

The ratio in the exponent is the ratio of the mean blob length to the free space in a measuring cell. The procedure of evaluating the data is as follows. First one determines in minimum tracks the number of blobs B and the total blob length LB per unit cell, while the mean blob length A in minimum tracks is obtained from the integral blob length distribution, and the mean grain diameter from the differential distribution. (The use of a blob length analyzer increases considerably the measuring speed.) From the measurements one obtains readily the following equations :

+ (1 - ao)wo] no, - Bo[ao + (1 - ad21

LB, = Bo[aod and

( 1.7.39) ( 1.7.40)

based on the assumption that clusters in minimum tracks contain just two grains.

1.

260

PARTICLE DETECTION

RESTRICTED IONIZATION

LOSS

FIG.10(a). Experimental values of function a, determining the percentage of blobs which are considered t o be grains.

.I

4

I

2

4

RESTRICTED

6

0 10

20

I

IONIZATION LOSS

Fro. lO(b). Experimental values of the reciprocal slope A of the exponential part of the integral blob length distribution. The dimensions of A are microns.

For tracks heavier than minimum ionization one uses the equation LB = B[ad

+ (1 - a).]

(1.7.41)

and because of (1.7.42)

1.7.

PHOTOGRAPHIC

261

EMULSIONS

one finds the total number of grains as given b y (1.7.43) Finally, in very dense tracks, the number of grains per cluster gc has to be corrected in accordance with Eq. (1.7.38). Equations (1.7.41) and (1.7.43) have to be modified, if tracks with dip angle >20° are analyzed, because all the lengths measured b y the analyzer are projected lengths, which have to be converted in true lengths.

ioJ 1

2

4

RESTRICTED

6

8 10

20

IONIZATION

I

LOSS

FIG. 10(c). Experimental values of the total blob length L g per unit length in microns per 100 microns. The abcissas for all three curves are restricted ionization loss. The curves were drawn as best fits t o the experimental points. The symbol 0 refers t o experiment A, t o experiment B.

+

This can be accomplished by multiplying the number of blobs by cos 0 and the blob lengths I' by sec 8. However if the integrated blob length distribution is plotted versus 1' sec 8, one would measure each blob too long by a n amount of d(sec e - 1); (see Della Corte133). Therefore the abscissa of the integrated distribution, from which the inverse slope is determined should be chosen to be 1' sec e - d(sec e - 1).Applying this correction Eq. (1.7.43) becomes w-d

The triangles and crosses in Fig. 6 represent dippingItracks of 40" and

262

1.

PARTICLE DETECTION

30" dip angle calculated in this way. The formula probably gives a value which is somewhat too high because of the possibility of including some single grains within the clusters. There are several reasons why the parameter n, is preferable to others, mentioned before. no is a sensitive parameter in the entire range of ionization. This can be seen from Figs. 10a, b, and c, representing the plot of the parameters: -a-, -A-, and L g as functions of the restricted ionization loss (cut off value 5 kev). For near minimum tracks the parameter -B- (not plotted here) increases rapidly with ionization loss, and, since -a- is near unity, while A and therefore w is small, the value of n, is essentially blob density which here is nearly equal to grain density. With increasing ionization loss -a- decreases rapidly while -Aincreases, so that the second term becomes more important. Even in the dense region some of the parameters, used in (1.7.41) and (1.7.43) are still sensitive to changes in ionization, so that n, is a sensitive parameter in the entire range. In the dense region subjective errors in blob density measurements might be considerable, but this fact does not affect the results greatly because the mean blob length, measured simultaneously by blob length analyzer compensat.es the error, as can be seen from Eqs. (1.7.41) and (1.7.43); since in dense tracks the value of -a- is small, the compensation will not suffer through fluctuations in the value of -a-. If the parameter no,defined by Eq. (1.7.43), is really identical with the true grain density, then no must be equal to n, = Pn,; where n1 is the total number of developable crystals per unit length and P is the probability of development defined by Eqs. (1.7.17) to (1.7.19). Therefore, n,, calculated from Eq. (1.7.43), must be a function of y = QZV,where u, the number of ionization acts per unit length, can be identified with the restricted ionization loss 4,. The experimental points in Fig. 6 represent experimental values found in two different experiments. Since the absolute value of y = q f u is not known, the value of y for the minimum track was found by normalization to the theoretical curve, while all other points were plotted by using the experimental values P = n,/nr and for y the value of the minimum track multiplied by g / g o , (ratio of restricted ionization values; cut off at 5 kev). The points ( 0 . . . experiment A, * * experiment B) lie fairly close to the curve within the error limits, with exception of the points referring to dipping tracks (A * . 40" dip angle, X . * 30" dip angle) which have been calculated according to Eq. (1.7.43a) and the very dense region ( 0 )which were corrected for the apparent loss of grains according to Eq. (1.7.38). The fact that measurements from two different experiments show agreement with a single theoretical curve indicates that n, is a parameter which is independent of development.

+

+

-

1.7.


263

The problem of finding a suitable model is b y no means trivial; there is a n urgent need for the standardization of ionization parameters in the medium and dense region. The present situation is quite involved, because the results of various authors do depend not only on the method of measurements, but also on the model used for the evaluation of experimental data. This dependence extends still farther to the problem of error evaluation. For instance, in O’Ceallaigh’s model, which assumes random distribution of crystals along the track, the standard deviation is simply given by the square root of the numbers counted, provided th a t systematic errors and irregularities in the emulsion constants can be neglected. For other models, however, the problem is more involved, because then, one has t o account not only for the statistical fluctuations of the parameters but also for the correlation between the fluctuations of each of the parameters involved. The error problem in emulsion work was studied in great detail by Blatt introduces fluctuation parameters for various authors. the evaluation of errors; the parameters are defined as the ratio of standard deviation t o the square root of the mean value of the respective parameters measured. Blatt’s fluctuation theory was experimentally verified by O’Brien for the case of the (H-D) model. I n connection with the problem of heavily ionizing particles, which ~ 4 ~recently includes particles near the end of their range, Alvial et ~ 1 . have proposed another method. This method is based on the measurement of track profiles; the tracks are projected on a screen and the distances of both borders of the track from a fiducial line are measured. The authors found in a preliminary investigation that the thickness of tracks for slow particles depends on the velocity of the particle and reaches a maximum near p = 0.1. The increase in thickness and the maximum is explained as being caused by short 6 rays, ionizing crystals in the immediate neighborhood of the tracks; for p 7 0.1 the track width diminishes, because the range of the emitted 6 rays is too short to reach crystals of the main track. If further measurements should confirm that the position of the maximum can be easily established, the method would be very valuable for mass determination of slow particles, since the position of the maximum varies with particle mass. Unfortunately it seems that the observability of the maximum depends on developing conditions and on the dip of the track; therefore, the measurement of each individual case affords a great amount of precalibration work. 123.136v146,148

G. Lovera, Nuovo cimento [lo]8, 1476 (1956). G. Alvial, A. Bonetti, C. C. Dilworth, M. Ladu, J. Morgan, and G. Occhialini, Nuovo cimento [lo] Suppl. No. 4,244 (1956). 148

149

264

1.

PARTICLE DETECTION

1.7.8. Photoelectric Method I n photoelectric methods the observer’s eye is replaced by photoelectric cells. In all measurements photomultiplier cells are used, and the image of the track is projected through a slit, in the image plane of the eyepiece of the microscope, onto the sensitive screen of the photocell. In most photoelectric devices the stage is motor driven and the output of the photocell is recorded, so that the function of the observer is limited to the task of keeping the track image in focus and in the center of the slit. The width of the slit is usually 2-3 p ; it should not be wider, because otherwise the photocell sees and records in addition t o the track segment a considerable amount of the background. However, the slit should not be too narrow, because then the adjustment of the track within the slit becomes difficult; furthermore the slit must be wide enough to accommodate also tracks of slow particles, which frequently suffer deviations from the initial direction. The length of the slit is determined by the purpose of the measurements. With short slits and high magnification the photocell acts as a blob and gap counter; in this case the measurements are relatively independent from background conditions and the depth position of the track. For long slits the photocell acts more as a densitometer, covering, but not resolving larger segments of the track; it is clear that this type of measurement depends greatly on the background and position of the track and it becomes necessary to perform accurate background measurements in order to evaluate the net photoelectric effect arising from the track alone. The length of the slit is somewhat limited by the dip of the track, because during the measurement the whole exposed track segment must be kept in focus. There is still another type of photoelectric measurement in which the profile of a track is measured by sweeping out the cross section of a track with oscillating prism or mirrors. A great number of authors have performed photoelectric measurements and details of measurements and devices used are given in the following papers.lS0-l68We will come back to this section in the chapter on mass measurements and will describe in greater detail the work of the Lund school, where the method has been applied with great success and has been constantly improved. The author warmly thanks Dr. A. Perlmutter for reading the manuscript and suggesting many style improvements. llr0

M. Blau, R. Rudin, and M. Lindenbaum, Rev. Sci. Znstr. 21, 978 (1950).

S. yon Friesen and K. Kristiansson, Arkiv Fysik 4,505 (1951). 152 M. Ceccarelli and G. T. Zorn, Phil. Mag. [7]43, 356 (1952). 161

153

C. Kayas and 0. Morellet, Comp. rend. 234, 1359 (1952).

P.Demers and R. Mathieu, Can. J . Phys. 31, 97 (1953). 15.5 L.Van Rossum, Comp. rend. 236, 2234 (1953). 154

M. Della Corte and M. Ramat. Nuovo cirnento [9]9,605 (1952). M. Della Corte, Nuovo cimento [lo] 4, 1565 (1956). 168 P.C. Bizzeti and M. Della Corte, Nuouo cimento [lo]7 , 231 (1958). 156

157

1.8. Special Detectors 1 A.1. The Semiconductor Detector* 1.8.1.1. Introduction. The junction region of a reversed-biased p-n diode is essentially a solid state version of the conventional gaseous ionization chamber. For the purpose of analogy, it is well to recall the essential features of a gas-filled ion chamber which are shown schematically in Fig. 1. The region between the plates of a charged parallel-plate capacitor is filled with a gas such as argon a t a pressure near one atmosphere. The externally applied voltage V establishes an electric field E = V / d ,where d is the interelectrode spacing. The field E is of sufficient magnitude to prevent recombination of the positive ions and electrons, but not large enough to permit gas multiplication. If a single a-particle,

f

i

PULSE OUT

FIG.1. Essential features of a gas-filled ionization chamber.

say, passes through the chamber it will lose energy by elastic and inelastic collisions with the argon atoms. The net effect of these interactions is the formation of a number of positive ion-electron pairs which are swept apart by the electric field. Frequent collisions with gas molecules preventj both the ions and electrons from obtaining enough energy from the field between collisions to produce secondary ion-electron pairs. As the electrons and ions drift apart, the collector electrode potential rises from zero to ne/C, where n is the number of ion pairs formed, e is the electronic charge, and C is the chamber capacitance. The rise-time and shape of the leading edge of the output pulse is dependent upon the interelectrode distance d, the mobilities p e and pi of the electrons and ions in the chamber gas, and the applied potential V . The decay time is determined by the RC time constant where R is an external resistor (see Fig. 1) and C is the chamber capacitance. A discussion of the principles of operation of a semiconductor detector

* Section 1.8.1 is by S. S.

Friedland and 265

F. P.

Ziemba.

266

1.

PARTICLE DETECTION

is aided by referring to Fig. 2. A p-n junction’ is formed close to one surface of a slab of high-resistivity p-type (boron-doped) silicon by a shallow diffusion of phosphorous. A reverse bias applied to the junction establishes a depletion region (space-charge region) on both sides of the junction. The thin phosphorous doped (n-type) region near the junction has a positive space-charge due to ionized donors, whereas the boron doped (p-type) region has a negative space charge due to ionized acceptors. This distributed dipole layer resembles the charged parallelplate capacitor of the conventional gaseous ionization chamber. If an a particle passes through the space-charge region, electron-hole pairs are produced by inelastic collisions with the silicon atoms. These carriers are swept apart by the electric field set up by the dipole layer, giving rise to an electrical pulse similar to that obtained in a gas chamber. Another variation of the semiconductor detector, often referred to as a “surface barrier counter,” has also been developed. These counters are made by evaporating a thin layer of gold (100-2OOOw) onto highresistivity n-type silicon (or germanium). A distributed p-type layer is formed by surface states at the interface between the metal and the semiconductor. 11,12 Positively charged ionized donors in the n-type material along with the p-type states form a dipole layer. The region in the semiconductor which is nearly stripped of conduction electrons is called a surface barrier. The charge distribution, potential gradient, barrier capacitance, and barrier depth can be calculated from Poisson’s equation and the Fermi-Dirac distribution of charge carriers. The results are 1 I. Van der Ziel, “Solid State Physical Electronics.” Prentice-Hall, Englewood Cliffs, New Jersey, 1957. J. W. Mayer and B. R. Gossik, Rev. Sci. Instr. 27, 407 (1956). J. W. Mayer, J. Appl. Phys. 30, 1937 (1959). a F. J. Walter, J. W. T. Dabbs, L. D. Roberts, and H. W. Wright, Oak Ridge National Laboratory, CF 58-11-99 (1958). F. J. Walter, J. W. T. Dabbs, and L. D. Roberts, Oak Ridge National Laboratory, ORNL-2877 (1960); Bull. A m . Phys. Soc. [11]3,181 (1958); 3,304 (1958); 6,38 (1960); 6, 22 (1960). F. J. Walter, J. W. T. Dabbs, and L. D. Roberts, Rev. Sci. In&. S1, 756 (1960). J. M. McKenzie and D. A. Bromley, Phys. Rev. Letters 2, 303 (1959); Bull. A m . Phys. Soe. [11] 4, 457 (1959). 7 E. Nordberg, Bull. Am. Phys. Soc. [11]4, 457 (1959). M. L. Halbert, J. L. Blankenship, and C. J. Borkowski, Bull. Am. Phys. Soc. [II] 6, 38 (1960). J. L. Blankenship and C. J. Borkowski, Bull. Am. Phys. Soc. [11] 6, 38 (1960). l o M. L. Halbert and J. L. Blankenship, Nuclear Instr. and Methods 8, 106 (1960). 11 W. Schottky, Z . Physik 118, 539 (1942). l2 R. H. Kingston, ed., “Semiconductor Surface Physics.” Univ. of Pennsylvania Press, Philadelphia, 1957.

1.8.

SPECIAL DETECTORS

267

similar to those to be described for the p-n junction. The nuclear characteristics of both the p--n diffused junction detector and the surface barrier counter are the same, since they are dependent upon the nature of the interaction of radiation with the semiconductor material. On the average, 3.5 ev of incident particle energy is required to produce one electron-hole pair in silicon (as opposed to 32 ev for a typical gas), To dat,e, all of the experimental evidence indicates that this value is independent of the particle type. Thus any particle losing energy E (electron volts) in the space-charge region produces n = &/3.5 electronhole pairs (as compared to &/32 ion-electron pairs for the gas chamber).

T-

SURFACE

CONTACT . r P TYPE SILICON BASE CONTACT D = DIFFUSION DEPTH Xp= WIDTH OF DEPLETION REGION I N P MATERIAL X,,= WIDTH O F DEPLETION REGION I N N MATERIAL d = X, + X, TOTAL WIDTH O F DEPLETION REGION

=

x= L

0

N TYPE S I L I C O N

FIG.2. Scheniatic diagram of a p-n junction under reverse bias.

Statistical arguments show that the fundamental device resolution limit is set by the characteristic fluctuation. One, therefore, theoretically expects and experimentally obtains a significant improvement in the energy resolution of the solid state chamber over that obtained with a gaseous chamber. The high carrier mobilities and drift velocities in silicon combined with the small width of the depletion region result in pulses with millimicrosecond rise times. Since the range of energetic particles in silicon is measured in microns as compared to centimeters in a gas, the physical size of the detector is several orders of magnitude smaller. For the p-n junctions under consideration the thickness of the depletion layer in microns is given approximately by d = (pV)l'2/3 where p is the resistivity of the p region in ohm centimeters and V is the applied reverse bias voltage. It is necessary that the incident particle lose all of its energy within the depletion region in order that there be a linear relationship between the pulse height and particle energy. It is obvious from the above result that the product pV be made as large as possible when the detectors are to be used as spectrometers for penetrating particles such as protons and electrons. The results of several workers using

268

1.

PARTICLE DETECTION

detectors with p = 10,000 ohm-cm and V = 400 volts are shown qualitatively in Fig. 3. The relative pulse height is linearly related to the electron energy to about 1 MeV, for protons to 10 MeV, a's to 40 MeV, heavy ions and fission fragments > 100 MeV. These results are consistent with the known range-energy relationships of these various particles in silicon. The typical noise level of amplifiers is shown to be near 20 kev. Resolutions of 0.3% for 5-Mev a particles have been obtained. It should be pointed out that if one wishes merely to count individual events (not I I I 1 1 1 1 1 ~ I l l l l l l l ~ I 1111111~ I 1 1 1 1 1 1 1 ~ I 1 1 1 1 ~

HEAVY

IONS

/

ELECTRONS

a .01

I lfllll'

I I 111111'

I I 1 1111J

I I I 111d

0.1 1.0 10 PARTICLE ENERGY IN

100 MEV

I ''I

1000

FIG.3. Relative pulse height versus energy of a semiconductor detector for various nuclear particles.

measuring the particle energy) then even minimum ionizing particles may be detected inasmuch as they will expend energy at approximately 0.35 kev/micron in silicon. Early attempts to use solid state devices as particle detectors date back to the work of Jaffe13and Schiller14who observed small changes in the conduction current of crystals irradiated with cr particles. Measurable pulses produced by individual & particles penetrating a AgCl crystal were first obtained by Van Heerden.16 Extending that work, McKayl8 studied conductivity changes in diamond under electron bombardment, and Ahearn17tested a large number of crystals for conduction pulses proG. Jaffe, 2.Physik 33, 393 (1932). H. Schiller, Ann. Physik [4] 81, 32 (1926). 16P. J. Van Heerden, "The Crystal Counter, A New Instrument in Nuclear Physics." North Holland Publ., Amsterdam, 1945. l6K. G. McKay, Phys. Rev. 74, 1606 (1948). 1 7 A. J. Ahearn, Phys. Rev. 76, 1966 (1949).

i

13

l4

1.8.

269

SPECIAL DETECTORS

duced by a particles. It was generally found that the crystal counter was not useful as a spectrometer because of poor resolution and polarization effects. Review articles by Chynowethls and Hofstadterlg summarize much of the work with crystal counters. McKay20 reported the response of point contact germanium rectifiers to a particles and suggested the use of p-n junctions. Orman et McKay,22and Airapetiants et measured the voltage pulses produced by germanium p n junctions struck by a particles. In 1955 Mayer and GossikL8used gold-germanium surface barrier counters as a-particle spectrometers. They found that the pulse height from such counters was proportional to the a-particle energy up to about 8 Mev and obtained good resolution from units with 0.8 to 16.0 mm2 active surface area. The Ge-Au surface barrier counter was developed ~,~ and Bromley16and Dearnaley and further by Walter et C L Z . , ~ ~ McKenzie Whitehead.25For good resolution, Ge-Au counters have had to be operated a t low temperatures. Room temperature operation was obtained with the introduction of silicon which has a larger energy gap, and, therefore, a much lower (reverse) saturation current. Si-Au surface barrier counters were developed by McKenzie and Bromley16Nordberg17 Blankenship and Borkowski,Yand Halbert and BlankenHalbert et d.,* ship.l o Such detectors have excellent properties a t room temperature, but until recently the fabrication has been most difficult. Use of a p-n junction diffused in silicon which resulted in an operational room temperature spectrometer was developed by Friedland et aLZ6 Further properties of the p-n diffused junction detector have been reported by a number of ~ o r k e r s . ~ ~ - ~ l ~

1

.

~

~

~

9

~

~

A. G. Chynoweth, Am. J . Phys. 20, 213 (1952). R. Hofstadter, Proc. IRE (Znst. Radio Engrs.) 38, 726 (1950). ,OK. G. McKay, Phys. Rev. 70, 1537 (1949). 2 1 C. Orman, H. Y. Fan, G . T. Goldsmith, and K. Lark-Horowitc, Phys. Rev. 78, 646 (1950). z* K. G. McKay, Phys. Rev. 84, 829 (1951). 28 A. V. Airapetiants and S. M. Ryvkin, Zhur. Tekh,Fiz. 2 7 , l l (1955); Soviet Phys., Tech. Phys. (Eng. Trans.) 2, 79 (1958). z 4 A. V. Airapetiants, A. V. Logan, N. M. Reinov, S. M. Ryvkin, and I. A. Sokolov, Zhur. Tekh. Fiz. 27, 1599 (1957); Soviet Phys., Tech. Phys. (Eng. Trans.) 2, 1482 (1957). zs G. Dearnaley and A. B. Whitehead, Atomic Energy Research Establishment, Harwell, Berkshire, United Kingdom, AERE-R 3278 (1960). 26 8.S. Friedland, J. W. Mayer, J. M. Denney, and F. Keywell, Rev. sci. Instr. 31, 74 (1960). 27 Report on the Seventh Scintillation Counter Symposium, Washington, D.C. February 25-26, 1960. Nucleonics 18 (5), 98 (1960). 28 Complete Proceedings of the Seventh Scintillation Counter Symposium, Washl9

270

1.

PARTICLE DETECTION

1.8.1.2. Junction Capacitance and Junction Width. The potential distribution due to the distributed dipole layer in the space-charge region of the plc junction may be calculated by solving Poisson’s equation with the appropriate boundary conditions. For sufficiently large-area diodes, edge effects may be ignored and the resulting one-dimensional problem is readily handled. The result so obtained allows for the determination of the capacitance and width of the space-charge region. The details of the calculation are omitted and may be found elsewhere.’ For a “Schottky-type potential barrier” where there is a sudden step from p type to n type, the thickness of the transition is d = x p x,; xp is the distance from the junction into the p-type material and xn is the distance from the junction into the n-type material. The thickness d is readily found to be given by

+

d = [2eeo(Vd

+ V)(N,, $- Nd)/eNoNd]112.

(1 A.1. I)

The acceptors and donors per unit volume are N , and Nd; ED

=

8.85 X 10-lafarad/m

and e is the relative dielectric constant; Va is the diffusion potential (built in voltage) and V is the applied bias. For the devices under consideration V >> Vd, Nd >> N,, and we obtain d

[2ee0V/eNd]’/~.

(1.8.1.2)

Since the total charge must be zero for over-all charge neutrality in ington, D.C., February 25-26, 1960. I R E Trans. on Nuclear Sci. NS-7,2-3 (1960). *9 G. L. Miller, W. L. Brown, P. F. Donovan, and I. M. Mackintosh, Bell Laboratory-Brookhaven National Laboratory Report BNL 4662 (1960). 3o G. F. Gordon, Univ. of California Radiation Lab. Rept. 9083 (1960). J. Beneveniste, Univ. of California Radiation Lab., private communication. a* H. Mann, Argonne National Laboratory, private communication. E. L. Zimmerman, “Comments on the Use of Solid State Detectors for Neutron Detection.” Solid State Radiations, Inc., Culver City, California, unpublished. 34T. A. Love and R. B. Murray, Oak Ridge National Laboratory, CF 60-5-121 (1960). 3 6 C. T. Raymo, J. W. Mayer, J. S. Wiggins, and 5. S. Friedland, Bdl. Am. Phys. SOC.[I11 6, 354 (1960). ae S. 6. Friedland, J. W. Mayer, and J. S. Wiggins, Nucleonics 18, 54 (1960). J. D. Van Putten and J. C. Vander Velde, Bull. Am. Phys. SOC.[11] 6, 197 (1960). 38 R. L. Williams, Bull. Am. Phys. SOC. [11]6, 354 (1960).

J. W. Mayer, R. J. Grainger, J. W. Oliver, J. 5. Wiggins, and S. S. Friedland, BuU. Am. Phys. SOC.[11] 6, 355 (1960). ‘O

P. F. Donovan, G. L. Miller, and B. M. Foreman, Bull. Am. Phys. SOC.[I11 6,355

(1960). 41

J. M. McKenaie, Bull. Am. Phys. SOC.[11] 6, 355 (1960).

1.8.

27 1

SPECIAL DETECTORS

the junction we must have NdXn

=

( 1.8.1.3)

NoXp.

Typical values for zn and r p are 0.1 micron and 0.5 mm respectively. The transition region capacitance per unit area is given by

c = [ccoeN,N,&(V,j

+ V)(Na+

Nd)]'/'

= tco/d.

(1.8.1.4)

The space-charge layer thus acts as a parallel plate capacitor with plate separation d. For a sudden step junction, one obtains the approximate expression C = [ee0eN~/2V]l/~. ( 1.8.1.5) In terms of resistivity and mobilities, Eqs. (1.8.1.2) and (1.8.1.4) take on the more convenient forms d = [2eeoVpspo]1'2 = ( + ) ( P V ) microns ~/' ZZ [Eeo/2VphP,p,]"'C (+)(pV)-"' 10' ppf/Cm2

c

(1.8.1.6)

where p. and ph are the mobilities of electrons and holes respectively with V expressed in volts and p in ohm-em. The above relationships have been combined by Blankenship27into a very useful nomograph. 1.8.1.3. Properties of the Semiconductor Detector. The response of a semiconductor detector is linearly related to the energy of the incident particle energy provided the range of the particle (in silicon) is less than the width of the depletion region. Equation (1.8.1.6) demonstrates that the depletion width is proportional to the product (pV)l/'. Detectors with resistivities up to 13,000 ohm-cm have been described and e ~ a l u a t e d . ' ~ Operating such a device with the not unreasonable bias of 750 v would lead to a depletion depth of approximately 1 mm which corresponds to the range of about a 15-Mev proton in silicon. Equation (1.8.1.6) shows that the capacitance is inversely proportional '~. the quantity pV therefore reduces the to the product ( P V ) ~Increasing capacitance, increases the output voltage V o= ne/C, and increases the signal energy E. = (ne)'/2C. The signal-to-noise ratio, however, does not in practice increase with bias voltage. The detector reverse leakage current increases monotonically with the applied voltage. At large voltages this current gives rise to a diode noise energy which increases more rapidly with voltage than the signal energy. The frequency distribution of the noise, the effects of surface leakage, and other matters relating to signal-to-noise ratios have not been studied extensively a t this time. At relatively low bias voltages, the detector noise level is usually below typical amplifier noise levels. This state of affairs has initiated a considerable effort to develop low noise preamplifiers for use

272

1.

PARTICLE DETECTION

with solid state detectors.27 Preamplifiers with noise levels as low as 7.5 kev have been reported. As discussed above, the capacity of an abrupt junction should vary with the voltage according to the relation W 2 C = constant. In a “graded” junction such as that obtained in a grown junction, the acceptor and donor densities N , and Na are slowly varying functions of position in the region of the junction. For a “graded” junction one expects the relation

I

10

100

REVERSE BIAS IN VOLTS

FIG.4. Approximate high-frequencyequivalent circuit of a semiconductor detector. The junction capacitance Cj and resistance Rj determine the pulse rise-time. The resistance Rb is due to the bulk silicon outside of the space charge region.

to be V I W = constant.2 Experimental measurements indicate that VfC = constant with 4 < f < $; typical data for capacitance versus reverse bias for typical detectors are shown in Fig. 4. The temperature dependence of the capacitance is negligible. An approximate equivalent circuit of a solid state detector is shown in Fig. 5. The radiation source is replaced by a charge generator charging the junction capacitance Cj through a resistance Rj which may be estimated by setting the time constant RjCj equal to the transit time of the carriers through the space charge region. Rise times in the millimicrosecond region are commonplace in contrast to the microsecond rise

1.8.

273

SPECIAL DETECTORS

times obtained with gaseous ion chambers. The development of lownoise wide-band amplifiers appears to be quite desirable. The resistor Ra is due to the bulk silicon outside of the space charge region and the ohmic contact with the p-type material. It may be reduced by: (i) having the wafer thickness comparable to the depletion depth; and (ii) a proper doping at the ohmic contact. The latter should also have the effect of producing a more uniform field within the space charge region, whence a more uniform collection efficiency and resolution in large-area detectors. Surface leakage current, space-charge-generated current, and diffusion current all contribute to the reverse current of a semiconductor deis ~ difficult t e ~ t o rIt. ~ ~ ~ ~ to determine the relative contributions in a par-

CJ

I

CHARGE GENERATOR

I

FIQ.5. The dependence of the junction capacity upon applied bias voltage. ticular device; however, the absence of a well-defined breakdown in many units indicates that surface leakage is usually the most important of the three sources. The temperature and voltage dependence of the reverse current are at present under considerable investigation. Surface effects will have to be minimized before any definite conclusions are drawn. 1.8.1.4. Experimental Results. The response of a semiconductor detector to various types of nuclear radiations is shown in Figs. 6 to 11. The energy range over which the device responds linearly to the different types of radiation and the resolution for each of the particle types are discussed and illustrated with experimental data. 1.8.1.4.1. HEAVYIONS AND FISSION FRAGMENTS. Fission fragments have relatively short ranges in silicon and there is no difficulty in obtaining depletion depths which are wide enough to ensure linearity with energy. The same remark applies to heavy ions such as C12. Figure 6 demonstrates that the device is linear for CI2 ions with energies to 120 M ~ v . ~The O kinetic energy spectrum of fragments from the spon42 J. H. Shive, “Semiconductor Devices.” Van Nostrand, Princeton, New Jersey, 1959.

274

1.

PARTICLE DETECTION

taneous fission of Cf262observed29with a p-n detector is in agreement with the accepted time-of-flight measurement^.^^ The detector appears to be “windowless” for fission fragments provided the phosphorous surface layer (n-type) is less than -0.1 p . The semiconductor detector does not exhibit the “ionization defect ’m7 characteristic of gaseous chambers and negligible columnar recombination appears to exist

ENERGY IN

MEV

FIG-.6. The relative pulse-height versus energy of a semiconductor detector for Cl* ions with energies from 30 to 120 Mev.

along the tracks of fission fragments even though carrier densities are -lozo ~ m - ~ . 1.8.1.4.2. PROTONS AND a PARTICLES. The proton response31 of a 5 mm X 5 mm area detector made from 10,000 ohm-cm silicon, and operating a t a reverse bias of 400 v, is shown in Fig. 7. The device is linear for protons to about 10 MeV. Energy-range relationships for protons in silicon show that the range of a 10-Mev proton is about 700 1.1. This agrees well with the calculated width of the depletion region. The resolution versus bias of a typical detector for 8.78-Mev a particles from Pb2I2is shown in Fig. 8. The poor resolution a t low bias voltages 4a

J. C. D. Milton and J. S. Fraser, Phys. Rev. 111, 877 (1958).

1.8.

SPECIAL DETECTORS

275

FIQ.7. The relative pulse-height versus energy of a semiconductor detector, 6,000 ohm cm, 400-v bias, for protons with energies from 2 to 12 MeV.

I

0

0

I I I 0 100 200 250 150 REVERSE BIAS (VOLTS)

50

FIQ.8. The energy-resolution for 8.78-Mev a particles and reverse current versus bias voltage of a semiconductordetector, 6000 ohm cm, 6 mm X 5 mm.

276

1.

PARTICLE DETECTION

is probably due to a combination of a poorer signal-to-noise ratio and a nonuniformity in collection efficiency over the sensitive area. The absence of a well-defined avalanche breakdown in the current-voltage characteristic would indicate considerable surface leakage. A maximum resolution of 0.3% for 5-Mev a! particles which has been obtaineds2 is currently limited by the noise level of the amplifiers. 1.8.1.4.3. ELECTRONS. Figure 9 shows that the detector response is linear for electrons to nearly 1 MeV. Using a calibrated charge-sensitive amplifier, Mann32shows that about 3.7 ev of incident electron energy

ENERGY IN

KEV

FIG.9. The relative pulse-height versus energy of a lo4ohm cm semiconductordetector for electrons with energies from 50 to 800 kev. Bias, 360 v; X, Pml47; 0, Agllom.

is required to produce one electron-hole pair in silicon. The result is in reasonable agreement with the results obtained for protons and a’s in silicon. The data are shown in Fig. 10. Internal conversion lines in CsI3’ are shown in Fig. 11. The K and L lines are distinctively resolved. The data were kindly supplied by C. S. Wu. 1.8.1.4.4. N E u T R o N S . ~ ~Since the semiconductor detector is an excellent device for observing heavy charged particles, it is obvious that its usefulness may be extended to include neutron detection by applying coatings which react with neutrons to produce heavy charged particles. Efficient thermal neutron detectors can be realized by BIO, Lis, and UZ35 coatings. Such devices are not directly useful as neutron energy spec-

1.8.

SPECIAL DETECTORS

277

trometers since the reaction energies are large compared to the incident neutron energy. A combination of bare and cadmium-covered detectors will, however, give some indication of the thermal neutron distribution. Threshold detectors based upon the Np239(n,f) and U238(n,f) reactions will be useful for high-energy neutrons. Neutron energy spectrometers based upon “proton recoil techniques,”* Lie (n,a) HS, and SiZ8(n,p) AlZ8

FIQ.10. The number of charges collected versus energy of a semiconductor detector for electrons with energies from 50 to 800 kev.

reactions hold considerable promise. I t is interesting to note that no coating is needed for the SiZ8(n,p) A128detector. Preliminary results of Love and on a Lie ( n , ~H3 ) neutron spectrometer are most encouraging. The promising ranges of usefulness of the variously coated semiconductor neutron detectors are illustrated in Fig. 12. The height of each curve is a rough indication of the degree of utility of each of the possible arrangements. 1.8.1.4.5. PHOTONS. The only experimental data available on the

* Refer to Section 2.2.2.1.

278

1.

PARTICLE DETBCTION

photon response of semiconductor detectors is some preliminary work with high-energy CosOgammas.36 The device is relatively insensitive to gamma radiation in this energy range (Compton effect) due t o the low-absorption cross section of silicon. The p-n detector is quite sensitive

K - LINE

2000

624 KEV

1500 w

i

0

z

3 1000

I-

0

u

500

BIASED SO THAT ZERO CHANNEL AT - 5 6

100

150 CHANNEL

200 NO

FIQ.11. The internal conversion lines in Cs1*7 aa measured with a semiconductor detector.

to photons with energies comparable to the gap energy in silicon (1.1 ev) and should not be exposed to light when used as a nuclear particle detector. 1A.1.4.6. HIGH-ENERGY PARTICLES. Recently, several laboratories have made investigations t o determine whether high-energy particles in

1.8.

279

SPECIAL DETECTORS

the minimum ionizing region can be measured with semiconductor detectors. The results obtained so far are quite e n ~ o u r a g i n g . ~ ~ Figure 13 shows the energy spectrum of a positive pion and proton beam with a momentum of 750 Mev/c from the Brookhaven Cosmotron obtained in a silicon junction detector. The resistivity of the silicon junction detector is 10,000Q-cm and it was operated a t a reverse bias

si 28cn.p) ( n, Li’

In.

AL~~SPECTROMETER

!/---

cx

INTEGRAL ~)H31NTEGRAL

i \

/I Li6 (h.oC)H3 N$”(n,f)

THRESHOLD

Cn.p) P

.dl d.i

i!o

Ib Ib2

1’0. I3 I2 I’O’

NEUTRON ENERGY-EV

!&.I

I107

SPECTROMETER

THRESHOLD

- RECOIL

Ibe

ilop

;do

4.5 MEV

0.26 MEV

FIQ.12. The relative utility of variously neutron sensitized semiconductor detectors versus the neutron energy.

of 100 v. At this momentum the pions are close to minimum ionization whereas the protons are twice minimum. The ionization losses for the pions and protons are found to be 110 kev and 200 kev, respectively, indicating a linear response, and the pion and protons are very clearly separated. 1.8.1.5. Conclusion. The small size of the semiconductor detector makes it possible to arrange a linear array of detectors in the focal plane of a spectrometer or a t several angles inside of a reaction chamber. Two 44 L. C. L. Yuan, Application of solid state devices for high energy particle detection. Intern. Conf. on Instrumentation for High Energy Physics, Berkeley, California, September, 1960.

280

1.

PARTICLE DETECTION

dimensional arrays can be assembled to obtain large-area a-survey instruments, for example, with low power requirements. A three-dimensional array along with an appropriate data handling system would obtain the equivalent of a “solid state cloud chamber.”46 The effects of radiation on semiconductor devices have been summarized in many report^.^^.^^ Radiation damage studies in semiconductor

300

302

>-

550

w 7T+ AND PROTON BEAM, 750

y,

IOK COUNTER, 1.4 CM DIAMETER,

2oc

100 VOLT BIAS.

Y

z

g5

?oo

a-

I50

LL

sJ! 0

> U

w-

IOC

z

100

W

50

-

)

1

I

I

I

I

20

30

40

50

60

I

m

I 80

90

I 100

CHANNEL NUMBER

FIG.13. 750 Mev/c positive pion and proton spectrum obtained in a Brookhaven Cosmotron beam. Number of counts per pulse-height analyzer channel is plotted versus the channel number which is a measure of the particle energy.

detectors is not available at the present. The data available a t present indicate that no changes in detector operation are observed after exposure to 10l214-Mev protons. The physical properties, low power requirement, wide range of linear relationship of pulse height versus particle energy, and high speed suggests that the semiconductor detector will soon become one of the basic operational devices in the field of radiation detection. S. S. Friedland, “The Solid State Cloud Chamber.” To be published. G. J. Dienes, Radiation effects in solids. Ann. Rev. Nuclear Sci. 2, 187 (1953). 41 F. J. Reid, The effect of nuclear radiation on semiconductor devices. REIC Report No. 10, Battelle Memorial Institute, Columbus, Ohio (1960). 46 *6

1.8.

SPECIAL DETECTORS

281

1.8.2. Spark Chambers* Frequently it is desirable to have good spatial resolution of high-energy particle interactions because so many modes of interaction are possible. Generally, the interesting processes also have a small cross section so that good time resolution is desirable too. Bubble chambers have been very effective in experiments where it is reasonable to expand the chamber and then have approximately ten charged particles incident on the chamber. If the chamber is large enough and if there are enough interesting events, then this is a good technique. The electromagnetic spectrometers which have been used to select the desired mass of the incident charged particles have helped to extend the use of bubble chambers. However, these will be considerably less effective at higher energies and new techniques will be desirable. Cloud chambers have been used in conjunction with scintillation counters to select interesting events and then the chamber is expanded to get tracks. Also diffusion chambers have been used in this manner, the lights being flashed to detect an interaction. The resolving time of such chambers is greater than 10 psec and background radiation is a rather severe limitation of these chambers. The scintillation chambers are being developed which have both good spatial and time resolution. For some experiments these have many desirable features. The chief disadvantages are the still rather small size and the limited flexibility for a variety of experiments. In some experiments' where only moderate spatial resolution is required, such as differential cross section scattering of antiprotons and K mesons, it has been quite practical to use arrays of scintillation counters and Cerenkov counters. I n these experiments the constraints due to twoparticle interactions have simplified the analysis. I n many cases it is now necessary to investigate specific details with good statistics. This frequently requires that rare events be selected from a background of many other interactions. The spark chamber used in conjunction with scintillation counter and Cerenkov counter telescopes is a device that permits both good spatial and time resolution of highenergy charged particle interactions. An early type of discharge chamber2 consisted of bundles of slightly "2. A. Coombes, B. Cork, W. Galbraith, G. R. Lambertson, and W. A. Wenzel, Phys. Rev. 112, 1303 (1958). M. Conversi and A. Gorrini, Nuovo cimento [lo] 2, 189 (1955);'M. Conversi, S. Focardi, C. Franzinetti, A. Gozzini, and P. Murtas, Nuovo cimento Suppl. [lo] 4, 234 (1956). __

* Section 1.8.2 is by

Bruce Cork.

282

1.

PARTICLE DETECTION

conducting glass tubes filled with one-half atmosphere of neon and placed between parallel metal plates. When a high-energy charged particle was transmitted through the tubes and plates a high electric field pulse was applied between adjacent plates. The tubes that transmitted the particle would then give a glow discharge and the light from the ends of the tubes could be photographed. These devices had a long recovery time, approximately one second, due to the electrostatic charges on the glass. 1 I

ji

i

4

FIQ.1. A parallel plate spark chamber. The plates are made of +-in. thick aluminum separated by a gap of in.

4

A similar device has been described by Cranshaw and DeBeer.3 However, they omitted the glass tubes, immersed the metal plates in air at one atmosphere, and applied a 20-kv pulse to the plates when a charged particle was transmitted. The efficiency for minimum ionizing particles was 99%, for a 3-mm gap. Then Fukui and Miyamotoe immersed the plates in an atmosphere of neon. They observed that minimum ionizing particles could be detected with nearly 100% efficiency and by applying a +psec pulse to the plates, the sensitive time was observed to be approximately 10 Fsec. Several other groups have built similar spark chambers. The details 8

T. E. Crawhaw and J. F. DeBeer, Nuovo cimento [lo] 6, 1107 (1957). S. Fukui and S. Miyamoto, Nuovo eimento [lo] 11, 113 (1959).

1.8.

283

SPECIAL DETECTORS

of a chamber6 built by Beall, Cork, Murphy, and Wenzel are given below. The chamber consisted of seven parallel plates of g i n . thick aluminum, separated by gaps of s i n . thickness, Fig. 1. The chamber has been filled with one atmosphere of argon or neon. This chamber has been tested in a beam of high-energy pions and protons a t the Bevatron.

Coincidence circuit

Discriminator 8 gote circuits

1--D

2i----Jy .

T h'.>'"."",. urntvnm

II

Thyratron 2

FIG.2. Block diagram of electronics, scintillators C , Cz, Cs and spark chamber.

The diagram, Fig. 2, shows a charged particle being detected by a scintillator coincidence telescope. The output of the coincidence circuit is used to operate a hydrogen thyratron which applies a 20-kv pulse, approximately 0.2psec, to alternate plates of the spark chamber. A battery supplies a dc clearing field between the parallel plates so that electrons produced in the gap between the plates can be swept away, thus reducing the sensitive time of the chamber. The efficiency of the chamber when filled with argon or neon is given by Fig. 3 for a clearing field of 0 v/cm and 270 mFsec delay after the traversal by a minimum ionizing charged particle. To determine the 6 E .Beall, B. Cork, P. G . Murphy, and W. A. Wenzel, UCRL-9313 (1960).

284

1.

PARTICLE DETECTION

sensitive time of the chamber for various values of clearing field, varying time delays were inserted in the trigger line to the thyratron. The efficiency as a function of delay time is given by Fig. 4. It is noted that the sensitive time can be made less than 0.5 psec. The recovery time of the chamber should be of the order of the deionization time of an inert gas at one atmosphere. The observed recovery time (Fig. 5) was long compared to the deionization time. This was measured by selecting a charged particle that was transmitted by the chamber, firing the chamber, and then selecting a second charged particle a t a predetermined time and again firing the chamber. One reason for the appar-

I

1

I

a

5

10

15

20

Pulse voltoge

1

25

(kv)

FIG.3. Efficiency of a single +in. gap in 1 atmos of argon or neon as a function of pulsed voltage across the t i n . gap.

ent long recovery time may be due to impurities in the gas. This is not a practical limitation for proton synchrotrons where the beam time is of the order of 100 msec. Typical photographs of the 6-gap chamber are given in Fig. 6. Minimum ionizing pions enter from the left. In Fig. 6(a) one pion interacted with the plate and a second pion entered during the sensitive time of the chamber. A second interaction is shown by Fig. 6(b) where one reaction product was scattered at an angle of 25 degrees, the second a t 36 degrees. When a magnetic field of B equal to 13 kg was applied parallel to the plates, the efficiency was still nearly 100% per gap. If a clearing field E of 80 v/cm was applied and the time of applying the hv pulsed electric field was delayed for 1psec, the electrons from the ion pairs were displaced an amount proportional to E X B, and the delay time. The photograph (Fig. 7) shows a displacement of the tracks for the above conditions of

FIQ.4. Efficiency of a single *in. gap in 1 atmos of argon as a function of delay in application of the high-voltage pulse. The zero of the delay axis is the time at which the particle passed through the chamber. I

I

3 100 E c 'p

c

80

0 V

$

60

8

Argon

A Neon

5

10

15

20

Time between partieter (maec)

FIG.5. Efficiency for a single gap to a spark on a second particle as a function of the time between particles. The clearing field was -40 v/cm. 285

286

1.

PARTICLE DETECTION

(b)

FIG.6. Typical photographs of the 6-gap chamber. In Fig. 6(a) one pion interacted with the aluminum plate and a second pion entered during the sensitive time of the chamber. Figure 6(b) shows two large angle scatters.

1.8.

SPECIAL DETECTORS

287

FIG.7. A magnetic field of 13 kg parallel to the plates and a clearing field of 80 v/cm cause a displacement of the sparks of ki in. if the high-voltage pulse is delayed for 1 usec.

288

1.

PARTICLE DETECTION

approximately 1 cm. Particles arriving “off time” could be detected by this means. Besides the good spatial and time resolution of the spark chamber, the chamber can be arranged in a manner that is appropriate to the particular experiment. For example, the quantity of light from the spark is so great that stereographic photography is easy from an extensive assembly of chambers. The plates can be made of metal or graphite, for example, to preferentially scatter polarized protons. It should be possible to make the plates of scintillator or cerenkov material so that the spark chamber and counter telescope are integral. Large solid angles for detecting short-lived particles can be obtained by this method.

2. METHODS FOR THE DETERMINATION OF FUNDAMENTAL PHYSICAL QUANTITIES 2.1. Determination of Charge and Size 2.1.1. Charge of Atomic Nuclei and Particles 2.1.1 .l. Rutherford Scattering.* Rutherford, in 1906, first noticed that the angular deflections experienced by a rays while passing through thin layers of air, mica, or gold were occasionally very large. He recognized that the large electrostatic field strengths required to cause such deflections could be produced only at very small distances from an electric charge. In 1911 Rutherford introduced his nuclear model of the atom, in which all the charge of one sign (now known to be positive) was concentrated into a central region, or nucleus, smaller than 10-l2 cm in radius, with an equal amount of charge of the opposite sign (now known to be the atomic electrons) distributed throughout the entire atom in a region whose effective radius is of the order of lo-* cm. It can be shown quite generally' that, when any unbound incident nonrelativistic particle (such as an CY ray) interacts with a target particle (such as an atomic nucleus) according to an inverse-square law of force (either attractive or repulsive), both particles must, in order to conserve angular momentum, traverse hyperbolic orbits in a coordinate system whose origin is at the center of mass of the interacting particles. When conservation of the sum of kinetic and potential energg is imposed as a third restriction on the orbits, it is found that the angle of scattering 6 in the center-of-mass coordinates is given by 5 =

b 6 -cot -* 2 2

(2.1.1.1.1)

Here the impact parameter x is the distance a t which the two particles would pass each other if there were no interaction between them, and b is the collision diameter defined by

["gb?;] ~

= lZzl

(2.818 X low1* cm)

(2.1.1.1.2)

1 R. D. Evans, "The Atomic Nucleus," pp. 12-19, 838-851. McGraw-Hill, New York, 1955.

*Section 2.1.1.1 is by Robley D. Evans. 289

290

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

where ze = charge on incident particle, Ze = charge on target particle, V = mutual velocity of approach, and Mo = reduced mass M 1 M 2 / ( M 1 M 2 ) of colliding particles. The absolute value of Zz is to be taken without regard to sign. The collision radius b/2 is the value of the impact parameter for which the scattering angle 0 is just 90" in center-of-mass coordinates, both for attractive and repulsive forces. For the special case of repulsive forces, as in the nuclear scattering of a rays, the collision diameter b is also equal t o the closest possible distance of approach, that is, to the minimum separation between the particles during a head-on collision. At the instant of minimum separation the particles are stationary with respect to one another, and therefore their initial kinetic energy (+)MoV2in center-of-mass coordinates is just equal to their mutual electrostatic potential energy. For this head-on collision x: = 0 and 8 = 180°. For other scattering angles 0, the minimum distance of approach pminfor repulsive forces is larger than b and is given by

+

pmin =

[

I ' )+: (

2 1 -k 4 1

=

5 [ + -11

1 sin(0/2)

(2.1.1.1.3)

I n center-of-mass coordinates there is no transfer of kinetic energy between the colliding particles. But in the laboratory coordinates there is an energy transfer and the target particle M , which was initially a t rest emerges from the collision with the kinetic energy

where the bombarding particle has mass M I and a n incident kinetic energy (+))n4',V2in the laboratory coordinates. Equation (2.1.1.1.4) is entirely a consequence of conservation of energy and linear momentum, and i t applies to all nonrelativistic elastic collisions regardless of the nabure of the forces between the colliding particles. Collisions between two nuclei will involve only the inverse-square Coulomb force whenever b is significantly larger than the combined effective radii of the interacting nuclei. If b is too small then specifically nuclear forces may become effective, some nuclear barrier penetration may occur, and Eq. (2.1.1.1.1) will become invalid. Such a deviation from the Rutherford scattering law, which is based on inverse-square Coulomb forces, is called anomalous scattering. It can be shown quite

2.1.

DETERMINATION OF CHARGE AND SIZE

29 1

generally that the classical Rutherford scattering relationships will be valid whenever (2.1.1.1.5)

in which 0 = V / c is the mutual velocity of approach in units of the velocity of light c, and X = h / M o V is the rationalized de Broglie wavelength of relative motion of the collidiiig nuclei. The cross sections for Rutherford scattering, that is, for scattering by inverse-square Coulomb forces, are the same in classical and in wavemechanical theory because the cross section is independent of Planck's constant h. Classically, the differential cross section du for Rutherford scattering between angles 6 and 6 d 8 is the area of a ring of radius x and width dx, or da = 27rxdx. This leads to the Rutherford scattering differential cross section 1 cm2 (2.1.1.1.6)

+

where dQ is the solid angle a t mean scattering angle 8 in center-of-mass coordinates, and 2, is the collision diameter as defined by Eq. (2.1.1.1.2). This marked preponderance of forward scattering is characteristic of long-range forces, such as the inverse-square interaction. I n the language of wave mechanics, these interactions involve interference between many partial waves whose angular momentum quantum numbers I extend from zero up to a t least 1 'V MoVx/h = x/X. Therefore the foreand-aft symmetry which characterizes interactions involving only one value of 1 is not seen in Rutherford scattering. Geiger and Marsden completed in 1913 a beautiful series of experiments which verified the dependence of Eq. (2.1.1.1.6)on the scattering angle 0, the incident a-ray velocity V , and t,he nuclear charge Ze. Figure 1 shows Geiger and Marsden's results for a-ray scattering from a thin foil of gold. For such a heavy target nucleus ill2>> M I , hence M ON M I , and the scattering angle 8 in center-of-mass coordinates is substantially equal to the scattering angle 8 in laboratory coordinates. The curve in Fig. 1 is proportional to l/sin4(8/2), as predicted by Eq. (2.1.1.1.6), and is fitted t o the arbitrary vertical scale a t 8 = 135". The agreement a t all angles shows that, under the conditions of these experiments, the only force acting between the incident CY rays and the gold nuclei is the inverse-square Coulomb repulsion. The Coulomb parameter, Eq. (2.1.1.1.5), has the value 22z/1378 N 36 for the collisions between Au and the 7.68-Mev a rays from RaC', and the closest distance of cm for 150" scatapproach in these experiments was p,,,,,, = 30 X tering. Therefore the positive charge in the gold atom is confined to s

292

2.


small central region which is definitely smaller than this, or about lo-' of the atomic radius. Using the 7.68-Mev RaC' a rays from a source containing RaB RaC, and thin scattering foils of Au, Ag, Cu, and All Geiger and Marsden showed experimentally for the first time that the nuclear charge Ze is approximately proportional to the atomic weight, and that 2 is about

+

MEAN ANGLE OF SGATTERlNG,8

FIQ. 1. Geiger and Marsden's relative differential cross section measurements for the single scattering of RaC' 01 rays by a thin foil of gold [from Evans'].

one-half of the atomic weight. Their experimental uncertainty in the absolute value of 2 was about 20%. For an accurate determination of 2, many experimental precautions must be observed. Especially the scattering foils must be so thin that the a rays lose only a small fraction of their energy by ionization and therefore have an accurately known mean velocity in the foil. Chadwick introduced in 1920 a n ingenious experimental arrangement which greatly increases the observable scattered intensity for any given angle, source, and thickness of scattering foil. The foil is arranged, as shown in Fig. 2, as an annular ring around an axis between the source and the detector. This annular geometry for the scattering body has subsequently been widely adapted to a variety of

2.1.

293

DETERMINATION O F CHARGE AND SIZE

*

other scattering problems, especially with neutrons. Chadwick’s precision a-ray scattering experiments with this arrangement gave the absolute value of the nuclear charge of Cu, Ag, and Pt as 29.3e, 46.3e, and 77.4e, with an estimated uncertainty of 1 to 2 %. This direct measurement was a welcome confirmation of the atomic numbers 29,47, and 78 which had in the meantime been assigned to these elements in 1914 by Moseley on a basis of their characteristic X-ray spectra. /

SOURCE

\

BAFFLE ORAY TO STOP DIRECT BEAM OF U RAYS

SCINTILLATION DETECTOR FOR 0 RAYS SCATTERED BETWEEN ANGLES AND 8 2

+

/#

ANNULAR RING OF SCATTERING FOIL

FIQ. 2. Chadwick’s arrangement of source, annular scatterer, and detector for increasing the intensity of (Y rays scattered between 61and 62, as used for his direct measurement of the nuclear charge on Cu, Ag, and Pt [from Evrtnsl].

Rutherford scattering is of course applicable to the scattering of any charged projectile (proton, deuteron, a ray, or fission fragment) under the restrictions imposed by Eq. (2.1.1.1.5). Modern scattering-experiment techniques with artificially accelerated projectiles usually provide for momentum or energy measurements on the scattered particles. t Then the energy of the Scattered particle is given by Eq. (2.1.1.1.4) and can be used to determine the mass number Mz of the scattering nucleus, while the charge Ze of the scattering nucleus determines the intensity of scattering according to Eq. (2.1.1.1.6).

2.1.1.2. Characteristic X-ray Spectra.* $ Following the proof of the existence of atomic nuclei by the a-ray scattering experiments, Niels Bohr in 1913 assigned the principal part of the atomic mass t o nuclei and introduced his quantum theory of the origin of atomic spectra. This step completed the basic concepts of the Rutherford-Bohr model of the

t See also Vol. 4, B, Part 9.

4 See also Vol. 1, Chapter 7.10. -

* Section 2.1.1.2 is by

Robley D. Evans.

294

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

nuclear atom. To the extent that the simple theory for hydrogen-like atoms is valid, the energy hv of characteristic X-ray quanta is given approximately by

where the principal quantum numbers are nl for the initial electron vacancy and n2 for the final electron vacancy, a = e2/hc ‘V & is the fine-structure constant, and mgc2 = 0.511 MeV. For characteristic X-ray lines of the K , series nl = 1, 722 = 2; for the Laseries n l = 2, n2 = 3. Equation (2.1.1.2.1) gives a reasonably good representation of the gross behavior of characteristic X-ray spectra, but of course does not give accurate values nor any information on the fine structure because it ignores the orbital ( E ) , spin (s),and total ( j ) quantum numbers, because it is nonrelativistic, and because it ignores screening of the nuclear charge by atomic electrons. Moseley in 1913 applied the then new principles of Bragg reflection to the study of X-ray lines and introduced a new era of X-ray spectroscopy by measuring the mean wavelength of the unresolved K , doublets for 21 elements from I d 1 to 4,Ag, and the mean wavelength of the . showed the unresolved La doublets for 24 elements from 40Zr to 7 9 A ~He existence of a linear relationship between the atomic numbers of the light elements, as previously assigned from chemical data, and v 1 / 2 for the characteristic K , and L, X-ray lines. The overlap between 4oZrand 47Ag oriented the La series with respect to the K , series, and thus permitted the use of the La series for bridging over and going above the rare earth group of elements. Moseley’s work was the first to show that a total of 15 places (Z = 57 to 71) had to be reserved for the rare earths, and the first to assign atomic numbers to the heavier elements, for example to gold (Z = 79). Screening of nuclear charge by atomic electrons. Moseley’s original data / the ~ K, are shown in Fig. 1. The plot of atomic number against v ~ for series does not pass through the origin but has an intercept of about unity on the scale of atomic number. If the nuclear charge Z is assumed to be the same as the atomic numbers which had been assigned to the light elements from chemical data, then Moseley’s data on the K , series have the form ( ~ V ) I /= ~ const X (Z - 1). (2.1.1.2.2) Moseley correctly interpreted this result as indicating that for the K , series the effective nuclear charge Ze is reduced to about, (2 - l ) e because

2.1.


295

of the screening of the nuclear potential by the potential due t o the other K , L, . . . electrons present in the ionized atom both before and after the X-ray transition. Similarly, Moseley’s data on the Laseries exhibit a substantially linear relationship given by (hv)’I2 = const X (2 - 7.4).

(2.1.1.2.3)

Under the same physical interpretation, the effective screening constant is about 7.4e for the La series. These screening constants are physically reasonable.

ATOMIC NUMBER

FIG. 1. Moseley’s original data (1914) showing the frequency Y of the K , and La X-ray lines. There is a uniform variation of ~ 1 ’ 2with integers 2 assignable as atomic numbers to the 38 elements tested, if aluminum is assumed to be 2 = 13 in accord with the chemical evidence.

It is concluded th at the atomic number is equal tBot,he number of elementary positive charges 2 in the atomic nucleus and hence also to the number of atomic electrons in the neutral atom, as had been proposed first by van den Broek in 1913. The approximately linear relationship between (hv) for the lines of any particular X-ray series and the atomic number 2, as illustrated by Eqs. (2.1.1.2.2) and (2.1.1.2.3),is commonly referred to as Moseley’s law. Identification of new elements. The new elements which have been produced by transmutation processes in recent years have no stable isotopes. But each of these new elements (2 = 43, 61, 85, 87, 93, 94,

2.

296

. . .)


does have a t least one isotope whose radioactive half-period is sufficiently long t o permit the accumulation of milligram quantities of the isotope. I n every case, the atomic number has been assigned first by combining chemical evidence and transmutation data, a t a time when the total available amount of the isotope was perhaps 10-lo gm or less. Confirmation of most of these assignments of atomic number has been made by measurement of the K and L series X rays, excited in the conventional way by electron bombardment of milligram amounts of the isotope.1'2 Measurements of the characteristic X-ray emission lines are regarded as conclusive evidence in the identification of any new element. X rays from radioactive substances. Whenever any process results in the production of a vacancy in the K or L shell of atomic electrons, the subsequent rearrangement of the remaining electrons is accompanied by the emission of one or more X-ray quanta of the K or L series, or by Auger electrons, or both. Such vacancies are always produced spontaneously in two types of radioactive transformations, electron capture and internal conversion. In electron-capture transitions it is generally more probable that a K electron will be in the vicinity of the nucleus and will be captured than that an L, M , . . . electron will be captured. The majority of the vacancies therefore are produced in the K shell. If 2 is the atomic number of the parent radioactive substance, then (2 - 1) is the atomic number of the daughter substance in which the electron vacancy exists and from which the X rays are emitted. For example, several isotopes of technetium (2 = 43) decay predominantly by electron capture, and the early identification of element 43 was aided by the observation of molybdenum (2 = 42) X rays which are emitted in the decay of these technetium isotopes. A number of transuranium isotopes decay by electron capture, in competition with a-ray emission, and can be identified unambiguously by the characteristic X rays which accompany the electron-capture transitions. Internal conversion is an alternative mode of deexcitation which always competes with y-ray emission. The nuclear excitation energy hv is transferred directly to a penetrating atomic electron, which is expelled from the atom with a net kinetic energy hv - B,where B is the initial binding energy of the electron. In the most common cases, internal conversion is more likely to expel a K electron than an L, M , . . . electron from the atom. Thus the majority of the vacancies are produced in the K shell of atomic electrons or in the L shell if the available energy hv is insufficient, to eject a K electron. Internal conversion transitions are always accom1L.E.Burkhart, W. F. Peed, and B. G. Saunders, Phye. Rev. 78, 347 (1948). W.F. Peed, E. J. Spiteer, and L. E.Burkhart, Phys. Rev. 76, 143L (1949). 95,

2.1.


297

panied by X-ray emission spectra. Because internal conversion involves no change in nuclear charge, the X-ray spectra are characteristic of the element in which the internal conversion and the competing y r a y transitions took place. The chemical identification of a number of radioactive nuclides among the transuranium elements ( Z > 92) has been made or confirmed by observations of the L series X rays which accompany the internal conversion of excited nuclear levels produced in the decay products, following a-ray or @-rayemission by the parent nuclide. For the measurement of these X-ray spectra it is frequently convenient to use a bent-crystal X-ray spectrometer.a In this way the fine structure of the L series X rays, which occur in the energy range of 10 to 20 kev for Z 2 80, can be resolved and studied in detail. The K series X rays of the heavy elements occur a t much higher energies, and are in the domain of 70 to 120 kev for Z 2 80. Measurements of the energy spectrum of the internal conversion electrons, with a &ray spectrometer, provides an additional method for determining the atomic number of the element in which the internal conversion transition takes place. If hv is the energy available, then a conversion electron from the K shell will have a kinetic energy hv - BK, where BK is the binding energy of a K electron, that is, the energy of the K edge. Similarly, a conversion electron ejected from the L shell will have a kinetic energy hv - BL, where BL is the energy of the L edge. Under sufficiently high resolution the fine structure of the L conversion electrons can be resolved into discrete groups from the L,LII, and LIIIelectronic levels. If hv is known, the absolute values of the energies of the conversion electrons serve to determine B K , BL, . . . and hence Z of the element. Even if hv is unknown, the energy difference between the conversion electron groups from a single nuclear transition can be used for a determination of 2. For example, the energy difference between the L-conversion electrons and the K-conversion electrons from the same nuclear transition is (hv - BL) - (hv - BK),which is equal to (BR - BL)and therefore to the energy of a K , X ray from the same element. cE Tables of the binding energies in kev of the various groups of K, L,M , N , and 0 electrons have been compiled by Hill et a1.7 and ~ t h e r s . ~ . ~ ]

G. W. Barton, Jr., H. P. Robinson, and I. Perlman, Phys. Rev. 81, 208 (1951). C. D.Ellis, Proc. Roy. SOC.AQQ,261 (1921);A101, 1 (1922). 6L.Meitner, 2. Phgaik Q, 131, 145 (1922). G.A. Graves, L. M. Langer, end R. J. D. Moffat, Phys. Rev. 88, 344 (1952). ’R.D.Hill, E. L. Church, and J. W. Mihelich, Rev. Sci. Znstr. 23, 523 (1952). a Y. Cauchois, J . phys. radium 13, 113 (1952). S. Fine and C. F. Hendee, Nucleonics 13(3),36 (1955). a

298

2.


Atomic Number, 2

FIG.2. Binding energy of the x electrons in the K , L, and M shells of the elements [from Evans”J1.

Figure 2 shows the variation wit.h Z of the binding energies of the most tightly bound groupsl the s electrons, whose binding energies correspond to the K, L I ,and MI edges.

2.1.1.3. Charge Determination of Particles in Photographic Emulsions.* The application of the most obvious method for charge deter-

mination-deflection in a magnetic field-to nuclear emulsion technique is fraught with several serious difficulties; these include the large amount of multiple scattering a particle suffers in the emulsion and the relatively short path in this dense medium. Two magnetic methods, namely the lo R. D. Evans, in “Handbuch der Physik-Encyclopedia of Physics” (S. Fliigge, ed.), Vol. 34. Springer, Berlin, 1958.

*Section 2.1.1.3 is by M. Bfau.

2.1.


299

sandwich method’’ and the magnetic deflection in the emulsion itself, have have treated by various authors. l-’% The “sandwich method ” consists in the measurement of deflections of particles traversing the air gap between two parallel emulsion sheets. The curvature of the trajectory in the magnetic field, existing in this gap, is determined by the angle between the particle’s exit and entrance directions in the two adjacent emulsions. Although the accuracy of this method could be increased by using wider gaps, the maximum separation is limited by the fact that it becomes increasingly difficult to follow a t,rack, which is not visible, over a considerable segment. Distortion of the emulsion may also impede the usefulness of this technique by causing large errors, especially for tracks with dip angles exceeding 10”. However, even if the results are inadequate for accurate charge determination, the method can be used to obtain the sign of the charge. The other method of magnetic deflection also requires distortion-free emulsions. It will yield accurate value of the charge only if the magnetic field is large, the path in the field long, and the mean scattering angle &, small compared to the magnetic deviation angle am.If the latter is defined by a, = ( t / p ) , where p is the radius of curvature of a trajectory which describes a path of circular arc t in a region where the magnetic field is H , then one obtains am = ( t H z / p ) ,where z is the charge of the particle and p is its momentum. If p can be determined independently, and if the particle trajectory is nearly perpendicular to the magnetic field, then z can be measured with considerable accuracy. However, the sign of the charge may be obtained from the direction of deflection, even if the momentum cannot be determined. It should be emphasized again th a t these measurements can be made only if multiple scattering does not, obscure the direction of the magnetic deviation a r n / a ~>c> 1. Since it can be shown that am/aAC H p dj, the magnetic deviation method is dependable only if H , t, or 0 are large. On the other hand, when 4 1, t a n d H must be very large to give a measurable value for am.The authors working in this field have adopted, as a general rule, that the sign of the charge of particles in the energy range 200-2000 Mev can be determined with 80%

-

I. Barbour, Phys. Rev. 76, 820 (1949). C. Franzinetti, Phil. Mag. 171 41, 86 (1950). a Y. Goldschmidt-Clermont and M. Merlin, Nuovo cimento 191 7 , 220 (1950). 4 C. C . Dilworth, S. J. Goldsack, Y. Goldschmidt-Clermont, and F. Levy, Phil. Mag. [7] 41, 1032 (1950).

C. C . Dilworth and S. J. Goldsack, Nuovo cimento [9] 10, 926 (1953). M. Merlin, Nuovo cimento Suppl. [lo] 2, 218 (1954). 1 M. Merlin and G. Someda, Nuovo n’mento [9]11, 73 (1954). 7s H. P. Furth, Rev. Sci. Znstr. 26, 1097 (1955). 6

0

300

2.


probability if the path length is at least 2 cm and the applied magnetic field is 30,000 gauss. Somewhat better results can be achieved, if diluted emulsion with a smaller content of silver bromide are used. This and the recent availability of large transient magnetic fields may well improve the conditions for magnetic methods in nuclear emulsions. There are other methods which can be used for the measurement of particle charge, but not the sign, of the charge. In the previous section, emphasis was placed on singly charged particles. It should be noted that the ionization of two particles of equal velocity is in the ratio of the squares of their charges. Therefore the grain density may be expressed by the relation g = f(z2j3). Hence, the grain density alone gives no information on z or j3 separately. Even if a large segment of track is available, the situation is not improved, since the rate of change of the grain density depends on j3, M , and z. Thus, one must supplement such observations with an independent method, such as scattering, which is a measure of momentum and charge (see Section 2.2.1.1).

Another important parameter for charge determination, closely related to ionization, is the production of 6 rays. The latter arise from collisions with atomic electrons when the energy transfer is greater than the average energy given to grains, forming the particle trajectory. Thus, the atomic electrons acquire considerable velocities, and hence are able to render several grains developable, thereby forming short trajectories which protrude from the original track. &ray measurements were first used to discriminate among charges of heavy primaries which were discovered in cosmic radiation experiments by Bradt and Peterss and Freier et aL9 The frequency of 6 rays increases with decreasing velocity, although at very low velocities the number of visible 6 rays decreases again. The latter decrease is mostly due to the smaller energy and therefore shorter residual range of the emitted 6 rays, a t very low velocities-near the end of the range of the multiply charged particle-there are no distinctly visible 6 rays, but the tracks seem merely somewhat thicker than the trajectories of singly charged particles. This phenomenon is known under the name of “thin down length”; the extension of the thin down length is a function of charge and mass of the particle, since both parameters determine the rate of velocity loss of the particle, The maximum energy which particles heavier than electrons can transfer to electrons is given by Em, = 2mJ2(1 - a2), where m. is the electron mass and B the particle

* H. L. Bradt and B. Peters, Phye. Rev. 74, 1828 (1948). OP. F‘reier, E. J. Lofgren, E. P. Ney, and F. Oppenheimer, Phys. Rev. 74, (1948).

1818

2.1.


301

velocity. If is small, the range of the ejected electron is not sufficient to produce a clearly visible trajectory. The &ray density depends not only upon velocity but also on particle charge, and it is necessary to know the exact relationship in order to evaluate the charge of particles traversing the emulsion. Bradt and Peters8*10-11 treated the problem of &ray frequency versus velocity and charge by applying a modified Mott12 equation, based on Rutherford's law for elastic-coulomb scattering. The number of 6 rays of energies between Emin and Em, is given by (2.1.1.3.1)

Here me is the electron mass and A is a constant, which depends on the composition of the emulsion used, and on the efficiency in observing 6 rays. Emin represents the smallest &ray energy which can be detected or is being considered in the count, and Em,, the greatest energy which the electron can obtain in the collision with a particle of velocity P, or the maximum energy which can be observed in an emulsion of given sensitivity. Bradt three to four grains protrudand Peters have chosen as criterion for Emin ing from the track while Freier et aL9 use a range criterion of 1.5 p for = the minimum length of the track, corresponding in both cases t80Emin 10 kev. Bradt and Peters calculated a family of &ray density versus residual range curves for particles of various charges and masses. The calculations are based on the assumption that the &-raydensity Na(zM) is proportional to the square of the charge, and inversely proportional to the particle velocity. Furthermore, Na(zM) is calculated with reference to the &ray densities in a-particle tracks, which, because of their relative abundance in cosmic radiation, are very well investigated. The &ray density is given by

where R is the residual range. Delta-ray counting is a quite difficult technique, as one can see, when looking at the microphotograph of multiply charged particles. Therefore it is easy to understand that the agreement is not too good if the results H. L. Bradt and B. Peters, Phys. Rev. 77, 54 (1950). H. L. Bradt and B. Peters, Phys. Rev. 80, 934 (1950). 12 N. F. Mott, Proc. Roy. SOC.(London) Al24, 425 (1929).

10

11

302

2.


of various author^^^-'^ are compared. The work of these authors does not clearly verify or disprove Bradt and Peters results. Dainton et al.,lb however, found in their experiments a distinct deviation of the &ray distribution from the Rutherford law. The &ray problem became increasingly more important in connection with investigations about the charge distribution of heavy primaries in cosmic radiation. The earlier experiments, performed especially by the Rochester and Bristol group, gave

-

G 14 g r o h (0)

-

1.5

-

x Dainton eta/ Rutherford W,,:

l5keV

.

FIG. 1. Comparison between experimental and theoretical results. (a) Number of 6 rays with four or more grains as a function of the particle velocity. (b) Number of d rays with residual range equal to or larger than 1.58 p . (The figure is taken from the paper by Tidman et aZ.9 KEY:X, Dainton et al.l6;0, Tidman et aZ.17

quite different results, which a t least in part were caused by differences in the measuring technique and interpretation of the &ray distribution. The &ray problem has been reinvestigated by Tidman et a1.l’ Their treatment of the &ray distribution function is similar as in the work of Bradt and Peters with the exception that the atomic electrons were assumed to be neither free nor a t rest. The authors treat the problem rigorously, considering separately close and distant collisions; they find a l / E z dependance for the former interactions and a l/E4 for the latter. They establish minimum criteria (for the slowest 6 rays accepted in the count) on a more realistic basis, by using experimental range-energy and S. 0. Sorensen, Phil. Mag. 171 40, 947 (1949). L. Voyvodic, Can. J. Research 28, 315 (1950). l6 P. Demers and L. Wasinkynska, Can. f. Research 31, 480 (1953). l 6 A. D. Dainton, P. H. Fowler, and D. H. Kent, Phil. Mag. [7] 43, 729 (1952). l7 D. A. Tidman, E. P. George, and A. J. Herz, Proc. Phys. Soc. (London) A66,1019 (1953). l3

l4

2.1. DETERMINATION

OF CHARGE AND SIZE

303

number of grains versus energy-relations, 18.19 and taking into consideration straggling and scattering effects. The results, which are based on partly theoretical and partly empirical considerations, can be expressed most conveniently by the curves given in Figs. 1(a) and 1(b). The curves compare the calculated results of these authors with the experimental data of Dainton et a1.16 for the grain criterion g 2 4 and the range criterion R 2 1.58 p , both corresponding to energies 2lFj kev. The curves refer to &-ray densities of singly charged particles which are plotted versus the particle velocity 0,. The agreement between experimental and theoretical values, especially in the range criterion case [Fig. l(b)] is truly remarkable, in view of the fact, that the &ray density is given in absolute units. Curve B in Fig, l(a) represents the &ray distribution according to Rutherford’s law and it seems evident that the experimental data are in much better agreement with the rigorous calculations of Tidman et al. (curve A ) . Also later investigations by Dainton and Fowlerz0on the &ray distribution in singly charged particle tracks are in good agreement. The results of measurements on more than 200 tracks of protons (with ranges between zero and 2800 p ) are presented in Table I; the data refer to the minimum criterion of at least 4 grains. The problem of charge identification in the case of heavy primaries is relatively easy, if the exposures were made in such conditions (altitude and geomagnetic latitude) that all primaries are of relativistic velocities. In this case the number of 6 rays, exceeding a certain energy Emi.[see Eq. (2.1.1.3.1)l depends only on the particle charge, and is given by N , = az2 b ; z is the charge of the particle and the constants a and b have to be determined by calibration experiments, which are best performed with the multiple scattering method.21J2Since the &ray densit,y in very heavy primaries is large, it is useful to abandon the grain criterion and to use a range criterion which includes only 6 rays of more than 8 p that protrude distinctly from the core of the The charge of relativistic primaries also can be determined by photometric opacity measurements alone, or in combination with scattering or &ray m e a ~ u r e m e n t s . * ~The - ~ ~photometric method essentially represents a measurement of the ionization, produced along the particle tra-

+

lsR. H. Hem, Phys. Rev. 76, 478 (1949). M. A. S. Ross and B. Zajac, Nature 164, 311 (1949). 20 A. D. Dainton and P. H. Fowler, Proc. Roy. SOC.(London) 221, 414 (1954). 2 1 0. B. Young and F. E. Harvey, Phys. Rev. 109, 529 (1958). 2 2 0. B. Young and F. W. Zurheide, Nuovo cimento [lo] 14, 90 (1959). 23 M. Koshiba, G. Shultz, and M. Schein, Nuovo cimento [lo] 9, 1 (1958). 24 B. Waldeskog and 0. Mathiesen, Arkiv Fysik 17, 427 (1960). 26 0. Mathiesen, Arkiv Fysik 17,441 (1960). 28 K. Kristiansson, 0. Mathiesen, and B. Waldeskog, Arkiv Fysik 17, 455 (1960). 19

304

2.


jectory; because of the high ionization density, this method is preferable to any other ionization method, based on blob or gap measurements. The method which has been developed to highest perfection by the Lund g r o ~ p , has ~ ~ the - ~ great ~ advantage of freedom from personal errors, but TBLE I. &Ray Density as a Function of Residual Range (Proton Track) ~

Residual range in microns

No. of 6 rays

Standard deviation (%)

0-100 100-200 200-300 300-400 400-500 500-600 600-700 700-800 800-900 900-1000 1000-1100 1100-1200 1200-1300 1300- 1400 1400-1 500 1500-1600 1600-1700 1700- 1800 1800-1900 1900-2000 2000-2100 2100-2200 2200-2300 2300-2400 2400-2500 2500-2600 2600-2700 2700-2800

0.01 0.02 0.03 0.06 0.12 0.22 0.28 0.43 0.61 0.71 0.66 0.87 0.81 0.92 0.96 0.84 0.84 0.97 1.01 0.94 1.03 1.06 0.87 0.90 0.97 0.83 0.99 0.96

60 40 40 30 20 15 14 13 11 10 10 9 9 9

8 8 8 8 8 8

8 8 8 8 8

8 8

the disadvantage to depend strongly on normalization and calibration measurements; also depth and dip corrections are more critical than in any other method. I n these e x p e r i m e n t ~ ~ 7the - ~ ~photometric instrument consisted of a Leitz microscope with a K.S. 53 objective; the place of the eyepiece is taken by an eyepiece of special construction. There is a slit in the image plane and above this slit a photomultiplier which registers the *'S. von Friesen and K. Kristiansson, Arkiv Fysik 4, 505 (1952). ** S. von Friesen and L. Stigmark, Arkiv Fysik 8, 127 (1954). 2 9 K. Kristiansson, Arkiu Fysik 8, 311 (1954). *O B. Waldeskog, Arkiu Fysik 10, 447 (1956).

2.1.


305

light passing through it; the dimensions of the slit, referred to the objective plane, were 54 p X 4.3 p. The slit is wide enough to cover not only the core of the track but also a great part of the 6 rays; some of the 6 rays are cut off by the slit, but the amount will be the same for all tracks of equal charge. The opacity or MTW (mean track width), as called by the Lund group, is measured along the track and is compared with background measurements, which are taken alternately after each measurement, at distances 10 p below and above the track. In these experiments only tracks lying in the middle of the emulsion and with dip angles less than 25' (unprocessed emulsion) were accepted for measurements. The dip correction is somewhat more complicated here than in the case of singly charged particles, since one has to consider separately the change in the light transmission through the nearly compact core and through the loose structure of the protruding 6 rays. Therefore special correction factors have to be introduced, which can be determined by calibration experiments. Charge calibration was performed by comparing the experimental results with those obtained by the &ray method (Na = uz2 b). Breakup events and interactions of heavy primaries with emulsion nuclei were used to obtain another independent correlation between mean track width and particle charge. The resolving power of the photometric method, concerning the discrimination between consecutive charges, depends of course on the error made in the mean track width determination. The error, as a function of track length, is given by

+

Here, n is the number of individual measurements per track, uo the standard deviation, and eo an error which is independent of track length and mostly due to irregularities in the emulsion. The error for a track passing through N pellicles is given by e ( N ) = e(L)N-''Z. The error can be made small, if the acceptance criteria are rigorous. Only tracks from the center part of the emulsion with a minimum length of 3 mm per pellicle are accepted, and the authors find that the method gives a better charge discrimination than the &ray method. The problem of charge determination of fast, but not relativistic tracks is much more difficult to solve. In this case not only the mean track width, but also the slope of the mean track width versus residual range curve has to be determined. Kristiansson et aL3I investigated this problem for heavy primaries with charges 6 5 z 2 26. They came to the conclusion that correct charge identification is possible if the following conditions 31

K. Kristiansson, 0. Mathiesen, and B. Waldeskog, Arkiv Fysik 17, 485 (1960).

306

2.


are satisfied: (1) the track length must be longer than 25 mm; and (2) the momentum of the particle with charge z must be smaller than a certain minimum value p , ; e.g., for magnesium p12 1.2 (Bev/c) per nucleon, 1.5 (Bev/c) per nucleon. This method can be used and for iron p26 together with the multiple scattering method, or in place of the latter, if the emulsion is distorted. For tracks of multiply charged particles ending in the emulsion, a combination of the range-energy relation, multiple scattering method (constant sagitta), 6 ray and ionization measurements may lead to correct charge determination if the conditions are favorable. Alvial et recommend for slow particles (e.g., hyperfragments and stable fragments, emitted from stars) a method, based on the count of slow 6 rays, which are observable only as blobs attached to the edges of the track. The fluctuations in the track width, caused by these attachments, are measured with a n eyepiece micrometer (Clausen micrometer). The authors claim that the integral number of these short 6 rays, measured from the end of the track up to a certain residual range R1is a charge-sensitive parameter, giving a distinct discrimination between consecutive charges. The method presumes the accurate measurement of the mean grain diameter and accepts as 6 rays only blob attachments, which cause a broadening of the track, equal or larger than 1.8 times the grain diameter. A somewhat similar method equally based on micrometric width measurements of short ending tracks was proposed by Nakagawa et The authors use for calibration purposes 01 particles of radioactive elements, and Be8, Lis and BEtracks emitted from stars. Using only flat tracks (dip angle 5 12.5') and very short cell lengths, they find that the track width increases proportional with > 1 (classical approximation), and scattering as a pure diffraction effect, zZ/137p > I] according to classical methods, and for fast particles [y = (zZ/137p) > 1 and ( b ) for y

g

200

cv

'S 0

v)

a c

I50

EARTH

3

w

2

100

50

0

2

4

6

0

NEUTRON ENERGY, MEV

FIO.12. Leakage neutron spectrum of a bare critical aeeembly measured by Stewart with the bulk emulsion technique.

care must be taken to insure that the spectrum observed Gas a minimum of contamination from neutrons riot proceeding directly from the source to the detector. I n this case Jezebel was operated out of doors at thirteen feet from the ground. Emulsions (200 Ilford C2 and El) were placed 2 9 L. Stewart, NucleaT Sci. and Eng. 8, 595 (1960).

478

2.


125 cm from the center of Jezebel; their small mass and volume contribute

little perturbation to the measurement. Some of the criteria for track analysis were that the range be greater than 3 p (maximum track lengths observed were of the order of 500p) and that the horizontal and dip angles of each track be less than 16". With the aid of computer techniques (IBM 704) the neutron spectrum shown in Fig. 12 was obtained from track analysis. The lower energy limit for this method is about 0.5 MeV. It is interesting to note the whole spectrum required only two nuclear plates exposed at the same time, thus minimizing errors th a t would occur if many separate irradiations were needed to measure various portions of the curve. Summary. Nuclear emulsions are continuously sensitive and can record many simultaneous events, but cannot establish simultaneity of two events in general. The time-integrated tolerance to background radiation is limited. A considerable time lag exists between the registering of a n event and its translation into physical knowledge. Small weight and volume and freedom from electronics problems are useful features. 2.2.2.1.4.4. Measurement of Neutron Energy by the Bubble-Chamber Method.* The hydrogen bubble chamber is another example of the bulk counter. Very high efficiency of detection may be anticipated because of the large mass of hydrogen that can be used. As in the case of nuclear emulsions, the bubble chamber may accommodate registry of a number of simultaneous events, the number being limited by increasing complication of analysis of the chamber photographs. Problems of recoil range measurements impose a lower limit on the neutron energy that may be studied which is rather higher than previously mentioned methods. There is of course no firm line of demarcation but neutron energies below 5 Mev do not lend themselves to easy measurement in the bubble chamber with the present state of the art. Conceivably the bubble chamber could be used with lower-energy neutrons by studying exothermic neutron-induced reactions in liquids other than hydrogen. Since bubble chambers are not continuously (or nearly so) sensitive, the apparent high efficiency of detect,ion may not materialize as more rapid accumulation of data compared with the other bulk counters. Cycling time may be of the order of 5-10 sec and longer so that relatively high efficiency will be realized only in conjunction with slowly pulsing sources of neutrons of comparable periods. It seems likely that automated data-processing equipment may be able to analyze proton-recoil spectra at a rate comparable to the rate of production, in which case the hydrogen bubble chamber should be able to produce neutron energy spectra at quite a significant rate.

* Refer to Section 2.2.1.2.3.

2.2.

DETERMINATION OF MOMENTUM AND E N E R G Y

479

It is instructive now to consider an actual application of the bubble chamber t o neutron spectra. The 4-in. Berkeley bubble ~ h a m b e r ~ Ois8 ~ ~ indicated in Fig. 13. In this bubble chamber the expansion system is external to the chamber. Dark-field illumination is used with stereoChamber recompression piston Expansion piston

Safety valve

Gaseous hydrogen Regulated

4 TO vacuum pump Fro. 13. Schematic diagram of the four-inch liquid hydrogen bubble chamber of Adelson, Bostick, Moyer, and Waddell.

photography a t 90" to the neutron beam. The illumination was provided by a xenon flash tube. Expansion and photography are suitably phased and synchronized with the accelerator supplying the neutrons. The expansion operation precedes the beam pulse slightly and requires about 4 msec. The limit of sensitive time is approximately 50 msec. Strobe so

H. E. Adelson, H. A. Bostick, B. J. Moyer and C. N. Waddell, Rev. Sci. Instr. 31,

1 (1960). *l

D. Parmentier, Jr., and A. J. Schwemin, Rev. Sci. Instr. 26, 954 (1955).

480

2.


flashing is done between 2 and 5 msec after the beam pulse. It, cannot, be delayed much longer because the bubbles grow too large and turbulence effects from the expansion become manifest. To evaluate the performance as a neutron spectrometer 14-Mev neutrons from bhe D-T reaction were used. The source was somewhat

‘’1

D -T

5/8” Collimator

I

En (MeV)

Fro. 14. 14-Mev neutron spectrum observed with the four-inch bubble chamber. The neutron spectrum has been somewhat distorted by thick-source and collimator effects.

thick and was estimated to furnish a neutron group centered a t 14.1 with a full width of 0.6 Mev at half-maximum. Using a collimator and suitable track criteria, the spectrum of Fig. 14 was obtained. Spectra obtained without a collimator had similar widths but more neutrons in the lowenergy tail. From this study it was estimated that resolution of the over-all system was about 10% for 14-Mev neutrons. It is estimated that

2.2.

DETERMINATION OF MOMENTUM AND ENERGY

48 1

track lengths and averages could be measured to about 1% and that the major distortion of the spectrum was introduced by the collimator. Summary. Hydrogen bubble chambers can provide accurate knowledge of neutron energy and are well suited to efficiently record data from pulsed machines operating a t rates comparable to the bubble chamber. The over-all amount of apparatus and personnel required is rather extensive compared with other methods. 2.2.2.1.5. DIFFERENTIAL SCATTERING: A LINEARMOMENTUM MEASUREMENT. In the first section we noted that the problem of momentum measurement was partly one of establishing the direction of the momentum vector be it linear or angular. I n the remainder of this chapter we shall discuss some specific examples of instrumentation used for momentum observations. The measurement of the differential scattering of neutrons is essentially the study of the probability of finding the neutron linear momentum vector pointing in some given angular interval after scattering. Techniques can be applied in several ways to measure the angular distributions of scattered neutrons. In a straightforward way one could use any of the recoil detectors discussed previously to measure the number of neutrons scattered at some angle by a scatterer with suitable shielding and collimation. If the detector is electronic, perhaps time-of-flight could be incorporated. We shall, however, consider another variant which will also lead us into the problem of measuring the spin direction. 2.2.2.1.5.1. Angular Distribution from Recoil Scatterer Measurements. If the recoil nucleus is light enough, the colliding neutron will impart significant kinetic energy. If the recoil nucleus is a constituent of the gas filling of a proportional counter, a signal will be produced in the counter proportional to the energy of the recoil. For all gas fillings, and for a monochromatic neutron beam, the amplitude of the signal will then be a measure of the scattering angle of the neutron. * The pulse-height spectrum gives the center-of-mass angular distribution on a cosine scale. Angular distributions of scattered neutrons from the hydrogen isotopes up to nuclei as heavy as neona2have been measured by this method. As a n illustrative example we shall consider the case of neutron-helium scattering as this will lead us naturally to the problem of spin analysis. One could of course observe the helium recoil by a variety of methods: liquid-helium bubble chamber, liquid-helium scintillator, cloud chamber, etc. * Strictly this is not true for the case of the hydrogen scatterer since the neutron is slightly heavier and this leads to double-valuedness in the scattering. Practically the mass difference between neutron and proton is often ignored. 32 H. 0.Cohn and J. L. Fowler, Phys. Rev. 114, 194 (1959).

482

2.


This method has been practiced by S e a g r a ~ eusing ~ ~ a proportional counter to observe the helium recoils. Helium is not a n especially tractable gas for counter fillings since it does not permit very much stable gas multiplication and has low stopping power. To some extent this situation is alleviated by using a mixture of helium and krypton. Signals from the krypton recoils are of course of much lower amplitude than those from helium. Boundary effects were diminished by collimating the neutron beam as shown in Fig. 15. Corrections need only be made for the back wall since all other recoils will stay within the active volume of the counter.

NEUTRON SOURCE

COLLIMATOR

PROPORTIONAL COUNTER

FIG.15. Schematic diagram of Seagrave’s proportional-counter arrangement for the study of neutron-He* scattering.

It is necessary to ensure that pulse distributions are reasonably independent of the physical position of the recoil trajectory in the counter. Seagrave obtained angular distributions of neutrons at energies of 2.6, 4.5, 5.5, 6.5, and 14 MeV. With the exception of the 14-Mev point these data, in the center-of-mass system, are exhibited in Fig. 16. The theoretical calculation is th at of Dodder and Gamme1.34It will be noted that this method does not lend itself to study of the forward scattering since the recoil signals grow smaller and the resolution will deteriorate. If a cloud chamber is the same method can be used a t considerably smaller forward scattering angles. J. D. Seagrave, Phys. Rev. 92, 1222 (1953). D. C. Dodder and J. L. Gammel, Phys. Rev. 88, 520 (1952). 36 D. F. Shaw, Proc. Phys. Soc. (London) A67, 43 (1955). 33

54

2.2.


483

Summary. Observations of the recoil signal in a proportional counter provide a simple and rapid technique for obtaining neutron-scattering angular distributions. Measurement of forward scattering is somewhat restricted. Quality of counter filling is quite important. 680 640

600 560

520

FIG.16. Examples of differential scattering data obtained by Seagrave with a proportional counter.

2.2.2.1.6. POLARIZATION. Our final consideration of the neutron beam is to ascertain its average polarization by recoil techniques. The approach to this question is rather less direct than in the previous remarks and we will briefly examine some of the relations involved from a pragmatic viewpoint. If one measures the probability or cross section for the scattering of a beam of initially unpolarized neutrons by a target having no intrinsic angular momentum (i.e., spin zero), the data may be represented by a

484

2.


general relation. This relation, of quantum-mechanical origin, is an expansion of the cross section in terms of angular momentum and spin of the neutron. It, has the form (center-of-mass system)36 u(0) = IAIz

+ IB12

(summed over spins)

(2.2.2.1.6)

where

A

1 2ik

{(Z

= -

+ l)[e2isr+- 11 +

Z[e2"l-

- l]}P~(cose) (2.2.2.1.7)

2

and (2.2.2.1.8)

The P's are Legendre polynomials. The unknown quantities are the 6's. They measure the strength of participation of a state of orbital angular momentum 1 and spin up (+), J = 1 &, or down (-), J = I - +, in the scattering process. For our purposes the 6's may be regarded as obtained by fitting the formulas to experimentally determined scattering cross sections. In practice, only a few values are required to reasonably fit the corpus of experimental data. Thus for neutrons of 10 Mev or less good fits are obtained for the set I = 0, 1, 2, on He4. If the 61% phases are determined, the polarization produced in the scattering can be computed from the relation3'

+

p(e) = -

AB* AA*

+ BA* + BB*

(2.2.2.1.9)

where n is a unit vector having the direction X kn.out. The phases for n-He4 scattering are becoming moderately well known. Seagrave has obtained a set from his data cited in the previous sections. Additionally, Dodder and Gamme13*have inferred n-He4 phases from p-He4 phases; their results are consonant with Seagrave's phases.? The latter set have been used by Levintov et ~ 2 1 to . ~ compute ~ a polarization map for n-He4 scattering. 2.2.2.1.6.1. Helium Polarimeter. There are many possible experimental techniques for the observation of the recoil He4's in polarization studies. We shall consider the method of Levintov et U Z . , ~ ~which is essentially a variant of the bulk-counter technique with additional geometrical constraints. Bundles of long, thin paraxial proportional counters filled with

t See the Appendix. 86

F. Bloch, Phys. Rev. 68, 829 (1940).

wJ. V. Lepore, Phys. Rev. 79, 137 (1950). We use the negative of Lepore's expression to conform to the Base1 Convention. *a I. I. Levintov, A. V. Miller, and V. N. Shamshev, Nuclear Phys. 3, 221 (1957).

2.2.

DETERMINATION O F MOMENTUM AND E N E R G Y

485

helium were employed to make a left :right measurement. Since there is a known and unique energy for the helium recoil for each angle of scattering, it is possible by pulse-height analysis of the proportional-counter signals to define the helium scattering angle. Geometry, in this case, improves the accuracy of angular knowledge. The counter pressure is set at that value where the helium recoils of desired angle just spend their range in the counter. Referring to Fig. 17 we see then that if signals of a certain maximum amplitude only are counted, then only recoils originating at one end of the counter, A , and having the appropriate energy corresponding to stopping a t B will register. Thus a target area of recoils is defined withDno window barrier between the target and counter, a desirable arrangement to save recoil energy.

f =

HELIUM RECOIL

FIG. 17. Helium-recoil counter-polarimeter of Levintov, Miller, and Shamshev. Detection of maximum-range particles is accomplished by high bias setting.

These refinements over the simple bulk counter, e.g., Seagrave’s, are necessary since we require not only the value of the angle of scattering, but a knowledge of whether the scattering was left or right. Complete 1eft:right symmetry of the neutron field and counters must be achieved or satisfactorily accounted for to avoid spurious data. It is possible to compute the counting rate of this device as a function of discriminator setting and obtain good experimental agreement. The preceding polarimeter, by virtue of using helium-filled counters, will require systems and problems tolerant of quite long pulse lengths (-10 psec). With scintillation techniques it is possible to construct highefficiency polarimeters wit8hcharacteristic pulse-times nearly a thousand times shorter. We briefly note a scintillation polarimeter under development by Perkins and Simmoris.”YThey have utilized the fact that a recoil He4 in 39 R. U. Perkins and J. E. Simmons, Lot; Alamov Scientific Laboratory (private communications).

2.

486


liquid helium generates a scintillation of rather short duration.40 I n principle pulse-height analysis of the scintillation value gives the angle of recoil but not the sign. In practice i t is found convenient to establish the angle and sign of scattering by detecting the neutron after its helium scattering with a high-efficiency organic scintillator. Thus the helium scintillation furnishes a fiducial signal which is then in coincidence with the organic scintillator. Evidently time-of-flight can be imposed so that neutrons of only the desired energy are measured. Such a fast system should be well adapted to “slavingff to a pulsed machine such as the cyclotron to further reduce background. Summary. Gaseous and liquid helium polarimeters can cover a broad spectrum of neutron energies: the former being relatively slow and the latter fast. Knowledge of the analyzing power of helium or other material is not very extensive or accurate yet. 2.2.2.1.7. APPENDICES. 2.2.2.1.7.1. Appendix A . Kinematics.* A few relations and some data useful for preliminary design considerations are given in this Appendix. The nonrelativistic kinematical relations41,42 between the incident particle and the elastic recoil are listed below. We use the following notation:

MI = rest mass of the incident, particle,

M z= rest mass of Eo

= El= Ez = 6 = $= 6’ =

$’

=

the target or recoil particle, laboratory energy of the incident particle, laboratory energy of the incident particle after collision, laboratory energy of the recoil particle after collision, laboratory scattering angle of the incident particle, laboratory scattering angle of the recoil particle, c.m. scattering angle of the incident particle, c.m. scattering angle of the recoil particle.

In terms of the abbreviations

* Refer to Appendix B. H. Fleishman, H. Einbinder, and C. 8. Wu, Rev. Sci. in st^. SO, 1130 (1959). B. Carlson, M. Goldstein, L. Rosen, and D. Sweeney, Los Alamos Scientific Laboratory Rept. LA-723 (1949). Unpublished. 42N. Jarmie and J. D. Seagrave, eds., “Charged Particle Cross Sections.” Los Alamos Scientific Laboratory Rept. LA-2014 (1956). ‘0

2.2.


487

the energies can be simply expressed as

z -- 2 c(1 - cos e’) E2

= 4c cos2 $

and

I n the last expression, use both signs if M I > M 2 ( D < l ) , in which case sin Omax = D. Use only the plus sign if M1 I M 2 ( D 2 1). The angles are related by the equations

D sin 2$ 1 - D cos 2$ 0’ = ?r - 2$, D sin(0’ - 0)

tan 0

=

and =

?r

-

el

=

=

sin 0

2+.

The cross sections transform from the laboratory system to the c.m. system in the ratios u(0’) --

u(0)

- sin 0 d 0 - C ( D z - sin2 0 ) I I 2 sin 0’ de’ Ei/Eo

u(8’) --

- sin $ d$

and u($)

sin e’de’

-

~. 1

4 cos $

Although the instruments described in this chapter have not operated in the extreme relativistic range some are capable of much extension. Moreover, if one wishes to do very precise work with neutrons even as low as 25 or 30 Mev it is necessary to use relativistic transformations. Blumberg and Schlesinger43 have compiled the relativistic kinematical relations. From this compilation we quote the following results (using the previously defined symbols for angles but the notation T O ,T O ,and T+ for the relativistic kinetic energies corresponding to 3 0 , El, and E 2 in the nonrelativistic case, respectively). The energies are given by the expressions

and

4 3 L. Blumberg and L. Schlesinger, “Kinematics of the Relativistic Two-Body Problem.” Los Alamos Scientific Laboratory Rept, LAMS-1718 (1955). Unpublished.

488

2.


where YI

and

= I

+ (TOlMlC2)

The angles are related by the equations cos

- yk2 tan2 + *’ = 11 + Y : ~ tan2 $

and tan 0 = Y2’(cY

sin $‘ - cos *’)

where

Finally, the recoil cross sections transform in the ratio

’ [(a)’ Mzc2 + 2 (“)]’” M

Y2

$2

-

[ + g2]1

(Y;2

1)1’2COS

*

The behavior of some of these relations is indicated in Fig. 18, Fig. 19, and Fig. 20, for the particular case of n-p scattering, 2.2.2.1.7.2. Appendix B. Neutron-Proton Cross Sections. Gamme144has considered the problem of generating a useful set of n-p cross sections for E , up to 40 Mev from the fragmentary experimental data now extant. He has found that the total cross section may be represented by uT(Eo) =

+

3~[1.206E (- 1.860

+ 0.09415E + 0.0001306E2)2]-’ + ~[1.206E+ (0.4223 + 0.1300B)2]-’

and the differential cross section in the center-of-mass system by

where EOis the laboratory neutron energy in Mev and b

=

2(E0/90)~.

4 4 J. L. Gammel, to be published in “Fast Neutron Physics” (J. B. Marion and J. L. Fowler, eds.), Vol. IV, Part 11. Interscience, New York, 1960. (I am indebted to the author for this advance communication.)

2.2.

10.000

DETERMINATION OF MOMENTUM AND ENERQY

489

F

1000

-> W

2

Y

I-

150

ic

10

I

I

10

00

1000

10,000

l00,000

To (MEV) FIG.18. Relation of recoil proton energy T#to incoming neutron energy T oin the laboratory system, for n-p scattering.

490

2.


1000

I

10

1000

100

l0,OOO

100,000

To (MEW FIG. 19. Cross-section transformation for t h e recoil proton from t h e laboratory system into the center-of-mass system, in n-p scattering.

2.2.


491

J8C I70 160

160

140

I30 120 I10

100

JI'

90 80

70 60

50 40

30 20 10

0

,

1

1

1

1

1

1

,

I

10

,

I

1 , 1 1 1 ,

I

1

I

,

,

,

,

100

,

L]

1000 TO

,,,,,,,,, l0,OOO

I l00.000

(MEV)

FIG. 20

+

FIQ.20. ReIation between the laboratory angle and center-of-mass system angle $' of the recoil proton, in n-p scattering.

2.

492


33

-

31

-

27

-

P

156 MEV

Z 7 MEV

4 I

W Iu)

a

- 4

- 2 20

-

I

I

- -

4 0 MEV

I

I

I

I

I72 MEV

- 12

I8 .-

I

I

I

4--I

I

0

- 15

215 MEV

QOMEV

0 >'IS

I

3OL 0

v I

t

30

60

I

I20

6

-

I

1800 30 6 C 90 120 C.M NEUTRON SCATTERING ANGLE 8' 90

12

150

Is0

180

FIG.21. Family of n-p scattering data collated with the calculations of Clementel and Villi.

FIG.22. Theoretical n-He4 phase shifts of Dodder and Gammel compared with the experimental values of Seagrave.

2.2.

493


Lo

I

I

I

I

I

1

0.8 0.0 0.6 0.4

0.2

P

o -0.2 -0.4 -0.6 -0.8

-I

.o 0

I

I

I

I

I

I

I

I

I

2

4

6

8

10

12

14

16

18

20

En (LAB) FIG.23. n-He4 polarization map inferred by Levintov, Miller, and Shamshev from the phases of Dodder and Gammel and of Seagrave.

Gammel and Thaler46and Clementel and Villi4Ehave also assembled n-p scattering data at higher energies and have analyzed them. For orientation some results of Clementel and Villi are displayed in Fig. 21. 2.2.2.1.7.3. A p p e n d i x C. Neutron-He4 Scattering. Design cross sections at any energy may be computed from the phase shifts according to Eqs. (2.2.6), (2.2.7), and (2.2.8). I n Fig. 22 are plotted the theoretical phase shifts of Dodder and Gamme134 together with experimental points of Seagra~e.~~ Polarization can be calculated from the phase-shift equation (2.2.2.1.9). Such calculations have been done by Levintov, Miller, and Shamshev and are shown in Fig. 23. For polarization at higher energies one may use a proton-alpha polarization map4’ shown in Fig. 24, and equate neutron and 46 J. L. Gammel and R. M. Thaler, in “Progress in Elementary Particle and Cosmic Ray Physics” (J. G. Wilson and S. A. Wouthuysen, eds.), Vol. 5, p. 155 North Holland Publ., Amsterdam, 1960. 48E. Clementel and C . Villi, Nuovo cimenlo [lo] 6, 1167 (1957). 47 J. L. Gammel and R. M. Thaler, Phys. Rev. 109,2041 (1958).

494

2.


E L A B( M e V )

FIG.24. Quasi-theoretical p-He4 polarization map calculated by Gammel and Thaler.

proton energies. This is not a strictly correct procedure but will provide a reasonable number. ACKNOWLEDGMENTS

I should like to thank N. Jarmie, L. Rodberg, L. Rosen, J. D. Seagrave, and L. Stewart for criticism.

2.2.


495

2.2.2.2. Time-of-Flight Method.* 2.2.2.2.1. INTRODUCTION. One of the most successful methods for determining the energy dependence of the interaction of neutrons with matter is the time-of-flight technique. t With this technique bursts of neutrons which have a wide distribution in energy are produced for a short interval of time a t a specific place, a neutron detector is placed a measured distance from the effective source of the neutrons, and the time between the production of the burst of neutrons and the detection of a neutron is measured. Knowing the time required for the neutron to travel over the measured path enables one t o calculate the velocity of the neutron and its energy. B y interposing samples between the source and the detector to absorb or scatter the neutrons, one can determine the effect of the sample on a neutron of a particular energy. It is this variation in energy of the interaction of the neutrons which gives information of value both in the study of the structure of the nucleus and in the study of the structure of solids and liquids. A summary of the important properties of the neutron and its interactions, is given in Table I, pages 496 and 497. Neutrons can only be produced in nuclear reactions. The reactions usually used for the production of neutrons are U236(n,f,2.43n); H2(d,n)He3; U(r,n). H3(d,n)He4;Li7(p,n)Be7; Beg(d,n)B'O; B e 9 ( ~ , n ) N 1Beg(y,n)Be*; 2; The energy of the neutron emitted in the nuclear reaction depends on the initial energy of the incident particle or photon, the particular nuclear reaction, the angle between the direction of the incident particle or photon, and the direction in which the neutron is emitted. The time-of-flight technique has been used t o study neutrons of enerev and as high as 5 X lo9ev. Because this energy gies as low as 2 X range is so wide, no single apparatus can be used to study the entire spectrum. The details of the neutron source and the experimental apparatus used t o measure the time of flight vary markedly with the energy of the neutron. It is therefore convenient to divide the entire energy range into three different energy intervals because of the differences in techniques used. These are shown in the tabulation. (1) Slow neutrons (a) Thermal (b) Epithermal (2) Fast neutrons (3) High-energy neutrons

E < 0.5 ev 0.5 ev < E lO4ev < E lOsev < E

t Refer to Section 2.2.1.3.1.

* Section 2.2.2.2 is by W. W.

Havens, Jr.

< 106 ev < lo8 ev

496

2.


The boundaries of these categories are somewhat arbitrary, and experiments which are designed primarily to measure the effects of epithermal neutrons can also be used to measure the effects of fast neutrons. The techniques used for time-of-flight measurements on high-energy neutrons are substantially the same as those used for fast neutrons, so category 3 will not be discussed in detail. Neutrons produced in nuclear reactions are usually fast, and it is therefore necessary to slow them down before their interactions can be studied. The slow neutron category is divided into two parts, thermal and epithermal, because the information obtained from studying neutrons with energies below 0.5 ev is primarily-although by no means exclusively-of interest in the study of the structure of liquids and solids, whereas the information obtained from studying neuTABLE I. Summary of Properties and Interactions of Neutrons Data

Property General

Mass, 1.00898; charge, 0; spin +; half-life 10-12 min., statistics, Fermi-Dirac

Wave length, energy, and velocity conversions

X = h/mv = h / . \ / m

~~

X(A.)

=

~~

3.96 X 108/v(meters/sec)

=

0.286/=

t(psec/meter) = l/u(meters/sec) X

lo8 = 253X(A)

3.96 X 10-at

= 72.3/.\/E(ev) E(ev) = 5.226 X 10-Dva(meters/sec) = 5226/t2 = 0.0818/X2(A) = 3.96 X 103/X(A) v(meters/sec) = 1O6/t = 1.383 X lo4 X = wavelength, cm; A(& = wavelength in 1 units; h = Planck’s constant, 6.624 X 10-81 erg sec; rn = mass of neutron in grams; v = velocity, cm/sec; v(meters/sec) = velocity in meters per E(ev) = kinetic second; E = kinetic energy in ergs = energy in electron vo1ts;a t = time of flight in microseconds per meter.

m)

~~

= 1 ev, = 0.286 d; a t E = 0.026 ev, X = 1.8 A; wavelength in this energy range is of the order of distance between atomic planes in crystals. Bragg diffraction occurs: nX = 2d sin e where n = order of reflection, d = distance between atomic planes, and 0 = angle of incidence with plane of atoms. Interaction with Pass through matter much more readily than charged particles. matter Practically no ionization produced. Fast neutrons knock protons from hydrogen-containing material. Prolonged irradiation may change color, thermal conductivity, or electrical conductivity. Bonds may be broken with decomposition of molecules. Owing to their magnetic moment, slow neutrons interact with electron magnetic moment of paramagnetic and ferromagnetic atoms.

Diffraction by crystals

At E

2.2.

DETERMINATION OF MOMENTUM AND ENERQY

497

TAEILH I. (Continued) Property Detection

Data Fast neutrons : recoil protons and nuclear reactions. Example: IP SP(n,p)A12* b Sias 2.3 min. 3.0 Mev

Slow neutrons: radioactivation of foils of In, Mn, Au, Ag, Rh, etc. Example: 8MnK6 (n,y)Mn66 Fe66 1.69 hr, 2.8 Mev

Counters lined with B or Li or proportional counters filled with B'OFa. Example: B10 n -+ Li' (Y L i ' + n + H * +(Y Fission Example: U2as n -+high energy fission fragments Photographic plates containing elements that become radioactive by interaction with neutrons. Scintillation counters arranged to detect y-rays emitted when neutrons are absorbed or to detect other nuclear reactions caused by neutrons.

+

+

+

Scattering Elastic nuclear scattering; inelastic nuclear scattering; resonant nuclear scattering; coherent crystal scattering (diffraction) ; procem, (n,n) reactions ferromagnetic scattering; paramagnetic Scattering; inelastic molecular scattering; neutron-electron scattering. Absorption process

The neutron is retained by nucleus and a photon or other particle is emitted: (n,y), (n,p) (%,a).Also (n,2n) and (n, fission).

One electron volt is the energy acquired by an electron when it is accelerated by a potential difference of 1 volt. The electron acquires an energy of E = Ve = 4.802 X 10-1°/299.8 = 1.602 X erg and a velocity given by E = #mu2or v = m m = 1/(2)(1.602 X 10-12)/9.107 X 10-28 = 5.931 X lo7 cm/sec Per mole of electrons, the energy would be (1.602 X 10-12)(6.023 X loz3) = 9.648 X 10" ergs/mole = 96,480 joules/mole = 23,055 cal/mole A neutron with an energy equivalent to one electron volt would have an energy of 1.602 X 10-12 ergs and a velocity of u = d(2)(1.602 X 10-12)(6.023 X 10z3)/1.00898 = 1.383 X lo6 cm/sec or 1.383 X lo4 meters/sec. ( I

trons of energies between 0.5 and lo6 ev is primarily of interest in the study of the structure of the nucleus. 2.2.2.2.2. THERMAL NEUTRONS. 2.2.2.2.2.1. Sources.* Among the important factors determining the experiments which can be performed

* Refer to Vol 5B, Section 3.2.1.4.

498

2.


with slow neutrons are the availability, intensity, and total number of neutrons emitted by the neutron source. The increase in the number of research reactors, as well as the increase in the neutron flux available with new reactors, will certainly increase the number and types of neutron experiments performed. Although some of the early work in the study of thermal neutrons was done with neutrons produced with radioactive sources and charged particle accelerators, practically all of the research in this energy range is now performed using nuclear reactors. No other source can give anywhere near the average intensity or total number of thermal neutrons that can be obtained from a nuclear reactor. Of course, this does not mean that it is impossible to perform ingenious experiments with thermal neutrons using natural radioactive sources or particle accelerators, and some outstanding experiments using these sources have been done recently when a reactor was not available. I n most cases, however, it would have been much simpler to perform the experiments if a reactor had been available. It has been suggested on many occasions that particle accelerators can be used to compete with reactors in all types of neutron studies. Although particle accelerators, especially those which naturally produce neutrons in bursts, have definite advantages for neutron studies with epithermal and fast neutrons, a simple order-of-magnitude calculation shows that they cannot compete with a reactor for studies with thermal neutrons. Although neutron production rates as high as 1019n/sec in the pulse are possible with a pulsed synchrocyclotron,l the more usual continuous production rate with a Van de Graaff accelerator or a Cockcroft-Walton machine is 10%/sec. This number of neutrons would be produced per second with a 300-pa beam of 2-Mev deuterons incident on a heavy ice target or a l-ma beam of deuterons of 250-kev energy incident on a zirconium-tritium target. The neutrons from the H2(d,n)He3reaction using 2-Mev deuterons would have an energy about 5 MeV, which would require a sphere of paraffin of about 10 cm radius to moderate the neutrons. Thus the neutron flux emitted from the sphere if all the neutrons were moderated and none were captured would be

Q

=

[1011/4s(10)2]= 8 X 107n/cm2/sec.

If the H3(d,n)He4reaction were used, the neutrons would have an energy of about 14 Mev and a sphere of approximately 20-em radius would be required to moderate the neutrons to thermal energies, reducing the flux 1.6 x 1O7n/crn2/sec. Since even a small a t the edge of the sphere to Q reactor of enriched uranium operating at 2 kw has a central neutron flux of about 10"n/cm2/sec and an edge flux greater than 101°n/cm2/sec, it is

-

J. Rainwater, W. W. Havens, Jr., J. S. Desjardins, and J. L. Rosen, Rev. Sci. 31, 481 (1960).

znstr.

2.2.

DETERMINATION OF MOMENTUM A N D ENERGY

499

obvious that the flux and the total number of neutrons available from a reactor is a t least two orders of magnitude greater than the flux available with a particle accelerator. Since the reactor flux of thermal neutrons is so much higher than can be obtained with particle accelerators, our discussion in this section will deal primarily with experiments which use reactors as a thermal neutron source. 2.2.2.2.2.2. Mechanical Chopper. The energy and time distribution of the neutrons emitted by a reactor are both continuous. Since a pulsed source is required for time-of-flight studies, a device called a chopper is usually used to produce the burst.* FIXED MASS O F FISSIONABLE MATERIAL SLIGHTLY LESS THAN C R I T I C A L \ ~

MASS OF FISSIONABLE MATERIAL SLIGHTLY LESS THAN CRITICAL MOUNTED ON ROTATING DISK

FIG.1. Schematic diagram of pulsed neutron source using two subcritical masses.

A collimating system is placed in the reactor shield to form a beam of neutrons as shown schematically in Fig. 2. A mechanical shutter (called a chopper because it chops the beam off in a short time interval) intercepts the continuous beam emitted by the reactor and thus produces the short bursts required. The detector for neutrons in the thermal region is usually a proportional counter or ionization chamber filled with boron trifluoride.

* Several methods of obtaining pulsed reactors for neutron spectroscopy have been suggested, both seriously and in jest. One device called the “dragon” would use a fixed mass of fissionable material just smaller than a critical mass and several subcritical masses on the rim of a wheel which rotated as shown schematically in Fig. 1. When the two subcritical masses were closest, the assembly would be above criticality and a large pulse of neutrons would be produced. The bursts could be very intense and also could be very short if the wheel were rotated at a sufficiently high speed. Although it might be practical, no one has as yet published any description of the operation of such a device. An atomic bomb gives a short burst of neutrons of very high intensity and bombs have been used for neutron spectroscopy.e Although the pulse of neutrons is extremely large, only one burst is available from each bomb and therefore few results have been obtained. * G. A. Cowan and A. Turkevich, Bull. Am. Phys. SOC.[2] 4, No. 1, p. 31 (1959).

500

2.


These devices make use of the boron disintegration reaction

B’O

+ n
100 near the binding energy of the neutron is usually less than 1 kev and sometimes less than 1 ev, so experiments which use particles of several kev energy cannot hope to resolve these levels. The measured neutron cross sections in the epitherma1 region are not only of great value for the study of nuclear structure but are also very important for their application to reactor physics. We will not deal with reactor physics applications, which are treated elsewhere,62but with the application to the study of nuclear structure. However, several millions of dollars have been saved in the design of production and experimental reactors because our knowledge of neutron cross sections at thermal energies is so extensive. One of the first contributions of neutron time-of-flight spectroscopy to the theory of nuclear structure was the experimental verification of the Breit-Wigner theory for isolated resonances. A plot of the experimental total cross section of cadmium,53which varies in magnitude from 5.7 to 7800 barns, is shown in Fig. 29 in the energy interval from 0.008 to 10 ev. The solid curve is the theoretical curve for EO = 0.178 ev, I’ = 0.114 ev, uo = 7800 barns, and u p = 4.9 barns, where EOis the resonance energy, 61 See, for example, K. Siegbahn, ed., “Beta- and Gamma-Ray Spectroscopy.” North-Holland Publ., Amsterdam, 1955. E. P. Wigner and A. Weinberg, “The Physical Theory of Neutron Chain Reactors.” Univ. of Chicago Press, Chicago, 1958. azL. J. Rainwater, W. W. Havens, Jr., C. S. Wu, and J. R. Dunning, Phys. Rev. 71, 65 (1947).

2.2.


539

ANL-fc7

I 3.01

0.I

I.o

FIG.29. Experimental and theoretical cross sections of the neutron resonance in the compound nucleus Cd1I4at a neutron kinetic energy of E = 0.178 ev.

r is the total level width, uois the cross section at the resonance, and up is the potential scattering cross section. The theoretical curve fits the experimental curve to within the experimental accuracy, even though the cross section varies by a factor of about 1500 and the energy varies by a factor of 1250. There are few processes in physics in which the theoretical curve fits the experimental data over so wide a range for both the ordinate and the abscissa. Several more experimental of the validity of O 4 R. E. Wood, Phys. Rev. 104, 1425 (1956); R. K. Adair, C. K. Bockelman, and J. M. Peterson, Phys. Rev. 76, 308 (1949); J. M. Peterson, H. H. Barschall, and C. K. Bockelman, Phys. Rev. 79, 593 (1950).

I

I

I

'

1.2

"23@

.:

J

...... I

I000

FIG.30. Backmound mlhtmntd noiintsl ncr ohnnnd fnr at the detector position waa viewed by large plastic scintillation detectors to detect (n,?) capture ?-rays. For the D T curves a sample having c1 = 475 bams/atom was also in a transmission setting. The plots show counts per 0.1 wec detection interval width for a 35.37-meter neutron path. [From J. L. Rosen, J. S. Desjardins, J. Rainwater, and W. W. Havens, Jr., Phys. Rev. 118,687 (1960).] ~

+

~~

2.2.

DETERMINATION O F MOMENTUM AND ENERGY

54 1

the application of resonance theory to the analysis of experimental data have been made and this theory is now regularly assumed to hold in determining the parameters of a particular resonance level. Energy level spacing and distribution of energy level spacings. A recent set of data showing the slow neutron resonance levels in the compound nucleus U2a9 in the energy interval from 100 to 1300 ev is shown in Fig. 30. The most striking feature of these data is the sharp peaks which occur. Each of these peaks represents a resonance level in the compound nucleus.

60/

FIG. 31. The sum E = 0 to E of the number of observed resonances in Uz88 to energy E. The slope of the curve yields the average level spacing d. The number of resonances contained in the indicated energy subintervals are shown in brackets. [From J. L. Rosen, J. S. Desjardins, J. Rainwater, and W. W. Havens, Jr., Phys. Rev. 118, 687 (1960).]

The actual positions of the levels are not of particular theoretical interest at the present time because the complexity of the nuclear many-body problem does not allow the observed positions of the levels to be compared with theory. However, the average energy level spacing and the distribution of energy level spacings are of theoretical interest. The average energy level spacing is obtained by plotting a histogram of the integral distribution of energy levels vs energy as shown in Fig. 31 for the UZa8 data given above. The slope of the curve provides the average level density or the inverse quantity, the average level spacing D. The dis-

542

2.


tribution of level spacings is obtained by determining the number of level spacings which are within a convenient energy interval AE, and plotting a histogram of the number of energy levels per energy interval AE versus the neutron energy. The convenient energy interval for the U238data shown in Fig. 30 is 5 ev and the dist,ribution of the 54 energy levels observed is shown in Fig. 32.

S

P

A

C

I

N

G

( c v )

FIG.32. The distribution of leveI spacings per 5 ev interval for the first 54 level spacings in U’38 from 0 to 1000 ev. The smooth curves represent the “repulsion” formula proposed by Wigner and an exponential function corresponding t o a random distribution of spacings. [From J. L. Rosen, J. S. Desjardins, J. Rainwater, and W. W. Havens, Jr., Phys. Rev. 118, 687 (1960).]

The energy level spacing and the distribution of energy level spacing were investigated theoretically by Bethe some time agob5and interest has recently revived.56The experimental results on the average energy level spacing and the distribution of energy level spacings is in good agreement with the trends predicted by theory, but the theory is not sufficiently well developed to make detailed comparisons. 5 7 Neutron spectroscopy gives information on the details of the decay H. Bethe, Revs. Modern Phys. 8, 82 (1957). C. Block, Phys. Rev. 93, 1094 (1954); T. D. Newton, Can. J . Phys. 34,804 (1956); N. Rosenzweig, Phys. Rev. 108, 817 (1957); J. M. B. Lang and K. J. LeCouteur, Proc. Phys. SOC.(London) A67, 486 (1954); S. Blumberg and C. E. Porter, Phys. Rev. 110, 786 (1958); N. Rosenzweig, Phys. Rev. Letters 1, 24 (1958); C. E. Porter and N. Rosenzweig, Ann. Acad. Sci. Fennicae, Ser. A , V I . Physcia No. 44 (1960). 6’See J. A. Harvey, Proo. Intern. Conf. on Nuclear Structure, Kingston, Canada, 1960, p. 659. University of Toronto Press, Toronto, 1960. 66

2.2.


543

of the compound nucleus. The neutron width rnis related to the phase of the nuclear wave function a t the nuclear boundary. Early theories of nuclear structure predicted that the neutron width of a resonance would be directly proportional to the square root of the energy of the resonance. However, wide variations from resonance to resonance are observed in the neutron widths and the fidependence does not hold in detail. A new concept had to be introduced to correlate the experimental data with

ev)llZ FIG.33. The distribution of reduced width amplitudes (F%a)i'zper 0.5 interval for 54 resonances in U'-'38from 0 to 1000 ev. The smooth solid curve represents the Porter-Thomas distribution ( V = 1) while the dashed curve corresponds to a random distribution of reduced widths ( v = 2 ) . [From J. L. Rosen, J. S. Desjardins, J. Rainwater, and W. W. Havens, Jr., Phys. Rev. 118, 687 (1960).]

the theory. This new concept assumed that the phases of the nuclear wave function a t the nuclear boundary for different resonances were statistically independent and random. After this new theoretical assumption was introduced, it was possible to compare the observed distribution of rnwith the distribution to be expected on this random phase basis. A comparison is shown in Fig. 33 of the distribution of neutron widths for U238with a Porter-Thomas distribution, which is the name given to the random phase distribution,

544

2.


The parameter Fn0is the neutron width reduced to a standard energy, usually 1 ev, to remove the l / d E dependence expected theoretically, i.e., rn0 = I',/dE. The agreement between the theoretical curve and the experimental data is very satisfactory. The total y-ray width I??, which is the probability for a compound nucleus to decay by emission of a y-ray, has been measured for many nuclei for transitions from a highly excited state of the compound nucleus t o the ground state of the compound nucleus. The total y-ray width is expected to be constant for a particular spin state in a specific isotope. Sufficient y-ray widths have recently been obtained6s so that detailed comparisons can be made between the experimental data and some of the predictions of nuclear theory. The agreement is only fair. Partial y-ray widths to a particular level in the compound nucleus have been observed recently and found to differ markedly from level to This variation has stimulated interest in measurements of partial y-ray widths.6DInvestigation of partial y widths will undoubtedly give us a better understanding of the y-decay process after neutron capture. The strength function, defined as S = F / D , where 2 is the average reduced neutron width and D is the average level spacing, is of theoretical interest since it represents the penetrability of the nuclear surface. This strength function is not a property of the nuclear energy level system but of the nuclear surface itself and is therefore of great importance in connection with the cloudy crystal ball model of the nucleus. The experimental values of the strength function for many isotopes have been compared with those predicted by various cloudy crystal ball models of the nucleus, and modifications of theory have resulted which give us a better picture of the nucleus. 2.2.2.2.4. FASTNEUTRON TIME-OF-FLIGHT TECHNIQUES. Slow neutron time-of-flight techniques have been used for many years, but the extension of these techniques to higher energies is just developing. The delay in this development is primarily technical. The flight times to be measured for fast neutrons are in the range of 10-7 to 10-9 sec, a range which has only recently become accessible because of developments in photomultipliers, scintillators, and amplifiers. Also, time-of-flight techniques have been slow to develop in the fast neutron region because monoenergetic sources of neutrons are readily available from Van de Graaff, Cockcroft-Walton, J. S. Desjardins, J. L. Rosen, W. W. Havens, Jr., and L. J. Rainwater, Phys. Rev. iao, 2214 (1960). s9 L. M. Bollinger, R. E. CotB, and T. J. Kennett, Phys. Rev. Letters 3, 376 (1959). Eo J. R.Bird, M. C. Moxon, and F. W. K. Firk, Nuclear Phys. 13,525 (1959);L. M. Bollinger and R. E. CotB, Bull. Am. Phys. SOC.[2]0, 294 (1960);D.J. Hughes, H. Palevsky, R. E. Chrien, and H. H. Bolotin, ibid. p. 295.

2.2.


545

and cyclotron accelerators, and therefore it is seldom necessary to use a time-of-flight analysis. Recently, however, interest has focused on the study of fast neutron spectra, particularly those resulting from inelastic neutron scattering, and it is primarily because of this problem that timeof-flight techniques have been developed. Interest in these spectra comes from the ever-increasing demand for good data on nuclear energy levels to check models of the nucleus and from the fact that inelastic scattering cross sections are needed for the design of intermediate and fast reactors. A study of the inelastic scattering of neutrons is particularly interesting in the study of the energy levels of heavy nuclei. To study heavy nuclei with charged particles, the energy of the particle must be sufficiently high to have a high probability of penetrating the barrier. When the energy of the particle is this high, the competition of various reaction processes may render the determination of the level structure extremely difficult. Neutron scattering experiments may be carried out at energies where inelastic scattering dominates other processes, and therefore the interpretation of the results should be very much simpler. Time-of-flight techniques developed for neutron spectroscopy have also found valuable application in charged particle spectroscopy. For instance, time-of-flight spectroscopy in conjunction with magnetic spectroscopy makes possible distinctions between particles of equal momentum but different mass. It has also been found profitable to use time-of-flight spectroscopy in order to eliminate background which comes from unknown sources. This is equivalent to applying a modulated signal to a particular device and using a detector which is tuned to the modulation frequency to eliminate noise. The relationship between the flight time and energy of the neutron for the flight path d measured in meters is t = 72.3 d / l / E , where t is measured in millimicroseconds, and E is in MeV. Table I1 gives the energy TABLE 11. Energy Resolution for Different Time Spreads

At(mpsec)

0.1

0.3

1.0

3.0

10.0

1 2

0.00087

0.0045

0.0017

0.009

0.0035

0.018

0.14 0.29 0.58

0.87

4

0.028 0.056 0.11

~

1.74 3.5

spread for the flight path of 1 meter corresponding to the given energy in a given uncertainty in time At. From this table, it is clear that, to obtain resolution at high energies, time resolutions approaching the limitations

546

2.


of presently available techniques must be used. At lower energies the time resolution is not important. In fact, for low energies the beam homogeneity and consequently the energy spread of the primary neutron beam is the important factor in determining the resolution. Thus if the spread in energy of the primary neutron beam is AE, then the spread in the energy of any neutron group excited by inelastic scattering will also be AE if the yield curve for exciting a particular level does not vary appreciably over this energy interval. Thus the conditions for optimum resolution for high-energy groups (corresponding to the excitation of levels just above the ground state of the residual nucleus) are different from those for low-energy groups (high-lying levels). For the lower end of the fast neutron spectrum a thin target is required and a short neutron flight path may be used, whereas for the higher energy end of the spectrum the target need not be thinner than the energy resolution expected from the fast time-of-flight measurements and the flight path should be as long as possible. Van de Graaff, Cockcroft-Walton, and cyclotron accelerators have been used for fast neutron time-of-flight studies. In all of these accelerators great care must be exercised to assure that the charged particle beam hits the neutron producing target for a very short period of time. In the Van de Graaff and Cockcroft-Walton machines, which usually accelerate a beam continuously, the beam must be modulated by an external device. The charged particle beam can be deflected past an aperture in the beam pipe of the accelerator by a pair of parallel plates to which an rf voltage has been applied. The beam deflection system used at Los Alamos is of this type and is shown schematically in Fig. 34. A typical value for the rf frequency is 4 mc/secel which gives a beam pulse of about 2 mpsec duration and 250 mpsec between pulses if the beam is forced to traverse the slit once in an rf cycle. In a cyclotron the phase bunching which results from the normal operation of the cyclotron is usually of a few millimicroseconds duration. This self-bunching has been used to form the pulse for fast timeof-flight studies. 6 2 When dealing with times as short as a few millimicroseconds, it becomes necessary to determine the exact time a t which the burst occurs. I n principle this can be determined from the rf deflection pattern, but in practice L. Cranberg and J. S. Levin, Phys. Rev. 105, 343 (1956). H. H. Landon, A. J. Elwyn, G. N. Glasoe, and S. Oleksa, Phys. Rev. 112, 1192 (1958); C. 0. Muehlhause, S. D. Bloom, H. E. Wegner, and G. N. Glasoe, ibid. 105, 720 (1956); R. Grismure and W. C. Parkinson, Rev. Sci. Znstr. 28, 245 (1957); G. F. Bogdanov, N . A. Vlasov, S. P . Kalinin, B. V. Rybakov, and V. A. Siderov, Intern. Conf. on Neutron Interactions with the Nucleus, New York, 1957. Columbia University Report CU-175 [U.S. Atomic Energy Commission Report TID-7547 (1957)l. 6*

2.2.

547

DETERMINATION O F MOMENTUM AND EN ERG Y

this is done by calibrating the time scale with a known structure which is readily identifiable, such as the y-rays from inelastic scattering, the elastically scattered neutrons, or the inelastically scattered neutrons from prominent levels which are well known from other studies. Another technique for determining the start time, which is particularly useful for fast neutron time-of-flight studies, is the associated particle method. This method is best described by using the reaction H3(d,n)He4for the production of neutrons. I n this reaction a neutron and an alpha particle are 390 Crn

8.3 Crn

3 C r n target

Scalterer

Defleclor plales

Plastic scinlillalor

Mu metal shield RCA 5519 photomultiplier

\

FIQ.34. Schematic diagram of physical layout of apparatus for measuring the inelastic scattering of neutrons by time of flight. [From L. Cranberg, Proc. 1st Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1965 4, 43 (1956).]

produced a t exactly the same time. The pulse from the alpha particle detector, which is close to the source, gives the start pulse for the timeof-flight measurement. Although the methods of establishing the start time differ markedly, all of them seem to be limited in accuracy to a few millimicroseconds. 2.2.2.2.4.1. Measurement of the Flight Time. Numerous devices are in use for the measurement of transit times of the millimicrosecond range. These include delayed coincidence arrangements with one or more channels and a special version of this principle embodied in a device called a

548

2.


chronotron. Lefevre and RussellG3have developed a chronotron which has gate widths of the order of 0.5 mGsec, which are stable to 10-4 mpsec. The systems usually used are not as complicated as the chronotron. A time-tovoltage conversion circuit usually suffices, and the pulse distribution is then fed into an ordinary commercial multichannel analyzer. In practice it is not the electronic timing system that limits the resolution which can be obtained but the size and jitter of the detector. 2.2.2.2.4.2. Detectors for Fast Time of Flight. Most of the detectors used for fast time-of-flight studies have been liquid or plastic scintillators, although the recent development of the solid state detector (see Section 1.8.1) may furnish another device which can be used as a detector for fast time-of-flight studies. At present the detector is the component which limits the time resolution in fast time-of-flight spectroscopy. In order to detect a large number of neutrons, it is desirable to have the detector as large as possible. However, if the detector is large, the difference in time between a neutron arriving a t the point in the detector nearest to the source and one arriving a t the furthest extremity of the detector can be large compared with the resolution time desired. For example, if a plastic scintillator which has a 2-cm length in the direction of the beam is used as detector, a 1-Mev neutron takes 1.4 mpsec to traverse the detector. This means that a neutron detected a t the front of the detector will be detected 1.4 mpsec before a neutron detected a t the rear of the detector, thus introducing a 1.4-mpsec time spread. In order to obtain better time resolution, it is necessary to use smaller detectors. Since the intensity decreases in direct proportion to the volume of the detector, very small detectors cannot be used with source intensities presently available. In order to improve fast time-of-flight spectroscopy, it is clearly necessary to increase the yield of pulsed neutron sources. by several orders of magnitude. 2.2.2.2.4.3. Recent Improvements in Fast Time-of-Flight Equipment. The effective yield of pulsed neutrons has recently been increased in Van de Graaff accelerators by pulsing the source of charged particles in the high voltage terminal of the machine, which allows the pulsed beam intensity to be increased because the average current accelerated remains low. The background is also decreased because ions are not accelerated except a t the times they are wanted. Another recently developed method for increasing the pulsed intensity from accelerators is the magnetic bunching system, which is used in conjunction with a regular fast time-of-flight system like that described by Cranberg.64 H. W. Lefevre and J. T. Russell, Rev. Sn'. Instr. 30, 159 (1959). s4L. Cranberg, Bull. Am. Phys. SOC.[ Z ] 6, No. 1 (1961). tj3

2.2.


549

The method used for the magnetic bunching originally suggested by Mobleya6is shown schematically in Fig. 35. The group of particles labeled (1) in Fig. 35 emitted by the pulsed accelerator is traveling with a velocity v , extends laterally over a distance d, and has a width w. If a target is placed in the path of this group of particles, the time At the group will take to strike the target will be At = d/v. For a 1-Mev beam which has a time spread of 2 X 10-9 sec in a regular time-of-flight arrangement, the lateral extension of the beam is 13.8 em. The group of particles is injected into a magnetic field which is shaped to cause the particles a t a larger disnEFLECTING PLATES

CHOPPING SLITS

TARGET

FIG.35. Schematic diagram of magnetic bunching system to be used with standard fast time-of-flight beam pulsing system.

tance from the center of curvature of the path of the particles in the magnet to travel on the circle of radius Rzwhile those particles closer to the center of curvature travel along a path of radius R1.If the path difference is exactly d in a deflection of 180°, the particles will leave the magnet a t exactly the same time. The geometrical width of the beam will be exactly the same as when it entered the magnet. Thus, in principle it is possible to shorten the pulse in time without loss of intensity, thereby increasing the burst intensity. The magnetic bunching system gives an order of magnitude improvement in the intensity for the same time-of-flight resolution, or conversely gives an increase in the time-of-flight resolution of a factor of 10 for the same intensity. The use of a time-of-flight system to eliminate extraneous background 6sR. C. Mobley, Phys. Rev. 88,260 (1952).

550

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is best illustrated by its recent use in the study of neutron polarization.66 The time-of-flight system is used to separate those neutron events of interest from a variety of background events. The background can be separated into three categories: (1) time-independent events, e.g., cosmicray counts, electronic noise pulses, and y-rays from neutrons which are moderated and radiatively captured in the general vicinity of the counter; (2) events which occur a t a given time within the rf cycle but at a different time from a neutron event of interest, e.g., neutrons or y-rays produced

I

r=o

CHA "€1

NUMBER

FIQ.36. Time-of-flight spectrum at 20" from the reaction Beg(d,n)B'O with a bombarding energy of 1.85 MeV, a fight path of 150 cm, and channel width of 0.594 X sec. Detector bias for recoil protons in t h e plastic scintillator was about 800 kev, and the deuteron burst width was 1.1 mpsec. The time scale proceeds linearly from left t o right,. The time scale can be made t o run in either direction in this as well as in any vernier instrument. [From H. W. Lefevre and J. T. Russell, Rev. Sci. Znstr. 30, 159 (1959).]

by the beam on the target or surrounding equipment; (3) events which occur at such a time within the rf cycle as to arrive a t the detector a t the same time as the event of interest. The time-of-flight system will decrease the background from time-independent events by a factor which is the ratio of the time duration of the pulse to the time between pulses. It will eliminate the background from events which occur a t a given time in the rf cycle but at a time different from the neutron events of interest. It will not, of course, eliminate the background from those events which occur at the same time as the event of interest. Time-of-flight as a background J. A. Baicker and K. W. Jones, Nuclear Phys. 17, 424 (1960); L. Cranberg, Phys. Rev. 114, 174 (1959).

2.2.


551

eliminator has been used in many studies.67Neutron time-of-flight data are improved considerably by using the pulse shape discrimination method developed by Brooks for the separation of pulses due to y-rays from those due to heavy particles in a scintillation detector.6s This system is described in the section on neutron detectors. 2.2.2.2.4.4. Applications of Fast Time-of-Flight Spectroscopy. Fast neutron time of flight has been used to study individual levels in nuclei and

6000

I‘

I

I

I

I

E LAST I C A L LY SCATTERED 7

‘-1

-

CHANNEL NUMBER INCREASING T I M E OF FLIGHT

FIG.37. Time spectrum obtained a t 90 degrees for neutron scattering at 2.45 Mev from gold. The “sample out” background has been subtracted. [From L. Cranberg and J. S. Levin, Phys. Rev. 103, 343 (1956).]

nuclear temperatures. An example of its use for the study of individual levels is shown in Fig. 36, taken from the work of Lefevre and Russell, where data obtained on the Beg(d,n)B’O reaction illustrate the resolution that can be achieved. When a heavy nucleus is studied by inelastic scattering, the level density is so large that it is impossible to resolve individual levels. Figure 37 shows the spectrum obtained by Cranberg and Levinegfor gold. As expected, the curve shows no peak due to nuclear energy levels and the results can be interpreted in terms of a nuclear temperature. However, 67 See, for example, Conf. on Neutron Physics by Time-of-Flight, Gatlinburg, Tennessee, November, 1956. Oak Ridge National Laboratory Report ORNL-2309

(1956). 88 F. D. Brooks, Nuclear Znstr. and Methods 4 , 151 (1959). e9L. Cranberg and J. S. Levin, Phys. Rev. 103, 343 (1956).

552

2.


new phenomena arise when new techniques are applied to old problems. Figure 38 shows the time spectrum of 2-Mev neutrons scattered from UZa8. Unexpectedly, the spectrum of U238 shows resonance structure. The large peak comes from the y-rays which are emitted when the neutron is captured; the small peak comes from structure in the UZ38 nucleus. It

d:l

Om

0.5

0.86

1.98

En(MEV)

FIG.38. Time spectrum of neutrons from the interaction of 2 Mev neutrons with Ups*.[From L. Cranberg, Proc. Intern. Conf. on the Neutron Interactions with the Nucleus, Columbia University Report CU-175 (1957).]

seems unlikely, both from general considerations of level density and the width of the observed peak, that a single level has been resolved. It is preferable to say that the levels centering about 1.98 Mev having a spread of 150 kev or less are excited with substantially higher cross section than adjacent levels. Fast neutron time-of-flight spectroscopy is capable of supplying more unique data on nuclear energy levels and nuclear temperatures. It is necessary to improve the resolution by an order of magnitude to obtain

2.2.


553

the resolution necessary to separate low-lying nuclear energy levels. Because of the physical limitations of the system, this improvement can only be obtained by increasing the intensity by an order of magnitude. New developments in accelerator techniques seem capable of providing the necessary increase in intensity, so fast neutron time-of-flight studies should prove a fruitful field of research. 2.2.2.2.4.5.Cross-Section Measurements in the kev Region. An apparatus has been developed to measure neutron cross sections in the kev region70 which competes with fast neutron spectroscopic equipment at its lower energy range and with epithermal neutron spectroscopic equipment a t its upper energy range. The apparatus is described here because the techniques used are the same as those used in fast neutron time-of-flight measurements. Epithermal neutron spectroscopic equipment uses a continuous energy distribution of moderated neutrons and determines the energy by time of flight, whereas fast neutron spectroscopy usually makes use of monoenergetic neutrons produced by a Van de Graaff machine or a cyclotron. The apparatus developed for studies in the kev region is a hybrid of both of these techniques. The method employs a pulsed Van de Graaff machine to produce bursts of neutrons of about lo-* sec duration from the Li7(p,n) reaction. By suitable choice of target thickness and proton bombardment energy, a spectrum of neutrons is produced a t 0' to the proton beam for covering the kilovolt range. These unmoderated neutrons constitute the source from which neutrons of a specific energy are chosen by time-offlight measurements. Thus the Van de Graaff supplies only a limited spectrum of neutrons whose precise energy is determined by time of flight. The resolution does not depend on the energy spread of the neutron beam, as it does when the Li7(p,n) reaction is regularly used to produce monochromatic neutrons for cross-section measurements. The resolution is obtained by timing the neutrons over a measured path using ultra fast timing techniques. The intensity of the neutron source is fairly low compared with the intensity of the neutron sources used for epithermal neutron spectroscopy. However, because no very low-energy neutrons are produced, a high repetition rate can be used. Average counting rates are comparable with those obtained with other neutron spectrometers in the energy range from 1 to 40 kev. The stray neutron background for this apparatus is considerably less than for other neutron spectrometers, because no neutrons are emitted in the backward direction from the initial direction of the proton when the bombarding energy is not more than 40 kev above the threshold of the Li7(p,n) reaction. 70

W. M. Good, J. H. Neiler, and J. H. Gibbons, Phye. Rev. 109, 926 (1958).

554

2.


The neutron bursts are produced by using a device which deflects the proton beam across a slit in the high-voltage terminal of an electrostatic generator. The pulsed proton beam is then accelerated and hits the target assembly. The voltage applied to the deflector has a frequency of 0.5 Mc. The resultant beam pulse has a time duration of less than see. For intermediate neutron energies, the detector that has been used is a 10-cm slab of Bl0 closely backed by a NaI(T1) crystal. A single-channel window is set over the full peak of the 480-kev y-ray from the Bl0(n,a,y)Li7reaction. The signal from the detector starts a time-to-pulse-height conversion circuit which is stopped by a signal from the target. The resultant voltage

I5

5

0 I

2

5

10

20

E, (kevl.

FIG.39. Total cross section of Ye9 as a function of neutron energy. [From W. M. Good, J. H. Neiler, and J. H. Gibbons, Phys. Rev. 109,926 (1958).]

is fed into a pulse-height analyzer to give the time-of-flight spectrum. The over-all resolution of the system described by Good et aZ.70 is 6 to 8 mpsec full width at half-maximum. Five different flight paths varying from 0.55 to 2.0 meters have been used to give the resolution desired for the particular energy neutron studied. An example of the results obtained in the measurement of the total cross section of Yes is given in Fig. 39. 2.2.2.2.5. TIMEANALYZERS FOR THERMAL AND EPITHERMAL NEUTRON SPECTROSCOPY. Time analyzers used in neutron spectroscopy range from a simple gate a t a variable delay time after the neutron burst to a complex electronic system equivalent to a modern computing machine. The complexity of the device depends on the speed required for the recording of the data, the counting rate, the repetition rate of the pulse, and the

2.2.


555

number of discrete intervals into which one would like to divide the time interval over which data can be taken. The phrase ‘(time gate” is used constantly in neutron time-of-flight instrumentation and means an electronically controlled switching circuit which permits signals from a neutron detector to pass through when the time gate is on or open and which prevents such signals from being transferred when the time gate is closed. This “time gate” is also referred t o as a (‘channel.” The complexity of the analyzer increases rapidly as the number of channels in the analyzer increases. One of the simplest time analyzers was that used by Baker a n d ’ B a ~ h e r , ~ ~ who pulse-modulated the ion source of a cyclotron. In this analyzer, a delay circuit w&s triggered at time t = 0 when the neutron puke was formed. The delay circuit was adjustable from 0 to 10,000 psec after the start of the time gate which pulse-modulated the cyclotron. The width of the time gate was adjustable from 10 to 1OOOpsec. The time delay circuits and the time gates used in a simple analyzer of this type can be unsymmetrical univibrators or phantastrons. Such circuits are fairly standard and are described in textbooks on electronic^.^^ * A single-channel analyzer of the type used by Baker and Bacher recorded only a small fraction of the available data, so multichannel analyzers for this purpose were quickly developed by adding channels in a straightforward manner. However, this simple method of adding channels caused the electronic circuitry to increase linearly with the number of channels, so neutron spectroscopists then resorted to modern digital computer techniques for recording the data. Block diagrams for simple analyzers are described in the early papers on neutron s p e c t r o ~ c o p yMuch .~~ better circuits than those described in these early papers are now available due to the remarkable advances in electronic techniques. 2.2.2.2.5.1. Multichannel Analyzers for Low Counting Rates. If the counting rate is low enough so that there is a very small probability of detecting two neutrons from one burst, then the time-of-flight analyzer can be very simple. If a multichannel pulse-height analyzer is available, the simpIest system to use is one in which the time a t which the neutron is detected is converted into a pulse height and the pulse height is then recorded by the pulse-height analyzer.

* See also Vol. 2, Part 7. C. P. Baker and R. F. Bacher, Phgs. Rev. 69, 332 (1941). J. Millman and H. Taub, “Pulse and Digital Circuits.” McGraw-Hill, New York 1956. 7 s J . H. Manley, J. Haworth, and E. A. Luebke, Phys. Rev. 61, 316 (1940); C. P. Baker and R. F. Bacher, ibid. 69, 332 (1941); J. Rainwater and W. W. Havens, Jr., ibid. 70, 136 (1946); J. Rainwater, W. W. Havens, Jr., C.S. Wu, and J. R. Dunning, ibid. 71, 65 (1947). 71 72

556

2.


Almost all of the time-of-flight analyzers for millimicrosecond time-offlight studies operate in this manner. Several time-to-amplitude converters for the millimicrosecond and microsecond range have been described in the literat~re.'~ If it is desirable to gather data for a time interval which is large compared to the length of the individual channels, then it is sometimes difficult to construct a time-to-voltage converter which is linear over the whole range. However, it is always possible to break the total time interval

ClRCUlT

1'5 RING COUNTER

7 1 41

%IDELAY

LOO'S RING COUNTER 10

1st

srwc

I

RESET

FIG.40. Brookhaven analyzer. [From W. A. Higinbotham, Proc. 1st Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1966 4, 56 (1956).]

up into several smaller intervals and construct time-to-voltage converters which are linear over a more limited range. If a multichannel pulse-height analyzer is not available, or if it is undesirable to tie up a multichannel pulse-height analyzer for the time-offlight work, then it may be desirable to construct a time analyzer using matrix circuits. The time-of-flight analyzer used at the Brookhaven National Laboratory, designed by Graham, Higinbotham, and Rankowitz and shown schematically in Fig. 40,is of this type. A start pulse from the photocell on a chopper or from the target of a pulsed accelerator a t time TOturns on the bistable trigger circuit and starts the "gated" 2-Mc oscillator. The oscillator frequency may be divided as shown to give F. Lepri, L. Mazzetti, and G . Stoppini, Rev. Sn'. fnstr. 26,936 (1955); W. Weber, C. W. Johnstone, and L. Cranberg, ibid. 26, 166 (1956); P. R. Orman, Nuclew Instr. 2, 95 (1958); P. R. Orman and F. H. Web, Proc. 6th Tripartite Instrumentation Conf., Chalk River, Canada. Report AECL-804 of Atomic Energy of Canada, Ltd., 1959.

2.2.

DETERMINATION O F MOMENTTJM .4ND ENERGY

557

channel widt,hs of 0.5 to 4 psec. Three ring c:ount>crsare employed. Thc first two ring counters are used to define trhe 100 channels. The t,hird ring is used t o delay the start of the sensitive period hy requiring that, a detector pulse be in coincidence with one of the stages of t,he slow ring. The delay may also be varied in steps of 10 channels by starting the “tens” ring counter in different configurations. The tens ring counter is made up of 11 bistable elements, ten of which are connected to the matrix. The eleventh stage eliminates any ambiguitJywhich might, arise due to cumulative delay in the several counting circuits. The sum of the numbers adjacent to the switches on the “tens” and the “hundreds” ring counters is the delay in unit#sof a channel width to the start of the 100 channel recording interval. If a pulse is received from the detector within the selected interval, the bistable trigger circuit is turned off and the ring counters store the channel number. Delay 1 is manually set to follow the last channel of interest and to precede the next To pulse. If a detector pulse has been received, the number stored in the ring counters is recorded. All circuits are reset by a pulse from delay 2. The frequency divider circuit and the units ring counter circuits are derived from the Chalk River scaler The slower ring counter circuits can be similar to those of Gatti.76 If an analyzer of the Brookhaven type were to be constructed a t the present time, it would be desirable to use recently developed beam switching tubes, such as the Haydu M-l0R rather than the ring counters. 2.2.2.2.5.2. Multichannel Analyzers f o r Higher Counting Rates. If the probability of detecting more than one neutron from one burst of the source is not small, then the time analyzer becomes much more complex. The starting system and time delay circuits can be the same as in the simple system, but the detection channels must have a much faster riseand-fall time and must be much more precisely timed than the detection channels used in analyzers which are not required to record more than one neutron per burst,. The detected neutron pulses occur randomly in time. If the edges of the successive time gates are not infinitely steep, some counts may be lost between channels or counted in two adjacent channels. Poor timing might, cause the channels to be unequal. The analyzers described in this section perform in a manner similar to the analyzers described briefly earlier. However, these circuits make use of B matrix in order to provide a large number of channels with a relatively small number of tubes. The first analyzer of this type was designed by deBoisblanc and McCol76N. F. Moody, W. 1). Howell, W. J. Battell, and R. H. Taplin, Rev. Sci. Znstr. 22, 439 (1951). 76 E, Gatti, Rev. Sn’. Znstr. 24, 345 (1053).

558

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lum.” A block diagram of the analyzer is shown schematically in Fig. 41. The shaded portion a t the top left represents a face of the reactor. The chopper, which has four slits, is shown next to the reactor. The detectors are located 16 meters distant from the chopper. A light source and photomultiplier tube generate a pulse at To which starts the variable delay. The variable delay is a commercially manufactured unit which uses a temperature regulated phantastron circuit and is stable to 1 part in 1000. The 100 adjacent counting channels start at the end of the delay period.

FIG.41. 100 channel analyzer, Phillips Petroleum Co. [From W. A. Higinbotham, Proc. 1st Intern. Conf. Peaceful Uses Atomic Energy, Geneva, 1955 4, 57 (1956).]

The crystal oscillator has a frequency of 8 Mc. The channel width selection unit contains binary frequency dividers to give channel widths of 0.5, 1.0, etc., psec. The electronic switch opens the gate circuit which permits pulses from the channel width unit to enter the units ring counter. The tenth stage of the units ring counter opens a gate permitting one pulse from the channel width unit to advance the tens ring counter one step. This insures that both ring counter circuits advance in phase. After one complete cycle of 100 steps, the gate is shut off and all circuits are reset. The two sets of ten leads form a raster or matrix with 100 intersections. A triple coincidence circuit is located at each junction. The third 77 D. R. deBoisblanc and K. A. McCollom, “ M T R Time of Flight Instrumentation,” Report No. IDO-16159 (1954). Obtainable from the U.6. Atomic Energy Commission, Office of Technical Services, Oak Ridge, Tennessee.

2.2.

DETERMINATION O F MOMENTUM A N D E N E R G Y

559

input is from the detector pulse shifter. Each coincidence circuit in succession switches the pulses from the detector into a channel counting circuit. Each coincidence circuit consists of two germanium diodes and three resistors. Each counting circuit consists of a glow discharge decade counter tube, a hot cathode gas tetrode, and a message register. A detector pulse shifter circuit is used in order to ease the stringent timing requirements mentioned a t the start of this section. Each pulse from the neutron detector-amplifier is delayed by an amount (not more than the duration of one channel) such that the corresponding pulse delivered to the coincidence circuits occurs in the middle of a channel interval. The circuit consists of a bistable trigger circuit. Pulses from the channel width unit, delayed by one-half channel interval, are connected to drive the trigger circuit in one direction. The amplifier lead is connected to drive it in the other direction. The normal state of the trigger circuit is determined by the pulses from the channel width unit. When a detector pulse is received, it throws the trigger circuit into the other state. The next pulse from the channel width unit throws the trigger circuit back to its normal state and a pulse is passed to the coincidence circuit. The figure shows circuits for recording the background counting rate and for recording all pulses received during the selected 100 channel interval. 2.2.2.2.5.3. Time-of-Flight Analyzers Using Digital Storage Techniques. The first analyzer to use digital storage techniques was described by Schultz et al.7aA fused quartz acoustic delay line was used as the storage or “memory” element. The analyzer has been used with a microwave cavity linear electron accelerator which produces neutrons by the (7,n) reaction. This accelerator can be synchronized from the time-of-flight circuits. This analyzer is shown in Fig. 42. The operation and circuits master pulse are very similar to those used by Hutchinson and S c a r r ~ tA. ~ ~ is generated automatically to start operation of the circuits. The master pulse modulates the 40-Mc oscillator and causes an acoustic pulse to travel through the 1000-hsec quartz delay line. The pulse is amplified and passed through a detector (or rectifier) circuit. It continues to recycle through the pass master pulse and reshaper units. The master pulse triggers a delay circuit which causes “channel marker” pulses and “clock” pulses to be generated for a time just less than 1000 psec. The master pulses pass through a frequency divider to trigger the liner accelerator. A detector pulse is time shifted so as to arrive in synchronism with a 500-kc channel pulse. The adder circuit causes a binary digit to be inserted in the corresponding channel. The binary digit continues to circu78 79

H. L. Schultz, G. F. Peeper, and L. Rossler, Rev. Sci. In&. 27, 437 (1956). G. W. Hutchinson and G. G. Scarrott, Phil.Mag. [7] 42, 792 (1951).

560

2.


late through the quartz line, the amplifiers, the reshaper, and adder circuits until another detector pulse is received in the same channel. Then the adder circuit adds one to the binary number stored in the channel interval; 21° counts may be stored in each channel interval, since the binary digits have 0.2-psec spacing. The information may be displayed on a cathode-ray tube oscilloscope : the horizontal sweep circuit is triggered by the master pulse. A vertical sawtooth waveform, triggered by the channel pulses, is applied to the vertical deflection plates. And the “information out ” terminal is coiinect,ed to the control grid of the cathode-ray tube. Schultz has designed

,

CIRCUIT

INFWATIW

OUT W K C

PULSES

OLZECTOR PULSE

I

OUANnZEA

FIG. 42. Five-hundred channel analyzer with acoustic delay line storage. [Froin

w. A. Higinbotham, Proc. 1st Intern. conf. Peaceful Uses Atornic Energrj, Geneva, 1,956 4, 59 (1956).]

an analog printing circuit, which also uses about 40 tubes: a delay circuit, triggered from the master pulse, is slowly varied from 0 to 1000 wsec. It selects the first channel pulse for a number of cycles, then the secoxid, and so -on. While the delay selects one channel in this manner, the digits stored in the selected channel operate gas tetrodes which control a set of relays. The relay contacts switch a set of resistors in such a manner as to generate a voltage proportional to the number in the channel. The successive channel readings are plotted 011 a moving (.hart by a pen recorder. I n most cases it is not possible to trigger the iieutroii source from the analyzer circuits, so the quartz line method cannot! be used. The remainder

2.2.

D E T E R M I N A T I O N OF MOMENTUM ANI) E N E R G Y

56 1

of the instruments to be described here may lw triggered hy a Y’,,signal. Schumannsohas designed and constructed an analyzer with 1024 Channels which uses magnetic core storage. The time is measured by counting pulses from a n oscillator. The number stored in the selected position is transferred to an arithmetic circuit. One is added to the number and it is transferred back into the magnetic core memory. The analyzer takes about 16 psec to record each pulse. I t stores up t o 216 counts per channel. It includes a cathode-ray tube display and has circuits for printing out the contents of the memory at the end of a run. Control

-

I

I

- Memory

I

I

I I

+ of the maximum pulse height are used in the channels i t is then not possible for a single electron of the energy being focused to multiply scatter from one crystal out to the walls or spectrometer baffles and back into the other crystal giving enough energy to trigger both discriminators. The only background observed is the random coincidence rate. Transmission and resolution tests of the intermediate-image pair spectrometer have shown that in favorable cases it, can yield pair lines, i.e., coincidence yield versus magnetic field current,, whose full width a t half-maximum is only O.G% in momentum. Expressing the pair transmission as t,he number of count,s por pair emit,t,edfrom n source t,he number * See d a o Vol. 2, Section l6

1 . 1 . 1 .:3.1.

D. E. Alburger, Rev. Sci. Instr. 27, (195(i). . H.Sliitis and K. Siegbahn, Arkiv Fysik 1, 339 (1949); P h p . Rev. 76, 1055 (1'349). H. Daniel and W. Bothe, Z. Naturforsch. 9a, 402 (1954).

-

0 2 4 6 8 012

INCHES

FIG. 3. Intermediate-image pair spectrometer. The light piping and coincidence detecting system ihas been improved over the design described in Rev. Sci. In&. 27, 991 (1956); see D. E. Alburger, Phys. Ref).111, 1586 (1958).

650

2.


at 2.5% resolution is 6 counts per lo6source pairs, and is lower by a factor of 100 a t 0.6% resolution. In external pair conversion using 2.76-Mev gamma rays from NaZ4and a uranium converter foil the yield at 3 % resolution is 1 count per 7 X lo8 gamma rays emitted by the source. Comparing with the figures mentioned above for Terrell’s uniform field instrument the yield per gamma ray of the intermediate-image pair spectrometer at about the same energy is 10 times greater even though the resolution is better by a factor of 4. However, comparing with the efficiency of 1 pair count per lo6 gamma rays for the 3-crystal NaI(T1) 250

I

I

I

I

I

I

RELATIVE MOMENTUM

FIG. 4. Nuclear and internal pair coincidence lines occurring in the F19(p,a)016 reaction as observed with the intermediate-image pair spectrometer. The numbers labeling the three peaks are the transition energies in MeV.

scintillation pair spectrometer discussed above it is seen that the intermediate-image spectrometer is less efficient than the 3-crystal instrument at 3 Mev by a factor of 7000, when the resolution of both is approximately 3 %. The usefulness of the intermediate-image pair spectrometer lies both in its ability to achieve higher resolution and in its capability of studying nuclear and internal pair emission. The latter types of transitions observed with the 3-crystal spectrometer using a plastic center crystal give rise to pair lines6 -30% wide a t 3 Mev and -17% wide a t 6 MeV. The spectrum’6 of nuclear plus internal pairs in the F19((p,a)Ol6 reaction taken with the intermediate-image instrument at 1.7 % resolution is shown in Fig. 4. investigation^'^ of nuclear reaction gamma rays have been made a t l9 See, for example, R. D. Bent, T. W. Bonner, and R. F. Sippel, Phys. Rev. 98, 1237, 1325 (1955); R. D. Bent, T. W. Bonner, J. H. McCrary, W. A. Ranken, and R. F.

2.2.


651

the Rice Institute using a similar instrument. It should be pointed out that in nuclear reaction work the Doppler shift of lines associated with fast transitions in recoiling nuclei must be corrected for and that the Doppler broadening of the lines may limit the resolution obtainable. In addition to the above-mentioned semicircular instrument proposed by Hornyak, several other pair spectrometer designs have been proposed which have not yet been tried. Jungerman20has suggested a multichannel homogeneous field pair spectrometer. This is essentially an axially focusing solenoid beta-ray spectrometer which uses a statistical-separation detector but which is also equipped with two sets of annular baffles which accept electrons over two angular ranges with respect to the axis. The possibilities that the positron and electron can pass through the same or different exit rings onto the statistical-separation detector increases the transmission over the use of one acceptance ring alone. Owing to the large distance along which the focused electrons cross the axis in a solenoid spectrometer the detectors must be either very large diameter flat crystals or small diameter long concentric cylindrical crystals. In either case the light piping problem is difficult. The iron-jacketed Slatis-Siegbahn intermediate-image spectrometer can also be used for pair measurement by piping the light from two statistical-separation crystal detectors out through a hole in the pole piece. A design of this type has been constructedz1at the Nobel Institute of Physics. Nielsen and Kofoed-HansenZ2have suggested that the multisection (6 orange ” beta-ray spectrometer could be applied as a pair spectrometer if the pole piece shapes were modified. The operation of each gap is analogous to that of the uniform field instrument. A novel pair spectrometer recently proposed by MalmforsZ3uses the time of flight of electrons in a magnetic field. The field produced in a ring-shaped magnet falls off as l/P. If the field is strong, i.e., 15,000 gauss, the electrons emerging from a source placed in the field will follow trochoidal paths and will drift away from the source with a velocity which depends on their energy but which is relatively independent of the emission angle. The time required to drift halfway around an annular Sippel, 99.710 (1955); R. D. Bent, T. W. Bonner, J. H. McCrary, and W. A. Ranken, 100, 771 (1955). 20 J. A. Jungerman, Rev. Sci. Instr. 27, 322 (1956). 31 J. Kiellman and B. Johansson, Arkiv Fysik 14, 17 (1958). 22 0. B. Nielsen and 0. Kofoed-Hansen, Kgl. Dansk. Videnskab. Selskab Mat.-fys. Medd. 29. No. 6 (1955). 2aK. G. MaImfors, in “Proceedings of t,he Rehovoth Conference on Nuclear Structure,” p. 506. North Holland Publ., Amsterdam, 1958.

653

2.

I)ETEH.MIS.\TION OF FUNDAMENTAL QUANTITIES

magnet will be -1 psec for electrons of a few MeV. Supposing that a pair is produced a t the source the two components will drift in opposite directions and if the energy is equally divided beti! een the two they will arrive a t the same time a t their respective detectors which are placed back to back halfway around from the source. The occurrence of a coincidence between the two vounters establishes the equal energy division and a measuremeiit of the time o f flight determines their energy. Evidently the mode of producing the pairs must be pulsed if a timing measurement is to be made. Using millimicrosecond timing techniques it should be possible t o derive the energy of the pair to an accuracy of better than 1%. The bunched beam of a cyclotron or a pulsed beam from a Van de Graaff accelerator could be used to induce nuclear reactions and the internal pair spectrum from a target would consist of coincidence pulses as a function of time following the bombarding pulse. No estimates of the pair transmission have been made although for single electrons the transmission is expected to be > 15% of 48.

2.2.3.5. Shower Detectors.* 2.2.3.5.1. INTRODUCTIOS. High-energy gamma rays or electrons may be detected and their energies measured by means of large scintillation or Cerenkov counters. It is also possible to use absorbers and radiators in combination with such large counters to utilize their capabilities most fully. In the case of gamma rays the quanta are made to materialize and the magnitude of their ensuing electromagnetic showers can be measured. If the counter is properly made the efficiency of detection can be as high as 100% and the energy of the gamma ray may be measured to *15% or better. This type of gammaray detector cannot at present compete with the well-known pair spectrometer as far as energy resolution is concerned but its efficiency greatly exceeds that of the pair spectrometer. A diagram of a typical practical shower detector is shown in Fig. 1. The counter shown in the figure may be used to detect either electrons or gamma quanta incident on the left and entering the porthole of the counter. Light produced in the crystal is detected by t,he photomultiplier a t the right. 2.2.3.5.2. NEEDFOR SHOWER DETECTORS. Why is such a detector needed? There are at least two answers to this questtion. (a) The efficiency of other methods of detection is low compared to that obtainable with shower detectors. I n high-energy experiments where the cross section is very small, e.g., in the Compton effect of the proton,

*Section 2.2.8.5 is by R. Hofstadter.

A-SILICONE

OIL

‘-PHOTOMULTIPLIER

GLASS WINDOW SCALE

INCHES

FIG. 1. This figure shows a typical large shower detector. In this wsr the detector itself is a NaI(T1) scintillation counter. The scintillation counter may easily be replaced by a Cerenkov light emitter with very few other changes. Between the NaI(T1) crystal and the photomultiplier lies a Lucite light coupling lens. The conical end shape of the scintillation counter is determined by the present crystal growing techniques and is known to be not optimum for light collection purposes. This counter has been used to detect electrons up to energies of 600 Mev with the results shown in Figs. 7, 12, and 13 of this paper. The counter was developed and used hy Knudscn and Hofstsdter.’!

654

2.

DETERMINATION O F FUNDAMENTAL Q U A N T I T I E S

no loss of efficiency can be tolerated. Hence the shower detector forms an important element in this experiment.’ (b) As the energies of man-made machines become higher and higher, the gamma rays and electrons appearing as bremsstrahlung, pairs, reaction products, etc., will also have very high energies. It will become increasingly difficult, if not impossible, to employ magnetic means to measure the energies or momenta of such particles. For example it requires something of the order of 100 tons of iron to provide a magnet which can bend 1.0-Bev electrons and still have a suitable solid angle and collection efficiency. If the energy is raised to 10 Bev the magnet will scale somewhere between the second and third power of the energy and its weight becomes of the order of the heaviest battleships ever built. To go on beyond this extreme value seems to be unreasonable at the present time. On the other hand a shower detector, of the kind we envisage, increases in size only logarithmically with the energy of the gamma ray or electron. For example, if a transparent material such as PbF2 could be developed in the form of a cylinder 80 cm long and 40 cm in diameter it would successfully contain a shower of energy 10 Bev. If a sodium iodide (Tl) scintillator is used for this energy its size would be about 1.2 m long and 1.0 m in diameter. Ninety per cent of a 10-Bev shower would be retained in such a crystal. These dimensions are rough and are merely presented to show the order of magnitudes involved. 2.2.3.5.3. METHOD OF OPERATION. When an energeticelectron orgamma ray* is permitted to enter a large crystal it develops a “cascade electromagnet,ic shower.”2 This shower is produced because the electron radiates electromagnetic quanta in any material it strikes. The quanta then produce electron-positron pairs. The electrons and positrons radiate again and so the chain is continued. The higher the energy of the incident electron the more intense is the subsequent shower. Thus the number of electrons and positrons will become greater and greater. The shower consists of electrons, positrons, and gamma rays all of which will be eventually degraded into very low-energy electrons and positrons (1.0 ev to 1.O MeV) . Now each of these electrons or positrons produces light, either by scintillation processes, or by cerenkov radiation; and the sum total of the light produced may be measured in conventional ways by a single photomultiplier or by a combination of photomultipliers. * Let us confine our attention to electrons, since gamma-measuring devices ultimately reduce t o electron detectors, and in fact, both detectors are very similar. ’L. B. Auerbach, G. Bernardini, I. Filosofo, A. 0. Hanson, A. C. Odian, and T. Yamagata, CERN Symposium, Geneva p. 291 (1956); see also I. Filosofo and T. Yamagata, ibid. p. 85. B. Rossi, “High Energy Particles.” Prentice-Hall, New York, 1952.

2.2.

DETERMINATION OF MOMENTUM A N D ENERGY

655

The total amount of light produced is expected to be proportional to the total track length2 of the particles appearing in the shower and hence proportional to the energy of the shower itself. In this way the height of the pulse produced by the photomultiplier will be proportional to the energy of the incident particle. We see that the basis of the method is the transformation, by electromagnetic means, of the kinetic energy of a single particle into that of a great many less energetic particles of smaller penetrating power. The less energetic particles can easily be contained in a relatively small volume. In actual practice the crystals will be surrounded by a shield carrying a small opening in one end permitting the entrance of the incident particles. The weight of this shield should not be underestimated although, of course, it does not approach the weight of a large magnet. The method outlined above was first employed by Kantz and Hofstadter3and by Fregeau and H ~ f s t a d t e rIt . ~has since been used by many authors, such as Cassels el CZZ.,~ Filosofo and Yamagata,6 Knudsen and Hofstadter.7 This method will not apply to heavier particles such as protons because heavy particles radiate very little compared to electrons and are stopped mainly by collision losses and nuclear reactions. 2.2.3.5.4.SIZEOF SHOWER DETECTOR. An important practical question arises immediately: how large does the detector have to be to measure the energy of the incident electron or gamma ray in order to obtain thereby a given accuracy? This type of question can be answered by reference to a diagram such as that shown in Fig. 2. In this case the data are taken from reference 3 and represent the amount of energy lost in a given ring of material in a tin* absorber when struck by electrons of 185 Mev initially traveling along the direction of the axis labeled “depth.” This axis represents the long axis of the specimen and the figure is considered to be a section of a cylinder obtained by rotation about the “depth” axis. The figure is then to be interpreted according to the following example. Consider a ring corresponding to an interval of depth in the specimen between 2 and 3 radiation lengths and lying between l and 2 radiation lengths in radius. (For

* Experiments have been carried out in tin and other metallic absorbers. Since a transparent material has to be used for an actual counter, tin is of course not suitable. NaI(T1) would give very similar results, however (see text). a A. Kants and R. Hofstadter, Nucleonics 12 (3), 36-43. 4 J. Fregeau and R. Hofstadter, see reference 3. 6 J. M. Cassels, G . Fidecaro, A. M. Wetherall, and J. R. Wormald, Proc. Phys. SOC. (London) A70,404 (1957). 6 T. Filosofo and T. Yamagata, CERN Symposium, Geneva p. 85 (1956). 7 A. Knudsen and R. Hofstadter (to be published).

656

2.


tin, one radiation length equals 1.22 cm.) This ring will absorb on the average 2.40 % of all the energy of the incident electxon of 185 MeV. Other rings correspond to other amounts of absorbed energy. By adding together all the energy absorbed in various rings one arrives a t the fractional amount of energy absorbed in a figure of revolution of whatever size one chooses in the diagram. If it is desired to select 90% of the original energy as a typical absorption figure, then this implies, e.g., a total depth of a tin cylinder of 10.5 radiation lengths and 12 radiation lengths in diameter. Other combinations are possible, of course, to secure 90% absorption. Such calculations are assisted materially by diagrams of t,he type indicated in Figs. 3 and 4 which apply to tin and lead respectively a t 185 MeV. Such “isoenergetic” curves permit one to choose the dimensions needed to absorb a certain amount of energy from the initial energy of

FIG.2. A block diagram showing the amount of energy lost in a given ring of material in a tin absorber for 185-Mev electron showers. The details of the figure are explained in the text.

185 MeV. For example it is not possible to choose a cylinder of tin less than 9 radiation lengths long, or one less than about 9 radiation lengths in diameter if one is to obtain a 90% absorption figure. I t is to be noted that these figures apply to tin (which incidentally is quite similar to NaI(T1) except for a density allowance which alters the scale in absolute centimeter units) at a certain energy, namely 185 MeV. At a different energy the behavior will be different. For example, as the incident energy is increased the peak of the shower absorption moves towards greater depths in the cylindrical specimen. However, from shower theory the qualitative features are probably valid a t even much higher incident energies. I t is unfortunate that, more data of the type shown in Figs. 2, 3, and 4 arc not available. However, atj 185 Mev data are given by Kantz and Hofstadter for several materials including (*arboil,nlumiiium, copper, tin, and lead. The size of the crystal, as we have seen above, is intimately related to the amount of energy absorbed in the cryst,al. The amount of energy is,

2.2.


657

FIG.3. Isoenergetic curves for 185-Mev electron showers in tin. Any point on a n isoenergetic curve corresponds to the same fraction of the energy of a shower absorbed by the material with the given outside dimensions.

I

Depth (Radiation lengths)

I

FIG.4. Isoenergetic curves for 185-Mcv clertron showers in Pb. This figure is sirnilnr to Fig. 3.

(358

2.


in turn, related to the question of the resolution of the crystal with respect to energy and to the linearity of the response of the shower detector. These two considerations can be understood easily from the discussion that follows. It is clear that if 100% of the energy is absorbed then the response of the detector will not suffer from straggling difficulties. In other words, if the transformation of light into pulse height is reasonably good (this point will be discussed later) every absorbed electron or gamma ray of a given energy will give rise to a pulse of the same magnitude. This means that the response function of the detector provides a unique pulse height for a given energy. I n actuality, due to straggling losses, the crystal detector is never as good as this. Instead, when the detector absorbs on the average only 90% of the incident energy, the 10% average loss results in a finite width of the pulse distribution. Occasionally a large quantum escapes from the back end of the crystal before interacting with it. For example, there is a finite probability that a 50-Mev quantum can escape, and if the incident energy is, say, 200 MeV, this escaping energy can produce a very large fluctuation in the pulse-height distribution. An approximate method of computing the pulse-height distribution has been given by Kantz and Hofstadter3and further calculations of this kind for NaI(T1) have been made by Knudsen and Hofstadter.7 We shall indicate below how such calculations are made. The statistical fluctuations in the “lost energy” will be approximated by a Poisson distribution. Take an example: consider a cylinder of lead fourteen radiation lengths long and fourteen radiation length‘s in diameter. A transparent Cerenkov crystal of PbF2 will behave in much the same way as lead. This specimen will capture approximately 90%, on the average, of a 200-Mev shower produced by an electron. This can be determined from Fig. 4. Thus a fraction of 10% of the incident energy escapes from the cylinder. We will assume that the radiation escaping from the greatest depths of the crystal will have the greatest penetrability. The energy of such radiation will lie at the minimum of the absorption cross-section curve.2 For this curve the abscissa is proportional to the gamma-ray energy and the ordinate to absorption coefficient. For lead the minimum occurs a t about 3.5 MeV. Thus the escaping 20 Mev will be made up in approximately 5.7 parcels of gamma rays, each corresponding to 3.5 MeV. Any actual distribution will not be as good as this because sometimes higher energy quanta will escape, since the absorption curve has a rather flat minimum. Now the distribution can be computed by applying Poisson statistics to the 5.7 gamma rays of 3.5 Mev. Such a calculation gives the curves in Figs. 5 and 6. Figure 5 shows the spread in energy in the response curve for Pb and PbF2for an incident energy of 185 In this case the peak of the curve is labeled 185 Mev and provides a nominal calibration figure. This

2.2.


659

Energy in Mev (Pulse height)

FIG. 5. This figure shows the results of a crude theory of the shape of the pulseheight distribution expected in lead and in PbF2for 185-Mev electrons. Because of the small nuyber of photoelectrons a t the emitting surface of the photomultiplier used with the Cerenkov material, PbF2, the statistical fluctuations of emission produce a width greater than t h a t expected from pure straggling in Pb. I n fact due to the occasional escape of gamma rays with energy larger than 3.5 Mev the actual pulse-height distribution may be slightly broader and will have a tail on the low-energy side.

explains why higher energies than the incident energy are apparently present in the beam. Figure 6 shows the corresponding calculations in NaI(T1) when 200 Mev and a 20% figure for the escaping energy are used and when different choices near the minimum of the gamma-ray absorption curve are employed as the quanta of escaping energy. It is observed that shapes vary but are not widely different. Using the minimum of the absorption curve the calculations give the solid line of Fig. 7.7 The dashed line shows the actual data obtained experimentally.? As expected, the actual curve shows a low-energy tail corresponding to the escape of a few energetic gamma rays. For a rough theory the check can be considered rather satisfactory. *

* Note added in proof: Recently a very important contribution to the technique of obtaining high resolution in large NaI spectrometers has been made by B. Ziegler, J. M. Wyckoff, and H. W. Koch, by choosing a combination of crystals in a suitable manner and selecting coincident events so t h a t a single annihilation photon escapes from the ’ front surface of the main spectrometer crystal. In this way a spectacular figure of 3; %

660

2.


I40 W

z 0 V

II.

40

0 0

=

20

n -

100 I20 140 160 180 200 ENERGY ABSORBED BY C R Y S T A L ( M E V )

70

FIG.6. The calculated line shapes are presented for straggling losses of 20% corrcsponding to twenty 2-Mev gamma rays and eight 5-Mev gamma rays for 200-Mev electron showers in NaI(T1). The calculations were made using Poisson statistics. This figure probably shows the extremes to be expected, except for the low-encrgy tail described in the legend t o the previous figure. The appropriate widths are shown in the figure and actually straddle the experimentally observed width, except for the tail. 2 . 2 . 3 . 5 . 5 . SIMILARITY RULE.Unfortunately the present detailed shower data have been obtained only a t 185 Mev and at lower energies. However, we may use shower theory t o help extrapolate the data to higher energies. For example, in shower theory2 (approximation B ) the behavior of all materials is the same if the lengths are expressed in radiat,ion unit,s and the energies are expressed in the units of “critical energy” = Eo.We define the critical energy as the collision loss per radiat,ion length of material for electrons of energy h’o. In actual practice this energy loss is the same as the collision loss per radiation length for electrons a t the minimum of the energy loss curve. If we apply this rule it is easy to convert the shower data of Kantz and Hofstadter to higher energies. We can use tin, as before, to represent the

energy resolution was obtained for 17.6 Mev gamma rays with a 9-in. diameter main crystal. The detection efficiency for 17.6 Mev gamma rays is stated to be about 7%. If this technique can be applied to high energies, which seems possible, qiiite good resolution can he achieved with crystal spectrometers.

2.2.

DETEIIMINATION O F MOMENTUM A N D ENERGY

140

66 1

c

CALIBRATION ENERGY ( M E V ) I I I I I 1 100 120 140 160 180 200 ENERGY ABSORBED BY CRYSTAL ( M E V )

I 80

Fro. 7. This figure is similar to Fig. 6 except that elcven 3.65-Mev gamma rays are cmploycd to arcount for the average energy loss. The energy 3.65 Mev lics a t thc minimum of the gamma-ray ahsorption curve for NaI(T1). The experimental curve is also shown In t h e figrirr.

iodine of NaI(T1) and can do similar things for carbon and carbon-bearing scintillators or Cerenkov materials. Thus NaI(T1) at 300 Mev is equivalent, to P b a t 185 by the similarity rule. Also 2.4-Bev electrons in carbon correspond to 6he Pb data at, 185 Mev. TTnfort,unately, we cannot extrapolate the data for heavy elements any further. I n any case this type of extrapolation must be considered only as a rough approximat,ion to the truth. 2.2.3.5.G. OTHERT H E o R E T I C i l L AIM. 1%.It. Wilson' has performed Some interesting Monte Carlo calculat,ioris OH t.he development of photon and electron-inhiated showers in lead. He has chosen to investigate incident, energies in the range 50-500 Mev. From these calculations rough est,imates can be made of the lengt,h of ,z shower detector only if the diameter of tmhecounter is very large. For pract,ic:aldiameters of Cerenkov or scintillat.ioii counters the Wilson estimates of the percentage of the shower retained in a sample cannot be made wit,h good accuracy. The reason is BR. R. Wilson, Phys. Rev. 86, 261 (1952).

662

2.


that radial spreading of the shower is considered in the theory only in the roughest approximation, by including effects of multiple scattering. Radial, i.e., sidewise, propagation of annihilation radiation and low-energy electrons is not considered in the theory. For information regarding the core of the shower, Wilson’s calculations may be very useful. An example of Wilson’s results is given in Fig. 8. This figure shows the number of electrons that would be counted in the core of a shower with paths in the

tFIG.8. Wilson’s* calculations for 20-500-Mev showers in lead. The figure shows the number of electrons expected in the core of a shower with paths in the direction of the shower. The abscissa gives the depth in lead of the detector when expressed in radiation lengths.

direction of the shower. The abscissa is the depth in lead in radiation lengths. The incident radiation consists of photons of the appropriate energies in Mev given in the figure. Other types of useful “transition”2 curves are also given in Wilson’s paper. A simple shower theory, neglecting radial spreading of the shower is also included in Wilson’s paper. The shower calculations of other authors have been compared by Wilson with the results of his own theory. For example the results of Rossi and Greisen9 and Arley’” are quoted by him. Some older shower calculations may still be useful to the reader and B. Rossi and K. Greisen, Revs. Modern Phys. 18, 240 (1941). Niels Arley, “On the Theory of Stochastic Processes and Their Application t o the Theory of Cosmic Radiation.” G.E.C. Gad, Copenhagen, 1943. lo

2.2.


663

for convenience, they are quoted For air showers radial spreads are estimated by Molihre, l6 Blatt,16 and Roberg and Nordheim.17 Experiments on radial spreads of copper (y,r]) threshold gamma rays were

gL L

3 2 MEV

79 MEV

> m a a

5

iL 5 0 MEV

r

9 5 MEV

r

1 2 5 MEV

V

c -I > 0 a

??A!%-b 3

8

0

8

16

24

32

0

8

16

24

32

40

VOLTS

FIG.9. The actual response, measured by Cassels et a1.6 of a eerenkov shower detect,or, to positrons in the energy range 32-125 MeV.

made by J. W. Rose1*in two materials: copper and lead. Transition curves for 330 Mev bremsstrahlung were also measured in various materials by Blocker, Kenney, and Panofsky.l9 2.2.3.5.7. EXAMPLES OF DATAOBTAINEDWITH SHOWER DETECTORS. Many investigators have employed large shower detector~’.~-~ and it is posH. J. Bhabha and W. Heitler, Proc. Roy. SOC.A169, 432 (1937). J. F. Carlson and J. R. Oppenheimer, Phys. Rev. 61, 220 (1937). 13 H. S. Snyder, Phys. Rev. 76, 1563 (1949). l 4 I. B. Bernstein, Phys. Rev. 80, 995 (1950). 16 G. MoliBre, Naturwissenschaften 30, 87 (1942); also W. Heisenberg, “Cosmic Radiation.” Dover, New York, 1946. 1.3 J. M. Blatt, Phys. Rev. 76, 1584 (1949). 17 J. Roberg and L. W. Nordhein, Phys. Rev. 76, 444 (1949). l 8 J. W. Rose, Phys. Rev. 82, 747 (1951). 1 9 W. Blocker, R. W. Kenney, and W. K. H. Panofsky, Phys. Rev. 79, 419 (1950). 11 12

064

2.


sible to give practical examples of the type of data that can now be obtained. Figure 9 t,aken from reference 5, shows the response of a &mnkov shower detector to positrons in i8heenergy range 32-125 Mcv. In this case Chssels et aL6 constructed their gamma spectrometer of “two right cylinders of Chance EDF 653335 glass, cach 5 in. in diameter and 4 in. long. These were placed in optical contact, to make a cylinder 8 in. in total length. All surfaces were polished and a 5-in. EM1 photomultiplier with a photocathode sensitivity of 37 pA/lumen was placed in optical contact. with one end.” An external lead convertor was used at the input end of the spectrometer and the pairs produced in the lead were detected by a thin count,er placed in coincidence with the shower detector. The purpose of

TRANSMISSION

3000

4000

5000

6000

WAVELENGTH (A)

FIG. 10. Physical properties of the Cassels shower detector. Details are given in the text.

this coincidence counter was to guarantee that the showers in the glass were generated near the axis of the counter. In this way the straggling is reduced and the resulting light pulses are more homogeneous. The glass was surrounded by a coaxial white shield and the photomultiplier was shielded magnetically by coaxial soft iron and mu-metal shields. The radiation entered the shower detector through a thin polished light-tight aluminum cap. Their results, as shown in Fig. 9, were observed with a 50-channel pulse analyzer and exhibit increasing pulse height as a function of increasing energy, as expect,ed. The glass used had a density of 3.9 grams/cm3 and a refractive index of 1.69. The radiation length of this material was 2.56 cm and its critical energy 16 MeV. Figure 10 shows some physical properties of the Cassels shower detector. The figure shows the transmission of light through 10 radiation

2.2.


665

lengths of glass as a function of wavelength. The response (quantum efficiency) of the photocathode and the spectrum of the Cerenkov radiation, are all given in arbitrary units. A type of glass possessing a higher response a t ultraviolet wavelengths would give still better performance. The data in Fig. 10 have been analyzed by Cassels et aL6 and plotted in Fig. 1 I . The mean pulse height of the spectrometer is proportional to the energy and the half-width is approximately proportional to the square root of the energy. The data in Fig. 9 clearly show that the gamma rays are detected with very nearly 100% efficiency. Similar results have been obtained by Knudsen and Hofstadter' who used a NaI(T1) scintillation counter. The dimensions of their counter are shown in Fig. 1. In Figs. 12 and 13 we see the various pulse-height spectra 30 1

10

c

/

0.15

Y 50

100 MeV

150

0

0.01

0.02 0.03 0.04 (MeV)-'

FIG.11. The spectra given in Fig. 9 have been analyzed to give the mean pulse heights and line widths plotted in this figure. The data are taken from Cassels et d 6

observed a t several energies for electrons between 125 and 600 Mev and also the pulse heights a t the maximum of the peaks as a function of the incident energy. Here again we see that practically 100% detection efficiency is realized and a very good linear response is obtained. Energy measurements are therefore quite simple after a calibration of the spectrometer has been made. The nearly 100% efficiency of the detector proves to be a great advantage. Other examples of line shapes are given in reference 6. 2.2.3.5.8. LIMITATIONS OF THE METHOD. One of the important advantages of the shower det,ect,or is the relative absence of systematic errors in obtaining an energy spectrum, since the efficiencyisnearly loo%, and since the whole spectrum is obtained simultaneously. Other advantages of a single counter in taking a spect.

a a t

m

4

a -

3

5

'3

w I

2

W v)

A

'

5)

n 0 0

100

2

ELECTRON

I0

ENERGY, MEV

FIG. 13. The figure shows the pulse-height peak values plotted against incident energy for the curves shown in Fig. 12. The relationship observed experimentally is linear within experimental error. This is a very desirable result from the point of view of calibration.

(3) The effective solid angle at the entrance of a shower detector will usually not be as large as one might think at first. This is because the incoming radiation should be kept near the axis of the counter in order to reduce geometrical discrepancies in the average amounts of low-energy radiations leaking out of the side walls of the counter. A diaphragm or a coincidence counting arrangement similar to that of Cassels and coworkers6 may be used to keep the desired radiation near the axis. (4)If the size of a shower detector is not large enough to retain a whole shower, it may still be used successfully when properly calibrated. In this case the pulse height will not increase as rapidly as demanded by the linear or proportional behavior. This feature of the detector is not neces-

668

2.

DETERMTNATI ON O F FUNDAMENTAL Q U A N T I T I E S

sarily a real limitation or disadvantage of this type of counter, but it is a point that, should be considered when choosing among various possible detectors. I t seems highly likely that a t very high energies 2.2.3.5.9. CONCLUSION. of electrons, positrons, or gamma rays the shower detector will prove to be one of the only economical types of counter available. It is desirable to search for new crystalline or glassy substances of high density and high optical transparency in the hope th at very good energy resolution with efficiencies of 100% can be realized without requiring excessively large volumes of the detecting material.

2.2.3.6. Gamma-Ray Telescopes.* Counter telescopes have been used successfully for the detection of gamma rays on many o ~ c a s i o n s . ~A- ~ typical arrangement is shown in Fig. 1. Sl, Sz, Sa, and Sq are countersfor instance, scintillation counters; C is a converter; and A l and A t are

A9

s4

FIG. 1 . A typical gamma-ray telescope. S1, S2, S3 and Sa are counters. C is a converter; A , and .4* absorbers. For their function see text..

absorbers. To identify the particle as a gamma ray one demands a coincidence between Sz, S s , and S d in anticoincidence with S1.Thus a neutral particle is converted into a charged particle between Sl and Sz, usually in C. This, of course, could also be a neutron which gives rise to a recoil proton between S1 and Sz, but the neutron background can be checked

* W. K. H. Panofsky, J. N. Steinberger, and J. S. Steller, Phys. Rev. 86, 180 (1952).

* G. Coeconi and A. Silverman, Phys. Rev. 88, 1230 (1952). L. J. KoeRter and F. E. Mills, Phys. Elev. BB, 651 (1956).

* Section 2.2.3.6

is by A. Silverman.

2.2.


669

easily by observation of the counting rate variation as the coiivcrter is varied. For instance, the vonverter is usually lead. If the lead converter is replaced by a n appropriate thickness of carbon or polyet hyleiie, the expected counting rate for gamma rays can be changed by a n order of magnitude without appreciably affecting the expected neutron rate. The absorbers -4,arid A , are not, in principle, necessary. I n general, their main function is to determine the efficiency of the gamma-ray detector

--

0.5

-

0.4

-

W

r

-

0.3 Q ..-u

r

w' 0.20.1 ..

I 01 0.

I

I

50

100

I

I50

I

200

2 0

E(MeV)

FIG. 2 . Efficiency ?(A')for the gamma-ray trlcscope shoun in Fig. 1. The curve nT refrrs to coincidenrcs hetarrn S? and S1 with St in anticoincidrnce. The rurvc n p is for (8: S , S , - 8,) The points arc measured cfhcicncirs using monorncrgetir gamma-rays of rnrrgy 100, 140, and 200 MeV. The two curves arc calculated from thr rxprrssions

+ +

as a function of gamma-ray cnergy. It is clear that, to detect a gamma ray in the above arrangement one of t,he pair of electrons formed between S1 and Sz must traverse S2A,S3A,S4. The total amount of material determines the minimum energy elect,ron required and thus determines a minimum energy gamma ray which can he detect.ed. In most applicat>ions,thc absolute efic*ic.iicy of t,he telescope must, tw determined for all energy gamma rays. This efIiciency may be determincd experimentally if monoenergetic gamma rays arc available or call be determined by cnlculabion. Figure 2 shows the efficiency versus gamma-ray

670

2.


energy for the particular telescope shown in Fig. 1.2In this telescope A 1 = s i n . Al; A z = 25in. Al; S1,S2, and S3 are 1-cm stilbene; and 5 4 is 1-cm NaI. The converter C = 1.5 rad lengths of Pb. The curve labeled nT is for coincidences between S2 and Sa in anticoincidence with S1. The curve labeled n F includes Sq in coincidence. The points at 100 MeV, 140 MeV, and 190 Mev are measured using monoenergetic gamma rays obtained by the technique described by Weil and M ~ D a n i e l The . ~ curves are fitted to the points using the empirical relationship V T ( E )= (0.50)[1 - e--(E-26)/40] V F ( E )= (0.45)[1 - e--(E--81)/43].

One can see that both curves have similar characteristics. They both rise rapidly from a threshold determined primarily by the absorbers and then

FIG.3. Efficiency of gamma-ray telescope of reference 5 as a function of converter thickness.

tend to saturate a t an efficiency approximately equal to 0.5. It is also possible to calculate the efficiency of such a telescope. Silverman and Stearnss calculated the efficiency of such a counter using the Monte Carlo shower calculations of Wilson.6 Koester and Mills3 used a Monte Carlo calculation programmed for the Illinois computer to make a similar calculation. J. W. Weil and B. D. &Daniel, Phys.Rev. 86, 582 (1952). A. Silverman and M. Stearns, Phys.Rev. 88, 1225 (1952). 8 R. R. Wilson, Phys. Rev. 86, 261 (1952).

2.2.


67 1

It is clear that such a telescope does not yield much information about the spectrum of gamma rays striking the telescope. In fact, for energies several times the threshold energy, the efficiency becomes independent of energy. A total absorption Cerenkov or scintillation counter whose output is proportional to the gamma-ray energy is very often a much more satisfactory detector. The efficiency of the counter is also influenced by the thickness of the converter C. The efficiency usually varies linearly with converter thickness for C 0.3 rad lengths, reaches a broad maximum a t 1-2 rad lengths, and then decreases as the converter becomes many radiation lengths thick. Qualitatively this behavior can be understood quite simply. Figure 3 shows the efficiency versus converter thickness for the counter telescope used by Silverman and st earn^.^ While the gamma-ray telescope is often a convenient and reasonably efficient instrument for gamma-ray detection, it has the following serious difficulties: (1) the efficiency of the counter is difficult to determine accurately; and (2) it gives very little information about the spectrum of the gamma ray under observation.

0.6 MeV, the Rayleigh scattering is negligible compared with Compton scattering. Leod Detector Housing

Leod Source Housing

43

I

125cm.

FIQ. 2. Arrangement of collimators (C), source, attenuators, and detector for minimizing secondary effects in narrow-beam ?-ray attenuation experiments down to transmission factors of -0.0002. Additional collimation of the primary beam is needed in the case of very dense attenuators, such as Ta (16.5 gm/cmg) [From Davisson and Evans, Phys. Rev. 81, 404 (1951)l.

Figure 3 shows the total mass-attenuation coefficients of Al, Cu, and Pb attenuators for photon energies from 0.01 Mev to 100 MeV. Each curve is the sum of the photoelectric, Compton, and pair-production coefficients, hence represents the total attenuation experienced by a wellcollimated beam of photons in narrow-beam geometry. The units of the p ) cm2/gm, and the attenuator total mass-attenuation coefficient ( ~ ~ / are "thickness" is to be expressed as its physical thickness times its density p, hence in units of (zp) gm/cm2. Then the fractional transmission 1/10of a homogeneous beam is I

- =

I"

exp[-((Cco/p)(w)I.

(2.2.3.7.1)

If the y-ray beam is a mixture of n1 photons having energy hvl, and mass 2

S. A. Colgate, Phys. Rev. 87, 592 (1952).

674

2.


FIG. 3. Total mass-attenuation coefficients (sum of photoelectric, Compton, and pair-production coefficients) for photons in Al, Cu, and Pb [From E ~ a n s 8 ~ ~ . . ]

attenuation coefficient ( p o / p )1, plus n2 photons having energy hvz, and mass attenuation coefficient ( p o / p )2, then the fractional transmission will be

where €1 and € 2 are the efficiencies of the detector for photons having energies hvr and hvz. Interpolation for other attenuators should not be made directly from Fig. 3 because each individual mass-attenuation process has a different

* R. D. Evans, “The Atomic Nucleus,” Chapters 23, 24, 25. McGraw-Hill, New York, 1955. 4 R. D. Evans, in “Handbuch der Physik-Encyclopedia of Physics” (S. Fliigge, ed.), Vol. 34, pp. 218-298. Springer, Berlin, 1958.

2.2.


675

functional dependence on 2. For attenuators other than All Cu, or P b accurate values of the mass-attenuation coefficients for the individual photoelectric, Compton, and pair-production interactions must first be calculated, and then summed to obtain the total mass-attenuation coefficient. Graphs and interpolation formula^,^,^ and tabulations6 of y-ray interaction coefficients are readily available. In narrow-beam geometry the detector should respond only to primary photons which pass through the attenuator without experiencing any collision. The elaborate experimental precautions indicated by Fig. 2 are required to minimize the response of the detector to singly scattered photons from the attenuators, from the edges of the defining apertures, and from the environs of the experimental setup, as well as to bremsstrahlung photons from recoil electrons, to coherent Rayleigh scattering, and t o multiply scattered photons. The chemical purity of the attenuators must be controlled also. Degradation of the primary photon spectrum by self-absorption within the source must be minimized by using sources of the highest available specific activity and smallest feasible mass. Narrow-beam experiments usually require a relatively strong source, of the order of 1 or more millicuries for nuclides which emit one photon per disintegration. If only weaker sources are available, the solid angle subtended by the detector as seen from the source may be increased either by enlarging the collimating apertures or by reducing the over-all source-to-detector distance. When this is done an increasing number of unwanted scattered photons will reach the detector, and the measured attenuation coefficient will be smaller than the true value of the total mass-attenuation coefficient. The magnitude of the error introduced by wider angle geometry can be estimated by making comparative or calibration measurements using photons of known energy. Convenient sources of essentially monoenergetic y rays include Hg203(0.279 MeV) , Aulg8 (0.411 Mev), CsI3’ (0.662 Mev), Zn@ (1.11 Mev), Coao(1.17 and 1.33 Mev), and Na24(1.37 and 2.75 Mev). Critical absorption edges can be utilized in some cases to obtain exceptionally accurate measurements of photon energy. The K edge of Pb is a t 0.0880 MeVJ and Fig. 3 shows that ( p ~ / p )changes discontinuously by a factor of 7.5 at this energy. Then a 0.0890-Mev y ray is very much more strongly attenuated by Pb than is a 0.0870-Mev y ray. The K edges of the elements adjacent to szPb are 0.0855 Mev for *IT1 and 0.0905 Mev for *3Bi, and there are corresponding discontinuities in their attenuation coefficients at these critical energies. Attenuators of T1, Pb, and Bi can then be used to bracket a photon energy which lies in the 2-3 kev domain between these critical absorption edges. For example, 6 Gladys White Grodstein, Null. Bur. Stundurds (U.S.) Circ. No. 685 (1957).

676

2.


a 0.0870-Mev y ray is strongly absorbed by T1, but only weakly by Pb or Bi; a 0.0890-Mev y ray is strongly absorbed by T1 and Pb, but only weakly by Bi. The K edges of the elements are shown in Fig. 2 of Section 2.1.1.2, and in selected cases the critical absorption method can be used, even with weak sources in broad-beam geometry, for photons whose energy is Iess than the K edge of uranium (0.115 Mev).

2.2.3.8. Detection and Measurement of Gamma Rays in Photographic Emulsions.* Gamma radiation, contrary to particle radiation, does not produce a specific effect in nuclear emulsions, but causes in the emulsion a general blackening, similar to visible light. I n gamma-ray exposures of low intensity it is possible to distinguish single beta-ray tracks, emitted ill secondary processes, and thus recognize the presence of gamma radiation. This method is applied in biological problems, but is more of a qualitative nature; it becomes important for problems of physics only for high-energy gamma radiation when electron pair production sets in and electromagnetic cascades start to develop. The study of these cascades, their multiplication, and the energy of individual electron pairs are used for the determination or, at least, estimation of the primary photon energy causing these phenomena. 2.2.3.8.1. ENERGY MEASUREMENT OF ELECTRON PAIRS.The most obvious method for the measurement of electron pairs is the multiple scattering method applied to both partners of the pair. The conditions for the scattering method are not very favorable if the pair energy is below 10 MeV, because of the frequency of large angular deviations which often cause the disappearance of one or both electron tracks from the emulsion layer. Energies of electron pairs in the region from 10 to 100 Mev can be determined with errors smaller than 30%, in general, if both tracks have path lengths exceeding 1 mm. The limitation of track length is a serious problem in the case of pairs in the Bev range, where large cell lengths in the scattering measurements should be used for good results. In these cases one tries to replace the ordinary method by the relative scattering method.' At high energies the two tracks of the pair are nearly parallel, and the separation distance of the tracks, measured a t regular intervals (cells), serves as a parameter for the energy determination of the pair. I n this type of measurement the stage noise is negligible, and M. Koshiha and M. F. Kaplon, Phys. Rev. 97, 193 (1955).

* Sect,ion2.2.3.8 is by M. Blau.

2.2.

DETERMINATION OF MOMENTUM A N D E N E R G Y

677

only grain and reading noise are present if occasional irregularities in t>hc emulsion, leading to spurious scattering, can be avoided. Iielative scattering measurements similar to ordinary scattering measurements can also be made with noise elimination, using two sets of cell lengths. The mean relative scattering angle in degrees per 100 p , ~~2 is given by E12 =

K

[(b3;;;)2+ (-')I*

P2P2C

(2.2.3.8.1)

where p$, and p!& refer to the two individual tracks. Variation of K , the scattering constant, with cell size and particle velocity has to be considered in the usual way. The relation between the individual values and p12/3~2 of the pair can be written in the form: 24

2 p12812c 5 PlPlC + p2P2c.

Relative scattering measurements can also be made on more than two tracks; for instance, electron pairs can be scattered with respect to tracks of fast mesons in highly collimated nuclear interactions. I n such cases p & h is found from the combination of ~ ~ $ 3and ~ 2paps. The distance between tracks used for relative scattering should be smaller than 20 p . Under favorable conditions relative scattering measurements give reliable results up to 20 t o 40 Bev pair energy.' 2.2.3.8.2. OPENINGANGLEOF PAIRS.The pair energy can be determined from the opening angle, assuming equipartition of energy for electron and positron. Borsellino2 describes the relation between opening angle w and pair energy E by the equation: (2.2.3.8.2)

Here m is the electron mass and cp.(a) the partition function, which depends only slightly on pair energy and atomic number Z of the atom in the field of which the pair is formed; a is the partition ratio; and cpl(a) % 1 for a = 0.5. cpz(a)deviates not very much from unity for values a # 0.5. According to relation (2.2.3.8.2)w is the most probable opening angle for a pair, with electron and positron having each the energy E / 2 . The validity of Eq. (2.2.3.8.2) has been confirmed up to energies of about 20 Bev, provided that deviations due to multiple scattering are considered in the measurements of the opening angle. The methods used in the determination of opening angles are essentially the same as in scattering measurements; one determines the separation between the two tracks in regular distances (cells) from the pair origin.

* A. Borsellino, Ph98. Rev. 88, 1023 (1953).

678

2.


Both measurements cease to give significant results when the separation distance becomes of the order of a grain diameter. METHOD.Fortunately there exists another 2.2.3.8.3. IONIZATION method, based on ionization measurements, which can be used at higher energies when scattering measurements fail to give reliable results. Observations made first by King3 and later confirmed and thoroughly investigated by perk in^,^ have shown that the grain density in highenergy pairs, near the origin of the pair, is smaller than twice the plateau value. The latter ionization density would be expected if both particles were to ionize independently from each other. The effect can be explained as a kind of “dipole effect,” and the phenomenon was treated mathematically by various authom6-9 In pairs of very high energy, and hence with extremely small opening angles, positron and electron move SO closely together that their separation is smaller than the impact parameter of distant collisions for relativistic singly charged particles. Thus a large number of electrons in the medium along the first few 100 p of the common path will receive energy from the mutual electromagnetic field of the pair, the latter being weakened by the destructive interference of the two fields of particles of equal but opposite charge. The grain density increases slowly with the distance from the origin and reaches the saturation value when the separation distance of the two partners is larger than twice the maximum impact parameter of a singly charged relativistic particle (-2.10W em). The effect is measured by the ratio R = ~[(dE/dR),,i~]/[2(dE/dR).in,l,]), where (dE/dR),in,l,is the energy loss of a singly charged relativistic particle. R can be calculated theoretically as a function of x, the distance from the pair origin, for pairs of various energies; in accurate calculations the influence of multiple scattering on the separation distance must be considered. A disadvantage of the method is the short extension of the dipole effect-100-200 p for pairs of ~ 1 0 ev-which ~ 1 impairs the statistical accuracy of the method. Weill et al. l o and Weill” have recently made an interesting observation which also might be useful for the energy determination of pairs. They found that the grain density in high-energy pairs D. T. King, unpublished data (1950). D. H. Perkins, Phil. Mag. [7] 46, 1146 (1955). A. E. Chudakov, Zzvest. Akad. Nauk S.S.S.R. 19, 650 (1955). 6 I. Mito and H. Ezawa, Progr. Theoret. Phys. (Kyoto) 18, 437 (1957). G . Yekutieli, Nuovo cimento [lo] 6, 1381 (1957). * H. Burkhardt, Nuovo cimento [lo] 9,375 (1958). J. Iwadare, Phil. Mag. [8] 3, 680 (1958). lo R. Weill, M. Gailloud, and Ph. Rosselet, Nuovo cimento [lo] 6, 413, 1430 (1957). R. Weill, Helv. Phys. Acta 31, 641 (1958). a

2.2.

679


remains below the saturation value (twice the plateau value) a t separation distances a t which the dipole effect should have ceased to exist. For pair energies of about loll ev the region of reduced grain density extends to distances of about 1 mm from the pair origin. The authors explain this phenomenon as a kind of geometrical effect: the two partners of the pair move so closely together that occasionally both cross the same grain of the emulsion, which would result in an apparent reduction of developable grains. This effect should cease to exist a t separation distances of the order of a grain diameter. The authors seem to find good agreement between experimental and calculated values, taking into account deviations due to multiple scattering. 2.2.3.8.4. GAMMARAYSBROM THE DECAY OF ?yo MESONS.The energy determination of gamma rays, electron pairs, respectively, is of great importance in connection with the problem of zro-mesonproduction and, energy distribution in high-energy interactions. The earliest data on &meson mass and half-life were derived from emulsion experiments by Carlson et a1.12 The authors searched the immediate neighborhood of stars (nuclear interactions) with minimum tracks for the presence of electron pairs and determined their energies by multiple scattering methods. The energy of the two gamma rays belonging to the same ?yo meson of total energy W has values between EI = &(V d W z - eo2) and Ez = $(W - d W z - E 0 2 ) , where eo is the rest energy of the ?yo meson. The angle 0 between the two gamma rays is given by

+

(2.2.3.8.3) where p is the velocity of the PO meson and r is the ratio of the two gamma-ray energies: r = EI/E2. The rest mass of the ?yo meson was found from €0 = 2 dm2 and the half-life from measurements of the finite path length of the ?yo meson before its decay into two gamma rays. The latter is given by the distance between the star centrum and the intersection of the two bisectors of the electron pairs. The search for electron pairs and the measurements necessary to correlate electron and y-ray pairs are quite lengthy and laborious. At higher energies, however, and hence in highly collimated shower cones, the experimental side of the problem becomes easier to handle, although correlation between electron pairs and individual T O mesons is not always possibIe if the shower density is very high. I n the shower cone immediately below the original interaction no electron pairs will be seen because of the large Lorenta time dilatation factor of the decaying high-energy d' mesons. Later, individual electron pairs will be visible, and still further l2

A. Carlson, I. Hooper, and D. T. King, Phil. Mag. [7] 41, 701 (1950).

680

2.


down in the emulsion stack the electromagnetic component starts to multiply and forms well-marked cores of electron showers. The experimental procedure for locating high-energy events consists in surveying the emulsion deep down in the stack for electronic showers, and tracing these showers back into the emulsion nearer the origin of the primary event. Such procedure presumes that the energy of the primary event is large, of the order of 5 x 10l2ev, and that the stack is well aligned and large enough to comprise more than 1 radiation unit; a radiation unit in a pure emulsion stack is 2.9 cm. A more effective procedure, first applied by Kaplon et aE.13and Kaplon and Ritsonl4 is the so-called emulsion-cloud chamber method, which uses instead of a pure emulsion stack a combination of emulsion and metal sheets, the emulsion being sandwiched between consecutive metal layers. The radiation length in an emulsion-metal combination is shorter, especially if metals of high atomic number are used. IEThe multiplication starts nearer the original event, the lateral spread is smaller, and hence the cores of electronic showers are more visible above the background of random tracks. According to Kaplon and RitsonI4 it is possible to detect showers in lead sandwiches from primary events of only 1Olo ev. A further advantage of a sandwich stack is the possibility of following the cascade down to lower energy values than in the case of a pure emulsion stack. The complete analysis of high-energy events (jets), however, often is not possible in combination stacks. Even in cases where the primary interaction takes place in an emulsion layer, large corrections have to be applied for secondary interactions occurring in the metal foils. Recently Duthie et al.'s have employed a n arrangement which combines the advantage of a pure emulsion stack with the improvements of sandwich stacks, in which case a multiple sandwich of lead and emulsion is placed underneath a large stack of pure emulsion. The number of T O mesons emitted in high-energy jets can be inferred from the number of electron cores, provided that each core is initiated by photons emitted from the decaying ?yo mesons. In the majority of experiments, corrections will have to be applied for photons arriving from outside the stack, for bremsstrahlung pairs, and for cascades from ?yo mesons produced in secondary interactions. M. Kaplon, B. Peters, and D. Ritson, Phys. Rev. 86, 900 (1953). M. Kaplon and D. Ritson, Phys. Rev. 88, 386 (1952). 0. Mmakawa, Y. N. Shimura, M. Tsuzuki, H. Yamanouchi, H. Aizu, H. Hasegawa, Y. Ishii, S. Tokunaga, Y. Fujimoto, 5. Hasegawa, J. Nishimura, K. Niu. K. Nishikawa, K. Imaeda, and M. Kazuno, Nuovo cimento [lo] 11, Suppl. No. 1, 125 (1959). l*J. Duthie, C. Fisher, P. Fowler, A. Kaddoura, D. H. Perkins, and I

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