Introduction to the Physics of Highly Charged Ions

Series in Atomic and Molecular Physics Introduction to the Physics of Highly Charged Ions H F Beyer GSI Darmstadt, Ger...

Author: Heinrich F. Beyer | Viateheslav P. Shevelko

221 downloads 649 Views 7MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

Series in Atomic and Molecular Physics

Introduction to the Physics of Highly Charged Ions

H F Beyer GSI Darmstadt, Germany and

V P Shevelko P N Lebedev Physical Institute Moscow, Russia

Institute of Physics Publishing Bristol and Philadelphia

© IOP Publishing Ltd 2003

c IOP Publishing Ltd 2003 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency under the terms of its agreement with Universities UK (UUK). British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN 0 7503 0481 2 Library of Congress Cataloging-in-Publication Data are available

Commissioning Editor: James Revill Production Editor: Simon Laurenson Production Control: Sarah Plenty Cover Design: Frédérique Swist Marketing: Nicola Newey and Verity Cooke Published by Institute of Physics Publishing, wholly owned by The Institute of Physics, London Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK US Office: Institute of Physics Publishing, The Public Ledger Building, Suite 929, 150 South Independence Mall West, Philadelphia, PA 19106, USA Typeset in LATEX 2ε by Text 2 Text, Torquay, Devon Printed in the UK by MPG Books Ltd, Bodmin, Cornwall


Contents

Preface List of symbols 1

Introduction 1.1 General remarks 1.2 Atomic masses, charges and sizes 1.3 Ions in nature 1.3.1 The Earth 1.3.2 The Sun 1.3.3 Cosmic x-ray sources 1.4 Ions in the laboratory 1.4.1 Plasma ion sources 1.4.2 Heavy-ion accelerators 1.4.3 Storage rings and ion traps 1.5 Visualization of single atoms 1.5.1 Scanning tunneling microscope 1.5.2 Single ions in magnetic traps

2

Radiation 2.1 Light and radiation 2.2 The electromagnetic spectrum 2.3 The distribution of radiation 2.4 Diffraction and interference 2.4.1 Diffraction 2.4.2 Interference 2.4.3 Diffraction at a single slit 2.4.4 Young’s double-slit experiment 2.4.5 The Heisenberg uncertainty principle 2.4.6 Fresnel lenses and zone plates 2.4.7 Bragg reflection, diffraction grating 2.4.8 Diffraction limited devices and the camera obscura 2.4.9 Massive particles as waves 2.4.10 The scanning electron microscope


Contents

vi 2.5

The Doppler effect

3

Spectroscopy 3.1 Spectral lines 3.2 The quantum nature of radiation 3.3 The photoelectric effect 3.4 Compton scattering 3.5 Mössbauer spectroscopy 3.6 Spectral-line analysis 3.7 The inner concept of atoms

4

Light and ion sources 4.1 Basic physical considerations 4.1.1 Elementary collisional and radiative processes 4.1.2 Statistical and collective behavior of particles 4.2 Bremsstrahlung 4.2.1 Radiation from accelerated charges 4.2.2 Longitudinal acceleration 4.2.3 Spatial distribution of bremsstrahlung 4.2.4 Spectral distribution of bremsstrahlung 4.2.5 Collisions 4.3 Synchrotron radiation 4.3.1 Angular distribution of the radiated power 4.3.2 Spectral distribution of synchrotron radiation 4.3.3 Insertion devices 4.4 Ion accelerators 4.4.1 General remarks 4.4.2 Acceleration of charged particles 4.4.3 Acceleration mechanisms 4.4.4 Focusing mechanisms 4.4.5 RFQ accelerators 4.4.6 Highly charged heavy ions 4.5 Ion cooler rings 4.5.1 Basic characteristics 4.5.2 Electron cooling 4.5.3 Stochastic cooling 4.5.4 Laser Cooling 4.6 Tokamak 4.6.1 Thermonuclear fusion 4.6.2 Conditions for a fusion reaction 4.6.3 The Tokamak configuration 4.7 Electron-cyclotron-resonance ion source 4.7.1 Basic operation principle 4.7.2 Magnetic configuration 4.7.3 Resonant heating


45 57 57 61 63 66 70 73 77 86 86 86 88 98 99 101 102 105 106 111 113 116 120 123 123 124 124 132 137 138 141 141 145 149 151 153 153 155 155 158 158 160 162

Contents

4.8

4.7.4 Electron supply 4.7.5 Enhancement of high charge states Electron-beam ion source and trap 4.8.1 Basic principle of operation 4.8.2 Step-by-step ionization 4.8.3 Ion heating and cooling

vii 163 163 166 166 168 169

5

Atomic structure 5.1 Classification of spectral lines 5.2 Coupling schemes 5.3 Selection rules 5.4 Transition probabilities and oscillator strengths 5.4.1 Transition probabilities of H- and He-like ions 5.5 Lifetimes 5.6 Autoionizing states and Auger decay 5.7 One-electron systems 5.8 Dirac equation: relativistic effects and the fine structure 5.8.1 Spin–orbit interaction 5.8.2 Nuclear finite-size correction 5.9 Magnetic effects and the hyperfine structure 5.10 QED effects and the Lamb shift 5.11 Many-electron systems 5.12 Transition energies and x-ray spectra 5.13 External fields 5.13.1 Polarizabilities 5.13.2 Electric field and Stark effect 5.13.3 Linear Stark effect in the hydrogen atom 5.13.4 Stark effect in H-like ions 5.13.5 Magnetic field and Zeeman effect 5.13.6 Zeeman effect in H-like ions 5.14 Quantum theory of line shape 5.14.1 Natural broadening of spectral lines 5.14.2 Doppler broadening 5.15 Absorption edges 5.16 Polarization of x-ray radiation

172 172 177 182 186 189 195 196 201 204 208 209 211 214 218 223 226 228 232 236 238 240 245 246 246 249 251 256

6

Atomic collisions 6.1 Collisional and photo processes in plasmas 6.2 Local thermodynamic equilibrium 6.3 Non-equilibrium plasma: the coronal limit 6.4 The principle of detailed balance 6.5 Photon emission and absorption 6.6 Excitation and de-excitation in collisions with electrons 6.6.1 Direct excitation 6.6.2 Resonant excitation

260 260 263 265 267 269 272 272 277


Contents

viii 6.7

7

Ionization and three-body recombination 6.7.1 Single-ionization processes 6.7.2 Direct ionization 6.7.3 Excitation–autoionization and the branching ratio coefficients 6.7.4 Resonant ionization via the capture of free electrons 6.7.5 Relativistic and QED effects 6.7.6 Inverse process: three-body recombination 6.8 Dielectronic recombination 6.8.1 Classification of the process 6.8.2 Dielectronic satellites 6.8.3 DR cross sections and rates 6.8.4 Dielectronic recombination experiments 6.8.5 Radiative recombination 6.8.6 Radiative recombination experiments 6.8.7 Radiative recombination at very low electron energies 6.9 Ion–ion collisions 6.9.1 General remarks 6.9.2 Experiments 6.9.3 Excitation 6.9.4 Electron capture 6.9.5 Heavy-ion collisions 6.9.6 Collisions between highly charged ions 6.9.7 Ionization 6.9.8 Inertial fusion driven by heavy ions 6.10 Ion–surface interaction and hollow atoms

284 287 287 290 295 295 297 298 300 300 303 306 308 308 310 311 316 318 319 320 322 327

Conclusion and further reading 7.1 Rydberg atoms and ions 7.2 Laser-produced plasma and related phenomena 7.3 Atomic many-electron processes 7.4 Recoil-ion momentum spectroscopy 7.5 Testing QED 7.5.1 Lamb shift 7.5.2 Hyperfine splitting 7.5.3 Bound-electron g-factor 7.6 Parity Violation

334 334 336 337 340 341 341 342 342 343

List of references for further reading

347

Atomic physics in chronological order

349


281 281 281

Preface

The physics of highly charged ions, or highly ionized atoms, is one of the most important and active domains of modern atomic physics which provides our basic information and understanding of atomic structure, short-wave radiation and electromagnetic interactions of these multiply ionized systems. During the last 10–20 years, the physics of highly charged ions began to be a very versatile tool for investigations of pure and applied physics including x-ray astronomy and astrophysics, x-ray lasers and lithography, heavy-ion induced fusion and nuclear physics, beam–foil spectroscopy and ion–surface interaction. Basically, three major events led to the development of this branch of physics: the discovery of x-ray radiation by W C Röntgen in 1895, the first spectroscopic measurements of highly charged ions made by I S Bowen, R A Millikan and B Edlén in the 1920s and 1930s and the discovery of the Lamb shift by W E Lamb and R C Retherford in 1947 which gave rise to the development of modern quantum electrodynamics. One has also to mention the progress achieved in the development of powerful laboratory sources of highly charged ions such as the Electron-Beam Ion Source (EBIS), Electron-Beam Ion Trap (EBIT) and accelerators which put, at the experimenter’s disposal, ions of the highest charge practically possible up to bare uranium ions. This textbook has been prepared to provide an overview of modern atomic physics with highly charged ions. It aims to serve as an introductory course on the subject for graduate and postgraduate students, as well as specialists in the field who will be able to find a reasonable mixture of fundamentals and practical applications for their research. We have attempted to cover the more important basic concepts without striving for completeness or generality. The student should be well prepared in electrodynamics and have some basic knowledge of atomic physics and quantum mechanics. The emphasis on multiple perspectives on similar subjects caused some overlap and repetition in some sections. We have also attempted to find a presentation that is a compromise between a basic introduction and a more stringent analysis of the subjects covered. This has meant deviating from the style usually prevailing in our own research. We hope that our readers will enjoy reading this book as much as we did writing it. We express our gratitude to numerous colleagues for helping us to select


material from their original research work to include in this book and for their valuable remarks. We are particularly indebted to A Müller who accompanied us from the start when the first vague plans for this book matured. It is our pleasure to acknowledge I L Beigman, V L Bychkov, J Kluge, D Liesen, R Neumann, V M Shabaev, I I Sobelman, H Tawara and A M Urnov. Special thanks go to Jim Revill of IOP Publishing for his patient cooperation during many delays in producing the manuscript. H F Beyer V P Shevelko Darmstadt and Moscow, July 2002


List of symbols

Fundamental constants = 299 792 458 m s−1 ~ = 1.054 571 596(82) × 10−34 J s ~c = 197.326 960 2(77) MeV fm e = 1.602 176 462(63) × 10−19 C m e = 0.510 998 902(21) MeV/c2 m p = 938.271 998(38) MeV/c2 u = 931.494 013(37) MeV/c2 c

= e2 /~c = 1/137.035 999 76(50) re = e2 /m e c2 = 2.817 940 285(31) × 10−15 m a0 = ~2 /m e e2 = 0.529 177 208 3(19) × 10−10 m Ry = m e e4 /2~2 = 13.605 691 72(53) eV kB = 8.617 342(15) × 10−5 eV K−1 µB = e~/2m e = 5.778 381 749(43) × 10−5 eV T−1

Velocity of light in vacuum Planck constant divided by 2π Conversion constant Elementary charge Electron mass Proton mass Unified atomic mass unit (mass of 12 C atom)/12

α

Fine-structure constant Classical electron radius Bohr radius Rydberg energy Boltzmann constant Bohr magneton

These constants are extracted from the set of constants recommended for international use by the Committee on Data for Science and Technology (CODATA) based on the ‘CODATA recommended values of the fundamental physical constants: 1998’ by Mohr P J and Taylor B N 2000 Rev. Mod. Phys. 72 351.

i


Notations α β

σ σ v τ χ ω ωp

Dirac matrix Relativistic factor, β = v/c0 Dipole polarizability = v/c0 Dirac matrix Beta function Relativistic factor, γ = (1 − β 2 )−1/2 Electric susceptibility Level width Auger width Radiative width Photon polarization vector Multipliciy Radiative rate coefficient Wave length Debye length Attenuation coefficient Reduced mass Magnetic moment Nuclear-charge density Binding radius Material density Cross section Rate coefficient Lifetime Magnetic susceptibility Wavefunction Angular frequency Plasmon frequency

A A b B B D E E E i,k E rel E cm

Atomic mass number Transition probability Impact parameter Magnetic flux density Magnetic inductance vector Dipole moment Electric field strength Energy Electronic binding energies of levels i and k Relativistic energy of the bound electron Particle energy in the center-of-mass frame

β β(s) γ a r κ λ λD µ µN ρ


Eκ f ik F g

I

In Iq j J k k L m mi M Mκ n ne ni nq N p P Pn (r ) P(θ ) q Q R s S S Sd T


Electric multipole transition Oscillator strength for transition i → k Force Statistical weight Lande factor Gaunt factor Nuclear spin quantum number Flux density Spectral line intensity Binding energy Ionization potential Single-electron total angular momentum Electron-current density Total angular momentum Wavenumber Photon momentum Single-electron orbital angular momentum Total orbital angular momentum Single-electron magnetic quantum number Ion mass Nuclear mass Magnetic multipole transition Principal quantum number Electron density Ion density Density of q-times ionized ions Number of atomic electrons Electron momentum Parity Radiation power Radial wavefunction Degree of polarization Ionic charge state Quadrupole moment of nuclear Photon energy Nuclear radius Internuclear distance Single-electron spin Total spin of the system Pointing vector Saha factor Optical saturation parameter Resonance strength for doubly excited state d Temperature

u U v W Wϕ Xq+ Z ZP ZT


Spectral energy density Scaled (reduced) electron energy Potential Velocity Probability Number of collisional events per second Work function q-times ionized atom Nuclear charge Projectile ion charge Target nuclear charge

Chapter 1 Introduction

Highly charged ions play a key role in many radiative and collisional processes occurring in laboratory and astrophysical plasmas and therefore they are a subject of detailed investigation in a special domain of atomic physics called the Physics of Highly Charged Ions. The radiation spectra of highly charged ions contain important information about plasma macro parameters such as electron and ion density and temperature, charge state distribution, polarization of x-ray radiation and provide an important diagnostic tool for investigations of laboratory and astrophysical plasma sources. Very specific radiative and collisional properties of highly charged ions are successfully used for many practical applications such as x-ray lithography, thermonuclear-fusion research, design of lasers operating in ultraviolet and x-ray spectral regions. In this introductory chapter, we recall some definitions basic to atomic physics and spectroscopy and give typical examples of ions which exist in nature and those created in laboratory ion sources. Historical dates, related to main discoveries and events in atomic physics including those in the physics of highly charged ions, are given in the appendix.

1.1 General remarks It was already the middle of the 1930s when it was discovered1 that neutral Sn atoms can lose more than 20 electrons in a laboratory plasma source—the vacuum spark. In those years it seemed that such highly charged ions were quite exotic and could hardly be found in nature. However, a principal discovery was made by B Edlén2 who used spectroscopic laboratory data for highly charged ions and explained the origin of many spectral lines in the solar corona as quantum transitions in 10–15 times ionized Ca, Fe and Ni atoms. These results clearly demonstrated the existence of highly charged ions in nature. 1 Bowen I S and Millikan R A 1925 Phys. Rev. 25 295. 2 Edlén B 1942 Z. Astrophys. 22 30S.


2

Introduction

Intensive observations of the solar and stellar spectra in the short-wavelength ˚ using rockets and satellites started shortly after the Second region, 1–2000 A, World War and provided a great impetus to the study of highly charged ions. For more than 50 years it has now been possible to measure the spectrum of the Sun in the ultraviolet (UV) and x-ray regions by means of spacecraft-based observations. These solar spectra showed transitions from highly charged ions up to Ni ions which have not been observed in the laboratory before stimulating further activity in experimental and theoretical atomic spectroscopy and diagnostics. Later, these studies were greatly aided by the development of new laboratory plasma sources. Novel techniques, involving recoil ions from ion–atom collision experiments or very highly ionized atoms from modern ion sources such as the EBIS (Electron-Beam Ion Source), the EBIT (Electron-Beam Ion Trap), the ECR (Electron Cyclotron Resonance) and storage rings, have been successfully tested. Recent advances in heavy-ion-beam technology has made it possible to produce ions from negative H− ions up to fully stripped uranium U92+ in a wide range of kinetic energies. The fundamental properties of highly charged ions are now widely used as a tool for both fundamental science and applied technology3.

1.2 Atomic masses, charges and sizes In a neutral atom, the number of positive (protons) and negative (electrons) particles is equal, i.e. an atom is electrically neutral. If under certain conditions an atom X loses q electrons, it becomes a positive ion Xq+ with a charge number q = Z − N, where Z is the nuclear charge number and N is the total number of electrons of the ion. If many electrons are removed from the atom, i.e. q 1, one has a highly ionized atom, or a multiply charged ion, or a highly charged ion. Every atom or ion constitutes a stable system of interacting charged particles with a positively charged nucleus surrounded by negatively charged electrons. The atomic nucleus consists of heavy particles—protons and neutrons. The proton and the neutron have nearly equal radii, of about 10−13 cm, and masses but they are approximately 2000 times heavier than the electron. Therefore the atomic mass is practically given by the nuclear mass, whereas the dimension of an atom is determined by the mean size of the outer-electron atomic shell which for neutral ˚ atoms is about (2–5) × 10−8 cm or 2–5 A. The mass of an atom or an ion, consisting of the nucleus and electrons, is given by the sum M = Np m p + Nn m n + Ne m e ≈ Np m p + Nn m n

(1.1)

because m e m p , where Np and Nn are the number of protons and neutrons, respectively, Ne is the number of electrons and m with corresponding index represents the mass of the component particle. Strictly speaking, the mass of 3 See Gillaspy J D 2001 Highly charged ions (topical review) J. Phys. B: At. Mol. Opt. Phys. 34 R93–130.


Atomic masses, charges and sizes

3

Table 1.1. Sizes, charges and masses of atomic particles. Particle

Radius (m)

Mass (kg)

Charge

Mass ratio

Electron Proton Neutron

2.82 × 10−15 8.47 × 10−16 8.47 × 10−16

9.109 × 10−31 1.673 × 10−27 1.675 × 10−27

−e +e 0

m n /m p = 1.0014 m p /m e = 1836.2 m n /m e = 1838.7

the bound nucleus in an atomic system is a little less than the sum of the separate components, i.e. Mnucl < Np m p + Nn m n . (1.2) The missing mass is called the mass defect of the nucleus and is associated with the interaction between nuclear components4. The masses and sizes of the electron, proton and neutron are given in table 1.1. For the electron, the classical radius is r0 = e2 /m e c2 , ascribed to the electron as a classical particle of spherically symmetric form with a charge e and rest energy m e c2 where m e is the electron mass and c the speed of light in vacuum. The atomic mass is the mass of an atom given in atomic mass units, 1 u 1.6601 × 10−27 kg. The atomic mass unit is defined as 1/12 of the mass of the carbon isotope 126 C. The atomic mass is a fractional number due to the natural abundance of different atomic isotopes on Earth. The mass number A is defined as the sum of the number of protons and neutrons: A = Np + Nn .

(1.3)

The atomic number is the number of protons of the chemical element given in the Periodic Table. The number of neutrons in the nucleus is the difference between its mass and atomic numbers. For example, the iron atom with 26 protons, 26 electrons and 30 neutrons has a mass number of A = 56. The weighted average over all iron isotopes on Earth yields the iron atomic mass of 55.84. To estimate the nuclear radius R of an atom or an ion with a mass number A ≥ 10, available experimental data give the following semiempirical expression: R = (0.836A1/3 + 0.570) fm

A = Np + Nn ≥ 10

(1.4)

where 1 fm = 10−15 m. For example, in the case of the uranium isotope 238 U with A = 238 this formula yields R = 5.75 fm close to the experimental value of R = 5.8604(23) fm. 4 See, e.g., Heyde K 1999 Basic Ideas and Concepts in Nuclear Physics 2nd edn (Bristol: IOP

Publishing).


4

Introduction

The electron charge is conditionally adopted as a negative value and defined as −e where e is called the elementary electric charge and equals e 1.602 × 10−19 C. The proton charge is equal to +e, and the neutron charge is zero, respectively (table 1.1). The charge of the nucleus is given by the number of its protons +Np e. The integer quantity Z = Np is called the nuclear charge number or sometimes erroneously the nuclear charge. As mentioned, the total electric charge of a neutral atom is zero: (1.5) Q A = +Np e + Ne (−e) = 0 as the number of electrons exactly equals the number of protons, Ne = Np . If this balance is violated, the atom is no longer electro-neutral, and its total charge begins to be positive Ne < Np or negative Ne > Np , and they say about positive or negative ions, respectively. In highly charged ions, one has Ne Np . As a consequence the Coulomb field of the ion’s nucleus is not very well shielded reflecting a specific property of highly charged ions which distinguishes them from those of neutral and weakly ionized atoms.

1.3 Ions in nature More than 90% of our universe is made of plasma—an ionized gas consisting of electrons, atoms, positive and negative ions and molecules. Ions can be found in various natural plasmas on Earth, in the Sun, in hot stars, in the interstellar space and in other objects. Radiation from natural plasmas and the atomic processes occurring in them are very important not only for our general knowledge but also for different practical applications. For example, it is necessary for the protection of electric transmission lines, predicting forest fires and for providing safety aviation flights. Spectroscopic measurements of aurora radiation allow one to obtain valuable information about the energy of the primary particles in the solar wind and to perform diagnostics on the electromagnetic state of the cosmos near the Earth. 1.3.1 The Earth Atmosphere. The Earth is surrounded by a mixture of gases called the atmosphere with a total mass of about 5 × 1018 kg. The gas composition of the Earth’s atmosphere is mainly represented by nitrogen (78%) and oxygen (21%) molecules. Atoms and molecules from nitrogen to xenon have nearly constant concentrations up to a height of 100 km, while the remainder of the molecules show a variable concentration in space and time. In the Earth’s atmosphere, positively charged ions can be created during thunderstorms or other kinds of electric discharges or as a result of the ionization of atoms and molecules in the upper atmosphere by cosmic rays. Because the Earth’s atmosphere mostly consists of the light atoms nitrogen and oxygen,


Ions in nature

5

the charge of these ions is not very high: q ≤ 5. Highly charged ions can exist in the plasma of cosmic objects, e.g. in the Sun or hot stars where the ion charge can be as high as q ≈ 30. Ions with a very high charge up to q = 92 can be artificially created in laboratory plasmas. Having relatively small concentrations, the variable constituents of the Earth’s atmosphere play an important role for life on the surface of the Earth. For example, water vapor is the source for all forms of precipitation and, in addition, is an active absorber and emitter of infrared radiation. It is the same for carbon dioxide, which besides being involved in the process of photosynthesis, also absorbs and emits infrared radiation. Ozone, mostly abundant in the region between 10 and 50 km above the Earth’s surface, is an effective absorber of UV radiation from the Sun and significantly shields the Earth from all radiation with ˚ wavelengths shorter than 3000 A. The upper atmosphere is that part of the Earth’s atmosphere above 100 km where the gas composition quickly changes with height and depends mainly on the solar activity. The upper atmosphere is always ionized and consists of relatively light ions. The main reason for its ionization is the ultraviolet radiation from the Sun and of electrons and protons from the solar wind and of highly charged ions (mainly oxygen ions) from cosmic rays. In the upper atmosphere, different elementary atomic processes take place such as inter-particle collisions, chemical reactions, ionization and dissociation. The Earth’s ionosphere is the ionized part of the upper atmosphere taking a height region between 100 and 350 km. It constitutes a weakly ionized plasma existing under the influence of the Earth’s magnetic field and the solar wind. In the ionosphere, ionization and recombination processes effectively take place. Ionization is caused by the solar ultraviolet radiation and by the solar wind consisting of electrons and protons. Radiation recombination takes place at heights around 1000 km. The main process is the dissociative recombination of molecular ions. At heights below 70 km, a three-body (ternary) recombination takes place between positive and negative ions. In the ionosphere, the electron density varies in a wide range up to n e = (3–5) × 105 cm−3 . The presence of the high-electron-density component in the ionosphere leads to a refraction of radio waves thus enabling the propagation of radio waves over very large distances. The Earth’s magnetic field is confined to a cavity called the magnetosphere which extends over about 10 Earth radii on the Sun’s side and about 1000 radii on the night side. The direct entry of the solar wind into the magnetosphere is prohibited. Within this region, plasma particles are confined in closed orbits by the Earth’s magnetic field, whereas those outside the region may drift to the edge of the magnetosphere. The inner region is called the plasmosphere5. The distribution of positive and negative ions over the height of the Earth’s atmosphere can be approximately summarized as follows: 5 See Akasofu S-I 1979 Dynamics of the Magnetosphere (Dordrecht: Reidel).


Introduction

6

200 km >600–1000 km ≈90–95 km

− − hydrate ions (H2 O)n H+ ; O− 2 , NO3 , HCO3 + + + NO , O2 , N2 O+ , N+ , H+ , He+ , He2+ , O2+ H+ Mg+ , Fe+ , Si+ , Na+ , Ca+ , Al+ , Ni+ .

Most probably, the presence of ions from heavy elements such as Mg+ , Fe+ , . . . , Ni+ is connected with the melting processes of incoming meteorites near an altitude of 90 km. Lightning. Lightning is a transient, high-current electric discharge in the air. A lightning stroke, or linear lightning, is similar to a laboratory arc discharge6 and is actually an oxygen–nitrogen plasma because N2 and O2 molecules are the main components of the Earth’s atmosphere. During the lightning discharge, a large amount of energy of 109–1010 J is released, and a plasma is created with a temperature of up to approximately 25 000 K, with an ionization degree of atoms and molecules of about 20%, an electron density of 1017–3 × 1018 cm−3 and a current of 20–30 kA. The stroke is several cm in diameter, several km in path length and has a duration of about 100– 1000 µs. Under these conditions, a lightning plasma consists mainly of atoms and ions such as O, Oq+ (q = 1–4), N, N+ and N+ 2 . The physics of the lightning discharge was considered by M Uman7 . Ball lightning. Ball lightning is a rare atmospheric phenomenon occurring during or after intense electrical activity in the atmosphere such as thunderstorm, tornado, volcanic activity or the launch of a vehicle. Ball lightning does not look like usual lighting but is a free flying electric ball very rarely seen and much more rarely photographed, which drifts through the air8 . Ball lightning is a local object most often observed during the summer months in the northern hemisphere. Some photographs of ball lightning are given in figure 1.1. Ball lightning is usually spherical (or oval) with a diameter ranging from a few centimetres up to a few metres, with various colors, mainly red, orange or acid blue. Ball lightning can also absorb light and it then appears grey or even black. Ball lightning usually has quite a long lifetime from about tens of seconds up to a minute. It can move horizontally with a speed of a few m s−1 in the atmosphere against the wind; it can also remain motionless in the air or may descend from a cloud. In contact with objects, these balls can explode releasing a large amount of energy of up to 106 –107 J. This property is most important reflecting the sense of the term ball lightning. Often, ball lightning is associated with radiation of a plasma. However, 6 Raizer Yu P 1991 Gas Discharge Physics (Berlin: Springer). 7 Uman M 1986 Lightning (New York: Dover).

Uman M 1987 The Lightning Discharge (San Diego, CA: Academic). 8 Singer S 1971 The Nature of Ball Lightning (New York: Plenum).

Barry J D 1981 Ball Lightning and Bead Lightning: Extreme Forms of Atmospheric Electricity (New York: Plenum). Stenhoff M 1999 Ball Lightning: An Unsolved Problem in Atmospheric Physics (New York: Kluwer/Plenum).


Ions in nature

7

Figure 1.1. Photographs of ball lightning: upper photograph, firework type with a 50 cm diameter; the photograph was taken from a distance of about 200 m; lower photograph, an oval type with a 35 cm diameter and estimated lifetime of about 10 s. From Singer S 1971 The Nature of Ball Lightning (New York: Plenum).

despite the numerous theoretical models proposed for the phenomenon, the mechanisms that cause ball lightning remain unclear. More than 100 (!) theories of ball lightning are known in the literature, including a quantum model, a plasma condensate of the Rydberg (highly excited) atoms and molecules, a chemical model and a polymer model. The most difficult and critical point of all theories is the ball lightning’s possibility to carry a very large uncompensated electrical charge which is of the order of a few Coulomb. So far ball lightning has not been simulated in the laboratory because it is necessary to fulfill, simultaneously, many, often contradictory conditions such as the creation of a high density of excited molecules exceeding 1018 mol cm−3 , a


8

Introduction

Figure 1.2. Photograph of an aurora in the Northern hemisphere (Kirovsk, Murmansk region, Russia). Photograph taken by J Wieser (GSI, Germany) on 1 September 2001. (See colour section.)

plasma temperature of less than 600 K, minimal radiation and thermal destruction. To obtain conditions close to those of the real ball lightning, the experiments are carried out with long-lived high-energy plasma formations titled plasma objects, the properties of which are very similar to those of ball lightning. Auroras. An aurora is a luminous phenomenon occurring in the Earth’s upper atmosphere at heights of about 100–120 km. Auroras are associated with the radiation of atoms and ions excited by fluxes of cosmic electrons and protons with an energy between 100 eV and 10 keV, mainly from the solar wind. The radiation spectra of the aurora range from the infrared over the visible and UV up to the x-ray region. Auroras occur in high latitudes of both hemispheres: in the northern hemisphere they are called the aurora borealis or northern lights; in the southern hemisphere, aurora australis or southern lights. Aurora lights are usually observed between 67◦ and 74◦ geographic latitudes but during periods of intense solar activity, auroras can also extend to the middle latitudes. The duration of aurora radiation can extend from a few minutes up to 1–2 hr. Aurora light can move rapidly across the sky or it can appear to stand still, flickering on and off. Auroras take many forms including curtains, arcs, bands and patches. The uniform arc is the most stable form of aurora with red or blue characteristic colors as can be seen in figure 1.2. The physical processes occurring in auroral phenomena are discussed in the book by J W Chamberlain9. The appearance rate and the radiation intensity of auroras correlate with 9 Chamberlain J W 1961 Physics of the Aurora and Airglow (New York: Academic).


Ions in nature

9

solar activity. The energetic electrons and protons arriving in the vicinity of the Earth as part of the solar wind are captured by the Earth’s magnetic field and conducted to the magnetic poles where they interact with atmospheric atoms and molecules. Therefore, such interactions occur in zones surrounding the Earth’s magnetic poles. The energetic electrons entering the atmosphere to altitudes of about 100 km, produce a shower of secondary and ternary electrons and those with a typical energy of about a few eV give the main contribution to the aurora luminosity. The most intensive radiation in the visible spectral region are the red and green lines of neutral oxygen and the violet and blue lines of the molecular nitrogen ion N+ 2 . These lines are associated with the strong forbidden lines of atomic oxygen. These visible spectral lines are very important because their intensities strongly depend on the flux and energy of incoming primary particles. The most probable process for exciting the red doublet and green line in oxygen is the dissociative recombination of the secondary and ternary electrons + with O+ 2 and NO ions: − ∗ ∗ O+ (1.6) 2 +e →O +O or

NO+ + e− → N∗ + O∗

(1.7)

because these molecular ions prevail in the Earth’s magnetosphere at heights below 200 km. As a result of reactions (1.6) and (1.7), one has the excited oxygen atoms in one of the terms belonging to their ground state. 1.3.2 The Sun The Sun is a plasma sphere and the nearest hot star to us which strongly influences the life on Earth and the state of the nearest cosmos. For us, the solar corona is the brightest x-ray source in the sky. The Sun mainly consists of H (91%) and He (9%) and of smaller fractions of ions of heavier elements (0.1%). By mass, the corresponding numbers are 71% H, 27% He and 2% others. The abundance of elements in the solar photosphere stay approximately constant for the whole volume of the Sun from the center to the solar corona and drop dramatically with increasing atomic number. High-Z elements, often studied in atomic-physics experiments, have little relevance to astrophysical spectra. The abundance of elements heavier than He is down by several orders of magnitude. There is a very distinct abundance peak at 56 Fe. The isolation of iron among other intermediate-Z elements makes its ions extremely important in x-ray radiation of the Sun. At the center of the Sun, the temperature is estimated to be 15.6 × 106 K and the density about 150 g cm−3 . Such a high temperature can only be sustained by nuclear reactions occurring in the center of the Sun. The age of the Sun (about 4.6 × 109 years) was deduced from the age of the oldest meteorites ever found on Earth and the assumption that the Sun has roughly the same age. The Sun’s age


10

Introduction

Figure 1.3. Photograph of solar eruptions taken on 2 November 1999 by SOHO space station (SOlar and Heliospheric Observatory), European Space Agency and NASA project, at a wavelength of 30.4 nm corresponding to the resonance line in He II. (See colour section.)

is about three times less than that estimated for the Universe according to the Big Bang theory. Investigations of the UV and x-ray radiation from the Sun have already given very important contributions to our knowledge about the physical processes and conditions in the hot solar plasma. These studies were possible because of the active development of the extraterrestrial investigations with the help of rockets and satellites. Analysis of the measured UV and x-ray spectra from the active solar regions, especially from solar flares (figure 1.3), has confirmed the existence of highly charged ions with a considerable abundance of almost all elements from H to Ni and from almost all stages of ionization. These spectra have provided information not only about the composition and physical conditions in the solar chromosphere and corona but also through their correct interpretation, about elementary collision processes taking place in the solar plasma. Therefore, to a large extent, solar physics is an essential ingredient to the physics of highly charged ions. The solar atmosphere is conventionally divided into four regions: the photosphere, chromosphere, solar corona and solar wind. The solar photosphere is the lowest part of the solar atmosphere consisting of a weakly ionized plasma and about 350 km thick. About two-thirds of the solar radiation originates from the photosphere as a continuum radiation with an effective temperature of 5830 K.


Ions in nature

11

The chromosphere lies above the photosphere, has a thickness of about 1500 km and a temperature of about 104 K. It is characterized by the emission of line radiation from atoms and ions, including H, He and other ions. The corona is the hottest low-density plasma layer with a plasma temperature near 2×106 K and a plasma density of 108 cm−3 . The corona is separated from the solar chromosphere by a thin intermediate layer in which the temperature jumps from 104 to 106 K. The solar corona is characterized by UV and x-ray radiation of spectral lines and a continuum in the wavelength range between 0.5 and 5 nm. The spectral lines are emitted by highly charged ions of the elements Fe, Si, Ni, Ca and O and the continuous spectrum is due to the bremsstrahlung caused by the interaction of charged particles. Outside the chromosphere, the corona expands into the everblowing solar wind. The solar wind is a collisionless plasma made up primarily of electrons and protons and, in smaller densities, He2+ (α-particles), highly charged ions of Oq+ , Siq+ , Sq+ , Feq+ and, probably, neutral Ne and Ar atoms. The solar wind represents an outflow of matter moving at supersonic speed which in the region of the Earth reaches about 400 km s−1 . Interaction of the solar wind with the Earth’s magnetosphere can lead to auroral display in the region of the poles. 1.3.3 Cosmic x-ray sources Our knowledge about cosmic objects in the Universe is mainly obtained from measurements of radiation spectra emitted by or transmitted through the plasmas of these objects. Usually, the radiation consists of continuum emission and a set of allowed and forbidden spectral lines. By the end of the 20th century, thousands of x-ray objects had been detected throughout the Universe. Because the Earth’s atmosphere absorbs x-rays very efficiently, x-ray telescopes and detectors must be carried high above it by spacecraft to observe objects that produce such electromagnetic radiation. The most recent all-sky survey conducted with the German/US/UK ROSAT mission, includes over 60 000 x-ray sources10 , only a fraction of which have been identified with previously known astronomical systems. In the UV and x-ray spectral ranges, the most intensive radiation comes from the binary stars, supernova remnants, the compact x-ray sources, galactic nuclei and quasi-stellar objects. This radiation is predominantly caused by highly charged ions present in these objects and signifies the occurrence of energetic phenomena in them. A proper interpretation of the short-wavelength spectra from these objects can give information about their structure, composition, energy balance, dynamics of mass flow, density and temperature distributions and other physical parameters. In the x-ray spectral range, cosmic objects of the hot Universe are seen at temperatures between millions and billions Kelvin. The spectra of many cosmic 10 Trümper J 1993 Science 260 1769.


12

Introduction

sources are dominated by a very strong continuum emission which arises from the hottest regions with the most energetic emission. Most of the information about cosmic x-ray plasmas comes from analysis of the shape and temporal variability of the continuum emission radiation. There are three important atomic processes responsible for continuum radiation in a hot astrophysical plasma: bremsstrahlung, synchrotron radiation and Compton scattering. Discrete emission spectra result from photon emission by atoms and ions with bound electrons in particular quantum states. For cosmic x-ray sources, the main characteristics of discrete spectra are primarily determined by the abundance of elements in the Universe which is almost the same as for the solar photosphere, i.e. the Universe predominantly consists of hydrogen and, to a lesser extent, of helium. However, neither of these elements has transitions at x-ray energies. Therefore almost all discrete features observed in the x-ray spectra of cosmic objects are caused by transitions in heavier ions. Thus, in the x-ray spectral range, we can see primarily K-shell transitions of C, O, Ne, Mg, Si, S, Ar, Ca and Fe. Emission lines from highly ionized Si and S have been detected in the early-type stars. The isolation of iron components among intermediate-Z elements, makes it prominent in cosmic x-ray spectra. In the vicinity of a hot star, the interstellar matter consists almost entirely of protons, i.e. of atomic hydrogen completely ionized by the star’s UV radiation. Such regions are called the H II regions. By far the greatest part of interstellar matter, however, exists in the form of neutral hydrogen clouds referred to as H I regions. Other components of the interstellar matter are grains of dust and cosmic rays consisting of high-energetic nuclei of atoms completely stripped by electrons. The electron temperature and density in various natural plasmas are given in table 1.2.

1.4 Ions in the laboratory Serious examples of man-made ions date back to simple arc discharges available towards the end of the 19th century when electricity started to be in general use. Before, as in ancient times, a fire was one of the most impressive phenomena over which homo sapiens gained control and flames might be regarded as the first primitive ion source. In modern times, the underlying physical processes for the creation of charged particles have been unveiled and technical devices which have shown an increasing level of sophistication built. Looking more carefully at these processes is the main subject of atomic physics as it developed from the beginning of the 20th century. The processes leading to the creation of charged particles may be classified by the way in which the external energy necessary to ionize neutral atoms or molecules in a gas or extracting them out of a solid material is applied. Collisions with charged particles (mainly with electrons) and also the interaction with radiation are the most important elementary processes prevailing in ion sources.


Ions in the laboratory

13

Table 1.2. Electron densities and temperatures in various natural plasmas. Plasma

Density (cm−3 )

Temperature (K)

Sun Center Photosphere Chromosphere Corona Solar wind (near Earth)

1025 1017 1011 –1014 6 × 107 –109 5

1.6×107 5830 104 2 × 106 4 × 105

Interstellar space H II regions H I regions Intergalactic space

1–3 1 × 10−4 1 × 10−6

104 100 3

Earth Outer magnetosphere Plasmasphere Ionosphere Metals

1–10 103 –104 105 –106 1022

104 104 250–3000 104

The average charge number of the ions produced as well as the effective fraction of charged particles in a source are the basic parameters used for classification. In technical applications, the power efficiency, that is the power of the electrical current associated with a stream of ions relative to the power input usually coming from an electrical power supply, also plays a role. According to the possible ways of generating highly charged ions in laboratory and the physical phenomena governing ionization processes, the ion sources can be divided into four main categories: plasma ion sources, heavy-ion accelerators, storage rings and ion traps. 1.4.1 Plasma ion sources As in natural sources, the majority of man-made ion sources are plasmas. The plasma state, very often titled the fourth state of matter, may be viewed as the result of successive heating of matter starting from a cold solid over liquid and cold vapor. The transition between a weakly ionized gas and a plasma is not sharply defined. Transport and the eventual separation of electrical charges give rise to internal electromagnetic fields that make the plasma a complicated but interesting subject of study. In a plasma environment with a huge number of particles, the average


14

Introduction

Figure 1.4. A map of temperature versus density for a number of astrophysical and laboratory plasmas.

particle velocity can be associated with a mean temperature. The frequency of collisional events is determined by the average distance a particle has to travel until it collides with one of its neighbors given by the particle density. Therefore, we can draw a map of density and temperature as in figure 1.4 in which important astrophysical and artificial plasmas occupy a certain region. The range in temperature and particularly in density is tremendous. Moderate to high charge states with appreciable intensity can be generated in the PIG (Penning Ion Gauge), the ECR (Electron Cyclotron Resonance) ion source, the LPIS (Laser-Produced Ion Source) and in the EBIS and EBIT (Electron-Beam Ion Source (Trap)). For the PIG, ECR and LPIS, the charge states q are in the range of 5 < q < 20. With these devices it is possible to produce one- to three-electron ions from atoms between neon and argon. The highest charges are currently available in the EBIS and EBIT where heavy atoms can also be stripped to few-electron states. They approach the limit q = Z of fully stripped ions otherwise only accessible by stripping accelerated heavy-ion beams. The most intense beams can be produced with the PIG source for q < 10 whereas high charge states can be produced in the ECR and EBIS. Penning Ionization Gauge (PIG). This is based on the ionization vacuum gauge invented in 1937 by F M Penning. It has a long tradition in accelerator technology and can be traced back to the early days of cyclotrons where it was used as an internal ion source. Later it became widely used mainly because of its simplicity and the high ion current available. Electron Cyclotron Resonance Ion Source. The Electron Cyclotron Resonance (ECR) ion source was first proposed by R Geller in 1969 and by H Potsma in 1970, and the first operational ECR source was presented by Geller


Ions in the laboratory

15

and his co-workers in 1971. Since then, the ECR ion-source technology for producing substantial amounts of highly charged ions has rapidly advanced11. Electron-beam ion source and trap. The principle of the Electron-Beam Ion Source (EBIS) was proposed by E D Donets in 1969. This device is now very efficient for the production of very highly charged ions12 . Ions trapped inside a dense electron beam are continuously bombarded by electrons and are sequentially ionized to high charge states. The physical parameters, like bombarding energy and confinement time, responsible for the production of very highly charged ions seem to be better under control in the EBIS than in other ion sources. That is why it has attracted many researchers and there is now an increasing number of these devices spread around the world. A very powerful extension of the EBIS concept was introduced by D A Knapp and co-workers in 1988 with the introduction of the ElectronBeam Ion Trap (EBIT) where ions with the highest charge states can now be produced and stored for very long times13 . The electron-beam ion trap developed at the Lawrence Livermore National Laboratory is actively used for various atomic-physics experiments including x-ray spectroscopy of few-electron ions and electron–ion cross-section measurements. The original EBIT has produced ionization states up to Ne-like U82+ as limited by the available beam energy of less than about 30 keV. Because of this limitation a second, high-energy trap, called the SuperEBIT that can run at an energy of more than 200 keV was built. Another well-known example for the trapping of positive ions in the spacecharge field of an intense electron beam, as in an EBIT, is the TOKAMAK—a toroidal chamber with an axial magnetic field. The abbreviation originates from the Russian TO(roidal’naya) KAM(era) s AK(sial’nym magnitnym polem). The idea for the TOKAMAK was suggested by I E Tamm and A D Sakharov in 1950 and experimental investigations of these systems were started in 1956. The laser ion source or laser-produced ion source14 is based on the plasma that is created when a solid surface is irradiated by intense laser light. When a pulsed laser beam is focused on a solid surface with a small spot typically less than 100 µm in diameter resulting in a high power density in excess of 108 W cm−2 , a hot plasma is created from which multiply charged ions can be extracted. The incoming light can penetrate the surface and the originated plasma only a small distance before reaching the critical electron density at which the plasma frequency equals the frequency of the laser light. The laser-produced plasma is a bright x-ray source which has been extensively studied15 . Conversion of laser light into x-rays can be very efficient 11 See Geller R 1996 Electron Cyclotron Resonance Ion Sources (Bristol: IOP Publishing). 12 Donets E D 1989 The Physics and Technology of Ion Sources ed I G Brown (New York: Wiley). 13 See Knapp D A 1995 Physics with Multiply Charged Ions ed D Liesen (New York: Plenum). 14 See, e.g., Moenke L 1989 Laser MicroAnalysis (New York: Wiley). 15 Hughes T P 1975 Plasma and Laser Light (London: Hilger). Radziemski L J and Cremers D A (ed) 1989 Laser-Induced Plasmas and Applications (New York: Dekker).


16

Introduction

for both continuum and line radiation. Total conversion efficiencies may exceed 50% whereas transfer into single spectral lines may reach about 5 × 10−4 of the incident laser energy. The plasma ion sources represent important tools for the study of highly charged ions. They have applications in pumping powerful shortwavelength lasers or for research into thermonuclear fusion plasmas. Extracted ions may be injected into particle accelerators like linear accelerators, cyclotrons and synchrotrons.

1.4.2 Heavy-ion accelerators Highly charged ions can be effectively created by stripping ion beams in accelerators. Stripping fast ions in an accelerator requires two steps. In a first step, ions with a low charge are injected and accelerated to an intermediate energy where they are stripped on penetrating a gas or thin solid foil. A fraction of the stripped ions can then be more efficiently accelerated further to high energies. Accelerator economy demands for the highest charge states possible, because the ion energy is proportional to the accelerating voltage multiplied by the particle’s charge. For a cyclotron, the energy is proportional to the square of the charge giving even a higher incentive to provide higher charge states from the source. From the very beginning, nuclear and high-energy physics have been the driving forces for the development of accelerators. Accelerator development, however, is closely linked to technological progress in other fields like highvacuum, radiofrequency or low-temperature techniques. More recently, atomic spectroscopy at particle accelerators has become an important technique since it is now possible to produce any desired ionization stage of virtually any element of the periodic table. A particle beam is a flow of a continuous stream or a bunch of particles that move along a straight or curved path defined as the longitudinal direction. The transverse velocity components and the spread in longitudinal velocities are generally small compared to the mean longitudinal velocity of the beam. Examples are the straight beams in linear accelerators and the curved beams in circular accelerators like betatrons, cyclotrons and synchrotrons. The light emitted by the ion beams in accelerators reveals a rich spectrum of lines ranging from the optical to the x-ray region. The identification of numerous lines and the study of the corresponding level structure represents one of the main applications of the beam–foil spectroscopy. The methods of beam–foil spectroscopy have been widely accepted initially at small accelerators with severe limitations on the accessible charge states. With the development of powerful heavy-ion accelerators, it is now possible to completely strip even the heaviest elements and to study high-Z few-electron systems.


Visualization of single atoms

17

1.4.3 Storage rings and ion traps Storage rings are versatile instruments for atomic and nuclear physics. The main purpose of these machines is the accumulation of a high ion current and the phase-space cooling providing high-quality ion beams for experiments. The pioneering developments, in particular those at the proton storage ring NAPM at Novosibirsk and at the Low-Energy Antiproton Ring LEAR at CERN, inspired many laboratories to build their own storage rings for various demands. Several small rings have become operational for nuclear and atomic physics with heavy ions. The purpose of ion traps is to confine charged particles to a small volume where they can be cooled to very low temperature and be used for experiments during a considerable amount of time. Owing to the detailed understanding of the operational principle and known systematic limitations electromagnetic traps have been turned into instruments for high-accuracy measurements. They have been applied for tests of quantum electrodynamics, fundamental symmetries and nuclear models and for metrology. The need to trap highly charged heavy ions is motivated by a further increased accuracy of mass measurements and extensions to new physical domains. Proposed measurements of the g-factor (the ratio of magnetic moment to mechanical one) and of the mass of heavy one-electron ions such as U91+ would test fundamental theory. Other applications like electron–ion and ion–atom interactions do not require the ultimate accuracies but explore the realm of highcharge–low-velocity collisions.

1.5 Visualization of single atoms 1.5.1 Scanning tunneling microscope A well-advanced method for imaging the density of electronic eigenstates in atoms is to use a Scanning Tunneling Microscope. With the scanning tunneling microscope it becomes possible to perform structural and spectroscopic imaging of atoms, molecules and surfaces on a scale down to atomic dimensions, i.e. it allows one to see single atoms on a surface with a resolution corresponding to about 0.1 nm. The technique was invented and developed by G Binnig and H Rohrer in 1982 at the IBM Zurich Research Laboratory, for which they received the Nobel prize in physics in 198616. The physics of the scanning tunneling microscope is based on a pure quantum-mechanical effect showing the wave nature of the electron. In classical physics, no electrons exist above the surface of a solid because reflection at the sharp boundaries of surfaces confines the particles. According to quantum mechanics, electrons behave like waves which leads to the existence of an electron 16 Binnig G and Rohrer H 1982 Phys. Rev. Lett. 49 57. Binnig G and Rohrer H 1986 The scanning tunneling microscope Sci. American 253 40.


18

Introduction

˚ 2 ) of the undisturbed double-row structure of D2 Figure 1.5. STM image (128 × 128 A organic molecules on pure graphite with intra-molecular contrast. The bright stripes are formed by the phenyl rings with attached dimethylamine groups of two adjacent rows of c D2 molecules. From Fritz J et al 1995 Surf. Sci. 329 1613. 1995 Elsevier.

cloud above the surface with exponentially decreasing probability to find electrons at a distance z from the surface. This effect is called tunneling because the electrons appear to be digging tunnels beyond the surface boundary17. The scanning tunneling microscope allows one not only to perform a surface topography but also to determine the kind of atoms and molecules located on a surface. The spectrum of applications in physics, chemistry, biology and other fields of science and technology is large, including, for example, the imaging of nucleic acids and viruses. The main advantage of the scanning tunneling microscope is the presence of a strongly focused electronic tunneling current with a small energy of the same order as in chemical reactions. No lenses or sources of light or electrons are needed. It uses free electrons existing on a metal surface. In comparison with electron microscopy, scanning tunneling microscopy handles the sample more gently during the imaging procedure, thus significantly reducing the danger of causing damage. Images of molecules. Scanning tunneling microscopy allows one to investigate the structure and orientation of individual molecules and of twodimensional aggregations of molecules adsorbed on a substrate. This is of special importance for the investigation of epitaxially grown monolayers of organic molecules on solid surfaces with respect to possible applications in molecular electronics18 . Figure 1.5 shows an image of monolayers of D2 organic molecules, having the sum formula C20 N19 N5 , on the basal plane of highly oriented pyrolytic graphite. The angle between a single D2 molecule and D2 rows is about 78◦. Single atoms on a metal surface. Adsorbed atoms or molecules and their 17 See Wolf E L 1989 Principles of Electron Tunneling Spectroscopy (New York: Oxford University

Press). 18 Fischer P, Port H and Wolf H C 1992 Z. Naturforsch. a 47 643.


Visualization of single atoms

19

Figure 1.6. Spatial image of the eigenstates of a quantum corral consisting of 48 Fe adatoms on a Cu surface. The average diameter of the ring (atom center to atom center) is c 14.26 nm. From Crommie M F et al 1993 Science 262 218. 1993 American Association for the Advancement of Science.

electronic states might be recognizable via local density-of-state effects in elastic scanning tunneling spectroscopy. The local density of states in a separate atom or in a group of atoms, positioned on a metal surface, can also be measured by means of a scanning tunneling microscope. When electrons are confined to a volume with a size approaching the de Broglie wavelength, their behavior is dominated by quantum-mechanical effects. Figure 1.6 shows a scanning-tunneling-microscope image of 48 Fe atoms individually positioned as a ring (corral) on a Cu surface. The ring has a mean radius of 7.13 nm and the spacing between neighboring Fe atoms is between 0.88 and 1.02 nm. One can see a strong modulation of the local density of states inside the corral confining the electrons. The details of this confinement mechanism are not yet completely understood. 1.5.2 Single ions in magnetic traps Ions can also be trapped in electromagnetic fields because of their charge. Single ions (or even one ion) can be imaged in miniature traps such as the Paul trap or the Penning trap by using a laser light to stimulate fluorescent radiation. The lowest temperature achievable is defined by the Doppler limit and lies in the mK range. For example, a single Mg+ ion with a laser detuning near resonance transition 3S1/2 –3P3/2 can be cooled to a temperature below 10 mK. These low temperatures can be reached within a fraction of a second. The large size of the trap allows one to use a large solid angle for detecting the fluorescence


20

Introduction

Figure 1.7. Two, three, four and seven 24 Mg+ ions confined by the dynamical potential of the Paul trap and crystallized into an ordered structure in a plane perpendicular to the symmetry axis of the trap. The average separation of the ions is 20 µm. From Walther H c 1994 Atoms in cavities and traps Adv. At. Mol. Phys. 32 379. 1994 Academic Press.

radiation either with a photomultiplier or by means of a photon-counting imaging system. When a relatively small number of ions are trapped, phase transitions can also be observed in two modifications: in a chaotic cloud phase and in an ordered crystalline structure depending on the degree of laser cooling. The Paul trap allows one also to observe such crystalline ion structure where the mutual Coulomb repulsion is compensated by the external dynamic trap potential (figure 1.7). Similar phenomena of crystallization of charged particles are observed also for aluminum microparticles (20 µm diameter, 105 times the elementary charge), Be+ and Hg+ ions. In general, the dynamic and static properties of the lasercooled ions in the Paul trap can be effectively used for the study of few-body non-equilibrium phenomena.


Chapter 2 Radiation

In the introduction we have addressed several radiative phenomena without explaining their physical foundations in detail. For a deeper physical understanding we are now going to recall electromagnetic wave phenomena. In this chapter we will recall basic definitions and discuss some radiation physics which will prepare the stage for the later chapters.

2.1 Light and radiation Light is a fundamental feature and has attracted philosophers since ancient times. For the Pythagoreans light was something that originated from a body and caused vision by entering the eye. Socrates and Plato reversed this idea in that the eye searches for objects by sending out light rays. In modern times, such a philosophical dispute appears strange and can only be understood by the tremendous size of the speed of light that prohibited a scientific proof in ancient times. It was at the beginning of the 19th century when it was generally agreed that light was a wave phenomenon with some similarities to water and sound waves. In 1887 Michelson and Morley established that the velocity of light was independent of the Earth’s movement. This fundamental observation was used later by Einstein to develop his theory of light. The understanding of the general properties of radiation is central to the physics of atoms. Interaction with light is also an important tool for investigating ions and atoms. Radiation is the way in which energy is transmitted through space from one point to another without the need for any connection or medium between these two places. The terms light, radiation, rays and waves characterize the same phenomenon and are often used as synonyms. Electromagnetic waves, i.e. periodically fluctuating electric and magnetic fields are matterless patterns, series of events that happen repeatedly. Physically they are described by transverse waves periodic in time and space and are characterized by the wavevector k, their period T , wavelength λ and the amplitudes E 0 of the electric field and B0 of the magnetic field, respectively. An


22

Radiation

Figure 2.1. A linearly polarized electromagnetic plane wave.

example of a linearly polarized electromagnetic wave is illustrated in figure 2.1. The wavelength is the distance between two successive wave crests which is related to the period in time and to the wavevector by c = λ/T

|k| = 2π/λ.

(2.1)

where c is the speed of light, i.e. the wavepacket group velocity of light. The electric and magnetic fields are not independent but are linked to one another. Maxwell ’s equations, representing the fundamental mathematical framework of radiation, require a changing electric field to be accompanied by a magnetic field1 . Both fields are propagating with the same finite speed c. For light in vacuum it is the highest speed possible in nature.

2.2 The electromagnetic spectrum White light is a mixture of many colors. In 1672 this was demonstrated by Isaac Newton who separated the colors of white light with a prism and joined them again with a second prism. Colors are determined by the wavelength. In experiments similar to this well-known example, different colors may be sorted by processes depending on wavelength like refraction or diffraction of light forming the basis of spectrographs. An illustration of the large range of wavelengths of electromagnetic radiation is contained in figure 2.2 showing a frequency and wavelength scale spanning 20 orders of magnitude. The radiation involved ranges from radio waves over infrared and visible light to x- and gamma-rays. As an illustrative comparison, a number of objects are included that are of similar size to the respective radiation. Whereas radio waves can exceed the biggest human buildings gammaray wavelengths correspond to the size of atomic nuclei. Also given in the figure are examples of typical instruments for measuring radiation of different 1 See for instance: Jackson J D 1999 Classical Electrodynamics 3rd edn (New York: Wiley).


The electromagnetic spectrum

Gamma rays

1 023

23

Gamma-ray detector Ge(i)

1 0-14

nucleus 1 021

X rays

1 019

Visible

1 017

1 015

1 0-12

atom

1 0-10

1 0-8

1 0-6

Infrared

bacteria 1 013

1 011

1 09

1 0-4

Prism spectrograph

1 0-2 bug

1 00

1 05

1 03

1 02

1 04

Wavelength (m)

1 07 Frequency (Hz)

Radio

human

Radiotelescope sky scraper

Figure 2.2. The electromagnetic spectrum ranging from radio waves to gamma-rays. The frequency and corresponding wavelength scale is related to the size of various objects. Examples of different detection schemes are also given. (See colour section.)


24

Radiation

400

500

600

700

λ (nm) Figure 2.3. The colors of visible light. (See colour section.)

wavelengths. The continuous, and more prominently, the line radiation emitted by stars, galaxies and other distant objects can reveal a lot of information about the composition, structure and motion. That is why the emission patterns as a whole are sometimes referred to as the cosmic barcode. Despite the large range in wavelength and different roles it is the same kind of radiation which altogether comprises the electromagnetic spectrum. The visible portion ranges only from about 400 to 700 nm with the maximum of the human eyes’ sensitivity in the middle of that range at about 550 nm. The components of visible light may be classified in more detail. At high wavelengths there is infrared radiation passing through various colors, known from the rainbow, to the near and far ultraviolet region. A continuous visible spectrum is shown in figure 2.3 with an approximate wavelength scale attached to it. Within this wavelength region, electromagnetic radiation is the physical stimulus that gives rise to our perceptual experience of vision. A white light source such as our Sun or a light bulb emits waves of all wavelengths within the visible region. When white light is incident on a surface, waves of some wavelength are absorbed and others are reflected. It is the wavelength-dependent reflectance that gives us the perception of the color of an object. The dominant wavelength at which most of the light is reflected is called the color or hue of the surface. In addition to the dominant wavelength there are two other perceptually important factors. These are the luminance or brightness, which is given by the amplitude of the wave, and purity or saturation. Purity is defined as the range of wavelengths that are present in a given light wave. The wider that wavelength range is, the less pure the light is. For instance pastels are not very pure colors. As opposed to the physiological and psychological color experience mediated by the human eye spectroscopists have spent much effort in filtering out a single color from a broad spectrum, i.e. to produce monochromatic radiation. Good examples of monochromatic light sources built on purpose are lasers radiating very pure light of almost a single wavelength. Yet another scale to be introduced has to do with the wave–particle dualism of radiation, which will be discussed in more detail in section 3.2. Radiation with a higher frequency is said to be more energetic and the scheme of photon energy E x is introduced which is related to the radiation frequency ν and wavelength λ



25

Table 2.1. Adopted terminology for the short-wavelength range of the electromagnetic spectrum. Name

Wavelength (nm)

Vacuum ultraviolet (VUV) Schumann UV Extreme ultraviolet (EUV) XUV Soft x-rays Hard x-rays Gamma radiation

30–200 100–200 10–100 2–30 0.1–4 0.01–0.1 ≤ 0.01

by the relation E x = hν = hc/λ

(2.2)

where h denotes Planck’s constant and c the velocity of light, respectively. Numerically equation (2.2) may be rewritten using the nm for the unit of wavelength and the electron volt (eV) as the unit of energy, respectively and inserting the numerical value of Planck’s constant. The electron volt (eV) is the most common unit of energy in atomic physics. It is the amount of energy an electron acquires when it falls through an electrical potential of 1 V. Other related units used for electromagnetic radiation are summarized here with their respective conversions. λ(nm) = 1239.8/E x(eV) λ−1 (cm−1 ) = 8065.8 × E x (eV)

(2.3)

ν(MHz) = 2.4180 × 10 × E x (eV). 8

The radiation spectra of highly charged ions lie in the short-wavelength regions which are given in table 2.1. The limits are not always sharply defined and are somewhat arbitrary. Historically the classifications originate from the techniques necessary to observe the radiation spectra or from the origin of the radiation. For example, vacuum-ultraviolet radiation having wavelengths below 200 nm is absorbed in air. That is why it can only be measured in evacuated spectrographs. Between about 100 and 10 nm no suitable window materials exist to transmit the radiation. The XUV region connects the x-ray region with the ultraviolet region. The discrimination between x- and gamma-rays very often refers to the radiation’s physical origin. Whereas gamma-rays are emitted from excited nuclei x-rays are due to the excitation of electrons in the electron cloud of an atom. X-rays have wavelengths of more than 0.01 nm and are further subdivided into soft and hard x-rays where the border between both is marked by the copper x-rays not far from 0.1 nm. The high-wavelength end of the soft x-ray region is usually put near the carbon x-rays close to 4 nm.


26

Radiation

400

500

600

700

λ (nm) Figure 2.4. The two yellow sodium D lines observed in emission (top) and absorption (bottom). (See colour section.)

Besides the continuous spectra such as the one shown in figure 2.3, line spectra are also observed as narrow intensity profiles if recorded as a function of wavelength. A simple and impressive example is the yellow lines emitted by hot sodium vapor as illustrated in figure 2.4. The spectral lines appear near 589 nm and are characteristic of the element sodium. An interesting finding is that at the very same wavelengths dark lines occur if sodium atoms are put between a white light source and the spectrograph recording the continuous spectrum of the light source. Actually many of the spectral lines that we know very well today were first observed in the solar spectra as Frauenhoffer lines after Joseph von Frauenhoffer (1787–1826) who first classified these absorption spectra. There are now three different kinds of spectra: continuous, emission line and absorption line spectra. The empirical observations of how these different spectra can be observed were summarized by Gustav Kirchhoff (1824–1887) in 1859 and are referred to as Kirchhoff’s laws: (i) A hot solid or liquid body or a sufficiently dense gas emits a continuous spectrum of radiation. (ii) A hot gas of low density emits a radiation spectrum consisting of single lines characteristic of the atoms constituting the gas. (iii) A cool, thin gas absorbs from a continuous distribution of light only at discrete wavelengths producing dark absorption lines superimposed on the continuous spectrum. The absorption lines occur at the same wavelengths as the emission lines of the very same gas at higher temperature. Emission and absorption of light by an atom is schematically depicted in figure 2.5. In absorption, radiation of suitable energy or wavelength impinging on an atom raises an electron from a lower to a more energetic upper state. In the reverse process spontaneous emission of a spectral line occurs if an atom undergoes a transition from an excited initial state into a final state of lower energy. This latter process can be enhanced in a so called stimulated emission if the excited atom is irradiated by light of the same energy as that of the spontaneously emitted radiation. Extensive use of stimulated emission is made in a laser (light amplification of stimulated emission of radiation) where radiation



27

2

1 absorption

spontaneous

stimulated emission

Figure 2.5. The basic interactions of radiation with atomic particles: absorption, spontaneous emission and stimulated emission.

of the proper wavelength repeatedly interacts with atoms of the laser medium in a cavity. Astrophysical examples of the absorption lines are the Frauenhoffer lines observed in sunlight. There are hundreds of absorption lines which are characteristic of the chemical elements present in the Sun. Furthermore numerous ionized atoms also contribute, each state of ionization having its characteristic line spectrum. As in the case of the sodium D lines characteristic for neutral sodium there are distinctive line patterns characteristic of the ionization stage. As a matter of fact, similar patterns are observable if the number of electrons present in different ions stays the same. In early spectroscopic investigations the spectra of a given chemical element were numbered by Roman numbers starting with I for the spectrum of the neutral atom, II for the singly ionized ion, III for doubly ionized etc. This fashion is still in use and Roman numerals are attached to the sign of the chemical element of the ion. However, there is also the well-known notation for elements and ions. A neutral atom is designated by the symbol of the chemical element in the Periodic Table. For positive (or negative) ions the number of missing (or additional) electrons is added as a superscript. For example, a neutral neon atom is denoted as Ne, three-time ionized Ne atom as Ne3+ and a negative atom as Ne− . In general, positive ions are denoted as Xq+ , q = Z − N, where Z is the nuclear charge number and N is the total number of electrons in the atom or ion. There is a correspondence between the chemical and spectral notation. For example, the neutral Fe atom is written as Fe I or Fe0+ , the Fe25+ ion as Fe XXVI and so on. Ions with a fixed number of electrons N arranged in an increasing order of the nuclear charge Z belong to the so-called isoelectronic sequence of the


28

Radiation

corresponding atom A and are termed A-like ions or [A] ions. If N = 1 one has H-like ions, [H]; if N = 2, one has He-like ions, [He], and so on. For example, the Ne atom and the ions Na+ , Mg2+ , Al3+ , . . . , have N = 10 electrons and belong to the Ne isoelectronic sequence and are termed Ne-like ions. If the number of electrons in a positive ion Xq+ is much less than the nuclear charge number, N Z , the ion is called a highly charged ion, or multiply charged ion, or highly ionized atom. As there is no strict definition, sometimes ions with q > 5 or Z > N/2 are already called highly charged. In any case, the main feature of highly charged ions is the long-range unshielded (uncompensated) Coulomb field of the nucleus which prevails in all interactions of these ions with photons and atomic particles: electrons, atoms, molecules and other ions. Further down the road in this book we will investigate the radiational processes of ions. Here, we try to roughly identify interesting spectral regions on the wide scale introduced here. For that purpose we confine ourselves to processes in the electron cloud of an ion or atom. Spectral lines occur throughout the entire electromagnetic spectrum. Usually, electronic transitions between the lowest orbitals in the lightest elements, like hydrogen or helium, produce ultraviolet and visible spectral lines. This spectral range can also be crowded by many lines originating from transitions between outer orbitals in heavy elements with many different stages of ionization, each having its own characteristic set of line spectra. Inner-shell transitions of heavy elements produce less closely spaced spectral lines in the x-ray region. The highest energy involved in a radiative absorption or emission process is set by the ionization energy of the heaviest ion abundant in nature which, in addition, is highly ionized with only one electron still present, that is U91+ . The ionization potential for this example is 132 keV and the transition energy between the innermost shells is about 100 keV.

2.3 The distribution of radiation All objects emit electromagnetic radiation. The total amount as well as its distribution with wavelength is determined by the object’s temperature. The temperature is a measure of the amount of internal microscopic vibration of an object: the hotter the object the higher the frequency of the vibrating constituents and the more energy radiated per time interval—usually designated by the radiation intensity. A perfect thermal emitter is called a black-body emitter. This is an ideal concept of an object that absorbs all radiation that falls on it. In equilibrium with its surroundings it must re-emit the same amount of energy that it absorbs. No real body behaves like a black-body radiator but very often the behavior of a hot object can be approximated by that of a black body quite closely. To find the spectral distribution of black-body radiation one has to resort to the laws of optics, thermodynamics and statistics. From classical electrodynamics J W S Rayleigh


The distribution of radiation

29

E 2, N2

A21

B12

B21

E 1, N1 Figure 2.6. Two energy levels E 1 and E 2 with respective occupation numbers N1 and N2 are connected by the transition probabilities B12 , A 21 and B21 .

(1842–1919) and J H Jeans (1887–1946)2 derived their radiation formula which correctly describes the frequency dependence of the energy density per frequency interval u(ν) = dW/dν for low frequencies ν: u(ν) =

8πν 2 kB T c3

(2.4)

where kB denotes Boltzmann’s constant and T is the temperature. This formula cannot be correct for high frequencies because of the ν 2 dependence which leads to a divergent integrated energy density. The correct description for all frequencies is Planck’s radiation formula: u(ν) =

c3

8πhν 3 . exp khν − 1 T B

(2.5)

In deriving this formula Einstein made several assumptions: light energy is emitted and absorbed as single packets of radiation energy called photons. Their energy is given as E = hν (2.6) where h denotes Planck’s constant. From figure 2.5 we already know the three different fundamental processes: absorption, spontaneous emission and stimulated emission. In thermal equilibrium, there must be equal numbers of transitions per time interval leading from the lower state of the atom to the upper one and vice versa. Each of the individual transitions is characterized by its transition probability per time interval 2 Rayleigh J W S 1900 Phil. Mag. 10 539. Jeans J H 1905 Phil. Mag. 10 91.


30

Radiation

as given in figure 2.6. Thermodynamic equilibrium then reads B12 u(ν)N1 = A21 N2 + B21 u(ν)N2

(2.7)

where N1 and N2 denote the number of atoms in the lower and upper state, respectively. The transition probabilities A and B are known as Einstein coefficients. The corresponding electron energies are denoted as E 1 and E 2 , respectively. According to Boltzmann’s law the number density of atoms in a certain energy state decreases exponentially with that energy according to exp(−E 2 /kB T ) N2 . = N1 exp(−E 1 /kB T )

(2.8)

Combining equations (2.7) and (2.8) yields u(ν) =

A21 B12 exp(hν/kB T ) − B21

(2.9)

if we use E 2 − E 1 = hν. At infinitely high temperature, T → ∞, the energy density has to diverge, u(ν) → ∞, requiring B12 = B21. From the low-frequency behavior (2.4) one gains A21 8πhν 3 = . (2.10) B12 c3 Together with (2.9) this gives Planck’s radiation law in the form of equation (2.5). The black-body spectrum can be given in terms of the radiative flux density or the spectral radiance Iν by the conversion Iν dν = u(ν) dν c

1 1 2 4π

(2.11)

noting that there are two degrees of freedom for a transverse wave having two planes of polarization. Inserting equation (2.5) into equation (2.11) yields −1 hν −1 Iν dν = hc−2 ν 3 exp dν. kB T

(2.12)

This may also be written as Iλ dλ =

hc02λ−5

hc0 exp λkB T

−1

−1 dλ

(2.13)

giving the wavelength dependence of the black-body radiation. Figure 2.7 shows these black-body curves for different temperatures, the thermal energies being kB T = 0.5, 5 and 50 eV, respectively. As the thermal energy increases the maximum of the distribution shifts to lower wavelengths.


The distribution of radiation

31

Photon energy (eV) 103

102

101

100

Radiance (W cm-2 nm -1)

101 1 109 kT=50 eV

107 105 kT=5 eV

103 101

kT=0.5 eV

10-1 10-3

100

101 102 Wavelength (nm)

103

104

Figure 2.7. Spectral distribution of the electromagnetic power emitted from black-body emitters at three different temperatures according to Planck’s radiation law.

This is indicated by the dotted line connecting the positions of the maxima at different temperatures. The shift of the maximum is expressed by Wien’s law T × λmax = constant

(2.14)

where λmax denotes the wavelength at which the intensity distribution becomes a maximum. Equation (2.14) can be derived directly from equation (2.13) by finding the zero of the first derivative with respect to the wavelength λ. The total amount of energy radiated also depends on temperature. Upon integration of Planck’s curve one finds for the power radiated per unit area into the half solid angle or the total energy flux that Wtot = σ T 4

(2.15)

known as the Stefan–Boltzmann law. The constant σ amounts to σ = π 2 kB4 /(60~3c2 ) = 5.670 × 10−8 Wm−2 K−4 . It is possible to build radiation sources which closely resemble a blackbody radiator by using heated filaments. Often they are used in the infrared as broad-bandwidth sources. Practical limits arise in the laboratory for constant temperatures above 2000 K or kB T = 0.17 eV. For this example the maximum of the emission occurs near 1500 nm. At short wavelengths (UV, soft x-rays) at least a 100 times higher temperature is required. Natural sources on Earth are not hot enough for their radiation curve to peak in the x-ray region. Astrophysical objects


32

Radiation

may well have temperatures so high that they can efficiently emit x-rays. Our Sun peaks in the green, however the tails of the radiation distribution still have a substantial flux to show radiative features in the far UV. The black-body curve can be used to measure the temperature of a radiation source. If the source is known to follow the black-body curve, it is sufficient to find the peak wavelength λmax or the integrated radiation power Wtot . Equation (2.14) or (2.15), respectively, then are used to find the temperature. Sometimes an effective temperature is assigned to a source even if it does not closely follow the Planck radiation distribution. In such a case one has to be very careful because the temperature is merely used as a qualitative parameter. Ionization of atoms in a plasma requires hot (fast) particles of some minimum temperature. Broad-band heating is not usually sufficient for highpower ion and light sources. To make more efficient use of overall power one has to use non-thermal distributions. Human inventions to produce substantial amounts of energetic radiation like x-ray generators are designed to radiate only in a narrow frequency band and therefore cannot be described by the laws of a black-body radiator.

2.4 Diffraction and interference 2.4.1 Diffraction Diffraction is the deviation of light from the laws of geometrical optics as the deflection of a wave passing an edge or narrow gap3. Usually these effects are small for visible light and one must look for them carefully in order to observe them. Also, most sources of light have an extended area so that a diffraction pattern produced by one point of the source will overlap that produced by another. Usual sources of light are not monochromatic and patterns for the various wavelengths overlap making the effect less apparent. The effects we are dealing with here are due to the wave nature of electromagnetic radiation. In figure 2.8 the example of a diffraction pattern observed behind a small hole is recalled. The concentric bright rings of light alternate with dark regions. Although the intensity rapidly falls off radially there is still an appreciable amount of intensity spread out into the geometrical shadow. This spreading can be explained using van Huygen’s principle: All points on a wavefront can be considered as point sources for the production of spherical secondary wavelets. After some time the new position of the wavefront will be the surface of tangency to these secondary wavelets. In the example, all points in the diaphragm are sources of new spherical waves spreading out into the space between the two screens. 3 See for instance: Born N and Wolf E 1980 Principles of Optics 6th edn (Oxford: Pergamon).


Diffraction and interference

33

inc ide nt ligh t le ho

ll wa

n ee scr

Figure 2.8. Diffraction of light at a small hole.

2.4.2 Interference Interference is the ability of a number of waves to cancel or reinforce each other. Mathematically this means adding wavevectors taking into account the phase of the wave. For two sine waves, such as those depicted in figure 2.1 with the same amplitude, there will be a complete cancellation if the phase difference of the two waves equals π. This is called destructive interference. If the phase difference is zero maximum enhancement or constructive interference is observed. The striped pattern of figure 2.8 is the result of constructive and destructive interference of the van Huygens wavelets emerging from the small hole. For the interference to take place it is necessary that the interfering waves overlap for a time that is long compared to the wave period. This condition, called coherence, is maintained, for instance, when light emitted from one point of a light source is joined again at a different place. As the waves have a finite length in space and time their path difference must be smaller than that length in order to interfere. The superposition of infinitely long waves with similar amplitude and frequency produces regions of destructive and constructive interference resulting in a beat phenomenon. Using many waves of slightly different frequency or wavenumber k it is possible to generate any form of beat. This is the idea of the Fourier transform breaking the light into its component parts of the spectrum. The wave may be expressed as (x, t) = (2π)−1/2

ϕ(k) exp[i(kx − ωt)] dk

(2.16)

where k = 2π/λ is the wavenumber and ω = 2πν is the angular frequency. The


34

Radiation 20

incident radiation

Intensity

10

defracted radiation k 0

θ r w

-10 -20

kw

Figure 2.9. Diffraction of parallel light at a slit. The intensity distribution is plotted as a function of the transverse wavenumber k multiplied by the slit width w.

function ϕ(k) is the Fourier transform of (x, t) and is given as ϕ(k) = (2π)−1/2

(x, t) exp[−i(kx − ωt)] dx.

(2.17)

An example application of the Fourier transform in optics is the diffraction of light when it passes through narrow slits. The ideas represented here can be applied to all forms of wave diffraction. 2.4.3 Diffraction at a single slit Let us suppose that parallel light is shone on a slit in an opaque screen as sketched in figure 2.9. The transmission function (x) for this arrangement is a simple band pass that lets light through at the slit but nowhere else. The frequency content of this is given by the Fourier transform ϕ(k) = (2π)−1/2

w/2 −w/2

exp[−ikx] dx

(2.18)

where w denotes the width of the gap. As illustrated in figure 2.9 the deflection of the radiation by an angle θ is related to a change of the incident wavevector k0 by an amount k in the transverse direction. The light intensity on the screen as plotted in the figure is given as I (k) ∝ |ϕ(k)|2 = ϕ(k)ϕ(k) 2sin2 kw 2 = k2π


(2.19) (2.20)

Diffraction and interference 15

incident radiation

35

Intensity

10 5

1

-5

2

1&2

0

diffracted radiation k

1

2

-10 -15 kw

Figure 2.10. Diffraction of parallel light at a double slit. The intensity patterns shown at the right are observed when (i) slit 1 is open, (ii) slit 2 is open and (iii) both slits are open.

where w denotes the width of the gap. The intensity distribution as a function of the deflection angle θ can be obtained if we set k = k0 sin θ =

2π sin θ. λ

(2.21)

The intensity distribution appears as a bright region centered around the slit which rapidly falls off towards either side of the slit. The diffraction minima occur at angles nλ sin θn = ± n = 1, 2, 3, . . . . (2.22) w From this we note that the width of the intensity distribution increases with the wavelength relative to the width of the slit. If white light is used the diffraction pattern becomes colored since for each wavelength the diffraction minima and maxima occur at slightly different positions. 2.4.4 Young’s double-slit experiment Interference effects can be well documented in the laboratory observing the transmitted light through a double slit. The historical experiment performed by Thomas Young (1773–1829) in 1801 provided crucial evidence in support of the wave theory of light. We summarize some of the more important details of this experiment and our present-day interpretation of it. When the screen of figure 2.9 is altered to have two parallel slits, with widths w and distance d apart, we obtain the situation illustrated in figure 2.10. If only one of the two slits is open we have the case of a single slit discussed before with the two individual intensity patterns ‘1’ or ‘2’ shown in the figure. A dramatic change is observed if both slits are open. The resulting intensity pattern ‘1 & 2’ is not the sum of the two single-slit curves. Instead a very pronounced


36

Radiation

diffraction pattern is observed representing closely spaced minima and maxima which gradually decrease the further from the center of the two slits you go. The center maximum occurs at a position between the two slits where it was dark in the case of only one open slit. Again, the intensity pattern can be reproduced by calculating the Fourier transform, this time of the double-slit band pass. If the waves are in phase before entering the slits the intensity on the screen can be expressed as I (k) ∝ |ϕ1 (k) + ϕ2 (k)|2 ∝ |ϕ1 (k)|2 + |ϕ2 (k)|2 + ϕ1 (k)ϕ2 (k) + ϕ1 (k) ϕ2 (k)

(2.23)

where the functions ϕ1 (k) and ϕ2 (k) are given by equation (2.18) with the integration path over slits 1 and 2, respectively. 2.4.5 The Heisenberg uncertainty principle The diffraction experiments may also be discussed considering the particle aspect of radiation. Then we talk about single wavepackets or photons having initial momentum p = ~ k impinging on a slit. In Young’s double-slit experiment, let us now suppose that the rate of photons is very low and there is only one photon at a time traversing the experimental setup. Although there is no possibility for two photons to interact with each other, the same oscillatory intensity pattern still appears on the screen. What has happened? Can the photon simultaneously go through both slits and interfere with itself? The finding is interpreted in a probabilistic fashion. As long as we do not know through which of the two slits the photon has passed, the probability of finding it at a certain spot on the screen is given by the oscillatory probability distribution shown in figure 2.10. If one tries to identify the path of the photon the diffraction pattern is lost4 . This has been impressively demonstrated in the following way5 . If the light source is emitting unpolarized radiation the photons going through the left or right slit, respectively, can be marked by inserting a left-circular polarizer before one of the two slits and a right-circular polarizer before the other. Because now the two pathways become (in principle) discernible, the oscillatory pattern on the screen disappears and the sum of the single-slit profiles is observed. The oscillatory pattern shows up again if a third polarizing filter is placed between the two slits and the second screen, to scramble up the information about which photon went through which hole. Now, once again, it is impossible to tell which path any particular photon arriving at the second screen took through the experiment. We also may wish to reinterpret the experiment with a single slit. As we send a photon through a slit with a finite width w its position coordinate is measured with an uncertainty x = w. After the passage through the narrow slit we are not certain where it goes exactly. Its deflection angle has a probability distribution as 4 Storey P, Tan S, Collett M and Walls D 1994 Nature 367 626, and references cited therein. 5 Scully M O, Englert B-G and Walther H 1991 Nature 351 111.



37

depicted in figure 2.9. The width of the distribution is approximately given by the angle of the first diffraction minimum of equation (2.22) with n = 1. Therefore the uncertainty in the wavenumber is k = k0 λ/x = 2π/x.

(2.24)

The actual width is bigger because of the presence of the higher diffraction orders. Therefore we may write k · x ≥ 2π

p · x ≥ h

or

(2.25)

where h is Planck’s constant. In this simplified treatment the momentum uncertainty p and the uncertainty on the position x are taken as the maximum uncertainties. If one chooses to define the s as the most probable uncertainties referring to normal distributions, we obtain a still smaller limit p · x ≥

~ 2

.

(2.26)

This is referred to as the Heisenberg uncertainty principle after Werner Heisenberg (1901–1976) who first introduced these ideas in 1927. This had a great impact on the development of the physics of particles and atoms. The consequence of equation (2.26) is that we cannot gain complete knowledge of a system. Once the position is determined within some limit x the momentum has an uncertainty p according to equation (2.26). We cannot make both uncertainties small simultaneously. Because h is so small these uncertainties are not observable in the ordinary world of macroscopic experience. Phenomena at the smallest scale, however, are highly coupled to the observing system. The act of observation changes the system irrevocably. This is fundamental to the modern physics of the microcosmos. These uncertainties are inherent and have nothing to do with the skill of the observer. Another form of the uncertainty principle is obtained by noting that a plane wave is represented by the function ∼ exp(−ikx − ωt) where x and t appear in a symmetric way. The wavepackets can be represented in the space coordinate x and in time t. In place of equation (2.26) one may, therefore, write down tω ≥

1 2

or

tE ≥

~ 2

.

(2.27)

This means that we have to measure for a sufficiently long time to determine an energy or frequency with some accuracy. The frequency given at a sharp instance is meaningless, we have to wait at least one period T to define a frequency. 2.4.6 Fresnel lenses and zone plates The Fresnel lens was invented in 1822 by Augustine Fresnel. His lenses were first used on the French coast as a lightweight and less-expensive alternative to the old,


38

Radiation

Figure 2.11. A Fresnel zone plate such as used for microfocusing of soft x-rays. Note the decreasing width of the concentric rings with diameter.

bulky lighthouse lenses. Later they became widely used not only in lighthouses, but also in overhead projectors, big screen TVs and stage lighting for instance. The lens works by diffraction rather than by refraction as in a conventional lens. A Fresnel zone plate consists of concentric rings that are finely spaced at the outside and coarsely spaced at the center. Thus the diffraction angle increases with the radius so as to produce a point focus. Figure 2.11 shows the layout of a zone plate used for microfocusing. The lens works by constructive interference at the focus of light rays passing neighboring zones. Those contributions that would result in destructive interference are blocked by the dark rings. Calculating the optical path differences of the light rays one finds, for the radius of a zone, rm2 = 2mλ(1/a + 1/b)−1

(2.28)

where a and b denote the distance of the source and the focus from the zone plate and λ is the wavelength of the radiation, and m is an integral number of wavelengths. Equation (2.28) has a similar form as the standard lens equation. The width of the individual zones becomes smaller with increasing radius conserving the area of the zones. It has to be noted that the focal distance strongly depends on the wavelength producing chromatic images with white light. In the far ultraviolet and x-ray region, one has to use diffraction rather than refraction for focusing. In x-ray astronomy the diffraction properties of zone plates are used in the water window spectral range, i.e. for wavelengths between 2.4 and 4.5 nm and at harder x-rays around 0.3 nm. Fresnel zone plates are employed as imaging elements in transmission x-ray microscopes and in scanning-type x-ray microscopes6. A scheme with a CCD (charged coupled 6 Aristov V V and Erko A I 1994 X-Ray Microscopy vol IV (Chernogolovka).

Michette A G, Morrison G R and Buckley C J 1992 X-Ray Microscopy vol III (Berlin: Springer).



39

t sof s y a r x sam

ple

l sne Fre plate e n zo

D CC en e r c s

Figure 2.12. Scheme of a transmission x-ray microscope.

device) as a photon detector is sketched in figure 2.12. To increase the resolving power of zone plates, the outermost zone width has to be reduced. Optimizing zone plates for the use at shorter wavelengths also leads to an increase of the aspect ratio. The resolution is determined by how finely one can make the outermost rings, and how accurately one can place them over the zone-plate diameter. Zone plates with outermost zone widths as small as 20 nm can be built employing electron lithography. With these optics, it is then possible to generate focal spot sizes as small as 0.4–0.5 nm in the range of 50–100 keV radiation energy, which is the smallest focused spot of electromagnetic waves of any wavelength. In x-ray microscopy the efficiency of Fresnel lenses is increased by partly satisfying the Bragg condition (see later) for the incident and the diffracted wave which is used for imaging. A considerable increase can be obtained by tilting the zones and increasing the thickness of the zone plates. 2.4.7 Bragg reflection, diffraction grating As we have seen, diffraction effects become important when the wavelength of the light is of the same order of magnitude as the diffracting objects such as a narrow gap for visible light. Such structures can also be represented by the regular lattice of a crystal. For these the wavelengths of x-rays are comparable to the spacing d of lattice planes. As illustrated in figure 2.13, x-rays of a certain wavelength λ may constructively interfere when partially reflected between surfaces that produce a path difference equal to an integral number m of wavelengths. This condition is defined by Bragg’s law, mλ = 2d sin θ.

(2.29)

where the angle θ is the grazing or Bragg angle. Equation (2.29) forms the basis for x-ray spectrometry developed by William Henry Bragg (1862–1942) and William Lawrence Bragg (1890–1971). The underlying physical process of the


Radiation

40

d

Θ

Figure 2.13. Bragg reflection of x-rays on the regular structure of a crystal lattice.

Bragg diffraction is the scattering of the incident radiation on the individual atoms of the crystal. Coherent radiation, however, does not originate from each atom independently. Instead, the waves radiated from each atom are correlated through their common origin from one x-ray source and the phase-sensitive scattering process. This way scattered radiation is produced that is weak or zero in a random direction but very strong in a particular direction. Whereas the scheme in figure 2.13 works fine in getting a grasp of what is happening, more complicated structures require Fourier analysis. The latter was introduced in its basic form when light diffraction at a slit or double slit was discussed. Mathematically, the diffraction pattern is always given as the Fourier transform of the scattering power of the sample. A perfectly ordered crystal, analyzed in three dimensions, will produce a diffraction pattern of isolated spots of constructive interference. The intensity of each diffracted beam and its relative phase are determined by the physical properties of the scattering centers in each unit cell of the crystal plus the nature of the three-dimensional packing of the unit cells within the crystal. Applications of these ideas include the structural analysis of single crystals as introduced by Max von Laue (1879–1960), Walther Friedrich (1883–1968) and Paul Knipping (1883–1935). Diffraction for crystalline powders was first used by Peter Debye (1884–1966) and Paul Herrmann Scherrer (1890–1969). Diffraction at regular structures has an important application in spectrographs7. In the x-ray region single crystals are used to disperse a wavelength spectrum. In the UV and visible range, ruled gratings both in transmission and in reflection are used. 2.4.8 Diffraction limited devices and the camera obscura Diffraction effects may limit the performance of optical instruments. This is usually the case if other aberrations of an imaging device are small. Despite its simplicity the camera obscura is the ideal instrument to observe these effects8. 7 Batterman B W and Cole H 1964 Dynamical diffraction of x-rays by perfect crystals Rev. Mod. Phys. 36 681–716. 8 Young M 1972 Pinhole imagery Am. J. Phys 40 715.



41

Figure 2.14. The camera obscura used by Frisius to investigate the solar eclipse of the year 1544. Reproduced with permission of Gernsheim Collection, Harry Ransom Humanities Research Center, The University of Texas at Austin.

The working principle of the camera obscura was known to astronomers and writers in ancient times. It was reinvented several times in the Middle Ages and it was in use in the 15th and 16th centuries. One of the oldest pictures, reproduced in figure 2.14, is the drawing by the astronomer Gemma Frisius in his book De Radio Astronomica et Geometrica from 1545. He used a hole in his darkened room to study the solar eclipse of the year 1544. The term camera obscura, dark room, is credited to Johannes Keppler (1571–1630). After photographic film became available, the lensless camera was termed a pinhole camera by Sir David Brewster (1781–1868) in the 1850s. The pinhole camera is a lighttight box with a small hole on one side and photographic film at the other. The image of a distant point in the camera obscura is simply made by the shadow of the wall with the hole in it and may be constructed by simple geometrical optics. Obvious advantages of the camera obscura are freedom from linear aberrations, a very large depth of field and a wide angular range. However, the light collection efficiency is poor which is of minor importance if the light source is bright. For a large hole diameter, the image of a distant point is a disc with the same size as the hole. For a small hole of diameter D, we have to consider, in addition, the diffraction pattern generated with a circular hole such as that illustrated in figure 2.8. The diameter of a diffraction disc is s = 1.22

λr D

(2.30)

where λ denotes the wavelength and r is the distance between the entrance hole and screen. The numerical factor of 1.22 applies for a circular aperture, for a slit it would be 1 in accordance with (2.22). Equation (2.30) relates to the ability to distinguish two closely spaced small points in an image usually represented by the resolving power. The latter is


42

Radiation

Figure 2.15. Auroral x-ray image taken with an x-ray pinhole camera during a stratospheric balloon flight in northern Sweden at local midnight. Figure reproduced with permission, courtesy T Freeman. (See colour section.)

defined as the reciprocal of the smallest angular difference of two points that are still distinguishable. A criterion was chosen by Lord Rayleigh to define the limit of resolution of a diffraction-limited optical instrument. It is the condition that arises when the center of one diffraction pattern is superimposed with the first minimum of another diffraction pattern, produced by a point source equally bright as the first. In this limit a dip of 27% appears between the two maxima giving rise to the impression of two features. If we consider both the geometrical-optics limit and the diffraction limit, a minimum in the image size of a distant point source is obtained when s = D or 2 Dopt = 1.22λr.

(2.31)

For a given dimension r of the camera obscura, Dopt represents the optimum hole diameter giving the highest resolving power. Reversely, for a given hole diameter equation (2.31) yields the optimum distance to observe an image with great detail. In these considerations it was tacitly assumed that we observe the far-field diffraction patterns discussed earlier. The general note that the aperture has to be of a comparable size with the wavelength in order to observe diffraction phenomena is not quite complete. If the wavelength is much smaller than the hole diameter we have to go sufficiently far from the aperture according to equation (2.31). Considerations similar to those for the pinhole camera hold true for a number



43

of optical instruments like the microscope, the telescope, our eyes, or the prism spectrograph. Modern examples of pinhole cameras are often used in the UV and x-ray region where lenses cannot be employed. The observation screen is replaced by a suitable imaging detector such as a charged coupled device (CCD). X-ray sources investigated this way include plasma devices containing a high concentration of highly charged ions and astrophysical x-ray sources together with the auroral x-ray imaging. The image shown in figure 2.15 was taken by an x-ray pinhole camera, flown on a stratospheric balloon from northern Sweden9. The area of the sky shown is approximately 100 km across, and it was taken during a period of an intense visual aurora, near local midnight. The bright area in the southwest corner of the image measures ∼20 km across, and moved in a matter of minutes across the field of view of the camera. 2.4.9 Massive particles as waves We have seen that light exhibits the properties of both particles and waves. In 1923 Louis de Broglie (1892–1987) suggested that matter in general had to be described as a wave. He started with Einstein’s theory of relativity and combined it with Planck’s ideas of a quantized energy. In special relativity the relation between energy E and momentum p is expressed as E 2 = m 2 c4 + p2 c2 .

(2.32)

If this is applied to particles of light which have zero rest mass, one finds p = E/c, which relates the momentum and energy of light. Considering light of frequency ν to be composed of photons having energy E = hν, the momentum of a photon is given as hν h p= = (2.33) c λ relating the momentum and wavelength of the photon. De Broglie extrapolated these relations in postulating that ‘Wave and particle characteristics always appear together, so that all particles animated by a momentum p are characterized by an associated wave such that λ = h/ p’. This finding and its later experimental confirmation was the basis for the development of quantum mechanics formulated by Erwin Schrödinger which require that all physical objects are really matter waves described by their wavefunction. Because of the small numerical value of Planck’s constant entering equation (2.33), the wavelengths of macroscopic objects are much below the size of atoms or even nuclei and therefore wave behavior is not observed in every-day life. For the electron, the wave nature was experimentally confirmed in 1927 by C J Davisson, C H Kunsman and L H Germer in the USA and by G P Thomson in Scotland who found diffraction and interference phenomena similar to those 9 Substorms observed by balloon-borne Instruments, Geostationary Particle Detectors, and the VIKING UV Imager Auroral x-ray imaging—Sweden 1986.


Radiation

44

Table 2.2. Various particles and their typical kinetic energies together with their corresponding De Broglie wavelengths. Particle

Kinetic energy

De Broglie wavelength

slow electrons in cathode ray tubes cold neutrons thermal neutrons Bose-Einstein condensate, Na ultrasonic (350 m s−1 ) I2 accelerated protons accelerated uranium ions

100 eV 10 keV 0.0002 eV 0.02 eV 4 × 10−14 eV 6 eV 100 keV 200 GeV

0.12 nm 0.012 nm 2.0 nm 0.2 nm 30 µm 0.004 nm 0.09 pm 0.005 fm

observed with light. Later quantitative results with slow neutrons provided strong support for matter waves10. In table 2.2 we compile the De Broglie wavelengths of various particles. Most of the time, they are comparable to or smaller than the wavelengths of x-rays. For a long time, matter waves were the domain of subatomic particles whereas beams of heavy particles were associated with a high momentum and consequently with a very small De Broglie wavelength. With the successful slowing down and cooling of atomic particles this has changed dramatically. Diffraction and interferometry has been demonstrated and applied with ultra-cold heavy atoms and even for molecules11. The Heisenberg uncertainty principle has also been extended to include matter waves. If one limits the amount of space the matter wave can occupy, then the spread of velocities for the matter wave is increased. Since the product of the spread in location and the spread in motion has a lower limit, one can know both to some accuracy or know one very well and the other not very well at all or vice versa but one cannot know everything perfectly. 2.4.10 The scanning electron microscope One obvious feature of the scanning electron microscope is the three-dimensional appearance of the specimen image. This is a direct result of the large depth of field of the scanning electron microscope. This field is due to the method in which the image is generated with a fine electron beam scanned over the surface and with the detected secondary electrons providing the signal. 10 Zeilinger A, Gähler R, Shull C G, Treimer W and Mampe W 1988 Rev. Mod. Phys. 60 1067. 11 Andrews M R et al 1997 Observation of interference between two bose condensates Science 275

637. Boch I, Hänsch Th and Esslinger T 1999 Atom laser with a cw output coupler Phys. Rev. Lett. 82 3009. Bordé Ch et al 1994 Molecular interferometry experiments Phys. Lett. A 188 187.


The Doppler effect

45

The other feature of the scanning electron microscopy is its spatial resolving power which is related to the smallest detail that a microscope can resolve. The resolving power of electron microscopes is orders of magnitude better than that of an optical microscope because the wavelength of the probing beam is orders of magnitude smaller. The principal limit on what size can be resolved is set by the wavelength. The resolving powers of high-quality light microscopes are limited by the wavelength of the imaging light to about 200 nm. Scanning electron microscopy uses electrons with energies near 5 keV. The De Broglie wavelength of such an electron beam is approximately 0.02 nm. In practical applications the resolution limit of ordinary electron microscopes is 1 nm because construction details determine resolving power. The column of a scanning electron microscope contains an electron gun and electromagnetic lenses operated in such a way as to produce a very fine electron beam, which is focused on the surface of the specimen. The beam is scanned over the surface in a series of lines and frames called a raster, just like the electron beam in an ordinary television set. The raster movement is accomplished by means of small coils of wire carrying the controlling current. For each raster element, the specimen is bombarded with electrons over a very small area. Several different secondary processes may be caused by the impinging electrons. They may be elastically scattered off the specimen, with no loss of energy. In inelastic collisions they may ionize part of the target atoms giving rise to secondary electrons together with the emission of x-rays. In photoabsorption events the absorbed electrons may give rise to the emission of visible and UV light. All these effects can be used to derive a scanning signal thereby providing an image. The most commonly used method, however, is image formation by means of the low-energy secondary electrons.

2.5 The Doppler effect The Austrian physicist Christian Doppler (1803–1853) enunciated that sound waves from a moving source would be compressed or expanded changing the apparent frequency of the sound. In 1842, Doppler introduced the basic equation relating frequency and the relative movement of sound source and observer. To prove his theory, Doppler conducted a unique experiment. He arranged for a train to pull repeatedly, at different speeds, a freight car with trumpeters playing on top of it. One musician with good hearing recorded the height of the tones as they appeared when the train moved closer or further away. The results of that experiment confirmed Doppler’s theory stupendously well. A modern visualization of the Doppler effect is shown in figure 2.16 for an aircraft hitting the sound barrier, the strongly compressed sound waves are in the forward direction. Doppler later tried to prove that his theory also applied to light but was unable to fulfill his goal. Instead, another scientist, A H Fizeau, generalized


46

Radiation

Figure 2.16. Breaking the sound barrier can be a very spectacular effect. There are many analogies between sound and light waves discussed in the text. The photograph is courtesy US Navy, photograph by Ensign John Gay. (See colour section.)

Doppler’s work and discovered that the Doppler effect also applies to light. The optical Doppler effect is the apparent shift in color, observed if the light-emitting object and the observer have a relative velocity. This is different from sound waves, where it matters whether the observer or the source moves. For sound waves, it is always necessary to have a medium that lets us unambiguously define the two different velocities of source and observer plus the sound velocity. For electromagnetic radiation there is no way of telling whether the source or the observer moves as there is no distinguished frame of reference. Qualitatively, the observed change in wavelength is explained in figure 2.17. Successive wave crests in the direction of an approaching source will be observed closer together because the source moves between the times of emission of one wave crest and the next. An observer in front of the moving source will measure a smaller wavelength than normal or blueshifted radiation. In the backward direction the waves appear expanded increasing the observed wavelength which is said to be redshifted. Looking sideways at the source yields approximately the same wavelength as for a source at rest. Now let us look more quantitatively at the Doppler effect and derive the acoustic Doppler formula. For this purpose let both the source and observer move in the same direction as sketched in figure 2.18. Both speeds v1 and v2 are assumed to be less than the sound speed c. After a time T , the source has traveled a distance v1 T . In the same time interval a wavefront has traveled a distance cT . If T is interpreted as the period of a sinusoidal sound wave the


The Doppler effect

47

Light source

at rest

λ

moving

Figure 2.17. Light waves emitted from a point source that is at rest (left) or that moves (right) with a constant velocity from left to right.

v1

v2

t=0

t=T v1T

λ cT

Figure 2.18. The Doppler effect with sound waves. After a time T the source has moved a distance v1 T and a wavefront a distance cT . The detector moving with a velocity v2 will measure the time interval between two wavefronts according to the relative velocity c − v2 .

effective wavelength will be λ = (c − v1 )T.


48

Radiation

The corresponding period that the moving observer ‘hears’ is T = λ /(c − v2 ). The measured frequency then reads: c − v2 ν c − v1 1 − v2 /c ν. = 1 − v1 /c

ν = T

−1

=

(2.34)

Equation (2.34) contains the two limiting cases, v1 = 0 or v2 = 0, obtained when either the source or the detector is at rest. In the two cases two different velocity dependencies apply. Because it is the speed relative to the expansion speed of the wave that matters, the Doppler effect is easier to observe for sound waves than for light. To derive the Doppler formula for electromagnetic radiation we have to use special relativity for the necessary transformations12. As for the sound waves, let us consider a linear movement between light source and observer. We assume a light source emitting a periodic signal with a time period T as indicated by the parallel lines in the space–time diagram of figure 2.19. A wavefront of the radiation travels with the speed of light c, whereas the observer moves relative to the source with a velocity v indicated by his/her world line. The time and space differences between the reception of two successive wave crests can be noted from the figure as t =

cT c−v

x =

and

vcT . c−v

(2.35)

For the relativistic transformation of the time interval into the observer’s frame of reference we have to use the Lorentz transformation as will be summarized later. The period in the observer’s reference system reads:

2

t = T = γ (t − v/c x) = T

1+β 1−β

1/2 (2.36)

with the usual abbreviations β = v/c

γ = (1 − β 2 )−1/2 .

and

(2.37)

With ν = T −1 equation (2.36) yields the frequency transformation ν = ν

1−β 1+β

1/2 .

(2.38)

For an observer approaching the source, instead of receding from it, the sign of β has to be reversed. This result can be extended to include cases where the light emitted from fast sources is observed at an angle different from 0 or π covered


The Doppler effect

49

t

vt x=

t=T

x=

ct

t

t=0

x

x

Figure 2.19. Space–time diagram for a light source and for a moving observer. The time (t) and path (x) differences between the observation of two successive wave crests are indicated.

by equation (2.38). Here, we summarize the relativistic transformations including those for the angles θ and : Lorentz–Einstein transformations x = γ (x − vt) y = y z = z t = γ (t − vx/c2 ) cos θ + β cos θ = 1 + β cos θ 1 ∂ = 2 ∂ γ (1 − β cos θ )2 ν ν = γ (1 − β cos θ )

(2.39)

The relativistic transformations have been confirmed experimentally for electromagnetic radiation ranging from radio waves up to high-energy gammarays. One of the most important implications of the Doppler effect is its influence 12 See for instance: French A P 1968 Special Relativity (Cambridge, MA: MIT).


50

Radiation

Figure 2.20. A Hubble diagram of distance versus velocity. The velocity of distant galaxies is increasing with distance. A slope of H0 = 72 km s−1 Mpc−1 is shown together with lines deviating by ±10%. From Freedman W L et al 2001 Astrophys. J. 553 47–72. c 2001 University of Chicago Press.

on astronomical studies. Astronomers use Doppler shifts to calculate how fast stars and other astronomical objects move toward or away from Earth. In 1914, Vesto Slipher discovered that the spectral lines of several nebulae were shifted to longer wavelengths. Edwin Hubble (1889–1953) was the first who interpreted this as the consequence of an expanding universe13 . The exciting fact is that the cosmological redshift does not result from motion through space but rather from the expansion of space. Hubble found that the observed redshift and thus the deduced receding velocity is approximately proportional to distance. This relation is plotted in figure 2.20 containing data taken from the Hubble Space Telescope14 . The factor of proportionality, the Hubble parameter H0, has been frequently re-evaluated. One difficulty is the accurate calibration of the distance scale using nearby galaxies of known distance. Further complications can arise from the interpretation of spectral data. Superimposed on the main radiation from a receding galaxy can be the light of other astronomical objects of different velocity 13 Freedman W L 1998 Sci. Am. (March). 14 Freedman W L et al 2001 Final results from the Hubble Space Telescope key project to measure

the Hubble constant Astrophys. J. 553 47–72.


The Doppler effect

51

3.0 2.5 relativistic

λ′/ λ

2.0 1.5

classical

1.0 0.5 0.0 -1

-0.5

0

0.5

1

β Figure 2.21. The Doppler shifted wavelength for colinear observation calculated with the correct relativistic and with the classical formula.

such as rotating stars. One consequence of an expanding universe is that it must once have been smaller. In its limit, this points towards a time when the universe was a singular point of zero size. The initial explosion of this singularity in a fireball of hot and dense energetic radiation is referred to as the Big Bang. Assuming a constant expansion, the age of the universe can be inferred from the value of the Hubble parameter. With a corresponding uncertainty due to the scatter of the data, seen in figure 2.20, Hubble’s constant has been located near 72 km s−1 Mpc−1 implying a value for the age of the universe near 14 billion years. A different kind of frequency shift results not from relative motion but rather it is associated with very strong gravitational fields and is therefore known as the gravitational redshift. Astrophysically relevant velocities range from the revolution velocity of the Earth around the Sun of about 30 km s−1 (β = 10−4 ) to the receding velocity of the farthest galaxies having a velocity of about 104 km s−1 (β = 0.03). With the invention of particle accelerators, the emission characteristics of fast sources, as given by the theory of special relativity, can be put under a rigorous test in the laboratory for even higher velocities. Figure 2.21 shows the calculated relative change in the wavelength λ /λ = ν/ν as given by the Doppler formula (2.38) for observation in line with the direction of movement of a fast light source. For comparison, the curve that is obtained when the time dilatation is neglected revealing large differences as the velocity approaches the speed of light or |β| → 1 is also plotted. Already in 1907 Einstein had suggested measuring the transverse Doppler shift, i.e. the wavelength shift observed under 90◦, as a test of special relativity. The angular dependence of the wavelength as obtained from equation (2.39) is


Radiation

52

90

wavelength (nm)

600

120

60

500 400 30

300

θ

200 100

β = 0.25

0

210

source: λ = 486 nm

0

330

240

300 270

Figure 2.22. Angular dependence of the optical Doppler effect: A monochromatic source emitting a bright cyan light travels with 25% of the speed of light from left to right. As the observer circles around the source he/she will observe the color Doppler shifted. The laboratory wavelengths are given as a polar diagram. (See colour section.)

given by

λ = γ (1 − β cos θ )λ.

(2.40)

This is visualized in figure 2.22 as a polar diagram for a source emitting light of cyan color corresponding to a wavelength of 486 nm. The velocity of the source is set to β = 0.25. As the observer circles around the source he/she will see the colors changing from blue to violet in the forward and from green to red in the backward direction. At 90◦ relative to the source movement, there remains a shift given by λ = γ λ which would give a direct measure of the Lorentz factor γ . A systematic difficulty arises from the strong variation in the wavelength near θ = π/2 which requires an accurate determination of the observation angle. This difficulty is circumvented by observing the light source with two detectors placed at opposite directions θ and π − θ , respectively, as sketched in figure 2.23. From equation (2.40) one obtains for the mean of the two wavelengths in the laboratory system λ (θ ) + λ (π − θ ) = γ λ. (2.41) 2 This property was used in a fundamental test conducted by H E Ives and


The Doppler effect

λ (θ) + λ (π–θ) =λ 2γ

λ

53

( θ)

λ θ

λ

–θ) π ( π–θ

Figure 2.23. The Doppler color-mixing rules: for opposite directions of emission the arithmetic mean of the observed wavelengths, divided by the Lorentz factor γ , equals the emitter wavelength.

G R Stilwell in 193815 using an atomic-beam apparatus to observe the Dopplershifted Hβ line of atomic hydrogen at 0◦ and at 180◦, respectively. As in this experiment the velocity of the source was varied, the authors could verify the approximately quadratic velocity dependence of γ for small values of β: γ = (1 − β 2 )−1/2 ≈ 1 + 12 β 2 . In a modern variant of this experiment colinear laser spectroscopy in an ion storage ring was used employing accurately known optical transition in 7 Li+ ions16 . Such precision tests would check deviations of time dilatation from that given by the theory of special relativity in a form which may be written as γ = γ (1 + δα β 2 + · · ·)

(2.42)

where γ denotes a new hypothetical time dilatation factor. The experiment may also be related to speculations on an anisotropy of space and the question of a suitable reference system. In the sketch in figure 2.24 the Earth is assumed to move relative to a hypothetical universal frame anchored somewhere in the interstellar space. Relative to the Earth and laboratory, the ions move at a speed corresponding to β = 0.064. In the experiments cited here, an upper limit of the test parameter of equation (2.42) could be ascertained to δα ≤ 8 × 10−7 . The relativistic transformations also affect the angular distribution of the light intensity in the laboratory. For a light source, that, in its own frame of reference, emits its radiation isotropically, the solid angle transformation ∂ /∂ from (2.39) gives the laboratory angular distribution. This is illustrated in figure 2.25 for three different emitter velocities β = 0.2, 0.4 and 0.6, respectively. When the source velocity approaches the velocity of light the intensity is strongly 15 Ives H E and Stilwell G R 1938 J. Opt. Soc. Am. 28 215–26. 16 Grieser R et al 1996 Hyperfine Interact. 99 145.


54

Radiation

S(t,x,y,z) v S(t,x,y,z) v S(T,X,Y,Z)

Figure 2.24. Speculative universal interstellar reference system. 90 120

∂Ω / ∂Ω

4.00

60

3.00 30 2.00

β = 0.6

1.00

β = 0.2

β = 0.4 0

0.00

210

330

240

300 270

Figure 2.25. Polar plot of the relativistic solid-angle transformation for three different velocities.

boosted in the forward direction. For β ≈ 1, ∂ /∂ of equation (2.39) can be expressed as 4γ 2 β→1 (2.43) ∂ /∂ ≈ (1 + γ 2 θ 2 )2 describing a forward cone of intensity as depicted in figure 2.26. The cone containing 50% of the intensity has a width of θ ≈ 1.3/γ and the maximum blueshift corresponds to the forward laboratory wavelength of


The Doppler effect

1.3 / γ

55

β ≈1

Figure 2.26. When the speed of the light source approaches the speed of light, the intensity is boosted into a narrow forward cone.

H+

γ

x D

t

Figure 2.27. The speed of gamma-rays can be determined by a time-of-flight method: cγ = x/t. A movable gamma-ray detector D registers the time difference between a bunch of protons, H+ , and the arrival of a gamma-ray.

λ (θ = 0) = λ/(2γ ). Such an extreme situation is encountered in experiments at high-energy accelerators such as the proton synchrotron (PS) or the super proton synchrotron (SPS) at the European high-energy laboratory CERN. The forward boost of the intensity can be turned to advantage by measuring the speed of light for highenergy gamma-rays emitted in flight from very fast projectiles. The scheme for such an experiment is shown in figure 2.27. A high-energy particle produces a gamma-ray flash in a fast nuclear reaction with target material. The gamma-rays are predominantly emitted in the forward direction. Because the primary ions occur as regular bursts the gamma-ray flashes with the same time structure. This gives the possibility of measuring the speed of the gamma-rays with a time-offlight method using a movable detector. Such an experiment was conducted17 at CERN making use of the radiative decay of a pion: π 0 → γ1 + γ2 which decays on a time scale of a few 10−16 s. The pion itself had an energy in excess of 6 GeV or γ > 45. The experiment excludes an increase in the velocity of light by more than 10−4 of the source velocity. Thus the experimental findings support the relativistic addition of two velocities β= 17 Alväger et al 1966 Arkiv Fysik 31 145.


β 1 + β2 1 + β1 β 2

(2.44)

56

Radiation

for a case where one velocity is the light velocity β1 = 1 and the other velocity is β2 > 0.999 75. Given the fact that the Doppler shift is accurately tested in many experiments, it represents a precision tool for the diagnostics of moving light sources. Besides the radar gun used by the police, there is a wide variety of (more useful) applications including also ionized plasmas. The Doppler effect also plays a key role in the laser cooling of ions in storage rings and in traps. For the spectroscopy of fast heavy ions, the Doppler formulae (2.39) form the basis for deriving wavelengths in the emitter frame of reference.


Chapter 3 Spectroscopy

The pattern of observed lines of absorption or emission is an intrinsic property of an atom and its state of ionization. The lines may appear as an emission spectrum or as absorption lines depending on whether an atom or ion emits light radiation or absorbs it. Spectroscopic investigation of the emission or absorption atomic spectra allows one to gain detailed information about the light source and its properties.

3.1 Spectral lines The investigation of the manner in which matter can emit and absorb radiation is known as spectroscopy. This entails the analysis of spectra—the splitting of radiation into its components using spectrometers as tools. For a continuous source, eventually characterized as a black-body emitter, the visible portion reveals the familiar rainbow of colors. In addition, there are light sources emitting narrow well-defined emission lines which are characteristic for the material. By changing the condition of excitation, the intensity of a particular spectral line changes but not its wavelength or frequency. Regarding a whole spectrum of lines as an entity, the pattern of relative intensity may depend on the mode of excitation; however the position of the lines on the wavelength scale remains unchanged. In chapter 2 we learnt about the example of the yellow sodium D lines representing the fingerprint of the element sodium. In general, the pattern of lines observed is an intrinsic property of the element and its state of ionization. The lines may appear as an emission spectrum or as absorption lines, i.e. dark lines in a continuous spectrum as, for instance, that observed from the Sun, explained by the presence of gases in the outer layer of the Sun and in the Earth’s atmosphere. The empiric Kirchhoff’s laws describe the conditions for the formation of the three different spectra which are continuous, emissionline and absorption-line spectra. Systematic investigation of the spectra enables investigators to perform an elemental analysis spectroscopically which is the only practical method for astrophysical light sources. In figure 3.1 the visible solar


Spectroscopy

58

Figure 3.1. Visible portion of the solar spectrum crowded by Frauenhofer lines. (See colour section.)

10000

Intensity

8000 6000 4000 2000 0 654

655

656 657 Wavelength (nm)

658

659

Figure 3.2. Absorption spectrum of the Sun near the Hα line of hydrogen. From Solar Atlas, Debouille et al (1972, 1981).

spectrum is shown revealing many of the Frauenhofer lines. They are marked with the atom or molecule from which they originate. A very prominent line occurring in the red is the hydrogen Hα line. A narrow wavelength region around the Hα line is shown in figure 3.2 in higher resolution. As a matter of fact, the element helium was discovered in the Sun 30 years before it was discovered on Earth (1895). Most of the spectral lines observed in the solar spectrum, however, are due to iron contributing more than 100 of the Frauenhofer absorption lines. Neutral iron with its 26 electrons already produces a rich line spectrum. In addition the iron atoms appear in numerous stages of ionization. The electronic structure of an ionized iron atom is completely different from that of neutral iron. There are 25 different ionization stages from Fe1+ to Fe25+ each producing a new rich line spectrum. Altogether the ions produce an enormous number of spectral lines. An image of the Sun can be focused on the slit of a spectrograph thus


Spectral lines

59

Figure 3.3. Diagram of the Solar Extreme-ultraviolet Rocket Telescope and Spectrograph (SERTS), reproduced with permission.

Figure 3.4. Soft x-ray images of the Sun taken from the Yohkoh satellite at two different times as indicated showing the change in solar activity. Reproduced with permission of ISAS and NASA. (See colour section.)


60

Spectroscopy

Figure 3.5. Coronal loop structure revealed in the soft x-ray image of Yohkoh overlaid with the Fe XV 28.4 nm contours; see, also, Brosius J W 1997 et al Astrophys. J. 477 969. Reproduced with permission of ISAS and NASA.

producing images in the light of different spectral lines originating from various ions in their particular ionization stages. A diagram of the Solar Extremeultraviolet Rocket Telescope and Spectrograph (SERTS)1 is depicted in figure 3.3. Data from that mission revealed over 240 lines in the active Sun spectrum from 57 different ions. Multiple lines are observed for the various ionization states of iron in the wavelength range of 17–45 nm for both the active and quiet Sun. The data are primarily used for temperature and density diagnostics and to derive differential emission distributions for both quiet and active solar features. Extended observations were made in the soft x-ray region from the Yohkoh satellite. In figure 3.4 two soft x-ray images of the Sun are displayed demonstrating the drop in solar intensity after a period of 3.5 years. Simultaneous 1 Brosius J W, Davila J M, Thomas R J, Saba J L R, Hara H and Monsignori-Fossi B C 1997 The structure and properties of solar active regions and quiet-Sun areas observed in soft x-rays with Yohkoh/SXT, and in the extreme ultraviolet with SERTS Astrophys. J. 477 969.


The quantum nature of radiation

61

Yohkoh and SERTS observations were obtained for flights in 1993 and 1995. The combined data show that SERTS images in the hot, coronal Fe XV and XVI regions corresponding to temperatures in excess of 2×106 K show nearly identical loop structures as those seen in the simultaneous Yohkoh soft x-ray telescope images. This is illustrated in figure 3.5 for the overlaid data from Fe XV.

3.2 The quantum nature of radiation The very specific frequencies of spectral lines cannot be explained by the wave nature of radiation. Interaction of light with matter on the atomic scale does not proceed in a continuous way. Rather it occurs stepwise as suggested by the discrete lines observed experimentally. This is taken into account in Einstein’s explanation of the photoelectric effect assuming that light travels as individual wavepackets of electromagnetic energy called photons. The apparent contradiction in the wave–particle duality is resolved by describing all particles including photons as wavepackets as briefly discussed in section 2.4. Mathematically they are represented by the same functions ψ(x, t) and ϕ(k, t) of equations (2.16) and (2.17) which were used in analyzing the light diffraction phenomena. They are representations of the particle in configuration and momentum space, respectively. These considerations formed the basis for the development of wave mechanics by Erwin Schrödinger (1887–1961). In this theory the functions ψ(x, t) and ϕ(k, t) are solutions of a wave equation which may be written as dz 1 d2 z = (3.1) dx 2 c2 dt 2 where the function z(x, t) is factorized as z(x, t) = ψ(x)ζ(t).

(3.2)

In equation (3.2) the function ψ(x) contains the dependence on the space coordinate, which, for simplicity, is taken here in one dimension x only. The function ζ (t) describes the time dependence. This ansatz leads to the differential equation −

~2

ψ + V ψ = Eψ (3.3) 2m known as the time-independent Schrödinger equation. In equation (3.3), = d2 /dx 2 is the delta operator and m is the particle mass. The quantities V and E denote the potential energy and total energy, respectively. The solutions of the Schrödinger equation describe the states of atoms, a subject to which we will return later in chapter 5. Time-dependent effects come into play when a beam of particles or photons collide with a sample of other particles or atoms as in the basic scattering experiment drawn in figure 3.6. Mathematically the incoming particles are


Spectroscopy

62

or

ect

det

ed tter

sca incident beam

sample

n

iatio

rad

Θ

undeflected radiation

Figure 3.6. A beam of particles interacting with a sample.

represented by plane waves. The effect of the target particles is to diminish the intensity of the plane waves. As a result of the scattering new spherical waves are created originating from the scattering center. The spherical waves will not usually be equally intense in all directions but rather show an angular intensity variation depending on the nature of the scattering event. Two principal classes of scattering can be observed depending on the relation between the incoming plane wave and the outgoing spherical wave. In coherent scattering the outgoing spherical waves have the same frequency as the incoming plane waves and there is some fixed phase relation, at the scattering center, between the two sets of waves. This is fulfilled for elastic scattering. Everything that does not fall into this category is referred to as incoherent scattering. If the frequency is changed the particle’s momentum and energy will be changed. So this class includes inelastic scattering. In many scattering experiments most of the incoming particles traverse the target undeflected and only a very small fraction of particles gives rise to a systematic scattering pattern manifesting itself in a pronounced angular and energy dependence. If the incident particles are photons we have to deal with the four basic interactions of light with matter as pictorially represented in figure 3.7: • •

• •

Sometimes the scattering of a photon occurs in such a way that the final photon has the same energy as the incident photon. This process is known as Thomson scattering. The incident photon may give all its energy to one atomic electron. The transferred energy is used to ionize the atom in a single event and the excess energy is carried away by the electron as kinetic energy. Such an interaction of electromagnetic radiation with matter is known as the photoelectric effect. An incident photon may collide elastically with an atomic electron transferring momentum and energy to the target electron. This kind of interaction is known as Compton scattering. If the photon energy is sufficient it may be used to produce a pair of particles such as an electron and a positron both acquiring some kinetic energy. This kind of interaction is referred to as pair production.


The photoelectric effect Thomson Scattering

Photoelectric Effect on

ot

incident photon

d re

photoelectron

ph

te

at

sc

incident photon

electron

atom

Pair Production

Compton Scattering

nucleus atomic electron incident photon

63

e-

Compton electron incident photon

sca

tte

red

e+

ph

oto

n

Figure 3.7. The basic interactions of light with matter.

The coherent scattering of a photon by a free electron was first investigated by J J Thomson in terms of classical electrodynamics. An incident wave accelerates a target electron by an amount that is proportional to the electric field amplitude of the incoming wave if the velocity of the electron is small compared with the speed of light. The accelerated electron itself becomes a radiation source with the same frequency as the incident wave. For a single scattering event there is a phase difference between the incident and the scattered wave which amounts to π for forward scattering.

3.3 The photoelectric effect In 1887, Heinrich Hertz (1857–94) discovered that light could release electrons from a metal plate. The characteristics of electron release from metal plates were studied by Hallwachs in 1888 thus establishing the photoelectric effect. Einstein gave an explanation for the results of the photoelectric experiment. He postulated that light had particle properties as well as wave properties. Today this postulate is an integral part of our understanding of electromagnetic radiation. The principle of the photoelectric effect is sketched in figure 3.8. If one shines a beam of light on the clean surface of a metal, electrons will be ejected from the metal. The light has to exceed a certain energy to remove electrons from the metal surface. If the light has more than the minimum energy required, then the extra energy will be given to the ejected electrons as kinetic energy. The photoelectric effect is, therefore, observed only for photon energies greater than


64

Spectroscopy

ultraviolet light blue light red light

fast electrons

slow electrons

no electrons

Figure 3.8. The photoelectric effect on a metal surface. The energy of red light is not sufficient to liberate electrons from the solid. Once the photon energy exceeds the minimum energy necessary for freeing electrons, the excess photon energy is taken by the electrons as kinetic energy.

the binding energy of at least some of the more weakly bound atomic electrons. The photoelectric effect is the dominant interaction of photons with matter in the photon-energy range 1–500 keV. Metals conduct electricity and permit the electrons to move on them freely. Usually the metal plates are made up of alkali metals which have just one electron in the outer energy level. The reason is that they need low energies to eject electrons from the atoms. The total energy W necessary for the release of an atomic electron may be thought of as consisting of two parts. In the first part an inner electron is raised to the conduction energy band, and in the second, a conduction-band electron is ejected from the metal. Thus W is the sum of the atomic binding energy and the work function characteristic for the metal. Sometimes the ionization of isolated atoms by absorption of photons is referred to as the inner photoelectric effect as opposed to the outer photoelectric effect involving the solid-state environment. If the incoming photon has enough energy to completely remove an electron from the plate an electric current can be measured in the circuit shown in figure 3.9. A variable retarding potential is applied between the photocathode and the anode of a glass vacuum-tube photocell. The current to or from the anode while it is being maintained at a specific potential is read from a sensitive electrometer that is capable of indicating currents to picoamperes. For any


The photoelectric effect

65

pA

light

V

Figure 3.9. The vacuum-tube photocell and its outer circuit for the study of the photoelectric effect.

given photon frequency ν there is a critical retarding potential, Vs which is just sufficient to stop all of the electrons leaving the photo cathode from reaching the anode, regardless of the angle of their initial direction of emission. This stopping potential can be found as E ν = hν = W + eVs .

(3.4)

The energy is provided by a photon of visible light or of shorter wavelength. Thus the stopping potential Vs provides a measure of the photon energy E ν with an unknown but fixed offset W . Usually light of several different specific frequencies are used from which the common constant W can be determined. In an experiment a particular wavelength of light is selected, for instance by a suitable filter. Then the retarding potential at which the forward current stops is to be found. Therefore we regard Vs as a function of the light frequency ν and note from equation (3.4) that it is a linear relation with slope h/e. There will be a threshold frequency, below which no current will be observed for any positive value of the retarding potential. These characteristics of the photoelectric experiment are shown in figure 3.10. Light of a single wavelength behaves as if it consisted of separate particles, photons, all with the same energy, with each ejected electron being the result of a collision between one photon and one electron in the metal. Higher intensity light means only that more photons are hitting the metal per time interval and more electrons are being ejected, not that there is more energy per photon. The energy of the outgoing electrons depended on the frequency of light used. Experiments made with x-rays and a variety of materials gave the following results. For any one element, the absorption coefficient depends strongly on photon energy varying over several orders of magnitude if the photon energy is varied from 1 keV to 1 MeV. Furthermore, the variation of the absorption


66

Spectroscopy

Figure 3.10. Characteristics of the photoelectric experiment conducted with the arrangement shown in figure 3.9. The dependence of the photo current I on the retarding potential V (left) is shown for two different light intensities but the same frequency. The stopping potential Vs increases linearly with the frequency (right).

coefficient is not smooth but rather it has smooth sections interrupted by discontinuities. In the smooth sections, the absorption coefficient decreases rapidly as the photon energy increases, being approximately proportional to the −7/2 power of the photon energy for energies below 500 keV, where the photoelectric effect dominates. The discontinuities are caused by steps in the absorption coefficient as the photon energy increases. There is a systematic dependence on the atomic number Z of the photon energy at which those discontinuities are observed. These findings are in strong support of the shell structure of atoms with definite shell binding energies as systematically explored by Henry Moseley (1887–1915). A qualitative explanation is that the probability of absorption increases with the number of electrons capable of taking over the photon’s energy, i.e. those electrons with binding energy smaller than the photon’s energy. Because the heavier elements have more electrons they show an increased absorption. Photons with less energy than the edge can ionize only the outer electrons from the target atoms but photons with higher energy can ionize both outer and inner electrons. Therefore the photons on the high energy side of the edge are much more strongly absorbed.

3.4 Compton scattering A very convincing demonstration of the particle nature of light is the Compton effect, named after Arthur H Compton (1892–1962) who discovered and explained it in the years 1919–23. It is the interaction of an incident photon with a quasi free electron in which the wavelength of the scattered photon is changed in contrast to Thomson scattering where the photon wavelength remains unchanged. Compton scattering is entirely explained as a collision process between a photon and an electron where the energy and momentum are conserved as in collisions


Compton scattering

67

Q, p

θ

Q0, p0

ϕ

E, P

e-

Figure 3.11. Energies and momenta involved in Compton scattering.

between billiard balls. The energy of the electron, recoiling in a certain direction with a specific energy, is provided by a reduction in the energy of the photon. As in the photoelectric effect a target atom is required for the recoil, in order to conserve the momentum vector. The situation is explained in figure 3.11 which specifies the energies and momenta involved. Let the incident photon have an energy Q 0 and a momentum vector p0 . After scattering, the photon deflected by an angle θ has the changed energy Q and momentum p whereas the electron gains energy E and momentum P. It is assumed that the electron initially has zero kinetic energy. In practice available electrons are bound to atoms, so that the approximation of a free, stationary electron is that the electron binding energy is much less than the photon energy. Under these assumptions momentum and energy conservation, respectively, can be expressed as p0 − p = P Q 0 − Q + mc2 = E.

(3.5)

Noting that the photon energy and momentum are simply related as p = Q/c we can write equations (3.5) as (Q 0 − Q)2 + 2(Q 0 − Q)mc2 + m 2 c4 = E 2 Q 20 − 2Q 0 Q cos θ + Q 2 = c2 P 2 .

(3.6)

Subtracting the two equations yields

or


2Q 0 Q(1 − cos θ ) − 2(Q 0 − Q)mc2 = 0

(3.7)

1 1 1 − cos θ − = . Q Q0 mc2

(3.8)

68

Spectroscopy

0o

Intensity

45o

90o

135o

70.0

70.5 71.0 71.5 Wavelength (pm)

72.0

Figure 3.12. Compton’s experiment showing the wavelength shift of incoherent scattering at various scattering angles. Coherent scattering with unchanged wavelength is also observed. The increase in shift with scattering angle cannot be explained classically. It logically follows from the collision kinematics involving photons. Compton A H 1923 c Phys. Rev. 22 409. 1923 AIP.

Using Q = hc/λ leads to the Compton wavelength shift λ = λ − λ0 =

h (1 − cos θ ). mc

(3.9)

The interesting result of equation (3.9) is that the wavelength shift is independent of the wavelength of the incident photon. The shift vanishes in the forward direction and increases with the scattering angle θ . The maximum wavelength shift is observed in the backward direction and amounts to λC =

h = 2.426 × 10−12 m mc

referred to as the Compton wavelength.


(3.10)

Compton scattering 0

1

100 keV 0.5

dσe d

/ re2

69

1000 keV 30

60

90

θ

120

150

180◦

Figure 3.13. Angular dependence of the Compton scattering at three different photon energies as indicated. In the limit of vanishing photon energy the curve is determined by Thomson scattering.

The angular dependence of the wavelength shift (3.9) was observed by Compton in his original measurements2 reproduced in figure 3.12. There one sees the contribution of the coherently scattered radiation with no wavelength shift plus the Compton scattered radiation with the shift increasing with the angle relative to the direction of the incident photon. In low-frequency limit the Compton electron recoils approximately perpendicularly to the direction of the incident light and Compton scattering merges smoothly into classical Thomson scattering. The recoil electrons have been observed and the angles θ and ϕ could be related. Following equations (3.5) the electron kinetic energy can be expressed as %(1 − cos θ ) 1 + %(1 − cos θ ) cos ϕ = (1 + %) tan(θ/2)

E kin = Q 0 − Q = hν0

(3.11)

% = hν0 /mc2. The probability of Compton scattering is usually small but at energies around 1 MeV it is the most common interaction. Applying relativistic theory of the electron to Compton scattering Klein and Nishina obtained a general expression for the probability of the process expressed as the differential cross section dσe = re2 [1 + %(1 − cos θ )]−2 ( 12 + 12 cos2 θ ) d % 2 (1 − cos θ )2 × 1+ (1 + cos2 θ )[1 + %(1 − cos θ )] 2 Compton A H 1923 Phys. Rev. 22 409.


(3.12)

70

Spectroscopy

where re = e2 /mc2 = 2.818 × 10−13 cm2 is the classical electron radius. The Klein–Nishina formula 3 which is plotted in figure 3.13, is in excellent agreement with experiments.

3.5 Mössbauer spectroscopy The Mössbauer effect, after R L Mössbauer (1929–), is the recoil-free emission of gamma radiation from radioactive nuclei embedded in a solid material. Since the gamma emission is recoil-free, the emitted radiation can be resonantly absorbed by ground-state nuclei also in a solid. As the nuclear transitions are very sensitive to the local environment of the radiating atoms Mössbauer spectroscopy can be employed as a sensitive probe of the different environments in a solid material.

Q>Q0

M0 Q0

M0 Q0

Absorption v

M

Emission

Q 0, the equation has finite and continuous solutions for arbitrary values of E and . If the energy E < 0, there are only solutions for the discrete eigenvalues of the energy given as µe4 Z 2 En = − 2 2 . (3.41) 2~ n Here n is an integer and n ≥ + 1. Neglecting the relative difference between the reduced mass µ and the electron mass m which is 1/1836 for hydrogen, one has En = −


Z 2 Ry n2

(3.42)

The inner concept of atoms

81

i.e. the same expression (3.35) as in the Bohr model. The quantity n in (3.41) is called the principal quantum number. For a given value of n, the orbital quantum number can take the values 0, 1, 2, . . . , n − 1 corresponding to 2 + 1 states differing by the magnetic quantum number m. The quantity m can take the values m = 0, ±1, ±2, . . . , ± in accordance with the operators of the angular momentum and its z-component: 2 ψ = ( + 1)ψ z ψ = mψ.

(3.43)

Thus, solving the quantum-mechanical Schrödinger equation can show which electron states are occupied. For the H-like ion state nm, the energy level values depend only on the principal quantum number n not on and m. These states are said to be n 2 -fold degenerate because there are n 2 = 1 + 3 + 5 + · · · + 2n − 1 states differing in quantum numbers and m. The independence of the energy E n on m is explained by the fact that in the central field all directions in space are equivalent and the energy, therefore, cannot depend on the orientation of the angular momentum . The independence from is a pure property of the Coulomb field and does not take place in the general case, even for a central-symmetrical field. The ionization energy of the hydrogen atom, i.e. the energy required for detachment of an electron from the atom, equals 13.5984 eV but not 1 Ry = 13.6057 eV because of the difference between the reduced mass in hydrogen µ ≈ 0.999 45 m e and the electron mass m e . The radial wavefunctions for the discrete spectrum of the hydrogen-like atom are expressed in terms of the generalized Laguerre polynomials: n! dn−m ex x −m n−m e−x x n (n − m)! dx (n − − 1)! 2Z 3/2 − naZr 2Zr 2+1 2Zr 0 Rn (r ) = − e Ln+ 2n[(n + )!]3 na0 na0 na0 m Lm n (x) = (−1)

(3.44)

where a0 is the Bohr radius. The functions Rn (r ) are orthogonal and normalized: ∞ Rn (r )Rn (r )r 2 dr = δnn .

(3.45)

At large distances, the functions Rn decrease exponentially: Rn (r ) ∼ exp(− |E n |r ) E n = −Z 2 Ry/n 2 r → ∞.

(3.46)

0

H (r ), Z = 1, are related The radial wavefunctions for the hydrogen atom Rn to those for H-like ions with the charge number Z by Z H (r ) = Z 3/2 Rn (Zr ). Rn


(3.47)

Spectroscopy

82

r (a.u.) 4

2

0

1

r 2|Rn(r )|2 0

r (a.u.) -4

-2

0

2

4

Figure 3.20. The radial distribution of the electron cloud of a hydrogen atom in its ground state.

The first few radial functions for the hydrogen atom are: R10 = 2e−r 1 1 1 R21 = √ e−r/2r R20 = √ e−r/2 1 − r 2 2 2 6 2 2 2 2 −r/3 R30 = √ e 1− r + r 3 27 3 3 8 −r/3 r 4 R31 = √ e R32 = √ e−r/3r 2 . r 1− 6 27 6 81 30

(3.48)

The orbital angular-momentum quantum numbers = 0, 1, 2, 3, 4 are also designated by the letters s, p, d, f, g. For example the state with n = 3 and = 2 is labeled 3d. 2 (r ), i.e. the probability density of finding the The radial distribution r 2 R1s electron between two spheres of radii r and dr from the origin, is shown in figure 3.20. This is now a more realistic image of the atom with its electron cloud described by the probability density as given by the square of its respective


energy or probability density (a.u.)


83

0.1 0

3s 2s

-0.1 -0.2

V(r)

-0.3 -0.4 -0.5

1s -20

-10

0

10

20

r (a.u.)

Figure 3.21. The Coulomb potential of the hydrogenic nucleus and the first three radial probabilities with = 0 (s orbitals).

wavefunction. Although the electron orbits are no longer strictly localized the probability distribution takes its maximum near the Bohr radius for the n = 1, = 0 ground state. One can also define quantum-mechanical averages or expectation values. For instance the expectation value of the radius is given as rn = n|r |n ∞ 2 2 r Rn r dr =

(3.49)

0

=

1 a0 [3n 2 − ( + 1)] . 2 Z

(3.50)

Examples for radial distributions of excited states are included in figure 3.21. Also included on the same scale is the form of the Coulomb potential. As the principal quantum number n increases the charge becomes more extended and the number of maxima increases. So far we have shown examples for the spherically symmetric = 0 wavefunctions. Let us now consider the angular functions Ym (θ, ϕ). They can be expressed in terms of the associated Legendre polynomials Pm : eimϕ Ym (θ, ϕ) = &m (θ )'m (ϕ) 'm (ϕ) = √ 2π (2 + 1)( − m)! m P (cos θ ). &m (θ ) = (−1)m 2( + m)!

(3.51)

Here, it is assumed that m ≥ 0. For m < 0, one has &,−|m| = (−1)m &,|m| .


(3.52)

Spectroscopy

84

n = 2, = 0, m = 0

n = 2, = 1, m = 0

n = 2, = 1, m = ±1

n = 3, = 0, m = 0

n = 3, = 1, m = 0

n = 3, = 1, m = ±1

n = 3, = 2, m = 0

n = 3, = 2, m = ±1

n = 3, = 2, m = ±2

Figure 3.22. The spatial distribution of the electron cloud of a hydrogen atom in various eigenstates according to the quantum numbers n, , m as indicated. Cuts of the probability density |nm (θ, ϕ, r )|2 through the x, z plane are shown.

The functions Ym are orthogonal and normalized: 2π π Y∗ m (θ, ϕ)Ym (θ, ϕ) sin θ dθ dϕ = δll δmm . 0

(3.53)

0

The expressions for the first functions &m with = 0, 1 and 2 are:

&10 = 32 cos θ &1,±1 = ∓ 34 sin θ &00 = 12

&20 = 52 ( 32 cos2 θ − 12 ) &2,±1 = ∓ 15 4 cos θ sin θ √ &2,±2 = 14 15 sin2 θ.

(3.54)

The spatial distributions of the electron clouds in the hydrogen atom



85

Figure 3.23. The shape of electron orbitals in a Cu atom obtained experimentally using a combination of x-ray diffraction and electron microscopy. Zuo J M et al 1999 Nature 401 c 49. Reproduced with permission 1999 Arizona State University. (See colour section.) 2 (r ) in the ground represented by the functions |ψ(θ, ϕ, r )|2 = |Ym (θ, ϕ)|2 Rn and a few excited states are shown in figure 3.22. The image of a many-electron atom is the image of its outermost electrons or the image of the averaged probability that the electrons will be at various places. The three-dimensional map of electron orbitals of Cu atoms and their bonds with neighboring atoms in a cuprite (Cu2 O) compound is shown in figure 3.23. The image was produced using a combination of x-ray diffraction and electron microscopy6. These orbitals look just like the drawings shown in figure 3.22 and represent an excellent confirmation of the quantum-mechanical theory.

6 Zuo J M et al 1999 Nature 401 409.


Chapter 4 Light and ion sources

In this chapter we are going to dig a little bit deeper into the atomic-physics principles underlying powerful sources of radiation and of highly charged ions. Various experimental aspects and some technical details will also be sketched.

4.1 Basic physical considerations In this section we will learn about ion formation, ionization and recombination collisions and we will study the basics of plasma formation. Part of our understanding is based upon our ability to track down individual elementary collisional and radiative processes1. Mostly, however, the description of the collective behavior of the particles present in the physical environment of an ion source is statistical in nature. Here we will introduce both concepts which provide the physical foundations for the operation of ion sources2 . The principles of the related plasma physics are treated more thoroughly in the respective literature3. 4.1.1 Elementary collisional and radiative processes In light and ion sources we often have to deal with the formation of a plasma consisting of neutral and charged particles. These are neutral atoms, electrons and ions of eventually several different charge states. Many different elementary interactions between individual plasma constituents may be important for the creation of charged particles from the neutral state and for the ignition and sustainment of the plasma conditions. 1 Sobelman I I, Vainstein L A and Yukov E A 1995 Excitation of Atoms and Broadening of Spectral

Lines (Berlin: Springer). 2 Wolf B H 1995 Handbook of Ion Sources (New York: Chemical Rubber Company). 3 See, for instance, Goldston R J and Rutherford P H 1995 Introduction to Plasma Physics (Bristol:

IOP Publishing).


Basic physical considerations

87

Direct and inverse processes The excitation and ionization of heavy particles is usually dominated by electron impact whereas interactions among heavy particles are less likely. The processes of interest are the following: •

Ionization and three-body recombination: Xq+ + e−

•

X

(4.1)

X

q+ ∗

+ e−

(4.2)

X

(q+1)+

+ e−

(4.3)

where ~ω is the photon energy. Dielectronic recombination and autoionization: Xq+ + e−

•

−

where Xq+∗ denotes an ion in an excited state. Radiative ionization and recombination: Xq+ + ~ω

•

+ e− + e

where Xq+ refers to a q-times ionized atom and the double arrow indicates that the process may proceed in both directions. Collisional excitation and de-excitation: Xq+ + e−

•

(q+1)+

X

(q−1)+ ∗∗

∗

→ X(q−1)+ + ~ω

(4.4)

where X(q−1)+∗∗ denotes a doubly excited state. Emission and absorption: Xq+

∗

X

q+

+ ~ω.

(4.5)

Each of the reactions (4.1)–(4.5) represents a pair of direct and inverse processes. The probability as expressed by the respective cross section for the individual process can be related to its reverse by simple formulae. In the case of dielectronic recombination, one electron is resonantly transferred to an excited ionic state in a first step by a simultaneous excitation of a bound electron. In a second step the doubly excited state stabilizes by a radiative decay below the autoionization limit. Collisions Some of the elementary processes proceed via a binary collision which may be viewed in the same way as the basic scattering experiment discussed on page 61. The probability for excitation or ionization to occur is proportional to the electron density n e and to the collision velocity v, W = ne σ v


(4.6)

Light and ion sources

88

where σ represents the excitation or ionization cross section. The cross section may be identified as the circular area that has to be hit by the collision partner in order for the interaction under consideration to occur with unit probability. In a plasma the electrons usually move at a much higher speed than the heavy particles and the collision velocity can be set equal to the electron velocity. Generally the electrons are not monoenergetic. That is why one has to average over their energy distribution f (E) in the following way: W = n e vσ ∞ vσ (E) f (E) dE. vσ =

(4.7) (4.8)

E min

The quantity vσ is referred to as the rate coefficient. To characterize the average distance between collisions, the mean free path used is defined as the inverse of the product of density and cross section, (n e σ )−1 . 4.1.2 Statistical and collective behavior of particles Thermodynamic equilibrium If one adopts the statistical approach for a macroscopic description of a plasma a great reduction results in the amount of information to be handled. For a complete description of a complex system of many different particles one would have to know the space and velocity distribution functions for all constituents. If the shapes of these distributions are known, the average quantities related to the moments of these functions would suffice to characterize a plasma completely. The most important mean quantities are densities, particle currents or velocities, pressure or temperature and heat flux. The distribution function for one particle species is given as the density of the particle in the phase space dn (r, v, t)/dr dv. The first four moments correspond to • • • •

density, mean velocity, momentum flow and energy flow.

If the particles are in thermodynamic equilibrium the electron velocity v is distributed according to the Maxwell distribution (3.19) which for three Cartesian velocity components can be written as dn e = n e f (v) dv = n e f (vx , vy , vz ) dvx dvy dvz = n e m 3/2 (2πkB T )−3/2 m 2 2 2 (v + vy + vz ) dvx dvy dvz . × exp − 2kB T x


(4.9)


89

E Ekin

gs

Es Eo

go

T1 < T2

< T3

0

ns

Figure 4.1. Principle of thermal excitation and ionization. The density n s of excited states as governed by the Boltzmann distribution (4.11) is illustrated for three different temperatures.

In terms of the kinetic energy E of free electrons the distribution (4.9) can be rewritten as dn e = n e f (E) dE = n e (2π)

−1/2

(kB T )

−3/2

E exp − kB T

dE.

(4.10)

According to statistical mechanics there is a distribution of excited states with energy levels E s given by the Boltzmann distribution, gs Es − E0 ns (4.11) = exp − n0 g0 kB T where n s and n 0 denote the density of atoms in an excited or the ground state, respectively. The gs and g0 are statistical weights, i.e. the multiplicity of a state according to its quantum-mechanical degeneracy. Equation (4.11) is visualized in figure 4.1 for three different temperatures. Thermal excitation into quantum states of increasingly high excitation energy becomes more likely with increased temperature. Eventually some of the bound electrons may receive sufficient energy to reach the region of positive energy states of the first ionization continuum. Excitation of the bound electrons in the singly ionized ions will occur at still higher temperatures eventually leading to


90


doubly ionized ions. This process of thermal multiple ionization will continue until a distribution of charge states is reached that corresponds to the given electron temperature. To determine the density of ionized atoms n (q) in various charge states q, we can start from the Boltzmann distribution. The problem reduces to the calculation of the statistical weight of an ionized atom in its ground state plus a free electron. For a single free electron we have to consider the number of quantum cells h 3 in the phase space. From statistical mechanics one can find that an electron with mass m occupies a volume of (2πmkB T )3/2 in momentum space. In real space the average volume of a free electron is n −1 e . Combining all phase-space factors and including a factor of 2 for the electron spin, we end up with the Saha distribution of degrees of ionization: n (q+1) g (q+1) E (q) mkB T 3/2 −1 (4.12) = (q) 2 n e exp − kB T n (q) g 2π ~2 where the superscript (q) refers to the charge state of an ion and correspondingly E (q) denotes the ionization potential of the q-times ionized atom. Plasma parameters (1) Charge neutrality. If a macroscopic volume of the plasma can be regarded as a closed system the number density of free electrons is balanced by the number of positively charged ions. Confining our discussion to positively charged ions we can write the general condition of charge neutrality as

qn q = n e (4.13) q

where the sum runs over all charge states q and n q denotes the number density of q-times ionized atoms. Locally, however, there might be small deviations from charge neutrality. If we introduce, for example, a positive charge into a gas of electrons, the electrons attracted by the Coulomb force would tend to gather around the point charge so as to screen it but tending to move away from the positive charge by diffusion. This is schematically illustrated in figure 4.2. There are characteristic time and size scales for these processes. The speed of any perturbations from the equilibrium condition will determine the appropriate theoretical approximation. A rough classification may be given as follows. Perturbation frequency

Approximation

Low Medium High

Boltzmann distribution equation of motion (inertia) particles at rest

For a slow perturbation the particles will have sufficient time to respond in a collective way given by the statistical approach of the Boltzmann distribution. At



field +

91

nt

curre

diffusion current

ne(r)

r

U(r)

Figure 4.2. Illustration of Debye screening showing one half of a Debye sphere. An equilibrium is established between thermal diffusion and field currents. The curves show the variation of the movable charge density δn e (r ) and of the electrostatic potential U (r ) with distance r .

higher frequencies we have to include the inertia of the particles and to solve their equation of motion. Only at very high frequency particles can no longer respond to an external disturbance and can be assumed to be at rest. (2) Plasma frequency. Space-charge oscillations occur at the natural frequency of the plasma. These oscillations are stationary if one can neglect the ion motion. Let us return to the positive point charge in the gas of electrons as depicted in figure 4.2. The mobile electron charges attracted by the Coulomb force will be accelerated towards the disturbing positive charge. Because of their inertial masses the electrons will move beyond equilibrium. The electric field created by the displaced electrons acts as a restoring force giving rise to oscillations. We will calculate the frequency of these oscillations in a very idealized simplified situation assuming the absence of magnetic fields. Because of their high mass the ions cannot respond to the wave fields as readily as the electrons. So we assume the ions to be at rest representing a stationary background. The electrons, in this picture, move only in the direction along the electric field neglecting all magnetic contributions. We start from the small (local) change δn e in the electron density which


92


assumes a value of n e = n e∞ + δn e

(4.14)

where n e∞ denotes the undisturbed electron density far away from the introduced perturbation. Now we can set up the basic equations: e E me −eδn e ∇E = %r %0 n˙e + ∇(n e v) = 0 v˙ = −

equation of motion

(4.15)

Poisson equation

(4.16)

continuity equation

(4.17)

where %r denotes the relative dielectric constant and %0 the permittivity of the vacuum, respectively. In the continuity equation (4.17) the term n e v = je /e is the current density, where v denotes the electron velocity. Upon differentiating of equation (4.17) with respect to time and subsequently inserting (4.15) and (4.16), we arrive at e2 n e δn e . (4.18) δn¨ e + %r %0 m i Equation (4.18) describes a harmonic oscillation of the density δn e with the characteristic angular frequency ωpe given by 2 = ωpe

e2 n e %r %0 m e

(4.19)

known as the electron plasma frequency. There also exists an ion plasma frequency given as q 2 e2 n i 2 ωpi = . (4.20) %r %0 m e Because the corresponding ion oscillations are more rarely encountered the electron plasma frequency is often simply called the plasma frequency. The space-charge oscillations at the natural frequency ωpe are called Langmuir oscillations after Irving Langmuir (1881–1957). The frequency ωpe does not depend on the temperature and thus can be used as a density diagnostic if escaping waves of this characteristic frequency are observed. In practical units equations (4.19) and (4.20) can be cast into f pe [Hz] = 8980n e [cm−3 ]1/2 −3

fpi [Hz] = 210(n i [cm

]/A)

(4.21) 1/2

(4.22)

where A denotes the atomic mass number. Coupling external waves into a plasma is efficient at frequencies below the electron plasma frequency fpe . (3) Debye screening. We will now calculate the characteristic distance at which screening by the electronic charge cloud is effective. The length scale is set by the dynamic equilibrium of thermal diffusion and electric-field-induced



93

currents. We again assume that the local perturbation in the charge density is small: δn e n e∞ . Introducing an electric potential U the electric field vector is given as E = −∇U. (4.23) The potential U is assumed to vanish at infinity. In the statistical average the density of electrons in the potential U will be distributed according to the Boltzmann factor exp(−W/kB Te ), where W is the potential energy of an electron. Within this approximation it is assumed that there are many electrons contributing to the charge averaging and that time constants are large. Starting with equation (4.14) we can write the relative density distribution as δn e W ne (4.24) =1+ = exp − n e∞ δn e∞ kB Te eU (4.25) = exp + kB Te eU ≈1+ . (4.26) kB Te The expansion (4.26) of the exponential is valid because we assume that δn e

n e∞ and hence eU kB Te . The Laplace equation is obtained by combining equations (4.16) and (4.23), eδn e ∇ 2U = . (4.27) %r %0 Solving equation (4.26) for U and inserting it in equation (4.27) yields ∇ 2 δn e =

e2 n e δn e . %r %0 kB T

(4.28)

The solution of this differential equation is an exponential decay (or growth) with the distance r , r (4.29) δn e (r ) = exp − λD where the characteristic decay length is given by λ2D =

%r %0 kB Te . e2 n e

(4.30)

λD is called the Debye length or the Debye–Hückel length after Peter Debye (1884–1966) and Erich Hückel (1896–1980) who first calculated λD for electrolytes in 1923. (4) Magnetic fields. The energetic charged particles will follow trajectories which are determined by the electromagnetic fields present in a plasma and which can be calculated by solving Maxwell’s equations. Very often a strong magnetic field is applied in order to confine the plasma to a small volume and/or to increase



94

the path of the particles thereby increasing their chance of undergoing ionizing collisions. With a given magnetic flux density B a particle of charge q moving at a velocity v will exert the Lorentz force F = qv × B.

(4.31)

Particles moving along the direction of the magnetic field will not be influenced by the magnetic field whereas those having a perpendicular velocity component v⊥ will spin around the magnetic field lines. Setting the particle’s centripetal force equal to the Lorentz force yields the orbital radius which is also called the gyration radius: mv⊥ ρ= . (4.32) qB The corresponding angular frequency or (cyclotron or Larmor frequency) is ωc =

qB . m

(4.33)

Equations (4.32) and (4.33) can be summarized as ρe [µm] = 3.3

Te⊥ [eV]1/2 B[T]

(ATi⊥ [eV])1/2 ρi [µm] = 14 q B[T]

fce [GHz] = 28B[T] (4.34) q f ci [GHz] = 15 B[T] A

specifying the quantities in their units as indicated. Because of the opposite charge of ions and electrons they also have opposite senses of rotation. This can be of importance when one wants to couple energy into the plasma via electromagnetic waves. The electrons, for instance, can be efficiently heated by microwaves tuned to the electron-cyclotron frequency fce . This is a resonant process called electron cyclotron resonance (ECR) but it can only work when the applied radiation is transported into the plasma volume. In order for this to happen the electron density has to be below a critical value such that the corresponding plasma frequency ωpe stays below the frequency of the applied microwaves. Besides the resonant heating a non-resonant microwave discharge also exists at higher pressures. For the formation of gyration discs of the charged particles in the magnetic field, the probability (as defined by equation (4.7)) for a violent collision has to be small within one period of revolution. Of course this limitation occurs at different pressures for ions and for electrons. Evolution of high charge states: a simple example To ionize an atom from its neutral state a certain amount of energy has to be supplied that is above a threshold given by the ionization energy or the ionization potential. The latter is defined as the energy necessary to move an electron from a



Arq+

4000 ionization potential (eV)

95

3000

2000

1000

0

0

5

10 charge state q

15

Figure 4.3. Ionization potentials of Arq+ ions. The two steps are located at those charge states where the L and K shells start to become depleted.

bound orbital of the atom to infinity. Such an ionization process has to be repeated many times to multiply ionize an atom to a high charge state by successive electron removal. Within this chain of processes the minimum energy for single electron removal does not stay constant but increases with the ion charge. This is illustrated in figure 4.3 for the case of an argon atom. The steps observed in the ionization potential as a function of the charge state can be attributed to the shell structure of the atom. The smaller step occurs after all eight electrons from the M shell have been removed and the more tightly bound L electrons have to be removed as next. The biggest increase, however, occurs when all electrons are removed besides two remaining K-shell electrons. The ionization energy plotted in figure 4.3 is valid only for the removal of one electron at a time. If we want to remove all electrons at once we have to sum all energies which results in a very substantial amount of energy. Table 4.1 gives some numerical examples for single and multiple ionization energies for an argon and for a uranium atom, respectively. The examples show that there is a huge energy necessary to remove the inner electrons of an atom especially for a heavy atom. If one wants to multiply ionize an atom in a single step, this would require substantially higher energies than for multi-step ionization. One of the most important ionization processes is single-step ionization by electron impact. The probability per unit time for a collisional ionization is given by equation (4.7) into which we have to insert the appropriate ionization cross section σi . This cross section depends on the impact energy and on the particular charge state and electronic configuration of the considered ion. The cross section has a maximum when the bombarding energy reaches approximately three times the ionization potential and the cross section decreases considerably with increasing ionic charge. An analytical approximation which has often been used to estimate


96


Table 4.1. Numerical examples of ionization potentials for single and multiple ionization. Ion

Ionization potential (eV)

Ar

q = 0→ 1 15 → 16 17 → 18 0 → 18

16 918 4 430 14 000

U

q = 49 → 50 0 → 50 91 → 92 0 → 92

2 860 48 200 132 000 610 000

the ionization cross section is the Lotz formula4: σi =

N

i=1

ai r i

ln(E/I ) {1 − bi exp[−ci (E/I − 1)]} EI

(4.35)

where E denotes the electron kinetic energy and I the ionization potential, respectively. The sum runs over all subshells considered. For a hydrogen- or helium-like ion only the K shell is to be taken into account, N = 1. The quantity ri is the number of equivalent electrons in the subshell and ai , bi and ci are individual constants. The ionization cross sections for many ion species have been measured. In figure 4.4 we display the cross sections for the successive removal of the first few electrons from an argon atom. Note the decrease in cross section with increasing charge state and the shift in the maximum to higher energy. For simplicity we assume that single-step ionization is the only collisional process to be considered and the electrons are monoenergetic. As we will see later this condition is nearly fulfilled in electron-beam ion sources. Under this assumption the progressive increase in the charge state from the neutral state to the fully ionized one can be estimated. This process is governed by a set of rate equations for the number Nq of ions in charge state q, which can be written as dNq j = [Nq−1 (t)σq−1 − Nq (t)σq ] dt e

(4.36)

where j denotes the electron current density. The two terms in equation (4.36) represent the gain and loss in the population of a particular charge state due to ionizing collisions. Using the Lotz formula the rate equations are solved and the result is displayed in figure 4.5 for argon atoms and for electrons having a kinetic energy of 10 keV and a high current density of 1000 A cm−2 . The upper part of 4 Lotz W 1968 Z. Phys. 216 241–7.



97

-16

2

Cross section (cm )

10

q=1 q=2

-17

10

q=3 q=4 q=5

–

q+

e + Ar

(q+1)+

→ Ar

+ 2e

–

-18

10

1

10

2

3

10 10 Electron energy (eV)

Figure 4.4. Cross sections for the ionization of Arq+ ions by electron impact as a function of the collision energy. The data were measured in a crossed-beam experiment. From c Müller A et al 1980 J. Phys. B: At. Mol. Phys. 13 1877, 1980 IOP.

the figure shows all charge-state fractions in the time range between 1 µs and 1 s and the lower part demonstrates the evolution of a charge-state spectrum with a dominating neutral fraction at the beginning and a nearly pure bare fraction after 1 s. The fixed bombarding energy of 10 keV is far beyond the energy necessary for maximum ionization of the ions with low charge. For the K-shell electrons there is a better match with the cross section maximum leading to the efficient production of bare ions after a time of 1 s. The relatively high peak at q = 16 is a consequence of the energy dependence. The time τq necessary to build up the charge state q starting from the neutral state may be estimated as

τq−1 =

q−1

n e ve σq

(4.37)

q =0

where one has to average over the velocity distribution. For a monoenergetic beam of electrons we can set n e ve = j/e.



98

1

fraction

0.8

Argon

q=0

18

E = 10 keV j = 1000 A/cm2

0.6

16 17

1 2 3

0.4

15 4 5 6 7 8 10 121314 9 11

0.2

-5 10

-4 10

-3 10

-2 10

-1 10

0 10

time (s) 18

charge state

15

10

5

0 0.5

1

0.5

1

0.5

1

0.5

1

fraction Figure 4.5. Evolution of high charge states by successive electron removal by electron impact. The example is calculated for argon and an electron energy of 10 keV and a current density of 1000 A cm−2 . The charge-state distributions are shown at t = 1 µs, 100 µs, 10 ms and at 1 s.

4.2 Bremsstrahlung Bremsstrahlung, a German word meaning breaking radiation, is the electromagnetic radiation emitted by accelerated charged particles. In particular it is attributed to the radiation caused by deceleration. Although bremsstrahlung refers to the whole electromagnetic spectrum it is mostly studied in the x-ray region. Because the radiation intensity drops proportionally with m −2 with increasing mass m of the fast particle it is mostly observed with light particles especially with electrons. In thick solid targets, notably used as anodes in x-ray tubes, incident energetic electrons can be completely stopped. The mechanism for the energy loss is the successive collisions with the target atoms. Very often the contribution of the fast electron colliding with an atomic electron can be neglected and the scattering in the Coulomb field of the atomic nucleus is the dominating


Bremsstrahlung

99

effect for Bremsstrahlung production. The effect of the atomic electrons is merely to shield the charge of the nucleus. A basic understanding of the bremsstrahlung processes can be obtained starting from classical electrodynamics5 and its theory is well documented in the literature6. 4.2.1 Radiation from accelerated charges The electromagnetic fields generated by moving charged particles can be derived from Maxwell’s equations which we recall as follows. ρ %0 ∇·B=0

∇·E=

(4.38) (4.39)

∂B ∇×E= − ∂t 1 ∂E j ∇×B= 2 + 2 . c ∂t c %0

(4.40) (4.41)

Using the potentials A and φ defined through B=∇×A

(4.42)

∂A E = −∇ ·φ− ∂t

(4.43)

and assuming the Lorentz gauge, ∇· A=−

1 ∂φ c2 ∂t

(4.44)

we obtain the wave equations 1 ∂2 A ρβ = c%0 c2 ∂t 2 1 ∂ 2φ ρ ∇ 2φ − 2 2 = − . %0 c ∂t

∇2 A −

(4.45) (4.46)

For the electric current density we have used j = ρv = ρβc. Solutions of the wave equations (4.45) and (4.46) have the form 1 ρ(x)v A(t) = d3 x (4.47) c R ret 1 ρ(x) d3 x. (4.48) φ(t) = c R ret 5 Jackson J D 1999 Classical Electrodynamics 3rd edn (New York: Wiley). 6 Pratt R H and Feng I J 1985 Atomic Inner-Shell Physics ed B Crasemann (New York: Plenum) and

references cited therein.



100

charge q at time t’ dA

dr = c dt’

β(t’ )

radiation emitted at time t’

charge q at time t

R(t’)

n Observer Figure 4.6. Parameters used to calculate the potential of a moving charge.

The integrations have to be performed by taking the charge distribution and the velocities at the retarded time tret = t −

R(tret ) c

at the instant when the radiation was emitted from the moving charge. For a small volume element d3 x = d Adr according to figure 4.6 we have a charge increment dq = ρ d A dr + ρn · v dt d A = ρ(1 + n · β) dr d A

(4.49)

where n denotes the unit vector. The second term of (4.49) originates from the charge that moves into the volume element during the time dr/c. Inserting dr d A from equation (4.49) into the integrals (4.47) and (4.48) we obtain the Liénard– Wiechert potentials: q β A= (4.50) R 1 + n · β ret 1 q . (4.51) φ= R 1 + n · β ret They describe the potentials of a moving charge with all charge parameters taken at the retarded time. The electromagnetic field at the observer can be gained by inserting the potentials (4.50) and (4.51) into the defining equations (4.42) and


Bremsstrahlung

101

(4.43). After some simplification using basic vector relations the fields read: ˙ q n × [(n + β) × β] n+β + (4.52) E(x, t) = −q (1 + n · β)3 γ 2 R 2 ret c (1 + n · β)3 R ret

1 B(x, t) = [E × n]ret . c

(4.53)

The first term of equation (4.52) is independent of the acceleration and describes the near field. For low velocities it approaches the familiar Coulomb field falling off as R −2 . This contribution, which becomes small at high velocities, can also be derived by Lorentz transformation of the static Coulomb field. The second term depending on the acceleration β˙ describes the far field as it falls off much slower than the near-field term. In the following we will only discuss the accelerationdependent far-field term. The energy flux radiated by the moving charge is given by the Poynting vector dt dtret c 2 E [(1 + β · n)n]ret = − 4π ret

Sret = S

(4.54)

taken at the retarded time. For the power radiated into the solid-angle element d towards the observer, i.e. into the direction −n, one can write dP = − n · Sret R 2 d c 2 E (1 + β · n)ret R 2 . = 4π ret

(4.55)

With the far-field term of equation (4.52) this gives ˙ 2 c q 2 {n × [(n + β) × β]} dP = d 4π c2 (1 + n · β)5

(4.56)

where we have dropped the subscript ‘ret’ for the sake of simpler notation. Equation (4.56) gives the angular distribution of the radiated power. 4.2.2 Longitudinal acceleration Now we will investigate the case where the acceleration is parallel to the particle’s velocity, β˙ % β or β˙ × β = 0. Furthermore a coordinate system is assumed where the particle is at the origin at the instant of the emission of radiation and the direction of emission is given by



102

the polar angle ϑ and the azimuthal angle φ. In Cartesian coordinates the unit vector is given as n = (sin ϑ cos φ, sin ϑ sin φ, cos ϑ).

(4.57)

The velocity and the acceleration are assumed to be in the z direction as β = (0, 0, β) ˙ β% = (0, 0, β˙% ). Equation (4.56) can now be written in the form ˙ 2 q 2 [n × (n × β)] dP% = d 4πc (1 + n · β)5 Z 2rc mc2 β˙%2 sin2 ϑ = 4πc (1 − β cos ϑ)5

(4.58)

where we have used the classical charge radius defined as rc =

q2 Z 2 e2 = . mc2 mc2

(4.59)

The numerical values of rc for electrons and protons, respectively, are: rc,e = 2.818 × 10−15 m

rc,p = 1.535 × 10−18 m.

4.2.3 Spatial distribution of bremsstrahlung Equation (4.58) reveals an angular distribution of the radiated power which is symmetric around the electron-beam direction and which vanishes in the axial direction. The sin2 ϑ term dominates at low velocities or in the particle’s rest frame where the emission pattern shows the familiar dipole pattern. Equation (4.58) may also be derived by a Lorentz transformation of the sin2 ϑ distribution into the laboratory system. Figures 4.7 and 4.8 show the respective patterns of the radiated power when viewed from the frame of the moving particle or from the laboratory frame, respectively. For large velocities the radiation is boosted into the forward direction. The maximum of the power is observed at an angle ϑmax given by (1 + 15β 2)1/2 − 1 (4.60) cos ϑmax = 3β or 1 if β → 1. (4.61) ϑmax ≈ 2γ In the case of relativistic particles with a high value for the Lorentz parameter γ the emission is concentrated close to zero degree with ϑmax given by equation (4.61)—compare, also, with figure 2.26 on page 55.


Bremsstrahlung

103

x

β β

y

z

Figure 4.7. Spatial distribution of bremsstrahlung viewed in the frame of the moving particle.

For the thick targets used in x-ray generators we may estimate the emission pattern using equation (4.58) as a starting point. Electrons incident on the cathode in an x-ray tube will be slowed down by successive collisions with the atoms of the cathode material. Each collision results in a deceleration and a chance for emission of radiation. Sommerfeld calculated the total radiated energy W of one particle by making simplifying assumptions. He assumed that a radiation pulse of very short duration is emitted as a result of the electron being uniformly decelerated along a straight line coincident with the projectile’s initial direction. Taking dt = dβ/β˙% as the time increment, the integration over the stopping time



104

x

β β

y

z

Figure 4.8. Spatial distribution of bremsstrahlung viewed in the laboratory frame. The example is for β = 0.5.

can be expressed as

dP% (ϑ, β ) dβ d β˙% 0 β 2 2 Z rc mc β˙% dβ 2 sin ϑ = 5 4πc 0 (1 − β cos ϑ) 2 2 2 Z rc mc β˙% sin ϑ [(1 − β cos ϑ)−4 − 1]. = 16πc cos ϑ

W (ϑ) =

β

(4.62)

The angular distribution of the radiated power from a thick target as given by equation (4.62) is illustrated in figure 4.9. It is again a distribution symmetric


Bremsstrahlung

105

β=0.3 β=0.9 β=0.1 ×

0.01

β

β

Figure 4.9. The theoretical emission pattern of solid-target bremsstrahlung as given by the approximate equation (4.62). The curve for β = 0.9 has been multiplied by 0.01. It clearly reveals the strong forward boost at high particle velocity.

around the particle beam showing the typical forward boost at high projectile velocity. The condition for acceleration along the line of the incident primary particle velocity β is nearly fulfilled for very light target material and for thin targets. For other cases the directional changes due to multiple collisions in the target can be appreciable. As a consequence some intensity will also be emitted along the line of the incident electron beam. 4.2.4 Spectral distribution of bremsstrahlung For a first estimate of the distribution of the radiated power over the radiation frequency, let us keep the simplifying assumption of the previous section. To simplify even further let us for a while assume a non-relativistic particle beam. For small velocities β 1 the power radiated per solid angle is given, compare with equation (4.56), as c 2 2 dP = R E d 4π

with E =

q β˙ sin ϑ. cR

(4.63)

For a very short duration of the deceleration process and hence of the radiation pulse we write

β˙ = δ(t0 − t)β ˙ = β. βdt

(4.64)

The electric field strength in the frequency domain can be formally expressed through its Fourier transform given as ∞ −iωt ˜ dω (4.65) E(ω)e E(t) = π −1/2 0


106

Light and ion sources ˜ E(ω) = π −1/2

∞ −∞

E(t)eiωt dω

(4.66)

where the integration only runs over positive frequencies which are physically meaningful. Equations (4.65) and (4.66) can now be expressed as q sin ϑ βδ(t0 − t) π 1/2 c R q sin ϑ ˜ E(ω) = 1/2 βeiωt0 . π cR E(t) =

This gives for the energy emitted per angular frequency interval c R2 dW 2 ˜ = | E(ω)| dω d dω 4π π q 2 (β)2 2π sin3 ϑ dϑ = 4π 2 c 0 2rc mc2 (β)2 = 3π 2 c

(4.67) (4.68)

(4.69)

which is independent of the frequency. The bremsstrahlung spectrum is a flat distribution extending from zero photon energy up to the limit set by energy conservation, i.e. the point where the whole kinetic energy of the particle is converted into photon energy,

~ωmax = E kin.

(4.70)

Examples of measured intensity spectra are shown in figure 4.10. Here the spectral intensity distribution is plotted given as the cross section multiplied by the photon energy. The results were obtained using thin solid targets for which the probability of multiple photon emission by a single electron is small. 4.2.5 Collisions Whenever an electron is deflected in the Coulomb field of a target nucleus, such as that sketched in figure 4.11, the electron accelerates with the possibility of the emission of radiation. As the trajectory of elastic scattering can be calculated so also can the acceleration along the electron’s path for which the characteristic radiation may be determined from classical electrodynamics. Because photon emission is a quantum process a full quantum-mechanical calculation is desirable. Here we will only give a very simplified and approximate approach whereas the quantum-mechanical description of bremsstrahlung will be resumed in chapter 5. Following equation (4.56) and assuming a reference frame in which the electron is momentarily at rest, the emitted power may be expressed as dP rc mc2 2 2 ˙ sin ϑ. = |β| d 4πc


(4.71)

Bremsstrahlung

107

Figure 4.10. Thin-target bremsstrahlung spectra represented by the cross section (in mbarn ster−1 ) multiplied by the photon energy. The curves are obtained for different target materials. In the case of the silver target characteristic x-rays are superimposed on the bremsstrahlung. From Quarles C A 1997 Accelerator-Based Atomic Physics Techniques c and Applications ed S M Shafroth and J C Austin (New York: AIP) 1997 AIP.

+Ze b -e

θ

Figure 4.11. An electron scattered in the Coulomb field of an atomic nucleus.

Integrating over the full solid angle yields the Larmor formula: P=

2 rc mc2 2 ˙ . |β| 3 c

(4.72)

The equation of motion for an electron in the Coulomb field of a nucleus with


108


charge Z e can be written as Z e2 (4.73) r2 where r denotes the distance between the electron and the nucleus. Inserting equation (4.73) into equation (4.72) yields the radiated power along the electron path as m|β|c =

2 Z 2rc3 mc3 . (4.74) 3 r4 For small deflections the electron–nucleus separation can be approximated by P=

r 2 = s 2 + b2 where b denotes the impact parameter defined in figure 4.11 and s is the path length measured from the point of closest approach of the two particles. Noting the time increment dt = ds/(βc), the total energy radiated in one scattering event is obtained by integrating along the path, 2 Z 2rc3 mc2 ∞ ds W (b) = 2 2 2 3 β −∞ (s + b ) =

π Z 2rc3 mc2 . 3βb 3

(4.75)

The radiation cross section may be given as ∞ W (b)b db. σ Wrad = 2π

(4.76)

bmin

For the minimum impact parameter we take the de Broglie wavelength bmin =

~ mβc

.

Equation (4.76) can now be evaluated to gain σ Wrad = =

2π 2 Z 2rc3 mc2 3β

∞ bmin

db b2

2π 2 Z 2rc3 m 2 c3

. (4.77) 3~ Equation (4.77) can be used to estimate the power density of bremsstrahlung in plasmas. If n e and n i , respectively, denote the number densities of electrons and ions the power density is given by dPbr = σ ve Wrad n e n i dV 2π 2rc3 m 3/2 c3 (kB Te )1/2 ≈ . 31/2 ~


(4.78)

Bremsstrahlung

109

Ee, pe

ϑe P

coincidence

Eo, po T

ϑk

Ek, k

Figure 4.12. The principle of coincidence measurements for investigation of the elementary process of bremsstrahlung.

In equation (4.78) we simply used ve = (3kB Te /m)1/2 as an average collision velocity rather than averaging over a Maxwell velocity distribution. In spite of the simple approach used for deriving equation (4.78), it is already close to the correct quantum-mechanical results. Usually the Gaunt factor is applied to correct the classical result for relativistic and quantum-mechanical effects. Here a factor of about 3/4 is needed to compensate for the deficiency. Inserting numerical values for the constants in (4.78) we arrive at dPbr ≈ 1.7 × 10−32 W cm−3 (kB Te [eV])1/2 n e [cm−3 ]n i [cm−3 ]. dV

(4.79)

Investigating the elementary bremsstrahlung process one considers both the emitted photon and, simultaneously, the decelerated electrons7 . Experimentally this entails coincidence measurements as schematically illustrated in figure 4.12. From such a measurement the following knowledge is gained: • • •

the energy E 0 and momentum p0 of the incoming electron, the energy E e and momentum pe of the scattered electron and the energy ~ω and momentum ~ k of the emitted bremsstrahlung photon.

In addition to the momenta and energies specified it would also be desirable to determine the spin orientation of the electron and the polarization of the photon. With such a set of correlated data on an event-by-event basis stringent tests of theory are possible. Various cuts through the multiparameter space are interesting as, for instance, the angular distribution of the emitted photon intensity for a fixed outgoing electron direction or the energy distributions for fixed photon and electron directions. With the coincidence technique it has also been possible to separate the photons emitted from electron–nucleus and electron–electron scattering. 7 Nakel W 1994 Phys. Rep. 243 317.



110

NNB

QFEB

– hω

e-

P

–hω P

T

T

SEB

AB

T2

h–ω

eP

P T1

Figure 4.13. Schematic classification of collisional processes responsible for continuous x-ray production in ion–atom collisions.

Furthermore, the process of two-photon bremsstrahlung has been investigated by simultaneously measuring the energies of the two correlated photons emitted. For ion–atom collisions matters become more difficult because of the large number of different processes which, in principle, can contribute to the formation of continuous x-ray spectra8,9,10 . A classification of the most important collisional phenomena contributing to bremsstrahlung emission in ion–atom collisions is summarized in figure 4.13. Nucleus–nucleus bremsstrahlung (NNB) can occur when the projectile nucleus is scattered in the Coulomb field of the target nucleus. Compared to electron bremsstrahlung, the intensity is, however, much reduced for massive projectiles because the deceleration is a factor m p /m e less: for protons, for instance, the electric field |E| is about ∼ 10−3 less and the radiated power is down by a factor ∼ 10−6 . Another reduction comes from the fact that the two charges have the same sign rendering the dipole part of the radiation small. The corresponding dipole (E1) part of the cross section scales as Zp Zt 2 − (4.80) σ (E1) ∝ mp mt where the indices p and t represent the projectile and target, respectively. The next higher order, quadrupole radiation, is further suppressed by a factor α 2 , where 8 Schnopper H W et al 1972 Phys. Lett. A 47 61. 9 Jakubassa D H and Kleber M 1975 Z. Phys. A 273 29. 10 Ishii K 1995 Nucl. Instrum. Methods B 99 163.


Synchrotron radiation

111

α ≈ 1/137 is the fine-structure constant. The quadrupole contribution scales like σ (E2) ∝

Zp Zt + 2. 2 mp mt

(4.81)

The endpoint of the NNB spectrum is given by

~ωNNB ≤

m t Ep . mt + mp

(4.82)

Quasi-free electron bremsstrahlung (QFEB) is due to the scattering of a target electron in the Coulomb field of the projectile. It is dominant when the projectile velocity is much higher than the orbital velocity of the electron. In the impulse approximation the electron is treated as a free electron. Secondary-electron bremsstrahlung (SEB) is the process in which a target electron is ionized by a binary collision with the projectile and the released electron subsequently emits a photon upon scattering in the field of a second nearby target nucleus. The maximum photon energy for this process is determined by the maximum kinetic energy of the ionized electron,

~ωSEB ≤ 4

m e Ep . mp

(4.83)

Atomic bremsstrahlung (AB) is emitted when in a projectile–target interaction the target electron is excited to a continuum state and the electron subsequently radiatively decays to a lower state. If the lower state is a continuum state the process is called radiative ionization. Atomic bremsstrahlung dominates in the spectral range ~ωAB ≥ ~ωSEB . (4.84) In figure 4.14 the x-ray spectrum produced in collisions between U90+ ions and Ar atoms is displayed. The advantage of this experiment is the extraordinarily high charge of the projectile and the very low target gas density making secondary bremsstrahlung processes unlikely. The spectrum, therefore, is dominated by quasi-free electron bremsstrahlung. The full curve in figure 4.14 represents a theoretical calculation based on the relativistic Born approximation with Coulomb and Elwert corrections.

4.3 Synchrotron radiation In the context of bremsstrahlung we have analyzed the radiation emitted by accelerated charges restricting ourselves to the case where the acceleration is in line with the velocity vector of the moving charge. In contrast, synchrotron radiation is the electromagnetic radiation that is emitted when the charge is accelerated perpendicularly to its velocity vector. Synchrotron radiation is


112


Figure 4.14. Bremsstrahlung spectrum for U90+ ions colliding with argon atoms at low density. The x-ray continuum is dominated by quasi-free electron bremsstrahlung. From c Ludziejewski T et al 1998 Hyp. Int. 114 165, 1998 Baltzer.

emitted from natural and from laboratory plasmas with a magnetic field. It plays a major role in high-energy circular accelerators where the bending of a beam of charged particles means a radial acceleration associated with photon emission11,12,13 . 11 Jackson J D 1999 Classical Electrodynamics 3rd edn (New York: Wiley). 12 Turner S (ed) 1990 Synchrotron Radiation and Free-Electron Lasers CERN Accelerator School

CERN 90-03 (Geneva: CERN). 13 Wiedemann H 1995 Particle Accelerator Physics II (Berlin: Springer).



113

4.3.1 Angular distribution of the radiated power We have already calculated the general expression (4.56) for the angular distribution of the power emitted from accelerated particles. In the case of perpendicular acceleration, β˙ ⊥ β

˙ =0 or ββ

we can calculate the vector products of equation (4.56) using the same coordinate system as in section 4.2.2. The vectors β and β˙ are then represented by β˙⊥ = (β˙⊥ , 0, 0).

β = (0, 0, β)

Now the radiated power per solid-angle element reads: ˙ − β(1 ˙ + n · β)]2 q 2 [(n + β)(n · β) dP⊥ = d 4πc (1 + n · β)5 2 Z 2rc mc2 β˙⊥ (1 − β cos ϑ)2 − (1 − β 2 ) sin2 ϑ cos2 ϕ = . 4πc (1 − β cos ϑ)5

(4.85)

The maximum of such a distribution is located at an angle ϑmax = arccos

5β 2 − 2 3β

≈ ( 73 )1/2 γ −1

β → 1.

(4.86)

In figure 4.15 the distribution given by equation (4.85) is plotted in real space where we have used the scaled angle γ ϑ to obtain the x and y coordinates. The x coordinate is assumed to lie in the plane of the orbiting particle. For highly relativistic particles the synchrotron radiation is collimated in a narrow cone in the forward direction. The power distribution can then be approximated by 2 2rc mc2 β˙⊥ dP⊥ 1 + 2γ 2 ϑ 2 (1 − 2 cos2 ϕ) + γ 4 ϑ 4 = γ6 . d πc (1 + γ 2 ϑ 2 )5

(4.87)

The intensity is confined to small opening angles characterized by the root-meansquare of the emission angle ϑ, 2 1/2

ϑ

=

P⊥−1

≈ γ −1

dP⊥ 2 ϑ sin ϑdϑdϕ d

for β → 1.

1/2

(4.88)

Synchrotron radiation is emitted tangentially from a circulating particle as sketched in figure 4.16. The space inside the accelerator circle is completely dark whereas outside the ring radiation is detectable only from an observing



114

γϑ

-1 x (o

rbita

l pla 0 ne) =γ

ϑ co

sϕ

1

y ( -1 pe rp en dic ula r

0 pla ne )=

γϑ

1 sin ϕ

r rad. powe

ϕ

Figure 4.15. Emission characteristic of synchrotron radiation according to equation (4.85).

Figure 4.16. Tangential emission characteristic of synchrotron radiation in a circular accelerator.

position very close to the plane of the circulating particle. Because of the forward collimation, light is received only from a relatively small spot along the particle trajectory for a given detector position. At existing synchrotron-radiation laboratories, this is taken into account in the beam-line design which has to follow the tangential directions indicated by the arrows in figure 4.16. The total radiated power P⊥ can be obtained by integration of equation (4.85) over the whole solid angle. For comparison we will do the same for the case of longitudinal acceleration as given by equation (4.58). The results are: 2rc mc2 4 2 γ β˙⊥ 3c 2rc mc2 6 2 γ β˙% . P% = 3c

P⊥ =


(4.89) (4.90)


115

If we express β˙⊥ and β˙% in terms of the time derivatives of the respective momenta, equations (4.89) and (4.90) can be written as P⊥ = and P% =

2rc 2 2 γ p˙ ⊥ 3mc

(4.91)

2rc 2 p˙ . 3mc %

(4.92)

In deriving these equations we have used the identities p˙ ⊥ = mcβ˙⊥ γ and p˙% = mcβ˙% γ 3 , respectively. As is obvious from equations (4.91) and (4.92), the same accelerating force, i.e. the same time derivative of the momentum, generates a factor γ 2 higher radiation power for the transverse acceleration compared to the longitudinal acceleration. This has important practical consequences for the radiation losses in accelerators. Commonly they are negligible in linear accelerators but they can be substantial in circular machines. Assuming a circular accelerator with a given bending radius ρ and a magnetic inductance B one can use β˙⊥ = β 2 c/ρ and Bρ = mcβγ /q to obtain the total radiation power as P⊥ =

2 B 4ρ 2 q 6 . 3 c3 m 4

(4.93)

Because of the q 6 /m 4 scaling the synchrotron power is much reduced for heavy particles. For comparison the numerical values for protons and for fully ionized uranium are related to those for electrons in the following way: 1 P⊥ (H+ ) ≈ ≈ 8.8 × 10−14 P⊥ (e− ) 18364 926 P⊥ (U92+ ) ≈ ≈ 2.0 × 10−15. P⊥ (e− ) 183642384 It is also interesting to consider the energy loss per turn in a circular accelerator by integrating the power (4.89) over the revolution period yielding ds W0 = P⊥ βc 4π β 3γ 4 rc mc2 (4.94) = 3 ρ where ρ denotes the radius of the bending magnets. In table 4.2 some basic parameters for high-energy accelerators specifying the calculated energy loss per turn and the total radiated power which is the number of particles N circulating in the ring times the power per particle P⊥ are listed. For the very high γ parameters involved the energy loss can only be kept at a tolerable level by building huge accelerators such as the Large Electron


116


Table 4.2. Examples of energy loss and synchrotron-radiation power for high-energy accelerators. ESRF, European Synchrotron Radiation Facility (Grenoble); LEP, Large Electron Positron ring (CERN); LHC, Large Hadron Collider (CERN); RHIC, Relativistic Heavy Ion Collider (Brookhaven). Accelerator

Particle

γ

ρ(m)

W0 (MeV)

N(1012 )

N P⊥ (MW)

ESRF LEP LHC RHIC

e− e± H+ Au79+

12 000 200 000 7 500 110

130 4200 4200 600

0.9 3100 0.005 8 × 10−6

3.5 2 300 0.06

0.19 15 0.005 6 × 10−9

A

Det.

B 1/γ

1/γ

ρ

Figure 4.17. Geometrical constraints for the detection of synchrotron radiation at a circular accelerator. Light is only received from a small section of the particle trajectory between points A and B.

Positron ring (LEP) at CERN with a circumference of 26.7 km. The numbers for the energy loss and the radiated power are only approximate estimates because it was assumed that each accelerator is a perfect circle. In practice, the lattice has several straight sections without any beam deflection. The total radiated power ranges from an impressive 15 MW for the LEP down to 6 mW for the Relativistic Heavy Ion Collider at Brookhaven operated with bare Au79+ ions.

4.3.2 Spectral distribution of synchrotron radiation Returning to the emission characteristic sketched in figure 4.16 we recall that only a detector outside of the ring and in the plane of the orbiting particles will be capable of receiving any synchrotron radiation. For a given detector position only light that is emitted from a small arc will be detected as explained in figure 4.17. The temporal width of the corresponding light flash is the difference in the travel


Synchrotron radiation times of the particle and radiation which may be estimated as ρ 2ρ 1 −1 ≈ 3 . t = tp − tγ ≈ γc β γ c

117

(4.95)

The time t corresponds to a typical angular frequency ω = 1/t ≈

γ 3c ρ

(4.96)

at which the radiation spectrum will start to decrease. Because of the periodic motion of the particle the frequency spectrum will consist of harmonics of the revolution frequency. However, natural broadening and the small spacing given by the revolution frequency will result in a quasi-continuous spectrum extending from the visible to the gamma-ray region. The time-varying electric field at the observation point will determine the total radiation energy for the passage of one particle which is obtained starting from equation (4.55) as follows. ∞ ∞ dW dP =− dt = − Sret nR 2 dt. (4.97) d −∞ d −∞ Using explicit Fourier transform techniques in analogy to section 4.2.4 the observed energy per unit solid angle and per unit frequency interval is obtained as 3rc mc2 ω2 2 γ 2ϑ 2 d2 W 2 2 2 2 2 = γ (1 + γ ϑ ) K 2 (ξ ) + K 1 (ξ ) dωd 4π 2 c ωc2 1 + γ 2ϑ 2 3 3 ω (1 + γ 2 ϑ 2 )3/2 (4.98) ξ= 2ωc 2γ 3 c . ωc = 3ρ The functions K ν are modified Bessel functions of the second kind. The first term in the second bracket of equation (4.98) accounts for the contribution from σ polarization with the electric-field vector in the plane of the orbiting particle and perpendicular to the deflecting magnetic field. The second term is due to the π polarization for which the electric-field vector is perpendicular to that of the σ polarization and perpendicular to the propagation direction of the radiation. In figure 4.18 the radiated energy according to equation (4.98) is plotted as a function of γ ϑ for two different values of the normalized radiation frequency ω/ωc . With increasing radiation energy the forward emission cone shrinks substantially. Integrating equation (4.98) over all frequencies and dividing by the revolution period 2πρ/(βc) yields the average radiation power: 5 γ 2ϑ 2 7rc mc2 c γ5 dP 1+ . (4.99) = d 32π ρ 2 (1 + γ 2 ϑ 2 )5/2 7 1 + γ 2ϑ 2



118

d2 W/dωd (normalized)

1 0.8 0.6

ω/ωc =

2

0.5

0.4 0.2

0

0.5

1

γϑ

1.5

2

2.5

Figure 4.18. Angular distribution of the synchrotron-radiation energy at two different values of the light frequency ω.

dP /d (normalized)

1 0.8 0.6

0.4 0.2

total

⊥ 0

0.5

1

γϑ

1.5

2

Figure 4.19. Angular distribution of the frequency-integrated radiation power showing the contributions of the two polarizations ⊥ and %.

In figure 4.19 the radiation power is plotted as a function of γ ϑ showing the two polarization contributions and their sum. When the curves are integrated over the angle one finds an intensity ratio of Pσ : Pπ = 1 : 7 The spectral distribution averaged over all angles is obtained from equation



119

0.8

F(ω/ωc )

0.7 0.6 0.5 0.4

S(ω/ωc )

0.3 0.2 0.1 0.00

0.25

0.50

0.75

1.00

ω/ωc

1.25

1.50

1.75

2.00

Figure 4.20. The universal functions S and F as defined in equations (4.101) and (4.102). They characterize the angle-integrated intensity and photon-flux spectrum, respectively.

(4.98) via

2 βc dP d W = d dω 2πρ dω P⊥ ω = S ωc ωc

where S( ωωc ) is a universal function evaluated as follows. 9 31/2 ω ∞ ω ω ω S = d . K5 ωc 8π ωc ω/ωc 2 ωc ωc

(4.100)

(4.101)

The function Srad (ω/ωc ) is plotted in figure 4.20. The area under the curve is unity and it is divided into equal parts at the frequency ω = ωc . From equation (4.100) the number of photons emitted per unit frequency and time is P⊥ ωc d2 N ω = S d~ω dt ωc (~ω)2 ω P⊥ ω (4.102) F = ωc (~ω)2 where F( ωωc ) is a universal function (also shown in figure 4.20) that characterizes the photon flux as a function of the frequency. The total photon rate integrated over the whole spectrum is given by ∞ 2 15 31/2 P⊥ d N d~ω = . (4.103) N˙ γ = d~ω dt 8 ~ωc 0



120

electron beam

radiation fan bending magnet

electron beam

Figure 4.21. Synchrotron radiation emitted from a bending magnet.

Multiplying by the revolution period and using the corresponding expressions for P⊥ and for ωc , the total number of photons emitted by one particle over one revolution is Nγ =

2πρ ˙ 5π Nγ = 1/2 αγ ≈ 0.066γ c 3

(4.104)

where α = e2 /(~c) is the fine-structure constant. The total number of photons depends only on the Lorentz factor γ . 4.3.3 Insertion devices In the previous sections we have worked out the characteristics of the synchrotron radiation that occurs whenever a beam of electrons is deflected by a dipole bending magnet as schematically illustrated in figure 4.21. It is possible to significantly increase the radiation power by installing linear arrays of short dipole magnets with alternating polarities such as that sketched in figure 4.22. The multiple changes of particle direction in such wigglers cause a very bright continuous emission of short-wavelength radiation. Undulators are technically the same devices as wigglers but their magnetic field strength is lower and their magnets are spaced closer together. The lower magnetic field narrows the cone of radiation allowing radiation originating from each individual magnet of the array to interfere constructively. From the interference patterns brilliant quasimonochromatic radiation is produced. The behavior of wigglers is similar to that of bending magnets and the resulting synchrotron radiation can be calculated with the formulae given in the previous sections. This is so because the radiative output is given by the incoherent superposition of radiation from each excursion of the particle beam. For undulators coherence effects that give rise to line radiation have to be considered.



121

Figure 4.22. Synchrotron radiation emitted from an undulator.

We now assume a periodic vertical undulator field given as 2π z B y = B0 sin λu

(4.105)

with λu denoting the magnet period. Under certain conditions the particles in the beam will conduct a harmonic oscillation with the horizontal elongation 2π λu sin z (4.106) x=K 2πγ λu where K is the undulator field parameter: K =

ceB0λu . 2πmc2

(4.107)

When K 1 the assumption of simple harmonic motion is justified with the transverse acceleration x˙ being parallel to the x direction. The Doppler-shifted wavelength observed in the laboratory can be shown to be K2 λu 1+ (4.108) + γ 2ϑ 2 λ1 = 2γ 2 2 where it is again assumed that the particle is moving close to the speed of light. For K > 1 the motion is no longer harmonic and the electron describes a lobe in the frame of the fast particle beam. With increasing K the longitudinal components of this motion become bigger and harmonics of the fundamental frequency ω1 = 2π/λ1 are generated at wavelengths K2 λu 2 2 1+ n = 1, 2, 3, . . . . (4.109) +γ ϑ λn = 2 2nγ 2



23

10

FEL's 20

10

Undulators 17

10

Wigglers

14

10

Bending magnets

-1

-2

Brightness (photons s mm mrad

-2

(0.1 % bandwidth)-1)

122

11

10

8

10

X-ray tubes

Rotating-anode X-ray tubes

5

10

1900 1920 1940 1960 1980 2000 2020 Year

Figure 4.23. The brightness of x-ray sources as a function of time.

The harmonics appear because of the time difference between the particles and the photons that occurs when traversing one period of the undulator. From the finite number Nu of undulator periods arises a natural line width λn that is determined by 1 λn = . λn n Nu

(4.110)

The success of the insertion devices, wigglers and undulators, is strongly related to the race for increased output power of modern synchrotron light sources. Figure 4.23 shows the brilliance of x-ray sources as a function of time since the discovery of x-rays. The steep rise in recent decades is mainly due to the use of insertion devices in high-energy storage rings. Whereas the first-generation synchrotron light sources were built for other purposes, mainly nuclear and particle physics, the second-generation light sources were dedicated machines


Ion accelerators

123

Table 4.3. Third-generation synchrotron light sources and some of their operating parameters. Name and location of institute

ESRF Grenoble France

APS Argonne USA

SPring-8 Harima Japan

Beam energy Beam current Circumference Beamlines Operational since

6 GeV 200 mA 844 m 56 1994

7 GeV 100 mA 1 104 m 68 1996

8 GeV 100 mA 1 436 m 61 1997

relying on bending magnets. The third-generation synchrotron light sources were built around insertion devices, i.e. they provide several straight sections for the insertion of undulators and the machine is optimized for this type of operation. Table 4.3 lists some of the parameters for third-generation synchrotron light sources.

4.4 Ion accelerators 4.4.1 General remarks Historically the motivation for the development of accelerators was the need to extend the energy range and versatility of particle beams beyond the capabilities provided by radioactive sources. From the very beginning nuclear and high-energy physics were the driving force behind the development. For instance, the pioneering work of Ernest Rutherford on atomic scattering (1906) and nuclear reactions (1919) were based on experiments with natural αparticles but it was 1932 when Cockroft and Walton observed the first mancontrolled nuclear reaction. The development is also closely linked to the progress made in other fields like high-vacuum, radiofrequency, microwave and low-temperature techniques which have all found their way into accelerator technology. Accelerated particles directed onto a target are often used for the generation of secondary radiation. The x-ray radiation discovered in 1895 by Wilhelm Conrad Röntgen is an early and prominent example of the production of secondary radiation. Today there are accelerators dedicated to the fabrication of secondary particles like neutrons, mesons, neutrinos or exotic heavy nuclei. Synchrotron radiation from light particles circulating in an accelerator has already been discussed in the previous section. It is a wide and still expanding field with many applications. The beginning of atomic physics with particle beams may be dated back to the famous experiments in 1913 by James Franck and Gustav Hertz who studied the excitation of atoms by electron impact. Atomic collision


124


and spectroscopy experiments at accelerators have evolved into a powerful technique since with modern accelerator facilities it is now possible to produce any ionization stage of virtually any element of the Periodic Table. Here some of the basic ideas and principles14,15,16 behind the major developments are summarized and some of the technical solutions are introduced and discussed mainly in the context of heavy charged particles. 4.4.2 Acceleration of charged particles All accelerators have a few features in common: they consist of a chargedparticle source, a vacuum tube in which to accelerate them and an electric field source providing the accelerating force. Electrons can be set free by thermionic emission from heated filaments. The basic principles of ion production by means of collisions, for instance in arc discharges, have already been discussed in section 4.1. The region in which the particles are accelerated must be kept at high vacuum to prevent them from being scattered out of the beam and getting lost through collisions with gas atoms or molecules. The various types of accelerators are distinguished by the configuration of the electric fields. In a linear accelerator the path of the particles is a straight line and the final kinetic energy of the particles is proportional to the voltage integrated along that line. In a cyclic accelerator the trajectory of the particles is bent by an appropriate magnetic field to form a spiral or a closed curve that is approximately circular. In this case one makes use of the fact that the particles repeatedly traverse the same accelerating device. If the electric potentials in a circular accelerator were static the particle would return to the same potential after one revolution without gaining any kinetic energy. That is why all cyclic accelerators use time-varying acceleration fields. For high-energy particles the total distance traveled in a cyclic accelerator may be huge eventually reaching millions of kilometres. Tiny deviations from the desired trajectory occurring each turn could cumulate and dissipate the beam. Therefore it is necessary to focus continually the beam by precisely shaped magnetic fields. 4.4.3 Acceleration mechanisms All acceleration and beam-guidance design can be based on the (invariant) Lorentz force F = q E + q(v × B)

(4.111)

where the first part gives rise to an increase in the kinetic energy and the second part to a deflection of the beam. In the absence of a magnetic field or if 14 Reiser M 1994 Theory and Design of Charged Particle Beams (New York: Wiley). 15 Wiedemann H 1993 Particle Accelerator Physics (Berlin: Springer). 16 A rich source of authentic material are also the proceedings of numerous CERN Accelerator Schools.


Ion accelerators

125

Es

a)

+ –

b)

U=

B(t) Es

Es(t) B(t)

s c)

Figure 4.24. Basic mechanisms for the acceleration of charged particles in electromagnetic fields: (a) an electrostatic field, (b) an induced electric field by a time-varying magnetic field as in a transformer, and (c) time-varying electric and magnetic fields in a cavity.

it is constant over time there might still be a static electric field E indicated in figure 4.24(a). This is the basis for electrostatic accelerators. There is, however, the additional possibility of exploiting time-varying fields as manifested in Faraday’s law: ∇×E=−

∂ B. ∂t

(4.112)

All modern high-energy accelerators function by this principle. Figures 4.24(b) and (c) show two different topologies making use of varying electromagnetic fields. Figure 4.24(b) may be viewed as a type of transformer where the particle beam effectively forms a single secondary coil. As the magnetic flux through the core of the magnet increases an azimuthal electric field is induced that drives the charged particles to higher energies. In the configuration of figure 4.24(c) a time-varying azimuthal magnetic field is considered which might be established in an RF cavity. The periodic change of this magnetic field induces an electric field on the axis of the cavity. It is the business of a proper equipment design to synchronize this rf field with the passage of the beam pulse.



126

+

+

+ + + + + + + + + + high-voltage + dome + + + + + + + + + + –

+ + + + + + +

+ +

ion source

belt

+ + + + + + +

acceleration tube

ion beam Figure 4.25. The principle of the Van-de-Graaff accelerator. A fast moving belt is continually transporting electric charge towards the terminal maintaining a high voltage. Charged particles are accelerated from an ion source with a high potential down through an acceleration tube to ground potential.

Direct voltage accelerators The simplest type of accelerator consists of a particle source mounted on one end of an insulated and evacuated tube plus a high-voltage generator connected to both ends of the tube such that the charged particles are accelerated from the source towards the other end of the tube. Such a linear accelerator usually uses the applied voltage only once. Despite the simple principle the task becomes technically demanding when voltages in excess of 1 MV are to be applied. The first successful experiments with artificially accelerated particles were conducted by J D Cockroft and E T S Walton in 1932 when they split the Li nucleus by the impact of protons with a kinetic energy of 710 keV. They used a four-stage high-voltage cascade generator made of large rectifiers and high-voltage capacitors. Such Cockroft– Walton generators are still in use at injectors for larger accelerators or as small industrial accelerators used, for instance, for ion implantation. Another development from that period is the belt-charged high-voltage generator invented by R J Van de Graaff in 1931. As shown in figure 4.25 electric


Ion accelerators

127

charge is extracted by field emission from a biased comb of sharp needles and sprayed onto an isolated endless belt via a corona discharge. The motor-driven belt transports the electric charge inside a spherical dome where it is stripped off by a second set of needles connected to the sphere. An ion source within the terminal produces positive ions which are accelerated as they move down through the accelerating column of electrodes. A carefully designed Van-deGraaff accelerator enclosed in a vessel pressurized with an insulating gas can be reliably operated at potentials exceeding 20 MV. Most of the proton and heavy-ion Van-de-Graaff accelerators still in use are two-stage tandem accelerators. They provide more than twice the energy that could be achieved by applying the high voltage only once. In the tandem accelerator negative ions are accelerated towards the positive terminal voltage where some of their electrons are stripped off by passage through a thin foil. The positive ions then traverse the same potential once again in the second part of the acceleration tube. Radiofrequency accelerators Direct voltage accelerators are limited by the maximum voltage that is technically feasible. In radiofrequency accelerators one makes use of a resonant acceleration by alternating fields as suggested in figure 4.24. This can be achieved in various topologies and configurations in which the bunches of charged particles and the accelerating electric fields have to be matched in their phases. Circular machines The betatron is an accelerator only useful for electrons—hence the name. It is based on the transformer principle. As the electrons are accelerated on their circular orbit the guiding field in the vicinity of the electrons has to be increased. The condition for a constant radius R during acceleration, known as the Wideröe condition, is ¯ B(R) = 12 B(R) (4.113) requiring that the magnetic flux density near the orbit be one-half the average flux density through the orbit. In the betatron the vacuum chamber can be made in the shape of a torus so as to accommodate the electron beam. The focusing and synchronization of the beam energy with the magnetic field are accomplished by the geometry of the main magnet. The poles of the magnet are shaped so as to cause the magnetic field to slightly decrease with radius near the orbit. This has a focusing effect on the circulating particles because any particle that deviates from the main path will experience a restoring force towards the proper path. The corresponding transverse oscillations are known as betatron oscillations because they were first analyzed for the betatron; however, they are of fundamental importance for all cyclic accelerators. The acceleration in the betatron lasts for one rising quarter of the sinusoidally varying magnetic field. A bunch of electrons


128


ion source

RF input

vacuum chamber

D’s

ion beam

Figure 4.26. Outline of the classic cyclotron. The ions are accelerated by an appropriate rf field in the gap between the two D-shaped electrodes. A constant guiding magnetic field is perpendicular to the drawing plane.

is injected just when the field passes through zero and is deflected out of the orbit when the field has reached its maximum. The cyclotron first built by Lorentz and Livingston 1932 borrows its key features from the fact that the orbits of ions in a uniform magnetic field are isochronal. Actually this is only fulfilled with non-relativistic particles as can be seen if we consider the cyclotron frequency (4.33) including the relativistic increase of the particle mass, qB ωc = . (4.114) γm If the kinetic energy is small compared to the rest mass the Lorentz parameter is γ ≈ 1 and the frequency is not dependent on the velocity. The isochronism makes it possible to accelerate a particle many times by applying an rf voltage with the fixed frequency ωc . As shown in figure 4.26 an ion source is centered in a vacuum chamber that has the shape of a short cylinder situated between the poles of a large magnet producing a homogeneous magnetic field parallel to the cylinder axis. The cylinder is split into two D-shaped halves between which the accelerating voltage is applied. Within the Ds there is no rf field hence the trajectories are semi-circles inside the Ds. When the frequency of the voltage equals the revolution frequency of the particles they are accelerated whenever they cross the gap as the voltage changes sign after one-half revolution. When the relativistic increase of the mass becomes too large the orbital frequency starts to decrease and the particles get out of phase. This is why the cyclotron is best suited for ions and not for electrons. In the classical cyclotron protons were accelerated to energies above 20 MeV having currents of 5 mA. These proton


Ion accelerators

~

RF

++ ion source

–

+

–

+ drift tube

129

– ion beam

Figure 4.27. Schematic diagram of the Wideröe rf linac. Charged particles are accelerated in the gaps between drift tubes which serve to shield the ions during the phase when the field is reversed.

beams were used for the synthesis of radioisotopes. In synchrocyclotrons or phasotrons the frequency of the accelerating voltage changes as the ions are accelerated thus higher energies became manageable. Another variant is the sector-focused cyclotron employing a constant acceleration field but a guiding field that increases with orbit radius. Limitations in energy then mainly arise from the fabrication costs of the huge magnets used in sector cyclotrons. The rf linac The operating principle of the rf linear accelerator or linac, as it is known, is based on resonance acceleration between drift tubes as illustrated in figure 4.27. The principle had been proposed in 1924 by Ising, and Wideröe demonstrated the principle in 1928 by building a 25 kV oscillator at 1 MHz. Although the principle is simple, special caution is required to expose the particles only to the accelerating positive half waves of the accelerating field. The decelerating effect of the electric field during the intervals when it opposes the motion of the particles is prevented by hiding the ions inside the electrically conducting drift tubes where the electric field is zero. If their length is properly chosen the particles cross the gap between adjacent drift tubes when the field produces an acceleration. The lengths of the drift tubes are proportional to the speed of the particles that pass through them. At high frequency the Wideröe structure becomes inefficient due to the dissipation of electromagnetic energy. To solve this problem L Alvarez proposed enclosing the structure in a common tank to form a resonant cavity. This Alvarez structure was very successfully applied to the acceleration of protons and heavy ions and is still in use as an injector for booster synchrotrons and other highenergy accelerators. The operating frequency of proton linacs is typically around 200 MHz. Because of the necessary high oscillator power the rf linacs are operated in a pulsed mode thus lowering the average power requirements. In the Alvarez linac the average voltage gradients along the beam direction are typically in the range of 1–2.5 MV m−1 whereas the peak values within the accelerating gaps may be as high as 10 MV m−1 . In these structures focusing is also needed because of the presence of slight defocusing radial field components in the gaps and because of the repulsive space-charge forces acting among ions


130


Figure 4.28. View inside an Alvarez structure as part of the UNILAC at GSI Darmstadt. Photograph courtesy A Zschau.

in a beam bunch. Based on the strong-focusing principle quadrupole lenses are usually employed in the cylindrical drift tubes of the linac. Such a configuration requires a minimum injection velocity equivalent to β = v/c > 0.04 for effective operation. A view inside an Alvarez structure is shown in figure 4.28. The example shown is part of the heavy-ion accelerator facility17 at the GSI Darmstadt. Phase stability One might expect that small deviations from the magnitude of the accelerating voltages in rf accelerators would cause the particles to lose the synchronism with the fields necessary for the device to operate properly. It has been proved, however, that rather intensive beams of ions can be accelerated in a stable fashion even if they cross the accelerating gaps not exactly at the intended times. Although the principle of phase stability can also be applied to circular accelerators we will discuss it here for a linac. It turns out that stable operation can be achieved only when the phase of the rf field is adjusted so as to use the rising field of the sine wave. Then, if an ion arrives early at the gap it will experience a smaller acceleration than the average particle and if it arrives late it will receive a larger kick allowing it to catch up with the majority of the ions in the bunch. As a 17 Angert N 1998 Proc. of the 6th Europ. Part. Acc. Conf., Stockholm, p 125.


Ion accelerators

131

result stable longitudinal oscillations occur; these are also known as synchrotron oscillations because they were first studied in detail for synchrotrons. Let us assume a cavity wave with a longitudinal electric field on an axis that has the form E az = E 0 cos ϕ

(4.115)

where ϕ represents the phase of the particle with respect to the maximum of the electric field. We further assume that the velocity of the ion is an increasing function of the distance s along the accelerator and the phase velocity of the wave increases with distance so that the particle in the center of the bunch is always synchronized with the wave. The synchronous particle then has a velocity equal to the phase velocity of the wave and its phase ϕ0 stays constant. The equation of motion for each particle can be written as mc

d(βγ ) = q E 0 cos ϕ + q E sz dt

(4.116)

where q E sz refers to the space-charge force originating from the Coulomb repulsion between the ions. With E = E − E c we denote the difference in kinetic energy between a non-synchronous and synchronous particle. A solution neglecting the space charge of the differential equation (4.116) can be readily worked out:

E E rf

2 = ϕ cos ϕ0 − sin ϕ + C

(4.117)

with E rf2 = β 3 γ 3 mc2 q E 0 λrf where λrf = 2πc/ωrf denotes the rf wavelength and C is an integration constant depending on the initial conditions E i , ϕi given by C = sin ϕi − ϕi cos ϕ0 + π

E i E rf

2 .

(4.118)

For each value of C equation (4.117) gives a possible trajectory in the phase plane E–ϕ. Some of these trajectories are plotted in figure 4.29. Choosing ϕ0 < 0 results in a stable particle motion provided the initial conditions are within the separatrix. Particles inside the separatrix move on closed curves in a counterclockwise direction. Particles with their initial energy and phase outside the separatrix cannot be trapped and accelerated by the rf wave. The separatrix, also referred to as the rf bucket, separates the stable from the unstable trajectories. Figure 4.29



132

2 ϕo

ϕo

E az

– ϕo

0

E / Erf 0.5

- 1.5

-1

- 0.5

0.5

1

1.5

ϕ

- 0.5 Figure 4.29. Particle trajectories in the E–ϕ plane according to equation (4.117). The example is plotted for a synchronous phase ϕ0 = −π/6. The broken curve represents the separatrix which separates stable from unstable trajectories.

was plotted for an asynchronous phase ϕ0 = −π/6 suitable for acceleration in a linac. At the expense of acceleration, the size of the rf bucket can be increased by decreasing the asynchronous phase down to ϕ0 = −π/2 for capturing particles from an injector that is not phase matched. Typically the asynchronous phase starts at a low value at injection and is gradually increased along the accelerator. 4.4.4 Focusing mechanisms Transverse stability For stable operation at high energies and high intensities it is necessary to have methods for both longitudinal and transverse focusing. Subsumed under accelerating methods, longitudinal focusing and bunching abilities have been shown to be closely related to the rf phase of the accelerating field. As for the transverse motion of the particles, it has been empirically found at cyclotrons that the guiding field has to decrease slightly with radius in order not to lose the beam. The requirement for a gradient in the guiding magnetic field in the radial direction can be deduced theoretically. This is based on the existence of restoring forces in the horizontal (x) and vertical (y) directions when the field index lies within the limits ρ ∂ By 0

Introduction to the Physics of Highly Charged Ions

Introduction to physics of highly charged ions

Highly Charged

Highly Charged

Physics of Atoms and Ions

Physics of Atoms and Ions

Physics of Atoms and Ions

Introduction to Mathematical Physics

Introduction to mesoscopic physics

Introduction to Mathematical Physics

Introduction to Mathematical Physics

Introduction to statistical physics

Introduction to statistical physics

INTRODUCTION TO THEORETICAL PHYSICS

Introduction to health physics

Introduction To Chemical Physics

Introduction to Nonlinear Physics

Introduction to Mathematical Physics

Introduction to statistical physics

Introduction to Nonlinear Physics

Introduction to the Physics of Landslides

Introduction to mathematical physics

Introduction to Statistical Physics

Introduction to Statistical Physics

Introduction to plasma physics

Introduction to Mathematical Physics

Introduction to Energy Physics

Introduction to Particle Physics

Introduction to theoretical physics

Introduction to Statistical Physics

Electro-diffusion of ions

Introduction to the Physics of Highly Charged Ions

Introduction to physics of highly charged ions

Highly Charged

Highly Charged

Physics of Atoms and Ions

Physics of Atoms and Ions

Physics of Atoms and Ions

Introduction to Mathematical Physics

Introduction to mesoscopic physics

Introduction to Mathematical Physics

Introduction to Mathematical Physics

Introduction to statistical physics

Introduction to statistical physics

INTRODUCTION TO THEORETICAL PHYSICS

Introduction to health physics

Introduction To Chemical Physics

Introduction to Nonlinear Physics

Introduction to Mathematical Physics

Introduction to statistical physics

Introduction to Nonlinear Physics

Introduction to the Physics of Landslides

Introduction to mathematical physics

Introduction to Statistical Physics

Introduction to Statistical Physics

Introduction to plasma physics

Introduction to Mathematical Physics

Introduction to Energy Physics

Introduction to Particle Physics

Introduction to theoretical physics

Introduction to Statistical Physics

Electro-diffusion of ions

Recommend Documents