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

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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: Fr´ed´erique 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

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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

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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 M¨ossbauer 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

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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

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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

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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 R¨ontgen in 1895, the ﬁrst spectroscopic measurements of highly charged ions made by I S Bowen, R A Millikan and B Edl´en 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

© IOP Publishing Ltd 2003

material from their original research work to include in this book and for their valuable remarks. We are particularly indebted to A M¨uller 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

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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

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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

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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

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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

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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 Edl´en2 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 Edl´en B 1942 Z. Astrophys. 22 30S.

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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.

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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).

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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,

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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).

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Introduction

6

<70–80 km ≈85–200 km >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).

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Ions in nature

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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

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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).

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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

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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.

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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 Tr¨umper J 1993 Science 260 1769.

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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.

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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

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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

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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).

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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.

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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.

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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.

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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

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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.

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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

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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).

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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.)

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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

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The electromagnetic spectrum

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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.

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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

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The electromagnetic spectrum

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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

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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

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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.

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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.

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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

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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).

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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

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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

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(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

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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.

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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,

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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).

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Diffraction and interference

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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

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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.

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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

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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

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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 Schr¨odinger 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.

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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, G¨ahler 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, H¨ansch Th and Esslinger T 1999 Atom laser with a cw output coupler Phys. Rev. Lett. 82 3009. Bord´e Ch et al 1994 Molecular interferometry experiments Phys. Lett. A 188 187.

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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

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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

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The Doppler effect

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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.

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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

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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).

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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.

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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

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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

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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.

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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

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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 Alv¨ager et al 1966 Arkiv Fysik 31 145.

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β 1 + β2 1 + β1 β 2

(2.44)

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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.

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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

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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

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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.)

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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.

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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 Schr¨odinger (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 Schr¨odinger 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 Schr¨odinger 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

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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.

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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

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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

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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

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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

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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

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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.

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(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.

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(3.12)

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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 M¨ossbauer spectroscopy The M¨ossbauer effect, after R L M¨ossbauer (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 M¨ossbauer 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
v

M Figure 3.14. The elementary absorption and emission processes involving recoil of the emitting atom.

The kinematics of absorption and emission processes are illustrated in ﬁgure 3.14. Let us assume a free atom with a rest mass M0 which initially is at rest and which has a ground and excited state separated by an energy Q 0 . A photon of energy Q is completely absorbed by the atom which will receive a momentum kick M v . Conservation of energy and momentum requires

Using yields

E = M0 c2 + Q = M c2 Q = M v p= c M0 c2 = M0 c2 + Q 0 .

(3.13)

E 2 = (M0 c2 )2 + (cp)2 = (M0 c2 )2 + Q 2

(3.14)

(M0 c2 )2 = (M0 c2 + Q)2 − Q 2 .

(3.15)

3 Klein O and Nishina Y 1929 Z. Phys. 52 853.

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M¨ossbauer spectroscopy

71

Combining equation (3.15) with the last one of equations (3.13) gives the initial photon energy Q0 Q = Q0 1 + (3.16) 2M0 c2 which must be slightly greater than the energy Q 0 of the atomic transition in order to supply the kinetic energy Q 20 /(2M0 c2 ) to the atom. The corresponding velocity can be obtained from equations (3.13) to be β =

Q v Q = 2 = . c Mc M0 c2 + Q

(3.17)

A set of equations similar to (3.13)–(3.15) can be written down for the emission process. Instead of equation (3.16) the energy of the emitted photon is Q0 Q = Q0 1 − . (3.18) 2M0 c2 The energy of the emitted photon is smaller than the transition energy Q 0 whereas an absorption would require a photon energy that is even higher than Q 0 . This may lead to the suppression of resonance absorption of the emitted radiation in the source material. For optical or ultraviolet spectra the recoil energy shift is very small because the photon energy of a few eV is a tiny fraction of the rest mass energy amounting to several GeV. The emission and absorption proﬁles are also much broader than the recoil shift and resonant absorption is frequently observed in optical spectroscopy. Nuclei emit and absorb radiation in much the same way as atoms do. The main difference, however, lies in the magnitude of the recoil energy involved in the emission or absorption of a photon. Gamma-ray lines at the same time have much higher energies and smaller line widths. Similar to the atomic shell, nuclei have excited states, some of which are accessible from the ground state by photon absorption. Especially when the excited states of the absorber have long lifetimes the range of photon energies that will resonantly excite absorption is extremely narrow. For resonant nuclear absorption to be observed, a high fraction of the energy of the source radiation must be within this range. Such a source may consist of excited nuclei from the same isotope as the absorber. The excited nuclei may be decay products of appropriate parent nuclei. For mercury nuclei 198 Hg which can be regarded as free the suppression of resonance absorption was observed and it was brought back by virtue of the Doppler effect, i.e. moving the source relative to the absorber. In a solid environment the recoil momentum and energy go into lattice vibrations. The temperature dependence of the radiation absorption led Rudolf M¨ossbauer 4 to be the ﬁrst to realize that a photon could be emitted with the entire solid taking up the recoil instead of the individual atoms. This process is 4 M¨ossbauer R 1958 Kernresonanzﬂureszenz von Gammastrahlung in 191 Ir Z. Phys. 151 124–43.

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Spectroscopy radioactive source γ detector

vibrator sample

master oscillator

data acquisition

relative intensity

control electronics

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 velocity (mm / s)

1.5

2.0

Figure 3.15. Principle of a M¨ossbauer spectroscopy experiment. The absorption proﬁle is obtained by sorting the counts registered by the gamma-ray detector into velocity bins deduced from the vibrator.

favored at low temperatures where the recoil energy cannot couple to the crystal’s phonon spectrum. Now the mass M0 in the term Q 0 /2M0 c2 of equations (3.16) and (3.18), respectively, can be replaced by the macroscopic mass of the crystal bulk. The energy lost to the recoil in this situation is negligible and the emitted photon may resonantly excite the absorber. As a result, nuclear spectroscopy can be achieved with the resolution limited by the natural width as determined by the lifetime uncertainty of the excited nuclear state. Often, the energy levels are scanned by moving the radioactive source repetitively towards and away from the absorber. Through the Doppler shift, the energy of the gamma-rays arriving at the absorber are varied according to the velocity modulation. The basic ideas of such a scheme are sketched in ﬁgure 3.15. Among all the elements, the most efﬁcient recoil-free resonant absorption occurs for the iron isotope 57 Fe which is produced through electron capture of 57 Co as shown in the nuclear decay scheme in ﬁgure 3.16. The energy of the resonant I = 3/2 level is very narrow corresponding to 7 × 10−9 eV or about 5 parts in 1013. Approximately 90% of the 57 Fe nuclear excited state decays through the intermediate level to produce 14.4 keV gamma radiation. These gamma-rays can then be absorbed by stable 57 Fe nuclei in the sample.

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Spectral-line analysis

272 d

73

57 27Co electron capture 8.7 ns

136 keV, I=5/2

89 ns

14.4 keV, I=3/2

stable

0, I=1/2

57 26Fe

Figure 3.16. Simpliﬁed nuclear decay scheme of 57 27 Co showing the production of the 14.4 keV M¨ossbauer line in 57 Fe. 26

This extraordinarily high resolution allows one to detect tiny changes in the emitting or absorbing nucleus or in the light quantum itself. Usually the resolution is high enough to resolve details of the small changes in the gammaray energy caused by the ﬁelds of the electron shell of an atom or of a solid. The link between the M¨ossbauer spectrum and the electron structure of the sample can be exploited in the study of many types of materials. Fields in which M¨ossbauer spectroscopy has been applied include solid-state physics, surface physics, metallurgy, chemistry, biochemistry and geology. Redshifts of the order of 10−15 caused by the Earth’s gravitational ﬁeld could also be measured in this way.

3.6 Spectral-line analysis As we discussed in previous sections, the lines appearing in the electromagnetic spectra of laboratory or astrophysical light sources are characteristic of the atoms and ions from which the radiation is emitted. The spectra of many atoms and ions are well known. A familiar spectral pattern may be recognized in an environment under study allowing the composition of the light source to be identiﬁed. Very often the lines known to originate from a particular ion appear to be shifted in wavelength. These shifts may be caused by the Doppler effect as discussed in section 2.5. The size of the shift may be used to extract the velocity of the ions in the direction of the line of sight. The line intensity depends (in part) on the number of ions present contributing to the emission or absorption for a particular transition. For the discussion of the temperature dependence let us, for a moment, assume that all

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Spectroscopy

ions are in their ground state because the temperature of the environment is too low for a substantial amount of ions to be excited into at least the ﬁrst excited state. As a result, no transitions are observed relating to energy differences other than those connected with the ground state. For the example of hydrogen that would mean that no Balmer lines can be observed. There are no ions present with the right orbital in order to absorb or emit photons in that spectral range. If the temperature is raised, more and more energy becomes available for collisional excitation of the atoms or ions. At some temperature the spectral lines connected with the ﬁrst excited state become the most prominent lines. As the temperature is further raised lines relating to the second excited state become the most intensive and so on until the available energy is sufﬁcient to ionize the particle. Because of the radiative decay of the excited ions they only stay temporarily in their excited state. There will be an equilibrium between the collisional excitation and radiative de-excitation determining the average number of ions in a particular state. Knowing the mathematical description of these processes and how they depend on the temperature it is possible to use line intensities or ratios of line intensities as a thermometer. For astrophysical objects like a distant star this is the only way to measure their temperature. But there is still another source of information, namely the line shape, that is the distribution of the intensity over the frequency or wavelength. Even if all electronic orbitals in an atom or ion had inﬁnitely narrow energies we would observe spectral lines to be broadened with a ﬁnite width. The broadening is caused by the environment in which the emission or absorption process takes place. There are a number of physical processes that may give rise to broadening. The most important one is caused by the Doppler effect. Emitting particles are not at rest, they are in a random thermal motion giving rise to a wavelength shift for each individual photon emitted. The size of that shift is determined by the particle velocity vector relative to the observer at the instant of the emission. Because there are many particles viewed at the same time one observes as a net effect the superposition of emission events involving many different velocity vectors. This is visualized in ﬁgure 3.17 The velocity also has a distribution with rather small values most of the time; large velocities rarely occur. This is why most of the spectral intensity is still observed at its nominal location as if there was no Doppler shifts. As one deviates from this wavelength position the intensity falls off smoothly as determined by the velocity distribution. Mathematically the line shape caused by the Doppler effect can be easily obtained for thermal emitters in a thin plasma. Very often their velocities follow a Maxwell distribution. For such cases the number of gas atoms or ions with a velocity component v in the direction of observation falling in the interval between v and v + dv is described as 2kB T m 0 v2 n(v) dv = N (3.19) exp − πm 0 2kB T

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Intensity

Spectral-line analysis

Frequency Figure 3.17. The random distribution of the particle velocity both in absolute value and in direction causing line broadening by means of the Doppler effect. (See colour section.)

where kB T denotes the thermal energy of the emitting atoms assumed to be in thermal equilibrium and kB the Boltzmann constant. According to section 2.5, the velocity v is associated with a frequency shift which is approximately νv/c for non-relativistic velocities. Under these conditions the intensity distribution becomes m 0 c2 (ν − ν0 )2 ID (ν) ∝ exp − . (3.20) 2kB T ν02 This is a Gauss distribution with a full width at half maximum of kB T 1/2 2ν0 2 ln 2 . νD = c m0

(3.21)

The Doppler proﬁle (3.20) may be convoluted with the natural line proﬁle to give a more realistic description. The natural line proﬁle is an intrinsic property of the atom under consideration. As illustrated in ﬁgure 2.6 the spontaneous transition probabilities Ai f for transitions between an initial state i and a ﬁnal state f may be summed up to obtain the total transition probability for the state i

τi−1 = Ai = Ai f . (3.22) f

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Spectroscopy

relative intensity

1 0.1 0.01

Voigt

Lorentz

0.001

Gauß -3

-2

-1

0

ν/∆ν

1

2

3

4

Figure 3.18. The convolution of a Gauss (ID (ν)) with a Lorentz (IN (ν)) curve yields the Voigt proﬁle (IV (ν)).

A number N of atoms initially in the state i will decay exponentially with time N = N0 exp(−t/τ ).

(3.23)

The mean lifetime τ is equal to the second moment of the exponential (3.23) which might be written as t¯ = t = τ. (3.24) Interpreting τ as the uncertainty t justiﬁes the application of Heisenberg’s uncertainty relation to obtain the corresponding natural width in energy or frequency units: νN = 1/(2πτ ). (3.25) The form of the frequency distribution over an atomic energy level may be derived as the Fourier transform of the electric ﬁeld of a damped oscillator to yield IN (ν) ∝

1+

1 ν−ν0 νN /2

2 .

(3.26)

The convolution of the Lorentz proﬁle (3.26) and the Gauss proﬁle (3.20) ∞ IN (ν − ν )ID (ν ) dν (3.27) IV (ν) = −∞

is known as the Voigt proﬁle. This is graphically shown in ﬁgure 3.18 for the case where the natural width equals the Doppler width. The wings of the resulting Voigt proﬁle are determined by the Lorentz shape whereas the more central region is dominated by the Gauss shape. Other physical processes resulting in a broadening of spectral lines include collisional broadening and presence of electric and magnetic ﬁelds. In the ﬁrst case the collision frequency is very high so that there is not a high probability

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The inner concept of atoms

77

for the emission or absorption process to occur without it being disturbed by a colliding charged particle another ion or an electron. Outer electric or magnetic ﬁelds change the energy orbits of the emitting ion or cause new energy levels to show up. If the energy levels are only slightly shifted or split, this again is observed as a spectral line broadening.

3.7 The inner concept of atoms In previous sections we have already made extensive use of the general concept of the states of a physical system and the possibility of a transition from one such state to another in a ‘quantum jump’. If the initial and ﬁnal states have different energies then the conservation of energy requires the release or capture of energy. In the case of an atom the energy may be carried by a photon as the subject of spectroscopy. Understanding quantum-mechanical state transitions is closely related with the study of atomic and nuclear behavior. The experimental basis for the development of the early atomic models were the systematics of the line spectra of atomic hydrogen. It was empirically found that ratios of observed wavelengths are very close to some rational numbers. Johann Balmer (1825–98) found the wavelengths in the optical spectrum of hydrogen to follow the formula 1 1 1 (3.28) = RH − 2 λ n 2f ni integer

with n f = 2, n i ≥ 3 and the proportionality constant RH . Later Johann Rydberg (1854–1919) found that the Balmer formula was only a special case of the more general equation (3.28) with different integer numbers n f characteristic for the different line series observed. Before the quantum-mechanical foundations were laid, the planetary model of an atom suggested by Ernest Rutherford (1871–1937) in 1911, had already been accepted on the basis of his experiments on α-particle scattering. According to this model, an atom is a system consisting of a dense, positively charged nucleus and negatively charged electrons rotating around it in circular or elliptic orbits, similar to planets rotating around the Sun. However, the laws of classical mechanics cannot explain the stability of atoms and their line spectra. Actually, according to classical electrodynamics, the acceleration of electrons around a nucleus should be accompanied by radiation. Radiation losses, in turn, should decrease the electron kinetic energy and, as a consequence, should lead to the electron falling into the nucleus but this does not take place in reality. In the period between the discovery of these basic atomic attributes and quantum theory, little was known as to why the electrons in atoms are not instantly attracted into the nucleus. After all, an electrostatic potential between the nucleus and the electrons exists. What kept them from falling inward, thereby causing the atom to collapse?

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Spectroscopy

78

Lyman series Balmer series

Paschen series

13.2 eV

12.8 eV

13.1 eV

10.2 eV

12.1 eV

n=1

Brackett series

n=2 n=3 Pfund series n=4 n=5 n=6

Figure 3.19. The Bohr atomic model showing the six innermost orbitals and the ﬁrst ﬁve spectral series.

In 1913 Niels Bohr (1885–1962) explained the stability of an atom by introducing his postulates5 : • •

•

Electrons in atoms are allowed to occupy only certain discrete orbits representing the stationary quantum states with energy E n . A non-interacting atom does not radiate. An atom only radiates when it makes a transition from one stationary state to another. The energy ~ω of the emitted photon is given by the energy difference between the initial and ﬁnal state ~ω = E i − E f . For electrons in high quantum states with large orbital radii the laws of quantum physics approach those of classical physics. This correspondence principle requires that the frequency of the emitted radiation equals the revolution frequency for high quantum numbers n.

Figure 3.19 visualizes the Bohr atomic model with circular orbits around the nucleus. Also included are the observed line series. Using the laws of classical physics plus Bohr’s postulates one can derive the atomic binding energies for the 5 Bohr N 1913 On the constitution of atoms and molecules Phil. Mag. 26 1. Bohr N 1913 The spectra of helium and hydrogen Nature 92 231.

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The inner concept of atoms

79

different quantum states in a straightforward way. For a stationary state there is a balance between centrifugal and Coulomb force which may be expressed as m er ω =

e2 r2

(3.29)

where ω denotes the circular frequency of the electron, r its orbital radius and m e its mass. The total energy given as the sum of potential and kinetic energy can be written as E = E kin + E pot 1 e2 m er 2 ω2 − 2 r e2 1 4 = − = − (e m e ω2 )1/3 . 2r 2

=

(3.30)

From a comparison of ~ω = hc/λ = E i − E f with the empirical Rydberg formula (3.28) the term energy En = −

RH hc n2

n = 1, 2, 3, . . .

(3.31)

is obtained. For high n the energy difference between two consecutive states becomes 2RH hc n 1. (3.32) E n = n3 The corresponding angular frequency ω = 4π Rc/n 3 is set equal to the classical orbiting frequency that appears in equation (3.30). Then we have En = −

RH hc 1 = − (e4 m e ω2 )1/3 2 n 2

with ω = 4π

RH c n3

(3.33)

from which we can calculate the Rydberg constant RH =

e4 m e . 4π ~3 c

(3.34)

Thus the orbital energies are given by En = −

Ry n2

with Ry =

e4 m e = 13.605 691 72(53) eV 2~2

(3.35)

where Ry is the Rydberg energy. N Bohr obtained the Rydberg constant for the ﬁrst time through other atomic constants which differed from experimental value, at that time, by only a few per cent. In the derivation of this formula the ﬁnite mass of the nucleus was neglected. In a reﬁned calculation the motion of the nucleus has to be taken into account by replacing the electron mass by the reduced mass

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Spectroscopy

80

m e /(1 + m e /M), where M denotes the mass of the nucleus. With the aid of equation (3.30) the radius of an atomic orbit becomes r n = a0 n 2

with a0 =

~2 e2 m e

= 0.529 177 249(24) × 10−10 m.

(3.36)

The electron’s orbital radius a0 for the hydrogen ground state is known as the Bohr radius. Using equations (3.33) and (3.36) we can calculate the angular momentum for the various n states as || = m e vn rn = m ern2 ωn = n ~.

(3.37)

This quantization of the angular momentum is fundamental to the quantum mechanics of atoms. The Bohr model with sharp electron orbits is in contradiction to the uncertainty principle known from free particles. In a pure wave mechanical approach the atomic electron is described by its wavefunction which is a solution of the Schr¨odinger equation (3.3). For the motion of a particle with the reduced mass µ = m e M/(m e + M) in a Coulomb ﬁeld with its −Z e2 /r potential for hydrogen and hydrogen-like ions, the time-independent Schr¨odinger equation has the form Z e2 ~2 ψ = Eψ. (3.38) − − 2µ r The wavefunction ψ, being the solution of this equation, describes the stationary states with deﬁnite values of the total energy E. Equation (3.38) can be solved analytically factorizing the wavefunction into angular and radial parts: ψnm (θ, ϕ, r ) = Ym (θ, ϕ)Rn (r )

(3.39)

where Ym (θ, ϕ) denote the spherical harmonics and denotes the orbital quantum number. The radial part Rn (r ) satisﬁes the equation Z e2 ( + 1) 2µ 1 d 2 dR r − R = 0. (3.40) R+ 2 E+ r 2 dr dr r2 ~ r If E > 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 = −

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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 Schr¨odinger 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

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(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

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energy or probability density (a.u.)

The inner concept of atoms

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| .

© IOP Publishing Ltd 2003

(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

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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.

© IOP Publishing Ltd 2003

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).

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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

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(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

© IOP Publishing Ltd 2003

(4.9)

Basic physical considerations

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

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Light and ion sources

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

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Basic physical considerations

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

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Light and ion sources

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

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Basic physical considerations

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–H¨uckel length after Peter Debye (1884–1966) and Erich H¨uckel (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

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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

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Basic physical considerations

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

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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.

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-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 M¨uller 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.

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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

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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.

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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 Li´enard– 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

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(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

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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.

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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

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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

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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

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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

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(4.71)

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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

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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 ~

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(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.

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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.

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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

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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).

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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

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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⊥ =

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(4.89) (4.90)

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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

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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

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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

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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

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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

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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.

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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

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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

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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 R¨ontgen 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

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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.

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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.

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+

+

+ + + + + + + + + + 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

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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 Wider¨oe 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

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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

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Ion accelerators

~

RF

++ ion source

–

+

–

+ drift tube

129

– ion beam

Figure 4.27. Schematic diagram of the Wider¨oe 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 Wider¨oe 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 Wider¨oe 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

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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.

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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

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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<− <1 (4.119) Boy ∂ x where ρ denotes the bending radius. This method is referred to as weak or constant-gradient focusing and uses a constant slightly decreasing ﬁeld around

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Figure 4.30. View along the axis of a large magnetic quadrupole lens at the heavy-ion synchrotron SIS at GSI Darmstadt. Photograph courtesy A Zschau.

the circumference of the ring with severe demands on the tolerances of the ﬁeld. As a consequence of the necessarily large apertures very big and expensive magnets are needed. In the strong or alternating-gradient focusing scheme, ﬁrst proposed by Christoﬁlos, the difﬁculties of weak focusing are avoided by installing a periodic sequence of focusing and defocusing elements along the circumference of the accelerator, the overall effect of which is a focusing. Initially this was achieved by alternating the gradient ﬁeld superimposed on the dipole ﬁeld of the deﬂection magnets, hence the name. This has an analogy in geometrical optics where a set of focusing and defocusing lenses have an overall focusing effect. For two lenses with focal lengths f 1 and f 2 separated by a distance d, the resulting focal length f is given by the lens equation 1 1 1 d = + − . f f1 f2 f1 f2 For a pair of lenses with f 2 = − f 1 , for instance, the resulting focal length d/ f 12 is always positive. Although defocused by a single lens the effect of a series of lenses is focusing. In particle–beam optics quadrupole lenses combined in doublets or triplets are often employed. As an example, ﬁgure 4.30 shows a photograph of a large magnetic quadrupole lens in use at the heavy-ion accelerator facility at the GSI Darmstadt. The periodic arrangement of the magnetic elements, consisting of dipoles,

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quadrupoles and sextupoles, around the circumference is called a lattice. If the focusing elements are not superimposed on the bending magnets the lattice is known as a separated-function lattice. This facilitates the modular construction of large accelerators and their theoretical description. A popular approximate description of the particle motion through a periodic lattice is: d2 y 1 p + K (s)y = 2 ρ p0 ds

(4.120)

where s and y denote the longitudinal and transverse coordinates in a coordinate system moving on the intended mean trajectory. The quantity y can represent the horizontal or vertical coordinate, yh or yv , respectively and was chosen in this manner to simplify notation. The focusing strength K (s) depends on the particular lattice with characteristic differences observed for the various segments containing bending magnets, quadrupole lenses, drift tubes or combined-function elements. For simplicity the bending radius ρ is assumed to be constant and p denotes the deviation of the momentum from its mean value p0 . Emittance A general solution of the equation of motion (4.120) can be expressed as y(s) = C(s)y0 + S(s)y0 + D(s)

p p0

(4.121)

where the functions C(s), S(s) and D(s) are the cosine-like, the sine-like and the dispersion function altogether referred to as the principal trajectories. For zero momentum deviation, p = 0, a real solution of equation (4.120) is y(s) = a y β(s)1/2 cos(ψ(s) + δ) y (s) = −a y β(s)−1/2 [sin(ψ(s) + δ) − 12 β (s) cos(ψ(s) + δ)]

(4.122)

where the prime denotes the derivative with respect to the longitudinal coordinate s. Equations (4.122) describe betatron oscillations with a betatron amplitude a y β(s)1/2 and with a betatron phase ψ(s) = ds/β(s), where β(s) is known as the beta function which is an intrinsic property of the lattice. In the y–y plane, equations (4.122) give a parametric representation of an ellipse sketched in ﬁgure 4.31. The phase-space ellipse may be viewed in two different ways: ﬁrst, for a single particle as it moves along its trajectory; or second, at a ﬁxed position for many particles with different phases occupying an area in the y–y plane given exactly by the same ellipse as in the single-particle case. Because of the random distribution of particles in y and y one has to specify the fraction of ions considered, for instance, one standard deviation. According to Liouville’s theorem, the emittance deﬁned as % = a 2y stays constant for a ﬁxed velocity. The normalized emittance βγ % is an invariant.

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y'

y

Figure 4.31. Beam particles, each represented by a dot, are assumed to follow a normal distribution in phase space. The ellipse as deﬁned in (4.122) marks two standard deviations.

Another important operating parameter is the tune or betatron wavelength 1 ds (4.123) Q h,v = 2π β(s) which may be different in the horizontal and vertical directions. Due to unavoidable imperfections in manufacturing the lattice, small disturbances may build up periodically at certain positions along the circumference if the machine is operated at one of the betatron resonances located at k Q h + Q v = m

with k, , m = 0, ±1, ±2, . . . .

(4.124)

In particular, the lower resonances have to be avoided for a stable operation. There are tune shifts in circular accelerators caused by the space charge of the beam particles which are of fundamental importance for the achievable intensities. Momentum compaction The path length between two points of the beam-transport line in an accelerator may depend on the particle momentum as suggested by the dispersed orbit of particles with ﬁnite momentum offset. Changes in the momentum imply changes in the path length around the ring and imply different revolution frequencies. The fractional frequency change is related to the fractional momentum change which can be written as ω 1 p 1 = − 2 (4.125) ω P0 γ2 γt

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where the ﬁrst term within the brackets originates from the relativistic velocity change with the change in momentum. The second term, γt−2 , is the relative change of the orbit length divided by the fractional change of the momentum. For a kinetic energy equal to the transition energy (γt − 1)mc2 the bracket in equation (4.125), also known as momentum compaction, changes sign. For particles above the transition energy the momentum compaction is negative, i.e. a particle with a higher energy needs more time for one revolution than a particle with lower energy. This is because the dispersion function is such that particles with a higher energy are sent on an orbit with a larger average radius and path length compared to the ideal orbit. Intuitively, the negative momentum compaction may be identiﬁed with the highly relativistic region where changes in the particle momentum causes only minor changes of the velocity which is already near the speed of light. By special design, using an oscillating dispersion function, one can make the momentum compaction vanish. In such an isochronal ring a small increase in velocity is just balanced by a longer path. Phase stability in circular accelerators and the modern synchrotron Most of the aspects discussed so far have been incorporated in today’s synchrotrons. In the synchrotron the guiding magnetic ﬁeld increases with energy keeping the orbit stationary as in the betatron. Acceleration is provided through an rf voltage via a gap or cavity. In principle the accelerating section in the synchrotron may be viewed as a small linear accelerator which provides a rate of acceleration that is adjusted to the rate of strengthening of the guiding magnetic ﬁeld. The discovery of phase stability along with the revolutionary strongfocusing principle made it possible to build synchrotrons with comparatively small magnets and to reach high energies. By continually correcting the trajectory of errant particles the entire beam is stabilized and it becomes possible to accelerate the particles uniformly to high energies without dispersing them. Below the transition energy the phase of the rf ﬁeld has to be tuned so as to use the ascending part of the sine wave (as in the linac) whereas above the transition one has to use a descending part. In order to accelerate a beam across the transition point, therefore, requires a phase shift in the accelerating rf ﬁeld. This can be achieved in a reasonably short time without losing many particles from the beam. In ﬁgure 4.32 a small section of the synchrotron SIS at the GSI Darmstadt which is used for acceleration of heavy ions up to uranium and energies ranging up to 2 GeV/u is shown. In the center of the picture is an rf section whereas the other large structures are bending magnets and quadrupole lenses. The maximum magnetic ﬁeld that can be used in synchrotron magnets is limited by the saturation of iron components. So, if the energy is to be increased one has to increase the bending radius ρ noting that the magnetic rigidity, Bρ, is proportional to the particle momentum. As a consequence limitations mainly arise from the construction costs of huge accelerators plus their energy consumption during operation. A way to reduce the power used by synchrotrons is the

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Figure 4.32. Part of the heavy-ion synchrotron SIS at the GSI Darmstadt. Photograph courtesy A Zschau.

introduction of superconducting magnets and cavities as for the electron–proton collider HERA at the DESY laboratory in Hamburg or for LEP and LHC at CERN. 4.4.5 RFQ accelerators A new rf linac structure was proposed in 1970 by two Russian scientists, I M Kapchinskii and V A Teplyakov18, which revolutionized accelerator design. This rf structure has become known as the radiofrequency quadrupole linac. Its electric ﬁelds are generated by an rf voltage applied to four long electrodes the cross section of which is schematically shown in ﬁgure 4.33. The pure quadrupole focusing geometry is modulated by shaping the edges of the four vanes, as the electrodes usually are called, into precisely machined ripples. As a consequence of the shaping, accelerating longitudinal electric ﬁelds are produced on the axis. But unlike the drift-tube linac the particles are not shielded when the ﬁeld reverses polarity. Instead, they drift through a region where the four vanes form a symmetric geometry with purely focusing ﬁelds during times when the electric ﬁeld is reversed. Once the electrodes are machined the RFQ structure reveals an extraordinary ﬂexibility, compactness and ease of operation. The only adjustable operation parameter is the rf power. 18 Kapchinskii I M and Teplyakov V A 1970 Linear ion accelerator with spatially homogeneous strong focusing Prib. Tekh. Eksp. 119 17.

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– +

+ –

Figure 4.33. Cross section through the electrodes of an RFQ accelerator.

injected beam

accelerated beam

Figure 4.34. Longitudinal cut through an RFQ accelerator showing schematically its combined bunching, focusing and acceleration capabilities.

The effect on a DC beam injected into the structure along its axis is that it can be focused, bunched and accelerated simultaneously as indicated in ﬁgure 4.34. No other type of accelerator can accomplish these functions within such compact dimensions. The typical RFQ linac is only 1–3 m long and 0.3–1 m in diameter and it can accept high-current beams of rather low energy of a few tens of keV which are accelerated with only minor losses to energies greater than 1 MeV. For practical applications variants of the RFQ accelerator can be made to carry ions with a current as high as 0.5 A. Therefore the RFQ design has entered fusion research using intense ion beams. Other applications include compact injectors for larger accelerators, dedicated accelerators for cancer therapy, production of shortlived radionuclides used for positron-emission tomography, activation analysis and ion implantation in metals and semiconductors. 4.4.6 Highly charged heavy ions The economics of ion acceleration makes it desirable to use projectiles with a high charge state as the accelerating force is proportional to the particle’s charge. For accelerator design, therefore, one needs accurate knowledge of the charge-state distributions for ion beams after they have traversed solid or gaseous targets. Attainable ﬁnal beam energies and intensities depend on the speciﬁc accelerator design and on the position where ion strippers are implemented in an accelerator facility. Charge-state data are also required for planning experiments and stripping processes are of fundamental interest in the study of collisional interactions of projectiles with target atoms in which they lose or gain electrons. The charge-state spectra can be measured with existing bending magnets in an accelerator beam line or with dedicated magnetic spectrometers. A typical

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Figure 4.35. A typical ion-optical arrangement for the measurement of charge-state distributions of heavy-ion beams after passage through a stripper target.

experimental arrangement is sketched in ﬁgure 4.35. After traversing the target the emerging charge states are dispersed in the dipole magnet and the ions following different trajectories are focused on the plane of a position-sensitive particle detector by means of a suitable ion-optical arrangement. Examples of charge-state distributions for uranium ions at three different ion energies are displayed in ﬁgure 4.36. With increasing energy the mean charge state is shifted to higher values. The process can be modeled similarly to the ionization in an ion source as sketched in section 4.1 In general, however, the theoretical description is complicated because of the many different collisional processes which contribute simultaneously. Processes to consider include: • • • • • •

excitation, ionization, radiative and collisional de-excitation, Auger decay, resonant and non-resonant capture and radiative electron capture.

In a many-electron system all processes involving many different states are important and a precise model of the ion population would require knowledge of a large number of different cross sections. That is why the theoretical predictions in the many-electron case remain largely uncertain. Because of the persistent small uncertainties it is also difﬁcult to explain the greater effectiveness of solid strippers over gas strippers.

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1.4 MeV/u 0.1

fraction

0 11.4 MeV/u

0.1 0 0.1

940 MeV/u 0 30

40

50

60

70

80

90

charge state Figure 4.36. Equilibrium charge-state distributions of uranium ions at three different c energies. From: Scheidenberger C et al 1998 Nucl. Instrum. Methods B 142 441. 1998 Elsevier.

The situation becomes clearer for very heavy ions at velocities when substantial amounts of bare ions are produced. The number of ionic states to consider are drastically reduced. At high energies it is enough to include only ions with zero, one, two and three electrons. The excited states eventually formed during collisions also decay rapidly to the ground state in very heavy ions. When the decay times of the excited states become small compared to the mean time between collisions one can neglect ion excitation altogether and it is justiﬁed to assume that all ions are in their ground state. Then the evolution of charge states inside a target can be modeled by an ionization balance between a few states only19,20 . Such a model has been applied to measurements21,22 on relativistic gold ions as demonstrated in ﬁgure 4.37 for 1 GeV/u Au ions penetrating a Ni target. The shapes of the curves for the highest charge states depend on the energy dependence of the cross section for ionization and for radiative and non-radiative electron capture. The dominance 19 Anholt R and Gould H 1986 Relativistic heavy-ion–atom collisions Adv. At. Mol. Phys. 22 315. 20 Eichler J and Meyerhof W E 1995 Relativistic Atomic Collisions (San Diego, CA: Academic). 21 Scheidenberger C and Geissel H 1998 Nucl. Instrum. Methods B 135 25. 22 Geissel H and Scheidenberger C 1998 Nucl. Instrum. Methods B 136–8 114.

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Figure 4.37. Charge-state evolution in solid Ni targets for incident Au69+ ions at a speciﬁc energy of 1 GeV/u. From: Geissel H and Scheidenberger C 1998 Nucl. Instrum. Methods c B 136–8 114. 1998 Elsevier.

of the fraction of bare ions at ultrarelativistic velocities is favored through the relativistic deformation of the electromagnetic potential giving rise to a relativistic increase in the ionization of inner-shell electrons. The charge state inside the target also strongly inﬂuences the stopping power for fast ions as it is dominated by the energy transfer from the fast projectile to the target electrons.

4.5 Ion cooler rings 4.5.1 Basic characteristics The atomic properties of highly charged heavy ions can be studied in collision experiments using ion beams and an interaction target. In such a single-pass experiment severe limitations can arise: • • •

from the short time of observation from insufﬁcient control over beam quality and from a low event rate with low-density atomic, electron or photon targets.

These drawbacks can be widely circumvented when the experiment is carried out in an ion beam circulating in a storage ring23,24,25 . Such devices are 23 Wolf A 1999 Atomic Physics with Heavy Ions ed H F Beyer, and V P Shevelko (Berlin: Springer) p 3. 24 Franzke B 1992 Third European Particle Accelerator Conference vol 1, ed H Henke, H Homeyer and C Petit-Jean-Genaz (Singapore: Editions Frontiers) p 367. 25 Reistad D 1998 Proc. of the 6th Euro. Part. Acc. Conf. Stockholm, p 141.

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synchrotron accelerators with a magnetic lattice and an rf-cavity operated at a moderate energy. Because heavy particles do not show notable self-cooling because they do not sufﬁciently dissipate synchrotron radiation, special beam cooling devices are installed which allow ion beams with low momentum spread and small size and divergence to be prepared. A large ensemble of ions following a closed orbit along the storage ring can be observed for an extended period of time which is determined by the storage time of the ion beam. The beam lifetime depends on the dominant loss processes like charge-changing collisions with residual-gas atoms or with particles of an internal target or scattering among beam particles. Observed beam lifetimes range from a few seconds to several hours. Typical residual gas pressures are 10−10–10−12 mbar. From the peculiarities of a storage ring necessary for operation valuable diagnostics are also derived supporting the internal experiments. Among the particle diagnostics are the number of stored ions together with their velocity and momentum spread. Through their high repetition rate by which the ions traverse the storage ring corresponding to a high ion current it also becomes possible to observe interactions with rather thin targets. One motivation for the development of beam cooling techniques was the need for high luminosities for the study of rare events in colliding beams of, for instance, protons and antiprotons. Antiparticles produced by bombarding targets with high-energy beams have inherently large emittances and momentum spreads. That is why their phase-space volume had to be reduced before they could be used in high-energy collision experiments. Electron cooling was ﬁrst proposed by Budker26 and applied to protons and antiprotons at the NAP-M storage ring at Novosibirsk. Stochastic cooling invented by van der Meer and co-workers at CERN27 paved the way for the discovery of the vector bosons W and Z. Basic experience on beam cooling and the operation of cooler rings was gained at the test facilities at Novosibirsk, CERN and at Fermilab near Chicago. Several cooler storage rings for heavy particles were developed for various demands. Some of the rings are used for nuclear and atomic physics with heavy ions. Table 4.4 lists existing cooler rings which started operation between 1987 and 1993. Rings with a circumference C are sorted in table 4.4 according to their magnetic rigidity which may be expressed as, cf equation (4.32), Bρ[Tm] = 3.11βγ

A q

(4.126)

where A denotes the atomic mass number and q the ionic charge of the stored particle, respectively. This gives the translation of the Bρ values into ion energies for a given ion species. For protons with A/q = 1 the nominal range of energies would be from 90 to 4500 MeV. Some of the accelerators listed operate with 26 Budker G I et al 1976 Part. Accel. 7 197. 27 M¨ohl D, Petrucci G, Thorndahl L and van der Meer S 1980 Phys. Rep. 58 75.

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Table 4.4. Operational ion cooler rings with their main characteristics. Facility CRYRING MSL, Stockholm TSR MPI, Heidelberg ASTRID ISA, Aarhus Cooler IUCF, Bloomington TARN II INS, Tokyo CELSIUS Svedberg Lab., Uppsala LEAR, LEIR CERN, Geneva ESR GSI, Darmstadt COSY FA J¨ulich SIS GSI, Darmstadt

Bρ[Tm]

C[m]

Typical ion

Main purpose

1.4

51.6

Ar18+

Atomic and molecular physics

1.7

55.4

Cu26+

Atomic and molecular physics

1.9

40.0

Mg+

3.6

86.8

H+

Atomic and molecular physics Nuclear and accelerator physics

6.1

77.8

H+

Accelerator and atomic physics

7.0

82.0

H+

Nuclear physics

7.0

78.0

Pb54+

Beam preparation for LHC

10.0

108.4

U92+

Atomic, nuclear and accelerator physics

11.7

184.0

H+

Nuclear physics

18.0

216.8

U92+

Nuclear physics injector for ESR

protons and light ions only and are dedicated to nuclear research. For heavy-ion atomic physics the kind of injector is important as it determines the available ion species. The Test Storage Ring (TSR) in Heidelberg and the Heavy-ion synchrotron (SIS) in combination with the Experimental Storage Ring (ESR) at Darmstadt are part of larger accelerator facilities. The injector of the TSR is a 12MV tandem van-de-Graaff accelerator plus an rf postaccelerator. The accelerator facility at GSI consists of the linear accelerator UNILAC as an injector for SIS which is connected to the ESR. An outline of ESR is shown in ﬁgure 4.38. There it is possible to store highly charged heavy ions up to bare U92+ or even rare and radioactive nuclides produced at the fragment separator. CRYRING employs a

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Figure 4.38. Outline of the Experimental Storage Ring (ESR) at GSI Darmstadt. Highly charged heavy ions from the heavy-ion synchrotron (SIS) or from the fragment separator can be injected into the ESR. (See colour section.)

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cryogenic EBIS source, CRYEBIS, delivering highly charged ions which are preaccelerated in an RFQ accelerator. The injected ions are further accelerated inside CRYRING operated as a synchrotron. ASTRID mainly uses singly charged ions from a smaller ion source plus electrostatic preaccelerator. In a heavy-ion storage ring (exempliﬁed in ﬁgure 4.38) the ions periodically cycle around a closed orbit with a typical frequency of about 1 MHz. The orbit length is given by the size of the deﬂection and focusing elements plus the intermitting spaces required for rf cavities, diagnostic and experimental installations. There are two large drift sections where the dispersion is small which are occupied by the electron cooler and by the internal gas jet, respectively. Both installations are extensively used by experiments. At the electron cooler ion– electron interactions like radiative or dielectronic recombination can be studied. Collisions of highly charged ions with gas atoms are investigated at the internal gas jet. Both the electron and gas target represent tunable x-ray sources providing insight into the structure and dynamics of highly charged few-electron systems. The two straight sections are also well suited for colinear laser spectroscopy which makes use of the superposition of a laser and an ion beam. In the electron cooling device, which will be discussed next in more detail, there is a third overlapping beam of electrons. 4.5.2 Electron cooling In electron cooling28,29,30 , a cold, i.e. a monoenergetic and parallel, electron beam is superimposed on a hot ion beam. The two beams have the same average velocity vi = ve and statistical deviations of individual particle velocities may be interpreted as a beam temperature. An arrangement for an electron cooler is shown in ﬁgure 4.39. A beam of free electrons is formed from a thermal cathode inside an electron gun from where they are electrostatically accelerated. After the interaction region (2.5 m for the ESR cooler) the electrons are decelerated and dumped into a collector. The electron beam is guided by longitudinal magnetic ﬁelds provided by solenoidal and toroidal conﬁgurations. Electron cooling works through the damping of the relative ion velocities in the gas of cold electrons viewed in the comoving frame of reference in which the ∗ = 0. This is visualized in average longitudinal electron velocity vanishes, ve% ﬁgure 4.39. The damping or friction force is introduced via Coulomb collisions between electrons and ions. In the theoretical description it turns out that different scaling behaviors of the cooling force are valid depending on whether the ion ∗ 2 1/2 . velocity is smaller or larger than the rms electron velocity ve⊥ The friction force may be estimated by considering the slowing down of 28 Poth H 1990 Phys. Rep. 196 135. 29 Parkhomchuk V V and Skrinsky A N 1991 Rep. Prog. Phys. 54 919. 30 Franzke B et al 1992 Third European Particle Accelerator Conference ed H Henke, H Homeyer

and C Petit-Jean-Genaz (Singapore: Editions Frontiers) vol 1, p 444.

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gun

collector

electrons

vi = ve ions

comoving frame

Figure 4.39. An electron cooling device, here the ESR cooler, guides a cold stream of electrons so as to overlap with the hot ion beam. The inset visualizes the situation from the comoving frame of reference: The electrons, represented by small dots, are nearly at rest whereas the ions show a random velocity distribution as represented by the arrows.

charged particles in a sea of cold electrons expressed by the Bethe–Bloch equation Fc =

4πq 2e4 n e dE =− LC dx m e ve2

(4.127)

where n e denotes the electron density and L C is the Coulomb logarithm given as bmax ≈ 10. (4.128) L C = log bmin The minimum and maximum impact parameters, bmin and bmax , can be set equal to the distance of closest approach for head-on collisions and to the Debye shielding length, respectively. From equation (4.127) a rate for momentum cooling can be deduced, τc−1 =

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4πq 2e4 n e η dp = 2 LC pdt γ m e m i ve3

(4.129)

Ion cooler rings

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where the Lorentz transformation into the laboratory system has already been carried out. It is also taken into account that cooling takes effect only over a small fraction η of the ring circumference. From equation (4.129) one expects a scaling of the cooling time, τc ∝ A/q 2 which makes cooling increasingly efﬁcient for highly charged ions. There are still additional effects to consider that enhance the cooling rate: • • •

anisotropic ﬂattened electron-velocity distribution magnetization of the electron beam and magnetic expansion of the electron beam.

The ﬂattening of the velocity distribution in the longitudinal direction is a result of the electrostatic acceleration adding a constant energy to each electron rather than a constant velocity increment. After an appropriate transformation from the laboratory into the comoving system the electron density may be expressed as an anisotropic Maxwell distribution with a transverse electron temperature Te⊥ , given by the temperature of the hot cathode in the electron gun, and with a longitudinal electron temperature reduced to Te% =

2 Te⊥

Te⊥ 2β 2 γ 2 mc2

(4.130)

where β and γ denote the usual relativistic parameters and m e the electron mass, respectively. The term magnetized electron beam refers to the freezing of the transverse motion of the electrons in the strong magnetic guiding ﬁeld. For distant collisions with the impact parameter larger than the Larmor radius and the interaction time larger than the inverse cyclotron frequency, the ion effectively interacts with a Larmor disc as a whole enhancing the friction force. Only at small impact parameters is the effect of the magnetic ﬁeld small. The non-magnetic force ∗ 2 1/2 but dominates remains below the magnetic cooling force when vi∗ ve⊥ 2 ∗ 1/2 . At low and high relative velocities the cooling force for vi∗ ve⊥ approximately scales as vi% vi% < ve% (4.131) F% ∝ v −2 v > v .

.

i

i

e⊥

When the electron beam starts in a strong longitudinal magnetic ﬁeld B% in the electron gun and the ﬁeld is adiabatically decreased before the interaction region the transverse electron temperature decreases by the same factor as the ﬁeld decreases and the cross section of the beam increases. This is a consequence of Liouville’s theorem requiring kB T⊥ = constant. B%

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(4.132)

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Light and ion sources 1

10

0

||

F (eV/m)

10

-1

10

-2

10

-3

10

3

10

4

10

5

10

6

10

7

10

relative velocity (m/s) Figure 4.40. The longitudinal cooling force measured for several fully stripped heavy ions. The electron density in the comoving frame is 1 × 106 cm−3 . From: Winkler T et al 1996 c Hyp. Int. 99 277, 1996 Kluwer Academic, reprinted with permission.

Such an adiabatic expansion of the electron beam is implemented in most of the low-voltage electron coolers of the cooler rings listed in table 4.4 and expansion factors range from 8 to 100. High-resolution measurements of dielectronic-recombination resonances beneﬁt very much from the reduced electron temperature which can be as low as the cathode temperature divided by the expansion factor. The cooling force can be measured if a suitable counter force is applied. One method derives the cooling force as a function of the relative velocity by observing the velocity shift of the ion beam as a result of the balance between the cooler drag force and the external force from an induction linear accelerator. In the voltage-step method the terminal high voltage of the cooler is suddenly changed and the response of the ion beam is observed as it adjusts its velocity to that of the electrons. This method does not work for very small voltage steps because the ion beam would then be dragged too fast with the electron beam. Heating the beam with calibrated rf noise at the same time as it is cooled can yield the cooling force from the measured beam velocity distribution. The latter can be obtained from an analysis of the Schottky noise spectrum. Figure 4.40

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-4

δp/p

10

U

92+

360 MeV/u

-5

10

-6

10

I e= 250 mA 1

10

10

2

10

3

10

4

10

5

10

6

10

7

number of stored ions Figure 4.41. Measured momentum spread of a U92+ beam stored in the ESR as a function of the ion current. The abrupt step occurs when the ions start to form an ordered chain. c From: Steck M et al 1996 Phys. Rev. Lett. 77 3803. 1996 APS.

shows the result of cooling force measurements carried out at the ESR with two different methods. At high relative velocity the voltage-step method was used whereas at low velocities calibrated noise was used. Ultimately the ion temperature of a cooled beam can approach the electron temperature. However, there are additional heating processes in the storage ring. For very low residual-gas pressures and without a gas target the remaining limitation comes from intra-beam scattering which increases with the ion current. In ﬁgure 4.41 the measured momentum spread of a U92+ beam stored in the ESR is plotted as a function of the ion current31. The behavior at high currents reveals the expected balance between cooling and intra-beam scattering. Around an ion number of about 103 there is a sudden step that is attributed to a linear Coulomb ordering32 of the ions, which can no longer pass each other and hence do not show self-heating due to scattering. 4.5.3 Stochastic cooling Electron cooling works best for ion beams that are not too hot to begin with and the cooling becomes less efﬁcient for ion beams with large emittances. Stochastic cooling33,34 is a complementary method that is well suited for hot ion beams and 31 Steck M et al 1996 Phys. Rev. Lett. 77 3803. 32 Hasse R W 1999 Phys. Rev. Lett. 99 3430. 33 van der Meer S 1985 Rev. Mod. Phys. 57 689. 34 M¨ohl D, Petrucci G, Thorndahl L and van der Meer S 1980 Phys. Rep. 58 73.

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Figure 4.42. Principle of stochastic cooling of betatron oscillations. The displacement sampled at the pickup is ampliﬁed and sent with the appropriate phase to the kicker applying a corrective deﬂection.

not too high a number of stored particles. That is why it is used for the cooling of rare and exotic particles like antiprotons or radionuclides produced in nuclear reactions. At GSI stochastic precooling of radioactive nuclei is implemented in the ESR. The concept of stochastic cooling is based upon the idea that it is possible to detect the displacement of individual particles of a stored beam from their intended orbit and to correct it with an appropriate feedback system. Such a feedback system, schematically shown in ﬁgure 4.42, may be designed to damp the radial betatron oscillations. Let us ﬁrst consider a single stored particle. We assume there is an electrostatic pickup probe consisting of two insulated electrodes at a certain location along the ring. If the particle deviates from the central orbit it will induce a difference signal on the pair of pickup plates. This signal is proportional to the displacement from the central orbit. It will be ampliﬁed and sent to a kicker also consisting of two electrodes and situated an odd number of quarter wavelengths of the betatron oscillation downstream of the pickup. The kicker deﬂects the particle by an amount that is proportional to the offset sensed in the pickup and which has the right polarity so as to decrease the betatron amplitude. The signal path length must, of course, be adjusted to the particle’s orbit path length in order for the signal to arrive at the same time at the kicker as the particle. As this process is repeated for many successive revolutions the particle loses all its transverse energy which is dissipated in the kicker system. In reality there are many particles being sampled in the pickup. They all perform betatron oscillations with some randomly distributed phase which leads to a reduction in the pickup signal. Due to the ﬁnite number of particles in the

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sample and the stochastic nature of the oscillations there are ﬂuctuations in the sampled centroid position with respect to the equilibrium orbit. Because of the statistical ﬂuctuations in the beam, the sample of ions measured in the pickup does not stay the same after many revolutions and mixing with other beam particles occurs. On the path between kicker and pickup the unwanted mixing opposes the cooling because the correction at the kicker is now applied to a group of ions that are not completely identical to those sampled in the kicker. A desired mixing between kicker and pickup serves to randomize the sample so that all beam particles take part in the cooling process. From an analysis of the coherent cooling and the incoherent heating mechanisms the cooling rate can be calculated. The cooling rate depends on the ampliﬁer gain showing a maximum. For optimum gain setting the cooling rate is 1 2Ws = τs Ni (Ms + Us )

(4.133)

where Ni denotes the number of stored ions and Ms and Us are mixing and thermal-noise factors, respectively. The bandwidth Ws of the cooling system has to be large for efﬁcient cooling. For highly charged ions the signal levels are high and the noise factor Us is small compared to singly charged ions. Cooling betatron oscillations works in both transverse directions but it can also be applied in the longitudinal direction. One possibility, known as Palmer cooling, is to place the pickup probe at a place along the ring where the dispersion function is large and to let the transverse information translate into a longitudinal kick. In the notch-ﬁlter method the ampliﬁed pickup signal is sent through a ﬁlter that shifts the phase by π at all harmonics of the intended revolution frequencies. Thus samples traveling too fast are decelerated while those traveling too slow are accelerated. 4.5.4 Laser Cooling The technique of laser cooling35,36,37 is closely related to the experimental schemes used in colinear laser spectroscopy of stored ions. For this purpose straight drift sections of the storage ring are used for the light–ion interaction. The interaction length relative to the ring circumference can be as high as η ≈ 0.2. In order to be laser cooled the ions must have a simple term scheme with a closed transition between a ground and an excited level so as to avoid optical pumping. Laser cooling of ions in electromagnetic traps has been used for a long time. In a storage ring a laser beam of suitable frequency is directed parallel or antiparallel to the ion beam. The ions that are in resonance with the laser beam can absorb photons. Each absorbed photon transfers a momentum ~ω/c to the ion. This momentum transfer occurs unidirectionally as explained in ﬁgure 4.43. When 35 Schr¨oder S et al 1990 Phys. Rev. Lett. 64 2901. 36 Hangst J S et al 1991 Phys. Rev. Lett. 67 1238. 37 Petrich W et al 1993 Phys. Rev. A 48 2127.

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p = 0

p= po

unidirectional absorption

isotropic emission

Figure 4.43. Principle of laser cooling. Momentum is transferred unidirectionally from the laser beam to the atom during the absorption process. When the excited atom decays back to its ground state the recoil momentum is at a random direction.

the excited ion spontaneously decays back to the ground level it again receives a recoil momentum ~ω/c. This momentum, however, is oriented in a random direction. When the emission is isotropic there is only a negligible momentum after many spontaneous emission processes. As a result there is a net radiation pressure force directed along the laser beam. This force depends on the ion velocity through the Doppler shift. To use the spontaneous force for laser cooling one has to establish a stable point on the curve of the force as a function of the ion velocity as illustrated in ﬁgure 4.44. This means that the force must be zero at the stable point and have a negative gradient with respect to the ion velocity. This can be achieved by applying a constant auxiliary force, for instance by an rf ﬁeld or by simultaneous electron cooling, or by a second counterpropagating laser. In ﬁgure 4.44 the cooling force is schematically shown as a function of the velocity for the case of a constant auxiliary force Faux and for the case of a second laser, respectively. The average force resulting from repeated cooling, i.e. scattering, cycles resembles the absorption proﬁle which can be expressed as F = ~ k

S/2 − Faux 1 + S + (2ω/ )2

(4.134)

where and S denote the spontaneous decay rate and the optical saturation parameter, respectively. The velocity dependence enters through the Doppler shift giving the frequency detuning from the resonance frequency ω0 in the ion rest frame as ω = (1 − β)γ ω − ω0 . (4.135) The laser cooling rate is derived from the laser force gradient as 2 ∂F 1 =− . τc m i ∂v v=v ∗

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(4.136)

Tokamak F(v)

153

a)

0

v

vo

F(v) b) 0

v1 v2

v

Figure 4.44. Laser cooling force, (a) in the case of a constant auxiliary force and (b) in the case of a second counterpropagating laser. The dots mark the points for stable operation.

Laser cooling obviously works only in the longitudinal direction. Indirect cooling in the transverse directions can also be observed through collisional coupling via intra-beam scattering or dispersive coupling.

4.6 Tokamak A tokamak is a toroidal plasma-conﬁnement device invented in the late 1950s by the Russian scientists I Y Tamm and A D Sakharov. The acronym tokamak is derived from the Russian words toroidal’naya kamera s aksial’nym magnitnym polem. The plasma is conﬁned here by magnetic ﬁelds rather than by the material walls. The magnetic ﬁelds in a tokamak are produced by a combination of currents ﬂowing in external coils and currents ﬂowing inside the plasma itself. The physics and technology of the tokamak are being developed as a major route to controlled thermonuclear fusion as a long-term energy supply38,39,40,41 . 4.6.1 Thermonuclear fusion Fusion is the energy source that powers the Sun and stars and requires extraordinarily high pressures and temperatures which are difﬁcult to achieve on 38 Wesson J 1987 Tokamaks (Oxford: Clarendon). 39 Hazeltine R D and Meiss J D 1992 Plasma Conﬁnement (Redwood City, CA: Addison-Wesley). 40 Gross R A 1984 Fusion Energy (New York: Wiley). 41 Cordey J G, Goldston R J and Parker R R 1992 Progress toward a Tokamak fusion reactor Phys. Today 45 22.

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8

1

H H

2

6

3

H

4 7

2

Li 238

U

0 -2

56

0

Fe

50 100 150 200 atomic mass number A

250

Figure 4.45. Excess energy per nucleon as a function of the nuclear mass number. The high excess energy observed for light elements is the basis for the exothermic thermonuclear fusion.

Earth. Fusion occurs when two light atomic nuclei, like hydrogen, collide with enough energy to overcome their Coulomb repulsion and merge, forming a new type of nucleus thereby releasing a tremendous amount of energy. The key to nuclear energy is the equivalence of energy and mass. Since the binding energy of a nucleus is quite large it is discernible as a difference in mass between the nucleus and the sum of its constituents. Because of this, the binding energy is also called the mass defect. In ﬁgure 4.45 the mass defect per nucleon is plotted as a function of the atomic-mass number of nuclides. It is positive for light elements and has a minimum around iron. The most suitable reaction occurs between the nuclei of the two heavy isotopes of hydrogen: deuterium (2 H) and tritium (3 H), 2

H + 3 H → 4 He + n + 14.1 MeV

(4.137)

where n denotes the neutron. Eventually reactions involving just deuterium or deuterium and helium (3 He) may be used. In the absence of the tremendous gravitational forces present in the Sun and stars, the conditions for controlled fusion on Earth can be created using magnetic forces to conﬁne the fusion fuel while heating it by a variety of methods. Ignition occurs when enough fusion reactions take place for the process to become selfsustaining. To achieve this, the fusion fuel must be heated to temperatures high enough (about 100 million Kelvin) to overcome the natural repulsive forces of nuclei and kept dense enough and conﬁned for long enough to withstand energy losses. Because of quantum-mechanical tunneling, there is already a ﬁnite fusion

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probability at kinetic energies below the Coulomb barrier. At very high impact energies the interaction time is short, and the kinetic energy of the relative motion is so large that the energy involved in the fusion reaction is only a small correction. As a result the probability of reaction decreases with higher impact energy, at high energy. This is why there is a given temperature at which the fusion reaction rate is a maximum. 4.6.2 Conditions for a fusion reaction The D–T fusion reactions occur at a sufﬁcient rate only at a very high temperature corresponding to kB T = 10 keV, other reactions require even higher temperatures. The hot plasma must be well isolated away from material surfaces in order to avoid cooling the plasma and releasing impurity atoms that would contaminate and further cool the plasma. In the Tokamak system, the plasma is isolated by magnetic ﬁelds. The efﬁciency of the magnetic insulation is measured by the energy conﬁnement time. This is the characteristic time scale for plasma cooling after the source of heat is removed. The density of fuel ions must be sufﬁciently large for fusion reactions to take place at the required rate. The fusion power generated is reduced if the fuel is diluted by impurity atoms released from the surrounding material surfaces or by the accumulation of helium ash from the fusion reaction. As fuel ions are burnt in the fusion process they must be replaced by new fuel and the helium ash must be removed. As a ﬁgure of merit the product of plasma density n, temperature T and energy-conﬁnement time τ can be used. At a plasma temperature of 10 keV, the Lawson criterion requires nτ > 1014 cm−3 s. (4.138) In the tokamak plasma, this condition is satisﬁed at rather low densities of the order of 2–3 × 1014 cm−3 and energy conﬁnement times of 1–2 s. This can be contrasted with inertial conﬁnement where, by laser or particle bombardment, a pellet is compressed to very high density at a conﬁnement time of the order of a nanosecond42. 4.6.3 The Tokamak conﬁguration In ﬁgure 4.46 the general layout of the Joint European Tokamak (JET) which started operation in 1983 is shown. It has a major radius of 2.96 m, a minor radius of 1.0 m and the toroidal magnetic ﬁeld is 3.5 T and it uses an input power of 42 MW. A view inside the vacuum vessel of this tokamak is shown in ﬁgure 4.47. In a tokamak, the magnetic lines of force are helices that spiral around the torus. The helical magnetic ﬁeld has two components: (i) a toroidal component, which points around the torus; and (ii) a poloidal component directed ‘vertically’ 42 See Lindl J D 1998 Inertial Conﬁnement Fusion (New York: AIP).

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Figure 4.46. Conﬁguration of the Joint European Tokamak (JET), illustration courtesy of EFDA-JET.

around the machine. Both components are necessary for the plasma to be in stable equilibrium. The toroidal ﬁeld is produced by coils that surround the toroidal vacuum chamber containing the plasma. As a consequence of the zero divergence of the magnetic ﬁeld, ∇ B = 0, the ﬁeld is necessarily weaker on the outside of the torus than it is on the inside. The net I × B Lorentz force is directed outwards, forcing the torus to expand. To counteract these hoop forces a poloidal vertical ﬁeld is required. It is generated by a toroidal electric current that is forced to ﬂow within the conducting plasma. The force on the system resulting from the toroidal current crossing this vertical ﬁeld is to force the loop of the torus to contract. Finally, the system is left in equilibrium. For heating the plasma to high temperatures, the following methods are used, each providing a power input in the 10–20 MW range: •

Ohmic heating and current drive. Currents of several mega amperes ﬂow in the plasma and deposit a few mega-watts of heating power.

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Figure 4.47. View inside the vacuum vessel of the JET Tokamak showing the divertor at the bottom and a remote handling manipulator. Photograph courtesy of EFDA-JET.

•

• • •

Neutral beam heating. Accelerated beams of deuterium or tritium ions are injected into the plasma. In order to penetrate the conﬁning magnetic ﬁeld, the accelerated beams must be neutralized. In the plasma, the beams become ionized and the fast ions transfer their energy to the plasma. Radiofrequency heating. Energy is given to the plasma at those locations where the radio waves are in resonance with the ion rotation. Current driven by microwaves. Microwaves at a few GHz are used to accelerate the plasma electrons to generate a plasma high current. Self-heating of plasma. The α-particles produced when deuterium and tritium fuse remain within the plasma’s magnetic trap. Their energy continues to heat the plasma to keep the fusion reaction going. When the power from the α-particles is sufﬁcient to maintain the plasma temperature, the reaction becomes self-heating—a condition referred to as ignition.

In a conﬁned fusion plasma, the ion Larmor radius ranges from millimetres at the edge to centimetres in the hot core, and the ion Larmor frequency is on the order of 100 MHz–1 GHz. When some additional effect acts to make the radius of the particle orbit vary more slowly than the gyrofrequency, the guiding centers drift. When this occurs all the particles move collectively. This is called ﬂuid drift motion. Since transport in conﬁned plasmas is not yet fully understood and it strongly affects the performance of fusion devices it remains an important

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outstanding problem in plasma physics. It is also related to the attainable energy density in the plasma as limited by the onset of violent instabilities. A ﬁgure of merit in fusion research is the energy gain Q which is deﬁned as the ratio between fusion power and heating power that would occur if the plasma contained tritium. Over the past few decades Q has increased from 10−7 in 1965 to 1 in 1995 when the long-sought goal of energy break even, Q = 1, was achieved. Current experiments conﬁne plasmas with volumes of 100 m3 at temperatures up to 30 keV. A next major step in the development of fusion power is the construction of a facility to study the physics of a burning plasma with inﬁnite Q. The proposed International Thermonuclear Experimental Reactor (ITER) is expected to produce energy at the level of a small fusion power plant and to address the key technical challenges involved in making fusion a practical energy source. Although fusion plasma fuel is made of light elements, much of the diagnostic work concentrates on heavy impurity ions. The latter have fatal consequences on the possible performance of a plasma device as they tremendously cool the plasma. For an efﬁcient fusion reactor the concentration of heavy contaminants has to be low enough for the effective atomic number of the plasma to be between 1 and 2.

4.7 Electron-cyclotron-resonance ion source The Electron-Cyclotron-Resonance (ECR) ion source was invented in 1969 and since developed by R Geller43 and his co-workers. It is a magnetic plasmaconﬁnement device which emerged from fusion research. In the ECR ion source the plasma electrons are heated to high energies by the use of microwaves while the ions remain relatively cold. Since its beginning the ECR technology has steadily improved and expanded to many laboratories. While there is still an ongoing lively development, today’s ECR ion sources are very reliable low-cost machines delivering ion currents of many different elements of gases and metals. The main ﬁelds of application are: • • •

ion sources for large accelerators, small experimental research facilities and singly charged ions for material processing.

4.7.1 Basic operation principle The conﬁguration of an ECR ion source is schematically presented in ﬁgure 4.48. It’s working principle is based upon the following components: •

trapping magnetic-ﬁeld conﬁguration,

43 Geller R 1996 Electron Cyclotron Resonance Ion Sources and ECR Plasmas (Bristol: IOP Publishing).

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r |B|=const Bradial microwaves

ions ECR surface gas

Baxial

axial solenoid

radial multipole

z

Figure 4.48. Conﬁguration of an ECR ion source. The plasma electrons are trapped in a minimum-B-ﬁeld structure and are heated by microwaves in resonance at the electron-cyclotron magnetic surface. (See colour section.)

• • •

electron heating by microwaves at electron cyclotron resonance, a supply of atoms as plasma fuel and a supply of plasma electrons.

For the ECR ion source a plasma-conﬁning magnetic ﬁeld is usually generated by a conﬁguration that is a superposition of an axial mirror ﬁeld and a radial multipole ﬁeld. Along the axis the ﬁeld varies as indicated in the ﬁgure with a minimum at the center of the source. For the radial hexapole ﬁeld assumed for the present case, the variation with the radius is also indicated. Such a conﬁguration, usually referred to as a minimum-B structure, has the capability of trapping charged particles for a long time. Figure 4.49 shows a view inside and parallel to the axis of an ECR ion source in operation44. The burning plasma shows the hexapolar symmetry of the magnetic conﬁguration. A metallic chamber is used as a plasma container and as a multi-mode cavity. The energy source for igniting and maintaining the plasma is resonant electron heating by a microwave generator coupled to the source by a wave guide. Many of the more powerful sources operate at frequencies between 10 and 18 GHz for which commercial devices developed for satellite communication are available. The microwave wavelength, λ = 3 cm for f = 10 GHz, has to be smaller than the dimensions of the plasma chamber. A thin gas introduced into the plasma vessel serves as the plasma fuel. It is essential for the operation of the ECR source to have a sufﬁcient supply of cold electrons which can be heated up in the ECR process. 44 Vamosi J et al 1997 Light Spectroscopy and X-ray Measurements on the new 14.5 GHz ATOMKI-

ECRIS (Proc. 13th Int. Workshop on ECRIS, Texas A and M University, College Station, Texas, USA).

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Figure 4.49. Axial view inside an ECR source showing the burning ECR plasma with its hexapole modulation. Picture courtesy S Biri, ECR Ion Source Laboratory, ATOMKI, Debrecen, Hungary.

For the production of highly charged ions through successive ionization by electron impact, it is important to maximize, on average, the product of the electron density, the electron velocity and the conﬁnement time, n e ve τc . At the same time, the density of the neutral atoms has to be kept low enough to avoid excessive electron capture processes. From the majority of the ECR ion sources ion beams are extracted whereas only in rare cases is the device solely used as an ion trap. The ion current derived from the source represents the ion loss rate. This implies a difﬁcult compromise to optimize a source for high-current, highcharge operation. On the one hand there must be a long conﬁnement time for efﬁcient ionization and on the other hand there must be a high loss rate (= short conﬁnement time) so as to give a high current. This is why all ECR ion sources need the operation parameters which affect the performance of the ECR plasma to be carefully tuned. As opposed to the electron-beam ion source, one only has indirect and limited control over the essential source parameters like the electron velocity or energy which all have some distribution. 4.7.2 Magnetic conﬁguration ECR ion sources may be classiﬁed by their speciﬁc magnetic-ﬁeld conﬁguration that is used to realize the minimum-B structure which functions as a magnetic mirror for hot electrons. For the good conﬁnement that is required for the production of highly charged ions, it is essential to achieve a large mirror ratio Bmax /Bmin . Typical mirror ratios range from 2 to 7. Table 4.5 summarizes the magnet technologies used in the prototype sources developed at CEA Grenoble. The ﬁrst ECR ion source delivering substantial amounts of highly charged ions was called superMAFIOS. Its strong solenoidal and hexapolar ﬁelds were all

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Table 4.5. Principal magnetic ﬁeld conﬁgurations used in the prototype ECR ion sources developed at CEA Grenoble. Hexapole

Prototype ion source

Warm

superMAFIOS cryoMAFIOS miniMAFIOS CAPRICE neoMAFIOS

Cold

Solenoid Perm Warm

x

Cold

Perm

Yoke

x x

x x x x

x x

x x

Solenoids

Microwave Biased Probe Ion Extraction Gas Inlet

Hexapole

Iron

Figure 4.50. MiniMAFIOS ECR ion source.

produced by conventional copper coils consuming much electrical power. The power consumption was subsequently reduced and the source was miniaturized by implementing superconducting coils and permanent magnets, respectively. The miniMAFIOS source sketched in ﬁgure 4.50 uses copper coils for the solenoidal ﬁeld and permanent magnets for the hexapolar ﬁeld. In the CAPRICE source an iron yoke was added. With the two frequently used materials for permanent magnets, SmCo5 or NdFeB, magnetic ﬁelds can be achieved such that the resonant frequency corresponds to 10 or 14 GHz, respectively. For these two frequencies microwave generators are readily available at reasonable cost. A low-power source such as the neoMAFIOS employs only permanent magnets. Hence only the microwave generator consuming electrical power remains. The magnetic arrangement of such an economic source is displayed in ﬁgure 4.51.

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Figure 4.51. Compact version of ECR ion sources use a magnetic conﬁguration realized entirely with permanent magnets. (See colour section.)

4.7.3 Resonant heating The plasma in the ECR ion source is enclosed in a magnetic volume where the magnetic isobars, the regions of equal |B| ﬁeld, form concentric closed surfaces. At one of these surfaces deﬁned by, cf (4.33), ωce =

e |BECR | = ωrf me

(4.139)

the electrons are in resonance with the rf ﬁeld. Electrons passing through the ECR surface at the right phase are accelerated by the transfer of electromagnetic energy perpendicular to the magnetic ﬁeld. The most efﬁcient heating is reached by launching a right-hand circularly polarized microwave along the main ﬁeld axis. The electric-ﬁeld vector of the wave rotates in the same sense as the electrons gyrate around the magnetic ﬁeld lines. For a left-hand polarized wave there is no resonance. For the heating to be efﬁcient the plasma electron density has to stay below the critical density n c at which the plasma electron frequency equals the cyclotron frequency. From equation (4.19) we ﬁnd the critical density scaling as n c ∝ ωrf2 . Hence a high electron density requires a high microwave frequency and accordingly a high magnetic ﬁeld. The strong electron heating in the transverse direction, where T⊥ T% , results in a very efﬁcient trapping between the high mirror ﬁelds at the ends of the plasma chamber. The enhanced trapping of the Larmor discs also known as mirror plugging can be understood from the conservation of angular momentum and energy in the absence of collisions. At the high-ﬁeld regions near the ends there

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Figure 4.52. Bremsstrahlung spectrum from an ECR ion source demonstrating the presence of electrons with high kinetic energy. From: Baru´e C et al 1992 Rev. Sci. Instrum. c 63 2844. 1992 AIP.

is only a small longitudinal thermal energy and the electrons oscillate between the mirror regions as they are reﬂected by the dipole force F = −µ∇ B, where µ denotes the magnetic moment of the circular currents. The electron energies in the ECR can exceed 100 keV as inferred from x-ray measurements. Such highenergy electrons may collisionally ionize inner-shell electrons of heavy materials present in the plasma giving rise to characteristic x-ray emission. There also is a pronounced electron bremsstrahlung spectrum extending beyond 100 keV as demonstrated in ﬁgure 4.52. 4.7.4 Electron supply To obtain a high electron density that is sufﬁcient for an effective ionization donors of cold electrons have to be present. The ﬁrst stage, i.e. the region towards the microwave inlet, serves as a source of both singly charged ions and slow electrons. A specially designed plasma cathode can add a constant supply of electrons as does a low-energy electron beam from an electron gun. Besides these external sources, internal electron sources are possible. Electrons ionized off the atoms can be accelerated and further ionize atoms and ions in the plasma. Wall coatings with a high secondary electron yield, such as SiO2 , contribute to the positive balance of the electron density. The introduction of a negatively biased metal disc helps to reduce plasma-electron losses. All these sources of electrons balance electron losses from the conﬁned plasma. 4.7.5 Enhancement of high charge states As already discussed, a high microwave frequency along with a high magnetic ﬁeld with a large mirror ratio generally ensures operation with a high output current of ions in intermediate to high charge states. Very high magnetic ﬁelds

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30

27+

intensity (arb.)

Au 20

10

0

rf pulse 0

100 time (ms)

200

Figure 4.53. Typical afterglow pulse of highly charged ions appearing at the end of the rf c pulse. From: Melin G et al 1994 Rev. Sci. Instrum. 65 1051. 1994 AIP.

are obtained using superconducting coils and high-power ECR sources will run at increased frequencies up to 30 GHz. There are a number of more speciﬁc upgrade technologies several of which are related to the electron supply methods mentioned earlier. Here they are shortly summarized: • • • • • • • •

electron-beam injection, biased electrodes, wall coatings, gas mixing, multi-frequency heating, light-ion ECR heating, pulsed afterglow operation and pulsed magnetic ﬁelds.

The use of a lighter carrier gas mixed with the atomic species of interest has received special attention. The enhanced output current is explained by a combination of several effects: a reduction in the mean ion charge lowering electron losses, a sympathetic cooling through a mass effect in ion–ion collisions, an increase in the electron density through the higher ionization rate for the lighter gas and an increase in the plasma stability. The gas-mixing effects can also be related to the use of several frequencies one of which would be tuned as to heat the light ions via ion-cyclotron resonance. The main point for more than one frequency, however, is due to the fact that there is more than one ECR surface. Thus electron heating takes place at an extended area moving towards volume heating rather than surface heating thereby increasing the overall performance. An ECR ion source can be operated very well in pulsed mode by pulsing the rf generator. The source can be tuned to release very intensive bursts of highly

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Mass-to-charge ratio 1

2

3

4

5

Current (arbitrary units)

Pb

O H 6

6

7

27+

8

O

9

10 11 12 13

2+

3+

+ 8

10

12

14

16

18

20

22

24

26

28

Dipole current

Figure 4.54. Typical afterglow pulse of highly charged ions appearing at the end of the rf pulse. From Tinschert K et al 1999.

charged ions just at the end of the rf cycle as shown in ﬁgure 4.53 for Au27+ ions45,46 . The duration of the ion pulse is of the order of 0.5–2 ms. This afterglow effect is a consequence of the electrostatic trapping of the ions in the space-charge potential well of the magnetically conﬁned electrons. After the breakdown of the discharge electrons leave the plasma volume and are no longer replaced. Because of charge neutrality the ions have to leave as well producing the strong ion pulse observed. A better control of the pulsed release of highly charged ions can be gained by pulsing the magnetic ﬁeld as well47 . This way the pulse shape of the extracted current can be controlled. Figure 4.54 shows the charge-state spectrum of Pb ions in the afterglow pulse of a CAPRICE source tuned for this type of operation48. The pulsed mode ﬁts the needs for injection into rf accelerators with a correspondingly low duty cycle very well. ECR ion sources operating in the pulsed afterglow mode are in use or under construction at virtually all the major heavy-ion accelerator facilities.

45 Melin G et al 1994 Rev. Sci. Instrum. 65 1051. 46 Sortais P 1992 Rev. Sci. Instrum. 63 2801. 47 M¨uhle C et al 1994 Rev. Sci. Instrum. 65 1078. 48 Tinschert K et al 1999 Proc. of the 14th Int. Workshop on ECR Sources (ECRIS99, CERN, Geneva).

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4.8 Electron-beam ion source and trap 4.8.1 Basic principle of operation The operating principle of the electron-beam ion source (EBIS) was proposed and successfully demonstrated by Evgenii Donets and his collaborators49. It is now a powerful source of very highly charged ions at low velocities. Positive ions trapped inside a very dense electron beam are continuously bombarded by electrons and thereby successively ionized to very high charge states. Physical parameters like the electron density and energy and the conﬁnement time can be controlled more directly than in other ion sources. The electron-beam ion trap (EBIT) developed at the Lawrence Livermore National Laboratory (LLNL) is a powerful extension to the EBIS concept50,51,52 . In the EBIT it is possible to generate ions of the highest charge states and to store them for extended periods of time. A high-energy version of the EBIT called the superEBIT can be operated near 200 kV which is sufﬁcient to ionize the K shell of uranium53. The EBIT was originally designed for internal experiments for the spectroscopic investigation of highly charged ions. Later, ions were also extracted for external experiments including the trapping in an electromagnetic trap under ultra-high vacuum. Both, the EBIS and EBIT employ a high-density electron beam which is launched along the axis of a strong magnetic ﬁeld forming a radial space-charge well for ions. Cylindrical electrodes surrounding the electron beam can be individually biased in order to determine the electrical potential along the electron beam. There is a large voltage drop along the gap between the electron gun and the ﬁrst electrode to accelerate the electrons to a high kinetic energy. In addition, the end electrodes have a relative positive bias so as to conﬁne trapped ions in the longitudinal direction. The length of the trap is typically around 1 m for the EBIS and about 2–3 cm for the EBIT. The high magnetic ﬁeld of up to about 5 tesla is produced by a long superconducting solenoid in the EBIS or by a pair of superconducting Helmholtz coils in the EBIT. The high magnetic ﬁeld serves to compress the electron beam down to diameters of below 100 µm reaching current densities up to several thousand A cm−2 . Figure 4.55 shows the EBIS scheme used at Kansas State University for a large variety of atomic-physics experiments54. Atoms from the residual gas introduced through a leak valve are ionized by electron impact. It is also 49 Donets E D, Ilushchenko V I and Alpert V A 1969 Premi´ere Conference sur les Sources d’Ions (Saclay, France) p 625. 50 Marrs R E, Beiersdorfer P and Schneider D 1994 Phys. Today October 27 51 Schneider D 1996 Hyp. Int. 99 47. 52 Schneider D H G et al 1999 Atomic Physics with Heavy Ions ed H F Beyer and V P Shevelko (Berlin: Springer) p 30. 53 Marrs R E 1996 Rev. Sci. Instrum. 67 941. 54 St¨ockli M P 1997 Accelerator-Based Atomic Physics Technics and Applications ed S M Shafroth and J C Austin (New York: AIP) p 67.

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Figure 4.55. Schematic view of an electron-beam ion source and typical electrical potentials along the electron beam. The gate between seed and main trap is opened at the beginning of a production cycle ﬁlling the main trap with seed ions which continue to be ionized until they are expelled from the source by a sudden change of the main trap c potential. From: St¨ockli M P et al 1992 Rev. Sci. Instrum. 63 2822. 1992 AIP.

possible to inject singly charged ions from an external ion source. The singly charged ions are trapped in the electron beam where they are stepwise ionized to consecutively higher charge states until equilibrium with electron-capture and other loss processes is reached. The performance of the ion conﬁnement depends on the dynamics of the space-charge compensation of the electron beam by ions produced from the residual gas. With increasing degree of space-charge compensation cold ions are better conﬁned whereas hot ions are lost. The ions may be extracted by a suitable change in the electrode potentials. Various operation modes have been developed and optimized for speciﬁc experimental demands. In a leaky mode there is a steady current of ions which is due to ions which have enough kinetic energy to escape from a moderately deep central-trap potential. The highest charge states are obtained in the batch mode where low-charge ions are ﬁrst trapped in a seed trap as shown in ﬁgure 4.55. A production cycle starts with the transfer of ions from the seed trap to the main trap by lowering the separating gate potential. After the necessary conﬁnement time for the desired charge state the main trap potential is suddenly raised expelling the ions from the trap. Figure 4.56 shows an outline of the high-energy superEBIT which is a rather compact laboratory device. The use of Helmholtz coils leaves enough space for viewing ports to observe the x-ray radiation resulting from the collisional processes inside the small source volume.

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Figure 4.56. Outline of the high-energy electron-beam ion trap, superEBIT, developed at LLNL. An intense electron beam is compressed by the magnetic ﬁeld produced by a pair of superconducting coils. From: Marrs R E, Elliott S R and Knapp D A 1994 Phys. Rev. c Lett. 72 4082. 1994 APS.

4.8.2 Step-by-step ionization As discussed in section 4.1 a number of atomic collision processes govern the evolution of high charge states. For lighter elements like argon single-ionization processes dominate and the balance equations might be simpliﬁed as in equation (4.36). Recombination and capture processes become increasingly important for heavy ions and the development of the charge states may be formally written as q−1 Z

dn q I I = n q n e ve σq,q − n q n e ve σq,q dt q=0

+

q=q+1

Z

q=q+1

C n q n 0 vq σq,q −

q−1

C n q n 0 vq σq,q

q=0

RR RR − n q n e ve σq,q−1 − + n q+1 n e ve σq+1,q

nq τc

(4.140)

I represents the cross section for ionization from charge state q to q where σq,q C RR and σq,q the corresponding capture cross section and σq,q−1 the cross section for radiative recombination, respectively. The last term takes into account a ﬁnite conﬁnement time τc . With the general decrease in the cross section with the binding energy the time necessary to increase the charge by one unit also increases accordingly. This becomes even more pronounced when approaching an atomic

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Figure 4.57. Evolution of charge states of uranium under the impact of 200 keV electrons. In the calculation, stepwise ionization and radiative recombination is considered. From: Knapp D A 1995 Physics with Multiply Charged Ions ed D Liesen (New York: Plenum), c 1995 Plenum.

shell, especially the K shell, which brings about a large step in the ionization cross section. As a consequence, in the process of generating a hydrogen-like ion, for instance, a vast amount of time is spent in the last ionizing step of the helium-like ion. For a practical estimate of the charge-state distribution of very highly charged ions, therefore, the last several steps in the ionization process are most important. An enhanced population of a single charge state can be reached when the electron energy is tuned just below the ionization potential of an electron shell. For heavy ions the charge-loss processes are dominated by radiative recombination. A calculation of the charge evolution applied for the superEBIT is shown in ﬁgure 4.57. There, single-step ionization and radiative recombination was considered for uranium ions bombarded with electrons with a kinetic energy of 200 keV. After a period of about 10 s equilibrium among the dominating charge states, U88+ . . . U91+ is reached.

4.8.3 Ion heating and cooling The long ionization times necessary for the creation of highly charged ions imply correspondingly long times for the ions to spend inside the trap. Keeping the ions conﬁned requires attention to the ions thermal energy. A number of different processes may contribute to the heating of the ions, such as heating by elastic electron–ion collisions, sudden charge changes in the ions in the potential well or instabilities in the electron plasma. For the EBIT collisional heating dominates.

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Figure 4.58. Evaporative cooling of highly charged ions by an admixture of low-charged ions. The low charged ions boil off from the trap more rapidly than the highly charged ions. From: Knapp D A 1995 Physics with Multiply Charged Ions ed D Liesen (New c York: Plenum), 1995 Plenum.

It’s rate can be expressed as 0.442q 2 jen q ln(ρmax /ρmin ) dE q = eV s−1 cm−3 dt AE e

(4.141)

where n q denotes the number density of q-times ionized ions and A their atomic mass number whereas ρmax and ρmin are the maximum and minimum impact parameters, respectively. The heating rate per ion can reach several keV which has to be compared with the trapping potential depth of less than 1 keV. To keep highly-charged ions in the trap sufﬁciently strong cooling mechanisms are therefore needed. For the EBIT an evaporative cooling technique has been developed55 which works in a way similar to the gas-mixing effect in the ECR ion source. The principle of evaporative cooling is illustrated in ﬁgure 4.58. Two ion species with very different ionic charges are simultaneously trapped with a trapping voltage of Vtrap . The potential energy of the ions is proportional to their charge q but they will have a common ion temperature Ti if they are allowed to interact for a sufﬁciently long time. Assuming a Boltzmann distribution the different ions will leave the trap with different loss rates qeVtrap −1 (4.142) Rloss(q) = τcomp exp − Ti where Vtrap is the effective potential depression in the charge compensated beam and τcomp the mean compensation time. From equation (4.142) we note that 55 Levine M A et al 1988 Phys. Scr. T 22 157.

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Figure 4.59. X-ray spectrum from the superEBIT operated at an electron energy near 200 keV. The high-energy lines are due to radiative recombination of hydrogen-like and bare uranium ions with electrons. From: Marrs R E, Elliott S R and Knapp D A 1994 c Phys. Rev. Lett. 72 4082. 1994 APS.

low charge ions are preferentially lost which is why light ions are an efﬁcient coolant for heavy highly-charged ions. The light ions are rapidly ionized to their maximum charge state while still being cold enough to cool down the heavy ions. The light ions can be derived from the residual gas, an external beam of neutral atoms or singly charged ions. The lowest mean ion temperature of trapped highlycharged ions is limited by the trapping potential Vtrap as implied by equation (4.142). In the superEBIT a beam of neutral neon atoms was injected for the evaporative cooling of uranium ions as indicated in ﬁgure 4.56. The method produced impressive results on the production of few-electron uranium ions which were detected by the measurement of x-ray spectra, an example of which is shown in ﬁgure 4.59. The prominent x-ray lines are due to radiative recombination of electrons with hydrogen-like and bare uranium ions populating the ﬁnal electron shells as indicated in the ﬁgure.

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Chapter 5 Atomic structure

In this chapter, the atomic structure of highly charged ions including energy levels, transition probabilities, radiation spectra and the behavior of ions in electric and magnetic ﬁelds is considered.

5.1 Classiﬁcation of spectral lines As we learned in chapter 3, each atom or ion has its own set of stationary quantum states with a characteristic set of discrete energies or electronic binding energies E k . If a bound electron makes a transition from an upper level E 1 to a lower one E 0 , a photon is emitted with an angular frequency ω and wavelength λ given by

~ω = E 1 − E 0

λ = 2πc/ω.

(5.1)

The emission of photons following transitions between bound states leads to the formation of spectral lines or discrete spectra. The binding energies of a hydrogen-like system with its central Coulomb ﬁeld Z e2 V (r ) = − (5.2) r depend only on the principal quantum number n and are given by the Rydberg formula (3.42) with the reduced-mass correction, En = −

Z2 1 Ry 1 + m e /M n2

(5.3)

where Ry denotes the Rydberg energy, m e and M the electron and nuclear masses and Z the nuclear charge, respectively. Expression (5.3), following from Bohr’s postulates, coincides with the eigenenergies obtained from the exact solution of Schr¨odinger’s equation for a one-electron system. For a hydrogen atom (Z = 1) in its ground state n = 1, equation (5.3) yields an ionization energy E 1 (H) ≈ 13.5984 eV. The difference

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in 0.007 eV between E 1 (H) and the Rydberg energy originates from the inﬂuence of the ﬁnite nuclear mass of the hydrogen for which the ratio MH /m e ≈ 1836.2. In the case of many-electron systems, the binding energies cannot be described by the simple formula (5.3) because of the electric and magnetic interactions between the atomic electrons and determining the binding energies becomes quite a complicated problem even for two-electron systems such as He and He-like ions. For many-electron ions, the energy levels and wavelengths are well described by quantum-mechanical and QED approaches which include the interactions in high-order approximations as will be discussed later in this chapter. Alkali atoms (Li, Na, . . . ) represent a certain exception as they have one valence electron outside a spherical atomic core approximating a simple oneelectron system. The atomic core of an ion with N electrons consists of the nucleus and a remaining N − 1 electrons. The energy levels of the alkalis are described by a formula similar to (5.3) but additionally depend on the orbital quantum number of the state in the form E n = −

Ry 1 2 1 + m n eff e /M

n eff = n − (n, )

(5.4)

where n eff denotes the effective quantum number, which is not generally an integer and (n, ) is the quantum defect depending on the n and quantum numbers. The –values are largest for s electrons, decrease with increasing and, for a given , are nearly independent of n. In a many-electron ion Xq+ , an orbital electron feels a certain effective potential Veff (r ) created by the atomic core, i.e. by the nucleus and other electrons. Due to the screening of the nuclear charge by the core electrons, the potential Veff (r ) is not a pure Coulomb one but approaches it at small and large distances r: Z e2 r →0 Veff (r ) ≈ − r (5.5) Z eff e2 (Z − N + 1)e2 ≡− r →∞ Veff (r ) ≈ − r r where Z denotes the nuclear charge and N the total number of atomic electrons. The effective nuclear charge number Z eff is deﬁned as the net charge number of the core: Z eff = Z − N + 1 = q + 1 (5.6) where q is the ionic charge number. For neutral atoms, the effective charge number is unity, Z eff = 1; for H-like ions it coincides with the nuclear charge number, Z eff = Z ; and for highly charged ions, one has Z N, i.e. Z eff ≈ Z . Most atomic characteristics can be scaled by Z eff which is very useful for many practical applications. In table 5.1, we summarize the characteristics of highly charged ions and the way in which they approximately scale with the nuclear charge number Z eff ≈ Z . In the case of low-charge ions, the

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Table 5.1. Approximate scaling behavior of the radiative and collisional characteristics of highly charged ions with the nuclear charge Z. The quantities α, m e and m p denote the ﬁne-structure constant and the electron and proton mass, respectively. Atomic radius Ionization potential Transition energy (frequency) Fine-structure (multiplet) splitting Hyperﬁne splitting Lamb shift Oscillator strength Radiative transition probability Radiative lifetime Relative Doppler width ω D /ω Autoionization probability Static dipole polarizability External electric ﬁeld strength for ionization Photoionization cross section Photorecombination rate coefﬁcient Bremsstrahlung cross section Electron-impact excitation and ionization cross sections Electron-impact excitation and ionization rate coefﬁcients Electron capture cross section Excitation and ionization cross sections of neutral atoms by ions Ion temperature

Z −1 Z2 Z2 α2 Z 4 (m e /m p )α 2 Z 3 α5 Z 4 Z0 Z4 Z −4 Z1 Z0 Z −4 Z3 α 1 Z −2 Z2 α3 Z 2 Z −4 Z −3 Z a , a = 1–5 Z2 Z2

corresponding scalings can be obtained by substituting the nuclear charge number Z with the effective charge number Z eff . Because of the different dependencies upon Z observed in the quantities listed in table 5.1, their relative importance strongly depends on Z . For instance, the size of a highly charged ion decreases as 1/Z as the nuclear charge Z increases. This means that the size of H-like uranium Z = 92 is about 100 times smaller than that of the neutral hydrogen. The inﬂuence of strong Coulomb forces inside the highly charged ion separates out its energy levels clearly. For example, the leading term in the radiative corrections responsible for the Lamb shift increases in proportion with Z 4 , whereas the electronic binding energies scale only as Z 2 . Therefore, the relative Lamb shift contributions are much enhanced at high values for the nuclear charge. The energy levels are often labeled according to the so-called L S-coupling scheme which will be treated in section 5.2. Within this description, an electron term is denoted as 2S+1 LJ (5.7)

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Table 5.2. Notation for electric and magnetic transitions. Notation

Transition

Example

E1 M1

Electric dipole Magnetic dipole

E1M1 E1M2 E2M1 E2 M2 2E1

Two-photon E1 + M1 Two-photon E1 + M2 Two-photon E2 + M1 Electric quadrupole Magnetic quadrupole Two-photon electric dipole

2 1 P1 → 1 1 S0 in [He] 2s1/2 → 1s1/2 in [H] 2 3 S1 → 1 1 S0 in [He] 2 3 P0 → 1 1 S0 in [He] 2p3/2 → 1s1/2 in [H] 2p3/2 → 2p1/2 in [H] 2p3/2 → 2p1/2 in [H] 2 3 P2 → 1 1 S0 in [He] 2s1/2 → 1s1/2 in [H] 2 1 S0 → 1 1 S0 in [He]

2E2 2M1 2M2

Two-photon electric quadrupole Two-photon magnetic dipole Two-photon magnetic quadrupole

where L and S are the quantum numbers for the orbital and spin angular momenta of an ion, respectively, and J is the total angular-momentum quantum number which, according to the general rule for quantum vector addition J = L + S, can take the values |L − S| ≤ J ≤ L + S. (5.8) The superscript 2S + 1 shows the multiplicity of the term. For instance, the terms with S = 0 form singlet levels and the terms with S = 1 form triplet levels. The quantum numbers of the total orbital angular momentum L are labeled by the letters S, P, D, F, G, H, . . . corresponding to the values of L = 0, 1, 2, 3, . . .. The non-alphabetical order of the ﬁrst four letters S, P, D, F has historical reasons: these correspond to the initials of the words Sharp, Principal, Diffuse and Fundamental according to the spectral series ending on these terms. In some cases, the spectral notation of the L S term (5.7) is complemented by a right-hand superscript of e or o specifying the even or odd parity of the electronic conﬁguration, respectively, 2S+1 e,o LJ .

(5.9)

The parity refers to a fundamental symmetry to be dealt with in section 5.3. Here, we give tables of the different notation used in atomic and x-ray spectroscopy to deﬁne atomic-energy levels and spectral lines. This notation should enable the reader to understand the material given later. The notation used for electric, Eκ, and magnetic, Mκ, multipole transitions are given in table 5.2 where κ is the multiplicity of the transition. The notation adopted in x-ray spectroscopy is listed in table 5.3. There, K, L, . . . , Q shells correspond to the

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Table 5.3. Notations for atomic shells used in x-ray spectroscopy. Shell

Atomic notation

Shell

Atomic notation

K L1 L2 L3 M1 M2 M3 M4 M5 N1 N2 N3 N4 N5

1s1/2 2s1/2 2p1/2 2p3/2 3s1/2 3p1/2 3p3/2 3d3/2 3d5/2 4s1/2 4p1/2 4p3/2 4d3/2 4d5/2

N6 N7 O1 O2 O3 O4 O5 O6 O7 P1 P2 P3 P4 Q1

4 f5/2 4f7/2 5s1/2 5p1/2 5p3/2 5d3/2 5d5/2 5f5/2 5f7/2 6s1/2 6p1/2 6p3/2 6d3/2 7s1/2

Table 5.4. Notations for the basic spectral lines in H- and He-like ions. Sequence

Transition

Notation

H H He He He He He He He

2p1/2 → 1s1/2 2p3/2 → 1s1/2 2s1/2 → 1s1/2 2p1/2 → 1s1/2 2p3/2 → 1s1/2 1s2p 1 P1 → 1s2 1 S0 1s2p 3 P2 → 1s2 1 S0 1s2p 3 P1 → 1s2 1 S0 1s2p 3 S1 → 1s2 1 S0

Lyman-α2 Lyman-α1 M1 Kα2 Kα1 w, resonance line x, magnetic quadrupole M2 transition y, intercombination transition z, forbidden line

electronic conﬁgurations with the principal quantum numbers n = 1, 2, . . . , 7, respectively. Transitions with a change in the total spin |S| = 1 are called intercombination transitions. Transitions which are not allowed by the selection rules, to be worked out in section 5.3, are called forbidden transitions. The basic spectral transitions in Hand He-like ions and those corresponding to the dielectronic satellites are given in tables 5.4 and 5.5. The spectral lines of He-like ions and their satellites are usually identiﬁed using Gabriel’s notation1. The satellite lines and parent lines are 1 Gabriel A H 1972 Mon. Not. R. Astron. Soc. 160 99.

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Table 5.5. Notation for the transitions in Li- and He-like ions in the j j -coupling scheme. Key

Transition

a

(1s2p23/2 )3/2 –1s2 2p3/2

b c d e f g h i j k l m n o p q r s t u v w x y z

(1s2p23/2 )3/2 –1s2 2p1/2 (1s2p1/2 2p3/2 )1/2 –1s2 2p3/2 (1s2p1/2 2p3/2 )1/2 –1s2 2p1/2 (1s2p1/2 2p3/2 )5/2 –1s2 2p3/2 (1s2p1/2 2p3/2 )3/2 –1s2 2p3/2 (1s2p1/2 2p3/2 )3/2 –1s2 2p1/2 (1s2p21/2 )1/2 –1s2 2p3/2 (1s2p21/2 )1/2 –1s2 2p1/2 (1s2p23/2 )5/2 –1s2 2p3/2

(1s2p1/2 2p3/2 )3/2 –1s2 2p1/2 (1s2p1/2 2p3/2 )3/2 –1s2 2p3/2 (1s2p23/2 )1/2 –1s2 2p3/2 (1s2p23/2 )1/2 –1s2 2p1/2 (1s2s2 )1/2 –1s2 2p3/2 (1s2s2 )1/2 –1s2 2p1/2 (1s2s2p3/2 )3/2 –1s2 2s1/2 (1s2s2p1/2 )1/2 –1s2 2s1/2 (1s2s2p3/2 )3/2 –1s2 2s1/2 (1s2s2p3/2 )1/2 –1s2 2s1/2 (1s2s2p1/2 )3/2 –1s2 2s1/2 (1s2s2p1/2 )1/2 –1s2 2s1/2 (1s2p1/2 )1 –(1s2 )0 (1s2p3/2 )2 –(1s2 )0 (1s2p3/2 )1 –(1s2 )0 (1s2s)1 –(1s2 )0

often used for x-ray diagnostics of high-temperature laboratory and astrophysical plasmas and will be explained in sections 5.12 and 6.8.2.

5.2 Coupling schemes In addition to the conservation of angular momentum known from classical mechanics, two new features show up in the quantum-mechanical description of an atom. The ﬁrst is quantization which we already discovered in the Bohr model in the form of the quantization of the orbital angular momentum (3.38) taking only integral multiples of ~. The second important feature is the spin—the

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sz

z m

1 + 2

s

0

0

−

1 2

|| = ( + 1)

|s| = s(s + 1)

Figure 5.1. Directional quantization of the orbital and spin angular momenta.

Vector model s

µ

µs

s spin-orbit precession

j

Figure 5.2. The vector model of the angular momenta of the electron.

intrinsic angular momentum carried by particles like the electron. The spin of the electron is always ~/2. The angular momenta can be explored by introducing operators of the square of the angular momentum L 2 and of the projection L z of the angular momentum to any preselected direction z in space. For a one-electron system it is easy to show that the stationary electronic states are eigenstates of the 2 operator with eigenvalues ( + 1)~2 and of the z operator with eigenvalues m ~, where m can take the integer values m = −, − + 1, . . . , − 1, . In the semiclassical vector model, illustrated in ﬁgures 5.1 and 5.2, the angular√ momentum vector of length ( + 1)~ rotates around the z-axis but only at those angles √ to the axis which satisfy the condition that z = m ~. For the spin vector of length s(s + 1)~, showing a similar behavior, the corresponding m s values can be ±1/2.

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The directional quantization manifests itself through the magnetic moment µ related to the angular momentum of the electron. Because of the negative charge of the electron the magnetic moments are always antiparallel to the angular momenta as indicated in ﬁgure 5.2. For a classical Bohr orbital, the magnetic moment of the corresponding circular electric current is e~ = −nµB 2m e

(5.10)

e~ 1 = ea0αc 2m e 2

(5.11)

µ = −n where µB =

is the Bohr magneton which is the magnetic moment of the n = 1 state of atomic hydrogen. For the spin of the electron, the ratio between magnetic moment and anguar momentum, given by the gyromagnetic ratio, is a factor of two bigger than for the orbital motion, a fact which can not be explained classically. We will dwell more on that in section 5.9. The presence of magnetic moments lead to a magnetic energy and a corresponding splitting of the energy states in magnetic ﬁelds. They also interact magnetically with each other leading to the spin-orbit precession indicated in ﬁgure 5.2. The precessional motion with a certain value of the z component means that the x and y projections are not known in accordance with the Heisenberg uncertainty principle. In a many-electron system the coupling scheme or vector coupling shows how the orbital i and spin si angular momenta of the individual atomic electrons are coupled into the total angular momentum J of the whole system depending on the magnetic and electrostatic interactions between the electrons in the atom. In the zero-order approximation, the Coulomb interaction between the electrons and the nucleus and the electrostatic interaction among the electrons themselves is represented by a central-symmetrical ﬁeld of the atomic core similar to Veff (r ) introduced in the previous section. In this approximation, the total energy is represented by the energy of the electronic conﬁguration n 1 1 , n 2 2 , . . . , n N N which is given by the set of the principal n and orbital quantum numbers of all atomic electrons. The inclusion of other interactions such as the non-central part of the electrostatic interaction and the magnetic interactions, i.e. the spin–orbit interaction, ﬁrst of all, leads to the splitting of the atomic level into different sublevels. The relative position of these sublevels can be described by vector addition of the individual angular momenta i and si of different electrons in the system to the total momentum J . As we have seen there is only one possibility in combining the angular momenta and s of a one electron system giving the total angular momentum j = + s. If one has two non-equivalent electrons with angular momenta 1 , s1 and 2 , s2 , there are four possibilities for coupling these vectors to the total momentum J which is conserved. Depending on the way in which the vectors

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are added, one has four types of vector coupling: 1 + 2 = L,

s1 + s2 = S,

L+S= J

L K coupling: 1 + 2 = L,

L + s1 = K ,

K + s2 = J

j K coupling:

1 + s1 = j1 ,

j1 + 2 = K ,

K + s2 = J

j j coupling:

1 + s1 = j1 ,

2 + s2 = j2 ,

j1 + j2 = J .

L S coupling:

(5.12)

Two equivalent electrons having the same quantum numbers n and obey the Pauli principle after Wolfgang Pauli (1900–1958), which states that no two fermions, i.e. particles with a spin of ~/2, can exist in identical energy quantum states. That is why the two equivalent electrons have to differ in their spin quantum number. As a consequence, only L S and j j couplings are possible because all electrons participate symmetrically. Each type of coupling scheme is characterized by the relative values of the different types of interactions. The frequently used L S coupling, also called Russell–Saunders coupling, is justiﬁed when the electrostatic interaction Ves is much larger than the spin–orbit interaction Vso , Ves Vso . In the L S scheme, a strong electrostatic interaction couples the momenta i and si of all states contributing to a given L S term with the total angular L and spin S angular momenta:

L= i S= si (5.13) i

i

and the vectors L and S sum to give the vector of the total momentum J = L + S which is conserved. In L S coupling, the numbers L and S in addition to the total angular momentum J are also good quantum numbers. L S coupling is appropriate for neutral atoms with a nuclear charge number Z 20 and lowcharged ions in not very excited states. The opposite case to L S coupling is j j coupling. With increasing nuclear charge, we approach the situation where

.

Ves Vso because the electrostatic interaction increases approximately with Z while the spin–orbit one increases as Z 4 (section 5.8.1). The j j coupling is realized for heavy atoms and highly charged ions, and is also used in nuclear physics. With increasing nuclear charge, a smooth transformation from L S to the j j coupling occurs as shown in ﬁgure 5.3 for the transition energies in He-like ions. As can be seen, for low Z values, very small energy splitting of the 3 P0,1,2 terms is observed whereas at high nuclear charges, the levels are grouped according to their total angular momentum J .

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Figure 5.3. Energy differences (in eV) scaled over the nuclear charge Z for 1s2s 3 S1 –1s2p transitions in helium-like ions as a function of the nuclear charge.

In j j coupling, the deﬁnitions of total momenta L and S lose their meaning because the spin–orbit interaction strongly mixes the momenta i and si of different electrons, and one can only speak about the total momentum ji of the individual electron which is conserved and coupled into the total atomic momentum J . In j j coupling, the total angular-momentum quantum number for an electron is = j ±1/2 since the electron-spin quantum number is s = 1/2. The j -values are usually shown as a right subscript below and the total conﬁguration is written in the form: s1/2 , p1/2 , p3/2 , d3/2 , d5/2 , f5/2 , f7/2 , . . . similar to designation of L S terms. Notations in j j coupling are given in table 5.5 for two- and three-electron ions. The L K and j K coupling schemes are quite rarely used. j K coupling is also called j l coupling and is applied when the spin–orbit interaction of the electrons of the atomic core is larger than the electrostatic interaction of these electrons with the excited electron. j l coupling appears in the spectra of excited states of ions with a noble-gas conﬁguration (Ne-, Ar-like, etc) with closed atomic shells and other cases when one of the electrons is, on average, at a large distance from the atomic core. In L S coupling the designation 2S+1 L J has been introduced earlier. For example, for the two-electron conﬁguration npn p, one has the singlet (S = 0)

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and the triplet (S = 1) terms: 1

S0 , 1 P1 , 1 D2 , 3 S1 , 3 P0,1,2 , 3 D1,2,3 .

Designating the terms in the other coupling schemes is more complicated. For the same conﬁguration npn p one has: L K coupling, level designation L[K] J : S[ 12 ]0,1 , P[ 12 ]0,1, P[ 32 ]1,2 , D[ 32 ]1,2 , D[ 52 ]2,3 j K coupling, level designation j [K] J : 1 1 3 1 1 3 3 3 3 5 2 [ 2 ]0,1 , 2 [ 2 ]0,1 , 2 [ 2 ]1,2 , 2 [ 2 ]1,2 , 2 [ 2 ]2,3

j j coupling, level designation [ j1 j2] J : [ 12 12 ]0,1 , [ 12 32 ]1,2 , [ 32 12 ]1,2 , [ 32 32 ]0,1,2,3. For a given total angular momentum J , the number of sublevels is the same for all types of coupling scheme. For example, in our case the number of sublevels with J = 1 equals 4. When the electrostatic and spin–orbit interactions are of the same order of magnitude, i.e. Ves ≈ Vso , the energy levels are described by the intermediate coupling scheme. Then the total wavefunction of the system is found by expanding it in terms of the wavefunctions ϕ given in ‘pure’ L S or j j coupling:

= Ci ϕi (L S( j j )). (5.14) i

This way the problem is reduced to determining the Ci expansion coefﬁcients. Intermediate coupling is used when a ‘pure’ coupling scheme cannot be applied. In general, which type of angular-momentum coupling, or vector addition is applied, is a question of which interaction is larger—the electrostatic or spin– orbit one.

5.3 Selection rules According to the Bohr postulates, considered in section 3.7, an atom or ion radiates a photon only if it makes a transition from an upper stationary state to a lower one. The properties of emitted radiation such as its intensity, multiplicity and polarization are deﬁned by the selection rules for radiation. The selection rules are formulated on the basis of quantum mechanical principles and describe the change in quantum numbers of an atom or ion undergoing a transition and reﬂect the energy and momentum conservation laws of the total system comprising atom and photon2. The bound electron which makes a transition is 2 See in detail Sobelman I I 1992 Atomic Spectra and Radiative Transitions (Berlin: Springer).

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often called the optical electron; it can belong to the outermost or inner shell of an atom or an ion. Exact and approximate selection rules should be distinguished. The ﬁrst follow from the conservation laws and the properties of the angular parts of the operators describing the transition matrix elements for the electric and magnetic interactions. The exact selection rules are independent of the coupling scheme of the angular momenta and formulated for exact quantum numbers—parity P, the total angular momentum quantum number J and its projection M. The exact selection rules for a change of the total angular momentum J , its projection M and the parity P read: J + J ≥ κ,

(5.15)

M = 0, ±1, . . . , ±κ for Eκ transitions (−1)κ P = (−1)κ+1 for Mκ transitions

(5.16)

J = 0, ±1, . . . , ±κ

(5.17)

where κ is the multiplicity of the transition. For dipole transitions, one has κ = 1, for quadrupole ones κ = 2, and so on. The notations for electric Eκ and magnetic Mκ transitions are given in table 5.2. The selection rules (5.16) reﬂect the conservation of the total angular momentum of the system consisting of atom and photon: J = J + κ.

(5.18)

Here J and J are the angular momenta of the initial and ﬁnal atomic states and κ is the photon angular momentum: | J | = J (J + 1)~ | J | = J (J + 1)~ |κ| = κ(κ + 1)~ (5.19) where ~ is the Planck’s constant divided by 2π and J and J are the quantum numbers of the total angular momenta. The parity of the atomic state characterizes the symmetry, or transformation behavior, of the corresponding wavefunction (r) upon an inversion transformation of the coordinates r → −r: (−r) = ±(r).

(5.20)

If the wavefunction does not change sign, the wavefunction is called even or of even parity. If the negative sign holds, one has a wavefunction of odd parity or, simply, an odd wavefunction. In the approximation of a centrally symmetric ﬁeld the wavefunction of an electron is given by the product of the radial Rn (r ) and angular Ym (θ, ϕ) functions (r) = Rn (r )Ym (θ, ϕ) (5.21) where Ym (θ, ϕ) is a spherical harmonic.

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Since in spherical coordinates, the reﬂection r → −r corresponds to the transformation r →r θ →π −θ ϕ →ϕ+π (5.22) and the radial part Rn (r ) does not change sign, the parity is totally deﬁned by the angular part of the transformation: Ym (π − θ, ϕ + π) = Ym (θ, ϕ)(−1)

(5.23)

i.e. by the factor P = (−1). For a many-electron system described by a centrally symmetric ﬁeld, the total wavefunction is a product of the single-electron functions (5.21), and, therefore, the total parity is given by P = (−1)

i i

.

(5.24)

Thus, the atomic state is even, if the sum of angular momenta i i is even and it is odd, that the parity is deﬁned by the arithmetical if the sum is odd. We note sum i i but not the vector sum i i . The approximate selection rules are formulated for a change in the quantum numbers of the initial and ﬁnal states described in a particular coupling scheme. In the L S-coupling scheme one has additional selection rules: for electric Eκ transitions: L = 0, ±1, . . . , ±κ

L + L ≥ κ

S = 0

(5.25)

for magnetic Mκ transitions: L = 0, ±1, . . . , ±(κ − 1)

L + L ≥ κ − 1

S = 0, ±1, . . . , ±(κ − 1)

S + S ≥ κ − 1.

(5.26)

For electric-dipole E1 transitions (κ = 1), also called optically allowed transitions, one has: L = 0, ±1 S = 0 (5.27) P = ±1 and L = 0 → L = 0 transitions are forbidden by these rules. For H-like ions, the selection rules for dipole transitions nm − n m read: = − = ±1

m = m − m = 0, ±1

(5.28)

and for transitions nl j − n l j between the ﬁne-structure components, respectively, j = ± 1/2 (5.29) j = j − j = 0, ±1

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where m and m are the magnetic quantum numbers deﬁned as projections of the angular momenta and , respectively, and j is the total angular momentum. There are no limitations on the change in the principal quantum numbers n and n . For electric quadrupole E2 transitions, the approximate selection rues are: L = 0, ±1, ±2

L + L ≥ 2

(5.30)

i.e. transitions between S terms (L = L = 0) and between S and P terms (L = 0, L = 1) are forbidden. A 2p3/2 –2p1/2 transition in H-like ions is an example of an E2 transition. For all electric Eκ transitions, the selection rules for spin quantum numbers S and S are the same: they allow transitions between terms with equal multiplicity 2S + 1. The 1s2p 3 P1 –1s2 1 S0 transition in He-like ions (y-line) is called an intercombination line because it is forbidden on change of spin; and the 1s2p 3 S1 – 1s 2 1 S0 transition (z-line) is forbidden because it is forbidden on both change of spin and orbital momenta L = 0. The x and y lines are widely used in plasma diagnostics and the determination of polarization of x-ray lines (see section 5.16). In a many-electron system, according to (5.15)–(5.17) the exact selection rules for the electric dipole E1 radiative transition L S J M–L S J M read: J = J − J = 0, ±1

J + J ≥ 1

M = M − M = 0, ±1

(5.31)

P = ±1. Magnetic-dipole M1 and quadrupole M2 transitions play a key role in hightemperature low-density plasmas and are very important for plasma diagnostics, for example, 3 P2 –1 S0 transitions in He- and Be-like ions. For He-like ions, the M2 decay 1s2p 3P2 –1s2 1 S0 competes with the E1 decay 1s2p 3P2 –1s2s 3 S1 for ions with a nuclear-charge number Z > 20. Electric-dipole E1 transitions between hyperﬁne-structure levels with the quantum numbers F and F obey the following selection rules: F = F − F = 0, ±1

F + F ≥ 1.

(5.32)

Here F are the quantum numbers of the total angular momentum F = J + I where J and I denote the total angular momentum of all atomic electrons and of the nucleus, respectively. Electric-dipole E1 transitions between two F-components of the same level J are forbidden by the parity selection rule but M1 and E2 transitions are allowed. For these two cases one has: M1 transitions :

F = 0, ±1

F + F ≥ 1

E2 transitions :

F = 0, ±1, ±2

F + F ≥ 2.

(5.33)

Selection rules also exist for transitions between states described by other types of coupling schemes such as L K , j K or j j couplings.

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5.4 Transition probabilities and oscillator strengths Radiative transition probabilities per time, or decay rates, of atoms and ions are very important atomic characteristics deﬁning the lifetimes of excited states, widths and intensities of spectral lines. The line intensity is the total energy radiated per time and per source volume. The intensity I10 of a spectral line resulting from atomic transition 1 → 0 from an upper state 1 to a lower state 0 is given by (5.34) I10 = n i1 A10 ~ω where A10 is the radiative transition probability per time, n i1 is the density number of ions in the level 1 and ω = (E 0 − E 1 )/~ is the transition frequency. The radiative probability of the resonance 2p → 1s transition in hydrogen is equal to A10 = 6.26 × 108 s−1 that gives an atomic scale of transition probabilities. In general, however, the transition probabilities critically depend on the ion charge and the type of transition as discussed in the next section. As we found in the previous section, there are two main types of radiative transition: electric multipole Eκ and magnetic multipole Mκ transitions. Within the framework of quantum electrodynamics (QED), the probability per unit of time of a one-photon transition from an initial excited state |1 to a ﬁnal state |0 is given by e2 ω d A10 = |0|α ∗ · e−ik r |1|2 d (5.35) 2π ~c where c denotes the speed of light, ω = (E 1 − E 0 )/~ the frequency of the emitted photon, its polarization vector, α the Dirac matrix, k the photon momentum and d denotes the solid-angle element for the radiation. Here, |1 and |0 are the relativistic wavefunctions obtained from the Dirac equation (section 5.8). In the non-relativistic approximation for describing the electric-dipole E1 transitions, the probability is simply obtained by utilizing the replacement: e−ik r → 1

α ∗ e−ik r → p ∗ /m e c

p → −im e r

where p is the operator of the electron momentum. Now |1 and |0 denote the non-relativistic wavefunctions obtained from the Schr¨odinger equation. The nonrelativistic probability for the dipole-allowed transition has the well-known form: A(E1) =

4 e 2 ω3 |0|r|1|2 3 c2

(5.36)

i.e. it is deﬁned by the electric-dipole matrix element 0|er|1. The transition probabilities for the higher multipole transitions, i.e. the electric multipoles Eκ and the magnetic multipoles Mκ, with κ > 1, can be obtained from (5.35) using the higher terms in the power-series expansion e−ik r ≈ 1 − ikr + 12 (ikr)2 + · · · .

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In the general case, the formulae for the corresponding transition probabilities A(Eκ) and A(Mκ) are quite complicated3. (Mκ) Let A(Eκ) and A10 denote the probabilities for a transition 1–0 of the 10 respective Eκ or Mκ modes. Then one can introduce the dimensionless oscillator (Eκ) (Mκ) strengths f 01 and f 01 for electric and magnetic transitions through the relations (Eκ)

g1 A10

(Mκ) g1 A10

(Eκ)

= g0 A0 (E/Ry)2 f01 =

(Mκ) α g0 A0 (E/Ry)2 f01 3 9 −1 2

A0 = α Ry/~ ≈ 8.033 × 10 s

(5.37) (5.38) (5.39)

where E denotes the transition energy in Ry units, g the statistical weight and α the ﬁne-structure constant. Oscillator strengths have positive and negative values depending on the direction of the transitions: for transitions from lower states to upper states, they are positive; they are negative for transitions from upper to lower states. For both Eκ and Mκ transitions, an order-of-magnitude estimate of the oscillator strength gives f (κ) (2a0 ω/c)2κ−2 (αE/Ry)2κ−2

(5.40)

where ω denotes the transition frequency and a0 the Bohr radius. For optically allowed dipole transition n 0 0 –n 1 1 , one has g0 (E/Ry)2 f 01 [s−1 ] g1 Q 01 E 2 R f 01 = |1 − 0 | = 1 3(20 + 1) Ry ∞ 2 R 2 = m Rn0 0 (r )Rn1 1 (r )r 3 dr m = max(0 , 1 )

A10 = 8.033 × 109

(5.41) (5.42) (5.43)

0

where g denotes the statistical weight, Q 01 the angular coefﬁcient for transition 0–1 and Rn (r ) the radial wavefunctions with normalization: ∞ Rn (r )Rn (r )r 2 dr = δnn . (5.44) 0

Here n and are the principal and orbital quantum numbers, respectively. Dipole oscillator strengths f and transition probabilities A for some atoms and ions are given in table 5.6. 3 See Akhiezer A I and Berestetskii V B 1965 Quantum electrodynamics Interscience Monographs

and Texts in Physics and Astronomy (New York: Wiley). Rose E M 1961 Relativistic Electron Theory (New York: Wiley).

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Table 5.6. Dipole oscillator strengths f 01 and transition probabilities A 10 for atoms and ions. Atom, ion

Transition 0–1

f 01

A 10 [s−1 ]

H H H H H He He He He N5+ N5+ N5+ N5+

1s–2p 1s–3p 2s–3p 2p–3s 2p–3d 1s2p 1 P1 –1s2 1 S0 1s2 1 S0 –1s3p 1 P1 1s2s 1 S0 –1s2p 1 P1 1s2s 1 S0 –1s3p 1 P1 1s2 1 S0 –1s2p 1 P1 1s2 1 S0 –1s3p 1 P1 1s2s 1 S0 –1s2p 1 P1 1s2s 1 S0 –1s3p 1 P1

0.416 0.0791 0.435 0.0136 0.606 0.276 0.0734 0.376 0.151 0.674 0.144 0.078 0.364

6.26 × 108 1.67 × 108 2.24 × 107 6.31 × 106 6.46 × 107 1.80 × 109 5.66 × 108 1.98 × 106 1.34 × 107 1.81 × 1012 5.16 × 1011 2.06 × 107 2.69 × 1010

The dipole oscillator strength f (n 0 −n 1 ) and radiative probability A(n 1 −n 0 ) for transitions between levels with the principal quantum numbers n, i.e. averaged over the orbital quantum numbers , are given by f (n 0 − n 1 ) =

n 1 −1 n

0 −1 1

f (n 0 0 − n 1 1 ) n 20 =0 =0 1

A(n 1 − n 0 ) =

1 n 21

= ±1

(5.45)

= ±1.

(5.46)

0

n

1 −1 n

0 −1

A(n 1 1 − n 0 0 )

1 =0 0 =0

For example, the transition probability for transition n 1 = 2 → n 0 = 1 in hydrogen is (see table 5.6): A(2 → 1) = 34 × 6.26 × 108 = 4.70 × 108 s−1 while A(2p–1s) = 6.26 × 108 s−1 , i.e. the average A-value for transition n = 2 → n = 1 does not coincide with the A(2p–1s) value. For transitions n 0 → n 1 between highly excited hydrogenic states with the quantum numbers n 0 , n 1 1 and n = n 0 − n 1 1, Kramers’ formulae are used: 16 A0 4 2 A(n 1 − n 0 ) = √ Z eff (5.47) n0, n1 1 3 n 1 n 0 (n 21 − n 20 ) 3 3 where the constant A0 is given in (5.39), Z eff = Z − N + 1 is the effective charge of an ion and N is the total number of electrons.

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A similar expression can be written for the average oscillator strength: 32 A0 f (n 0 − n 1 ) = √ 3 3

n0n1 n(n 0 + n 1 )

3 n 0 , n 1 1.

(5.48)

In H-like ions, the total radiative decay probability A(n) of the state n to all lowlying states, including the n = 1 ground state, can be estimated in the Kramers approximation as A(n) =

n−1

n =1

3 A0 4 n − 1/n A(n − n ) = 5 Z eff ln 2 n

n 1.

(5.49)

The accurate quantum-mechanical expression of A(n) for arbitrary hydrogenic states with n ≥ 2, reads4 : A(n) =

8 A0 4 Z [ln(2n − 1) − 0.365] 3 eff

n ≥ 2.

(5.50)

For the nearest excited n = 2 state in hydrogen, the A(n)-value given by (5.50) and the exact one are 4.89 × 108 and 4.70 × 108 s−1 , respectively. 5.4.1 Transition probabilities of H- and He-like ions For highly charged H- and He-like ions, the formulae for radiative probabilities A for electric and magnetic transitions between the ground and lowest excited states can be obtained in a closed analytical form. These formulae are useful because they reﬂect the dependence of the A-values on the nuclear charge and transition energies. The formulae given here were obtained on the basis of relativistic calculations using equation (5.35). H-like ions The 2s state in H-like ions has the same parity as the 1s ground state, therefore it can decay to the ground state either by a magnetic dipole M1 or a twophoton electric-dipole 2E1 transition. For M1 transitions, the leading term of the relativistic transition probability A(M1) is5 : A(M1) (2s1/2 − 1s1/2 ) =

α 9 Z 10 m e e4 2.46 × 10−6 Z 10 [s−1 ] 972 ~3

(5.51)

where Z denotes the nuclear charge of the ion and α the ﬁne-structure constant. 4 Chang S E 1985 Phys. Rev. A 31 495. 5 References on the formulae for transition probabilities, experimental and theoretical values for A in

H-like ions can be found in Beyer H F, Kluge H-J and Shevelko V P 1999 X-Ray Radiation of Highly Charged Ions (Berlin: Springer).

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Table 5.7. The decay rates A (in s−1 ) of the 2s1/2 state in H-like ions as a function of the nuclear charge Z. Z 2

8 9 16 18 28

Experiment 491+95 −140 520(21) 525(5) 2.21(22) × 106 4.22(28) × 106 1.37(13) × 108 2.82(20) × 108 4.606(38) × 109

Theory 526.61

2.1552 × 106 4.3699 × 106 1.3964 × 108 2.8590 × 108 4.6368 × 109

Fully relativistic calculations for the two-photon 2E1 transition read: 1 + 3.9448(α Z )2 − 2.040(α Z )4 −1 [s ] 1 + 4.6019(α Z )2 (5.52) with an accuracy of 0.05% for the nuclear charge number 1 ≤ Z ≤ 92. In table 5.7, experimental decay rates of the 2s1/2 states in H-like ions are compared with theoretical data represented the sum A = A(M1) + A(2E1) . Two-photon transitions take place via intermediate virtual states. We note that both probabilities A(2E1) and A(M1) for the 2s state increase with the nuclear charge Z more rapidly than the probability A(E1) for the electric-dipole transition 2p–1s, where A(E1) ∼ Z 4 . For ions with Z > 40, the M1 transitions represent the main contribution to the radiative decay of the 2s state. The ratio A(M1) /A(E1) strongly increases with Z and for H-like uranium (Z = 92) reaches the value of about 4 × 10−3 . Calculated transition probabilities in H-like ions for transitions n = 2 → n = 1 are given in table 5.8. Besides showing dependences of transition probabilities on the ion charge Z , table 5.8 also illustrates the importance of the magnetic-dipole M1 transitions for ions with Z > 40 for when the transition probabilities A(M1) become larger than the probabilities A(2E1) for two-photon electric-dipole transitions. A(2E1) (2s1/2 − 1s1/2 ) = 8.22943Z 6

He-like Ions He-like ions are the simplest many-electron systems well suited for the study of the additional compilation arising from the electron–electron interaction. At low nuclear-charge numbers, the energy-level pattern of He-like ions is characterized by two independent singlet and triplet terms well described in the framework of the L S coupling scheme. The change in the energy-level pattern with

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Table 5.8. Calculated M1, 2E1 and E1 transition probabilities (in s−1 ) for transitions between n = 2 and n = 1 states in H-like ions as a function of the nuclear charge Z . Z

2s1/2 –1s1/2 M1

2s1/2 –1s1/2 2E1

2p1/2 –1s1/2 E1

2p3/2 –1s1/2 E1

1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 22 24 28 30 34 38 42 46 50 54 58 62 66 70 74 78 82 86 90 92

2.4946 × 10−6 2.5559 × 10−3 1.4744 × 10−1 2.6192 × 100 2.4406 × 101 1.5121 × 102 7.0694 × 102 2.6895 × 103 8.7423 × 103 2.5100 × 104 1.5580 × 105 7.3003 × 105 2.7845 × 106 9.0777 × 106 2.6149 × 107 6.8154 × 107 1.6357 × 108 7.7347 × 108 1.5525 × 109 5.5106 × 109 1.7050 × 1010 4.7295 × 1010 1.2003 × 1011 2.8303 × 1011 6.2741 × 1011 1.3198 × 1012 2.6541 × 1012 5.1345 × 1012 9.6037 × 1012 1.7443 × 1013 3.0881 × 1013 5.3459 × 1013 9.0760 × 1013 1.5151 × 1014 1.9468 × 1014

8.2291 5.2660 × 102 5.9973 × 103 3.3689 × 104 1.2847 × 105 3.8348 × 105 9.6657 × 105 2.1526 × 106 4.3614 × 106 8.2015 × 106 2.4453 × 107 6.1554 × 107 1.3688 × 108 2.7686 × 108 5.1965 × 108 9.1804 × 108 1.5427 × 109 3.8637 × 109 5.8231 × 109 1.2238 × 1010 2.3636 × 1010 4.2657 × 1010 7.2824 × 1010 1.1868 × 1011 1.8595 × 1011 2.8162 × 1011 4.1409 × 1011 5.9328 × 1011 8.3060 × 1011 1.1391 × 1012 1.5331 × 1012 2.0287 × 1012 2.6427 × 1012 3.3931 × 1012 3.8251 × 1012

6.2649 × 108 1.0028 × 1010 5.0772 × 1010 1.6048 × 1011 3.9181 × 1011 8.1252 × 1011 1.5054 × 1012 2.5684 × 1012 4.1146 × 1012 6.2721 × 1012 1.3009 × 1013 2.4110 × 1013 4.1145 × 1013 6.5935 × 1013 1.0054 × 1014 1.4728 × 1014 2.0872 × 1014 3.8719 × 1014 5.1061 × 1014 8.4378 × 1014 1.3190 × 1015 1.9723 × 1015 2.8443 × 1015 3.9800 × 1015 5.4291 × 1015 7.2458 × 1015 9.4897 × 1015 1.2225 × 1016 1.5521 × 1016 1.9453 × 1016 2.4100 × 1016 2.9547 × 1016 3.5884 × 1016 4.3202 × 1016 4.7260 × 1016

6.2648 × 108 1.0027 × 1010 5.0764 × 1010 1.6043 × 1011 3.9163 × 1011 8.1198 × 1011 1.5041 × 1012 2.5654 × 1012 4.1084 × 1012 6.2604 × 1012 1.2975 × 1013 2.4022 × 1013 4.0950 × 1013 6.5538 × 1013 9.9797 × 1013 1.4596 × 1014 2.0648 × 1014 2.8154 × 1014 2.0206 × 1014 3.2559 × 1014 1.2834 × 1015 1.9073 × 1015 2.7317 × 1015 3.7935 × 1015 5.1318 × 1015 6.7872 × 1015 8.8017 × 1015 1.1218 × 1016 1.4079 × 1016 1.7427 × 1016 2.1301 × 1016 2.5740 × 1016 3.0776 × 1016 3.6434 × 1016 3.9502 × 1016

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Figure 5.4. Energy-level and decay schemes for the low-lying states of H- and He-like heavy ions.

increasing nuclear charge has already been exempliﬁed in ﬁgure 5.3 for the 1s2p conﬁguration. We complete this illustration in ﬁgure 5.4 by showing an energylevel and decay scheme for H-like and He-like heavy ions. In pure L S coupling, all intercombination transitions with a change of the spin quantum number by S = 1 are forbidden. However, in highly charged ions, the selection rule S = 0 is violated due to the inﬂuence of electromagnetic interactions which rapidly increase with increasing nuclear charge Z . Consequently, the intensities of the intercombination and forbidden lines increase as well. In this case one has to use the intermediate coupling scheme. For example, for the intercombination transition 2 3 P1 –1 1 S0 , the intermediate coupling mixes the 2 3 P1 and 2 1 P1 states and the transition probability A(2 3 P1 –1 1S0 ) increases with increasing ion charge: A(He) = 1.79 × 102 s−1 , A(Ne8+ ) = 5.43 × 109 s−1 , A(Fe24+ ) = 4.42 × 1013 s−1 and A(U90+ ) = 2.99 × 1016 s−1 , respectively. Merely as a label we are using the L S-term designation also in case of high-Z ions where j j coupling would be more appropriate. In this way we avoid a change in labeling at an arbitrary point along the isoelectronic sequence. The E1-decay rates of the 1s2p 1P and 3 P states All three 2 3 P0,1,2 levels in He-like ions decay to the 2 3 S1 state by optically allowed E1 transitions (see ﬁgure 5.4). This is the dominant decay route for low nuclear-charge numbers. The 2 3 P1 state is mixed with the 2 1 P1 and higher 1snp 1 P1 states by the spin–orbit interaction (section 5.8.1). The mixing is very small in neutral helium but increases rapidly with increasing the nuclear charge

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number Z . The allowed decay mode for the 2 3 P1 state is the electric-dipole E1 transition to the 2 3 S1 state. The 2 1 P1 level, however, can directly decay to the ground state by electric-dipole E1 radiation. The 2 3 P0,2 levels are not mixed with the singlet system and they have lifetimes ∼10 times higher compared to the 2 3 P1 level. The 2 3 P0 level can decay to the ground state only by two-photon, E1 + M1, or threephoton, 3E1 transitions which are negligible in the whole range of Z . The 2 3 P2 state can also decay to the ground state by the magnetic-quadruple M2 transition. Theoretical analysis shows that the transition probabilities are small for the lowZ systems but since A(M2) scales as Z 8 it becomes the dominating depopulation mode of the 2 3 P2 state for ions with Z > 20. The magnetic-quadrupole M2 transition 2 3 P2 → 2 1 S0 is also possible but its probability remains low for all Z values. The 2 3 P2 state decays to the ground state by M2 transition for which the transition probability can be estimated by the formula (Z ≥ 30): α7 Z 8 me4 (E/Z 2 Ry)5 3 [1 + 0.28(α Z )2 ] 1215 ~ 8 0.037Z [1 + 0.28(α Z )2](E/Z 2 Ry)5 s−1

A(M2) (2 3 P2 –1 1 S0 ) =

Z ≥ 30. (5.53)

where 1 Ry 13.606 eV. The total decay of the 2 3 P2 state consists of two branches: 2 3 P2 → 2 3 S1 (E1) and 2 3 P2 → 1 1 S0 (M2) transitions. The 2 1 S0 → 1 1 S0 transitions The 1s2s 1 S0 state decays to the 1 S0 ground state by a 2E1 two-photon emission. An accurate extrapolation formula for the decay rate including relativistic corrections has the form: 1.539 Z − 0.806 389 6 1+ A(2E1)(2 1 S0 –1 1 S0 ) = 16.458 762Z 6 Z (Z + 2.5)2 0.6571 + 2.040(α Z )2 −1 s . − (α Z )2 (5.54) 1 + 4.6019(α Z )2 A non-relativistic power-series expansion of (5.54) yields: 4.8383 11.293 (2E1) 1 1 6 + (2 S0 –1 S0 ) 16.458 762 Z 1 − A Z Z2 25.63 87.15 − + + · · · s−1 . Z3 Z4

(5.55)

In general, the relativistic effects decrease the decay rates although they increase the 2 1 S0 –1 1 S0 energy separation.

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194

The 2 1 S0 level can also decay to the 2 3 P1 state by a spin-forbidden E1 transition but with much lower probability because the 2 1 S0 level is close to the 2 3 P1 level in energy. We note that for He-like ions with Z ≤ 6, the 2 1 S0 level lies lower than the 2 3 P1 level. The 2 3 S1 → 1 1 S0 transitions The 2 3 S1 state decays to the ground 1 S0 state by a relativistically induced M1 transition or by a 2E1 two-photon emission. The M1 transition probability is three to four orders of magnitude higher than that for 2E1 decay for all Z and can be estimated by 25 9 10 me4 α Z (E/Z 2 Ry)3 3 9 3 ~ 0.397 Z 10 (E/Z 2 Ry)3 s−1

A(M1)(2 3 S1 − 1 1 S0 ) =

Z ≥ 10.

(5.56)

Forbidden transitions like the Mκ and intercombination transitions are very important for the identiﬁcation of x-ray lines in solar and laboratory plasma spectra. Very accurate calculations of the transition probabilities for the low-lying states in He-like ions are those given by the corrected non-relativistic6 and fully relativistic7 calculations. For He-like U90+ , one has the following theoretical values for transition probabilities A (in s−1 ): A(2E1) (2 1 S0 –1 1 S0 ) = 7.24 × 1012 A(M1) (2 3 S1 –1 1 S0 ) = 1.21 × 1014 A(E1) (2 1 P1 –1 1 S0 ) = 5.00 × 1016 A(M2) (2 3 P2 –1 1 S0 ) = 2.06 × 1014 A A

(E1)

(E1M1)

3

1

(5.57)

16

(2 P1 –1 S0 ) = 2.99 × 10 (2 3 P0 –1 1 S0 ) = 5.61 × 109 .

Approximate radiative decay probabilities in He-like ions for transitions between terms of the n = 1 and n = 2 levels as a function of the nuclear charge Z were obtained on the basis of sophisticated calculations and ﬁtted in the form: A = a(Z − b)c [s−1 ]

(5.58)

where a, b and c denote the ﬁtting parameters, given in table 5.9, and Z the nuclear charge. 6 Drake G W F 1971 Phys. Rev. A 3 908. 7 Johnson W R, Plante D R and Sapirstein J 1995 Adv. At. Mol. Opt. Phys. 35 255.

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Table 5.9. Fitting parameters a, b and c (5.58) for transition probabilities between n=1 and n = 2 levels in He-like ions. Z is the nuclear charge. Transition

Type

Range of Z

a

b

c

21 P1 − 11 S0 21 P1 − 21 S0 21 P1 − 23 S1 21 S0 − 23 S1 21 S0 − 23 P1 21 S0 − 11 S0 23 S1 − 11 S0 23 S1 − 11 S0 23 P1 − 11 S0 23 P1 − 11 S0 23 P1 − 23 S1 23 P1 − 23 S1 23 P2 − 23 S1 23 P2 − 23 S1 23 P2 − 11 S0 23 P2 − 21 S0 23 P0 − 23 S1

E1 E1 E1 M1 E1 2E1 M1 2E1 E1 E1 E1 E1 E1 E1 M2 M2 E1

Z < 80 Z < 30 Z >4 2 < Z < 50 10 < Z < 20 Z >2 4 < Z < 80 Z < 25 6 < Z < 20 Z ≥ 20 Z < 50 Z ≥ 50 15 < Z ≤ 20 Z > 20 Z < 80 Z ≥ 20 Z < 50

9.6 × 108 1.5 × 104 0.66 1.8 × 10−8 10.8 16.5 1.8 × 10−6 9.2 × 10−10 551.0 6.48 × 108 1.33 × 107 1.1 × 106 3.73 × 109 5.3 × 108 0.038 1.23 × 10−9 1.33 × 107

−0.5 −2.14 1.18 1.25 0.62 0.8 0.56 1.0 2.57 4.13 −0.57 18.25 10.97 5.29 0.69 17.5 −0.57

4.0 3.0 6.0 7.0 2.0 6.0 10.0 10.0 8.0 4.0 1.0 2.0 1.0 4.0 8.0 8.0 1.0

5.5 Lifetimes The radiative lifetime τk of an excited state k is the time during which the number of excited particles decreases e times (e 2.7183) due to spontaneous transitions to the low-lying states. The lifetime τk is deﬁned as −1

τk = Aik (5.59) i≤k

where the quantities Aik denote the transition probabilities of all possible decay channels including forbidden and intercombination transitions; the equality sign refers to the autoionization states. According to equation (5.59), the lifetime for the n = 2 state in hydrogen is τ (n = 2) = 2.13×10−9 s. For an arbitrary hydrogenic n-state, the lifetime τn can be written in a closed analytical form using probabilities Aik from equation (5.50): τn =

3n 5 − 1) − 0.365]

4 [ln(2n 8 A0 Z eff

n≥2

(5.60)

where Z eff = Z − N +1 denotes the effective charge of an ion and the constant A0 has been deﬁned in equation (5.39). As can be seen, the lifetimes of excited states

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rapidly increase with the principal quantum number n. For the lowest excited n = 2 state in the hydrogen atom, equation (5.60) gives τ (n = 2) = 2.03×10−9 s. For a speciﬁc hydrogenic state n, )= 0, the accurate quantum-mechanical expression reads8 : τn =

3n 3 4 4 A0 Z eff

( + 1/2)2

n ≥ 2, )= 0.

(5.61)

For 2p resonance state in hydrogen, the lifetime τ (2p) = 1.60 × 10−9 s while equation (5.61) gives τ (2 p) = 1.68 × 10−9 s. For excited states n-states with n < 25 in H-like ions, the accurate calculations of lifetimes and dipole matrix elements are presented in the works9. Most of the data on lifetimes τ for positively charged ions have been obtained by employing the beam–foil time-of-ﬂight method covering a wide range of ion charges10. For highly charged ions, ion traps were also used including ion radiofrequency (RF) traps, heavy-ion storage rings and electron-beam ion traps (EBIT). Precision lifetime measurements are required for systematic checks of the theoretical approaches, especially for ions with one or two electrons outside closed shells. Beam–foil time-of-ﬂight measurements of forbidden transitions (2E1, M1, M2) in H- and He-like ions have been carried out with an accuracy ranging from 0.1% for light ions up to a few per cent for heavy ions. Some recent data on lifetimes of highly charged ions are presented in table 5.1011. The radiative lifetimes of excited ions with a noble-gas conﬁguration are of particular interest because these levels are employed in certain short wavelength laser schemes, astrophysics and thermonuclear fusion studies. Some beam–foil measurements of lifetimes are reproduced in table 5.11. Figure 5.5 represents results for the lifetimes of the 3p 3 D3 and 3d 3F3 levels in Ne-like ions which, among ions of other isoelectronic sequences, are of interest for developing efﬁcient short wavelength lasers.

5.6 Autoionizing states and Auger decay The autoionizing states of an atom or an ion are the bound states with two or more excited electrons and a total binding energy higher than the ionization potential of the ground state. For example, the ionization potential of Li, with the ground 8 Chang S E 1985 Phys. Rev. A 31 495. 9 Omidvar K 1982 Phys. Rev. 26 3053.

Omidvar K 1983 At. Data Nucl. Data Tables 28 1. 10 See Curtis L and Martinson M 1999 Atomic Physics with Heavy Ions ed H F Beyer and V P Shevelko

(Berlin: Springer) p 197. 11 References on experimental and theoretical works on ion lifetimes for forbidden and intercombination transitions can be found in the works by Tr¨abert E 2000 Physica Scripta 61 257 and 2001 Physica Scripta T 92 444.

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Figure 5.5. Scaled lifetimes (Z − 9)τ for the excited 3p 3 D3 and 3d 3 F3 levels of Ne-like ions as a function of the nuclear charge Z: symbols, experimental data; full curves, theoretical values. From Ando K, Zon Y, Kambara T, Nakai Y, Kanai Y, Oura M, Awaya c Y and Tonuma T 1996 Phys. Scr. 53 33. 1996 Royal Academy of Sciences, Stockholm.

Table 5.10. Theoretical and experimental lifetimes τ (in s) of excited states in H- and in He-like ions. Ion

State

Experiment

Theory

O6+

2 3 P2 2 3 P1 2 3 P2 2 3 P1 2 3 P0 2 2 S1/2 2 1 S0 2 2 S1/2 2 1 S0 2 1 S0 2 1 S0 2 3 P2 2s1/2 2p1/2 3 P0

12.10(20) 1.52(8) 10.44(15) 0.531(20) 9.48(20) 3.487(36) 156.1(1.6) 217.1(1.8) 34.08(34) 39.32(32) 15.33(60) 1.24(11) 4.49(8) 54.4(3.4)

12.2 × 10−9 1.6 × 10−9 10.4 × 10−9 0.52 × 10−9 11.0 × 10−9 3.497 × 10−9 154.3(0.5) × 10−12 215.45 × 10−12 33.41(14) × 10−12 39.63(16) × 10−12 15.245 × 10−12 1.08 × 10−12 4.35 × 10−12 57.31 × 10−12

F7+

Ar17+ Ni26+ Ni27+ Kr34+ Br33+ Nb39+ Ag45+ Ag46+ U90+

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Table 5.11. Experimental and theoretical lifetimes (in ps) of 2p5 3s 1 P1 and 3 P1 states in Ne-like ions. From Nielsen J 1997 J. Opt. Soc. Am. B 14 1511. 1P

3P

1

1

Ion

Experiment

Theory

Experiment

Ne0

1470 ± 100 1870 ± 180 1300 ± 100 320 ± 50 580 ± 60 100 ± 15 110 ± 15 46 ± 10 28 ± 8 18 ± 7 14.5 ± 1.0 12 ± 3 13 ± 1 10 ± 2 8±2 6.5 ± 2.0

1547

29 600 ± 1000 31 700 ± 1600 29 800 ± 2000 6000 ± 1200 10 600 ± 500 1900 ± 190

Na+ Mg2+ Al3+ Si4+ P5+ S6+ Cl7+

Ar8+ Se24+ Br25+ Kr26+ Rb27+ Sr28+ Y29+

346 131 63 36 21 15 11

0.60 0.53 0.47 0.42 0.38 0.34

130 ± 30 52 ± 2 49 ± 13 27 ± 1 34 ± 12 30 ± 5 19 ± 4

Theory

5623 1820 637 248 137 52 27

16 0.28 0.24 0.21 0.19 0.16 0.14

state conﬁguration 1s2 2s, is I2s = 5.4 eV. This means that all the excited states of the type 1s2 2p, 1s2 3s, . . . , 1s2 n lie below the ionization limit. If the inner 1s-electron is now excited into the states 1s2s2 , 1s2s2p, . . . , 1s2sn, its binding energies, counted from the ground 2s state, are larger than 47.5 eV. The way in which the autoionization state is created may vary appreciably, e.g. by ionization or capture of the inner-shell electron, by simultaneous excitation of two or more outermost electrons or by excitation of the inner-shell electron. In all cases, the formation of the autoionizing state is related to the creation of one or more electron vacancies in the inner shells. The autoionizing state is not stable and may decay via autoionization or radiative channels. Autoionization is a non-radiative decay, or stabilization, followed by ejection of one or more electrons into the continuum and an increase of the ion charge. In the case of a doubly excited autoionizing state

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Figure 5.6. Schematic representation of the autoionization process. The given example is for KLL Auger decay of a doubly excited Li-like ion into He-like ion and a free electron: [Li] → [He] + e− .

[Xq+ (γ1 n)]∗∗ , the two possible stabilization channels can be written as autoionization: [Xq+ (γ1 n)]∗∗ → [X(q+1)+ (γ0 )]∗ + e− (E e , ) radiation: [X

q+

(γ1 n)]

∗∗

→ [X

q+

∗

(γ0 n)] + ~ω

(5.62) (5.63)

where n and denote the principal and orbital quantum numbers of the excited bound electron, γ1 and γ0 the sets of quantum numbers of the ion before and after decay, E e and are the kinetic energy and the angular momentum of the free electron, respectively. The process (5.62) is called Auger or autoionization decay and the emitted electron is called the Auger electron. The energy, released by the transition of a bound electron into a lower state, is used to ionize a second electron into the continuum; therefore, autoionization is a two-electron transition. A free electron can be emitted if the following condition for the atomic energies is satisﬁed: E e = E(γ1 n) − E(γ0 ) > 0.

(5.64)

A schematic representation of the possible creation of the autoionizing state in a Li-like ion followed by KLL Auger decay is presented in ﬁgure 5.6. In an intermediate step, the K-shell electron is excited to the L shell and a doubly excited autoionizing state is prepared. In a ﬁnal step, one L electron makes a transition to the K shell and the energy, released due to the L-K transition, is used for ionization of another L electron. As a result of this Auger decay, one ﬁnally has a He-like ion and a free electron. The radiative decay (5.63) via the transition γ1 n → γ0 n is associated with the emission of a photon with an energy slightly less than that of the parent transition γ1 –γ0. In this way a satellite line in the vicinity of the parent line is produced. The decay of autoionization states also satisﬁes a certain set of selection rules. For non-radiative transitions via different interactions these are given in table 5.1212. There, the notation L = L − L, S = S − S, etc is used where 12 See Feldman P and Novick R 1967 Phys. Rev. 160 143.

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Table 5.12. Selection rules for autoionization transitions; α is the ﬁne-structure constant, m e and m p are electron and proton masses, respectively. Interaction

L

S

J

P

Order of magnitude

Coulomb Spin–orbit Spin–other-orbit Spin–spin Hyperﬁne

0 0, ±1 0, ±1 0, ±1, ±2 0, ±1, ±2

0 0, ±1 0, ±1 0, ±1, ±2 0, ±1

0 0 0 0 0, ±1

0 0 0 0 0

1 α4 α4 α4 α 4 (m e /m p )2

the angular quantum numbers L S J and L S J correspond to the discrete and continuum states, respectively. The parity P of the state is deﬁned by the sum taken over all electrons. i i In ﬁrst-order non-relativistic perturbation theory, the probability Aa of the autoionization decay from the state 1 to the state 0 per second is given by the matrix element: 2π |0 |V |1 |2 δ(E 1 − E 0 ) (5.65) Aa =

~

where V is the interaction operator, coupling the discrete and continuum states and may be either an electrostatic, 1/r12 , or a magnetic one. Calculated autoionization probabilities for doubly excited 2s2 , 2s2p and 2p2 conﬁgurations in He-like ions are given in table 5.13 as a function of the nuclear charge Z . They have been calculated using an expansion over the parameter 1/Z 13 . For the 2p2 3 P1 term, the probabilities Aa = 0 for all Z . On neutral atoms and ions of low charge, the autoionization probability Aa is of the order of 1013–1014 s−1 and is weakly dependent on the ion charge q. The radiative probability Ar increases with the ion charge approximately as Ar ∝ q 4 and Ar Aa for ions with q ≥ 10. Therefore, stabilization of the autoionizing states takes place via Auger or non-radiative decay in the case of low charged ions and via radiative decay in highly charged ions. The total lifetime τ of metastable autoionizing state is given by τ = (Aa + Ar )−1

(5.66)

where Aa and Ar denote the total probabilities per time for autoionization and radiative decay, respectively. Finally we note that autoionization processes play a key role in the physics of highly charged ions, especially in resonance electron–ion collisions such as resonant excitation, single and multiple ionization and dielectronic 13 See Shevelko V P and Vainshtein L A 1993 Atomic Physics for Hot Plasmas (Bristol: IOP Publishing).

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Table 5.13. Calculated autoionization probabilities A a (in 10−13 s−1 ) for He-like ions as a function of the nuclear charge number Z . 1.92-5 means 1.92 × 10−5 . 2s2p Z

1P

4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54

20.3 20.3 20.3 20.3 20.2 20.2 20.2 20.2 20.1 19.9 19.8 19.6 19.3 19.0 18.7 18.3 18.0 17.6 17.3 17.0 16.7 16.5 16.2 16.0 15.8 15.7

1

3P

0

1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36

3P

1

1.36 1.36 1.36 1.37 1.37 1.39 1.42 1.47 1.56 1.68 1.85 2.07 2.33 2.64 2.97 3.32 3.67 4.01 4.33 4.64 4.92 5.17 5.40 5.61 5.79 5.96

3P

2

1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36 1.36

2p2

2s2 1S 0

3P

34.9 34.7 34.6 34.5 34.4 34.2 34.0 33.7 33.3 32.9 32.4 31.9 31.4 30.9 30.4 29.9 29.5 29.1 28.8 28.5 28.2 27.9 27.7 27.5 27.4 27.2

1.92-5 2.44-4 1.42-3 5.36-3 1.52-2 3.53-2 6.95-2 1.19-1 1.82-1 2.49-1 3.12-1 3.59-1 3.86-1 3.93-1 3.84-1 3.64-1 3.39-1 3.11-1 2.83-1 2.57-1 2.32-1 2.11-1 1.91-1 1.74-1 1.58-1 1.44-1

2

3P

0

6.70-4 4.60-3 2.16-2 7.61-2 2.20-1 5.49-1 1.22+0 2.41+0 4.27+0 6.72+0 9.46+0 1.21+1 1.44+1 1.63+1 1.79+1 1.83+1 1.73+1 1.66+1 1.66+1 1.55+1 1.51+1 1.47+1 1.45+1 1.42+1 1.40+1 1.39+1

1D

2

37.4 37.4 37.3 37.3 37.1 36.8 36.2 35.0 33.1 30.6 27.9 25.3 22.9 21.0 19.5 19.1 20.0 20.8 21.4 21.9 22.3 22.6 22.9 23.1 23.3 23.5

1S

0

1.23 1.43 1.55 1.66 1.78 1.92 2.11 2.35 2.64 3.00 3.41 3.88 4.37 4.88 5.39 5.87 6.33 6.74 7.11 7.45 7.74 8.00 8.23 8.44 8.62 8.78

recombination. These processes are considered in chapter 6. To a large extent, x-ray diagnostics of hot laboratory and astrophysical plasmas is based on the spectroscopy of autoionizing states14 .

5.7 One-electron systems In studying the atomic structure of positive ions, one- and few-electron highly charged ions are of particular interest since their rigorous theoretical description 14 Gabriel A 1971 Highlights in Astronomy ed C de Jager (Dordrecht: Reidel) pp 486–94.

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Atomic structure

is possible thus allowing systematic study of fundamental laws. In the non-relativistic approximation, the energy levels in neutral hydrogen and H-like ions (He+ , Li2+ , Be3+ , . . .) can be found from the Schr¨odinger equation which can be solved analytically and yields the Rydberg formula (5.3). In H-like ions, spectral series such as Lyman, Balmer and others are similar to those in the neutral hydrogen spectrum but they scale as E ∝ Z 2 and the wavelengths scale as λ ∝ Z −2 , so that for high-Z ions the corresponding spectral lines are shifted to the extreme UV and x-ray spectral regions. For example, the wavelength for the transition n = 2 → n = 1 (resonance line) in the hydrogen ˚ i.e. about 262(= 676) times ˚ while for H-like Fe25+ it is 1.78 A, atom is 1216 A 91+ shorter. For H-like uranium ions U , the Lyman series in the spectral region of E ≈ 100 keV and the Balmer series in the range E = 15–35 keV have been observed15. The dependence of the energy on the ﬁnite nuclear mass in equation (5.3) is related to the motion of the nucleus around the center of mass of the nucleus and atomic electrons. This means that different isotopes have energy levels shifted relative to each other. Isotopes are atoms or ions from the same chemical element which have different nuclear masses due to the different number of neutrons in the nucleus. The effect of energy-level shifting is called the isotope shift. For example, as follows from equation (5.3), the isotope shift of the n level in deuterium (M = 2m p ) relative to the hydrogen level (M = m p ) is E = −

Ry n2

mp 2m p − m e + 2m p me + mp

≈−

Ry m e n 2 2m p

(5.67)

where m p is the proton mass. The corresponding spectral lines in deuterium will be shifted towards shorter wavelengths: 1 + m e /m p >0 (5.68) λ = λH − λD = λH 1 − 1 + m e /2m p ˚ which for the n = 2 → n = 1 transition gives λ = 0.35 A. In heavy atoms and highly charged ions the isotope shift can be quite substantial because it also strongly depends on the correction related to the ﬁnite size of the nucleus (section 5.8.2) which in highly charged ions is of the order of α 3 Z 4 /n 3 and can be comparable in size to the ﬁnite-mass correction which is of the order of (m e /m p )Z 2 /n 2 . The non-relativistic Schr¨odinger equation does not describe the ﬁne structure of the level, i.e. the splitting on the quantum numbers j = ± 1/2 where is the orbital quantum number. This splitting is described by the relativistic analog of the Schr¨odinger equation, the Dirac equation (section 5.8), which takes relativistic effects into account. For H-like ions, the Dirac equation also has an analytical 15 St¨ohlker Th 1996 Phys. Bl. 52 42. St¨ohlker Th et al 1997 Hyperﬁne Interactions 108 29.

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solution with the eigenvalues now given by E nl j = E n + E nl j + · · · Z2 Ry n2 3 α2 Z 4 1 − = − Ry j + 1/2 4n n3

En = − E nl j

(5.69)

where E n is the non-relativistic part of energy, E nl j is the ﬁne-structure splitting, α ≈ 1/137 is the ﬁne-structure constant; dots in the ﬁrst line of equation (5.69) mean the higher-order corrections to the energy in powers of α 2 Z 2 . As seen from equation (5.69), the states with the same quantum number j but different have equal energies, e.g. the 2s1/2 and 2p1/2 states in H-like ions have the same energy. This -degeneracy in the point Coulomb ﬁeld is removed in the framework of QED theory which includes the interaction of the electron with its own electromagnetic radiation ﬁeld. For a hydrogenic n level, the ﬁne structure is illustrated in ﬁgure 5.7. After inclusion of the relativistic effects, the level n splits into two components which in addition are shifted. The scales for splitting and shifting are of the same order of magnitude: ∼α2 Z 4 . According to equation (5.69), the energy difference between two components, j = ± 1/2, is given by δ E n =

α2 Z 4 n 3 ( + 1)

Ry.

(5.70)

For example, for the 1s and 2p states one has, for the splitting δ E and the shift E (in units of α 2 Z 4 Ry), δ E2 p =

1 E 2 p3/2 = − 64 . (5.71) Another very important consequence following from equation (5.69) is the strong Z 4 -dependence of the ﬁne-structure splitting. In the neutral hydrogen atom, the splitting between the ﬁne-structure components of the n = 2 level (the doublet 2p3/2 − 1s1/2 and 2p1/2 − 1s1/2 ) is about 0.36 cm−1 . This doublet of spectral lines is unresolved in the emission spectra of low-temperature plasmas (T ≥ 1 eV) because the Doppler width (section 5.14.2) of the corresponding resonance lines is about two orders of magnitude larger. In the spectra of highly charged ions, the ﬁne-structure splitting increases more rapidly with Z than the relative Doppler width ωD /ω ∝ Z , and the ﬁne-structure components of the resonance lines are usually well resolved even at high temperatures T ≥ 106 K. For example, in H-like Ca19+ ions the wavelengths of the resonance doublet are ˚ respectively, and these lines are well resolved in the emission 3.018 and 3.024 A, spectra of laboratory and astrophysical plasmas.

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1 16

E 1s1/2 = − 14

5 E 2 p1/2 = − 64

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Atomic structure

Figure 5.7. Diagram of the ﬁne-structure shift and splitting of the nl level in H-like ions.

5.8 Dirac equation: relativistic effects and the ﬁne structure The non-relativistic Schr¨odinger equation for a one-electron system is the secondorder differential equation which is the analytical solution of the wavefunction and energy E (see Section 3.2). However, the equation does not account for two very important effects: (1) the existence of the electron spin ~/2; and (2) the relativistic dependence of the electron energy on its momentum. In 1928, Dirac deduced his ﬁrst-order differential equation for the wavefunction of one-electron systems (H-like ions) which is a relativistic analog of Schr¨odinger’s equation16: H = E

H = −eϕ + β m e c2 + α(c p + e A)

(5.72)

where ϕ and A denote the scalar and vector potentials of the external electromagnetic ﬁeld and m e the electron rest mass. Here α and β are 4 × 4 matrices and α is the Dirac matrix associated with the Pauli spin matrices σ : I 0 0 σ (5.73) β= α= 0 −I σ 0 0 1 0 −i 1 0 σx = σy = σz = (5.74) 1 0 i 0 0 −1 and I denotes a unit quadratic two-row matrix. The matrices α and β satisfy the conditions: α2 = 1 αβ + βα = 0 β 2 = 1. (5.75) 16 See Sobelman I I 1992 Atomic Spectra and Radiative Transitions (Berlin: Springer).

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The Dirac electron energy corresponds to the relativistic relation: E = c(αx px + α y p y + αz pz ) + βm e c2

p2 = p 2x + p 2y + pz2

E = p c + (m e c ) . 2

2 2

2 2

(5.76) (5.77)

Since α and β are 4 × 4 matrices, the solution of the Dirac equation is a vector wavefunction with four components: 

 '1 '  = 2 '3 '4

(5.78)

where 'i denotes the wavefunction components. The corresponding transition matrix elements for the Dirac wavefunctions are given by |V | =

4

'∗i |Vik |'k .

(5.79)

i,k=1

In the ﬁrst order expansion on v/c → 0, the Dirac equation has the form ( p + e A/c)2 − eϕ + µB σ B = (E − m e c2 ) (5.80) 2m e where µB = e~/2m e is the Bohr magneton and B is the magnetic-induction vector, B = rot A. This is the Pauli equation which is a non-relativistic analog of the Dirac equation and the fundamental equation in non-relativistic theory. It differs from the Schr¨odinger equation by the term −µB σ B due to electron spin. According to the Pauli equation, an electron behaves as a particle with intrinsic angular momentum s=

~ 2

σ

(5.81)

and an intrinsic magnetic moment 2 e − µB s = − s. ~ m This is equivalent to

s µ =g µB ~

g = −2.

(5.82)

(5.83)

The ratio g of the electron magnetic moment to the angular momentum, measured in units of µB and ~, respectively, is called the electron g-factor. In the Pauli approximation, the wavefunction is the two-component spinor ('3 = 0, '4 = 0) which corresponds to the spin components s = ±1/2.

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For a point nucleus of charge Z , the Coulomb potential is given by −eϕ = −Z e2 /r

A=0

(5.84)

and equation (5.72) has an analytical solution for each component of the wavefunction . The corresponding eigenvalue E rel which an electron can take in the n state is E rel = m e c

2

1+

αZ √ n − |κ| + κ 2 − α 2 Z 2

κ = (−1) j ++1/2( j + 1/2)

2 −1/2 (5.85) (5.86)

where j is the total angular momentum quantum number including the electron spin and α is the ﬁne-structure constant. The electron energy E rel in (5.85) includes the rest-mass energy of the electron. Expansion of (5.85) in powers of α 2 Z 2 and the substitution E nκ = E rel − m e c2 gives the series (5.69) given earlier (see the book by Sommerfeld A 1934 Atomic Structure and Spectral Lines (New York: E P Dutton & Co)). Similar to the Schr¨odinger equation for a one-electron system in a central Coulomb ﬁeld, the Dirac equation has an analytical solution for the wavefunction in both the bound and continuum states. In the case of the bound states, one has 

 A1 · g(r )  A · g(r )  = 2  iB1 · f (r ) iB2 · f (r )

(5.87)

where the functions Ai and Bi (i = 1, 2) depend on the angular momenta and the spin projection of the electron. The functions g(r ) and f (r ) are the radial Dirac wavefunctions, called the large and small components, respectively, and these depend on the principal quantum number n and the Dirac angular-momentum quantum number denoted the κ number: = |κ + 1/2| − 1/2.

(5.88)

The solution (5.87) of the one-electron Dirac equation is often written in the form: g(r ) · κm (r) = (5.89) i f (r ) · −κm (r) where (r) is the spin-angular spinor depending on the angular and spin momenta of the electron17. 17 See Rose E M 1961 Relativistic Electron Theory (New York: Wiley).

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In the Dirac equation, the two radial functions, g(r ) and f (r ), satisfy the coupled ﬁrst-order differential equations: κ Z d E − m e c2 + r · g(r ) = +α r · f (r ) (5.90) dr r ~c r κ Z d E − m e c2 − r · f (r ) = −α r · g(r ) (5.91) dr r ~c r and the normalization condition (g 2 + f 2 )r 2 dr = 1.

(5.92)

It is also possible to write uncoupled second-order equations for the functions g(r ) and f (r ) which are the Schr¨odinger-type equations18: −1 α2 ( + 1) 1 d 2Z 2Z dR 1+ E + r2 + R − 2 R − 2 4 r dr r r dr r −2 α2 dR 2Z α2 Z − R = E R 1 + E + r + − +1 2 r3 4 r dr (5.93) where E = E − m e c2 , and the R(r ) are the Schr¨odinger-like wavefunctions. The upper quantity − refers to the case j = + 1/2 (large component, g(r )), and + 1 to the case j = − 1/2 (small component, f (r )). The solutions for the functions g(r ) and f (r ) are expressed in terms of the hypergeometric functions F(a, b, c). At the non-relativistic limit α Z → 0, and with all distances r except small r , one has f (r ) → 0

αZ → 0

gn (r ) → Rn (r )

(gn − Rn )/Rn ∼ (α Z )2

(5.94)

where Rn (r ) is the non-relativistic Schr¨odinger radial wavefunction. However, small values of r are often the most important in Dirac theory and, for small r , i.e. Z e2 /r E + m e c2 , the Dirac radial functions g(r√ ) and f (r ) are √ expressed in terms of the Bessel functions J2γ ( 8Zr ) and J2γ +1 ( 8Zr ). At small r , the matrix elements 1/r a are very important in relativistic quantum theory; they are expressed via integrals of the type 2 2 −a 2 (g + f )r r dr (g f )r −a r 2 dr a ≥ 2. (5.95) 18 See Slater J C 1960 Quantum Theory of Atomic Structure vol II (New York: McGraw-Hill).

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5.8.1 Spin–orbit interaction In section 5.2 we have already noted the importance of the electron spin, the intrinsic angular moment s which is independent of the electron’s movement in space. The eigenvalue of the square of the spin s2 is given by s2 = s(s + 1)~2 = 34 ~2

(5.96)

and its z-component can have only two values sz = ± 12 . Beside the mechanical angular moment s, an electron has an associated magnetic moment µ given by equation (5.83). Because of the electron magnetic moment, an additional interaction arises between the electron and the magnetic ﬁeld. The magnetic ﬁeld, represented by the magnetic-induction vector B, is caused by the electric ﬁeld of the nucleus, represented by the electric-ﬁeld vector E. For the magnetic energy of a magnetic dipole we can write Vso = − µ · B 2 = µB s · B = µ B σ B

~

(5.97)

The term Vso , called the spin–orbit interaction, appears in the Pauli equation (cf (5.80)). The spin–orbit interaction can be expressed in terms of the more accessible Coulomb potential Z e2 U (r ) = − (5.98) r if we re-interpret the magnetic ﬁeld as a motional electric ﬁeld via B=−

1 [ E × v] c2

(5.99)

where v denotes the electron velocity. We further use the relation −e E = −∇ · U (r ) = −

dU r dr r

(5.100)

for the force acted on the electron, plus the expression = m e [r × v]

(5.101)

for the electron angular momentum. Inserting (5.99)–(5.101) into equation (5.97) yields the spin–orbit interaction as Vso =

1 1 1 dU × s. 2 m 2e c2 r dr

(5.102)

We note that expression (5.102) has been multiplied by a factor of 1/2 accounting for the electron acceleration in the external electric ﬁeld (the so-called Thomas– Frenkel factor). The spin–orbit interaction comprises the scalar product of the

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spin s and orbital vectors which is why it is called the spin–orbit interaction. This interaction has a pure quantum-mechanical relativistic nature and vanishes when v/c → 0. The spin–orbit interaction depends not only on and s but also on the total angular momentum j = + s. Actually, j 2 = 2 + s 2 + 2 · s j ( j + 1) − ( + 1) − 3/4 2 j 2 − 2 − s 2 = ~ ·s = 2 2

(5.103)

since s2 = s(s + 1)~2 = 3/4~2 for the spin quantum number s = 1/2. Taking into account that, for H-like ions, U (r ) = − r −3 n

Z e2 r

Z e2 dU = 2 dr r

1 Z3 = 3 n ( + 1)( + 1/2) a03

(5.104)

one has for the spin–orbit interaction and corresponding energy correction, respectively, Vso = E so =

Z e2 ~2 j ( j + 1) − ( + 1) − s(s + 1) −3 r 2m 2e c2 2

(5.105)

j ( j + 1) − ( + 1) − s(s + 1) α 2 Z 4 Ry 2( + 1)( + 1/2) n3

(5.106)

where α denotes the ﬁne-structure constant and a0 the Bohr radius. Since r −3 ∝ Z 3 , the spin–orbit interaction Vso ∝ Z 4 . The relativistic corrections, associated with the dependence of the electron mass on velocity, and the spin–orbit interaction (5.106) deﬁne the ﬁne structure of the atomic levels given by (5.69). The proportionality of the spin–orbit interaction Vso ∝ Z 4 is very important in the case of many electron highly charged ions where it plays the main role among the other magnetic interactions (see the Breit operator in section 5.11).

5.8.2 Nuclear ﬁnite-size correction The Dirac equation allows one to calculate the correction to the energy related to the ﬁnite size of the nucleus which in highly charged ions can be rather large. In the case of a homogeneously charged nuclear sphere with radius r0 , the nuclear

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potential is often written in the form  2 2  Z e r  − 3− r ≤ r0  2r0 r0 V (r ) =   Z e2  − r ≥ r0 r

(5.107)

The numerical solution of the Dirac equation gives the following correction to the energy due to the nuclear ﬁnite-size correction, i.e. the energy difference with the use of a point nucleus −Z e2 /r or a charged sphere (5.107), respectively: E FS =

2α 3 Z 4 Ry FFS πn 3

(5.108)

where FFS denotes the reduced nuclear ﬁnite-size correction. The values of r0 can be expressed in terms of the root-mean-square (rms) radius by (5.109) r 2 1/2 = 3/5r0 . The values for the root-mean-square radius can found in the reference19. For estimating r 2 1/2 , the empirical formula (1.4) can also be used. Another simple model also used for the nucleus charge density is the twoparameter Fermi distribution ρF (r ) =

ρ0 1 + e(r−r0 )/a

(5.110)

where r0 denotes the half-density radius. For ns1/2 states, the accurate ﬁnite-size correction for H-like ions with charge Z = 1 − 100 is approximated within 0.2% by the following expression20: 2γ (α Z )2 αZ R 2 E FS (ns1/2 ) = [1 + (α Z ) f ns (α Z )] 2 m e c2 (5.111) 10n n (~/m e c) where γ = 1 − (α Z )2 and f 1s1/2 = 1.380 − 0.162α Z + 1.612(α Z )2 f 2s1/2 = 1.508 + 0.215α Z + 1.332(α Z )2 and R is the effective radius given by "1/2 ! 3 1 3 r 4 5 2 2 r 1 − (α Z ) − . R= 3 4 25 r 2 2 7 19 AufMuth P, Heilig K and Stendel A 1987 At. Data Nucl. Data Tables 37 455.

Fricke G et al 1995 At. Data Nucl. Data Tables 60 287. 20 Shabaev V M et al 2000 Physica Scripta T 86 7.

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Table 5.14. Nuclear ﬁnite-size correction function FFS for the ground and excited states in H-like ions, equation (5.108). Z

1s1/2

2s1/2

2p1/2

10 20 30 40 50 60 70 80 90 100

0.018 0.027 0.041 0.060 0.093 0.144 0.236 0.398 0.711 1.343

0.018 0.028 0.043 0.065 0.105 0.171 0.300 0.544 1.060 2.214

0 0 0 0.001 0.003 0.007 0.017 0.044 0.117 0.332

For example, in H-like uranium U91+ (1s1/2 ) one has: FFS = 0.84, E FS = 200 eV, the ionization energy I1s = 131 814 ± 2 eV (see table 5.16). The reduced nuclear ﬁnite-size corrections FFS for the lowest states in H-like ions are given in table 5.14.

5.9 Magnetic effects and the hyperﬁne structure The hyperﬁne interaction between the atomic nucleus and the orbital electrons strongly affects the energy-level structure and transition probabilities in atoms and ions. The hyperﬁne interaction is the interaction between the magnetic and electric moments of the nucleus and the magnetic ﬁeld generated by the bound electrons at the nucleus. The hyperﬁne interaction leads to the hyperﬁne splitting (hfs) of the energy levels in atoms and ions; the scale of this structure is about three orders of magnitude less than the ﬁne-structure splitting caused by the spin–orbit interaction (section 5.8.1). The well-known 21 cm transition between hyperﬁne levels of the ground state in hydrogen plays a key role in astronomy and astrophysics. The hfs of the energy level into components occurs in atoms and ions with a non-zero magnetic dipole moment µ )= 0, or with a non-zero electric quadrupole moment Q )= 0 of the nucleus. Each component is characterized by a set of quantum numbers J I F M F , where J = L + S denotes the total, orbital plus spin, angular momentum of the atomic electrons, I the nuclear spin momentum, F = J + I the total angular momentum of an atom and M F the projection of F on the quantum axis. The number of splitting sublevels is 2I + 1 if I < J , and 2 J + 1 if I > J . The hfs E is deﬁned by the hyperﬁne interaction matrix element VHF ∝

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|I · J | and is given by21 E = 12 AC + BC(C + 1) − 43 B I (I + 1)J (J + 1) C = F(F + 1) − J (J + 1) − I (I + 1)

(5.113) (5.114)

where A and B are the dipole and quadrupole hfs constants, respectively. The third term in (5.113) is responsible for level shifting and is usually omitted in the expression for hfs. Determining the constants A and B for heavy atoms and highly charged ions involves relativistic and nuclear ﬁnite-size effects. In the case of highly charged H-like ions with the nuclear charge Z 1, the splitting constants are given by the following equations A=

gI m e α2 Z 3 Ry j ( j + 1)( + 1/2) m p n 3

(5.115)

B=

3Q/a02 Z3 Ry 8I (2I − 1) j ( j + 1)( + 1)( + 1/2) n 3

(5.116)

where a0 denotes the Bohr radius, m p the proton mass and Q the nuclear electric quadrupole moment. The quantity Q has the dimension of an area (a02 ) and is a measure of the deviation of the charge distribution from spherical symmetry. The dimensionless quantity g I is called the gyromagnetic ratio or the nuclear g-factor and is deﬁned, in analogy to the electron g factor of equation (5.83) by µ I = gI µN ~

(5.117)

where µ denotes the nuclear magnetic dipole moment, I the nuclear spin and µN the nuclear magneton: µN ∝ µB /1836. (5.118) The magnetic interaction, corresponding to the terms in (5.113) proportional to B, is nonzero if the nuclear spin quantum number I ≥ 1 and if the electron angular momentum j ≥ 1. Experimental data on nuclear spin, nuclear dipole and quadrupole moments are given in the work22. Usually, the quadrupole terms in (5.113), depending on Q, are found to be small compared with the magnetic dipole term and the hfs splitting is represented in the form: E = 12 AC

(5.119)

where the constants C and A have been given in (5.114) and (5.115), respectively. For highly charged ions with one optical electron outside a spherical core, the inclusion of relativistic and QED corrections leads to the following form of 21 Sobelman I I 1992 Atomic Spectra and Radiative Transitions (Berlin: Springer). 22 Raghavan P 1989 Atom. Data Nucl. Data Tables 42 189.

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Table 5.15. Experimental (exp.) and theoretical (theor.) values for the hyperﬁne splitting E (in nm) of the ground 1s 1 S1/2 -state in heavy H-like ions. Calculations are taken from Shabaev V M et al 2000 Physica Scripta T 86 7; experimental data and corresponding references can be found in Beiersdorfer P et al 2001 Phys. Rev. A 64 032506. Ion

Transition F − F

165 Ho66+

4–3

572.64(15)

566.3(1.7)

185 Re74+

3–2

456.05(30)

451.0(1.7)

187 Re74+

3–2

451.69(30)

446.6(1.7)

203 Tl80+

1–0

385.822(30)

384.0(2.1)

205 Tl80+

1–0

207 Pb81+

1–0

209 Bi82+

5–4

E, exp.

382.184(34) 1019.7(2) 243.87(4)

E, theor.

380.2(2.1) 1020.3(4.5) 243.(1.3)

the hfs23 : E =

1 2 AC['(α Z )(1 − δ)(1 − %) + x rad ]

(5.120)

where '(α Z ) is the one-electron relativistic factor, δ and % denote the nuclearcharge and magnetization-distribution corrections and x rad is the one-electron QED correction. Available experimental data for the hfs of the ground state in heavy Hlike ions are compared with theory in table 5.15. One can see that most of the measurements have been carried out with a very high relative accuracy of about 10−4 . However, a comparison of these results with the advanced theoretical calculations shows that theory and experiment still disagree on the level of a few nm, which is explained by the uncertainty of the nuclear magnetization distribution correction, the so-called Bohr–Weisskopf effect24. As can be seen from table 5.15, the hfs in very highly charged ions can reach the optical regime. This circumstance is used in high-resolution laser spectroscopy25 and other methods to measure the atomic structure so accurately that information about the nucleus, e.g. the g-factors, can be obtained. Laserspectroscopy experiments with highly charged ions became feasible by the advent of heavy-ion cooler rings described in chapter 4, which, compared to usual accelerators, have a much larger interaction time and interaction time and the quality of the ion beam. An example of the Experimental Storage Ring at GSI in Dramstadt was shown in ﬁgure 4.38 indicating a laser beam colinear (parallel or anti-parallel) to the direction of the ion beam. 23 Shabaev V M 1994 J. Phys. B 27 5825. 24 See Shabaev V M 2001 Phys. Rev. Lett. 86 3959. 25 See Demtr¨oder W 1998 Laser Spectroscopy (Berlin: Springer).

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2p3/2 2s1/2 2p1/2

j =3/2 j =1/2

- 29.6 keV - 34.1 keV - 34.2 keV

75 eV

Ly-α

U91+

M1

n=1 470 eV

j =1/2 Bohr

Dirac

1s1/2

- 131.8 keV

QED

Figure 5.8. Level and decay scheme for the low lying states of hydrogen like U91+ with binding energies as indicated by the numbers.

5.10 QED effects and the Lamb shift In atomic theory of spectra for neutral and weakly ionized atoms, relativistic effects are considered as small corrections in contrast to the case of highly charged ions for which relativistic and quantum electrodynamic (QED) effects strongly depend on the ion charge. The development of modern QED started in 1947 when Lamb and Retherford experimentally showed that the 2s1/2 and 2p1/2 levels in neutral hydrogen are nondegenerate and that the levels are separated by a very small but measurable value E ≈ 1058 MHz ≈ 4.4 × 10−6 eV, the so-called Lamb shift. Experimental studies of bound-state QED have spread out to cover different atomic states and ions of considerable nuclear charge. Figure 5.8 shows the level and decay scheme of the low lying states of hydrogen like U91+ employing atomic models of increasing sophistication when going from left to right. The Dirac theory correctly describes the ﬁne-structure splitting of the n = 2 states with the total angularmomentum quantum numbers j = 1/2 and 3/2. The further splitting is caused by the interaction of the electron with its own electromagnetic radiation ﬁeld and is well explained by QED theory. The splitting between the 2s1/2 and 2p1/2 states occurs because the radiative shifts are much smaller for p states than for s states. Although the term Lamb shift was originally used only for the 2s1/2 − 2p1/2 energy splitting in hydrogen, it is now commonly used for the shift of isolated levels in atoms and ions, e.g. the Lamb shift of the 1s1/2 , 2s1/2 , 2p1/2 . . . levels. The Lamb shift for one-electron systems is deﬁned as the difference between the true (experimental) binding energy and the Dirac eigenvalue calculated for a point nucleus and disregarding all QED effects. In the deﬁnition of the Lamb shift for many-electron systems (He-, Li-like ions etc), one has to be very accurate because there is no well-deﬁned analog to the Dirac equation for these ions, even for He-

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like ones as the simplest multi-electron system. Signiﬁcant progress has been achieved in accurately measuring the energy levels in heavy few-electron ions including the 1s Lamb shift in H-like U, Bi, and Au, the hyperﬁne structure of H-like Bi, Ho, Re and Pb and the ground-state energies of He-like ions. These experimental data provide a unique possibility of testing QED effects in strong nuclear ﬁelds26 . The main contribution to the total Lamb shift E is given by the sum of the so-called electron self-energy (SE), vacuum-polarization (VP) and nuclear-ﬁnite size (FS) corrections. For H-like ions, the Lamb shift is usually presented in the form: α (α Z )4 F(α Z )m e c2 F(α Z ) = FSE + FVP + FFS (5.121) π n3 where F(α Z ) is a dimensionless slowly varying function describing the sum of the different corrections as explained later. The leading QED corrections are the self-energy and vacuum polarization corrections which constitute a large part of the Lamb shift in highly charged ions. Both corrections are of the order of α5 Z 4 m e c2 . The SE corrections originate from the emission and reabsorption of a virtual photon by an electron. The VP effect describes the Coulomb interaction of the electron and the nucleus when the exchanged photon excites a virtual electron–positron pair in the vacuum. In QED theory, the different interactions between the electrons, ions and photons are represented by Feynman diagrams. For a one-electron Lamb shift, the Feynman diagrams for the main contributions are shown in ﬁgure 5.9. For states with angular momentum )= 0, F(α Z ) ≈ 1 whereas F(α Z ) ≈ ln[(α Z )−2 ] for the s-states. Therefore, the Lamb shift is largest for the ground state and strongly increases (∼ Z 4 ) with nuclear charge Z . The calculated Lamb shift contributions E SE , E VP , E FS for the 1s level in H-like ions is plotted in ﬁgure 5.10 as a function of the nuclear charge. Table 5.16 shows these contributions for the 1s state in H-like uranium relative to the electron rest energy. It can be seen that the main radiative contributions arise from the self-energy (SE) and vacuum-polarization (VP) corrections, while the main non-QED correction is due to the nuclear ﬁnite-size effect. The total theoretical value of the ground-state U91+ ion is estimated to be E(1s) = (131 814 ± 2) eV with the accuracy mainly limited by the higherorder QED corrections which are expected to be about a few eV. Also included in table 5.16 are two experimental results for the 1s Lamb shift obtained at the ESR storage ring27,28 . Sophisticated QED theory predicts a value of 463.95 ± 0.50 eV for the 1s Lamb shift, while the experimental results can be combined to give 470 ± 10 eV, E =

26 See the review papers by Mohr P J, Plunien G and Soff G 1998 QED corrections in heavy atoms Phys. Rep. 293 227–369. Eides M I, Grotch H and Shelyuto V A 2001 Theory of light hydrogenic atoms Phys. Rep. 342 63–261. 27 Beyer H F et al 1995 Measurement of the ground-state Lamb shift of hydrogen-like uranium at the electron cooler of the ESR Z. Phys. D 35 169. 28 St¨ohlker Th et al 2000 1s Lamb shift in hydrogen-like uranium measured on cooled, decelerated ion beams Phys. Rev. Lett. 85 3109.

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Figure 5.9. Feynman diagrams for the one-electron Lamb shift: SE, self energy; VP, vacuum polarization; and FS, nuclear ﬁnite size.

Table 5.16. Theoretical contributions (in eV) to the 1s ground-state energy, and the Lamb shift in H-like U91+ . Experiment: Beyer H et al 1995 Z. Phys. D 35 169 and St¨ohlker T et al 2000 Phys. Rev. Lett. 85 3109. Theory: Yerokhin V A and Shabaev V M 2001 Phys. Rev. Lett. 64 062507. Contribution One-electron Dirac eigenvalue (point nucleus) One-electron SE One-electron VP One-electron nuclear ﬁnite size Nuclear recoil Nuclear polarization One-electron second-order QED Total energy Total theoretical Lamb shift: Yerokhin et al 2001 Experiment: Beyer et al 1995 St¨ohlker et al 2000

Value [eV] −132 279.9 355 −88.6 198.8 0.5 −0.2 ±2 −131 814±2

463.95±0.50 470±16 468±13

i.e. the theoretical accuracy is about one order of magnitude higher than the experimental one. Experimentally, the Lamb shift of the ground state in H-like ions has been derived from measurements of the photon energies of K x-ray lines. The x-ray

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Figure 5.10. Correction contributions (5.121) to the 1s Lamb shift in H-like ions as a function of the nuclear charge Z. From Mokler P and St¨ohlker Th 1996 Adv. At. Mol. Opt. c Phys. 37 297. 1996 Academic Press.

spectra are induced by radiative recombination and electron capture into excited states by bare ions. The difference between the measured and the Dirac transition energies approximately gives the 1s1/2 Lamb shift because the shifts of higher states are small. The results of a similar measurement27 with bare uranium ions in the electron cooler are displayed in ﬁgure 5.11 where the Lyα2 and Lyα1 lines are shown. The other method makes use of electron capture in the gas target of a storage ring. A comparison of existing experiments with theory is given in ﬁgure (after ﬁgure 5.11) for all atomic numbers. Plotted is the Lambshift F(α Z ) scaled according to equation (5.121). Extraordinarily high experimental accuracy

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Atomic structure

Figure 5.11. Measured Lyα lines of U91+ ions in the electron cooler of the ESR storage ring. The intensity appearing on the low-energy side of the peaks is caused by delayed c cascade feeding. From Beyer H F et al 1975 Z. Phys. D 35 169. 1975 Springer.

±3.5 ppm has been obtained for atomic hydrogen29. At intermediate nuclear charge Z , a few measurements reached the 2% level of accuracy. The uranium measurements cited in table 5.16 are also close to that level.

5.11 Many-electron systems The main difference between the atomic levels in H-like ions and many-electron systems is that in the latter the interaction potential V (r ) is no longer the Coulomb potential which removes the degeneracy of the atomic levels on the orbital quantum numbers and resolves the transitions n–n with different and n = 0. The radiation spectra of many-electron systems are usually deﬁned by the transitions of one of the outer-most electrons while the interaction with other (core) electrons is reduced to screening the nuclear charge. However, in some cases, transitions involving inner-shell electrons may be very important, i.e. in dielectronic and autoionization processes (chapter 6). In few-electron highly charged ions, Z 1, N Z , where N is the total number of electrons, the interaction between the bound electrons and the nucleus plays the main role and many characteristics can be obtained by an expansion in power of the 1/Z parameter. The main dependencies of some quantities on Z are given in table 5.1 but they mainly refer to H-like ions. In many-electron systems these dependencies can take other forms. For example, in Li-like ions with three electrons, transitions with n − n and n = 0 are resolved and the 29 Udem Th et al 1997 Phys. Rev. Lett. 79 2646.

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resonance energies E(2s–2p) and corresponding oscillator strengths f (2s–2p) are approximately given by E(2s–2p) ≈ 0.0707 Z − 0.120 f (2s–2p) ≈ 1.35 Z −1 + 2.20 Z −2

(5.122)

i.e. E ∼ Z and f ∼ Z −1 . Here Z denotes the nuclear charge, and E is given in atomic units, (1 a.u. of energy is 2Ry ≈ 27.212 eV). If an ion contains two or more electrons, the inter-electron Coulomb interaction becomes very important even in the case of two-electron He-like ions. The non-relativistic Schr¨odinger equation for the N-electron system H = E

H=

N

i=1

pi2 Z e2 − 2m e ri

+

e2 ri j

(5.123)

no longer can be solved analytically because of the presence of the Coloumb electron–electron interaction terms e2 /ri j . In 1929, G Breit (1899–1981) introduced an operator for the atomic Nelectron system taking account of relativistic corrections of order up to (v/c)2 into the Schr¨odinger equation where v is the electron-orbit velocity. For a twoelectron system, the Breit operator takes the form30: H = H0 + H1 + H2 + H3 + H4 + H5 1 Z e2 Z e2 e2 H0 = ( p12 + p22 ) − − + 2m e r1 r2 r12 1 H1 = − ( p4 + p24 ) 8m 3e c2 1 e2 1 (r12 · p12 )(r12 · p2 ) p1 · p2 + H2 = − 2 2m 2e c2 r12 r12

(5.124)

Z πe~2 πe~2 [δ(r ) + δ(r )] − δ(r12 ) 1 2 2m 2e c2 m 2 c2 e2 ~ 1 2 Z [r1 × p1 ] − 3 [r12 × p1 ] + 3 [r12 × p2 ] s1 H4 = 2m 2e c2 r13 r12 r12 e2 ~ 1 2 Z + [r2 × p2 ] − 3 [r21 × p2 ] + 3 [r21 × p1 ] s2 2m 2e c2 r23 r12 r12 8π e2 ~2 1 3(s1 · r12 )(s2 · r12 ) H5 = 2 2 − s1 · s2 · δ(r12 ) + 3 s1 · s2 − 2 3 me c r12 r12

H3 =

30 See Bethe H A and Salpeter E E 1977 Quantum Mechanics of One- and Two-Electron Atoms

2nd edn (New York: Plenum).

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where p denotes the electron-momentum operator and s is the electron-spin operator. The term H0 represents the usual Hamiltonian in the non-relativistic Schr¨odinger equation, and the remaining terms H1, H2 , . . . , H5 are associated with relativistic effects. The Hamiltonians H1 and H2 reproduce the dependence of the electron mass on its velocity and the retardation of the electromagnetic interaction, respectively. The term H3 is called the Darwin term or the contact term. All three terms, H1, H2 and H3 do not contain spin operators and they are not, therefore, responsible for the level splitting. The spin-dependent Hamiltonians H4 and H5 describe the magnetic interactions—the spin–own orbit, spin–other orbit, spin–spin and orbit–orbit interactions. The most important is the spin–orbit interaction, H4, because it has the highest dependence on the nuclear charge: H4 ∝ Z 4 . One can see that even for two-electron atomic systems, solving the Breit equations is quite complicated. Usually, they are solved numerically on the basis of perturbation theory using a non-relativistic Hamiltonian and Hartree–Fock or Coulomb wavefunctions. The analog Breit–Dirac Hamiltonian is much more complicated and can be written approximately as N 2

e2 Z e αi pi + βi m e c2 − + + HB (5.125) H= ri ri j i) = j i=1

αi α j (αi · ∇ i )(α j · ∇ j )ri j (5.126) + HB = − e 2 ri j 2 i) = j

where α and β are the Dirac matrices and H B is the Breit operator. The two terms in HB account for the magnetic interaction between the electrons and the retardation effects in the Coulomb electron–electron interaction which arise because of the ﬁnite speed of light. Experimental investigations of the atomic structure of highly charged ions are strongly facilitated by theoretical results based on sophisticated calculations. Modern theories are characterized by a deep understanding of the many-electron problem including non-relativistic, relativistic and QED effects, on one side, and are inﬂuenced by the rapid development of computers and computer codes, on the other side. In many important cases, the theoretical accuracy approaches or even supersedes that of the experimental data. For example, the QED corrections to the 1s ground-state Lamb shift of H- and He-like highly charged ions can be now calculated with a high precision not yet achieved by experimental techniques. But complications certainly arise when the number of electrons increases. At present, many methods and computer codes are available for calculating the atomic structure of many-electron systems, i.e. the wavefunctions, energy levels, transition probabilities, etc. These include the MCHF method (multiconﬁguration Hartree–Fock), MCDF (multi-conﬁguration Dirac-Fock), the MBPT (many-body perturbation theory), the HULLAC (relativistic multi-

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conﬁguration Hebrew University Lawrence Livermore atomic code), the CCE (coupled-cluster expansion), the Cowan code, the Grant code, the MZ code (Zexpansion method) and the RRPA (relativistic random-phase approximation). Almost all of these methods for treating many-electron systems are basically nonrelativistic and the relativistic effects are usually incorporated within the Breit– Pauli or Breit–Dirac Hamiltonian. The progress in recent years in accounting for the relativistic, QED and electron-correlation effects in many-electron systems is described in review31. In the many-electron ion system described in L S coupling (section 5.2), the energy term depends on the quantum numbers for orbital L, spin S and total angular momentum J . If L ≥ S the term splits into 2S + 1 different components. The quantity 2S + 1, determining the number of components, is called the multiplicity of the term. If L < S it splits into 2L + 1 components. In this coupling scheme, the energy splitting between neighboring levels is given by the Land´e interval rule: E j, j −1 = E j − E j −1 = a(L S) · J

(5.127)

giving a splitting proportional to J . The multiplet splitting constant a(L S) depends only on the quantum numbers L and S. If a > 0 the multiplet is called normal. In this case the component with the smallest possible value, J = |L − S|, has the lowest energy value. If a < 0 the multiplet is called inverted. The component with the greatest possible value J = L + S has the lowest energy value. In multi-electron highly charged ions with an increase in nuclear charge Z , a smooth transition from L S to j j coupling takes place. The level structure of highly charged ions usually corresponds to an intermediate type of coupling. For example, in He-like ions the Land´e rule for the 1s2p 3 P0 triplet is violated as is shown in ﬁgure 5.12 for the energy-level diagrams of the n = 2 levels in the He-like Ne8+ (Z = 10) and W72+ (Z = 74) ions. In the case of highly charged ions Z N, the electron–nucleus interaction signiﬁcantly exceeds the interaction between electrons. As a result, electron conﬁgurations with the same set of principal quantum numbers have rather close energy levels. For transitions with no change in the principal quantum numbers, n = 0, the relative energy values (E n − E n )/E n decrease as Z −1 . The corresponding frequencies of transitions n − n scale as Z 1 , while the frequencies for transitions n − n , n )= n scale as Z 2 . For example, in Belike ions there are many-electron conﬁgurations with the same sets of principal quantum numbers, e.g., 1s2 2s2 , 1s2 2s2p, 1s2 2p2 and so on. For transitions with n )= 0, such as 1s2 2s3p − 1s2 2s2 , with an increase in the ion charge Z , the corresponding wavelength decreases as Z −2 and for transitions with n = 0 (1s2 2s3p − 1s2s3s), the wavelength decreases as Z −1 . QED theory of two-electron ions is more complicated than the one-electron 31 Lindgren I 1992 Recent Progress in Many-Body Theories ed J Ainsworth (New York: Plenum) vol 3, p 245.

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Figure 5.12. The multiplet structure of the excited n = 2 levels (conﬁguration 1s2 2s and 1s2p) of the He-like ions with the nuclear charge Z = 10 and Z = 74.

system because of the necessity of including corrections caused by electron– electron interactions. Despite these difﬁculties, substantial progress in the theory of two-electron systems has been achieved and the theoretical accuracy is approaching that of one-electron systems32 . The energy of the He-like system can be presented as the sum of two independent one-electron contributions and a two-electron contribution. The latter includes the effects from the electron– electron interaction such as one- and two-photon exchange diagrams, self-energy and vacuum-polarization screening diagrams and higher-order corrections. The corresponding set of Feynman diagrams are shown in ﬁgure 5.13. The main part of the two-electron contribution is given by the one-photon exchange diagram. A differential method was established at the SuperEBIT Electron-Beam Ion Trap at Lawrence Livermore National Laboratory which exploits radiative recombination transitions into the vacant 1s shell of bare and H-like ions (see also ﬁgure 4.59). That method has allowed direct measurements of the two-electron contributions to the ground-state binding energies of He-like ions with nuclear charge 32 ≤ Z ≤ 83. Improved experimental uncertainties in the SuperEBIT33 and in next-generation EBIT-type34 ion sources would provide an experimental sensitivity for the ground-state QED of H- and He-like ions which is beyond the lowest-order Lamb shift. 32 Mohr P J, Plunien G and Soff G 1998 QED corrections in heavy atoms Phys. Rep. 293 227–369. 33 Marrs R E, Elliott S R and Knapp D A 1994 Phys. Rev. Lett. 74 4082. 34 Marrs R E 1999 Nucl. Instrum. Methods B 149 182.

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Figure 5.13. Feynman diagrams representing one- and two-photon contributions to the Lamb shift of two-electron ions: (a) and (b) two-photon exchange diagrams; (c) and (d) self energy; (e) and (f ) vacuum polarization.

5.12 Transition energies and x-ray spectra The inﬂuence of electron correlation, relativistic and QED effects on the electron binding energy strongly increases with the nuclear charge number Z . For ions with more than one electron, i.e. He-, Li-like ions, . . . , the difﬁculties involved in calculating the wavefunctions and energy levels increase signiﬁcantly due to the additional interactions between the atomic electrons. Several sophisticated theoretical approaches are applied to highly charged many-electron ions: the uniﬁed method, the Z -expansion method, Multi-Conﬁguration DiracFock (MCDF), the Conﬁguration-Interaction (CI) method, relativistic ManyBody Perturbation Theory (MBPT), the relativistic generalization of the random phase approximation and others (see section 5.11). Relativistic conﬁgurationinteraction calculations of energy levels for the n = 2 states in Li-like ions with nuclear charges in the range 10 ≤ Z ≤ 92 are presented in table 5.17. Calculated 2s–2p transition energies are compared with available experimental data and other calculations in table 5.1835. Finally, we note that the radiation spectra of highly charged many-electron 35 For details, see Yerokhin V A et al 1999 Phys. Rev. A 60 3522.

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Table 5.17. Calculated atomic energy levels in atomic units (1 a.u. = 2 Ry) for the n = 2 states in Li-like ions as a function of nuclear charge. The energy values are counted from the ionization limit. Z

2s1/2

2p1/2

2p3/2

10 15 20 26 32 42 54 64 74 82 90 92

102.794 72 238.820 49 432.049 40 740.569 71 1 134.508 2 1 988.013 7 3 358.248 5 4 816.925 6 6 601.107 5 8 296.781 1 10 270.053 10 812.432

102.210 80 237.871 87 430.727 75 738.783 68 1 132.233 2 1 984.849 2 3 353.845 1 4 811.290 1 6 594.002 8 8 288.303 1 10 260.106 10 802.122

102.203 28 237.820 70 430.541 64 738.196 90 1 130.794 6 1 980.223 1 3 340.160 1 4 782.478 0 6 538.779 5 8 199.689 8 10 122.138 10 648.563

ions are much richer in lines compared with the spectra from neutral and weakly ionized atoms for two main reasons. First of all, highly charged ions contain a large number of lines associated with forbidden transitions, the decay probabilities of which strongly increase with ion charge (section 5.4). The second reason is associated with the presence of the dielectronic satellite lines in the spectra of highly charged ions considered in section 6.8. These satellite lines are located above the ﬁrst ionization limit and exist only in non-hydrogenic ions, i.e. those with two or more electrons. For highly charged ions, the intensities of the satellite lines markedly increase when the ion charge increases and are comparable with the intensities of the resonance and other strong lines; therefore, the satellites are very important for plasma diagnostics. In table 5.19, calculated values of transition energies, radiation Ar and autoionization Aa transition probabilities are given for He- and Li-like iron ions36. ˚ there are It can be seen that in a very small spectral interval λ ≈ 0.05 A, 23+ many spectral lines corresponding to transitions in Li-like Fe and He-like Fe24+ ions. The intensities of these lines are very sensitive to the plasma macroparameters such as the electron density and temperature or the distribution over ionization stages. Therefore, to identify the spectra correctly it is necessary to know the wavelengths and line intensities of these lines very accurately. The spectral resolution obtained in spectroscopic investigations of laboratory and astrophysical plasmas is comparable with the accuracy of theoretical calculations and lies in the range of λ/λ ≈ 10−4 –10−5 for the wavelength ˚ The high resolution for identifying the satellite spectra is a very λ = 1–10 A. 36 From Nielsen J 1988 At. Data Nucl. Data Tables 38 339.

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Table 5.18. The 2s1/2 –2p1/2 transition energies (in eV) for Li-like ions as a function of the nuclear charge Z. From Chen M H et al 1995 Phys. Rev. A 52 266. Z 10 15 18 20 26 28 30 32 40 42 47 50 54 60 70 74 80 82 90 92

Rel. CI 15.8888 25.813 35.963 48.600

61.907 86.11

119.82

270.80 280.74

RMBPT 15.8885 25.812 31.868 35.964 48.602

57.389 61.911 81.04 86.12 107.92 119.84 139.29 176.56 193.33 220.99 230.70 270.69 280.68

QED

MCDF 25.806

31.865 35.960

52.951 57.384 61.906 81.03

35.957 48.597

61.908 86.129

99.43 107.90 119.82 139.25 176.44 193.33 220.93 230.68 270.60 280.48

Experiment 15.8887(2) 25.814(3) 31.866(1) 35.962(2) 48.599(1) 48.602(4) 52.950(1) 57.384(3) 61.902(4)

119.901

86.10(1) 99.438(7) 107.911(8) 119.97(10)

272.320 282.568

280.59(9)

important advantage which is exploited for x-ray plasma diagnostics. Examples of typical x-ray spectra from highly charged ions are given in ﬁgures 5.14–5.17 for astrophysical and laboratory plasmas, mainly for He-like ions and their satellites, i.e. Li-like ions. Designations of lines are given in table 5.5. According to the plasma conditions, the spectra have different patterns. The intercombination (y), magnetic-quadrupole (x) and forbidden (z) lines can be of comparable intensity as the resonance line w. The satellite lines are the result of transitions 1s2 n–1s2pn with n ≥ 2. At n = 2, the satellite lines are the most intense and differ strongly from the resonance line w, the wavelength of which is usually adopted from sophisticated calculations as a reference line. Some satellites can also be formed as a result of electron-impact excitation of Li-like ions, for example, the intensity of the q-satellite is mostly determined by excitation of Li-like ions. By comparing the relative line intensities, it is possible to determine such plasma parameters as the electron temperature and density, the ionization stages and the deviation of the electron-velocity distribution from a Maxwell function.

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Table 5.19. Calculated wavelengths λ, radiation A r and autoionization A a transition ˚ Here probabilities for He- and Li-like Fe ions in the spectral range of 1.85130–1.89759 A. the ‘+’ sign corresponds to j = + 1/2 and the ‘−’ sign to j = − 1/2. In the last column, a key letter is given according to the Gabriel’s notation explained in table 5.5. Transition

˚ λ (A)

A r (s−1 )

A a (s−1 )

Key

1s+ 2p+ (1) → 1s2 (0) 1s+ 2p+ 2p+ (1/2) → 1s2 2p− (1/2) 1s+ 2s+ 2p+ (3/2) → 1s2 2s+ (1/2) 1s+ 2p+ 2p+ (1/2) → 1s2 2p+ (3/2) 1s+ 2p+ 2p+ (1/2) → 1s2 2s+ (1/2) 1s+ 2p+ 2p+ (3/2) → 1s2 2p− (1/2) 1s+ 2p− (1) → 1s2 (0) 1s+ 2p+ 2p+ (3/2) → 1s2 2p+ (3/2) 1s+ 2p− 2p+ (1/2) → 1s2 2p− (1/2) 1s+ 2p− 2p+ (3/2) → 1s2 2p− (1/2) 1s+ 2s+ 2p− (1/2) → 1s2 2s+ (1/2) 1s+ 2p− 2p+ (1/2) → 1s2 2p+ (3/2) 1s+ 2p− 2p+ (3/2) → 1s2 2p− (1/2) 1s+ 2p+ 2p− (1/2) → 1s2 2p− (1/2) 1s+ 2p− 2p+ (5/2) → 1s2 2p+ (3/2) 1s+ 2s+ 2p− (3/2) → 1s2 2s+ (1/2) 1s+ 2p− 2p+ (3/2) → 1s2 2p+ (3/2) 1s+ 2s+ 2p− (1/2) → 1s2 2s+ (1/2) 1s+ 2p+ 2p− (1/2) → 1s2 2p+ (3/2) 1s+ 2s+ 2s+ (1/2) → 1s2 2p− (1/2) 1s+ 2s+ 2s+ (1/2) → 1s2 2p+ (3/2)

1.85130 1.85263 1.85670 1.85704 1.85753 1.85833 1.86048 1.86277 1.86343 1.86367 1.86420 1.86790 1.87073 1.87308 1.87352 1.87456 1.87523 1.87562 1.87758 1.89298 1.89759

4.78+14 1.13 + 13 1.12 + 12 2.41 + 14 1.78 + 14 8.44 + 12 4.42 + 13 6.15 + 14 5.40 + 14 3.22 + 14 3.18 + 14 1.65 + 14 3.64 + 10 2.06 + 13 3.37 + 13 1.59 + 13 1.01 + 13 4.97 + 12 8.10 + 10 9.49 + 12 9.83 + 12

0 2.64 + 13 1.06 + 14 2.64 + 13 7.79 + 13 3.47 + 13 0 3.47 + 13 1.02 + 12 1.25 + 14 3.84 + 13 1.02 + 12 7.94 + 11 2.47 + 11 2.25 + 13 7.10 + 11 7.94 + 11 2.65 + 10 2.47 + 11 1.42 + 14 1.42 + 14

y n s m t b w a d k j l g i e u f v h p o

5.13 External ﬁelds The basic properties of atoms and ions as particles with atomic structure are revealed in interactions with external electric and magnetic ﬁelds and surrounding media such as gases or plasmas. In particular, the interaction with external electromagnetic ﬁelds is very important because the external ﬁelds inﬂuence the atomic energy-level structure and radiation transition probabilities and, therefore, the radiation and absorption properties of particles in plasmas. Investigations of the splits and shifts in atomic levels in electric and magnetic ﬁelds have conﬁrmed the basic statements follow from quantum theory. Later, the results of these investigations have been utilized in different physical applications, mainly for the ionization of atomic systems in strong electric ﬁelds and atoms and ions from highly excited Rydberg states. Among varied applications, one

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Figure 5.14. X-ray spectrum of the resonance w line of He-like Fe24+ ions and its satellites obtained in the PLT tokamak. The β-line is a sum of line intensities for Be-like c ions. From: Bitter M et al 1979 Phys. Rev. Lett. 43 129. 1979 APS.

can mention ionization by electric ﬁelds of atomic states selectively populated by laser radiation, the splitting and shifting in spectral lines in electromagnetic ﬁelds created by laboratory plasma sources37, the radiation of highly excited atoms in the interstellar medium38, the inﬂuence of strong electric and magnetic ﬁelds created on the surface of the neutron stars39 on atomic structure. In our considerations below we will use the symbol E for the electronic ﬁeld strength and the atomic unit E0 of the electric ﬁeld strength and B0 of the magnetic ﬂux density:

E0 =

e 5.142 × 1011 V m−1 a02

(5.128)

B0 ≈

m 2 e3 Ry = e3 ≈ 1.7 × 103 T µB c ~

(5.129)

where a0 denotes the Bohr radius, µB = e~/2m e the Bohr magneton and T tesla units. The values of E0 and B0 are quite high and are usually not reached in the laboratory.

37 Griem H 1974 Broadening of Spectral, Lines in Plasma (Berlin: Springer). 38 Seaton M J 1983 Rep. Progr. Phys. 46 167. 39 Garstang R H 1977 Rep. Phys. Progr. 40 105.

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Figure 5.15. Spectra of He-like Fe24+ ions and associated satellites recorded on the solar ﬂare at two different times. Te is the electron temperature. From Grineva Yu I et al 1973 c Sol. Phys. 29 441. 1973 Elsevier.

5.13.1 Polarizabilities In an external electric ﬁeld strength E and of angular frequency ω, the energy E a of an atom or an ion in the state a is given by40 : E a = E a0 −

1 1 βa (ω)E 2 − γa (ω)E 4 + O(E 6 ) 2! 4!

(5.130)

where E a0 < 0 denotes a ﬁeld-free atomic level and βa (ω) and γa (ω) the dynamic dipole and hyper polarizabilities, respectively. For the ground state, only coefﬁcients with even powers in E are non-vanishing. For degenerate states, there are also terms with odd powers of E in the expansion (5.130) (see section 5.13.3). The dipole polarizability βa (ω) has the same dimension as a03 and is 40 Bethe H A and Salpeter E E 1977 Quantum Mechanics of One- and Two-Electron Atoms 2nd edn (New York: Plenum).

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Figure 5.16. X-ray spectra of Ca18+ and Ca17+ ions in a laser-produced plasma recorded with a resolution of λ/λ = 3 × 10−4 . From Chichkov B N et al 1981 Phys. Lett. A 83 c 401. 1981 Elsevier.

Figure 5.17. X-ray spectra of Cr22+ and Ca21+ ions of a tokamak plasma (T-10, Kurchatov Institute) recorded with a resolution of λ/λ = 3×10−4 . From Bryzgunov V A c et al 1982 JETP 55 1095. 1982 MAIK (Nauka Moscow).

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given in second-order perturbation theory, the hyper polarizability γ (ω) has a dimension like a07 /e2 and is given in fourth-order perturbation theory; here a0 is the Bohr radius and e the electron charge. The dipole moment of the atomic system induced by the external electric ﬁeld is given by the series expansion: D(ω) = β(ω)E + 16 γ (ω)E 3 + · · · .

(5.131)

At zero frequency ω = 0, the corresponding values of β(0) and γ (0) are called the static polarizabilities. In this case, expression (5.130) reduces to the shift in a constant electric ﬁeld: E a = E a0 −

1 1 βa (0)E 2 − γ (0)E 4 + O(E 6 ). 2! 4!

(5.132)

For the ground-state hydrogen atom, at ω = 0 one has the expansion: E(1s) = − 12 − 94 E 2 −

3555 4 64 E

+ −4908E 6 − 794237E 8 − 1.945 · 108E 10 − · · · (5.133) where the energy is given in atomic units. One has to account for higher orders in expansion on E for ﬁelds E ≥ 0.06E0, where E0 is the atomic unit of electric ﬁeld strength. The dipole polarizability is an important quantity used for determining the induced dipole moment of an atomic system, the oscillator strengths of atoms and ions, the energy-level shifts, Van der Waals constants and other characteristics describing the interaction of an atomic system with an external electric monochromatic ﬁeld. The polarizability βa (ω) of an atom or an ion in state a is given in second-order perturbation theory by

1 1 βa (ω) = + |k|Dz |a|2 E k − E a − ~ω − i0 E k − E a + ~ω + i0 a) =k

(5.134) where Dzis the z-component of the dipole moment of the atomic system, N Dz = e i=1 z i , E k are the atomic energies and ω is the frequency of the external ﬁeld. The sum and integral mean summation over discrete and integration over continuum states, respectively. The real and imaginary parts of the dipole polarizability can be expressed as Re βa (ω) = 2

|k|Dz |a|2

k) =a

Im βa (ω) = π

Ek − Ea (E k − E a )2 − (~ω)2

|k|Dz |a|2 δ(E k − E a − ~ω).

(5.135) (5.136)

k) =a

The static dipole polarizability βa (0) of the state a (5.135) at the ﬁeld frequency ω = 0 is given by

|k|Dz |a|2 βa (0) = 2 . (5.137) Ek − Ea k) =a

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The real and imaginary parts of the dipole polarizability are related to many physical quantities such as the photoionization and elastic photon scattering cross sections, the Van der Waals constants and the dielectric susceptibility of matter. For example, for the photoionization cross section one has: 4πω Im βa (ω) c

σ ph (ω) =

~ω ≥ I

(5.138)

where I denotes the binding energy of the atom and c the speed of light. The nonrelativistic static dipole polarizability of the H-like ion with the nuclear charge Z in the n state reads: Z βn (0) = Z −4 [n 6 + 74 n 4 (2 + + 2)] a03

(5.139)

which for the 1s state yields Z β1s (0) =

9 3 a . 2Z 4 0

(5.140)

Inclusion of the leading relativistic correction to (5.140) gives Z β1s (0) =

9 [1 − 28(α Z )2 /27] a03 2Z 4

(5.141)

where α Z denotes the relativistic correction parameter, α the ﬁne-structure constant, α 1/137 and Z the nuclear charge. The static polarizabilities β(0) of the ground-state atoms and ions strongly differ from each other depending on the atomic structure. For example, the static dipole polarizabilities for neutral H, He and U atoms are equal to 4.5, 1.38 and 137 a03 , respectively41. For highly charged ions, the β(0) values strongly decrease with the ion charge. Thus, for Rbq+ ions, the experimental data are β(q = 0) = 320, β(q = 1) = 12, β(q = 27) = 0.00082 a03 , respectively. The dynamic dipole polarizability of H-like ions scales as: Z H βn (ω) = Z −4 βn (ω/Z 2 )

(5.142)

H (ω) is the polarizability of the hydrogen atom. β H (ω) is expressed in where βn n terms of the hypergeometric functions and in the frequency region above the ﬁrst excited state has the resonant behaviour: π H βn (ω) = −4.9107 cot √ − 4.3105 ω > 3/8 a.u. (5.143) 1 − 2ω

where a.u. refers to the atomic unit of frequency amounting to 2Ry/~. The real part of the dynamic dipole polarizability β(ω) for the hydrogen atom in its 1s 41 The values for static dipole polarizabilities of atoms and ions can be found in the book by Shevelko V P 1997 Atoms and Their Spectroscopic Properties (Berlin: Springer).

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Figure 5.18. Dynamic dipole polarizability of H(1s) (in a03 ) as a function of frequency in a.u. (1a.u. = 2Ry/~). At resonance frequencies of 0.375, 0.444, 0.469, . . . , the polarizability changes sign.

state is displayed in ﬁgure 5.18. According to equation (5.143), it has a cotangent form and at frequencies 1 1 ωn = − a.u. n = 2, 3, 4, . . . (5.144) 2 2n 2 H (ω) has i.e. at ω ≈ 0.375, 0.444, 0.469, . . ., the dynamic polarizability βn 3 H resonances and changes sign. At ω = 0, β1s (0) = 9/2 a0 .

5.13.2 Electric ﬁeld and Stark effect The Stark effect was discovered by Johannes Stark (1874–1957) in 1913 and refers to the splitting and shifts of atomic energy levels in an external electric ﬁeld. In a static homogeneous electric ﬁeld E , the interaction energy of an atom or ion with this ﬁeld is given by V = −E · D = −E · D z

(5.145)

where E denotes the electric ﬁeld strength, Dz the z-component of the atomic dipole moment D if the z-axis is directed along the ﬁeld E : Dz = e

N

i=1

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zi .

(5.146)

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In ﬁrst-order perturbation theory, the energy shift of a certain level a is given by the matrix element: E a(1) = E a|Dz |a (5.147) where |a denotes the ﬁeld-free wavefunction of the state a, i.e. with E = 0. For states a of the same parity, the matrix element (5.147) equals zero (see section 5.3), and the energy correction is given in second-order perturbation theory by

|a|Dz |k|2 (5.148) E a(2) = E 2 Ea − Ek k) =a

where E a and E k denote the ﬁeld-free atomic energies. Equation (5.148) describes the quadratic Stark effect. The sum in equation (5.148) is related to the static dipole polarizability βa of an atom in the state a (5.137), therefore equation (5.148) can also be written in the form E a(2) = −

βa 2 E . 2

(5.149)

Here, the external ﬁeld E is supposed to be quite weak so that the energy (2) correction E a is less than the difference between the unperturbed energies: (2) E a E a − E k . For the hydrogen atom in an excited state n ≥ 2, the matrix element in (5.147) is non-zero, and the energy shift in the electric ﬁeld is described by the linear Stark effect. This is a consequence of the -degeneracy of the energy levels. Let us denote the unperturbed states by the set of quantum numbers γ J M where J and M are the total angular momentum and its projection to the z-axis, respectively. Then the second-order correction can be written as E γ(2)J M = E 2

|γ J M|Dz |γ J M|2 γJ

Eγ J − Eγ J

Since the matrix elements are given by √  J 2 − M2 γ J M|Dz |γ J M ∼ M  (J + 1)2 − M 2

.

J = J − 1 J = J J = J + 1

(5.150)

(5.151)

one has, for the energy correction, E γ(2)J M = E 2 (A + B · M 2 )

(5.152)

where A and B are constants independent of the magnetic quantum numbers M. Therefore, in a homogeneous electric ﬁeld, a level with total angular-momentum quantum number J splits into components |M| = J, J − 1, . . . .

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(5.153)

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234

All levels, except the one with M = 0, are doubly degenerate with respect to the sign of the angular-momentum projection. The levels with J = 0 and J = 1/2 do not split but they do shift. Therefore, equation (5.152) reﬂects some asymmetry in the splitting. In principle, equation (5.150) describes the general features of quadratic Stark splitting and shifting but it is quite complicated to calculate. However, in some cases it is possiible to make rough estimations of the Stark values, e.g. for the ground state and for two close and strongly interacting levels. (1) Ground-state levels. In this case, one can approximately replace the ionization potential I with the energy difference E γ J − E γ J , (5.150) then has the form:

E γ J M ≈ E 2 I −1 |γ J M|Dz |γ J M |2 = E 2 I −1 γ J M|Dz2 |γ J M. γJ

(5.154) (2) Two close levels. If the energy difference E γ J − E γ J is much less than the others, then approximately one has only one term: E γ J M ≈ E

γ JM

≈

E2

(E γ J −E γ J ) |γ

E2

(E γ J −E γ J )

J M|Dz |γ J M|2

|γ J M|D

(5.155) z |γ

J M|2

= −E γ J M .

If two levels are optically bound, i.e. their angular-momentum quantum numbers satisfy the selection rules for optical transitions, the square of the matrix element |Dz |2 in (5.155) can be replaced with the corresponding dipole oscillator strength f (γ J − γ J ): 2 3e2 J 1 J (2 J + 1) f (γ J − γ J ) E γ J M = −E γ J M = E 2 −M 0 M 2m e ω2 (5.156) where ~ω is the energy difference between the levels in question and the two-row coefﬁcient on the right-hand side is the angular 3 j -symbol42. Equations (5.155) and (5.156) have been derived by assuming the corrections are small compared to the initial splitting E γ J − E γ J . In principle, the internal interactions like the central Coulomb potential, electron–electron and spin–orbit interactions should be added to the interaction (5.145). They also lead to the splitting of levels γ J and γ J . In the two-level approximation, the diagonalization of the matrices for interactions leads to the following expression for the energies E 1,2 of states 1 and 2 in an external ﬁeld43 : %1 + %2 %1 − %2 2 ± E 1,2 = + |Dz |2 E 2 (5.157) 2 2 42 Theory of angular momenta of atoms and ions is presented in Varshalovich D A, Moskalev A N and Khersonsky V K 1988 Quantum Theory of Angular Momentum (Singapore: World Scientiﬁc). 43 See, e.g., Sobelman I I 1992 Atomic Spectra and Radiative Transitions (Berlin: Springer) p 175. Friedrich H 1998 Theoretical Atomic Physics 2nd edn (Berlin: Springer) p 367.

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Figure 5.19. Transforming the quadratic Stark effect into a linear one.

where %1,2 denote the unperturbed energy levels and |Dz | = |1|Dz |2|. If the electric ﬁeld is absent, E = 0, one has unperturbed energy levels: E 1 = %1

E 2 = %2 .

(5.158)

If the initial energy interval is much higher than the perturbation by the electric ﬁeld, i.e. %1 − %2 2|Dz |E , one has the quadratic Stark effect: E 1 = %1 +

|Dz |2 E 2 %1 − %2

E 2 = %2 −

|Dz |2 E 2 . %1 − %2

(5.159)

In the opposite case, %1 − %2 2|Dz |E , one has the linear Stark effect: E1 =

%1 + %1 + |Dz |E 2

E2 =

%1 + %1 − |Dz |E . 2

(5.160)

Therefore in large electric ﬁelds, the quadratic Stark effect transforms into a linear effect as reproduced in ﬁgure 5.19. This transformation usually takes place in quite a dense plasma with strong electric ﬁelds. If the electric ﬁeld is not homogeneous, it is necessary to add to the dipole interaction (5.145) higher-order multipole interactions such as the quadrupole one. For example, a quadrupole splitting occurs for levels with J )= 0 and J )= 1/2 and depends linearly on ﬁeld E . In the case of highly charged non-hydrogenic ions, calculating the Stark splitting for ﬁne-structure components also means including the electron–electron and spin–orbital interactions which makes the problem much more complicated44. 44 See Klimchitskaya G L and Labzowsky L N 1976 Sov. Phys.–JETP 43 278.

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5.13.3 Linear Stark effect in the hydrogen atom In the hydrogen atom, the energy levels undergo a splitting proportional to E , i.e. a linear Stark effect takes place because of the degeneracy of the levels with respect to the orbital quantum number : the -levels with the same principal quantum number n ≥ 2 have the same energy but the states with different parity (−1) are mixed. The matrix elements between the states nm and n m are non-zero nm|Dz |n m )= 0

(5.161)

if the magnetic quantum numbers are equal: m = m . For example, for the n = 2 state with four sublevels = 0, m = 0, 1 = 1, m 1 = 0, ±1, only one matrix element 2s0|Dz |2p0 is non-zero and this equals 2s0|z|2p0 = −3ea0

(5.162)

where e denotes the electron charge and a0 the Bohr radius and the attached ‘0’ denotes the m = 0 state. The ﬁrst-order correction energy E (1) is obtained by diagonalizing the matrix for the operator V = −E Dz for eigenstates with degenerate energies. In the two-level approximation, the linear Stark effect for hydrogen can be obtained from the general equation (5.157) and if %1 ≈ %2 = %, one has E 1 = % + |Dz |E E 2 = % − |Dz |E (5.163) where % denote the ﬁeld-free energy. For the level with n = 2, m = 0, one has, from equation (5.162), E 1,2 = % ± |2s0|z|2p0|E = % ± 3ea0E .

(5.164)

For other levels with m = ±1, n = 2, the dipole moment is zero, and, therefore, the level n = 2 splits into three components according to the solution of the corresponding secular equation. Similar considerations for the n = 3 levels lead to a splitting into ﬁve components which are symmetrical and linear on the ﬁled E . The 1s ground-state level is shifted according to the quadratic Stark effect (cf equation (5.132)) yielding, in atomic units: E 1s ≈ −0.5 −

β1s 2 9 E = −0.5 − E 2 2 4

E E 1s

(5.165)

where β1s = 9/2 a03 is the dipole polarizability of the hydrogen atom in the ground state. The splitting of the n = 2 and 3 levels is schematically displayed in ﬁgure 5.20 which also shows the possible radiative transitions from the split levels. As can be seen, the Hα line splits into 16 components (8π and 8σ

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Figure 5.20. Level splitting for n = 2 and 3 in an electric ﬁeld. Possible radiation transitions are shown by arrows.

components). The selection rules for magnetic quantum numbers m remain the same as in the case without an external electric ﬁeld (section 5.3), i.e. m = 0 m = ±1

π-components σ -components.

(5.166)

To perform the energy-level calculations for higher n states in the static electric ﬁeld E is quite complicated because the secular higher-order equations have to be solved. Usually, for this purpose, parabolic coordinates are used with ‘parabolic’ quantum numbers n 1 , n 2 and the magnetic quantum number m. The (usual) principal quantum number n is related to these by n = n 1 + n 2 + |m| + 1.

(5.167)

For a given n, the number |m| can take n different values: |m| = 0, 1, 2, . . . , (n − 1). For each |m|, the number n 1 takes the values n 1 = 0, 1, . . . , n − |m| − 1. In parabolic coordinates, the Schr¨odinger equation for hydrogen in a weak static electric ﬁeld E → 0, neglecting electron and nuclear spins, 1 2 1 − ∇ − + E · Dz (r) = E(r) 2 r

(5.168)

has been solved by applying a perturbation-theory expansion on E up to the 25th

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238

(!) order45. In atomic units the leading terms are given by E 0 = − 1/2n 2 E1 = E2 = E3 =

3 2 n(n 1 − n 2 )E 1 4 − 16 n [17n 2 − 3(n 1 − n 2 )2 − 9m 2 + 19]E 2 3 7 2 2 2 3 32 n (n 1 − n 2 )[23n − (n 1 − n 2 ) + 11m + 39]E .

(5.169) (5.170) (5.171) (5.172)

The ﬁrst-order correction (5.170) shows the linear part of the Stark effect and reveals that, for a given |m| value, the Stark levels are separated by (cf equation (5.164)) E 1 = E 1 (n, n 1 , |m|) − E 1 (n, n 1 − 1, |m|) = 3n E .

(5.173)

Because for a given n, the difference n 1 − n 2 can take the values n − 1, n − 2, n − 3, . . . , −(n − 1), the n level splits into 2(n − 1) + 1 = 2n − 1 components which agrees with the results for n = 2 and 3 considered before. According to (5.171), the quadratic effect depends on the absolute values of m and at large values of E removes the degeneracy with respect to |m|. The deviation from the linear dependence on E for hydrogen begins for ﬁelds of the order of E ∼ 0.1e/n 4a02 ≈ 3 × 1010/n 4 V m−1 . (5.174) Highly excited states with large principal quantum numbers n 1 are very sensitive to the external electric ﬁeld because their ionization potential I decreases as 1/n 2 and, for example, for hydrogen in the n = 100 state it is only I ≈ 1 meV. Therefore if the electric ﬁeld strength E is higher than a certain critical ﬁeld Ec deﬁned by46 Ec ≈ 6.43 × 1010 Z 3 /n 4 V m−1 (5.175) the electron can be ejected from the atom due to ﬁeld ionization. Similar features of the Stark effect, considered here for the hydrogen atom, can be found for highly excited Rydberg levels in other hydrogenic atoms, for example in alkali-metal atoms Li, Na, K, Rb and Cs, which have one valence electron outside the closed atomic core. A relatively weak electric ﬁeld allows one to study the properties of such Rydberg atoms, in particular the quadratic Stark effect, mixing of the n and states, and the interaction with other atoms47 . 5.13.4 Stark effect in H-like ions In calculating the Stark effect in highly charged ions, all relativistic and QED corrections, including the Lamb shift, should be taken into account. Two 45 Silverstone H J 1978 Phys. Rev. A 18 1853. 46 Kanter E P et al 1984 Phys. Rev. A 29 583. 47 See Gallagher T F 1988 Rep. Progr. Phys. 51 143. Zimmerman M L et al 1979 Phys. Rev. A 20 2251.

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main cases can be distinguished: -degenerate states with j < n − 1/2 (like 2s1/2 , 2p1/2 ), and non-degenerate states with j = n − 1/2 (like 2p3/2 ). For degenerate states, the linear Stark effect (5.147) takes place and since the dipole moment Dz ∝ ea0n/Z , the ﬁeld splitting scales as E (1) ∝ Dz E ∝

nE Z

(5.176)

where n denotes the principal quantum number, E the electric ﬁeld strength and Z the nuclear charge. For non-degenerate states, the level splitting is described by the quadratic Stark effect (5.148). The dipole polarizabilty for highly charged ions scales like βz ∝ n 6 /Z 4 , therefore the second-order correction is of the order of E (2) ≈ βz E 2 ≈

n6E 2 . Z4

(5.177)

(1) Degenerate states with j < n − 1/2. In a weak ﬁeld, when the ﬁeld splitting E F is smaller than the ﬁne-structure interval but still larger than the Lamb shift, i.e. (5.178) E L < E F < E FS there is a linear Stark effect resulting from level mixing in the states with = j ± 1/2. Taking into account the leading relativistic corrections in expansion of ∝Z leads to the following shift of the nl j -level48. In atomic units the latter reads (2n + j + 1/2)(α Z )2 P E nl j = E 1 − (5.179) (2 j + 1)(n + j + 1/2)n where m is the projection of momentum j and E P is the level shift in the Pauli approximation49. In atomic units the latter reads

3 2 nm E . (5.180) E P = ± n − ( j + 1/2)2 4 j ( j + 1) Z In the case of intermediate electric ﬁelds, E F ≈ E FS

(5.181)

the energy shift can be obtained numerically by diagonalization of the energy matrix. However, for |m| = n − 3/2, it is possible to obtain an expression for the ﬁeld shift analytically. With the main relativistic correction of the order of (α Z )2 , one has 3 2 3n 2 3n − 10n + 3n + 1 1 + (α Z ) E (5.182) E = n F 2Z 3n 2 (n − 1)(2n − 1) 48 Zapryagaev A S 1978 Opt. Spectrosc. 44 527. 49 Bethe H A and Salpeter E E 1977 Quantum Mechanics of One- and Two-Electron Atoms 2nd edn

(New York: Plenum).

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240 where

nF =

0 ±1

j = n − 1/2 j = n − 3/2 and = j ∓ 1/2.

(5.183)

In the case of strong electric ﬁelds, E F E FS

(5.184)

the non-relativistic value for E n reads: E n =

n 4 3n (n 1 − n 2 )E − [17n 2 − 3(n 1 − n 2 )2 − 9m 2 + 19]E 2 (5.185) 2Z 2Z

where n 1 and n 2 are the parabolic quantum numbers (5.167). In general, the j splitting in H-like ions can be large and the situation may be quite complicated because some components undergo the quadratic Stark effect (e.g. 2p3/2 ) and some (2s1/2 , 2p1/2 ) the linear one. (2) Non-degenerate states with j = n − 1/2. The energy-level shift for a H-like ion in an excited state with j = n − 1/2 is determined by the quadratic law: E nm = − βnm E 2 /2 j ( j + 1) − 3m 2 βnm = βnS − βnT j (2 j − 1)

(5.186) (5.187)

S(T)

where βnm is called the differential polarizability and βn is called the scalar (tensor) polarizability50. For the special case of |m| = j and α Z 1, the energy shift is given by51 : 3 2 nr 2 80n + 150n + 87n + 9 1 − (α Z ) (5.188) E n,|m|= j = E 2n(n + 1)(4n + 5)(2n + 1)2 where the non-relativistic correction E nr is n 4 2(n + 1)(4n + 5)E 2 . E nr = 2Z

(5.189)

5.13.5 Magnetic ﬁeld and Zeeman effect The Zeeman effect by which the atomic levels are split in a magnetic ﬁeld into separate components M of the total angular momentum J was ﬁrst observed by Pieter Zeeman (1865–1943) in 1896. The Zeeman effect and its peculiarities, related to the spin–orbit interaction, are well described in two 50 See Angel J R P and Sanders P H 1968 Proc. R. Soc. London A 305 125. 51 Zapryagaev A S and Manakov N L 1976 Nucl. Phys. 23 482.

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books52. Information about the Zeeman splitting of atomic levels and the corresponding spectral lines is very important for classifying radiation spectra, the angular-momentum coupling scheme of energy levels and g-factor values. In contrast to an electric ﬁeld, a magnetic ﬁeld completely removes the degeneracy of levels with respect to the projections M of the total angular momentum J . In the case of a weak magnetic ﬁeld B, the interaction potential of an atom with the ﬁeld has the form V = −µB

(5.190)

where µ is the total atomic magnetic moment made up by two parts, one electronic, one nuclear. Since the nuclear magnetic moment is about three orders of magnitude smaller than the electronic one, the magnetic moment of an atom in the γ J state can be written as (cf equation (5.83)) µ = −µB g J J /~

(5.191)

where J denotes the total electron angular momentum and g J is the gyromagnetic ratio or the total electron g-factor. If the z-axis is directed along the magneticinduction vector B, one has for the energy splitting, according to (5.190) and (5.191), E (1) = |V | = µB Bg J M. (5.192) Therefore, a magnetic ﬁeld causes to split the J level into 2 J + 1 components, M = 0, ±1, ±2, . . . ± J , i.e. the splitting is symmetrical. The total electron g-factor is of the order of 1 and can be calculated in the vector L S coupling (section 5.2). The operator of the electron magnetic moment has the form: µ = −µB (g + gs s)~ (5.193) where g = 1 is the g-factor for orbital momenta and gs = 2 for spin momenta, respectively, and therefore one has #

$ g J J = g (5.194) i + g s si = (L + 2S). i

i

Taking into account the general angular-momentum properties, L + 2S = J + S S =

S · J J= J (J + 1)

J ( J +1)−L(L+1)+S(S+1) J 2 J ( J +1)

(5.195)

52 Bethe H A and Salpeter E E 1977 Quantum Mechanics of One- and Two-Electron Atoms 2nd edn (New York: Plenum). Sobelman I I 1969 Introduction to the Theory of Atomic Spectra (New York: Pergamon).

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one has from (5.194), gJ = 1 +

J (J + 1) − L(L + 1) + S(S + 1) 2 J (J + 1)

(5.196)

where L and S denote the total orbital and spin angular-momentum quantum number of an atom. The quantity g J is called the Land´e factor. For some speciﬁc cases one has: g = 1 for S = 0, g = 2 for L = 0 and g = 3/2 for L = S. The Land´e factor can also equal zero, e.g. for the 4 D1/2 or 5 F1 levels. Therefore, these levels will not split in ﬁrst-order perturbation theory. From (5.192) it follows that the levels with J = 0 also split in second-order perturbation theory according to E (2) =

|γ J M|µB|γ J M|2 γJ

Eγ J − Eγ J

(5.197)

which are non-zero for J = M = 0. For the spectral lines in a magnetic ﬁeld, the σ -components (transition M = ±1) are observed in the direction of the z-axis and the σ - and πcomponents (transitions M = 0) in the direction perpendicular to the z-axis. These lines, can correspondingly be written, using equation (5.192), as

~ωπ = ~ω0 + µB B(g − g )M ~ωσ = ~ω0 + µB B[g M − g (M ± 1)].

(5.198) (5.199)

If the g-factors are equal, i.e. g = g , one has ωπ = ωσ

~ωσ = ~ω0 ± µB B.

(5.200)

This splitting occurs for instance when S = 0 (g = g = 1) and it is called the normal Zeeman effect. Splitting (5.199) is called the anomalous Zeeman effect because the splitting (5.200) followed from classical electron theory, while (5.199) was not explained by theory until electron spin was discovered. In a strong magnetic ﬁeld, the spin–orbit interaction A L S is considered to be small compared to the ﬁeld splitting and the mean values for the matrix element should be found for the given angular momenta L and S because in this case they are conserved separately. Then we obtain E = µB + A L S = µB B(M L + 2M S ) + AM L M S

(5.201)

where M L and M S denote the z-components of L and S, respectively, and A a certain constant. For radiative transitions between split components, the following selection rules hold: M S = 0 M L = 0, ±1 (5.202)

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400

p-electron

ML+2M S = 2

Energy E M [cm -1]

M = 3/2 1

200 1/2 2

P3/2

-1/2

0 P1/2

1/2 -1/2

-200

-3/2

2

0 0

-1

-2

-400

0

100

200

300

400

Magnetic induction B [Tesla] Figure 5.21. Splitting of the 2 P3/2 and 2 P1/2 components with a ﬁne-structure splitting E FS 100 cm−1 in magnetic ﬁeld B.

with the corresponding transition energies

~ωπ = ~ω0 + (A − A )M S M L ~ωσ = ~ω0 ± µB B + AM S M L − A M S (M L ± 1).

(5.203) (5.204)

From equations (5.203) and (5.204) it can be seen that in a strong magnetic ﬁeld the line γ L S–γ L S splits in a strong magnetic ﬁeld into a larger number of components compared to the normal Zeeman effect (5.200). This splitting in a strong magnetic ﬁeld is called the Paschen–Back effect. For a single electron outside closed subshells, the Breit–Rabi formula describing the magnetic-ﬁeld dependence of the ﬁne-structure doublet can be applied for the Zeeman effect, the Paschen–Back effect and the intermediate-ﬁeld region53. The energy-level dependence on the magnetic ﬂux density B has the form % E FS E FS 4M x + µB B M ± + x2 1+ E M (B) = − 2(2 + 1) 2 2 + 1 (5.205) µB B x = (gs − 1) E F S where E FS denotes the ﬁne-structure splitting between the upper and lower states in the ﬁeld-free case B = 0. M is the projection of the total angular 53 Bethe H A and Salpeter E E 1977 Quantum Mechanics of One- and Two-Electron Atoms 2nd edn (New York: Plenum) p 212.

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Figure 5.22. Energies of the hydrogen atom as a function of external magnetic ﬂux density B for excited states with n = 10–16. From Zimmerman M L, Kash M M and Kleppner D c 1980 Phys. Rev. Lett. 45 1092. 1980 APS.

momentum J , gs = 2 is the electron g-factor and the electron orbital quantum number. The plus sign corresponds to the sublevels belonging to the upper state and the minus sign to those of the lower state. A smooth transition from Zeeman splitting into the Paschen–Back effect, as calculated by equation (5.205), is shown in ﬁgure 5.21 for a 2 P term with a ﬁnestructure splitting of E FS 100 cm−1 . According to equation (5.205), in the Zeeman (B 100 T) and Paschen–Back (B 100 T) regions, the energies E M show a linear dependence on B while in the intermediate region of B, the dependence is nonlinear in character. The energy levels of a hydrogen atom in a magnetic ﬁeld have been calculated in a series of papers54. Calculated energies of highly excited hydrogen with principal quantum numbers n = 10–16 are presented in ﬁgure 5.22 showing many level crossings. The calculations performed by diagonalization of a zeroﬁeld spherical basis, agree well with experimental data for low magnetic ﬁelds55 .

54 See, e.g. Zimmerman M L et al 1980 Phys. Rev. Lett. 45 1092. 55 Zimmerman M L et al 1978 Phys. Rev. Lett. 40 1083.

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5.13.6 Zeeman effect in H-like ions In a weak external magnetic ﬁeld B in which the Zeeman splitting is much less than the ﬁne-structure interval: E nm E FS

(5.206)

the linear Zeeman effect for H-like ions reads56, in atomic units, 2n 1 − (α Z )2 /n 2 + 1 (1) m = ±( + 1/2) E nm = α Bmn 4n 2 − 1

(5.207)

where n and denote the principal and orbital quantum number, respectively, Z the nuclear charge and α the ﬁne-structure constant. If α Z 1, one obtains (1) E nm = α Bmn

2n(1 − (α Z )2 /2n 2 ) + 1 4n 2 − 1

α Z 1.

For H-like ions in the ground state with m = ±1/2, one has α Bm (1) 2 1 − (α Z )2 + 1 ≈ α Bm(1 − (α Z )2 /3). E 1,±1/2 = 3

(5.208)

(5.209)

On increasing the ﬂux density B, the quadratic Zeeman effect takes place and is given by57 (2) E 1s = −χ1s B 2 /2 (5.210) where χ1s is the dipole magnetic susceptibility. With leading relativistic corrections of order (α Z )2 , the quadratic effect (5.210) can be written in two different forms for two cases: |m| < j , i.e. levels with angular momenta j = ± 1/2 are mixed; and |m| = j when the levels are not mixed. In the ﬁrst case, one has: (α Z )2 n 4 a03 n(4n + 1) α Z 2 1+ χn j = 4 n 4Z 4 & 2 4 3 16n + 44n + 32n 2 + 7n α Z 2 m . (5.211) − 2 1− j 4(2n + 1)2 n In the second case, i.e. for states with |m| = j and j = n − 1/2, one obtains: (α Z )2 n 4 a03 n + 1 5n + 3 α Z 2 χn,|m|= j = . (5.212) − 4Z 4 n 2n + 1 n Therefore, the linear Zeeman shift E (1) for H-like ions is nearly independent of the nuclear charge Z and principal quantum number n and is of the order of E (1) ∼ α B (5.213) 56 Manakov N L and Zapryagaev S A 1976 Phys. Lett. A 58 23. 57 See Ovsyannikov V D and Khalyov K V 1999 JETP 89 444.

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where α denotes the ﬁne-structure constant. The quadratic Zeeman shift E (2) strongly scales with Z and n: E (2) ∼ α 2 B 2 n 4 /Z 2 .

(5.214)

The ratio of these two correction terms is given by E (2) /E (1) ∼ α Bn 4 /Z 2

(5.215)

i.e. for low-Z ions, high magnetic ﬁelds and large principal quantum numbers, the quadratic term E (2) might be of the same order as the linear term E (1) .

5.14 Quantum theory of line shape The x-ray spectra of highly charged ions contain plenty of additional spectral lines, compared to neutral atoms; and these should be resolved experimentally and identiﬁed theoretically with a very high accuracy to obtain information about the atomic characteristics and plasma parameters. The line proﬁles can be strongly broadened by the Doppler effect, external electric and magnetic ﬁelds (Stark and Zeeman effects), collisions with electrons and heavy particles. The inﬂuence of some of these effects on spectral line broadening is considered in this section. 5.14.1 Natural broadening of spectral lines The concept of natural broadening of spectral lines exists both in classical and quantum-mechanical theories of radiation. According to the classical treatment, if a system radiates photons, it loses energy and therefore its free oscillations would necessarily be damped. Since a damped oscillation is not monochromatic, it contains a whole set of frequencies that leads to spectral line broadening. The general theory of spectral line radiation is described in the books58. In Weisskopf’s59 classical theory, the shape of a spectral line is determined by three main factors: the radiation damping of the emitting atom, its heat motion and the effects of neighboring atoms. The line width due to the ﬁrst factor is proportional to the square of the frequency, ω2 , while for the other two it is proportional to the ﬁrst or lower powers of ω; therefore, the effect of radiation damping predominates in the case of x-rays. Classical theory treats the radiating atom as a damped harmonic oscillator radiating at an angular frequency ω0 with the displacement given by L(t) = L 0 e−t /2 cos(ω0 t) 58 Heitler W 1954 Quantum Theory of Radiation 3rd edn (Oxford: Clarendon). Sobelman I I 1969 Introduction to the Theory of Atomic Spectra (New York: Pergamon). Griem H R 1974 Spectral Line Broadening by Plasmas (New York: Academic). 59 Weisskopf V and Wigner E 1930 Z. Phys. 63 54. Weisskopf V 1933 Phys. Zeits. 34 1.

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(5.216)

Quantum theory of line shape

247

where L 0 denotes the initial amplitude of the oscillator. The energy decrease due to radiative loss and the ﬂux radiated by the oscillator are given by an exponential decay E = E(0)e−t E = E (0)e−t . (5.217) The quantity is called the radiation damping constant and in classical theory is given by 2e2 ω02 (5.218) = 3m e c3 where e and m e denote the electron charge and mass, respectively. Now we turn to the quantum-mechanical treatment of the spectral line shape. In the case of the two-level atom, considered in sections 2.3 and 6.5, the population n 2 (t) of the excited level 2 decreases exponentially as n 2 (t) = n 2 (0) e−A21 t = n 2 (0) e−t /τ

τ = 1/A21

(5.219)

if there is no external radiation ﬁeld, i.e. the Einstein coefﬁcients B12 = B21 = 0. Here A21 is the transition probability per time deﬁned in equation (5.41), and τ is a mean lifetime of the level 2. The ﬂux E radiated by the atom is proportional to the density n 2 of the excited state and, therefore, decays exponentially as it is given in equation (5.217) with the decay constant followed from equation (5.219) = A21 = 1/τ. The Fourier transformation of the radiation ﬂux in equation (5.217) in the time domain gives the expression for the frequency-dependent spectral line intensity, also called a line proﬁle, in the form: I (ω) =

1 . 2 2π (ω − ω0 ) + (/2)2

(5.220)

The function (5.220) is referred to as Breit–Wigner curve or Lorentz function. The integrated area of the function (5.220) is unity: +∞ I (ω) dω = 1 (5.221) −∞

and its full width at half maximum is which is called a line width (see ﬁgure 5.23). It is possible to estimate the ratio of the natural line width to its undamped frequency. In fact, 4π e2 1 1 ≈ 1.2 × 10−12 = ω0 3 m e c2 λ λ

(5.222)

where λ is the wavelength in cm. With increasing λ (the infrared region), the ratio /ω0 decreases and with decreasing λ (the short wavelength region)

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Atomic structure

Figure 5.23. Contour of a spectral line.

the ratio /ω0 increases. For example, in the visible spectral region where λ ≈ 5 × 10−5 cm, /ω0 ≈ 2 × 10−8 and for the x-ray region of λ ≈ 1 × 10−8 cm /ω0 ≈ 1 × 10−4 . The classical radiation width (5.218) gives the right order of magnitude for the lifetime (5.219) of excited states but only for strong resonance lines optically connected with the ground state two-level scheme but it does not describe the whole range of lifetime values. A quantum-mechanical treatment leads to a lineshape formula similar to (5.220) but with another meaning for . In quantum mechanics, the probability W (ω) per time of photon emission in the frequency interval ω, ω + dω due to electron transition from state b to state a is also given by the dispersion formula: dω W (ω) dω = Aba 2π (ω − ω0 )2 + (/2)2 +∞ W (ω) dω ω0 = (E b − E a )/~ Aba =

−∞ +∞

−∞

(5.223) (5.224)

dω =1 2π (ω − ω0 )2 + (/2)2

where Aba is the total radiative probability per time for a transition from level b to level a (section 5.4). Here, the total line width is given by the sum of all radiation widths of the initial and ﬁnal levels:

a = Aac b = Abe . (5.225) = a + b c≤a

e≤b

Therefore, the radiation width of any level is the sum of the radiation transition probabilities from this level to all low-lying levels or levels of the same energy, as in autoionization processes. The lifetime of the state b is again given by the

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quantity τb = b−1 which, for the ground state, means = 0 and τ = ∞. For transitions to the ground state, the radiation width is deﬁned by the widths of the upper levels. For the resonance line, i.e. for the nearest optically allowed transition to the ground state, the quantum-mechanical expression for the line width, as it follows from equation (5.225) and (5.41), can be written as res = Aba =

2e2ω02 f ab m e c3

(5.226)

where f ab is the optical absorption oscillator strength of the transition. Expression (5.226) differs from the classical formula (5.218) by the factor 3 f ab . For a H-like Z -charged ion, ω02 ∝ Z 4 and the line width scales as H res = Z 4 res

(5.227)

because the oscillator strength f ab has a weak dependence on Z ; here H refers to the width in the hydrogen atom. 5.14.2 Doppler broadening The width of a spectral line is usually much larger than the natural width and the line shape is much more complicated than the dispersion contour because of the inﬂuence of many factors such as the Doppler effect and the interactions between the radiating atom and the surrounding atomic particles—electrons, ions, atoms and molecules60. Let us consider the line broadening caused only by the Doppler effect neglecting other types of broadening. Doppler broadening arises as a result of the thermal motion of the emitting atoms or ions in a plasma. According to the Doppler effect (2.34), the observed angular frequency of light ω radiated by an ion moving with velocity v, relative to the observer, differs from the frequency ω0 observed when the ion is at rest, by ω = ω0 ±

v c

v c

(5.228)

where c is the speed of light, the signs ± correspond to the cases when the ion moves towards or away from the observer, respectively. If the radiating ions follow a Maxwell velocity distribution function, then the number N(v) of the ions with a velocity between v and v ± dv is given by 1 2kB Ti v 2 dv N(v) dv = √ exp − v0 = (5.229) v0 v0 Mi π 60 Sobelman I I 1969 Introduction to the Theory of Atomic Spectra (New York: Pergamon). Griem H R 1997 Principles of Plasma Spectroscopy (Cambridge: Cambridge University Press).

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where kB = 0.8617 × 10−4 eV K−1 denotes the Boltzmann’s constant and Ti and Mi the ion temperature and mass, respectively. The intensity distribution function for radiation between an angular frequency ω and ω + dω is given by ω − ω0 c I (ω) dω = N(v)dv = N c dω. ω0 ω0

(5.230)

Substituting (5.229) into (5.230) gives the normalized Doppler shape for the spectral line: 1 ω − ω0 2 I (ω) = √ exp − ωD πωD

ωD = ω0

v0 c

(5.231)

where the quantity ωD is called the Doppler width and is given by ωD = ω0

2ω0 v0 = c c

2 ln 2

kB Ti Mi

1/2 .

(5.232)

If the ion temperature Ti is in eV units and the ion mass in atomic units then the Doppler broadening can be estimated by the formula ωD 7.7 × 10−5 ω0

Ti /Mi .

(5.233)

For example, the Doppler broadening of the M1-line 2p 2P3/2 − 2p 2 P1/2 in the Ar13+ ion with Mi = 40 a.u. and a wavelength λ = 441 nm in a plasma with kB Ti = 300 eV about ωD ≈ 0.1 nm. The intensity distribution (5.231) is a Gauss–Cauchy function centered at a frequency ω0 . Its full width at half maximum is √ δ = 2 ln 2 ωD .

(5.234)

According to (5.232), the Doppler width is proportional to the frequency ω0 and to the square root of the ion temperature Ti . In highly charged ions, the ion temperature is proportional to Z 2 , therefore the relative Doppler width scales like: ωD ∝ Z. ω0

(5.235)

The general formula for a Doppler proﬁle (5.231) is valid for an arbitrary velocity distribution (not necessarily a Maxwellian one) provided the radiation wavelength λ is much smaller than the particle free path length L, λ 2π L.

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(5.236)

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251

5.15 Absorption edges The position of the maximum of a spectral line yields information about the transition energy, i.e. on the difference in the binding energies of the two electronic states between which the transition has occurred. However, in many cases, the binding energy of one speciﬁc state is of interest and then the xray absorption or photoionization of matter is used to measure the dependence of the absorption coefﬁcient on photon x-ray energy. If the photon energy is high enough, the photoabsorption spectrum comprises the characteristic ‘edges’ corresponding to the binding energies (K, L, M edges, etc) at which the absorption coefﬁcient ‘jumps’ to higher values (ﬁgure 5.24). The positions of these jumps give information on the binding energies of the inner electronic subshells in atoms, ions, molecules and solid state. When a light beam passes through a material, its intensity I0 decreases (is attenuated) because the photons are either absorbed or scattered. The dependence of the attenuated intensity I is described by Lambert’s law61 : I (x) = I0 e−µl x

(5.237)

where x is the path length and µl is the linear attenuation coefﬁcient with dimension cm−1 . Experiments show that the linear attenuation coefﬁcient deﬁned in the form (5.237) depends on the state of the material, i.e. whether it is a gas, liquid or solid. Therefore, another quantity is introduced—the so-called mass attenuation coefﬁcient µm through the Bouguer–Lambert–Beer exponential law: I (x) = I0 e−µm ρx

(5.238)

where ρ is the material density and µm = µl /ρ and now has a dimension of cm2 g−1 . The atomic attenuation coefﬁcient µa is often used to describe the attenuation per atom of the material. Then we have I (x) = I0 e−µa n

(5.239)

where n is the number of atoms per cm2 of a material layer. The quantity µa is in cm2 and is associated with the atomic photoabsorption (photoionization) cross section; it is equal to µl A (5.240) = µa = µm NA n where A is the atomic (mass) number and NA is the Avogadro constant: NA 6.022136 × 1023 atom mol−1 . The atomic attenuation coefﬁcient is actually the sum of two components: (5.241) µa = τa + σa 61 See Agarwal B K 1991 X-Ray Spectroscopy (Berlin: Springer).

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Figure 5.24. K and L edges in the attenuation coefﬁcients of Pt as a function of a photon wavelength in the x-ray region. The coefﬁcient µa is in cm−1 . From Agarwal B K 1991 c X-Ray Spectroscopy (Berlin: Springer) p 122. 1991 Springer.

where τa denotes the true atomic absorption coefﬁcient and σa the atomic scattering coefﬁcient; usually, τa σa . Figure 5.24 represents a typical example of the atomic and mass attenuation coefﬁcients as a function of the photon wavelength for platinum in the x-ray region. There are some jumps in the dependence in the vicinities of the binding energies of the K and L shells caused by the photoionization of the inner-shell electrons when a sudden change in the transmitted intensity is observed. These jumps are called absorption discontinuities or absorption edges. The positions of these jumps correspond to the energies required to eject K, L, M, etc, electrons from the atom. For other elements, the atomic attenuation coefﬁcients have similar dependencies but with edges lying at different characteristic wavelengths corresponding to the set of inner-shell binding energies. After each edge, the absorption coefﬁcient is proportional to Z 4 λ3 where Z is the nuclear charge of the atom and λ is the photon wavelength. Now we consider the quantum theory of the shape of the absorption edge. When an electron is removed from the inner atomic shell, the atom has a vacancy (a ‘hole’) in its inner shell and therefore it becomes excited because this vacancy can be ﬁlled by other electrons from upper shells, mainly via dipole or quadrupole transitions. From the quantum theory of the line shape considered in the previous section, it is known that every excited state has a ﬁnite (nonzero) level width. The

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Table 5.20. Experimental K - and L-level widths (in eV) for neutral atoms. Atom

K

LII

LIII

Fe Co Ni Cu Zn Ga

1.8 1.5 1.9 1.3 1.5 1.9

1.7 2.1 1.7 2.4 2.0 1.4

1.2 1.3 0.7 1.4 1.4 1.2

effective lifetime of such a state i with a hole is given by τi =

A−1 i

=

−1 Aki

(5.242)

k≥i

where Aki is the radiative transition probability from the upper level k to level i . From the quantum theory of line shape, the quantity i = 1/τi

(5.243)

characterizing the width of the i th level equals the FWHM value of the quantummechanical probability. Therefore, the term width of an atomic shell (K, L, M, . . .) is introduced which means the width of the atomic state with a hole in the corresponding shell (K, L, M, . . .). The closer the i th shell is to the atomic nucleus, the larger the number of possible transitions k–i is, and therefore the larger the i value is. In this respect, it should be K > L > M > · · · .

(5.244)

Experimental widths of the energy levels for some atoms are given in table 5.20; as can be seen, they are of the order of a few eV. Because the atomic state with a hole in the inner shell has a ﬁnite level width, the shape of the absorption edge is not sharp and abrupt but has an extended structure described by an arctangent curve when the electron is removed from the inner bound state i to the continuum state f due to the photoionization processes. The shape of the inner shell level i is given by (5.223) W (ω) dω = W (E) dE =

i dE i 2π (E i − E i0 )2 + (i /2)2

(5.245)

where E i0 is the energy corresponding to the distribution maximum and i is the width of the i th level. Let E f 0 denote the beginning of the positive-energy region

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Figure 5.25. Quantum-theory explanation of the absorption-edge shape near the ionization limit: (a) left, absorption of a photon with energy E to a level in continuum; right, the partial absorption coefﬁcient for the f level; (b) Total absorption coefﬁcient in the entire vicinity of the ionization limit E f 0 . From Agarwal B K 1991 X-Ray Spectroscopy (Berlin: Springer) p 143.

in the continuum consisting of a set of narrow levels f with widths much smaller than i : f i . If the photon energy is larger than the threshold energy E > E th where E th = E f 0 − E i (5.246) then the electron will make a transition from the inner level E i to one of the levels E f (ﬁgure 5.25). The partial absorption coefﬁcient τ f (E) for the f level is given by τ f (E) = C W (E i ) = C(E − E f ) = C

i /2π (E − E i0 f )2 + (i2 /2)2

(5.247)

where C is a certain constant and E i0 f = E f − E i0 . Each of the f levels will contribute equally to the total absorption, i.e. with the same width i and constant

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Figure 5.26. Experimental Pb K-absorption edge in the vicinity of the ionization threshold. c From Bearden A J 1960 Phys. Rev. Lett. 4 240. 1960 APS.

C, as shown in ﬁgure 5.25. The total absorption coefﬁcient τ (E) is deﬁned by integration of (5.247) over all possible E i0 f values:

∞

dE i0 f + (i /2)2 2 E i0 f 0 (E − E i0 f ) E − E i0 f ∞ C arctan = − π i /2 E i0 f =E i0 f 0 E − E i0 f 0 1 1 + arctan =C 2 π i /2

τ (E) =

C1 2π

(5.248)

where E i0 f 0 = E f0 − E i0 62 . The constant C is found from the condition C = τ (E = ∞). The arctangent shape of the absorption discontinuity is shown in ﬁgure 5.25(b). From the shape of the absorption edge it is possible to ﬁnd the binding or ionization energy and the width of the given level. By deﬁnition, the binding energy E i0 f0 is found from the inﬂection point of the arctangent curve as it is shown in ﬁgure 5.25(b). A typical example of the K-absorption edge for Pb is shown in ﬁgure 5.26 with a K-shell binding energy of I = 88.011 keV. 62 The detailed derivation of equation (5.248) was made in the work by Richtmyer F K, Barnes S W and Ramberg E 1934 Phys. Rev. 46 843.

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Atomic structure

5.16 Polarization of x-ray radiation It was found63 that an ion colliding with a beam of unidirectional electrons emits an anisotropic and polarized radiation. A polarization occurs when the radiation components polarized with the electric vector parallel or perpendicular to the electron-beam direction have different intensities, I% and I⊥ , respectively. The degree of linear polarization P(θ ) for the radiation emitted at an angle θ to the beam direction is given by P(θ ) =

I% (θ ) − I⊥ (θ ) . I% (θ ) + I⊥ (θ )

(5.249)

Thus, the polarization of spectral lines and bremsstrahlung might be quite sensitive to the presence of electron beams in plasmas, the so-called non-thermal or suprathermal electrons, or, more generally, to the presence of an electronvelocity distribution. In the case of highly charged ions, the polarization of x-ray lines is of special interest because these lines are more sensitive to beam excitation than the optical lines emitted by low-charged ions for which the inﬂuence of plasma electric and magnetic ﬁelds can mask the polarization effects. Investigating the polarized spectral lines emitted from a plasma is the subject of Plasma Polarization Spectroscopy64, the idea of which is based on the assumption that the observed polarization characteristics of ions can be explained in terms of the anisotropic velocity distribution of the plasma electrons. Fortunately, the spectroscopic diagnostic methods based on the analysis of line spectra or continuous radiation are, to some extent, supplementary, i.e. different electron energies are responsible for the discrete or continuous radiation of highly charged ions in a plasma. The high-energy photons with E > 50 keV are emitted due to bremsstrahlung of the non-Maxwellian electrons and provide the best tool for studying the high-energy ‘tails’ of the distribution function. At the same time, the main contribution to the polarization of x-ray lines is given by electrons with an energy E close to the corresponding transition energy in a highly charged ion, i.e. E ≈ 1–10 keV. The general quantum-mechanical theory for the linear polarization of photons emitted by ions is based on the photon-polarization density-matrix formalism65. The main mechanism leading to the polarization of x-ray lines in highly charged ions is the different population of the magnetic M-sublevels by the electron beam. A directivity of electrons in a plasma leads to an alignment of ions excited by these electrons that, in turn, causes a preferential population of magnetic M-sublevels belonging to an excited atomic level with the angular 63 Oppenheimer J R 1927 Z. Phys. 43 27. Percival C and Seaton M J 1958 Philos. Trans. R. Soc. London A 251 113. 64 See, e.g., Fujimoto T and Kazantsev S A 1997 Plasma Phys. Control. Fusion 39 1269. 65 Blum K 1981 Density Matrix Theory and Applications (New York: Plenum). Steffen R M and Adler K 1975 Interaction in Nuclear Spectroscopy ed W D Hamilton (New York: North-Holland) p 505.

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Figure 5.27. Calculated degree of linear polarization P in % at θ = 90◦ for the x, y and w lines of He-like Fe24+ ions as a function of the incident electron energy: full curve, calculated using the Coulomb–Born approximation with exchange (CBE); broken curve, c using the Distorted Wave Approximation (DWA). 1994 Springer.

momentum L. As previously mentioned, line emission from such levels is polarized and anisotropic. The degree of polarization of x-ray lines, including dielectronic satellites, has been mainly calculated for H-, He- and Li-like ions and include the following transitions: 2p3/2 → 1s1/2 1s1/2 2p1/2 , J = 1 → 1s21/2 , J = 0 1s1/2 2p3/2 , J = 1 → 1s21/2 , J = 0 1s1/2 2p3/2 , J = 2 → 1s21/2 , J = 0 1s1/2 2s1/2 , J = 1 → 1s21/2 , J = 0

1s2p(1 P)2s 2 P3/2 → 1s2 2s 2 S1/2 1s2s2p 4 P3/2 → 1s2 2s 2 S1/2

Lyα1 line the resonance line (w) the intercombination line (y) the magnetic quadrupole line (x) the forbidden line (z) q-satellite u-satellite.

Calculations and measurements of the degree of polarization P for Lyα lines in H-like ions performed for the solar ﬂare plasma showed relatively low values of P < 10%. In the laboratory plasma sources, the degree of polarization for the most prominent x-ray lines yield higher P-values up to 40–60% depending on

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1,0

He-like

0,8

Polarization

0,6

1

w( P1)

0,4 0,2 0,0

3

z( S1)

-0,2

3

y( P1)

-0,4 -0,6

3

x( P2)

-0,8 5

10

15

20

25

30

35

40

Nuclear charge Z Figure 5.28. Polarization degrees of the w, x and y and z lines in He-like ions as a function of the nuclear charge Z (between Ne10+ and Kr34+ ions) at an electron energy of 6800 eV and at an angle of θ = 90◦ . The curves give the calculated degrees of polarization whereas the experimental values are given by the full and open circles, i.e. two measurements for Fe24+ ion (Z = 26) performed with two different crystals on the Livermore electron-beam ion trap (EBIT) facility (from Beiersdorfer P et al 1996 Phys. Rev. A 53 3974 and 1997 Rev. Sci. Instrum. 68 1073). The full square gives the polarization values for the w line in c Sc19+ (Z = 21). From Henderson J R et al 1990 Phys. Rev. Lett. 65 705. 1997 APS.

the incident electron energy as illustrated in ﬁgure 5.27 for the resonance line of Fe24+ . In the case of highly charged ions, an ideal source for investigating x-ray line polarization is an electron beam ion trap (EBIT) where highly charged ions are excited by a monoenergetic electron beam. The EBIT data for x-ray line polarization comprise only a few measurements for He-like highly charged Sc19+ and Fe24+ ions at an electron energy of about 6.8 keV as shown in ﬁgure 5.28. As can be seen, the polarization of x, z and w lines has a weak dependence on the nuclear charge while the polarization of the y line reveals a strong dependence on the nuclear charge because the relativistic j j -coupling between 1s1/2 2p1/2 , J = 1 and 1s1/2 2p3/2 , J = 1 levels rapidly falls off with decreasing nuclear charge and strongly inﬂuences the excitation cross sections of the 1s1/2 2p3/2 , J = 1 level. These measurements showed good agreement with relativistic distorted-wave calculations and disagreement with the Coulomb–Born calculations. However, at higher electron energies, these two approximations are consistent with each other as displayed in ﬁgure 5.27. We note that the polarization of x-ray radiation is also possible in radiative

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recombination processes, synchrotron radiation, radiative electron capture, excitation and radiative-transfer excitation processes in heavy-ion atom collisions.

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Chapter 6 Atomic collisions

When electrons or ions collide with atomic or ionic targets, a large variety of atomic processes can occur. The main characteristics of these processes including the effective cross sections and the rate coefﬁcients are considerable in this chapter.

6.1 Collisional and photo processes in plasmas A plasma is an ionized gas, sometimes titled as the fourth state of matter, and consists of electrons, ions, atoms and molecules. The name plasma was given to this state of matter by Langmuir who was a pioneer in the study of ionized gases. In Greek, πλασ µα means ‘moldable substance’, or ‘jelly’. A plasma is characterized by two main properties. It contains charged particles (electrons, ions) but is electrically neutral as a whole and the motion of plasma particles is correlated, i.e. collective effects take place. Due to the inﬂuence of long-range Coulomb forces, all charged particles in a plasma may simultaneously interact with each other, i.e. undergo atomic collisions with electrons, atoms and ions until a certain degree of ionization is achieved. Since atoms and ions have an atomic structure, as considered in chapter 5, they can be excited into higher electronic states or ionized, decay into lower states emitting photons. All these elementary atomic processes including ionization, recombination and radiative processes, determine the plasma microparameters such as its temperature, density, degree of ionization and radiation spectra. The range of plasma temperatures and densities in different types of laboratory and astrophysical plasmas is very wide (see ﬁgure 1.4 for a temperature–density map in a plasma). As a rule, free electrons play the main role in a plasma because their low mass results in velocities that are much higher than those of heavy particles. Therefore collisions between electrons and massive particles are the most effective in a wide temperature range. Depending on the electron densities and temperatures, a plasma may be in local thermodynamic

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equilibrium (LTE), or may differ from LTE and then other, non-equilibrium types of distribution occur (see sections 6.2 and 6.3). The main plasma parameters, such as density, temperature and ionization degree, strongly depend on the interactions between the plasma particles and their interaction with radiation. These types of interaction are described by the elementary collisional and photo processes partly considered in section 4.1.1. Finding relations between the rates of the elementary processes and the intensities of the spectral lines allows one to develop reliable methods for spectroscopic and corpuscular plasma diagnostics. In this respect, atomic physics in plasma applications describes the universal dependencies of the different atomic processes on plasma temperature, density and other parameters and explains the behavior of a plasma in real conditions. The elementary processes can be divided into two main groups—radiative and collisional. Radiative processes are associated with the interaction between atoms or ions and photons, while the second group describes collisions among electrons, atoms, ions and molecules. This division into two groups is rather conventional. In reality, they are related to each other and uniﬁed by the Fermi conception1 in which an atom makes a transition during its interaction with an electromagnetic ﬁeld created by real photons or with a ﬁeld created by charged incident particles and described by equivalent or virtual photons. The last case is known as the Weizs¨acker–Williams principle of equivalent photons2. In atomic physics, the main elementary processes are the following. Photoexcitation by absorption of a photon and transition of an ion into an excited state: (6.1) Xq+ + ~ω → [Xq+ ]∗ where Xq+ refers to a q-times ionized atom and the asterisk denotes the excited atomic state. Photoexcitation processes are usually important for level populations in a low-density astrophysical plasma. Collisional excitation induced by electron impact: Xq+ + e− → [Xq+ ]∗ + e− .

(6.2)

Photoionization or the photoeffect with the ejection of an electron: Xq+ + ~ω → X(q+1)+ + e− .

(6.3)

Electron-impact ionization releasing an additional electron: Xq+ + e− → X(q+1) + 2e− .

(6.4)

As previously mentioned, electron-impact ionization together with excitation plays a key role in plasma processes. 1 Fermi E 1924 Z. Phys. 29 315. 2 See Jackson J D 1999 Classical Electrodynamics 3rd edn (New York: Wiley).

Heitler W 1954 The Quantum Theory of Radiation (Oxford: Clarendon).

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Three-body recombination is the inverse process to electron-impact ionization: Xq+ + e− + e− → X(q−1)+ + e− . (6.5) Three-body recombination processes are important in low-temperature and highdensity plasmas. Photorecombination or radiative recombination is the recombination of an electron and an ion under emission of a photon: Xq+ + e− → X(q−1)+ + ~ω.

(6.6)

If the ion Xq+ is completely stripped, q = Z , then three-body and radiative recombination are the only two possibilities for an electron and an ion to recombine fulﬁlling momentum and energy conservation. The process to be mentioned next needs at least one bound electron. Dielectronic recombination is a resonance two-step process involving the capture of a free electron with simultaneous excitation of a bound electron plus a subsequent radiative de-excitation: Xq+ (γ0 ) + e− → [X(q−1)+ (γ1 , n)]∗∗ → X(q−1)+ (γ2 , n) + ~ω

(6.7)

where γi (i = 0, 1, 2) denote the electronic core conﬁgurations involved. Dielectronic and radiative recombination processes are important in hightemperature plasmas as recombination mechanisms. Autoionization is an alternative decay mode of the doubly excited state in the dielectronic capture process (6.7) and leads to the emission of an electron (see section 5.6): Xq+ (γ0 ) + e− → [X(q−1)+ (γ1 , n)]∗∗ → Xq+ (γ2 ) + e− .

(6.8)

Resonant scattering is obtained if autoionization replaces the radiative decay in (6.7) resulting in the process Xq+ (γ0 ) + e− → [Xq+ (γ1 , n)]∗∗ → [Xq+ (γ2 )]∗ + e− .

(6.8a)

Bremsstrahlung or continuous radiation is caused by the deceleration of an electron in the Coulomb ﬁeld of an ion: Xq+ + e− (E 0 ) → Xq+ + e− (E 1 ) + ~ω

(6.9)

where E 0 and E 1 are the electron kinetic energies before and after the collision, respectively. Electron capture or charge exchange, arising in ion–atom or ion–ion collisions: Xq+ + Y → X(q−1)+ + Y+ .

(6.10)

In this case, the electron from the target Y makes a transition to the projectile ion Xq+ and creates a new ion X(q−1)+ after the collision. Electron capture is

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263

Figure 6.1. Diagram of direct and inverse atomic processes: AI, autoionization; and DR, dielectronic recombination. From Pradhan A K 2000 NIFS-PROC-44 (Nagoya: NIFS).

important for the population of excited states in highly charged ions colliding with neutral atoms. Figure 6.1 schematically shows how the collisional and radiative atomic processes are interrelated. Most of them may proceed from left to right or from right to left, i.e. there are direct and inverse processes marking as for instance excitation–de-excitation, photoionization–photorecombination, and so on. The probability of an elementary process is measured by the cross section σ which is deﬁned as the ratio of the event rate per target particle to the incident projectile ﬂux (see also equation (4.6)). The cross section has the dimension of an area and its atomic unit is πa02 0.88 × 10−16 cm2 , where a0 ≈ 0.53 × 10−8 cm is the Bohr radius. The cross sections of elementary processes involving highly charged ions strongly depend on the ion charge q because of the inﬂuence of the unscreened Coulomb ﬁeld of the nucleus. In most cases, the q-dependence of the effective cross sections has been approximated by power law σ ∝ q a , where a is a constant. For electron–ion and ion–ion collisions, a is a negative number, while for radiative and ion–atom processes a is positive (see table 5.1). In the physics of electronic and atomic collisions, the rate coefﬁcient σ v as introduced in section 4.1.1 of the process is often used. Multiplied by the number density of target projectiles the rate coefﬁcient shows how many events per second of a given atomic process take place.

6.2 Local thermodynamic equilibrium Local thermodynamic equilibrium (LTE) is reached if the local volume of a plasma is in thermodynamic equilibrium, i.e. if a plasma can be described by the following equilibrium distributions3: (1) Maxwell velocity distribution. In an LTE plasma at temperature T , the 3 Properties of a plasma in thermodynamic equilibrium were also considered in sections 2.3 and 4.1.2.

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distribution of particles of mass M over a velocity v distribution is given by a Maxwell function: dn p /n p = F(v, T ) dv (6.11) = F(vx , v y , vz ; T ) dvx dv y dvz M (v 2 + v 2y + vz2 ) . F(vx , v y , vz ; T ) = M 3/2 (2πkB T )−3/2 exp − 2kB T x where n p is the total number of particles and kB = 8.617 × 10−5 eV K−1 is the Boltzmann constant. In the case of an isotropic distribution, the function F only depends on the absolute value of the velocity v and therefore can be written as 3/2 2 Mv M (6.12) exp − F(v, T ) = 4πv 2 2πkB T 2kB T 2 E E 1/2 F(E, T ) = 1/2 (6.13) exp − kB T π kB T kB T where E = Mv 2 /2 is the particle’s kinetic energy. The isotropic distribution functions (6.12) and (6.13) are normalized to unity ∞ ∞ F(v, T ) dv = F(E, T ) dE = 1. (6.14) 0

0

The particle velocity distribution in a plasma is not necessarily isotropic. For example, if an external magnetic ﬁeld is applied to an electron beam, there can be a two-temperature Maxwell distribution function with longitudinal Te% and transverse Te⊥ electron temperatures, respectively: 1/2 2 2 m e ve% m e ve⊥ me me . (6.15) exp − − F(v, T ) = 2πkB Te⊥ 2πkB Te% 2kB Te⊥ 2kB Te% In electron-cooling devices, one can have temperature parameters kB Te⊥ ≈ 0.1 eV and kB Te% ≈ 0.001 eV so that Te% Te⊥ resulting in a ﬂattened distribution. If Te% = Te⊥ , one has a symmetric distribution. The electron-velocity distribution function can deviate from the Maxwell one (6.11) in other ways, function caused, for instance, by the presence of electric ﬁelds in the plasma, temperature gradients, or parametric instabilities4 . Of special importance are suprathermal electrons with velocities higher than those given by the Maxwell distribution. Although the relative number of such electrons could be as small as about 10%, they can lead to the polarization of x-ray lines and bremsstrahlung of highly charged ions representing an important source of information for plasma diagnostics. 4 See Lamoureux M 1993 Adv. At. Mol. Phys. 31 233.

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Non-equilibrium plasma: the coronal limit (2) Boltzmann distribution of atoms over atomic-energy levels E i : gi Ei − E0 ni = exp − n0 g0 kB Te

265

(6.16)

where n i denotes the number of atoms in the i th state, gi and g0 the statistical weights of level i and the ground-state level 0, respectively. (3) Saha distribution of atoms over degrees of ionization: G q+1 Iq n (q+1) S (6.17) = exp − Gq kB Te n (q) q 3 &3/2 Te m e kB Te 3/2 (6.18) = &= 2 S = 2n −1 e 2π ~2 4π 3/2 a03n e q Ry where Iq denotes the ionization energy of the ion Xq+ , Te the electron temperature, n e the electron density and a0 the Bohr radius. The factor S is called the Saha factor. The partition or statistical function G q is deﬁned as

(q) Ei − E0 . (6.19) Gq = gi exp − kB Te i

(4) Planck distribution. If a plasma volume is in thermodynamic equilibrium with the radiation ﬁeld, the energy density u ω of this ﬁeld per frequency interval is given by the Planck law for black-body radiation (see section 2.3): uω =

~ω3 π 2 c3

1 e~ω/ kB T − 1

,

(6.20)

where c is the speed of light.

6.3 Non-equilibrium plasma: the coronal limit In LTE with its sufﬁciently high electron density, equations (6.16) and (6.17) completely determine ion populations in the plasma. The equations, merely determined by electron density and temperature, are independent of the peculiarities of the various elementary collision processes. However, for plasmas with medium and low electron densities, these distributions are no longer valid and the level population can be estimated by solving the rate or kinetic equations accounting for all possible excitation, ionization and recombination processes. Usually, the solution of the kinetic equations is found numerically because of the large number of atomic levels and processes involved and eventually because of time dependent plasma density. To describe a plasma in a non-equilibrium state, one needs to apply a certain collisional-radiative model5. The solution 5 See, e.g., the SCROLL (Super Conﬁguration Radiative cOLLisional) model described in BarShalom A, Oreg J and Klapish M 1997 Phys. Rev. E 56 R70.

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of these kinetic equations can be simpliﬁed if the deviation from LTE is not very large and one has near-LTE conditions so that a perturbation method can be used to give a simpliﬁed description of the atomic dynamics6. Nevertheless, a Maxwellian distribution function is often used for calculations and explanation of experimental data. To make some qualitative estimates, we consider the simplest two-level atomic model and an arbitrary electron density n e . The balance equation in this case equating the number of excited and de-excited ions, can be written as n 0 n e σ01 v = n 1 n e σ10 v + n 1 A10

(6.21)

where n 0 and n 1 denote the ion densities in the ground and excited states, respectively, σ01 v and σ10 v the electron-impact excitation and de-excitation rate coefﬁcients, and A10 the radiative transition probability from level 1 to level 0 (section 5.4). Using the relation between excitation and de-excitation rate coefﬁcients (6.68), instead of the Boltzmann distribution (6.16) one has: E 1 −E 0 exp − A10 g1 n1 kT R= (6.22) = n0 g0 1+ R n e σ10 v i.e. the level population now strongly depends on the ratio R describing the deviation from LTE. However, the temperature dependence is still described by the Boltzmann function because the de-excitation rate coefﬁcients weakly depend on the electron temperature Te as discussed by Sobelman et al7 . If the upper level is the ionization limit, one has an equation for ionization– recombination equilibrium taking into account ionization and three different recombination processes: dielectronic, radiative and three-body recombination. The rate coefﬁcients for the ﬁrst two processes are proportional to the ﬁrst power of the electron density n e while for three-body recombination it is proportional to n 2e . Taking into account relation (6.120) between ionization and three-body recombination rates, one has, instead of the Saha distribution (6.17), gq+1 Iq n (q+1) S = exp − gq kB Te n (q) κdr + κrr R= (6.23) n e κr where κr , κdr and κrr denote the rate coefﬁcients of three-body, dielectronic and radiative recombination considered in sections 6.7.6, 6.8.3 and 6.8.5, respectively. Again, if the ratio R > 1, the distribution over the ionization degrees will differ from the Saha distribution and will strongly depend on the ionization and recombination rate coefﬁcients. 6 See, e.g., More R and Kato T 1998 Phys. Rev. Lett. 81 814. 7 See Sobelman I I, Vainshtein L A and Yukov E A 1995 Excitation of Atoms and Broadening of

Spectral Lines 2nd edn (Berlin: Springer).

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A plasma of very low-density is the opposite case to LTE. A good example of a low-density plasma is the solar corona with an electron density n e ≈ 108– 109 cm−3 which is much less compared to that of laboratory plasma sources. The low-density case is called the coronal limit. Using equations (6.22) and (6.68) and putting n e → 0, one has, for the level population, σ01 v n1 = ne n0 A10

(6.24)

which is now expressed in terms of the excitation rate coefﬁcients. The intensity of the spectral line, corresponding to the transition 1 → 0, can be written as I10 = n 1 A10 ~ω = n 0 n e σ01 v~ω

(6.25)

where ω is the angular frequency of the transition 1 → 0, i.e. the line intensity does not depend on the transition probability A10 . This is a very important feature of the coronal limit. Let us compare the values of the electron density in the coronal limit for neutral atoms and highly charged ions. According to equation (6.22), the coronal limit takes place if R 1

or

n e n max =

A10 σ10 v

(6.26)

where n max is the maximum possible electron density. As stated in table 5.1, the following scalings apply for the transition 4 , σ v ∝ (E)3/2 ∝ probability rate and rate coefﬁcient: A10 ∝ (E)2 ∝ Z eff 10 −3 7 . This means that for neutral atoms with Z eff , so that n max ∝ (E)7/2 ∝ Z eff Z eff = 1, we have n max ≈ 1016 cm−3 and for highly charged ions with Z eff ≈ 10, n max ≈ 1023 cm−3 . Respectively, for the ionization equilibrium in the coronal limit (see equation (6.23)) one has: σi v n (q+1) = (6.27) (q) κrr + κdr n where σi v is the ionization rate coefﬁcient. Therefore, in contrast to the LTE case, the degree of ionization in the coronal limit is independent of the electron density n e which is another important feature of the coronal limit.

6.4 The principle of detailed balance Each elementary process is invariant under time inversion and the probabilities of direct and inverse processes are related to each other by the principle of detailed balance which follows from the main principles of classical, quantum-mechanical and statistical physics. In classical physics, according to this principle, the

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probabilities W of direct i – f and inverse f –i transitions between the states i and f for an isolated system, are equal, i.e. Wi f = W f i .

(6.28)

In quantum mechanics, the principle of detailed balance follows from the symmetry of the quantum equations of motion relative to the inversion of time. The principle can be formulated in more detail for binary collisions of atoms, molecules and elementary particles. For example, it allows one to calculate the probability of the inverse process if the probability of the direct one is known and vice versa. This is quite important because sometimes it is possible to measure the effective cross section for only one of these processes, e.g., it is possible to determine de-excitation cross sections when excitation cross sections are known from experiment. Let us consider the general case of direct and inverse processes with a rearrangement of particles I and II in the initial state into particles 1 and 2 in the ﬁnal state, e.g., the electron capture process (6.10): I + II ↔ 1 + 2.

(6.29)

According to the quantum-mechanical deﬁnition of the cross section, the differential cross sections of direct (i f ) and inverse ( f i ) reactions have the form8: dσi f 4π 2 ˆ i m sI m sII |2 = 2 |−n f m s1 m s2 | S|n d f pi

(6.30)

dσ f i 4π 2 ˆ f m s1 m s2 |2 = 2 |−ni m sI m sII | S|n di pf

(6.31)

ˆ f denotes the element of the scattering S-matrix, m s the projection where i | S| of spin s; pi = µi vi , p f = µ f v f are the particle momenta, µi and µ f are the reduced masses of two initial and ﬁnal particles and d is a solid angle element. We recall the physical meaning of the S-matrix: it transforms the wavefunction of the total system in the initial state i into the wavefunction in the ﬁnal state f . The minus sign before the unit vector n f means that the particles in the ﬁnal f state move away from the coordinate origin. If the particles are not polarized, it is necessary to sum the cross sections over the spin projections in the ﬁnal state and average them over the spin projections in the initial states, which gives dσ i f 4π 2 1 = 2 d f pi (2sI + 1)(2sII + 1) 8 A derivation of the general formula connecting effective cross sections of direct, σ , and inverse, if σ f i , processes is given in Armenteros R, Barker K H, Batler C C and Cahon A 1951 Phil. Mag. 42

1113. Landau L D and Lifshitz E M 1977 Quantum Mechanics (New York: Pergamon).

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Photon emission and absorption ×

ˆ i m sI m sII |2 |−n f m s1m s2 | S|n

269 (6.32)

m sI m sII m s1 m s2

dσ f i 4π 2 1 = 2 di (2s + 1)(2s pf 1 2 + 1)

ˆ f m s1 m s2|2 . |−ni m sI m sII | S|n ×

(6.33)

m sI m sII m s1 m s2

Taking into account the symmetry properties of the S-matrix and the fact that the summation in equations (6.32), (6.33) runs over all spin projections, one obtains: dσ i f p2f

d f

(2sI + 1)(2sII + 1) =

dσ f i pi2 di

(2s1 + 1)(2s2 + 1).

(6.34)

Equation (6.34) reﬂects the principle of detailed balance for direct and inverse elementary processes and is valid for non-polarized particles. The principle of detailed balance is also used for relations between the total cross sections and the corresponding rate coefﬁcients of direct and inverse elementary processes. According to equation (6.34), the total cross sections σ01 and σ10 , i.e. the cross sections for direct 0 → 1 and inverse 1 → 0 processes, respectively, integrated over solid angles are related by equation: g0 · p02 · σ01 ( p0 ) = g1 · p12 · σ10 ( p1 )

(6.35)

where g denotes the statistical weight given by the number of possible quantum angular- and spin-momentum states, and p = µ · v is the momentum of the particles in the center-of-mass system. The principle of detailed balance (6.35) is valid for any pair of direct and inverse processes in the atomic system at thermodynamic equilibrium.

6.5 Photon emission and absorption If the photon energy ~ω of the radiation ﬁeld exactly coincides with the transition energy E of an atom, i.e. the energy difference between the ﬁnal and initial states is: ~ω = E ≡ E 0 − E 1 (6.36) then the atomic electron can make a transition from state 0 to a higher state 1, i.e. absorption of radiation occurs Xq+ + ~ω → [Xq+ ]∗ .

(6.37)

This process was partly considered in section 2.3. As an inverse process, the ion can emit a photon with a frequency given by (6.36) even without any external perturbation. This emission is called spontaneous emission which is described

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within the framework of quantum electrodynamics (QED) according to which spontaneous emission is caused by the interaction between the ion and the ‘vacuum’ state of the radiation ﬁeld. However, the excited ion Xq+ can also emit a photon with energy (6.36) under incoming photons if their frequency coincides with the transition energy. This is called stimulated emission, the most striking feature of which is that a stimulated photon is emitted in the same direction and with the same phase as that of the incoming photon, i.e. coherently. This feature lies at the basis of lasers or lasing9 . The probabilities of absorption and spontaneous and stimulated emission are called the Einstein coefﬁcients. The relation between all these coefﬁcients can be found from the detailed balance principle. Let us again consider the two-level atom in LTE with atomic densities n 1 and n 2 for levels 1 and 2, respectively. Three radiative processes can occur between the levels: absorption of radiation and spontaneous and stimulated emissions. All these processes are described by the Einstein coefﬁcients B12 , A21 and B21 , respectively, which are deﬁned so that the rate of change in the population numbers is dn 2 dn 1 = − = −B12 u ω n 1 + B21 u ω n 2 + A21 n 2 dt dt ω = (E 2 − E 1 )/~

(6.38)

where u ω denotes the energy density of the radiation ﬁeld per frequency interval. At equilibrium, one has dn 2 dn 1 =− =0 (6.39) dt dt and, according to the detailed balance principle, the number of ions, making transitions upward 1 → 2 and downwards 2 → 1, is equal. Taking into account the Boltzmann (6.16) and Planck (6.20) distributions, this can be written in the form: n 1 B12 u ω = n 2 (A21 + B21u ω ) g2 n2 = e−(E2 −E1 )/ kB T n1 g1 ~ω3 1 u ω = 2 3 ~ω/ k T B π c e −1

(6.40) (6.41) (6.42)

where kB denotes the Boltzmann constant, g the statistical weights and T the temperature of the system. From this set of equations, one has the following relations between the three coefﬁcients: g1 B12 = g2 B21 π 2 c3 B21 = A21 . ~ω 3

9 See, e.g., Elton R C 1990 X-Ray Lasers (New York: Academic).

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(6.43) (6.44)

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From (6.43), it can be seen that the probabilities for absorption and stimulated emission between levels 1 and 2 are proportional to each other independently of the temperature and strength of the radiation ﬁeld. In addition, all three probabilities should follow the same selection rules for radiation (section 5.3). Furthermore, two important characteristics can be obtained: the condition for saturation regime for the population of the upper level 2 and the photoabsorption cross section for the transition 1 → 2. From equations (6.39), (6.42) and (6.44) one obtains: n2 1 (6.45) = n1 + n2 (1 + g1 /g2 + A21 /u ω B12 ) which for A21 /u ω B12 1 gives the saturation condition for the maximum population of level 2: n2 g2 = . (6.46) n1 + n2 g1 + g2 For instance, for the 2p1/2 –2p3/2 transition in B-like Ar13+ it gives 2 n2 = n1 + n2 3

(6.47)

i.e. in the saturation regime, 66% of the ions will be in excited state 2. Now we estimate the laser power density IL (ω) per unit frequency to achieve the saturation. The IL (ω)-value is usually expressed in units of W cm−2 MHz−1 and is deﬁned as IL (ω) = c · u ω (6.48) where c is the speed of light. Therefore, the saturation condition A21 /u ω B12

is equivalent to A21 ~c IL (ω) ≥ c = 16π 2 3 (6.49) B12 λ where λ is the wavelength for the transition 1–2. Equation (6.49) is a very important condition used in laser spectroscopy10. For our example of the Ar13+ (2p1/2 –2p3/2) transition with λ = 441 nm, one obtains for the saturation laser power density: IL (ω) ≥ 5.1 × 10−3 W cm−2 MHz−1 . The two-level scheme considered here allows one to obtain an expression for the photoabsorption cross section for transition 1 → 2 in the form11: σ12 (ω) =

g1 π 2 c 2 λ2 g1 aω 2 = aω g2 4 g2 ω

(6.50)

where g denotes the statistical weight, λ the wavelength and aω the spectral probability of spontaneous emission, a dimensionless quantity normalized to aω dω = A21 . (6.51) 10 See Demtr¨oder W 1998 Laser Spectroscopy (Berlin: Springer). 11 See Sobelman I I 1992 Atomic Spectra and Radiative Transitions (Berlin: Springer).

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Here A21 is the radiative transition probability per time from level 2 to level 1. Equation (6.50) is valid for one-photon absorption cross sections for any kind of electric Eκ and magnetic Mκ radiation (for notation see table 5.2). The equation gives the value of the photoabsorption cross section at its maximum. Since aω,max ≈ A21 /ω, where ω is the total width of the upper state 2, one obtains max σ12 =

λ2 g1 A21 . 4 g2 ω

(6.52)

If the laser frequency width ωL ω, i.e. is much higher than the natural or Doppler line widths, one has to use the value ωL instead of ω in (6.52). For example, let us estimate the photoabsorption cross section at its maximum for the resonance transition 2s1/2 –2p1/2 in Li-like U89+ by radiation of a free-electron laser with energy width of ~ωL ≈ 2.8 × 10−3 eV. The radiative transition probability for this transition is estimated to be A(2p1/2 –2s1/2 ) ≈ 2.6 × 1010 s−1 ˚ = 280 eV. Therefore, the natural width of the and the wavelength λ = 44.3 A upper state is ~ωnat ≈ 1.7 × 10−5 eV ~ωL and the photoabsorption cross section at its maximum is estimated to be σ max (2s1/2 –2p1/2 ) ≈

˚ 2 2 1.7 × 10−5 eV (44.3 A) 3.0 × 10−16 cm2 . 4 2 2.8 × 10−3 eV

6.6 Excitation and de-excitation in collisions with electrons 6.6.1 Direct excitation For the elementary process of electron-impact excitation (6.2) the necessary excitation energy from level 0 to level 1 is taken from the kinetic energy of the collision partners. The excitation is only possible if the collision energy E exceeds a certain threshold set by the excitation energy E as E ≥ E = |E 0 − E 1 |.

(6.53)

In this case the collision reaction reads Xq+ + e− (E) → [Xq+ ]∗ + e− (E )

(6.54)

According to the energy conservation law, the scattered electron has the energy E = E − E ≥ 0. The probability of this process is described by the effective excitation cross section σ01 which depends on the incident electron energy E and the atomic structure of the target. However, there are some general properties characterizing the excitation cross sections and rate coefﬁcients. The ﬁrst concerns the behavior

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of the excitation cross sections near threshold (E ≈ E). In the case of neutral atoms, the excitation cross section tends to zero at threshold energy E ≈ E: σth (E) → 0

E → E

(6.55)

and has its maximum at about 2–3 threshold energies, i.e. E max ≈ (2–3)E. A typical example of the excitation cross section of optically allowed dipole transitions, i.e. those with a change in the orbital = ±1, and spin angular momenta S = 0 in neutral atoms, is shown in ﬁgure 6.212. In the case of positive ions, the long-range attractive Coulomb force causes the excitation cross section to be ﬁnite at threshold and, as a rule, also to have its maximum there. For dipole transitions in atoms and ions, the excitation cross sections falls off at high energies according to the Bethe formula: σ dip ≈

ln E A +B E E

E E

(6.56)

where A and B are constants. The Bethe constant B is associated with the dipole oscillator strength f (section 5.4) and A can be obtained from numerical calculations. For dipole transitions, the maximum excitation cross section of ions can be estimated by the semi-empirical Van Regemorter formula: dip

σmax ≈

2.90 f πa 2 (E/Ry)2 0

(6.57)

where f denotes the transition oscillator strength, a0 the Bohr radius; 1 Ry 13.606 eV and πa02 0.88 × 10−16 cm2 . For other types of transitions, the excitation cross section at high electron energies E E falls off approximately as −1 (6.58) σ ∼ E −3 for transitions with || )= 1, S = 0 E for intercombination transitions S = 1. Often, excitation cross sections σ01 are presented in a standard form: σ01 [πa02 ] = (x)/(g0 E)

x = E/E

(6.59)

where g0 denotes the statistical weight of the initial state, E the incident-electron energy in Ry units and x the scaled, reduced electron energy in transition-energy units. The quantity is termed the collision strength which is symmetrical for direct 0–1 (excitation) and inverse 1–0 (de-excitation) transitions, i.e. 01 = 10 .

(6.60)

12 Simple semi-empirical formula of electron-impact excitation cross sections for dipole transitions in neutral atoms is described in the paper by Fisher V, Bernshtam V, Golten H and Maron Y 1996 Phys. Rev. A 53 2425.

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Figure 6.2. Excitation cross sections for transitions Cs(6s–6p), Na(3s–3p) and H(1s → n = 2): experiment symbols and broken curve, experimental results, full curves semi-empirical ﬁt of theory. From Vriens L and Smeets A H M 1980 Phys. Rev. A 22 940. c 1980 APS.

The quantity has the dimension cm2 eV−1 . Excitation rate coefﬁcients deﬁned by equation (4.8) and obtained with a Maxwell velocity distribution also can be presented in a standard form: ∞ G(β)e−β G(β) = βeβ e−β x (x) dx (6.61) σ v = K g0 1 √ K = 2 π ~a0 ≈ 2.18 × 10−8 cm3 s−1 (6.62) x = E/E

β = E/kB Te

where the function G(β) is called the effective collision strength; Te denotes the

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electron temperature and kB the Boltzmann constant. Besides the direct excitation process, there is the inverse process called deexcitation or quenching: [Xq+ ]∗ + e− → Xq+ + e− .

(6.63)

The excitation cross sections of both direct 0 → 1 and inverse 1 → 0 transitions are related by the Klein–Rosseland formula in accordance with the principle of detailed balance (section 6.4): g0 · (E + E) · σ01 (E + E) = g1 E · σ10 (E) g0 σ01 ≈ g1 σ10 σ10 → ∞

E E E→0

(6.64) (6.65) (6.66)

where E denotes the incident electron energy and E the transition energy. The principle of detailed balance can also be formulated for excitation and deexcitation rate coefﬁcients. In a plasma under LTE conditions (section 6.2), the number of excitation and de-excitation events per second are equal, i.e. n 0 n e σ01 v = n 1 n e σ10 v

(6.67)

where n 0 and n 1 denote the atomic densities of the states 0 and 1, respectively. Using the Boltzmann distribution (6.16), one obtains the relation between excitation and de-excitation rate coefﬁcients: g0 σ01 v = g1 σ10 v exp(−β)

β = E/kB Te .

(6.68)

In highly charged ions, the excitation cross sections induced by electron impact decrease with increasing nuclear charge Z as σ ∝

1 Z4

(6.69)

if the scaled electron energy E/E is used. The corresponding excitation rates scale as 1 1 (6.70) σ v ∝ 4 Te ∝ 3 Z Z as a function of the scaled electron temperature Te /Z 2 (see table 5.1). Most experimental data on the electron-impact excitation cross sections of positive ions have been obtained using a crossed-beam radiation detection or a merged-beam electron-energy loss method. These measurements have been carried out for allowed dipole transitions by detecting the radiation resulting from collisions in a crossed-beam geometry. For example, in highly charged ions, absolute excitation cross sections have been measured mainly for the resonance s–p transitions in C3+ , N4+ , Al2+ , Si3+ , and Ar7+ ions. A typical example of the

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Figure 6.3. Absolute electron-impact excitation cross sections for the transition 2s–2p in C3+ : open circles, experiment; broken curve, close-coupling calculations; full curve, the same convoluted with the apparatus electron-energy resolution function. From Taylor P O c et al 1977 Phys. Rev. Lett. 39 1256. 1977 APS.

excitation cross sections for a resonance transition in C3+ is given in ﬁgure 6.3 showing the non-zero value of the cross section at threshold. However, with increasing nuclear charge number Z , the excitation cross sections for these transitions rapidly fall with Z as σ ∝ Z −4 which makes these measurements very difﬁcult because high target densities are necessary together with detection of short-wavelength radiation bearing in mind that the wavelength is also proportional to Z −4 . Measurements carried out by the merged-beam energy loss method, e.g. for Si3+ and Ar7+ , rely on the detection of electrons which have lost the appropriate amount of energy instead of photons emitted from excited states so that the absolute excitation cross sections for allowed and forbidden transitions can be measured. For example, this method was applied for measurements of the intercombination transitions in Li+ (1s2 1 S–1s2p 3P) and Kr6+ (4s2 1 S–4s4p 3P); in the latter case, the cross section is dominated by dielectronic resonances (section 6.8). The invention of new ion sources like the EBIT (Electron-Beam Ion Trap) open new possibilities in the investigation of the excitation cross sections of highly charged ions by measuring the x-ray emission from the trapped ions and obtaining the absolute values of the cross sections by normalization to theoretical

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radiative recombination cross sections performed, e.g., for Ti20+ and Ba46+ . New methods and techniques have also made it possible to detect sharp resonances in the excitation cross sections of highly charged ions by electron impact (see the next section). 6.6.2 Resonant excitation Along with direct electron-impact excitation, considered in the previous section, Xq+ (α0 ) + e− → [Xq+ (α1 )]∗ + e−

(6.71)

the ion Xq+ can be excited to the same upper state α1 by several intermediate processes; here α0 and α1 denote the sets of quantum numbers of the ion in the initial and ﬁnal states. In highly charged ions, the most probable intermediate process is resonant excitation which strongly changes the shape and value of the excitation cross section. Due to the long-range Coulomb force of the positive ion, a free electron can be captured by the ion into doubly excited autoionizing state (section 5.6), i.e. into the bound states lying above the ionization limit of the target ion: Xq+ (α0 ) + e− → [X(q−1)+ (γ )]∗∗

γ = αn

(6.72)

where α1 n denotes the set of quantum numbers of the ‘intermediate’ X(q−1)+ ion. Thus, a doubly excited state can be created by the capture of a free electron into the n state and simultaneous excitation of the target electron into the α1 state. The excited [X(q−1)+ ]∗∗ ion can decay and create another ion [Xq+ ]∗ in the α1 state as in ‘usual’ excitation (6.71), i.e. [X(q−1)+ (γ )]∗∗ → [Xq+ (α1 )]∗ + e− .

(6.73)

This excitation process via an intermediate autoionization state gives an additional contribution to the excitation cross section and manifests itself as a set of pronounced resonances on top of the smooth shape of the potential cross section. We recall that in highly charged ions, the autoionization state can also decay via the competitive radiative decay channel: [X(q−1)+ (γ )]∗∗ → [X(q−1)+ (γ )]∗ + ~ω.

(6.74)

This radiative decay is called dielectronic recombination (section 6.8). The branching between autoionization and radiative decays depends on the ion charge and its atomic structure. Radiative decay (6.74) takes place predominantly in highly charged ions, while autoionization (6.73) occurs mainly in low-charged ions. The relative probabilities of decay via these two competitive channels are described by the branching ratio coefﬁcients considered in section 6.7.3. If the autoionization (6.73) leads to excitation α0 → α1 with different quantum numbers, i.e. α0 )= α1 , such an additional excitation channel is termed

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resonant inelastic excitation. If the ion in the ﬁnal state has the same quantum numbers, i.e. α0 = α1 , the two reactions (6.71) and (6.73) are direct and inverse relatively to each other and the two-step process is called resonant elastic scattering. Now we consider the shape and value of the resonant excitation cross section. Electron capture (6.72) does not occur at all electron energies but only if the energy E of the free electron is ‘in resonance’ with one of the autoionization energies of the X(q−1)+ ion, i.e. E = E res ≈ E α1 − E α0 − (q − 1)2 Ry/n 2

n 1.

(6.75)

Condition (6.75) is valid within the energy-level width γ of the autoionization state γ . The dependence of the capture cross section on the electron energy E is described by a dispersion Lorentz-type formula: σc (γ |E) = σc (γ )φγ (E) γ /2π φγ (E) = (E − E res )2 + γ2 /4 where the Lorentzian proﬁle φγ (E) is normalized to unity: +∞ φγ (E) dE = 1. −∞

(6.76) (6.77)

(6.78)

The dependence of the factor σc (γ ) on the incident-electron energy E res can be found from the detailed balance principle valid for elastic excitation (α1 = α0 ) and autoionization as an inverse process: 1

E res · g(α0 ) · σc (γ ) = 2πτ0 · g(γ ) · Aa (γ , α0 ) Ry πa02 ~a0 τ0 = 2 = 2.419 × 10−17 s e 2

(6.79) (6.80)

where Aa (γ , α0 ) denotes the autoionization decay to the ﬁnal state α0 of the ion Xq+ . Thus, the resonant excitation leads to the appearance of a set of resonances at energies (6.75) on top of the ‘usual’, potential cross sections, as displayed in ﬁgure 6.4. Using the capture cross section formulae (6.76)–(6.79), the resonant excitation cross section can be written in the form: σres (α0 − γ − α1 |E) = σc (γ )φγ (E)B(γ , α1 )

γ = αnL S J

(6.81)

where B(γ , α1 ) is the branching ratio coefﬁcient (section 6.7.3): B(αn, α1 ) =

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Aa (αn, α1 ) Aa (αn) + Ar (αn)

(6.82)

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Figure 6.4. Relativistic Breit–Pauli calculations of the excitation cross section for the dipole transition 3s2 1 S0 –3s3p 1 P1 in Fe14+ (1 Mb = 10−18 cm2 ). From Grifﬁn D C et al c 1998 J. Phys. B: At. Mol. Opt. Phys. 31 3713. 1998 IOP Publishing.

and Aa (αn) and Ar (αn) are the total autoionization and radiative decay probabilities per time. Finally, the proﬁle of the resonant excitation cross section is described by the following Breit–Wigner function: σres (E) =

γ /2π 2πτ0 Ry2 g(γ ) Aa (γ , α0 )Aa (γ , α1 ) E res g(α0 ) (E − E res )2 + γ2 /4 Aa (αn) + Ar (αn)

(6.83)

where g denotes the statistical weight, γ the width of the intermediate state γ , Aa and Ar the autoionization and radiative transition probabilities, respectively; the quantities τ0 and E res are deﬁned in equations (6.80) and (6.75), respectively. In most cases, except for the crossed-beam experiments, separate excitation resonances are not resolved and one deals with cross sections averaged over resonances. The average resonant excitation cross section can be deﬁned as 1 E+D/2 σ¯ res = σres (E) dE (6.84) D E−D/2 where D is the distance between resonances, D ≈ 2(q + 1)2 Ry/n 3 . Using the relation between the autoionization probability and excitation cross section at threshold, (2 + 1)g( α)Aa (αn) =

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(q + 1)2 E res g(α0 )σ (α0 − αn) π ~n 3 πa02

n1

(6.85)

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Figure 6.5. Calculated excitation cross sections for the 2p6 –2p5 3s transition in Fe16+ as a function of the scale transition energy E/E where E denotes the excitation energy: 1, the ‘usual’ (potential) excitation cross section; 2, the sum of the usual and resonant cross sections averaged over the resonances. From Presnyakov L P and Urnov A M 1975 J. Phys. c B: At. Mol. Phys. 8 1280. 1975 IOP Publishing.

the average resonance excitation cross section can be presented in the form: σ¯ res (α0 − αn − α1 ) ≈ σ (α0 − αn)B(αn, α1 )

E α0 α1 ≤ E ≤ E α0 α . (6.86)

Thus, the resonant excitation cross section via an intermediate level n is an extrapolation of the potential excitation cross section for higher levels to the nearthreshold region13. There are cases where the resonant excitation contribution is much larger than the usual, potential excitation. One example is shown in ﬁgure 6.5 for the 2p6 –2p53s dipole transition in Ne-like Fe16+ ion. It can be seen that at nearthreshold energy, the resonant cross section is about ﬁve to seven times larger than the potential one. The contribution of the resonant scattering is very important in a low-temperature plasma where the excitation rate can be signiﬁcantly enhanced due to resonant excitation of highly charged ions.

13 See for details Shevelko V P and Vainshtein L A 1993 Atomic Physics For Hot Plasmas (Bristol: IOP Publishing).

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6.7 Ionization and three-body recombination 6.7.1 Single-ionization processes The ionization of atoms and ions by electron impact, Xq+ + e− → · · · → X(q+1)+ + 2e−

(6.87)

together with electron-impact excitation plays a key role in practically all kinds of laboratory and astrophysical plasmas. The dots in the reaction (6.87) indicate possible intermediate processes, which can lead to the release of one additional electron. In other words, electron-impact ionization can occur via direct and indirect processes. Direct ionization is associated with the ejection of an electron from an outer or inner target shell by direct incident-electron impact. Indirect ionization may occur as a result of Auger decay of an intermediate autoionizing state lying above the ionization limit of the Xq+ ion. Autoionizing states can be created in several different ways (section 5.6), for example by primary ionization of an inner-shell electron making a ‘hole’ in the target, by an excitation of an inner-shell electron into an autoionizing state or as a capture of the incident electron by the target into a short-lived state of a compound ion which can then decay by several autoionization transitions. The relative contribution of direct and indirect ionization processes to the total ionization cross section strongly depends on the incident-electron energy and on the atomic structure of the target ion. In some cases, direct ionization of the outermost electrons in the dominating process, in other cases, this contribution is very small compared to indirect ionization processes14. 6.7.2 Direct ionization Direct ionization of outer- or inner-shell electrons − (q+1)+ + e− Xq+ + e− 1 (E) → X 1 (E 1 ) + e2 (E 2 )

(6.88)

occurs if the energy E of the incident electron is higher than the binding energy I of the target electron shell. The energy conservation law for ionization has the form: I >0 E1, E2 ≥ 0 (6.89) E − I = E1 + E2 where E 1 and E 2 denote the energies of the scattered e1 and ejected e2 electron, respectively. The ﬁrst classical model adopted by Joseph John Thomson (1856–1940) in 191215 gave the following expression for the ionization cross section: 2 4u Ry πa 2 u = E/I − 1 (6.90) σ Th = N I (u + 1)2 0 14 Ionization processes of atoms and ions by electron impact are considered in M¨ark T D and Dunn G H 1985 Electron Impact Ionization (Berlin: Springer). 15 Thomson J J 1912 Philos. Mag. 33 449.

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where a0 denotes the Bohr radius, N the number of equivalent electrons in the target shell and u the reduced incident-electron energy. The Thomson cross section is a smooth function of the electron energy E and at high E decreases as σ Th ∝ u −1 u 1. (6.91) A more appropriate, quantum-mechanical consideration leads to an asymptotic behavior with an additional logarithmic energy term σ ∝ A/u + B ln u/u

u1

(6.92)

arising as a pure quantum-mechanical effect called the Bethe logarithm (cf (6.56)). The constant A is associated with the classical ionization cross section and the constant B, called the Bethe constant, is related to the corresponding photoionization cross section σ ph (ω): ∞ ph σ (ω) I dω ω≥I (6.93) B= πα I ω where ω denotes the photon angular frequency and α the ﬁne-structure constant. The ionization cross section has the following asymptotic behavior:   u 3/2 u → 0, for neutral atoms q = 0 σ ∼ u u → 0, for positive ions q ≥ 1  A/u + B ln u/u u → ∞ for all targets and assumes a maximum value of σmax ∝ I −2 ∝ q −4

at

u max ≈ 2, E max ≈ 3I.

(6.94)

With a Maxwell velocity distribution function F(v, T ) of the incident electrons, given by (6.15), the ionization rate coefﬁcient is given by σ v = σ (v)v F(v, Te ) d3 v ∞ σ (u) −βu 1/2 3/2 −β = K 0 (I /Ry) β e (u + 1) e du (6.95) πa02 0 √ K 0 = 2 π ~a0 /m e = 2.18 × 10−8 cm3 s−1 (6.96) β = I /kB Te

u = E/I − 1

(6.97)

where m e denotes the electron mass, Te the electron temperature and kB the Boltzmann constant. Correspondingly, the number of ionization events per second is Wi = n e σ v where n e is the electron density. Unlike the electron–ion excitation cross sections, the ionization cross section is equal to zero at the threshold for both neutral atoms and positive ions: σth = 0 at E = I . The description of the threshold behavior of the ionization cross section

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is much complicated by the strong inter-electron repulsion and the pronounced three-body character of the problem at low energy. The nature of the threshold behavior of ionization cross sections in a classical approximation was considered by Wannier16 who demonstrated that the ionization cross section for H-like ions at threshold follows the Wannier law: 1 1 100Z − 9 1/2 σ ∝ ua a=− + u→0 u = E/I −1. (6.98) 4 4 4Z − 1 Equation (6.98) gives the following a-values: for hydrogen atom, Z = 1, a = 1.127; for He+ , Z = 2, a = 1.056; for Z = 5, a = 1.021 and for Z → ∞, a → 1. Figure 6.6 displays the scaled cross sections of H-like ions from C5+ up to Ar17+ in comparison with calculations performed in the Coulomb–Born approximation with Exchange (CBE). Ionization of H-like ions is a pure case because only direct ionization contributes and indirect processes are absent. For a many-electron atom or ion, the contribution from ionization of the inner-shell electrons to the total cross section can be signiﬁcant reaching up to 50% and more. It is especially important for the ionization of heavy low-charged ions (Bi1+ , U4+ ) when the values of the ionization potential and binding energies of a few outer-most electrons are comparable in size. Cross sections for direct ionization and rate coefﬁcients for positive ions are often estimated by the semi-empirical Lotz formulae: ln(u + 1) cm2 u = E/In − 1 (6.99) u+1 (6.100) σL · v = 6.0 × 10−8 Nβ 1/2 (Ry/In )3/2 e−β f (β) cm3 s−1 β β = In /kB Te (6.101) f (β) = e |Ei(−β)|

σL = 2.43 × 10−16 N(Ry/In )2

where In denotes the binding energy of the target n N shell, N the number of equivalent electrons, Te the electron temperature and Ei(x) is the integral exponent. The function f (x) is ﬁtted to within 5% by the formula: 0.562 + 1.4x x x > 0. (6.102) f (x) = e |Ei(−x)| ≈ ln 1 + x(1 + 1.4x) The Lotz formula (6.99) was derived on the basis of numerical calculations for one-electron ions in the CBE and, therefore, produces errors for many-electron systems. However, the Lotz formulae are convenient and useful for estimating the ionization cross sections and rates with an accuracy of a factor of about two to three. As for the theory of electron-impact ionization, several basic approximations are now used for calculating single-ionization cross sections and rate coefﬁcients. 16 Wannier G H 1953 Phys. Rev. 90 817.

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Figure 6.6. Classically scaled (6.90) ionization cross sections of H-like ions as a function of the scaled electron energy; Iz is the ionization potential: symbols, experimental data obtained at EBIS (Electron Beam Ion Source) for ions from C5+ to Ar17+ ; full curve, CBE (Coulomb–Born approximation with exchange) results for Z = 128. From Donets E D and c Ovsyannikov V P 1981 JETP 53 466. 1981 MAIK (Nauka Moscow).

Each approximation has a name for the way in which the wavefunction of the ejected electron in the continuum is obtained. Among them, the Born plane-wave approximation (BPWA), the distorted-wave approximation (DWA), the Coulomb– Born approximation with exchange (CBE), K- and R-matrix approaches, and the close-coupling method (CC) should be mentioned. Theoretical calculations can provide a typical accuracy for the total ionization cross sections within (15–20)% with the exception of some particular cases where an accuracy of a few percent can be achieved. The total experimental accuracy for ionization cross sections has reached the level of around 5–10%. 6.7.3 Excitation–autoionization and the branching ratio coefﬁcients For atomic targets with three or more electrons (Li-, Be-like ions, etc), excitation of the inner-shell electrons to autoionizing states can signiﬁcantly contribute to the ionization process. This excitation is followed by autoionization decay, i.e. it leads to the production of an additional electron, and is termed excitation–

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autoionization (EA). For example, in collisions of Li-like ions with electrons, excitation and autoionization can occur via excitation of the 1s-electron into states with principal quantum numbers n ≥ 2: Xq+ (1s2 2s) + e− → [Xq+ (1s2snl)]∗∗ + e− → X(q+1)+ (1s2 ) + 2e− .

(6.103)

Therefore, similar to direct ionization (6.88), after the decay of a doubly excited state [Xq+ ]∗∗ , one has an additional electron in the ﬁnal channel. In this case, the total cross section can be presented as a sum of the Direct Ionization (DI) and Excitation–Autoionization (EA) cross sections:

( j) B aj σex (6.104) σtot = σDI + σEA = σDI + B aj =

Aaj m

m≤ j

j

Arj m +

m≤ j

Aaj l

−1

.

(6.105)

l≤ j

Here the quantity B aj is called the branching ratio coefﬁcient of the autoionizing state j of the target; Aa and Ar are the autoionization and radiative probabilities of the decay of the j th state into lower states, respectively, and σex is the electronimpact excitation cross section of the inner-shell electron into an autoionizing state. The branching ratio coefﬁcient (6.105) reveals the fractional decay probability via the autoionization channel, i.e. the probability of creating an additional electron. But stabilization of the j th state can also occur via the radiation channel followed by emission of a photon, and in our case of a Li-like ion, one obtains: [Xq+ (1s2s2p)]∗∗ → Xq+ (1s2 2s) + ~ω.

(6.106)

The branching ratio coefﬁcient for the radiation decay channel is given by the sum (cf (6.105)): B rj =

m≤ j

Arj m

Arj m +

m≤ j

Aaj l

−1

.

(6.107)

l≤ j

The autoionization probability Aa is of the order of 1013 –1014 s−1 and is weakly dependent on the ion charge q. The radiative probability Ar increases with ion charge approximately as Ar ∝ q 4 , therefore, Ar Aa for ions with a charge q ≤ 10, and for low-charged ions, one can roughly put B aj ≈ 1 so equation (6.104) becomes simpler:

( j) σex . (6.108) σ = σDI + j

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Figure 6.7. Electron-impact ionization cross sections of U16+ : full circles, crossed-beams measurements; broken curve, DWA calculation for direct ionization only, and the solid curve, same including a contribution from EA processes. Gregory D C et al 1990 Phys. c Rev. A 41 106. 1990 APS.

The EA processes are of particular importance for atoms and ions with a few outer electrons outside closed inner shells such as p6 , d10 or f14 because in these systems the excitation cross section of inner-shell electrons is proportional to the number of equivalent electrons in the shell. The strong contribution of indirect processes to electron-impact ionization is demonstrated in ﬁgure 6.7 for electronimpact ionization of U16+ ions. It can be seen that in the whole electron energy range considered, the direct ionization cross section is three to ﬁve times smaller than the experimental data and calculations taking account of EA processes.

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6.7.4 Resonant ionization via the capture of free electrons Besides EA, there are other important indirect processes leading to the net ionization, the so-called resonant ionization. This has a resonant character and takes place only at deﬁnite, resonance energies of the incident electron similar to the resonant excitation considered in section 6.6.2. Resonant ionization is a multi-step process leading to the appearance of narrow resonances on top of the direct ionization cross section of outermost and inner electrons. Again, the ﬁrst step in resonant ionization is the capture of a free electron by the target ion and the creation of an ion in a doubly excited autoionizing state. An excited ion then decays by the sequential or simultaneous emission of two electrons resulting in a net single ionization of the target. For example, for Li-like ions, one has Xq+ (1s2 2s) + e− → [X(q−1)+(1s2s2p3)]∗∗ → X(q+1)+ (1s2 ) + 2e−

(6.109)

Xq+ (1s2 2s) + e− → [X(q−1)+(1s2s3p3)]∗∗ → [Xq+ (1s2s2 )]∗ + e− → X(q+1)+ (1s2 ) + 2e− .

(6.110)

Similar reactions can be written for excitation of 1s electron into other states. The process (6.109) is termed Resonant-Excitation-Auto-Double Ionization (READI) and the process (6.110) is called Resonant-Excitation-Double Autoionization (REDA)17. The contribution of resonant ionization depends on ion charge and the electronic conﬁguration of the target. The largest contribution from REDA has been observed in the ionization of Fe15+ ions (ﬁgure 6.8) where REDA contributes up to 30% of the total ionization cross section. Finally, a general scheme for single ionization by electron impact is displayed in ﬁgure 6.9 showing the relative contribution of all the processes considered here. The total cross section contains a structure on top of the direct-ionization cross section caused by excitation-autoionization and resonant ionization. 6.7.5 Relativistic and QED effects With increasing ion charge, relativistic and quantum electrodynamic effects in electron-impact ionization such as electrostatic, magnetic and retardation become stronger and the laws valid for low- and middle-Z ions are violated. In calculations, these effects are included in the interaction by using the M¨oller interaction: 2 VM = (1 − α1 α2 )eiωr12 (6.111) r12 and in the wavefunctions by using the relativistic Dirac equation (section 5.8). Here r12 is the inter-electron separation, α1 and α2 are the Dirac matrices and ω is the angular frequency of the exchanged photon. The M¨oller operator describes 17 M¨uller A 1991 Physics of ion Impact Phenomena ed D Mathur (Berlin: Springer).

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Figure 6.8. Electron-impact ionization cross sections of Fe15+ : full circles, full and dotted lines, results of sophisticated calculations. Direct ionization cross sections were calculated by the Lotz formula (6.99). From M¨uller A 1999 Atomic Physics with Heavy c Ions ed H F Beyer and V P Shevelko (Berlin: Springer) p 272. 1999 Springer.

the electron–electron interaction and includes Coulomb, magnetic and retardation effects. If retardation and other relativistic effects can be neglected (ω → 0), it reduces to the pure Coulomb interaction. The results of ionization from the ground 1s and excited 2s states in highly charged H-like ions are shown in ﬁgure 6.10. It can be seen that for ions with ion charge q > 40–50, the ionization cross sections do not follow the classical Thomson scaling (6.90), i.e. σ I 2 against the scaled energy E/I , where I denotes the binding energy of the 1s or 2s states. There are only a few measurements of electron-impact ionization cross sections for very highly charged ions. In the case of H-like ions, they have been measured for ions ranging from Mo (nuclear charge Z = 42) up to U (Z = 92) using an extension of the ionization-balance technique18. These measurements are in quite good agreement with theoretical calculations including the M¨oller interaction and are substantially larger than those calculated with the pure Coulomb interaction. 18 Marrs R E et al 1994 Phys. Rev. Lett. 72 4082.

Marrs R E et al 1997 Phys. Rev. A 56 1338.

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Figure 6.9. General picture of the electron-impact ionization cross section showing contributions from direct ionization, excitation-autoionization, and resonant REDA and READI processes. From Defrance P 1995 Atomic and Molecular Processes in Fusion c Edge Plasmas ed R K Janev (New York: Plenum) p 153. 1995 Kluwer/Plenum.

The L-shell ionization cross sections have been measured for the several highly charged uranium ions ranging from U83+ (F-like) to U89+ (Li-like). Highly charged uranium ions were produced and trapped in the high-energy electronbeam ion trap at the Lawrence Livermore National Laboratory. Experimental ionization cross sections for Li- and Be-like uranium ions are displayed in ﬁgure 6.11 in comparison with calculations performed with and without the M¨oller interaction. Again, including the M¨oller interaction makes agreement with experiment much better than in the case with only the Coulomb interaction included.

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Figure 6.10. Cross section times the square of the binding energy for ionization of H-like ions from the 1s ground (left) and 2s excited (right) states against the scaled energy E/I . NR gives the non-relativistic results. The relativistic cross sections are labeled by the ion c charge. From Moores D L and Pindzola M S 1990 Phys. Rev. A 41 3603. 1990 APS.

6.7.6 Inverse process: three-body recombination An inverse process to electron-impact ionization (6.71) is three-body or ternary recombination − (q−1)+ Xq+ (γ0 ) + e− (α0 ) + e− 1 (E 1 ) + e2 (E 2 ) → X 1 (E)

(6.112)

where E 1 , E denote the electron energies of the scattered electron before and after the collision and γ0 and α0 sets of quantum numbers. The energy conservation law is the same as for the ionization process (6.89), i.e. E1 + E2 = E − I

I >0

(6.113)

where I denotes the ionization potential of the X(q−1)+ ion. The presence of the third particle (an electron) is necessary to fulﬁll the energy conservation law unless a photon is emitted serving the same purpose. So in capturing one electron (e2 ), the excess energy is deposited on to a neighboring electron (e1 ). Three-body

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Figure 6.11. Electron-impact ionization cross sections for Li- and Be-like uranium ions: full circles, experiment; broken curves, results with the Coulomb interaction only full curves, including the M¨oller interaction. From St¨ohlker Th et al 1997 Phys. Rev. A 56 c 2819. 1997 APS.

recombination is quite poorly investigated both experimentally and theoretically although it constitutes a very important charge-changing reaction at low electron temperatures and high plasma densities (see later). According to the principle of detailed balance (section 6.4), ionization σi and recombination σr cross sections are related by 2π 2 · k 2 · gq ·

dσi (α0 − γ0 , E 1 ) = E 1 · k22 · gq−1 · σr (γ0 , E 1 − α0 ) dE 1 k 2 /2m e = E k22 /2m e = E 2 (6.114)

where g denotes the statistical weight of the ions Xq+ and X(q−1)+ . We note that the differential ionization cross section dσi /dE 1 has a dimension of cm2 /eV, the total ionization cross section σi of cm2 and the recombination cross section σr

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has a dimension of cm4 s. The factor of 2π 2 in equation (6.114) is related to the normalization of the electron continuum wavefunctions to unity. The number of recombination events per second is given by Wr = n e κr [s−1 ] κr = n e σr (E)v1 v2 F(E 1 )F(E 2 ) dE 1 dE 2 [cm3 s−1 ]

(6.115) (6.116)

where n e denotes the electron density, κr the recombination rate coefﬁcient19 and F(E) is the electron-energy distribution function. We note that the number of recombination events due to three-body recombination is proportional to the square of the electron density, Wr ∝ n 2e , in contrast to the excitation and ionization processes for which W ∝ n e . This very important n 2e dependence makes three-body recombination the main recombination process in high-density plasmas. In a plasma with moderate and low electron densities, two-body recombination processes such as dielectronic recombination (section 6.8) and radiative recombination (section 6.8.5) prevail because their rates are proportional to the ﬁrst power on electron density n e . Now we ﬁnd the relation between ionization and three-body recombination rates. The density of recombining ions per second can be expressed as dn q /dt = −κr n q n e

(6.117)

with n q and n e being the ion and electron densities, respectively. Under equilibrium conditions, the rates of recombination and ionization have to be equal in accordance with the detailed balance principle: κr n q n e = σi vn q−1 n e

(6.118)

where σi v is given by (6.95). Using the Saha equation (6.17), one obtains, for the ratio of n q and n q−1 , Iq−1 n q−1 m e kB Te 3/2 −1 gq−1 (6.119) =2 n exp − e nq 2π ~2 gq kB Te where Iq−1 is the binding energy of the captured electron and kB is the Boltzmann constant. Finally, one arrives at the detailed balance equation for the rates in question: 3/2 n q−1 Iq−1 gq−1 2π ~2 κr = . (6.120) σi v = n e σi v exp − nq m e kB Te 2gq kB Te Therefore, knowing the ionization rate coefﬁcient σi v and its dependence on the electron temperature Te from theory or experiment, one can estimate the threebody recombination rate. For example, the use of the semi-empirical Lotz formula 19 Sometimes, the recombination rate coefﬁcient is deﬁned by equation (6.116) without the n factor, e but then κr has the dimension of cm6 s−1 .

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(6.99) or the classical Thomson formula (6.90) for the ionization cross section in (6.120) leads to identical results: κr ≈ n e

16π 2 e4 ~3 gq−1 N 2 2gq Iq−1 (kB Te )2 5m e

(6.121)

where N is the number of equivalent electrons in the shell by which the electron is captured. For three-body recombination of a bare nucleus to a ﬁnal state with principal quantum number n, one has: gq−1 = 2n 2

gq = 1

N =1

Iq−1 =

Z 2 Ry n2

(6.122)

where Z is the nuclear charge of the resulting H-like ion. Thus, for a speciﬁc nstate, the rate coefﬁcient for three-body recombination at low temperatures scales as20 n4 κr ∝ . (6.123) (Z Te )2 The total rate coefﬁcient for three-body recombination is then deﬁned by the sum over all possible n-states: κrtot =

n

max

κr (n)

n max =

n=1

Z 2 Ry . kB Te

(6.124)

The maximum principal quantum number n max is limited by the electron temperature Te because ions being in higher excited states n, such that In kB Te , will be re-ionized and will not therefore contribute to the net recombination. The total recombination rate is then equal to κr (Te ) = 5.8 × 10−27 cm3 s−1 Z 3 n e [cm−3 ](Te [eV])−9/2.

(6.125)

From (6.125) it follows that the three-body recombination process is important at low electron temperatures and high densities. The total rate coefﬁcient (6.125) obtained by the simple consideration shown here reveals the same dependence on density, temperature and nuclear charge as more accurate treatments do. Only the numerical factor of 5.8 in (6.125) is about twice as large. There are several collisional and radiative processes responsible for the stabilization of the ﬁnal charge state. The net decrease in the charge state can be caused by reactions which can be grouped into three categories. The ﬁrst is collisional recombination which is a three-body recombination populating the high n-states followed by collisionally induced transitions to more tightly bound states. Second, radiative recombination can effectively contribute to the 20 Bates D R et al 1962 Proc. R. Soc. London A 267 297.

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Figure 6.12. Total rate coefﬁcient for the recombination of U92+ ions with free electrons at electron density of n e = 108 cm−3 as a function of electron temperature. The individual contributions according to (6.126) are shown: dashes, the three-body recombination contribution κr ; dots, ædr involving radiative stabilization; dashes-dots, ærr due to radiative recombination; full curve, sum αtot of the previous three. From Beyer H F et al 1989 Part. c Accel. 24 163. 1989 Gordon and Breach.

population of low-lying bound states. The third mechanism is a three-body recombination followed by a stabilizing decay that is dominated by radiative transitions. The net effect of these processes is known as collisional–radiative recombination and the theory for it was ﬁrst laid out for hydrogenic plasmas by Bates and co-workers21. The total rate coefﬁcient for collisional–radiative recombination in a plasma with electrons and completely stripped ions can be represented in the form αtot = κr + κdr + κrr

(6.126)

where κr is the TBR rate given by (6.125), κrr is the radiative-recombination rate coefﬁcient (section 6.8.5) and κdr is the rate coefﬁcient for the radiative stabilization or dielectronic recombination to be considered in section 6.8.3. Three different contributions to the total rate coefﬁcient for collisional–radiative recombination of bare uranium ions as a function of the electron temperature Te at low electron density of n e = 108 cm−3 are shown in ﬁgure 6.12. 21 Bates D R 1975 Case Studies in Atomic Physics IV ed E W McDaniel and M R C McDowell (Amsterdam: North-Holland).

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6.8 Dielectronic recombination The dielectronic recombination process was ﬁrst considered theoretically in the early 1940s22 and the ﬁrst cross-section measurements were reported in 198323 employing crossed electron and ion beams. These early measurements suffered from low counting rates and large backgrounds because they were carried out mainly with ions in low charge states. The problems encountered in the past were due to the limited energy resolution in measurements involving intra-shell electron transitions and many closely spaced resonances which could not be resolved at that time. Now the importance of dielectronic recombination for the ionization balance in plasmas is well recognized, especially for the formation of excited states and for plasma cooling by radiation losses. The feasibility of producing highly charged ions and storing them in storage rings has opened a new area for detailed and precise studies of the recombination of ions and free electrons24. Besides its importance in the spectra of highly charged ions, dielectronic recombination has turned out to be a very sensitive tool for accurate atomic-structure studies. For example, a novel experimental method to extract the 2s1/2 –2p1/2 energy splitting in Li-like ions (Au75+, U89+ ) has been developed by measuring the dielectronicrecombination resonances associated with the capture of free electrons into highly excited (Rydberg) states with principal quantum numbers n > 20 where an extrapolation of the measured resonance energies up to n → ∞ leads to the sought value sensitive to the 2s-level Lamb shift. Another example is related to studying the isotope shifts in moderately short-lived radioactive isotopes: highprecision measurements of dielectronic recombination into lower states is used as an effective tool to determine the root-mean-square radius of the nucleus. 6.8.1 Classiﬁcation of the process Dielectronic Recombination (DR) is a resonant two-step process in which a free electron is captured by the incident ion Xq+ into an autoionizing state followed by a stabilizing radiation decay of the resulting ion in the second step: Xq+ + e− → [X(q−1)+]∗∗ → [X(q−1)+ ]∗ + ~ω.

(6.127)

The ﬁrst step is a double-electron process, often called dielectronic capture, through which one free electron is captured and another core electron is simultaneously excited forming a doubly excited state [X(q−1)+ ]∗∗ . This means that the DR process can occur if the incident ion Xq+ has one or more electrons, i.e. DR cannot occur with bare ions. In a second step, the ion in a doubly 22 Massey H S W and Bates D R 1942 Rep. Prog. Phys. 9 62. 23 Mitchell J B A et al 1983 Phys. Rev. Lett. 50 335.

Dittner P F et al 1983 Phys. Rev. Lett. 51 31. 24 See M¨uller A 1999 Atomic Physics with Heavy Ions ed H F Beyer and V P Shevelko (Heidelberg:

Springer) pp 272–90.

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Figure 6.13. Schematic representation of the dielectronic recombination process for a KLL resonance in a He-like ion.

excited state emits a photon and decays into a stable state below the ionization limit. Because two electrons are involved, the process is called dielectronic recombination. The capture process is possible provided the kinetic energy of the electron fulﬁlls the resonance condition (6.75) similar to the resonant excitation and ionization processes considered earlier but in a DR process, the energy of the free electron is less than the transition energy of the inner-shell electron. DR is also similar to radiative recombination: a free electron is captured in the Coulomb ﬁeld of the ion and a photon emitted, but in radiative recombination, a photon is radiated due to the transition of a captured electron from an upper state to a lower one, while in dielectronic recombination a photon is radiated due to the transition of a core electron. There is a competitive process for the decay of the doubly excited ion in equation (6.127)—the autoionization or Auger decay (section 5.6) associated not with the radiative transition but with a change in ion charge: [X(q−1)+ ]∗∗ → [Xq+ ]∗ + e− .

(6.128)

However, in highly charged ions, the decay via DR with photon radiation is a more probable process than Auger stabilization with emittance of a free electron. The DR process is schematically illustrated in ﬁgure 6.13 for a KLL resonance. A free electron is captured into the empty L shell of a two-electron ion and a K-electron is simultaneously excited into the same L shell so that the total energy remains the same. As a result, a Li-like ion is created in the KLL state which then decays radiatively. Alternatively the doubly excited KLL state may autoionize thus returning to its original charge state. The x-ray photons associated with the stabilizing decay of the doubly excited states compose the dielectronic satellites appearing in the vicinity of the corresponding parent lines.

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6.8.2 Dielectronic satellites In highly charged ions, rich radiation spectra of closely spaced x-ray lines originate from the radiative decay of autoionizing states which lie above the ionization limit and are created either by the capture of a free electron via dielectronic capture Xq+ + e− → [X(q−1)+ ]∗∗ (6.129) or by direct excitation of the inner-shell electron: Xq+ + e− → [Xq+ ]∗∗ + e− .

(6.130)

The lines arising from DR (6.127) are called dielectronic satellites. Let γ0 and γ1 denote the sets of quantum numbers characterizing an ion Xq+ in its initial and ﬁnal state, respectively. A dielectronic satellite to the transition γ0 → γ1 can occur if ionic states are simultaneously present with a spectator electron in the state n of the ion X(q−1)+ . Hence the satellite transition can be designated as γ0 n → γ1 n. For example, 1s2pn → 1s2 n transitions in Li-like ions are satellites to the resonance line 1s2p → 1s2 in He-like ions. The number of excited electrons in the autoionizing states can be more than two; for example, in Be-like ions the following autoionizing states are possible: 1s2s2 2p, 1s2s2p2, 1s2p3, . . . , 1snn n , . . . Gabriel’s notation for the basic spectral lines in H- and He-like ions and corresponding satellites were given in table 5.5. There is no special notation for ions from other isoelectronic sequences. In highly charged ions, the satellites are very close in wavelength to the parent line and their intensities increase approximately as q 4 so that some satellite lines have intensities comparable with the intensity of the resonance line. Therefore, one observes a large number of spectral lines of comparable intensity in a narrow spectral interval (see ﬁgures 5.13–5.15). At ﬁrst glance, it seems that the presence of a large number of spectral lines makes the identiﬁcation and analysis of the VUV and x-ray spectra quite difﬁcult. However, with high-resolution detection techniques (λ/λ ≈ 10−5 in the spectral ˚ the situation turns out to be most advantageous for x-ray interval λ ≈ 1–10 A), spectroscopy and x-ray plasma diagnostics. The point is that a limited spectral interval contains quantitative information not only about the ionic structure such as wavelengths, radiative and autoionization transition probabilities, rates of electron excitation and ionization—the micro parameters—but also about the plasma’s macro parameters such as electron and ion temperatures, densities, ionstate distributions and the presence of electron beams in plasmas. The absolute and relative intensities of the dielectronic satellites and their degree of polarization are very sensitive to the plasma’s macro parameters. This fact is the basis for x-ray diagnostics of high-temperature laboratory and astrophysical plasmas25 . 25 See, e.g., Gabriel A H 1971 Highlights of Astrophysics ed C de Jager (Dordrecht: Reidel). Presnyakov L P 1976 Sov. Phys.–Uspekhi 19 387.

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6.8.3 DR cross sections and rates The theory of DR, as a resonant process, is similar to that for resonant excitation considered in section 6.6.2 with the exception that in DR a photon is emitted in the ﬁnal channel instead of emission of an electron in the resonant excitation. Let us consider the DR process (6.127) for a particular resonance e− + |i → |d → | f + ~ω

(6.131)

involving the initial state |i of the target ion Xq+ , the state |d of the doubly excited ion [X(q−1)+ ]∗∗ and the ﬁnal state | f of the stabilized ion X(q−1)+ . The reaction is possible when the incident electron energy E satisﬁes the resonance condition (6.75): (6.132) E = E res = E d − E i within the natural width of the level |d, where E i and E d denote the total electronic binding energies of the initial and intermediate state, respectively. In the isolated resonance approximation, the DR cross section can be expressed as 2π 2 ~3 Wa (i → d)φ(E)ωd (6.133) σdr (E) = p2 where p denotes the electron momentum, p2 /2m e = E. The quantity Wa (i → d) represents the total probability per second for the dielectronic capture |i → |d and φ(E) is the Breit–Wigner proﬁle of the resonance whereas ωd denotes the ﬂuorescence yield of the doubly excited state. The dielectronic capture probability Wa (i → d) is related to the reverse autoionization probability Aa (d → i ) via the principle of detailed balance as (cf equation (6.79)): Wa (i → d) =

gd Aa (d → i ) 2gi

(6.134)

where gi and gd are the statistical weights. The function φ(E) is deﬁned as d /2π φ(E) dE = 1 (6.135) φ(E) = (E − E res )2 + d2 /4 and for the ﬂuorescence yield one has ωd =

r r = a + r d

(6.136)

using the radiative r and Auger a widths and the total width d of the doubly excited state |d. Combining equations (6.133)–(6.136), we obtain the following expression for the DR cross section: σdr (E) =

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π ~2 gd a (d → i )r (d → f ) . p2 2gi (E − E res )2 + d2 /4

(6.137)

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The natural width d is usually small compared to experimental linewidths. In this case one can use a delta function for the line proﬁle resulting in the following cross section: σdr (E) =

2π 2 ~2 gd a (d → i )r (d → f ) δ(E − E res ). p2 2gi d

(6.138)

This suggests the deﬁnition of a resonance strength Sd as the integrated cross section Sd ≡

σdr (E) dE =

2π 2 ~2 gd a (d → i )r (d → f ) d . d p2 2gi

(6.139)

The DR rate coefﬁcient, which is sometimes a more convenient quantity, is obtained by integrating over the velocity distribution of the electrons: κdr = σdr ve .

(6.140)

For plasma diagnostics, a Maxwell distribution is used for integrating (6.137) yielding κdr (Te ) =

2π ~2 m e kB Te

3/2

gd a r exp(−E res /kB Te ) 2gi d

(6.141)

where Te is the electron temperature. The DR rate coefﬁcient has the following approximate behavior:  −1/T e e (max) κdr ∼ κdr  −3/2 Te

Te → 0 (max) Te ≈ (E d − E i )/(1 + 0.02Z ) Te → ∞

(6.142)

where Z denotes the nuclear charge, E d − E i the energy difference deﬁned in (6.132). Within the theoretical framework outlined here, the calculation of DR cross sections and rate coefﬁcients entails evaluating the rates for autoionization and radiative decay discussed in chapter 5. In addition, electronic binding energies, determined with a high degree of accuracy, are required for the prediction of the location of the resonances. Depending on the speciﬁc electronic conﬁguration of a given ion, a tremendous computational effort can be involved especially if the number of states within a resonance feature is large26. 26 See M¨uller A 1999 Atomic Physics with Heavy Ions ed H F Beyer and V P Shevelko (Berlin: Springer) p 272. M¨uller A 1999 Phil. Trans. R. Soc. London A 357 1279.

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6.8.4 Dielectronic recombination experiments The main achievements in experimental techniques for ion sources and storage rings have been obtained on energy resolution, signal rate, background and high charge states. This is also reﬂected in the increasing number of high-resolution experiments at various experimental facilities. A large amount of data has been obtained in merged-beam experiments performed at storage rings with an ionic charge state ranging from 7+ up to 90+. At storage rings, the experiments make use of the gain in signal by repeated circulation of the ions and the electron cooler used as a cold electron target. At an electron cooler, DR resonances are measured monitoring the signal of those circulating ions which have changed their charge state. Figures 6.14 and 6.15 show examples for DR measurements using storage rings. The width of the observed lines is limited by the availability of intense cold electron targets. The study of very heavy few-electron ions is of particular interest. With DR measurements of such systems relativistic effects on the transition probabilities and the corresponding wavefunctions have been investigated. At the ESR storage ring at GSI Darmstadt, DR involving n = 0 transitions on Li-like Au76+ has been measured using the electron cooler as an electron target. Figure 6.14 shows the measured rate coefﬁcient for an energy range covering the processes Au76+ (1s2 2s) + e− → [Au75+ (1s2 2p1/2,3/2n j )]∗∗ → Au75+ + ~ω. The spectrum is dominated by the 1s2 2p3/2 6 j resonances. Very good agreement with a fully relativistic calculation is obtained including the DR processes plus the smooth background from radiative recombination. Figure 6.15 demonstrates the high resolution obtained with an adiabatically expanded electron beam. The n = 0 resonances were obtained with Na-like Fe ions stored and cooled at the Test Storage Ring (TSR) in Heidelberg. The data could be well ﬁtted with a theoretical calculation including the convolution with a very narrow velocity distribution corresponding to transverse and longitudinal electron-beam temperatures of Te⊥ = 15 meV and Te% = 0.15 meV, respectively. 6.8.5 Radiative recombination Radiative recombination (RR) is a non-resonant one-step process: X(q+1)+ + e− (E) → Xq+ (n) + ~ω

(6.143)

in which a free electron is captured into a bound state with principal quantum number n of the resulting ion Xq+ followed by the simultaneous emission of a photon ~ω. In an RR process, energy and momentum conservation is balanced by the emission of a photon. RR plays a key role in the determination of the ionization-recombination balance of highly charged ions in both electron–ion-beam experiments and in

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Figure 6.14. Rate coefﬁcient for the dielectronic recombination of Li-like Au76+ (1s2 2s) measured at the ESR of GSI. The full line represents a calculation taking into account n = 0 dielectronic resonances plus a smooth background from radiative recombination. c From Spies W et al 1992 Nucl. Instrum. Methods B 98 158. 1992 Elsevier.

high-temperature laboratory and astrophysical plasmas. The study of these processes also includes understanding the fundamental processes in reactions of free electrons with ions, the determination of the corresponding cross sections and rates in plasma diagnostics. Moreover, x-rays from RR of fully stripped ions and subsequent x-ray transitions originating from excitation cascades provide new possibilities for studying QED effects in heavy highly charged ions27 . Here, we recall some important relations and formulae involving RR cross 27 Liesen D (ed) 1994 Physics with Multiply Charged Ions (NATO ASI Series) vol B 348 (New York: Plenum).

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Figure 6.15. High-precision DR measurement for Fe15+ ions with an expanded electron beam at the TSR in Heidelberg. The lines correspond to n = 0 dielectronic transitions in Fe15+ . The dotted curve represents a theoretical calculation convoluted with a velocity distribution given by a transverse temperature of 15 meV and a longitudinal temperature c of 0.15 meV. From Linkemann J et al 1995 Nucl. Instrum. Methods B 98 154. 1995 Elsevier.

sections and rate coefﬁcients. RR is the inverse process to photoionization or the photoeffect: Xq+ + ~ω → X(q+1)+ + e−

(6.144)

considered in many review articles and books28. Photoionization σν and photorecombination σrr cross sections are mutually related by the detailed balance principle through the Milne formula: gq+1 · σrr =

(~ω)2 · gq · σν 2m e c2 E

(6.145)

where m e denotes the electron mass, g the statistical weight and E is the photoelectron energy. Both photoionization and photorecombination cross sections are expressed in terms of the transition dipole matrix elements similar to the radiative transition probabilities and oscillator strengths considered in section 5.4. 28 See, e.g., Bethe H A and Salpeter E E 1977 Quantum Mechanics of One- and Two-Electron Atoms (New York: Plenum). Sampson D H 1982 Atomic Photoionization (Berlin: Springer). Amusia M Ya 1990 Atomic Photoeffect (New York: Plenum).

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Dielectronic recombination The number of photoionization events per second is given by ∞ W = cn ω σν (ω) dω [s−1 ]

303

(6.146)

0

where n ω denotes the density of photons at a given angular frequency ω. The RR rate coefﬁcient κrr , averaged over a Maxwellian energy distribution of the incident electrons with a temperature Te , is given by ∞ σrr (u) −βu κrr = σrr v = K (I0 /Ry)1/2 β 3/2 u e du u = E/I0 (6.147) πa02 0 √ 2 π ~a0 −1 β = I0 /kB Te K = = 2.18 × 10−8 cm3 s (6.148) me where I0 is the binding energy of the resulting ion Xq+ and 1Ry = 13.606 eV. Photoionization cross sections and photorecombination rate coefﬁcients scale as σν (u)

u = ω/Z

Z eff = 1 (neutral atoms)

(6.149)

2 σν (u) Z eff

2 u = ω/Z eff

Z eff > 1 (positive ions)

(6.150)

−1 Z eff κrr (&)

2 & = Te /(Z eff Ry)

(6.151)

where ω denotes the photon angular frequency, Z eff = Z − N + 1 the effective ion charge. For colliding beams of electrons and ions, the recombination rate coefﬁcient (6.147) is usually presented by Kramers’ formula: σrr (n) = 2.10 × 10−22 cm2

I02 n E cm (I0 + n 2 E cm )

n1

(6.152)

where E cm is the energy in the electron–ion center-of-mass frame. The total RR cross-section is given by the sum over all possible Rydberg states which can contribute to the process σrr(tot)(E cm ) =

n cut

σrr (n, E cm ).

(6.153)

n=1

Here n cut is the limiting principal quantum number up to which recombination can be observed and is deﬁned from such experimental conditions as the external electric and magnetic ﬁelds. 6.8.6 Radiative recombination experiments Ion-accelerator facilities and storage-ring technology have opened a new era of electron–ion collision studies. Using heavy-ion storage rings supplied with electron coolers, very high energy resolution in electron–ion recombination rate

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and cross-section measurements can now be achieved with energy spreads of up to about 10 meV at low center-of-mass energies29. It has become possible30 to determine the characteristic energy spread of the electron beam by an independent method and to measure the RR cross sections or, more precisely, the RR rate coefﬁcients σ v resulting from the electron-velocity distribution function F(v, Te ) which is the Maxwell two-temperature distribution function introduced by equation (6.15). The transverse and longitudinal electron temperatures Te⊥ and Te% differ markedly. In modern electron coolers, typical temperatures are Te⊥ ≈ 0.1 eV and Te% ≈ 0.001 eV, respectively, and the two-temperature distribution (6.15) corresponds to the ﬂattened distribution. Experimental temperatures Te⊥ ≈ 0.02 eV and and Te% ≈ 0.0001 eV are as low as have been reported. Photorecombination processes involving highly charged ions were extensively investigated at the electron cooler in the Heidelberg test storage ring (TSR) with stored Li-like Si11+ , Cl14+ ions as well as Na-like Cl6+ , Fe15+ and Se23+ ions. Absolute cross sections and radiative (and dielectronic) recombination rates have been measured for center-of-mass energies E cm ranging from 0 eV up to several keV. In separate experiments at the GSI (Darmstadt) accelerator facilities, absolute rates for recombination of highly charged ions such as Au76+ and U28+ were determined with an energy spread as low as 10 meV. The RR rates for Ne10+ and Ar13+ have been measured at very low electron energies (<1 meV) using the ion storage ring CRYRING at the Manne Siegbahn Laboratory in Stockholm. The most striking feature of the photon-energy spectrum for RR with a bare ion into the ground state is the existence of a sharp edge corresponding exactly to the electron binding energy I1s . This makes the photons from RR very attractive for accurate x-ray spectroscopy. However, we note that before going to establish accurate spectroscopic methods one has to be able simply to observe the photons in the experiment. After 10 years, the radiative recombination with a bare, very heavy ions is the ﬁrst and still the only one process where the short-wavelength photons are measured. This method has been used for an accurate measurements of the 1s1/2 Lamb shift in highly charged, hydrogen like U91+ using free cooling electrons and a dense ion beam31 . The early measurements of the RR rate coefﬁcients carried out using the merged-beam technique showed quite good agreement between the experimental results and the theoretical calculations performed using the Kramers formula (6.152) with the Stobbe correction for the Coulomb wavefunction of a free electron32. The typical behavior of the RR rate is shown in ﬁgure 6.16 for RR of H-like F8+ , O7+ and C5+ ions. However, in later storage-ring experiments, when it became possible to reach 29 See Schippers S 1999 Phys. Scr. T 80 158. 30 See, e.g., M¨uller A 1999 Phil. Trans. R. Soc. London A 357 1279. 31 See St¨ohlker Th et al 2000 Phys. Rev. Lett. 85 3109. 32 Stobbe M 1930 Ann. Phys. 7 661.

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Figure 6.16. RR rates for H-like ions as a function of the relative electron–ion energy: circles ﬁlled, experimental results for F8+ ; triangles, experimental results for O7+ ; open circles, experimental results for C5+ ; full curves, Kramers formula (6.152) with Stobbe correction convoluted with the experimental velocity function (6.15). From Andersen L H c et al 1990 Phys. Rev. Lett. 64 729. 1990 APS.

very low relative electron–ion energies (up to 10−7 eV), such good agreement was only observed for low-Z ions. With the ion charge increasing, a signiﬁcant disagreement between experimental data and theoretical predictions was found in many cases including H-like ions and non-hydrogenic many-electron systems. In some cases of non-bare heavy ions, deviations from RR theory by even one to two orders of magnitude have been found in a low energy range. For example, in the case of RR of U28+ ions, the observed crosssection is a factor 20 to 50 higher than the RR expectation. It has been shown that very low-lying dielectronic resonances are the main contributors to this enhancement but even for bare ions, an RR rate coefﬁcient enhanced by a factor of four to eight was measured near zero relative energy (ﬁgure 6.17). A convincing explanation of these phenomena does not yet exist (see next section).

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Figure 6.17. Experimental recombination rate coefﬁcient of bare Ar18+ ions as a function of the electron–ion center-of-mass energy. The full curve represents the RR rate calculated with the classical Kramers formula and convoluted with the electron distribution (T⊥ = 0.2 eV, T% = 2 meV). From Uwira O et al 1997 Hyperﬁne Interact. 108 167. c 1997 Kluwer.

6.8.7 Radiative recombination at very low electron energies A large number of measurements of RR rates of very slow electrons, with relative energies 10−7 –10−2 eV, colliding with heavy ions in storage rings or in interacting merged beams show quite a large discrepancy, up to orders of a magnitude, with the calculated rates. Enhancement of the experimental rates has been found in many measurements of H-like ions with charge Z = 1, 2, 6, 7, 10, 14, 16, 18, 83 and some many-electron ions such as Au28+ , Pb53+ and others. A typical comparison of experimental and theoretical data at low electron energies is displayed in ﬁgure 6.17. As a possible explanation of this enhancement, the

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Table 6.1. The low-energy experimental (exp), theoretical (th) and excess κ RR rates for H-like ions (in units of 10−12 cm3 s−1 ) at energies E < E 0 . Z

κexp

κth

1 2 6 7 10 14 17 18 83

6.0 50 900 1 200 — 7 000 6 000 8 000 100 000

5.4 30 300 600 — 2 000 2 000 1 000 30 000

κ

E 0 [eV]

0.6 20 600 600 1 500 4 500 4 000 7 000 70 000

3 × 10−4 3 × 10−3 5 × 10−3 5 × 10−3 5 × 10−4 1 × 10−2 1 × 10−2 5 × 10−2 1 × 10−2

inﬂuence of the external magnetic ﬁeld on the RR process has been proposed: the magnetic ﬁeld conﬁnes the electrons and limits their transverse motion which might result in an apparently lower transverse electron temperature in an RR process. However, further study is needed before any ﬁnal conclusions can be made. In order to reveal the main features of the enhanced rates, the calculated RR rates κth are subtracted from the experimental values κexp and the ‘excess’ rate coefﬁcients κ deﬁned as κ = κexp − κth . (6.154) The values of the excess rates are constant at very low relative electron–ion energies limit and strongly increase with increasing nuclear charge Z : κ ∼ Z 2.8 , whereas the theoretical value of κth increases as: κth ∼ Z 2.2 . The data for absolute and excess RR rates of H-like ions are summarized in table 6.1. We note that the experimental data on RR rates at low energies have been obtained in different laboratories with different longitudinal and transverse electron temperatures so it is difﬁcult to pick out the inﬂuence of the conditions in the different electron coolers. Rather accurate experiments33 have been carried out to measure the RR rates of bare C6+ and Li-like F6+ ions showed the inﬂuence of the electron density n e and magnetic ﬁeld and carefully investigated for the ﬁrst time. It was found that the excess RR rates are not sensitive to the electron density n e but strongly depends on the magnetic ﬁeld strength B and electron temperatures T% and T⊥ : −1/2

κ ∝ T%

−1/2

T⊥

B 1/2 .

(6.155)

The observed dependence (6.155) could be the key to explain the enhancement 33 Gwinner G et al 2000 Phys. Rev. Lett. 84 4822.

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effect due to an increased density of slow electrons near the ion during their passage in the combined Coulomb ion ﬁeld and external magnetic ﬁeld. However, this problem still remains unresolved.

6.9 Ion–ion collisions 6.9.1 General remarks Ion–ion collisions, i.e. collisions between positive and negative ions, belong to the fundamental elementary processes occurring in many types of astrophysical and laboratory plasmas. The cross sections of ion–ion processes can be up to 10−15 cm2 and more, therefore, ion–ion collisions should be considered in plasma modeling, plasma diagnostics and problems of controlled thermonuclear fusion along with ion–atom and electron-impact processes. Following elementary ion–ion processes have been investigated in detail: excitation: (6.156) Xq1 + + Yq2 + → Xq1 + + [Yq2 + ]∗ electron capture: Xq1 + + Yq2 + → X(q1 −1)+ + Y(q2 +1)+

(6.157)

Xq1 + + Yq2 + → Xq1 + + Y(q2 +1)+ + e− .

(6.158)

and ionization:

Investigations of ion–ion collisions are of considerable interest and importance for several reasons. First, the diagnostics and modeling of nonequilibrium plasmas requires detailed knowledge of all the relevant atomic processes of which ion–ion collisions constitute a fundamental group. While dense gases or plasmas obey the laws of equilibrium thermodynamics and their properties can be discussed without recourse to atomic physics, most laboratory plasmas, as well as stellar and planetary atmospheres, are far from the local thermodynamic equilibrium considered in section 6.2. Second, thermonuclear fusion research requires data for ion–ion charge-changing collisions and, ﬁrst of all, electron capture and ionization processes because they are mainly responsible for the energy and charge losses in accelerated ion beams. In general, the main subjects to study ion–ion collisions can be brieﬂy formulated as follows: (1) Coulomb interaction between colliding particles, i.e. repulsion in collisions of ions with the same charge sign or attraction in collisions of negative ions with positive ones; (2) interaction of ion beams with plasma particles; (3) charge-changing processes, mainly electron capture and loss in ionaccelerator physics and thermonuclear research;

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(4) plasma neutralizers for injection of neutral H beams into plasma; and (5) intra-beam scattering inside an ion beam where ions interact with each other via charge-changing processes and therefore inﬂuence the energy and charge losses in the beam. The basic mechanisms arising in ion–ion collisions remain the same as in ion–atom collisions, however, several new fundamental aspects appear in the dynamics of ion–ion collisions. First, ion–atom collisions are usually described by the straight-line trajectories because of absence of a Coulomb interaction between a charged projectile and a neutral target atom. When two ions collide, the process is affected by long-range Coulomb repulsive or attractive forces at an internuclear distance R: q1 q2 e 2 Uc (R) = ± (6.159) R which are absent in ion–atom collisions and the relative trajectory of two colliding ions is described by a hyperbola. Another important difference is that the Coulomb interaction strongly affects the regions of potential-energy curve crossings as well as the interaction energies. However, these main differences are mainly important at low colliding relative velocities v < 1 a.u. ≈ 2.2 × 108 cm s−1 . Theoretical and experimental data on ion–ion collisions are limited because this domain is relatively young in atomic and accelerator physics. Only since 1977, experimental data on the effective cross sections for charge-exchange and ionization processes, involving singly and doubly charged positive and negative ions by the crossed-beams technique, become available34. In recent years, the impetus for accurate data required to describe such collisions including probabilities, effective cross sections and rate coefﬁcients has become even greater due to research in thermonuclear fusion using either magnetic or inertial conﬁnement35. A theoretical description of ionization, electron capture, excitation and other processes in heavy-ion collisions at non-relativistic and relativistic velocities can be found in the books36. In particular, ion–ion collisions in one- and twoelectron systems are well described by modern theoretical approaches, at least as far as total cross sections are concerned. For multi-electron systems, several experimental data sets are available but the theoretical descriptions of processes such as electron capture or ionization are still rare. Several processes in collisions between various negative ions have been investigated experimentally37. 34 Rinn K et al 1985 J. Phys. B 18.

Rinn K 1986 J. Phys. B 19 3717. 35 Lindl J D 1998 Inertial Conﬁnement Fusion (New York: AIP). 36 Janev R K, Presnyakov L P and Shevelko V P 1985 Physics of Highly Charged Ions (Berlin:

Springer). Eichler J and Meyerhof W E 1995 Relativistic Atomic Collisions (San Diego, CA: Academic). 37 See Melchert F 1999 Atomic Physics with Heavy Ions ed H F Beyer and V P Shevelko (Berlin: Springer) p 323.

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6.9.2 Experiments Compared to ion–atom collisions, experimental investigations of ion–ion collisions are quite scarce since the ion beams are very sensitive to the mutual repulsion within both the projectile and the target ion beams. Nevertheless, modern crossed-beam experiments with the use of electron-cyclotron resonance (ECR) ion sources (see chapter 4) overcome these difﬁculties and give detailed information about ion–ion interactions in a wide range of collision energies. Experimentally, each ion–ion collision event requires isolating ions with a given charge state. The important idea of investiging these collisions is to use the crossed-beams technique where two ion beams are made to intersect; this offers the experimental advantages of observing single collision events. In spite of the difﬁculties related to the use of a complex apparatus, the crossed-beam technique is widely used to measure the differential and total effective cross sections for ion–ion collisions. The colliding particles and the type of reaction should be properly deﬁned together with the range of accessible energies which spans seven or more orders of magnitude. If two ion beams cross at an interaction angle θ , the collision energy E cm in the center-of-mass frame (cm) is given by E cm = µ

E2 E1 + −2 M1 M2

E1 E2 cos θ M1 M2

(6.160)

where E i and Mi denote laboratory-frame energies and masses of beams 1 and 2, and µ the reduced mass µ = (M1 M2 )/(M1 + M2 ). The interaction angle θ determines the cm-energy range that is accessible for given laboratory-frame energies. Mutual Coulomb repulsion between the ions within their beams causes them to expand along their beam path. Under certain circumstances, a mutual deﬂection of both ion beams can also be important. To ensure experimental resolution, the beam diameter and divergence have to be limited which, as a consequence, limits the ion beam currents. For example, a beam of singly charged Ar+ ions with 5 keV energy, 5 mm diameter, and 1 deg maximal divergence cannot transport more than a few µA of ion beam current. This beam contains less than 107 ions cm−3 which is comparable to the particle density within an UltraHigh Vacuum (UHV) and this corresponds to a low density. Moreover, the ion beams travel through UHV along their complete trajectory, while they overlap only in the interaction region which has typically mm dimensions for a crossed-beam arrangement. This is why the ion–ion reaction signal is masked by the 102 –104 times more intense background contributions which arise from ionic collisions with the residual gas.

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6.9.3 Excitation Let us consider a relatively simple ion–ion process—excitation: Xq1 + + Yq2 + → Xq1 + + [Yq2 + ]∗ where the particles before and after the collision remain the same in contrast to electron capture and ionization processes where there are particles of different charge state in the initial and ﬁnal channels of the reactions. If two ions with charge numberss q1 and q2 collide, their relative trajectory is described by a hyperbola deﬁned parametrically. In collisions of two ions with the same charge sign, one has a repulsive interaction Uc (R) ≡ +q1 q2 e2 /R with a hyperbolic trajectory a(% 2 − 1) R R = a(% · chξ + 1) % = [(1 + (ρ/a)2 ]1/2

−1 + % · cos ϕ =

t = (a/v)(% · shξ + ξ ) a = q1 q2 /(µv 2 ). (6.161)

Here µ denotes the reduced mass of the colliding ions, v their relative velocity, and the ξ parameter runs from −∞ to +∞. The x and y components of the R vector are given by x = a(% + chξ ) y = a % 2 − 1shξ. (6.162) R2 = x 2 + y 2 In the case of attractive particles interacting by the law Uc = −q1 q2 e2 /R, the trajectory is also a hyperbola but described by other equations: a(% 2 − 1) R R = a(% · chξ − 1)

1 + % · cos ϕ =

t = (a/v)(% · shξ − ξ )

with x and y components of the R vector now being given by R2 = x 2 + y 2 x = a(% − chξ ) y = a % 2 − 1shξ.

(6.163)

(6.164)

If the Coulomb interaction constant η tends to zero which means that relative velocity is very large, v → ∞, or one of the colliding particles is neutral, q = 0, i.e. q1 q2 e2 E →0 (6.165) η= µv 2 ~v then the hyperbolic trajectory of the relative particle motion transforms into a straight line, i.e.

R = ρ 2 + v2 t 2 . (6.166)

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Here E denotes a transition energy in the ion Y . Actually, if one deﬁnes a time variable dR dR

t (R) = = (6.167) 2 vr (R) 2 2 µ (E − Uc (R) − L /2µR ) where L = µρv is the angular momentum of the system, vr is the radial velocity and E = µv 2 /2 the kinetic energy. Then, if η → 0, the Coulomb interaction Uc (R) → 0 and the relation R2 − ρ 2 dR

(6.168) t (R) = = 2E v 2 /R 2 ) (1 − ρ µ reproduces the straight-line trajectory (6.166). Excitation of ions in ion–ion collisions was investigated mainly theoretically for proton impact on highly charged ions: H+ + Xq+ → H+ + [Xq+ ]∗ .

(6.169)

For other colliding partners experimental data are practically absent. Excitation processes in ion–ion collisions are especially important for transitions between close atomic levels with transition energy E kB T where T is the plasma temperature. These transitions, which are usually dipole or quadrupole ones, are very important for plasma diagnostic purposes and for astrophysical problems. For such transitions the contribution from electron– ion and ion–ion collisions can be comparable in size. The importance of ion– ion collisions is concluded from ﬁgure 6.18 where the rate coefﬁcients for the transition 2s–2p in N4+ induced by electron (e) and proton (p) impact: e− , H+ + N4+ (2s) → e− , H+ + [N4+ (2p)]∗

(6.170)

are shown. As can be seen, for plasma temperatures T < 102 eV, the main contribution is given by electron impact; at high temperatures T > 103 eV, protons are mainly responsible for this transition. At intermediate temperatures T ≈ 103 eV, both light and heavy particles can contribute to the excitation rate coefﬁcients. Usually, the complex interaction V (r, R) between two colliding ions (or ion and atom) is presented as expansion on multipoles Vκ (R) in the form: V (r, R) =

∞

κ=1

Vκ (R) ≡

∞

Cκ R κ+1

(6.171)

κ=1

where the coefﬁcients Cκ reproduce the matrix elements of the target-electron radial wavefunctions R(r ) deﬁned in section 3.7: ∞ R0 (r )R1 (r )r æ+2 dr (6.172) Cκ ∼ 0

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Figure 6.18. Excitation rate coefﬁcients for transition 2s–2p in N4+ by electrons (e) and protons (p) as a function of plasma temperature: full curves, Born approximation; triangles, experiment. From Shevelko V P and Vainshtein L A 1993 Atomic Physics for Hot Plasmas c (Bristol: IOP Publishing). 1993 IOP Publishing.

so that for a dipole transition one has κ = 1, for a quadrupole, κ = 2, and so on. Reasonably accurate results are obtained with the use of a model interaction potential of the type VκM (R) = −Z P 0|r κ |1

Rκ (R 2 + R02 )κ+1/2

κ ≥ 1.

(6.173)

Here Z P is the projectile charge number and R0 is the effective, cut-off radius. The model potential VκM (R) coincides with the exact potential at small and large inter-nuclear distances R, i.e. ' κ R R→0 (6.174) VκM (R) ∝ R −κ−1 R → ∞. Usually, the value of R0 = n 0 n 1 /Z T (6.175) is adopted where n 0 and n 1 are the principal quantum numbers for the transition n 0 0 − n 1 1 in the target ion with the charge number Z T . For example, for the dipole transition n 0 0 − n 1 1 , = ±1, one has R C1 ∼ f 01 (6.176) V1 (R) = C1 2 2 3/2 (R + R0 ) where f01 denotes the optical oscillator strength (5.42). The typical behavior of the dipole (κ = 1) and quadrupole (κ = 2) model potentials (6.173) is shown in ﬁgure 6.19.

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Figure 6.19. Dipole (κ = 1) and quadrupole (κ = 2) potentials for transitions in H-like ions: full curves, exact potentials; broken curves, model potentials (6.173). From c Shevelko V P and Yukov E A 1985 Physica Scripta 31 265. Royal Academy of Sciences, Stockholm.

Using ﬁrst-order perturbation theory in the impact-parameter representation, the probability P of excitation in ion–ion collisions can be written as an integral over time t: +∞ 2 M −iE·t /~ Vκ (R)e dt (6.177) Wκ (ρ, v) = −∞

where ρ denotes the impact parameter and E is the transition energy. Then the excitation cross section σex (v) can be obtained by integration over hyperbolic trajectories corresponding to the repulsive (6.161) or attractive (6.163) Coulomb potentials, respectively38. At small relative velocities v, the cross section for excitation in ion–ion collisions is exponentially small: ∞ ZP ZT R0 E · µ1/2 π W (ρ, v)ρ dρ ∝ exp − σex (E cm ) = 2π 1/2 R0 E cm 0 E cm (6.178) where E cm denotes the energy in the center-of-mass frame, µ the reduced mass of the colliding particles, Z P the projectile ion charge and Z T the charge of the target ion. Cross sections of excitation transitions in highly charged ions induced by proton impact calculated by the close-coupling method and using normalized 38 See section 7.3 in the book by Pal’chikov V G and Shevelko V P 1996 Reference Data on

Multicharged Ions (Berlin: Springer).

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Figure 6.20. Cross sections for proton-induced quadrupole excitation in C-like Ne V ions as a function of the proton energy E in Ry units: crosses, normalized cross sections; full curves, close-coupling calculations for low energies; broken curves, interpolation of the full curves. From Pal’chikov V G and Shevelko V P 1995 Reference Data on Multicharged c Ions (Berlin: Springer). 1985 Springer.

Born approximation are shown in ﬁgure 6.20. As it is seen from the ﬁgure, the normalized excitation cross sections are in a good agreement with those calculated by the close-coupling method in the whole energy range considered.

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6.9.4 Electron capture Among different types of ion–ion electron-capture processes occurring in ion–ion collisions, the resonant reactions X(q−1)+ + Xq+ → Xq+ + X(q−1)+

(6.179)

are of particular interest for applications because they occur with the largest probability. For these processes, the resonance defect is zero: E res = 0. The resonant defect is deﬁned as the difference between the binding energies of the captured active electron before and after the collision: E res = I0 − I1

(6.180)

where I0 refers to the target ion or atom and I1 to the scattered projectile. Therefore, if the resonance defect is zero, E res = 0, one has a resonant electron capture. Theoretically, the cross section for reaction (6.179) can be obtained as an (2q−1)+ over integral of the energy separation (R) of the molecular system X2 inter-nuclear distances R. At low energies E q(q − 1) a.u. (1 a.u. = 2Ry ≈ 27.2 eV), the resonant electron-capture cross section decreases exponentially, while the ion–atom resonant cross section increases logarithmically. At large energies, E q(q − 1) a.u., the resonant cross section behaves similarly to the case of the ion–atom collisions with straight-line trajectories, i.e. decreases logarithmically as the relative velocity increases39 . The cross sections for resonant or quasi-resonant electron capture (E res ≈ 0) can also be large (up to 10−15 cm2 ) even for ion–ion collisions of positive ions undergoing Coulomb repulsion. In the case of non-resonant ion–ion electron-capture (E res )= 0) Xq1 + + Yq2 + → X(q1 +1)+ + Y(q2 −1)+

(6.181)

the collisional dynamics at relatively low energies is governed by a series of pseudocrossings of the initial molecular potential energy curve with those corresponding to different ﬁnal molecular states, the Landau–Ziner or Rosen– Ziner–Demkov models. Experimental investigations of ion–ion electron-capture collisions are restricted to low charged ions only. Electron capture between protons and singly charged helium atoms H+ + He+ → H + He2+ (6.182) represents the simplest ion–ion collision system as only one electron is involved. Together with another one-electron capture process He2+ + He+ → He+ + He2+ 39 See for details, Janev R K and Presnyakov L P 1981 Phys. Rep. 70 1.

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(6.183)

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Figure 6.21. Experimental cross sections for He+ +He+ collisions as a function of the center-of-mass energy: σi is the ionization cross section, σc is the electron-capture cross section, σ2+ = σi + σc . From Rinn K et al 1986 J. Phys. B: At. Mol. Phys. 19 3717. c 1986 IOP Publishing.

reaction (6.182) is of considerable interest in plasma physics and thermonuclear research and, besides, provides a unique test for the study of the long-range Coulomb interaction in the quantum three-body problem. Due to the speciﬁc properties of electron capture, at relatively small energies the cross section of process (6.182) is higher than that for ionization by proton impact: H+ + He+ → H+ + He2+ + e− .

(6.184)

The corresponding experimental cross sections for the two lightest ions are displayed in ﬁgure 6.21. Experimental and theoretical investigations are mainly restricted to electron capture between protons and singly charged ions H+ + X+ → H + X2+ .

(6.185)

The crossed-beam technique enables the capture cross sections σc and the total X2+ production cross sections σ2+ to be measured with relatively high accuracy. Among different theoretical calculations of electron-capture cross sections, the best description is given by the close-coupling calculations. At present all known

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Figure 6.22. Measured cross sections for electron capture (6.186) and ionization (6.187) in H+ + X+ collisions. From Melchert F 1999 Atomic Physics with Heavy Ions ed H F Beyer c and V P Shevelko (Heidelberg: Springer) p 323. 1999 Springer.

measurements of the electron-capture cross sections are in good agreement with theory at the center-of-mass energy interval E cm = 10–1000 keV. However, the measured capture cross sections for protons colliding with ions from the groupIIIB elements, such as Al+ , Ga+ , In+ and Tl+ , are in poor agreement with available theoretical predictions. In thermonuclear fusion plasmas, the fuel ions H+ or D+ can capture an electron from heavy impurity ions, leading to the loss of neutralized fuel and, furthermore, to an enhanced charge for the impurity ion which increases the radiative energy loss of the plasma. Cross sections for electron capture σc involving protons H+ + X+ −→ H0 + X2+ (6.186) and for ionization H+ + X+ −→ H+ + X2+ + e−

(6.187)

are plotted in ﬁgure 6.22. 6.9.5 Heavy-ion collisions Understanding the electron capture and ionization processes in collisions between identical heavy ions (called homonuclear collisions) is necessary for several aspects in controlled thermonuclear fusion research. Pellets of deuterium and tritium to be bombarded with intense GeV ion beams are accumulated in large

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storage rings (section 6.9.8). Within these rings, electron capture:

and ionization

Xq+ + Xq+ → X(q−1)+ + X(q+1)+

(6.188)

Xq+ + Xq+ → Xq+ + X(q+1)+ + e−

(6.189)

X+

at 1–10 keV relative occurs in collisions between the stored identical ions energies. These processes characterize the intra-beam Coulomb scattering. Two X+ ions are lost from the stored ion beam per one capture event (6.188) and one particle X+ per ionization event (6.189). For heavy-ion-fusion applications it is necessary to know σc and σi separately to obtain the total loss cross section σL which is given by (6.190) σL = 2(2σc + σi ) = 2(σc + σ2+ ). Here, the factor 2 appears twice because two identical particles are lost in a single collision and both ions simultaneously act as projectile and target particles. The electron-capture cross sections have been measured for collisions of identical heavy-ion pairs: X+ + X+ → X + X2+

X = Li, Na, Ar, K, Rb, Xe, Cs.

(6.191)

Estimating particle loss within storage rings due to ion–ion collisions as well as a better physical understanding require knowledge of both the capture (6.188) and ionization (6.189) channels individually. The cross sections for these channels have only been measured for a few ions X+ , plotted in ﬁgure 6.23. The closed Xe-like shell of Cs+ is responsible for the relatively small capture cross section (6.188), while the ionization cross section of Cs+ exceeds data for Xe+ and Bi+ ions. 6.9.6 Collisions between highly charged ions Ion–ion collision data required for various applications are sparse, especially for collisions between highly charged ions where they are practically absent. Development of powerful electron cyclotron resonance ion sources enable enhanced intensities of multiply charged ion beams to be produced so that crossed-beams experiments can be carried out with beams of these ions. Only a few experiments, in which at least one ion with a charge higher than one, have been carried out. Large cross sections for electron capture reactions can be expected for the resonant (E res = 0) and quasi-resonant (E res ≈ 0) ion–ion collisions, where E res is the resonant defect given by equation (6.180). The quasi-resonant electron-capture cross sections in two mutually reverse reactions: C2+ + B+ −→ C+ + B2+ 2+

B

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+

+

+ C −→ B + C

2+

(6.192) (6.193)

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Figure 6.23. Cross sections for electron capture (6.188) and ionization (6.189) in X+ + X+ collisions between identical ions: symbols, experiment, curves, calculated data. From Melchert F 1999 Atomic Physics with Heavy Ions ed H F Beyer and V P Shevelko c (Heidelberg: Springer) p 323. 1999 Springer.

with E res ≈ 0.77 eV have been measured (ﬁgure 6.24). As seen from ﬁgure 6.24, the use of the Rosen–Zener–Demkov model for reaction (6.192) predicts cross section data of about 6 × 10−16 cm2 for collision energies 0.5 keV ≤ Ecm ≤ 50 keV and under-estimates the experimental data by a factor of four (curve 3, left ﬁgure). Besides, neither were the calculated oscillations detected in the experiment. As for the inverse reaction (6.193), the qualitative description of the experiment can be obtained by including the capture into the individual singlet and triplet states of the resulting B+ ion (curves 2 and 3, right-hand ﬁgure). 6.9.7 Ionization Similar to the electron-capture considered in the previous section, experimental investigations of the ion–ion ionization collisions Xq1 + + Yq2 + → Xq1 + + Y(q2 +1)+ + e− are limited to low-charged ions with charges Z 1 , Z 2 < 3. Experimentally, the one-electron ionization cross section σi is usually obtained from the difference σi = σ2+ − σc

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(6.194)

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Figure 6.24. Cross sections for electron capture in C2+ + B+ collisions (left) and for electron capture in B2+ + C+ collisions (right). Experimental data are represented by the symbols. Curve 3, left: calculation in the Rosen–Zener–Demkov model. The other curves are theoretical results within the framework of non-adiabatic transitions: curve 1, total cross sections; curve 2, left, theoretical results with 20% of the incoming ions in the metastable C2+ (2s2p 3 P) state; curve 2, right, partial cross section for capture to the metastable B+ (2s2p 3 P) triplet state; curve 3, right, partial cross section for capture to the B+ singlet states. From Brandau C et al 1995 J. Phys. B: At. Mol. Opt. Phys. 28 L579. c 1995 IOP Publishing.

where σc is the electron-capture cross section, σ2+ is the total cross section for the production of a doubly ionized atom in the ﬁnal channel. For simple colliding partners H+ + He+ and He+ + He+ , measured ionization cross sections are shown in ﬁgure 6.25. They have been carried out using the crossed-beam technique. The theoretical data are mostly from close-coupling calculations and atomic- and molecular-orbital representations. We note here that even for these one- and two-electron systems, there is no agreement between the experimental results and theoretical calculations. In general, ionization of these systems is much less understood compared to electron capture. Theoretical methods for treating ion–atom ionizing collisions can be directly applied to ion–ion ionization processes if the Coulomb repulsion is taken into account in the description of the relative motion of the ions. However, it should be noted that the Coulomb repulsion has a non-negligible effect on the cross section only in the energy region of the cross section maximum and below, E cm ≤ Z P Z T a.u. (1a.u. = 2Ry = 27.202 eV).

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Figure 6.25. Ionization cross sections in H+ + He+ (left) and He+ + He+ (right) collisions. Experimental data correspond to the crossed-beam results. From Melchert F 1999 Atomic Physics with Heavy Ions ed H F Beyer and V P Shevelko (Berlin: Springer) c p 323. 1999 Springer.

The total X2+ production cross section for reactions X+ + X+ → X2+ + · · ·

(6.195)

measured for many colliding ions X+ are plotted in ﬁgure 6.26. The contribution to σ2+ from multiple-electron ionization reactions is mostly negligible in the investigated energy range. With the exception of a few Li+ data points, all cross sections σ2+ were measured at velocities less than an atomic unit and increase with the cm-collision energy. As can be seen, σ2+ increases with the atomic number. The production of doubly charged alkali ions requires the break-up of a closed electronic shell, which is 75 eV for a Li+ ion, and only 25 eV for a Cs+ ion; the latter, therefore, has a larger cross section. 6.9.8 Inertial fusion driven by heavy ions Ion–ion interactions are of signiﬁcant interest in solving the problems of heavyion fusion (HIF) or heavy-ion-driven-inertial-fusion (HIDIF). The idea of HIF was ﬁrst suggested in 1974, and is associated with the initiation of nuclear fusion reactions by the interaction of heavy-ion beams with solid targets, e.g. with deuterium–tritium (D–T) pellets. Due to the ion–ion processes occurring in the incoming beam, mainly electron capture and ionization (intra-beam scattering), the beam can lose its particles and energy producing undesirable ion-beam

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Figure 6.26. Experimental cross sections for the total X2+ production (6.195) in X+ + X+ collisions. From Melchert F 1999 Atomic Physics with Heavy Ions ed H F Beyer and c V P Shevelko (Heidelberg: Springer) p 323. 1999 Springer.

losses. Therefore, possible beam losses due to intra-beam interactions should be estimated and taken into account in practical applications. Nuclear fusion is the phenomenon that occurs when two or more light atomic nuclei as a result of a nuclear reaction, create new heavier nuclei and release additional energy as gamma photons or in the kinetic energy of the resulting particles40 . Nuclear and atomic reactions which release energy are exothermic reactions. Exothermic nuclear reactions were ﬁrst discovered in the 1930s, and since then they have been of great interest from both the scientiﬁc and technical points of view41. However, till now all attempts to construct an operating fusion reactor have failed. In fact, the idea that the fusion reactions constitute the main source of the energy of the stars, including the Sun, was proposed by H A Bethe in 1939 and since then it has become generally accepted. 40 The problem of the nuclear fusion was also considered in section 4.6.1. 41 See, e.g, Lindl J D 1998 Inertial Conﬁnement Fusion (New York: AIP).

Lindl J D 1995 Phys. Plasmas 2 3922. Labaune C, Hogan W J and Tanaka K A 2000 Inertial Fusion Sciences and Applications: State of the Art 1999 (Amsterdam: Elsevier).

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Figure 6.27. Effective cross sections (in cm2 ) for the D + T nuclear reaction, solid curve, and D + D nuclear reaction, dashed curve.

The reaction of a practical interest is the D + T exothermic nuclear process: D + T → α(3.5 MeV) + n(14.1 MeV)

(6.196)

where D represents deuterium, and T tritium, the two heavy hydrogen isotopes 2 H and 3 H, respectively, and α is the α-particle, He2+ . The kinetic energy released in the reaction, 17.6 MeV, is distributed between the neutron (14.1 MeV, 80%) and α-particle (3.5 MeV, 20%). As a result, one has high-energy neutrons. Another nuclear reaction of interest is D + D reaction: 3 He(0.82 MeV) + n(2.45 MeV) (6.197) D+D→ T(0.82 MeV) + p(3.02 MeV) where p denotes a proton. Both channels in (6.197) take place with an approximately equal probability. The reactions (6.196) and (6.197) have different effective cross sections and rates as reproduced in ﬁgures 6.27 and 6.28. The maximum cross section for reaction (6.196) is rather small: it is about 5 × 10−24 cm2 at relative energy E ≈ 10 keV and the corresponding rate of the reaction is at its maximum, σ v ≈ 10−15 cm3 s−1 at a temperature T ≈ 6 × 108 K ≈ 50 keV.

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Figure 6.28. Rate coefﬁcients (in cm3 s−1 ) for the D + T nuclear reaction, solid curve, and D + D nuclear reaction, dashed curve.

The reaction (6.196) or (6.197) can occur only if two nuclei can approach one another at a distance shorter than 10−13 cm to overcome the electrostatic Coulomb repulsion force because both particles are positively charged. Since the Coulomb force is proportional to the product of two particle charges, q1 q2 e2 , the reaction can occur with appreciable probability when two nuclei approach each other with not very high kinetic energy that corresponds to reactions between the lightest nuclei, in particular, the nuclei of the various isotopes of hydrogen and helium. The general aim of controlled thermonuclear fusion on the basis of D–T plasma conﬁnement is based on two main problems: (1) to achieve the fusion burning condition in the reactor: n i · Ti · τ > 5 × 1015 [cm−3 keV s] i.e. to reach the electron and ion density n e ≈ n i ≈ 1014 –1015 cm−3 , the plasma-conﬁnement time τ > 1 s and the plasma electron and ion temperature Te ≈ Ti ≈ 10–20 keV, (2) to achieve a high energy gain (10) of the sustained plasma burn. Heavy-ion accelerators have the potential to contribute to the enormously

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Table 6.2. Approximate characteristics of the ion beam and the pellet. Ion beam

Pellet

Species: Bi1+ , Bi2+ , U1+ , U4+ Energy: 20–40 MeV/u Density: 2 × 108 cm−3 Ion-beam radius: 2 cm Relative ion velocity: 1 × 107 cm s−1 Emittance: 5 × 10−5 rad m

D–T mixture: 50%–50% Outer layers: Pb, PbLi Diameter: 1 mm D–T mass: 1 mg Interaction time: 20–30 ns Compression: 103 –104 times

growing energy market of the 21st century. It is expected that the ignition of fusion targets and energy gain will be demonstrated within the next decade by the powerful laser facilities which are now under construction at Livermore, Bordeaux and Darmstadt. However, the development of a suitable driver with high efﬁciency and a high repetition rate capability still remains a challenging issue. Heavy-ion accelerators are considered to be the most promising option for such an inertial fusion driver. According to the HIDIF scenario, the kinetic energy of 10 GeV as a good compromise between enhanced space charge problems for lower energies and excessive range at signiﬁcantly higher energies has to be provided by a linear accelerator which ﬁlls a set of storage rings to accumulate a total beam energy of about 3 MJ (typically 1015 bismuth ions). The reference scenario for the HIDIF driver includes a total number of 6–12 storage rings arranged in two stacks delivering 144 beamlets which are compressed in induction bunches and guided to the target chamber (ﬁgure 6.29). The target is indirectly driven by conversion of the heavy-ion beam kinetic energy into soft x-rays. This requires two cylindrical converters made of Be foam and contained in a hohlraum (‘hollow space’) with thin gold walls which conﬁnes the x-ray ﬂux and provides spatial uniformity of the radiation (ﬁgure 6.30). The fusion pellet, ﬁlled with the deuterium–tritium fuel, absorbs the x-rays which leads to surface heating and ablation. Spherical shock waves driven by the ablation pressure compress and heat the fusion pellet. The conditions for the thermonuclear burn (1000-fold compression and 5–10 keV temperature) are reached in the center provided that the implosion is spherically symmetric within typically 1% precision. It is estimated that 3 MJ beam energy with 6 ns pulse length and 1.7 mm spot radius are necessary to reach the ignition condition. Average characteristics required for the ion beam and pellets are given in table 6.2.

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Figure 6.29. HIDIF scenario for heavy-ion inertial fusion. From 1999 GSI-Nachrichten 3 (Darmstadt) p 18.

Figure 6.30. Fusion target in HIDIF scenario. From 1999 GSI-Nachrichten 3 (Darmstadt) p 19.

6.10 Ion–surface interaction and hollow atoms Among the different topics related to highly charged ions, the interaction of slow ions with a surface constitutes a special problem because it is connected with the transformation of these ions during the interaction to the new physical objects termed Hollow Atoms. Hollow atoms are characterized by population inversion when almost all, or even all, atomic electrons are in highly-excited Rydberg states

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while a large number of vacancies in the inner shells remain empty, and by quite a short lifetime for this population inversion, typically about 10–100 fs. The term hollow atom was ﬁrst introduced by J P Briand and co-workers in 199042 during investigations of characteristic soft x-ray emission resulting from impact of Ar17+ ions on a gas-covered silver surface. Hollow atoms can be regarded as one of the most exotic creations of atomic collision physics involving highly charged ions. With modern advanced sources of highly charged ions such as the EBIS, EBIT and ECR (see chapter 4), investigation of the mechanisms for the production and decay of hollow atoms near the solid surface has become a subject of considerable interest. These investigations are mainly aimed at understanding the neutralization dynamics of highly charged ions as they approach the surface and penetrate into the solid. Highly charged incident ions carry a very high total potential energy, up to a few hundreds keV, and characteristic x-ray emission studies are particularly suited to illuminate the different radiative and collisional processes which lead to this transfer of this potential energy to the surface. Moreover, such experiments with the creation of hollow atoms allow one to observe the surface modiﬁcation caused by incident-ion collisions43 . The interaction dynamics between highly charged incident ions and a solid surface (usually an automatically clean metal surface) has evolved from experimental data on the total electron yields, electron emission statistics and fast Auger electron energy distributions, together with analysis of scattered projectiles and soft x-ray emission. An approximate scenario for the creation of hollow atoms is shown in ﬁgure 6.31. If a bare ion with a high ion charge q and small velocity approaches a metal surface, then according to the properties of electron-capture processes, the ion captures all Z electrons from the surface, which is an inﬁnite reservoir of weakly bound electrons, predominantly into highly excited states of the resulting neutral hollow atom, where Z is the ion nuclear charge number. For example, as was found experimentally, when 340 keV Ar17+ ions (Z = 18) collide with an Ag surface, all or most of the 18 captured electrons were located in the M and N shells while the K and L shells remained empty. Then with a ‘mantel’ of electrons, this large neutral ball undergoes a further interaction with the surface resulting in de-excitation via autoionization and loss of a large number of slow electrons, even larger than the highly charged incident ion charge. For example, in Th80+ + Au surface collisions, more than 300 electrons per ion were registered, i.e. more than the thorium nuclear charge Z = 90! When a hollow atom further approaches the surface so as to reach stage C of ﬁgure 6.31, it gets screened by the metal electron gas which causes further accelerated de-excitation. And ﬁnally (stage D), all remaining inner-shell vacancies are ﬁlled by the solid-state electrons which increase the emission of fast Auger electrons or soft x-rays depending on the projectile ion ﬂuorescence yield, 42 Briand J P et al 1990 Phys. Rev. Lett. 65 159. 43 Ban-d’Etat B and Briand J P 1999 Atomic Physics with Heavy Ions ed H F Beyer and V P Shevelko

(Berlin: Springer) p 360.

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Figure 6.31. Scenario for the impact of slow highly charged ions on a metal surface: A, a highly charged ion approaches the surface acquiring image-charge energy gain E im ; B, formation of HAs above the surface gives rise to autoionization; C, screening of the HAs by surface electrons causes further electron emission; D, relaxation of the HAs at or below the surface proceeds by electron and x-ray emissions. From Winter H and Aumayr F 1999 c J. Phys. B: At. Mol. Opt. Phys. 32 R39. 1999 IOP Publishing.

i.e. the hollow atom decays in the bulk of the surface. Certainly, this physical picture of creating hollow atoms is simpliﬁed because if the incident ion is not bare, the situation is much more complicated due to the other elementary processes which take place. A description of the surface charge exchange and relaxation is based on the classical over-barrier model44 which incorporates resonant multi-electron capture of metal electrons, resonant loss into unoccupied states of the conduction band and ultra-atomic Auger de-excitation. A highly charged ion with a charge number q and a small velocity vi (vi vF , vF being the Fermi velocity of the electrons inside the metal target) while approaching the metal surface induces a collective response of the metal electrons which at large distances R from the surface can be described by the classical image potential q 2 e2 Vim (R) = . (6.198) 4R The highly charged ions, accelerated by the image potential towards the metal surface, impose the lower limit to the projectile velocity corresponding to the upper limit for the highly-charged-ion–surface interaction time. Besides, the image interaction causes a shift in the projectile electronic states and decreases the height of the electronic potential barrier between the ion and surface which is 44 Burgdorfer J 1993 Fundamental Processes and Applications of Atoms and Ions ed C D Lin (Singapore: World Scientiﬁc).

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Figure 6.32. Electronic potential barrier between an Ar12+ ion and an Au surface at a distance R ≈ 50 a0 (dotted curves) and R ≈ 26 a0 (full curves), above Au surface, respectively. In the latter case, the potential barrier has decreased below the Fermi level of Au, where the electron capture becomes classically allowed. Here Wϕ denotes the work function of metal, E F the Fermi energy of the conduction band, and a0 the Bohr radius. 1 a.u. of energy corresponds to 2Ry ≈ 27.2 eV. From Aumayr F and Winter H 1994 Comm. c At. Mol. Phys. 29 275. 1994 Gordon and Breach.

formed by the projectile’s potential, its image potential and the image potential of the particular electron to be captured (ﬁgure 6.32). At a certain critical distance √ 2q Rc ≈ (6.199) Wϕ where Wϕ is the work function, the potential barrier between the metal and the projectile drops below the Fermi level of the metal and the ion starts to capture electrons resonantly from the conduction band into highly excited states of the projectile (Resonant Neutralization ﬁgure 6.33). If neutralization of a singly charged ion takes place at distances of a few atomic units, the resonant neutralization of a highly charged ion, according to (6.199), can start at larger distances but in any case, slow highly charged ions are completely neutralized before they touch a metal surface. The classical over-barrier model also predicts the principal quantum numbers n c of the highly excited states of the projectile in which resonant neutralization takes place: q − 0.5 −1/2 q nc = 1+ √ . (6.200) 8q 2Wϕ

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Figure 6.33. Energy diagrams describing (a) resonant neutralization, (b) intra-atomic Auger transition, (c) Auger de-excitation and (d) inter-atomic Auger transitions.

The resonant neutralization stops as soon as the captured electrons have screened the ion charge and, as a consequence, the potential barrier has moved up again above the Fermi level. With further approach of the projectile towards the surface, the over-the-barrier condition will be restored as the image interaction mentioned earlier and the screening of the projectile charge by the electrons already captured will shift the energy levels of the projectile upwards (image shift, IS, and screening shift, SS, as shown in ﬁgure 6.34), and again resonant neutralization can go on. Thus, the quantity n c in (6.200) denotes the highest n-value of the projectile which can be reached during the whole neutralization sequence. Further evolution of such a multiply excited projectile depends on competition between resonant neutralization and other kinds of electronic

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Figure 6.34. Stages of neutralization of a highly charged ion approaching a metal surface. Electrons captured via resonant neutralization (RN) can be emitted via autoionization (AI), promotion into vacuum due to screening and image shift (SS/IS) and ‘peeling off’ of all electrons which cannot stay bound to the projectile inside the solid. Furthermore, electrons can be recaptured into the solid via resonant ionization (RI). Different full curves correspond to different excited n states. From Aumayr F and Winter H 1994 Comm. At. c Mol. Phys. 29 275. 1994 Gordon and Breach.

transitions such as intra-atomic Auger transitions, Auger de-excitation and interatomic Auger transitions (ﬁgure 6.33). However, all electrons lost from the projectile are rapidly replaced by resonant neutralization and ﬁnally a fully neutralized hollow atom is formed. The complete de-excitation of these highly excited species to its neutral ground state, via the mentioned manifold electronic interactions, would require a time not available because of the upper limit set by image charge attraction. When such a highly excited hollow atom approaches the surface, electrons of a projectile with a Rydberg radius r = a02 n 2 /q larger than the screening length within the solid, λs = vF /ωp (ωp is the surface plasmon frequency) can be removed (ionized) from the projectile (‘peeling off’ effect). In other words, the metal electrons form a dynamic screening cloud around the highly charged ion’s projectile core which peels off those electrons have been captured earlier into outer projectile states. However, such removed electrons are rapidly replaced by electrons from the conduction band of the solid and a hollow atom in less excited state is formed. Further relaxation of the projectile within the solid will be ﬁnished when the inner-shell vacancies not yet recombined through a series of cascade processes and electron transfer between inner shells of projectile and metal are full up.

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These transitions result in the majority of the observed fast (projectile) Auger electrons, in competition with x-ray emission. De-excitation of this short-lived complex can take place via various electronic transitions between intra-atomic states in the particle or between the particle and metal. Besides their fundamental interest, this process is also of practical relevance for plasma–wall interactions in gas discharges including thermonuclear fusion experiments or ion-beam-activated material modiﬁcation. The same scenario is adopted for highly charged ion impact on semiconductor surfaces but the experiments carried out for the insulator surfaces have shown that image-charge formation involves physical mechanisms different from those for metal surfaces. In metals, the conduction band is divided by Fermi level into occupied and un-occupied zones with a work function of about 4– 6 eV, while in insulators the least-bound electrons form a completely occupied valence band with a larger binding energy (6–12 eV) which is separated by a band gap of forbidden states with a several eV width from the ﬁrst allowed states in the completely empty conduction band. The more important point is that the dielectric response of an insulator is described by a frequency-dependent dielectric function %(ω) which is quite different from that of a metal.

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Chapter 7 Conclusion and further reading

In this book, we have to present an introduction to the physics of highly charged ions and to help the reader to orient himself/herself in the ocean of information related to these interesting atomic systems. Our main attention has focused on the physical background of the atomic structure and radiative and collisional phenomena involving highly charged ions. Due to the limited length of this book, we were unable to consider many interesting topics in detail. Here, in conclusion, we will give a brief description of some prospective directions in the physics of highly charged ions, and the list of references for further reading.

7.1 Rydberg atoms and ions An atom or ion with one electron in a highly excited state with principal quantum number n n 0 is called a Rydberg atom or ion where n 0 refers to the principle quantum number of the ground state. The atomic properties of such a highly excited system are considerably different from those in the ground or low-excited state as shown in table 7.1. It is seen that some characteristics have a much stronger dependence on n than on the nuclear charge Z . The Rydberg system has a very large atomic size rn , small ionization energy In and very large cross sections for their collisions with photons, electrons and atoms. Being in highly excited states, these objects possess both quantum and classical properties in accordance with the correspondence principle considered in section 3.7. Also, the large value of the principal quantum number n 1 in many cases makes it possible to describe the Rydberg atoms in terms of the classical-mechanical approaches which markedly simplify the calculations. Neutral Rydberg atoms have been investigated quite well in laboratory plasmas and cosmic space (interstellar medium). In laboratory conditions, the states with quantum number up to n ≈ 500–1000 can be achieved, while the observed atomic quantum number in the interstellar medium reach values of n up to n ≈ 10001. 1 Beigman I L and Lebedev V S 1998 Physics of Highly Excited Atoms and Ions (Berlin: Springer).

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Table 7.1. Approximate scaling behavior of atomic properties of highly excited H-like ions with nuclear charge Z in a state with principal quantum number n. Numerical values are given for neutral hydrogen (Z = 1). a0 denotes the Bohr radius, v0 the atomic velocity unit, A 0 = 8.03 × 109 s−1 , 1 Ry = 13.606 eV. From Beigman I L and Lebedev V S 1998 Physics of Highly Excited Atoms and Ions (Berlin: Springer). Value Ionization potential, In Ionic radius, rn Cross section, πrn2 Orbital electron velocity, vn Frequency between nearby levels, ωn,n±1 Wavelength between nearby levels, λn,n±1 Period of classical motion, Tn Fine-structure correction, E f s Total radiation lifetime, τn

Expression

Unit

n = 10

n = 100

n = 1000

Z 2 Ry/n 2 a0 n 2 /Z π a02 n 4 /Z 2

eV cm cm2

1.4 × 10−1 5.3 × 10−7 8.8 × 10−13

1.4 × 10−3 5.3 × 10−5 8.8 × 10−9

1.4 × 10−5 5.3 × 10−3 8.8 × 10−5

Z v0 /n

cm s−1

2.2 × 107

2.2 × 106

2.2 × 105

2Z 2 Ry/(~n 3 )

s−1

4.1 × 1013

4.1 × 1010

4.1 × 107

2π c/ωn,n±1

cm

4.6 × 10−3

4.6

4.6 × 103

2π/ωn,n±1

s

1.5 × 10−13

1.5 × 10−10

1.5 × 10−7

−α 2 Z 4 Ry/n 3

eV

7.3 × 10−7

7.3 × 10−10

7.3 × 10−13

n5 3 A0 Z 4 ln(n/1.1)

s

8.4 × 10−5

17

3.7 × 103

Studies of highly charged ions Z 1 in the Rydberg states n 1 are quite limited; some of these were considered in chapters 5 and 6 in this book. However, during the year 2001, high-precision measurements of the dielectronic recombination (DR) of Li-like U89+ were carried out in the electron cooler of the ESR in Darmstadt in the center-of-mass energy range of 0–400 eV. These data mainly consist of two sets of peaks belonging to the high-n Rydberg resonances 1s2 2p1/2 n j , n ≥ 20 and 1s2 2p3/2 5 j DR resonances. These DR experiments allowed one not only to provide a precise determination of the DR energy peaks but also to measure the radiative and autoionization probabilities of doubly excited states. Another important topic concerning highly charged Rydberg ions is related to the determination of the atomic nuclear size, i.e. the high experimental resolution makes it possible to probe the nuclear parameters. Figure 7.1 shows the DR rate coefﬁcients for two isotopes of highly charged uranium. In calculations, the Fermi nuclear density distributions (5.110) were used with the root-mean-square (rms) radius of 5.86 fm for 238 U and 5.81 fm for 233 U isotopes, respectively. It can be seen that the two resonance patterns are shifted relative to each other by approximately 0.1 eV. Therefore, high-resolution DR spectra of the Rydberg states will provide an effective tool for determining

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Figure 7.1. DR rate coefﬁcient of U89+ in the vicinity of the 1s2 2p3/2 55/2 resonances for 238 U and 233 U isotopes: full circles, experimental data for 238 U isotope (with rms radius of 5.86 fm); dotted line and dark shaded area, theoretical prediction for the isotopic volume shift in 233 U isotope and with rms radius of 5.81 fm. The peaks are labeled with the letter, representing the Rydberg electron’s angular momentum, and the number denoting the total momentum J . From Brandau C et al 2002 GSI Scientiﬁc Report 2001 p 95 (GSI, c Darmstadt, ISBN 0174/0814). 2002 GSI (Darmstadt).

isotopic shifts in heavy highly charged ions2 . Among the prospective topics related to highly charged ions in Rydberg states, we have to mention the interaction of slow highly charged ions with a surface. This is connected with the transformation of highly charged ions during this interaction into hollow atoms, considered in chapter 6, when almost all captured electrons are created in highly-excited Rydberg states and, simultaneously, a large number of vacancies in the inner shells remain empty3.

7.2 Laser-produced plasma and related phenomena Highly charged ions can be created in a special kind of plasma called a laserproduced plasma. If an ultra intense laser beam with power P = 1020– 1022 W cm−2 is focused onto a solid target inside a vacuum chamber, an 2 See Brandau C et al 2002 High Rydberg resonances in dielectronic recombination of Pb79+ Phys.

Rev. Lett. in press. 3 For a topical review, see Winter H P and Aumayr F 1999 Hollow atoms J. Phys. B: At. Mol. Opt.

Phys. 32 R39.

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expanding plasma, consisting of ions, electrons, atoms and other particles, is created. Changing the laser parameters (intensity, pulse duration, size of the focal spot) and the atomic number of the target, the electron density, temperature and degree of ionization of the produced plasma can be varied over a wide range. This allows one to study ionization-recombination and radiation processes occurring in such a plasma, to develop the collisional-radiative schemes for short wavelength lasers (including x-ray lasers), to investigate the inﬂuence of high-dense laser ﬁeld on interactions in a plasma, to develop intense x-ray and black-body radiation sources with the radiative temperatures of about a few hundred eV, development of x-ray detectors, spectrometers, vacuum techniques and many others. The ultra dense laser-produced plasma is also of interest for the investigation of Inertia Conﬁnement Fusion (ICF) where heavy ions can also be involved. At present, the main directions of these investigations are: the development of table-top x-ray lasers4 , new laser sources for producing highly charged ions of heavy elements (Pb, Ta, Au ions with charge up to q ≈ 30)5 and the interactions of a laser-produced plasma with a solid surface (wall), the so-called plasma–wall interaction6.

7.3 Atomic many-electron processes The many-electron processes occurring in collisions of highly charged ions with electrons, atoms and molecules belong to one of the most interesting and challenging domains of modern atomic physics. Although singleelectron processes associated with one (optical) electron transition in the target or projectile are quite well understood, a basic study of many-electron processes requires the use of advanced experimental techniques and sophisticated theoretical methods which drastically differ from those used in traditional oneelectron approximations. In general, many-electron processes are very interesting in furthering our understanding of the nature of many-electron and many-particle transitions. Due to their strong Coulomb ﬁeld, highly charged ions, colliding with atoms or molecules, can ionize more than one target electron with rather large probabilities. For example, the experimental cross section for simultaneous ionization of N = 30 (!) electrons from a Xe atom colliding with 15.5 MeV/u U75+ ion is about σ30 ≈ 10−18 cm2 , compared to the single-electron ionization cross section of 10−14 cm2 (ﬁgure 7.2). Note that stripping a large number of electrons (n > 25) from the target is more effective due to electron capture by the incident ions instead of direct ionization. Figure 7.3 shows a comparison of the observed cross sections for Ar recoil 4 Rocca J J 1999 Table-top soft x-ray lasers Rev. Sci. Instrum. 70 3799. 5 Fournier P et al 2000 Novel laser ion sources Rev. Sci. Instrum. 71 1405. 6 Shevelko A P et al 2001 Structure and intensities of x-ray radiation in a laser plasma–wall interaction SPIE (International Society for Optical Engineering) 4504 in press (ISBN 0-8194-4219-4).

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Figure 7.2. Experimental cross sections for recoil-ion production in collisions of 15.5 MeV/u U75+ ions with Xe: crosses, direct ionization; circles, electron capture. From c Kelbch S et al 1985 J. Phys. B: At. Mol. Phys. 18 323. 1985 IOP Publishing.

atom production under various incident particles. It should be noted that heavy, highly charged projectile ions are by far more effective in creating highly charged recoil ions, compared with structureless electrons or protons. In turn, a many-electron projectile ion can easily lose two or more electrons in one collision with the target atom; for instance, in 1.4 MeV/u U10+ ion collisions with the N2 molecule, the experimental one-electron ionization cross section for uranium ion is σ1 ≈ 1.6 × 10−16 cm2 while the total ionization

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Figure 7.3. Comparison of experimental multiple-ionization cross sections of Ar atoms production in collisions with photons, electrons, protons and highly charged ions. From Tawara H and Shevelko V P 1999 Multiple ionization of negative and positive ions, neutral atoms and molecules under electron impact, data and database Int. J. Mass. Spectrosc. 192 c 75. 1999 Elsevier.

cross section is σ1 ≈ 4.0 × 10−16 cm2 indicating that the sum of double-, tripleetc ionization processes gives more than 50% contribution to the total projectile stripping process. Collisions of heavy ions with one another and with atoms and molecules are of interest in studying Heavy Ion Fusion (HIF) or Heavy-Ion Driven Inertial Fusion (HIDIF)7. These collisions often lead to fundamental effects such as energy and charge losses in ion beams, charge-state evolution, scattering of the initial beam conﬁguration in collisions with the rest gas and others. Multiple electron transitions are basically many-body phenomena governed by the correlation effects between the electrons involved that are in a sharp contrast with the single-electron transitions where electron–electron correlation, 7 See, e.g., Hofmann I and Plass G (ed) 1998 The HIDIF-study Report GSI-98-06 (Darmstadt).

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plays a minor role8 . In general, many-electron processes can play a key role and they have to be accounted for in different physical applications such as the projectile charge-changing reactions responsible for the beam lifetimes of stored ions in acceleration and storage devices, modeling plasma processes and the charge-state evolution of atoms exposed to an electron beam.

7.4 Recoil-ion momentum spectroscopy Detailed information about many-particle (or fragmentation) processes occurring in collisions of atoms and molecules with electrons, ions and photons can be obtained with the help of a new domain of atomic physics called Recoilion momentum spectroscopy established slightly more than a decade ago. This method is based on the simultaneous determination of three-dimensional momentum vectors Px , Py , Pz of all particles after collision that allows one to obtain very detailed information on the kinematics, the ﬁnal charge states and the change in internal electronic energies of the colliding partners. The quantum states of the projectile and target before collision are considered to be known and well prepared. Usually, the cooled target atoms are prepared with a typical temperature of T ≈ 1 µeV, therefore the target is called cold and the method is known as COLTRIMS: COLd Target Recoil Ion Momentum Spectroscopy9. The term recoil ion comes from the cold target ion created after the reaction. The recoil ion is often the most interesting object because its charge gives the multiplicity of the process and the momentum transferred to it by the incident projectile provides unique information on the dynamical mechanism of the fragmentation reaction. The use of magneto-optical traps (MOT) in the near future for producing an ultra cold target with a temperature of T ≈ 1 nK will lead to an increase in resolution by a factor of more than 10. Correspondingly, this investigation direction is known as MOTRIMS: Magneto-Optical Trap Recoil-Ion Momentum Spectroscopy. Together with reaction microscopes where, in addition, the momenta of several electrons are detected, COLTRIMS constitutes a very effective tool for the investigation of many-particle fragmentation reactions such as multiple ionization and electron capture, dissociation and single and multiple photoionization when relatively small momentum transfers between colliding partners dominate. As a novel space-imaging technique, reaction microscopes make it possible to measure three-dimensional momentum vectors simultaneously for recoiling 8 McGuire J H, Stratton J C and Ishihara T 1996 Atomic, Molecular and Optical Physics Reference Book ed G W F Drake (New York: AIP). McGuire J H 1997 Electron Correlation Dynamics in Atomic Collisions (Cambridge: Cambridge University Press) ch 40. Shevelko V and Tawara H 1998 Atomic Multi-Electron Processes (Berlin: Springer). 9 See Ullrich J et al 1997 Recoil-ion momentum spectroscopy J. Phys. B: At. Mol. Opt. Phys. 30 2917 (topical review).

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target ions, emitted electrons and other particles with extremely high resolution and completeness. For example, this technique enables one to measure projectile scattering angles as small as θ ≈ 10−9 rad, to separate electron–electron and electron–nuclear interactions, to determine the spatial distribution of the fragmentation particles and the projectile in single- and multi-electron transitions, to separate the contribution to double ionization caused by usual photoabsorption and Compton scattering and to explore many other reactions10. What is of special importance, is the unique combination of high resolution and an extremely large detection solid angle of nearly 4π which makes them ideally suited for multiple coincidence studies of all kinds of atomic collision processes.

7.5 Testing QED During the last decade, a series of fundamental experiments have been carried out with few-electron highly charged ions which can provide a good testing for quantum electrodynamics (QED). One can distinguish three main directions in these experiments: the Lamb shift, the hyperﬁne splitting and the bound-electron g-factor11. 7.5.1 Lamb shift Available data on the Lamb shift in H-like ions up to 238 U91+ (see section 5.10) showed that the present status of theory and experiment gives a basis for testing ﬁrst-order QED effects in the ﬁne-structure constant α on the level of about 5 %. At present, the best experimental and theoretical ground-state Lamb shift values in 238 U91+ are, respectively: L 238 91+ ( U ) = 469(13) eV E exp L 238 91+ ( U ) = 463.95(50) ± a few eV. E th

(7.1) (7.2)

Experimental uncertainty is expected to be improved by a factor of 10 at GSI Darmstadt. To test the second-order QED corrections, it is necessary to calculate all the α 2 corrections in Li-like ions where the Lamb shift values for the 2s–2p transitions are measured with a highest accuracy (0.15–0.20 %) or to improve experimental data for He-like ions at least by an order of magnitude. 10 See D¨orner R et al 2000 Cold target recoil ion momentum spectroscopy: a ‘momentum microscope’ to view atomic collision dynamics Rep. Progr. Phys. 330 95. Brabec Th and Kapteyn H (ed) 2002 Strong Field Laser Physics (Berlin: Springer) in press. Shevelko V P and Ullrich J (ed) 2003 Many-Particle Quantum Dynamics in Atomic and Molecular Fragmentation (Berlin: Springer) (in preparation). 11 See Shabaev V M et al 2001 Towards a test of QED in investigations of the hyperﬁne splitting in heavy ions Phys. Rev. Lett. 86 3959.

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7.5.2 Hyperﬁne splitting The most accurate experimental data on hyperﬁne splitting (hfs) in high-Z ions were obtained for H-like ions up to 209 Bi82+ (nuclear charge Z = 83) as discussed in section 5.9. However, at present investigations of the QED effects on hfs in these ions do not seem to be feasible. The reason is that the QED contribution to the hfs is of the same order of magnitude as the uncertainty of the Bohr–Weisskopf correction (the nuclear magnetization distribution correction) calculated in the single-particle model. To test QED and to achieve a better agreement with experiment on hfs in H-like ions, calculations of the Bohr–Weisskopf correction using many-particle models and the higher-accuracy experimental data on the nuclear magnetic moments are necessary. Another idea for testing QED is to measure the hfs with a high accuracy in Li-like ions and to reduce considerably the inﬂuence of the Bohr–Weisskopf effect by knowing the 1s-hfs value in the corresponding H-like ion from experiment12. For example, this method predicts the 2s hyperﬁne splitting in 209 Bi80+ to be 0.7971(2) eV which is much more precise than the theoretical single-particle nuclear model: 0.800(4) eV. The available experimental value is 0.820(26) eV. In general, the suggested method can provide a possibility for testing QED effects in the strongest electric ﬁeld available at present for experimental study. 7.5.3 Bound-electron g-factor The electron g-factor is deﬁned as the ratio of the electron magnetic moment to its angular momentum. The relativistic Dirac equation (section 5.8) gives the value g = −2 exactly for a free electron (here, we omit the minus sign). The QED corrections to this value due to the interaction of the electron with its own electromagnetic ﬁeld give a slightly larger value: g = 2(1 + α/2π + · · ·) ≈ 2 × 1.001 16.

(7.3)

The free-electron g-factor recommended by CODATA13 is: g = 2.002 319 304 373 7(82).

(7.4)

The g-factor of the bound electron depends on the ion charge. For the 1s1/2 electron in H-like ions, the simple relativistic correction gives (cf (5.209)): αZ 1 (7.5) g0 = 2 − (4/3) 1 − 1 − (α Z )2 ≈ 2(1 − (α Z )2 /3) where Z denotes the nuclear charge number. Taking account of QED, nuclearsize (NS) and nuclear recoil (rec) corrections, the bound electron factor is given 12 See in detail Shabaev V M et al 2000 QED and nuclear effects in high-Z few-electron atoms Phys.

Scr. T 86 7. 13 Mohr P J and Taylor B N 2000 CODATA recommended values of the physical constants: 1998 Rev.

Mod. Phys. 72 351.

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by g = g0 + gQED + gNS + grec

(7.6)

where g0 is deﬁned in (7.5). Sophisticated calculations of the g-factor in the form (7.6) for the groundstate electron in H-like ions were performed in the work14 and for individual ions the g-factor values are: g(12 C5+ ) = 2.001 041 589 9(9) 40

(7.7)

17+

g( Ar ) = 1.990 778 082(8) g(208Pb81+ ) = 1.738 282 7(11) g(238 U91+ ) = 1.659 208 9(27). The calculated value for 12 C5+ is in very good agreement with the result of highaccuracy measurements15: g(12C5+ ) = 2.001 041 596(5).

(7.8)

Moreover, these measurements provide a new independent determination of the electron mass16 : m e = 5.485 799 092(4) × 10−4 u (7.9) with an accuracy three times better than that recommended by CODATA: m e = 5.485 799 110(12) × 10−4 u.

(7.10)

Here, u denotes the atomic mass unit: 1u =

12 1 12 m( C)

= 1.660 538 73(13) × 10−27 kg.

(7.11)

Future investigations of the bound-electron g-factor in highly charged ions can provide not only a good test for the QED theory in the presence of a strong ﬁeld but also (which is of particular importance) a new determination of the ﬁnestructure constant α, nuclear magnetic moments and nuclear charge radii17.

7.6 Parity Violation Parity violation or parity non-conservation (PNC) in atoms and ions is a physical effect followed from the Standard Model of electroweak interactions predicted by 14 Shabaev V M and Yerokhin V A 2002 Recoil correction to the bound-electron g factor in H-like atoms to all orders in α Z Phys. Rev. Lett. 88 091801. 15 H¨affner H et al 2000 High-accuracy measurements of the magnetic moment anomaly of the electron bound in hydrogenlike carbon Phys. Rev. A 85 5308. 16 Beier Th et al 2002 New determination of the electron’s mass Phys. Rev. Lett. 88 011603. 17 See in detail Werth G et al 2001 The Hydrogen Atom ed S G Karsshenboim et al (Berlin: Springer) p 204.

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Weinberg and Salam18. PNC is a mixing of atomic parity of different eigenstates due to the interaction between electrons and nucleons in which virtual neutral massive particles called Z-bosons are exchanged. As a result, stable atoms are the eigenstates of not exact parity. For example, the mixing of S and P closely lying levels in H-like ions leads to the possibility of electric dipole E1 transition between S states, i.e., states of the same parity (see section 5.3). The level mixing considered is extremely small (on the level of 10−5 –10−6) and is very difﬁcult to be observed experimentally. If the wavefunction of the valence electron in an atom or ion is non-zero at the nucleus, (0) )= 0, this electron may interact with the unscreened nucleus by exchange of virtual Z -bozons (the bozon mass M Z = 92 GeV ≈ 100m p where m p is the proton mass), and the PNC effects caused by this interaction are proportional to Z 3 in neutral atoms and to Z 5 in highly charged ions. That is why to detect the PNC effect in heavy neutral atoms and few-electron highly charged ions are chosen as the most promising candidates. Intensive experimental and theoretical investigations of PNC have been performed for neutral Cs, Tl, Pb and Bi atoms and the most accurate experiments were carried out for Cs atoms19 and the highest accuracy of 0.3% was obtained in the work in note 2020. In the case of highly charged ions, the determination of the PNC effects has only been discussed theoretically so far mainly for H-like21 and He-like ions22. The general theory of PNC is given in the book23. To observe the tiny PNC effects in heavy neutral atoms experimentally, it is also necessary to have very accurate theoretical data on such atomicstructure parameters as the amplitudes for allowed radiative transitions, dipole polarizabilities and hyperﬁne splitting constants. The main theoretical uncertainty for evaluating the PNC matrix elements comes from the need to include electron– electron correlation effects24. In highly charged ions, the situation is much more complicated because calculating the PNC matrix elements requires the inclusion of not only the electron–electron correlation effects but also the QED 18 Weinberg S 1967 A model of leptons Phys. Rev. Lett. 19 1264.

Salam A 1968 Elementary Particle Theory: Relativistic Groups and Analyticity (Nobel Symposium No 8) ed N Svartholm (Stockholm: Almqvist and Wilsell). 19 Bouchiat M A and Bouchiat C 1997 Parity violation in atoms Rep. Progr. Phys. 60 1351. 20 Wood C S, Bennet S C, Cho D, Masterson B P, Roberts J L, Tanner C E and Wieman C E 1997 Measurement of parity nonconservation and anapole moment in cesium Science 275 1759. 21 Dunford R W and Lewis R R 1981 Analysis of weak neutral currents in hydrogenic ions Phys. Rev. A 23 10. Bednyakov I et al 1999 Standard model in strong ﬁelds: electroweak radiative corrections for highly charged ions Phys. Rev. A 61 012103. Zolotarev M and Budker D 1997 Phys. Rev. Lett. 78 4717. 22 Dunford R W 1996 Parity nonconservation in relativistic hydrogenic ions Phys. Rev. A 54 3820. 23 Khriplovich I B 1991 Parity Non Conservation in Atomic Phenomena (Philadelphia, PA: Gordon and Breach). 24 See, e.g., Johnson W R et al 2001 Vacuum-polarization corrections to the parity-nonconserving 6s–7s transition amplitude in 133 Cs Phys. Rev. Lett. 87 233001.

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Figure 7.4. Calculated polarization asymmetry coefﬁcient of one of the two photons emitted in two-photon decay in 1s2p 3 P0 –1s2 1 S0 transition in He-like U as a function of photon energy. In calculations, a mixing coefﬁcient δw = 5 × 10−6 and energy difference E(2 3 P0 –2 1 S0 ) = 1 eV were used. From Dunford R W 1996 Phys. Rev. A 54 3820. c 1996 APS.

corrections25. There are two main types of atomic PNC experiments: optical rotation and Stark interference. In the ﬁrst case, the interference between the parity-conserving amplitude and parity-violating amplitudes leads to a difference of refractive index for left- and right-circularly polarized (laser) light. A very small polarization rotation of the order of 10−7 rad has to be measured. In the second type of experiments, in addition, a dc electric ﬁeld is applied. Similarly, the PNC effects in highly charged ions are suggested to be observed by measuring a circular dichroism of a certain transition, i.e. the difference in transition decay rates for right- and left-circularly polarized light. Such measurements are quite difﬁcult to perform because the transition photon energy is in the x-ray region. Another option is a quenching-type experiment where the interference between hyperﬁne- and weak-quenched transitions is to be 25 See Labzowsky L N et al 2001 Parity-violation effect in heliumlike gadolinium and ruropium Phys. Rev. A 63 054105.

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measured but such experiments require the use of polarized ion beams. In highly charged ions, the expected asymmetry coefﬁcient of the radiative transition rate is of the order of 10−4 –10−5 which seems to be more promising than for heavy neutral atoms. The calculated coefﬁcients of asymmetry for the two-photon decay transition 1s2p 3P0 –1s2 1 S0 in He-like ions such as U90+ , Th88+, Pb80+ , Gd62+ are of interest for PNC investigations because they have a small energy separation between 2 3 P0 and 2 1 S0 levels which are mixed by the weak interaction. An example of the calculated polarization asymmetry coefﬁcient in the He-like U90+ as a function of photon energy is displayed in ﬁgure 7.4. The optimistic estimations show that for the observation of the PNC effect on the level of 10−4 , one needs about 107 events, and with reasonable beam intensity (∼1010 ions s−1 ) and realistic statistics, about 105 s of observation time is required. Such experiments can be realized at SIS/ESR at GSI in Darmstadt by using the beam–foil time-of-ﬂight technique.

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List of references for further reading

Chapter 1 Binnig G and Rohrer H 1982 Sci. Am. 253 40 Phillips K J H 1992 Guide to the Sun (Cambridge: Cambridge University Press) Singer S 1971 The Nature of Ball Lightning (New York: Plenum) Stenhoff M 1999 Ball Lightning (New York: Kluwer Academic) Tr¨umper J 1993 Science 260 1769 Uman M 1993 The Lightning Discharge (San Diego, CA: Academic)

Chapter 2 Hagedorn R 1980 Relativistic Kinematics (Reading, MA: Cummings) Haken H and Wolf H C 2000 The Physics of Atoms and Quanta: Introduction to Experiments and Theory 6th edn (Berlin: Springer) Nassau K 1983 The Physics and Chemistry of Color (New York: Wiley) Svanberg S 2001 Atomic and Molecular Spectroscopy 3nd edn (Berlin: Springer) Thorne A, Litzen U and Johansson S 1999 Spectrophysics (Berlin: Springer) Tipler P A 1982 Physics 2nd edn (New York: Worth)

Chapter 3 Heisenberg W 1930 The Physical Principles of Quantum Theory (New York: Dover) Loudon R 1973 The Quantum Theory of Light (Oxford: Oxford University Press) Lustig H 1961 The M¨ossbauer effect Am. J. Phys. 29 1 Schiff L I 1968 Quantum Mechanics 3rd edn (New York: McGraw-Hill) Uns¨old A and Baschek B 2001 The New Cosmos: an Introduction to Astronomy and Astrophysics 5th edn (Berlin: Springer)

Chapter 4 Bryant P J and Johnsen K 1993 The Principles of Circular Accelerators and Storage Rings (Cambridge: University Presss) French A P 1968 Special Relativity The MIT introductory physics series (New York: Norton)

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Goldston R J and Rutherford P H 1995 Introduction to Plasma Physics (Bristol: IOP Publishing) Jackson J D 1999 Classical Electrodynamics 3nd edn (New York: Wiley) Reiser M 1994 Theory and Design of Charged Particle Beams Series in beam physics and accelerator technology (New York: Wiley) Wesson J 1987 Tokamaks (Oxford: Clarendon) Wiedemann H 1993 and 1995 Particle Accelerator Physics vols 1 and 2 (Berlin: Springer) Wolf B H (ed) 1995 Handbook of Ion Sources (New York: Chemical Rubber Company)

Chapter 5 Aggarwal B K 1991 X-ray Spectroscopy (Berlin: Springer) Bethe H A and Salpeter E E 1977 Quantum Mechanics of One- and Two-Electron Atoms 2nd edn (New York: Plenum) Cowan R D 1981 The Theory of Atomic Structure and Spectra (Berkeley, CA: University of California Press) Drake G W F (ed) 1996 Atomic, Molecular and Optical Physics Handbook (New York: AIP) Friedrich H 1998 Theoretical Atomic Physics 2nd edn (Berlin: Springer) Griem H R 1997 Principles of Plasma Spectroscopy (Cambridge: Cambridge University Press) Heitler W 1954 The Quantum Theory of Radiation (Oxford: Clarendon) Landau L D and Lifshitz E M 1975 Quantum Mechanics (Oxford: Pergamon) Mohr P J, Plunien G and Soff G 1998 QED corrections in heavy atoms Phys. Rep. 293 227 Rose E M 1961 Relativistic Electron Theory (New York: Wiley) Sobelman I I 1992 Atomic Spectra and Radiation Transitions 2nd edn (Berlin: Springer) Thaller B 1992 The Dirac Equation (Berlin: Springer)

Chapter 6 Beyer H F and Shevelko V P 1999 Atomic Physics with Heavy Ions (Berlin: Springer) Eichler J and Meyerhof W 1995 Relativistic Atomic Collisions (San Diego, CA: Academic) Lindl J D 1998 Inertial Conﬁnement Fusion (New York: AIP, Springer) Mott N F and Massey H S F 1965 The Theory of Atomic Collisions (Oxford: Pergamon) Raizer Yu P 1991 Physics of Gas Discharge (Berlin: Springer) Sobelman I I, Vainshtein L A and Yukov E A 1995 Excitation of Atoms and Broadening of Spectral Lines 2nd edn (Berlin: Springer)

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Atomic physics in chronological order

In this appendix, we brieﬂy give some dates related to the history of atoms and ions from astronomy and laboratory physics, especially of highly charged ions which are the main subject of this book. To collect the dates, the materials from the books26 and review article27 were used. Obviously, the list is far from complete but gives some idea about important developments in atomic physics. Atom (from Greek atomos, atomon—indivisible) is a small particle of a size of the order of 10−8 cm, which conserves its chemical properties and consists of a nucleus and electrons. An atom can exist separately or in complex with other atoms, i.e. molecules. The atomic conception, i.e. hypothesis that all subjects consist of indivisible (structureless) particles—atoms—was suggested about 2500 years ago and is ascribed to the Greek philosophers Leucippus and Democritus. The modern conception of an atom as a complex, structured particle was established early in the 20th century, and is related practically with all ﬁelds of physics and many domains of chemistry as well. Fifth century BC Greek philosophers Leucippus and Democritus suggested an atomistic hypothesis of matter. They also established some ideas of the modern laws of mass and energy conservation. Sixth to third centuries BC The ideas of Leucippus and Democritus were developed further by the Greek philosopher Epicurus and his school, who believed in the materialism philosophy of life. First century BC Roman poet Lucretius in his book On the Nature of Things gave a detailed description of the ideas suggested by Leucippus, Democritus and Epicurus. Most of these principles in the modiﬁed form formed the basis of the modern atomic conception. 26 Lang K R 1980 A Compendium for the Physicist and Astrophysicist (Berlin: Springer). Series G W 1988 The Spectrum of Atomic Hydrogen, Advances (Singapore: World Scientiﬁc). Malina R F and Bowyer S 1989 Extreme Ultraviolet Astronomy (New York: Pergamon). 27 Urnov A M 1995 Solar spectroscopy in the Russian space programme; past results and future prospects J. Phys. B: At. Mol. Opt. Phys. 28 1.

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Seventeenth century New science started in experimental work by Italian scientist Galileo Galilei and English scientist Isaac Newton on classical mechanics and the corpuscular and wave nature of light. Signiﬁcant progress was also achieved in work of English scientists Robert Boyle and Robert Hooke and Dutch scientist Christian Huygens. R Hooke formulated a theory for the properties of matter (mostly gases) in terms of the movement and collisions of atoms. 1661 R Boyle gave a deﬁnition of the chemical element. 1800 Italian physicist Allessandro Volta and English scientist Michael Faraday established the basis of modern electrochemistry. As a result of their experiments, it was shown that the forces unifying atoms into a chemical complex are electric in nature. 1803 English chemist and physicist John Dalton in the book A New System of Chemical Philosophy made a ﬁrst attempt to unify atoms in chemistry, i.e. the ﬁrst general theory of the chemical complexes. 1834 M Faraday introduced the term ‘ion’ from Greek i´on—going, passing. 1859 English amateur R Carrington discovered Sun ﬂares visually (in white light). 1865 English physicist James Clerk Maxwell created A Dynamical Theory of Electromagnetic Field which uniquely uniﬁed all known laws of electricity and magnetism. 1869 Russian chemist D I Mendeleev discovered Periodic Table of chemical elements. There are about 90 elements in Nature; about 20 elements were created artiﬁcially. 1879 Austrian physicist Joseph Stefan discovered experimentally the black-body radiation. 1884 Austrian physicist Ludwig Boltzmann justiﬁed theoretically the black-body radiation law dicovered by J Stefan. 1885 Swiss teacher Johann Balmer derived analytical formula for four wavelengths of the atomic hydrogen in the visible spectral range. This series of spectral lines was called further the Balmer series. (Other spectral series were discovered in 1890.) J Balmer hypothesized about the existence of a more general formula for the wavelengths of an arbitrary element. 1887 German physicist Heinrich Rudolf Hertz discovered experimentally a phenomenon of photoeffect. 1890 Swedish physicist Johannes Robert Rydberg derived a formula for the wavelengths for many of the spectral series of elements mentioned by

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J Balmer. The formula contains the universal constant which was later called the Rydberg constant. 1895 German physicist Wilhelm Konrad R¨ontgen discovered x-rays in W¨urzburg and won the ﬁrst Nobel Prize in Physics in 1901. 1896 French scientist Henri Bacquerel discovered the phenomenon of radioactivity. 1897 English physicist Joseph John Thomson discovered electrons—negative corpuscules in cathode rays. He assumed that electrons are the component particles of any atom, i.e. that an atom is a structured particle. He also measured the ratio of electron charge to its mass, e/m e . 1900 German physicist Max Planck discovered a quantum character of light radiation, i.e. that radiated (or observed) light energy E can change only by discrete portions E = n · hν, where n is an integer number and h is a universal constant (Planck’s constant). 1902 English physicists Lord Kelvin (W Thomson) and J Thomson suggested a model for the atom known as the Thomson model, according to which ˚ with a uniformly an atom consists of a sphere with a radius of about 1 A distributed positive nucleus, and electrons inserted into it similar to grapes inserted into a pudding. 1904 Japanese physicist Hantaro Nagaoka suggested a model of an atom with a positively charged nucleus in the middle, around which negative electrons create rings similar to rings of Saturn. 1904 English physicist Charles G Barkla ﬁrst measured a polarization of x-ray spectral lines. 1905 Albert Einstein theoretically explained photoeffect using Planck’s hypothesis about the quantum character of light energy. 1906 American physicist Theodor Lyman discovered a spectral series of hydrogen in the UV spectral range (the Lyman series). 1910–1911 English physicist Ernest Rutherford suggested a planetary model of the atom where negatively charged electrons rotate around a positive nucleus similar to planets rotating around the Sun. The planetary model has a number of disadvantages and the most serious of them was that electrons should radiate continuous spectra (but not lines like as in experiments) and should ﬁnally fall into the nucleus. This discrepancy was improved by N Bohr in 1913 on the basis of his famous postulates. 1913 Danish physicist Niels Bohr signiﬁcantly improved the atomic model of E Rutherford on the basis of two main postulates and explained, qualitatively

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and quantitatively, the discrete spectrum of atomic hydrogen introducing the deﬁnition of a positive integer principal quantum number n. The Bohr model is still used in quantum physics. 1913 English physicist F Soddy introduced the term ‘isotope’ based on an investigation of the radioactivity of heavy elements. 1913 English physicist Henry Bragg invented the ﬁrst x-ray spectrometer. 1913–1914 English physicist Henry G J Moseley established the basis for qualitative and quantitative x-ray spectrochemical analysis by ﬁnding the relationship between the wavelength of x-ray spectral lines and atomic number. 1913–1923 Swedish physicist Manne Siegbahn carried out his classic measurements on the wavelengths of the x-ray spectra of chemical elements. 1915 German physicist Arnold Sommerfeld introduced two additional quantum numbers, orbital number and its projection m, and explained a normal Zeeman effect. 1920 E Rutherford predicted the existence of neutrons and deuterons. 1922 German physicists Otto Stern and Walter Gerlach experimentally proved the existence of a magnetic moment in atoms and conﬁrmed the fact, followed from quantum mechanics, that the projection of atomic moment is quantized in the direction of an applied magnetic ﬁeld. They performed the ﬁrst experiment proving the existence of the electron magnetic moment—spin. 1923 French scientist Lous-Victor de Broglie suggested that a moving particle has wave features (wave–particle dualism), i.e. a particle of mass m and velocity v behaves as a wave with a (de Broglie) wavelength λ = h/(mv) where h is the Planck constant. This prediction was excellently conﬁrmed experimentally in 4 years by investigating electron diffusion on a crystal lattice where electrons behave as waves with the de Broglie wavelength. 1923 Dutch physicists Dirk Coster and G von Hevesy discovered hafnium (Hf), the ﬁrst element to be identiﬁed by its x-ray spectrum. 1925 American physicists George Uhlenbeck and Samuel Goudsmit introduced a conception of the electron spin. 1925 Wolfgang Pauli formulated the exclusion principle for electrons: two electrons in one atom cannot possess the same set of four quantum numbers (n, , m, s). The Pauli principle gave an order for ﬁlling the electron shells in atoms and gave a qualitative explanation of D I Mendeleev’s Periodic Table of the elements.

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1925 American physicists I S Bowen and Robert A Millikan ﬁrst measured VUV spectra of Na-like ions up to Cl6+ in a condensed vacuum spark in the ˚ spectral range λ ≥ 200 A. 1926 Austrian physicist Erwin Schr¨odinger using de Broglie’s idea of the wave properties of a particle and work by Irish mathematician William Hamilton on classical Newton mechanics, presented the wave mechanics theory and introduced a differential equation for a particle wavefunction. 1927 German physicist Werner Heisenberg established the uncertainty principle according to which it is not possible to measure a coordinate and the velocity of a particle simultaneously with inﬁnite accuracy. For an electron, it reads as x · vx ≥ h/(2πm e ), where h denotes the Planck’s constant and m e the electron mass. 1928 English physicist Paul A M Dirac derived his famous relativistic wave equation for the electron (an analogue of Schr¨odinger’s equation) taking relativistic effects and electron spin into account. The Dirac equation is widely used for explaining ‘thin’ effects such as the Lamb shift or the ﬁne structure in highly charged ions but with an allowance for the effects of quantum electrodynamics. 1932 The new particle—the positron (e+ )—was discovered in cosmic rays, a particle similar to the electron but with opposite sign of charge. 1932 English physicist James Chadwick ﬁrst registered a neutron, a particle predicted by E Rutherford in 1920 with a mass close to that of the proton mass. 1939 Swedish physicist Benqt Edl´en discovered the satellite lines corresponding to transitions 1s2pn–1s2n of highly charged ions in a vacuum-spark plasma. 1942 H Flemberg carried out the ﬁrst measurements of x-ray radiation from the optically allowed transitions in He-like Mg10+ and Al11+ ions in a vacuumspark plasma. 1942 B Edl´en proved the existence of highly charged ions in the solar corona by explaining the origin of its spectral lines as forbidden transitions of highly ionized Ca, Fe and Ni atoms. 1946 First launch of the captured German sounding V2 rocket and UV radiation ˚ was registered by R Tousey’s group at the Naval from the Sun (2100–2900 A) Research Laboratory (USA). 1947 American physicists Willis Lamb Jr and Robert Retherford measured the energy difference between 2s1/2 and 2p1/2 levels in the hydrogen atom equal

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to the very small value of 4.4 ×10−6 eV. This energy difference (now termed the Lamb shift) is not described by the relativistic Dirac equation. The experiment made a revolutionary upheaval in quantum mechanics and gave rise to the development and formulation of modern quantum electrodynamics (QED) which provides a complete description of the interaction between a charged particle and the electromagnetic ﬁeld. 1952 First detection of the brightest emission line—the hydrogen Lα -line ˚ 1215.7 A—in the Sun’s spectra was performed by W A Rense’s group at the University of Colorado (USA). 1962 Discovery of the brightest steady cosmic x-ray source, Scorpius X-1, with a rocket payload, was made. 1964 The idea of an x-ray laser was suggested. 1964 American physicists Murey Gell-Mann and George Zweig made a hypothesis that all particles (adrons) consists of quarks—hypothetical material objects having a spin of 1/2, baryon charge of 1/3 and electric charge of +2/3e, +1/3e or −1/3e where e is the elementary charge. Thus a proton consists of three quarks with charges +2/3e, +2/3e and −1/3e (+e in sum). Although this hypothesis about quarks was introduced to explain spectroscopic regularities and dynamics of hadrons, they had not yet been observed in high-energy accelerators, cosmic rays, or in the surrounding media. 1965 First registration and identiﬁcation of spectral lines of Mg9+ , Fe8+ , Fe9+ and Fe16+ ions in the solar ﬂare on board of the cosmic station Electron-2 were performed. ˚ range for an 1967 First use of the x-ray (Schwarzshild) telescope in the λ ≈ 10 A x-ray image of the Sun was made. 1969 The ﬁrst Electron-Beam Ion Source (EBIS) was built in Dubna (Russia) by E D Donets. 1969 The Electron Cyclotron Resonance (ECR) ion source was ﬁrst proposed by R Geller and H Postma. 1969 First measurements of polarization (40% ± 20%) of the continuum xray radiation from solar ﬂares at 15 keV photon energy on board of the Intercosmos-1 satellite were carried out. 1970 English physicists Carol Jordan and Alan H Gabriel created the basis for modern x-ray plasma diagnostics. They have found that the relative intensities of the selective line pairs can be used to determine the local plasma temperature and density.

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1971 The UHURU satellite was launched which was the ﬁrst astronomical satellite to perform an all-sky survey at x-ray wavelengths. 1970–1972 First detection of the magneto-quadrupole M2-transition (x-line) and other characteristic lines (y, z and w) of Fe24+ ions as well as their ˚ range with dielectronic satellites in the solar ﬂare plasma in the 1.85–1.93 A λ/λ ≈ 10−4 resolution was made. 1972 German physicist E Spiller using computer simulation showed that by varying the layer thicknesses in a multilayer coating, a reﬂection up to tens per cent can be obtained in the VUV spectral range. 1972 E Chupp and collaborators detected for the ﬁrst time in the solar ﬂare the gamma rays—the extreme short-wavelength end of the electromagnetic spectrum—by an instrument on OSO-7 (Orbiting Solar Observatories) showing the presence of extremely energetic nuclear reactions. 1975 Discovery of the ﬁrst EUV star with a Berkeley EUV telescope ﬂown on the Soviet–American Apollo–Soyuz mission was made. 1976 German scientists R-P Haelbich and C Kunz manufactured the ﬁrst multilayer mirror with several periods of C–Au structure and recorded an enhancement of reﬂectivity in the VUV spectral range. 1977 Russian physicists Alexander V Vinogradov and Boris Ya Zel’dovich developed a comprehensive theory of x-ray mutilayer mirrors. 1978 Mission of the Einstein observatory with the ﬁrst satellite-borne x-ray telescope was performed which led to a gain in sensitivity and angular resolution by orders of magnitude. 1978 First observation of an extrasolar EUV star—the white dwarf HZ43—was performed on the Voyager planetary mission. 1982 Swiss scientists Gerd Binnig and Heinrich Rohrer invented a scanning tunneling miscroscope to investigate a surface structure of atoms and molecules. 1985 The ﬁrst laboratory x-ray laser was created at Lawrence Livermore National ˚ spectral Laboratory (USA) on Ne-like Se24+ ions in the λ = 182–263 A range. 1986 First creation of bare U92+ beams was carried out at BEVALAC accelerator at Berkeley. 1988 The principle of the Electron-Beam Ion Trap (EBIT) was introduced by D A Knapp.

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1990 The German satellite ROSAT (R¨oentgen Satellite) was launched which found 60 000 cosmic x-ray sources and 384 EUV sources of which only a dozen were known before. 1990 French physicist Jean Pierre Briand and co-workers introduced the term hollow atom—a highly-excited neutral atom with inner shells remaining transiently empty and created in collisions of slow highly charged ions with a metal surface. 1995 Eric Cornell and Carl Wieman observed Bose–Einstein condensates for the ﬁrst time—a unique state of matter in which all atoms exist in the same quantum state. 1999 L V Hau and co-workers observed, for the ﬁrst time, light pulses traveling at velocities of only 17 m s−1 through the cloud of ultra-cold sodium atoms at nanokelvin temperatures. 1999 Scientists from NIST/University of Colorado observed, for the ﬁrst time, a Fermi-degenerate gas of ultracold (290 nK) potassium atoms. 1999 Scientists from Arizona State University, using a combination of x-ray diffraction and electron microscopy, for the ﬁrst time imaged the electronic orbitals of copper atoms and their bounds with neighboring atoms in a Cu2 O compound.

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