THE INTERNATIONAL SERIES OF MONOGRAPHS ON PHYSICS SERIES EDITORS J.BIRMAN S F EDWARDS R FRIEND M.REES D SHERRINGTON G V E N E Z I A N O CITY UNIVERSITY OF NEW YORK U N I V E R S I T Y OF CAMBRIDGE UNIVERSITY OF C A M B R I D G E UNIVERSITY OF CAMBRIDGE U N I V E R S I T Y OF OXFORD CERN, GENEVA THE I N T E R N A T I O N A L SERIES OF M O N O G R A P H S ON PHYSICS 125 124 123 122 121 120 119 118 117 116 115 114 113 112 111 110 109 108 107 106 105 104 103 102 101 100 99 98 97 96 95 94 91 90 89 88 87 86 84 83 82 81 80 79 78 76 75 73 71 70 69 68 51 46 32 27 23 S Atzeni, J Meyer-ter-Vehn: Inertial Fusion C Kiefer: Quantum Gravity T Fujimoto: Plasma Spectroscopy K Fujikawa, H Suzuki: Path integrals and quantum anomalies T Giamarchi: Quantum physics in one dimension M Warner, E Terentjev: Liquid crystal elastomers L Jacak, P Sitko, K Wieczorek, A Wojs: Quantum Hall systems J Wesson: Tokamaks, Third edition G Volovik: The Universe in a helium droplet L Pitaevskii, S Stringari: Bose-Einstein condensation G Dissertori, I.G Knowles, M Schmelling: Quantum chromodynamics B DeWitt: The global approach to quantum field theory J Zinn-Justin: Quantum field theory and critical phenomena, Fourth edition R M Mazo: Brownian motion — fluctuations, dynamics, and applications H Nishimori: Statistical physics of spin glasses and information processing - an introduction N B Kopnin: Theory of nonequilibrium superconductivity A Aharoni: Introduction to the theory offerromagnetism, Second edition R Dobbs: Helium three R Wigmans: Calorimetry J Kübler: Theory of itinerant electron magnetism Y Kuramoto, Y Kitaoka: Dynamics of heavy electrons D Bardin, G Passarino: The Standard Model in the making G C Branco, L Lavoura, J P Silva: CP Violation T C Choy: Effective medium theory H Araki: Mathematical theory of quantum fields L M Pismen: Vortices in nonlinear fields L Mestel: Stellar magnetism K H Bennemann: Nonlinear optics in metals D Salzmann: Atomic physics in hot plasmas M Brambilla: Kinetic theory of plasma waves M Wakatani: Stellarator and heliotron devices S Chikazumi: Physics of ferromagnetism R A Bertlmann: Anomalies in quantum field theory P K Gosh: Ion traps E Simánek: Inhomogeneous superconductors S L Adler: Quaternionic quantum mechanics and quantum fields P S Joshi: Global aspects in gravitation and cosmology E R Pike, S Sarkar: The quantum theory of radiation V Z Kresin, H Morawitz, S A Wolf: Mechanisms of conventional and high Tc super-conductivity P G de Gennes, J Prost: The physics of liquid crystals B H Bransden, M R C McDowell: Charge exchange and the theory of ion-atom collision J Jensen, A R Mackintosh: Rare earth magnetism R Gastmans, T T Wu: The ubiquitous photon P Luchini, H Motz: Undulators and free-electron lasers P Weinberger: Electron scattering theory H Aoki, H Kamimura: The physics of interacting electrons in disordered systems J D Lawson: The physics of charged particle beams M Doi, S F Edwards: The theory of polymer dynamics E L Wolf: Principles of electron tunneling spectroscopy H K Henisch: Semiconductor contacts S Chandrasekhar: The mathematical theory of black holes G R Satchler: Direct nuclear reactions C Møller: The theory of relativity H E Stanley: Introduction to phase transitions and critical phenomena A Abragam: Principles of nuclear magnetism P A M Dirac: Principles of quantum mechanics R E Peierls: Quantum theory of solids Plasma Spectroscopy T A K A S H I FUJIMOTO Department of Engineering Physics and Mechanics Graduate School of Engineering Kyoto University C L A R E N D O N PRESS O X F O R D 2004 OXFORD UNIVERSITY PRESS Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Bangkok Buenos Aires Cape Town Chennai Dar es Salaam Delhi Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Sao Paulo Shanghai Taipei Tokyo Toronto Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Oxford University Press 2004 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2004 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, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer A catalogue record for this title is available from the British Library Library of Congress Cataloging in Publication Data (Data available) ISBN 19 8530285 (Hbk) 10 Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in India on acid-free paper by Thomson Press (India) Ltd PREFACE Throughout the history of spectroscopy, plasmas have been the source of radiation, and they were studied for the purpose of spectrochemical analysis and also for the investigation of the structure of atoms (molecules) and ions constituting these plasmas About a century ago, the spectroscopic investigations of the radiation emitted from plasmas contributed to establishing quantum mechanics However, the plasma itself has been the subject of spectroscopy to a lesser extent This less-developed state of plasma spectroscopy is attributed partly to the complicated relationships between the state of the plasma and the spectral characteristics of the radiation it emits If we are concerned with the intensity distribution of spectral lines over a spectrum, we have to understand the population density distribution over the excited levels of atoms and ions in the plasma Since the latter distribution is governed by a collection of an enormous number of atomic processes, e.g electron impact excitation, deexcitation, ionization, recombination, and radiative transitions, and since the spatial transport of the plasma particles and the temporal development are sometimes essential as well, it is rather difficult, by starting from these elementary processes, to deduce a straightforward consequence concerning the population distribution For certain limiting conditions of the plasma, e.g for the low- or high-density limit, several concepts like corona equilibrium and local thermodynamic equilibrium have been proposed, but they have been accepted on a rather intuitive basis This book is intended first to provide a theoretical framework in which we can treat various features of the population density distribution over the excited levels (and the ground state) of atoms and ions and give their interpretation in a unified and coherent way In this new framework several concepts, some of which are already known and some newly derived, are properly defined For these purposes, we take hydrogen-like ions (and neutral hydrogen) as an example of an ensemble of atoms and ions immersed in a plasma Following the first three introductory chapters, these problems are discussed in the subsequent two chapters The following three chapters are devoted to several facets which are useful in performing a spectroscopy experiment This volume concludes with a chapter treating several phenomena characteristic of dense plasmas This chapter may be regarded as an application of the theoretical methods developed in the first part of the volume The main body of this book is based on my half-year course given at the Graduate School, Kyoto University, for more than a decade This book is intended mainly for graduate students, but it should also be useful for researchers working in this field A reader who wants to obtain only the basic ideas may skip the chapters and sections marked with an asterisk vi PREFACE In writing this book I owe thanks to many colleagues and students First of all, Professor Otsuka is especially thanked for his careful reading of the whole manuscript and for pointing out errors, and giving me critical comments and valuable suggestions Professor Kato and M Goto provided me with their valuable unpublished spectra for Chapter Various materials in Chapters and were taken from publications by my former students, K Sawada, T Kawachi, and M Goto These and A Iwamae, my present colleague, created many beautiful figures for this book I am also grateful to Dr Baronova for her comments, which have made this book more or less comprehensive She also helped in some parts of the book If this book is quite straightforward for beginners, that is due to my students, Y Kimura and M Matsumoto, who gave me various comments and questions as students I would like to express my thanks to workers who permitted me to reproduce their figures in this book Professors Xu and Zhu even modified their original figure so as to fit better into the context of this book Mrs Hooper Jr who gave me permission to use a figure on behalf of her recently deceased husband The names of these workers and the copyright owners are mentioned in the reference section and the figure captions in each chapter CONTENTS List of symbols and abbreviations ix Introduction 1.1 Historical background and outline of the book 1.2 Various plasmas 1.3 Nomenclature and basic constants 1.4 z-scaling 1.5 Neutral hydrogen and hydrogen-like ions 1.6 Non-hydrogen-like ions 1 12 13 14 15 19 Therniodynaniic equilibrium 2.1 Velocity and population distributions 2.2 Black-body radiation 22 22 25 Atomic processes 3.1 Radiative transitions 3.2 Radiative recombination 3.3 Collisional excitation and deexcitation 3.4 lonization and three-body recombination *3.5 Autoionization, dielectronic recombination, and satellite lines *3.6 Ion collisions Appendix 3A Scaling properties of ions in isoelectronic sequence *Appendix 3B Three-body recombination "cross-section" 30 31 42 48 59 64 72 76 79 Population distribution and population kinetics 4.1 Collisional-radiative (CR) model 4.2 Ionizing plasma component 4.3 Recombining plasma component - high-temperature case 4.4 Recombining plasma component - low-temperature case 4.5 Summary and concluding remarks *Appendix 4A Validity of the statistical populations among the different angular momentum states *Appendix 4B Temporal development of excited-level populations and validity condition of the quasi-steady-state approximation 83 83 96 111 120 131 Ionization and recombination of plasma 5.1 Collisional-radiative ionization 5.2 Collisional-radiative recombination - high-temperature case 5.3 Collisional-radiative recombination - low-temperature case 150 151 157 163 134 136 viii CONTENTS 5.4 lonization balance 5.5 Experimental illustration of transition from ionizing plasma to recombining plasma Appendix 5A Establishment of the collisional-radiative rate coefficients Appendix 5B Scaling law *Appendix 5C Conditions for establishing local thermodynamic equilibrium *Appendix 5D Optimum temperature, emission maximum, and flux maximum 167 182 188 190 191 202 Continuum radiation 6.1 Recombination continuum 6.2 Continuation to series lines 6.3 Free-free continuum - Bremsstrahlung 205 205 207 211 *7 Broadening of spectral lines 7.1 Quasi-static perturbation 7.2 Natural broadening 7.3 Temporal perturbation - impact broadening 7.4 Examples 7.5 Voigt profile 213 214 218 219 224 233 *8 Radiation transport 8.1 Total absorption 8.2 Collision-dominated plasma 8.3 Radiation trapping Appendix 8A Interpretation of Figure 1.5 236 236 240 245 252 *9 Dense plasma 9.1 Modifications of atomic potential and level energy 9.2 Transition probability and collision cross-section 9.3 Multistep processes involving doubly excited states 9.4 Density of states and Saha equilibrium 257 257 261 266 277 Index 286 LIST OF SYMBOLS AND ABBREVIATIONS first Bohr radius atomic units autoionization probability for (p,nl') Einstein's A coefficient or transition probability for p —> q line absorption stabilizing radiative transition probability Einstein's B coefficient for photoabsorption and for induced emission b(p) population normalized by the Saha-Boltzmann value B z – (T e ) partition function BV(T) black-body radiation distribution or Planck's distribution function C(p, q) excitation rate coefficient E kinetic energy of an electron, energy of level EG energy of Griem's boundary level with respect to the ground state E(p, q) energy separation between level p and q Ei(–x) exponential integral f(u), f(E) ) electron velocity (energy) distribution function fqoqscillator strength for transition p —> qition p q fp,c oscillator strength for photoionization from level p h Fc hctric field strength of the plasma microfieldld F0 normal field strength F(q,p) deexcitation rate coefficient G scale factor for excitation or deexcitation rate coefficient g(p) statistical weight of level p g(E/R) ) density of states per unit energy interval ge degeneracy of electron (=2) gbb, gbt, gft Gaunt factor G(a) reduced density of states h Planck's constant, ratio of quasi-static broadening to impact broadening h Planck's constant divided by 2p H scale factor for radiative decay rate / scale factor for continuum radiation k Boltzmann's constant K scale factor for radiative decay rate \gx log^ Inx logex a0 au A a (p,nl )) A(p, q) AL Ar B(p, q) q q q hh h MULTISTEP PROCESSES WITH DOUBLY EXCITED STATES 273 FIG 9.8 Deexcitation rate coefficient of lithium-like aluminum ions in the recombining phase in a dense plasma Horizontal lines: the direct deexcitation rate coefficient for transitions between the singly excited lithium-like aluminum levels Thin lines: extended Griem's boundary and Byron's boundary for 3pq and 3dq doubly excited levels the DL deexcitation rate coefficient for 3d —> 2s; for 3p —> 2s; — • — • — the CR recombination rate coefficient which is common to the low-lying levels (Quoted from Kawachi and Fujimoto, 1990; copyright 1990, with permission from The American Physical Society.) boundary for the doubly excited (3dn/') and (3p«/') levels are shown with the thin lines The DL deexcitation rate coefficient is given with the thick solid and dashed lines The CR recombination rate coefficient is given with the dash-dotted line In this low electron temperature (remember that the reduced 274 DENSE PLASMA temperature in the present example is r e /z ~450K, where z is 10), the last loss mechanism is predominant at higher densities Figure 9.9 compares the calculated population distribution over the singly excited levels in a Boltzmann plot by the conventional CR model and that with these additional recombination and all the deexcitation processes included The difference is substantial, especially for low-lying levels This figure also includes the result of experimentally determined populations, indicating good agreement with the calculation which includes all the above processes Thus, this figure demonstrates the validity of the above theory FIG 9.9 The populations of a recombining lithium-like aluminum plasma in the Boltzmann plot A: result of calculation by the conventional CR model O: result of the CR model calculation with the DL deexcitation and CR recombination processes included 0A: experiment (Quoted from Kawachi et al., 1999; with permission from TOP Publishing.) MULTISTEP PROCESSES WITH DOUBLY EXCITED STATES 275 Decrease and disappearance of resonance contributions to excitation cross-section In Section 3.5 we introduced the resonance contributions to the excitation crosssection by taking the ls^2s transition in hydrogen-like ions as an example See eq (3.51) and Fig 3.21 See also Fig 9.10 Figure 9.11 is another plot of Fig 3.21; the resonance contributions, which appear as sharp peaks in the latter figure, have been averaged over energy, and in the present figure they appear as smooth crosssections added on the top of the direct excitation cross-section The energy range of Fig 3.21 is from 1.02 to 1.21 keV Contributions from the (3s, ri), (3p,«), and (3d, K) levels are shown separately It would be natural to assume that, in a dense plasma, the intermediate state, (3p,«) for example, may suffer electron collisions before it autoionizes Compare eq (3.51) with eq (9.5): the first lines are common and, in a dense plasma, instead FIG 9.10 Schematic energy-level diagram of hydrogen-like Is, 2s, and 3p levels with the accompanying doubly excited helium-like levels The doubly excited levels 3p« are populated by the dielectronic capture from the ground state It is depopulated by autoionization to the ground state and to 2s, the latter of which results in the resonance contribution to the excitation cross-section for Is —> 2s, stabilizing radiative transition and collisional excitation which results in the DL excitation of Is—>3p 276 DENSE PLASMA FIG 9.11 Another plot of Fig 3.21; the excitation cross-section for Is —> 2s, plus the resonance contributions through the doubly excited levels 3s«, 3p«, 3d« These contributions are lost owing to the development of the DL excitation process in these doubly excited levels The corresponding increase in the direct excitation of, say ls^3p, is expressed as extrapolation of the excitation crosssection below the excitation threshold down to extended Griem's boundary level Similar extrapolation is given to the direct ls^2s excitation crosssection (Quoted from Fujimoto and Kato, 1987; copyright 1987, with permission from The American Physical Society.) of the second line of eq (3.51), eq (9.5b) becomes dominant We define extended Griem's boundary again for the doubly excited levels, say (3p,ri) Then the flux of dielectronic capture into the levels lying higher than this boundary will enter into the ladder-like excitation chain to result in the DL excitation of 3p Thus, the corresponding part of the resonance contribution to the excitation cross-section Is —> 2s is lost Figure 9.11 illustrates this situation For a certain electron density the part of the resonance cross-section with energies higher than the energy of extended Griem's boundary is lost For each of the contributions, (3s, ri), (3p,ri), and (3d,ri),the energy positions of extended Griem's boundary levels are given, and the part of the resonance contributions at higher energies are lost This part instead contributes to the DL excitation of the core singly excited level In this figure, the excitation cross-section Is —> 3p is extrapolated to this energy, eq (9.10), and the excitation cross-section Is —> 2s is also extrapolated below the threshold to DENSITY OF STATES AND SAHA EQUILIBRIUM 277 account for the DL excitation introduced in the preceding subsection It is noted that, in the electron densities considered in this figure, ne < 1028 m~3, the energylevel structure and the excitation cross-section are almost unaffected See Fig 9.1 (b) for the former and Fig 9.4 for the latter: RQ > 100a0 for the present example 9.4 Density of states and Saha equilibrium Density of states In Chapter 2, in considering thermodynamic equilibrium, we derived the SahaBoltzmann equilibrium relationship In doing so, we implicitly assumed an isolated atom In the case of a hydrogen atom, the statistical weight of a level with principal quantum number p is 2p2 Thus, the number of states of a level over one principal quantum number (remember Fig 1.11 (a)) is The energy width corresponding to one principal quantum number is See eq (1.5) This quantity is understood as the energy width allocated to this level p, which is 2p2-fold degenerate Thus, we may define the number of states in a unit energy interval, or the density of states, g(p)dp/dE, or where we use the rydberg units of energy In Fig 9.12 the smooth curve in the negative energy region and the dotted curve connecting with it, represents eq (9.13) It is obvious that eq (9.13) diverges toward the ionization limit, or zero energy This is a natural consequence of our assumption of isolated atoms which has an infinite number of Rydberg levels In the preceding chapters the continuum states of electrons were approximated as free states An example is the Maxwell distribution of electron energies, eqs (2.2) and (2.2a) In this case, the density of states is given from eq (2.5a) with ge = and g(l) = as This density of states is inversely proportional to electron density, and is shown in Fig 9.12 in the positive energy region with the dash-dotted curves for several values of ne A strong discontinuity is seen at the zero energy This is the result of 278 DENSE PLASMA FIG 9.12 Density of states of neutral hydrogen in the energy region close to the ionization limit connecting to in negative energy: for an isolated hydrogen atom, eq (9.13); — • — • — for free electrons, eq (9.14); in positive energy, extending across the ionization limit to negative energies: the density of states on the basis of the ion sphere type model, eqs (9.22)-(9.26b) (Quoted from Shimamura and Fujimoto, 1990; copyright 1990, with permission from The American Physical Society.) our approximations which are inconsistent each other: i.e an isolated atom for negative energies and free electrons for positive energies We now remember that our atoms and ions are in a plasma; we assume a plasma made of protons and electrons In reality, the electron states with negative energies are affected by plasma particles as we have seen in Section 9.1, and the electrons with positive energies move in the Coulomb potentials of protons under the influence of other plasma particles If we intend to resolve the difficulty above, DENSITY OF STATES AND SAHA EQUILIBRIUM 279 these effects should be properly taken into account As we have noted already in Section 9.2, this poses enormous difficulties Instead, we adopt here a rather crude model We start with a model potential of the ion sphere type for an electron: and treat the electron motion classically To be consistent with eq (9.1), we are using here the convention that an electron is regarded to have a positive charge e We first consider an electron having a high energy E so that its trajectory is virtually a straight path inside the sphere Let p0 be the momentum and r0 be the distance of the closest approach to the proton We may define the angular momentum quantum number semiclassically Then we may assign to this angular momentum the number of states 2(27 + 1), i.e the number of the directions of the angular momentum (space quantization) times that of the electron spin We have the transit time We assign the energy width to this electron state To help justify this reasoning, see the discussions around eq (1.6) From the above we may define the density of states for this angular momentum The density of states for the present E is given by the summation of that over /: where lc is the cut-off angular momentum which is defined by eq (9.16) with r0 replaced by R0, For sufficiently high energy, eq (9.19) may be approximated to 280 DENSE PLASMA Here we have used eq (7.6) for R0 = pm and E =pQ/2m2, We have arrived at an expression that is exactly the same as eq (9.14) Thus, by following the above procedure we are able to obtain the "correct" density of states of free electrons For lower energies, we follow similar procedures; instead of the straight path we adopt hyperbolic trajectories for positive energies and elliptic trajectories for negative energies We calculate the density of states from the transit times of the electron over the sphere We define the units of energy and the dimensionless energy We rewrite eqs (9.13) and (9.14) in the form with Then \X\ 5//2G(o), which may be called the reduced density of states, is independent of «e Figure 9.13 shows the reduced density of states of eqs (9.13) and (9.14) with the thin dashed line for negative energy and the dash-dotted line for positive energy, respectively For positive energy, X> 0, the electron trajectory is hyperbolic as shown in Fig 9.14(a) From arguments similar to those leading to eq (9.19a) we obtain an analytical expression The result is plotted on Fig 9.13 with the solid line for X > It is noted that this tends to a finite value at the null energy For slightly negative energies, — < X< 0, the major axis of the elliptic electron orbit is so large that the orbit extends outside the ion sphere The circular orbit is absent because its radius is larger than the ion sphere radius, R0, See Fig 9.14(b) By following a similar procedure we obtain the expression DENSITY OF STATES AND SAHA EQUILIBRIUM 281 FIG 9.13 Reduced density of states See text for details (Quoted from Shimamura and Fujimoto, 1990; copyright 1990, with permission from The American Physical Society.) which is plotted in Fig 9.13 with the solid line for —1 < X < Q This curve continues smoothly from the curve, eq (9.24), at the null energy At X=—\ the radius of the circular orbit is equal to the ion sphere radius This is seen from eq (1.2), i.e n = ^/Ro/a0, and the energy is given from eq (1.1), i.e E(ri) = —Ra0/R0, For still lower energies, —2