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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 0 19 8530285 (Hbk) 10 9 8 7 6 5 4 3 2 1 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 1. Various materials in Chapters 4 and 5 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

1 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 1 12 13 14 15 19

2 Therniodynaniic equilibrium 2.1 Velocity and population distributions 2.2 Black-body radiation

22 22 25

3 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

4 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

5 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

6

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 – 1 (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 1 )) A(p, q) AL Ar B(p, q)

q q q

hh

h

x

LIST OF SYMBOLS AND ABBREVIATIONS

hm

electron mass, magnetic quantum number

H n

ion mass

n*

effective principal quantum number

ne n H p) n 0 (p)) n1(p)

electron density

z

principal quantum number, population of upper level

density of ions in the next ionization stage population (density) of level p

recombining plasma component ionizing plasma component n1, n parabolic quantum numbers 2 N perturber particle density, density of ground-state atoms p designation of a level, momentum of an electron PG Griem's boundary level /IB Byron's boundary level P R p (v) ) recombination continuum radiation power P L p (v) ) line radiation power PR+B(v) ) radiation power of recombination continuum and Bremsstrahlung r d(p, nl') dielectronic capture rate coefficient for (p, nl') r 0 (P), r 1 (p)) population coefficient R Rydberg constant, Radius of cylinder RD Debey radius R0 mean distance between perturbers, ion sphere radius Ry Rydberg units SCR collisional-radiative ionization rate coefficient S(p)) ionization rate coefficient tres response time of populations of excited levels tr1(p) relaxation time of population n(p) ttr(p) transient time for population of level p T(x, y) Stark profile with ion and electron broadening Te electron temperature Teo optimum temperature u excitation or ionization energy in threshold units v speed of an electron W three-body recombination flux, equivalent width W(b) field distribution function z ze is the nuclear charge of the next ionization-stage ion Z Ze is the charge of perturber particles Z(p) Saha-Boltzmann coefficient Zp(pq) Saha-Boltzmann coefficient with respect to the energy position of level p a fine structure constant aCR collisional-radiative recombination rate coefficient a(p) three-body recombination rate coefficient

LIST OF SYMBOLS AND ABBREVIATIONS

a d (l, nl')

dielectronic recombination rate coefficient into (1, nl')

b b(p)

FlF0

r

7 A DX DwD Dw1/2

n nv Kv

q q(Te,ne)

xi

radiative recombination rate coefficient coupling parameter full width at half-maximum, decay rate of energy shift of the line center lowering of ionization potential Doppler half-width half-width at half-maximum reduced electron density, ne/z7 emission coefficient absorption coefficient

reduced electron temperature, Te/z2

Wp,q(E)

correction factor to Saha equation impact parameter impact parameter of Weisskopf radius mean distance between perturbers real and imaginary parts of the impact broadening cross-section excitation or deexcitation cross-section photoabsorption cross-section for excitation p —> q photoionization cross-section radiative recombination cross-section ionization cross-section mean free time period of one revolution of the electron in level n period of one revolution of the electron in the first Bohr orbit transit time optical thickness atomic unit velocity autocorrelation function intensity or radiant flux or radiant power of emission radiation for transition p —> q ionization potential of level p central (angular) frequency of a spectral line collision strength

suffixF suffixH suffix0 suffix¥ suffix¥ + suffixB suffixD suffixE

free electron indicating the quantity for neutral hydrogen quantity in the low-density limit quantity in region II quantity in region III Byron's boundary Doppler relationship valid in thermodynamic equilibrium

p

Po Pm sr, si

sp,q(u), sp,q(E) sp,q(u) sp,e(v)

h

se,p(e) sp,c(u) T Tn

TB Ttr Tv

C

F(s), F(s)

F(p,q) x(p) w0

xii

LIST OF SYMBOLS AND ABBREVIATIONS

suffixG suffixH suffixIB suffixL suffixv suffixw

Griem's boundary Holtsmark ionization balance Lorentzian Voigt Weisskopf

CR DL FWHM l.h.s. LTE QSS r.h.s.

collisional-radiative dielectronic capture ladder-like full width at half-maximum left-hand side local thermodynamic equilibrium quasi-steady state right-hand side

1 INTRODUCTION 1.1 Historical background and outline of the book

The history of spectroscopy began more than three hundred years ago with the experiment by Newton in which sunlight was dispersed by a prism into light rays which bore the colors of a rainbow (Fig. 1.1). Later, mainly in the nineteenth century, when the instrument called the spectroscope was used to observe the spectra of radiation emitted from various plasmas, i.e. flames, the Sun and several stars, and later electric arcs and sparks (see Fig. 1.2), an enormous number of spectral lines were found as emission or absorption lines. As a result of the invention of the photographic plate, or of the spectrograph, spectroscopy developed into a science of very high precision in terms of wavelength of observed lines. Numerous attempts were made to find regularities manifested by these lines. In the beginning of the twentieth century the experimentally established laws governing the wavelengths, or the frequencies, of the lines characteristic of atoms and ions, together with the spectral characteristics of the black-body radiation, played an essential role in establishing quantum mechanics. Atomic spectroscopy, which deals primarily with wavelengths of spectral lines, is still actively studied to establish the energy-level structure of complicated atoms

FIG 1.1 The sketch of "the critical experiment" drawn by Newton himself. (By permission of the Warden and Fellows, New College, Oxford.)

2

INTRODUCTION

FIG 1.2 The "map" of various plasmas. NFR means the future nuclear fusion plasma; "laser" means laser-produced plasmas. The oblique line shows the scaling law for neutral hydrogen and hydrogen-like ions according to the nuclear charge z. See the text for details. and highly ionized ions. The intensities of these lines are of concern mainly from the viewpoint of determining the ionization stage of the ions emitting the line and the strength of the transition, i.e. the oscillator strength and the multipolarity. (These terms are explained in a subsequent part of this book.) Other characteristics of the spectrum, e.g. broadening of the lines, have less significance to atomic spectroscopists.

HISTORICAL BACKGROUND AND OUTLINE OF THE BOOK

3

The reason why characteristics except for the wavelengths are unimportant in atomic spectroscopy lies in the fact that these spectral characteristics are ephemeral rather than basic, or they are dependent on the conditions under which this particular plasma is produced and on the parameter values this plasma has. This very fact constitutes the starting point of plasma spectroscopy. Thus, plasma spectroscopy deals with these variable characteristics of the radiation emitted from the plasma in relation to the plasma itself, which is regarded as an environment of the atoms and ions emitting the radiation. It may be interesting to note that the intensity (the radiant flux or the radiant power is a more precise term; see later) has been a quantity which is difficult to determine experimentally. In particular, its absolute value could be determined only in favorable situations. However, developments in techniques, i.e. photomultipliers for the last half-century, and multichannel detectors with digital signal processing techniques in recent years, have enabled us to perform quantitative spectroscopy much more easily. This situation is favorable for the development of quantitative plasma spectroscopy as treated in this book. When we look at a plasma, laboratory or celestial as shown in Fig. 1.2, through a spectrometer, we find a spectrum of radiation emitted from this plasma. This is a pattern of spectral lines (and continuum), with varying intensities, distributed over a certain wavelength range. Figure 1.3 shows an example of the spectrum from a plasma. This plasma is produced from a helium arc discharge plasma streaming along magnetic field lines into a dilute helium gas. The distribution pattern of lines in terms of wavelength reflects, of course, the energy-level structure of atoms, or the composition of the plasma: what atomic species constitute the plasma. The spectrum of Fig. 1.3 is of neutral helium. Figure 1.4 is the energy-level diagram, called the Grotorian diagram, of neutral helium. It is straightforward to identify the lines in Fig. 1.3 with transitions each connecting two levels in this diagram. The thin solid lines show these identifications. We sometimes find that two plasmas having identical wavelength-distribution patterns show different intensity-distribution patterns. Figure 1.5 shows an example; both the spectra are of the resonance series lines of ionized helium (hydrogen-like helium), terminating on the ground state as shown in Fig. 1.6. The plasmas producing these spectra are essentially the same as that for Fig. 1.3. It may be said that the two plasmas in Fig. 1.5 are of the same composition but show different "colors". This difference might be attributed to different temperatures of these plasmas; a plasma with a higher temperature tends to emit intense lines having shorter wavelengths. This suggestion may be supported for two reasons: first, the higher the energy of electrons in the plasma the higher the energy of atomic states excited by them (see Fig. 1.6) and thus the shorter the wavelengths of the light originating from these states; second, the higher the temperature the shorter the peak wavelength of the black-body radiation, i.e. Wien's displacement law (see Chapter 2). We will see later (Chapters 5 and 8) whether our conclusion here is adequate or not.

FIG 1.3 An example of the spectra observed from a plasma. This is the near-ultraviolet part of the spectrum of radiation emitted from a helium plasma. Several series of lines of neutral helium and recombination continua are seen. (Plasma produced in the TPD-I machine, Institute for Plasma Physics, Nagoya. Quoted from Otsuka M., 1980 Japanese Journal of Optics (in Japanese), 9, 149; with permission from The Japanese Journal of Optics.)

HISTORICAL BACKGROUND AND OUTLINE OF THE BOOK

5

FIG 1.4 Energy-level diagram of neutral helium. The thin solid lines show the transitions corresponding to the emission lines in Fig. 1.3. The dashed lines show the transitions of Fig. 3.6. The dotted line shows the emission line which appears in Fig. 5.17(b). The dash-dot lines appear as emission lines in Figs. 7.4 and 7.6. When we look closely at a single spectral line, we sometimes find that its intensity is distributed over a narrow but finite wavelength region, with its peak displaced from the original position of the line where it is found under normal conditions. We can find several broadened lines in Fig. 1.3 in the longerwavelength region, as contrasted with the accompanying sharp lines. We also find prominent examples in Fig. 1.7. We may also find continuum spectra underlying the spectral lines. An example is seen in Fig. 1.3. Figure 1.7 shows another example. For certain plasmas like these examples the continuum is weaker than the lines but for some others it is even stronger than the lines. The plasma of Fig. 1.7 is produced from a hydrogen pellet (frozen hydrogen ice) injected into a high-temperature plasma. A dense plasma is produced from evaporated hydrogen. Several broadened lines, tending to a continuum, of neutral hydrogen atoms are seen. These lines correspond in Fig. 1.6 to the transitions

6

INTRODUCTION

FIG 1.5 Two spectra from plasmas produced under slightly different conditions. The spectral lines are the resonance series lines (1 2S —« 2 P) of ionized helium. (TPD-I. By courtesy of Professor T. Kato.) The asterisk shows the real peak position when the saturation effect of the detector is corrected for. terminating on the n = 2 levels. Of course, the transition energies are about onequarter for these lines, because Fig. 1.6 is for hydrogen-like helium ions, not neutral hydrogen in Fig. 1.7. As mentioned earlier, all these variable characteristics of the spectrum of atoms and ions are dependent on the nature of the plasma which emits the radiation. In other words, the spectrum contains information about the plasma: i.e. it is the fingerprint of the plasma. This notion constitutes the basis of plasma diagnostics or using the observed spectrum to infer the characteristics of the plasma, e.g. its temperature, density, and particle transport property over space. The first task of plasma spectroscopy would naturally be to find and establish the relationships between the characteristics of the emission-line (and continuum) intensities from a plasma and the nature of this plasma. Since a spectral line

HISTORICAL BACKGROUND AND OUTLINE OF THE BOOK

7

FIG 1.6 Energy-level diagram of ionized helium, where the different / and mi states are reduced to a single level specified by n. The series of transitions terminating on the ground state correspond to the emission lines in Fig. 1.5. The transitions terminating on the n = 1 level correspond to the emission lines in Fig. 1.7, except that this diagram is for hydrogen-like helium and Fig. 1.7 is for neutral hydrogen. is emitted by excited atoms or ions (see Figs. 1.4 and 1.6) its intensity is given by the number density of these atoms or ions. Here we have assumed that the identification of the spectral line, or the correspondence of the line to the upper and lower levels of atoms or ions, is established and that the transition probability is known for this transition. We have also ignored intricate factors, like polarization and reabsorption of radiation (see Chapter 8), both of which can affect the observed line intensity. Thus, the problem of the intensity distribution reduces to that of the population-density (called simply the population henceforth) distribution of atoms and ions over excited levels (and in the ground state, too). Once these relationships are established, several conventional concepts will turn out to be incorrect. For example, the interpretation mentioned above concerning the temperature and the "color" (Fig. 1.5) will be found to be too naive; sometimes a hot plasma may look more "red" than some colder plasmas having the same composition.

8

INTRODUCTION

FIG 1.7 A spectrum of neutral hydrogen atoms from a plasma produced by pellet (a solid hydrogen ice) injection into a high-temperature plasma. (Produced at the LHD in the National Institute for Fusion Science, Toki. By courtesy of Dr. M. Goto.) The primary objective of this book is to provide the reader with a sound basis for interpreting various features manifested by a spectrum of radiation emitted from a plasma in terms of the characteristics of the plasma. The first two chapters are intended so that the reader acquires the background necessary to proceed to the main part of the book developed in subsequent chapters. First, thermodynamic equilibrium relationships are discussed for the discrete-level populations, for the ionization balance, and for the radiation field. The subsequent chapter discusses the atomic processes important in plasmas, i.e. the spontaneous radiative transition and the transitions due to electron impact. It is pointed out that, for a pair of levels, a single parameter, the absorption oscillator strength, which gives the radiative transition probability, also determines the collisional excitation cross-section, although to a limited extent. Another important fact worth noting is that various features associated with high-lying excited states continue smoothly across the ionization limit to those associated with low-energy continuum states. This is a natural consequence of the continuity of the corresponding wavefunctions of the atomic electron and the ion (the continuum-state electron). Chapters 4 and 5 present a theoretical framework in which the experimentally observed population distribution is interpreted in terms of various characteristics of the plasma. In Chapter 4 we introduce the method known as the collisionalradiative (CR) model or the method of the quasi-steady-state (QSS) solution.

HISTORICAL BACKGROUND AND OUTLINE OF THE BOOK

9

By this method we treat in a coherent manner the population couplings among the excited levels (and the ground state) in the population formation by the collection of atomic processes. Figure 1.8 shows schematically the energy-level structure; p or q denotes a level and/> = 1 means the ground state. As an example of ensembles of atoms and ions immersed in a plasma and emitting radiation we take hydrogenlike ions (and neutral hydrogen) for the purpose of illustration. A discussion of the validity of this method, or of the QSS solution, is given in Appendix 4B. Then an excited-level population is expressed as a sum of the ionizing plasma component and the recombining plasma component. Figure 1.9 shows schematically the structure of the populations. For both the components the population distribution and its kinetics are examined in detail. Figure 1.10 is the "map" of the populations of these components, or the summary of our investigations in Chapter 4. Several characteristic population distributions, e.g. the minus sixth power distribution, and the corresponding population kinetics, i.e. the ladder-like excitation-ionization

FIG 1.8 Schematic energy-level diagram of an atom or ion with symbols used in this book.

FIG 1.9 The structure of the excited-level populations in the collisional-radiative model. The population n(p) is the sum of the ionizing plasma component n^(p) which is proportional to the ground-state population «(1) and the recombining plasma component n0(p) proportional to the "ion" density nz. Full explanations are given in Chapter 4.

FIG 1.10 The "map" of the excited-level populations of neutral hydrogen and hydrogen-like ions in plasma. This diagram is the summary of our investigations to be developed in Chapter 4, so that a reader who has started to read this book does not need to understand the details of this diagram. The abscissa is the (reduced) electron density and the ordinate is the principal quantum number of excited levels, (a) The ionizing plasma component; (b) the recombining plasma component. Griem's boundary pG, given by eq. (4.25), (4.29), or (4.59), divides the whole area into a low-density region and highdensity region. Byron's boundary />B, given by eq. (4.55) or (4.56), divides the high-density region into low-lying levels and high-lying levels. In each area, the name of the phase I, the population distribution, and the dominant population kinetics are shown for level p with which we are concerned. For the capture-radiative-cascade (CRC) phase in (b) the near-SahaBoltzmann population is for the high-temperature case. In practical situations of the ionizing plasma (a) Byron's boundary lies far below p = 2, and only the saturation phase with the multistep ladder-like excitation-ionization mechanism appears in the high-density region. (Quoted from Fujimoto, T. 1980 Journal of the Physical Society of Japan, 49, 1591, with permission from The Physical Society of Japan.)

HISTORICAL BACKGROUND AND OUTLINE OF THE BOOK

11

mechanism, are established. Two important boundaries, Griem's boundary and Byron's boundary, are derived for electron density and temperature, or for excited levels. It is noted that the strong population coupling among the excited levels and its continuation to the ionic (continuum) states play an essential role in determining the population distributions in both the components. In a plasma an ensemble of atoms or ions as a whole may be in a dynamical process of ionization or recombination, depending on the time history and the spatial structure of the plasma. In these processes, in addition to the direct ionization and recombination, excited levels play essential roles, too, and they affect the effective rate of ionization and that of recombination. This subject is examined in Chapter 5. An important finding is that the ionization process is associated with the ionizing plasma component of populations and thus the magnitude of the ionization flux is proportional to the excited-level population. A similar proportionality is also valid for the recombination flux and the recombining plasma component. In both the cases, the proportionality factors naturally depend on the parameters of the plasma. Thus, an emission line intensity is a measure of the ionization flux and that of the recombination flux, depending on the nature of the plasma. There is a class of plasmas in which the ionization flux and the recombination flux balance with each other, or plasmas in ionization balance. An important conclusion is reached for this class of plasmas: the ratio of the contributions to an excited-level population from the ionizing plasma component and from the recombining plasma component could be comparable in magnitude. This finding leads to another even more important conclusion: for many actual plasmas which are out of, sometimes far from, ionization balance, only one of the two components dominates the actual populations while the other gives a negligibly small contribution. Thus we have reached an important step for a correct explanation of the different spectra in Fig. 1.5. Another important finding is that, among plasmas in various states of ionization-recombination, a plasma in ionization balance gives rise to the minimum of radiation intensity. Plasmas out of ionization balance emit much stronger radiation. Following these two chapters of primary importance, we now turn to other facets which together constitute plasma spectroscopy. Chapter 6 treats the continuum radiation. The spectral characteristics are examined for the recombination continuum, and a smooth continuation is established for its intensity to those of the accompanying series lines, as is observed in Figs. 1.3 and 1.7. This continuation is interpreted as the continuation of the "populations" of the discrete states to the continuum states. The problems of broadening and shift of spectral lines follow in Chapter 7. We have already seen some examples in Figs. 1.3 and 1.7. This aspect is important for determining the atom (ion) temperature or the plasma density. Besides the Doppler broadening and Stark broadening in the quasi-static approximation, natural broadening and impact broadening is treated in a rather elementary way. The latter class of broadening is regarded as the relaxation of optical coherence.

12

INTRODUCTION

Chapter 8 treats the phenomena associated with radiation transport. We first examine how the absorption line profile develops in an absorbing medium. Then we describe how the observed intensity and profile of an emission line develops in a plasma when the plasma becomes optically thick to this line. We then consider the situation in which the excited-level population is controlled by a sequence of processes of emission and reabsorption of radiation, i.e. radiation trapping. We examine the phenomenon on the basis of two approaches which are complementary to each other. At this point we are able to give the correct interpretation to the spectra in Figs. 1.5 and 1.7. When our plasma is dense, various new phenomena may appear which are absent in "everyday" plasmas. Although part of this problem has already been discussed in the context of excited-level populations and line broadening, further discussions deserve a separate chapter. In Chapter 9 we investigate how the atomic state energy and the collision cross-section are affected by the screening of the Coulomb interactions by the plasma particles surrounding the atom or the ion. We then examine additional new excitation and deexcitation processes of ions involving the doubly excited states. Contributions from the resonance process to the excitation cross-section are also found to be affected in a dense plasma. Direct recombination of ions in an excited level can be important under certain conditions. Finally, we investigate modification to the density of atomic states over energy and its consequence incurred in the thermodynamic equilibrium relationship of the densities of atoms and ions in the plasma, a modification to the result of Chapter 2. 1.2 Various plasmas

Figure 1.2 shows various kinds of plasmas on the plane ne—Te, Its abscissa and ordinate are the most important parameters of a plasma: the number density of electrons, or simply the electron density, «e, in units of m~3, or in cm~3, and the electron temperature, Te, in units of K. Occasionally, kTe expressed as eV (electron volts: the energy of an elementary charge e accelerated by a potential of 1 V) is used: kTe= 1 eV corresponds to Te= 11,605 K. The abscissas and the ordinates of Fig. 1.2 are expressed in these units. So, a plasma is located somewhere on this plane according to its ne—Te values. The most modest plasmas are flames like candles, and those in internal combustion engines, which are produced and heated by chemical reactions. We have enormous numbers of plasmas produced by electric discharge. The class of glow discharges, which are produced in a lowpressure gas, includes many laboratory plasmas as well as plasmas encountered in our everyday life; an example is the plasma in fluorescent lamp tubes. According to the discharge current drawn, «e varies over several orders, but Te lies in a rather narrow range, and kTe is one through a couple of eV. Later in Chapter 4, we will encounter an example of this class of plasma. Many kinds of processing plasmas, which are used for the purpose of manufacturing, e.g. semiconductor devices, are produced by radio-frequency or microwave discharges in chemically active gases.

NOMENCLATURE AND BASIC CONSTANTS

13

They have similar parameter values. If an electric discharge is made continuously in a gas under atmospheric or even higher pressure, we have arc discharge plasmas. This kind includes many kinds of lamps for illumination, e.g. mercury discharge lamps. Interestingly, Te of these plasmas has a very narrow range around kTe = 1 eV. The plasma shown in Fig. 1.7 happens to be very similar to this class. When the heating of electrons by electric power input stops, the plasma decays in time or, in the case of a flowing plasma, in space. The plasmas in these decaying processes are called afterglows, and have low Te. In Chapter 4, we will find a few examples of this class of plasma. It will turn out that the plasma of Fig. 1.3 also belongs to this class. If an electric breakdown of a high voltage takes place in an atmospheric-pressure gas, we have a spark discharge, or even lightning. Owing to the sudden input of energy into a thin area of the gas, a plasma with a rather high Te is produced. In contrast to these "classical" plasmas we now have more "powerful" plasmas which have been developed in the last couple of decades. One of the motivations of these developments came from the possibility of realizing nuclear fusion reactions for a future energy source. One class of such plasmas is called the magnetically confined plasma. A high-temperature and moderate-density plasma is confined within the toroidal-shaped vessel made with a magnetic field. With the enormous progress in scientific and technological developments, plasma machines with the configuration called tokamak now produce plasmas close to the practical condition for nuclear fusion reactions. Helical configuration plasmas are also being vigorously investigated. In Fig. 1.2 the NFR region means the parameters of the future nuclear fusion reactor. Another class is high-energy-density plasmas which are produced by putting a vast amount of energy into a small volume of a gas or a solid in a very short time. A rather traditional approach is pinch plasmas, sometimes called a vacuum spark, a plasma focus, etc. Other more modern plasmas are produced by irradiating a solid or gas or even clusters by short-pulsed laser radiation. These plasmas occupy quite a large area of the parameters: the nature of a plasma strongly depends on the experimental conditions, and its parameters are different during laser irradiation and in the decaying period after that. These high-energy-density plasmas may be used as an x-ray light source or even as an x-ray laser source. Plasmas in nature should not be forgotten. It is sometimes said that more than 99 percent of the material in the universe is in the form of plasma. Just two examples are given Fig. 1.2. The Earth is surrounded by several layers of ionosphere. It starts at about 100 km above the Earth's surface and extends up to some 500 km. Another example is the solar corona surrounding the Sun; this plasma greatly inspired the development of plasma spectroscopy. 1.3 Nomenclature and basic constants

In this book the term plasma has dual meanings; the first is in the ordinary sense to express a material which is composed of electrons, ions, and some neutral atoms

14

INTRODUCTION TABLE 1.1 Basic constants. m = 9.109x 1(T31 M= 1.6605 xl(T 2 7 e= 1.6021 xl(T 19 c = 2.998 x l O 8 h = 6. 626 x 1(T34 ft =1. 0545 x 10~34 fc= 1.3805 x 10~23 e0 = 8.854 xlO~ 1 2 a = e2/2hce0= 1/137.0 flo = £0h2/Tmie2 = 5.292 x 10~u tf = e2/87re0a0 = 2.1799 x i(r18 = 13.605

[kg] [kg] [C] [m/s] [Js] [Js] [J/K] [C/Vm] [m] [J] [eV]

electron rest mass atomic mass units electron charge speed of light in free space Planck's constant Boltzmann's constant dielectric constant of vacuum fine structure constant first Bohr radius Rydberg constant

or even molecules. Several examples are shown in Fig. 1.2. The second usage is to express an ensemble of atoms or ions immersed in a plasma. The latter may sound strange, but it will turn out that this is rather natural. As we have already seen important parameters to characterize a plasma (in the first sense) are «e and Te. Since electrons are usually much more active than ions and neutrals in determining excited level populations, a plasma in the first sense sometimes means simply an electron gas having a certain ne and Te. Constants which are used in this book without explanation are given in Table 1.1. The Rydberg constant, which is virtually equal to the ionization potential of neutral hydrogen, is expressed as R. Figure 1.8 shows the schematic energy-level diagram of ions with several symbols. Suppose we are interested in the ion or the atom denoted with (z — 1). z indicates the charge ze of the ions in the next ionization stage, or roughly speaking, the effective core charge felt by the optical electron of the (z — 1) ion. Here we call the electron that plays the dominant role in a transition and emits radiation the optical electron. If we are treating singly ionized helium in Figs. 1.5 and 1.6, for example, (z — 1) is 1 (i.e. singly ionized) and z is therefore 2. nz indicates the density of the ions in the next ionization stage. For hydrogen-like ions (and neutral hydrogen), with which the dominant part of this book is concerned, z is equal to the nuclear charge, p or q is used to indicate a discrete state. nz_i(p), gz-i(p) and XZ-I(P) indicate, respectively, the population (in units of m~3), the statistical weight and the ionization potential (a positive quantity), of level p. Ez_i(p,q) is the energy difference between the lower level p and the upper level q, i.e. Ez_i(p,q) = Xz-i(p) — Xz-i(01.6 Non-hydrogen-like ions The energy-level structure of ions having many electrons is complicated, sometimes extremely complicated. For ions with a small number of electrons, however, it is rather simple, and may be regarded as a modification from that of a hydrogenlike ion, Fig. l.ll(b). Neutral helium and helium-like ions are good examples. See Fig. 1.4. Atoms and ions having one or two electrons outside of the closed shells are other examples. For these atoms and ions an excited level (denoted by p) is designated by the principal quantum number n of the excited electron, the sum of the orbital angular momenta Lh* of all the electrons, and the sum of the spin angular momenta Sh* This scheme of combination of the angular momenta is called the L— S coupling, which describes well the energy-level structure of these atoms and ions. A level is designated by n(2S+ l)L. The suffix (25* + 1) is called the multiplicity; S = 0 is a singlet, ^ is a doublet, 1 is a triplet, and so on. To levels with L = Q, 1, 2, 3 , . . . we assign the symbol S, P, D, F , . . . , respectively.1" Figure 1.4 carries this nomenclature. The level p= n(2S+l)L has the statistical weight g(p) = (2S+ 1)(2L + 1). This level is further split into the fine-structure levels, and each component is designated by the total angular momentum Jh* In the case of L>S, we have J=L — S, L S+l,...,L + S, These fine-structure levels are designated by n(2S+r>LJ. For example, in Fig. 1.4, the 2 3 P ("two triplet P") level consists of three closely lying levels 23P0, 23P1; and 23P2. The statistical weight of each of them is (2/+ 1), and their sum XX2/+ 1) is equal to (2S + 1)(2L+ 1). In the present example, g(2 3P) = 9. This designation is also adopted for hydrogen atoms and hydrogen-like ions. Figure l.ll(b) carries this nomenclature. In this case, L = / and S = ^. We defined the optical electron as the electron playing the dominant role in making a transition and emitting radiation. We also define z; this quantity indicates the effective core charge ze felt by the optical electron when it is at a large distance from the core. This happens to be equal to the roman numeral used to denote the ionization stage of a spectral line, e.g. HI, CIII, OV. * As noted above, the actual magnitudes of these angular momenta are \/L(L + l)fi, \/S(S + l)h, and \/J(J + 1), respectively. t These symbols stem from the early nomenclature of the series lines, "sharp", "principal", "diffuse", and "fundamental". These characteristics can be recognized in Fig. 1.3: in the longwavelength region the series of pairs of sharp and diffuse lines are seen, each originating from the S and D levels, respectively, in Fig. 1.4. Other lines in Fig. 1.3 are of the principal series, which originate from P levels in Fig. 1.4.

20

INTRODUCTION

The energy of a level p with principal quantum number n may be expressed as a modification of eq. (1.1) by

where the parameter 0 is introduced which accounts for the degree of completeness of the screening of the nuclear charge by the electrons other than the optical electron. If the screening is complete 0 equals zero. For the optical electron having an orbit penetrating deep into the core electron orbits, e.g. the s electrons, (see Fig. 3.2(b) later), the screening is not complete, and 0 is a positive quantity, usually smaller than 1. An alternative way of expressing the energy is

Here 6 (= n — «*) is called the quantum defect and n* the effective principal quantum number. Again for s states 8 is large and n* is appreciably smaller than n. For d and higher-/ states 8 is very small and n* is almost equal to n. An example is seen in Fig. 1.4. In the following even in the case of nonhydrogen-like ions p = 1 is understood to denote the ground state. As seen in Figs. 1.4 and 1.1 l(b) high-lying levels form a series of levels converging to the ionization limit. Their energies measured from the limit are given by eq. (1.1) or eq. (1.7) (or eq. (1.7a)) with large values of n. We call these levels the Rydberg levels (states). In atomic spectroscopy an emission (absorption) line and the corresponding transition is customarily written like: Hel A 318.8 nm (2 3 S—4 3 P); this means that this spectral line (one of the lines in Fig. 1.3) is of neutral helium (called the first spectrum) with wavelength 318.8 nm for transition with lower level 2 3S and upper level 4 3P. The lower level comes first. In the following we follow this convention. Figure 1.3 carries an alternative notation. Instead of nm, units of A are sometimes used; 1 A is 0.1 nm. The first prominent line terminating on, or, in the case of absorption, starting from, the ground state is called the resonance line. An example is the Hell A 30.3 nm (1 2 S—2 2 P) line shown in Fig. 1.5 and identified in Fig. 1.6. Another example is the transition in Fig. 1.4 of Hel (1 : S—2 :P) with the wavelength of 58.4nm. Finally, units of energy are mentioned. 1 au (atomic units) is equal to 2_R = 27.2eV. When R is used as units of energy (Rydberg units) energy is expressed as Ry. Units cm^1 is sometimes used to express a spectral line frequency (v), and equal to vie in the cgs units. 1 cm^1 is the energy difference corresponding to a transition wavelength of A= 1 cm in vacuum; i.e. 1 Ry= 1.0974 x 105cm^1.

REFERENCES

21

References

Several books are available for plasma spectroscopy in general and for its various facets which are treated in the later part of this book. A few of them are listed below. Cooper, J. 1966 Rep. Prog. Phys. 22, 35. Griem, H.R. 1964 Plasma Spectroscopy (McGraw-Hill, New York). Griem, H.R. 1997 Principles of Plasma Spectroscopy (Cambridge University, Cambridge). Huddlestone, R.H. and Leonard, S.L. (eds.) 1965 Plasma Diagnostic Techniques (Academic Press, New York). Lochte-Holtgreven,W.(ed.)l 968 Plasma Diagnostics (North-Holland, Amsterdam). There are excellent books on atomic structure and atomic spectra. Only a few are mentioned. Bethe, H.A. and Salpeter, E.E. 1977 Quantum Mechanics of One- and Two-Electron Atoms (Plenum, New York; reprint of 1957). Condon, E.U. and Shortley, G.H. 1967 Theory of Atomic Spectra (Cambridge University Press, London; reprint of 1935). Hertzberg, G. 1944 Atomic Spectra and Atomic Structure (Dover, New York). Shore, B.W. and Menzel, D.H. 1968 Principles of Atomic Spectra (John Wiley and Sons, New York). Thorne, A., Litzen, U. and Johansson, S. 1999 Spectrophysics (Springer, Berlin). White, H.E. 1934 Introduction to Atomic Spectra (McGraw-Hill, New York).

2

THERMODYNAMIC EQUILIBRIUM 2.1 Velocity and population distributions Maxwell distribution In this book we consider a plasma, in the first sense, as consisting of electrons, ions, atoms, and even molecules. If ne is high and Te is low so that the mean distance between electrons becomes comparable to or shorter than the de Broglie wavelength of electrons with thermal energy, A = h/'^fI^nnkT~e, quantum effects prevail, and the electron velocity distribution is given by the Fermi-Dirac distribution. In the following we assume the opposite, i.e. low «e and high Te:

Then, we have the classical Maxwell-Boltzmann distribution (called simply the Maxwell distribution)

which satisfies the normalization condition, ff(v)dv = 1. The average speed is v = ^/SkTe/irm, the root mean square speed is \/(v}2 = ^/3kTe/m, and the most probable speed is t>p = ^/2kTe/m. The corresponding energy distribution function is

with normalization, ff(E)dE=

1. The average energy is E = 3kTe/2.

Boltzmann and Saha-Boltzmann distributions In this book, the word "ions" is used to denote both ions and neutral atoms. However, the word "atoms" is used when it is more convenient to distinguish atoms and ions in adjacent ionization stages. It is well known from statistical mechanics that, in thermodynamic equilibrium, the ratio of the number of ions per unit volume in two different energy levels, or the population density ratio, is given by the Boltzmann distribution

where we have assumed levels/? < q in ionization stage (z — 1). See Fig. 1.8. Actual level schemes are shown in Figs. 1.4, 1.6, and l.ll(b). If the ions in this ionization stage have many levels including p and q, we may plot the populations per unit statistical weight in a semilogarithmic plot. This plot is called the Boltzmann plot,

VELOCITY AND POPULATION DISTRIBUTIONS

23

and we obtain a straight line, the slope of which corresponds to Te. We will see examples later in Chapters 4 and 5. Equation (2.3) applies to levels of ions in a particular ionization stage. If we take a hydrogen-like ion (z — 1), each of these levels corresponds to a level as shown in Fig. 1.6. The above thermodynamic relationship may be extended to higher energies across the ionization limit to the continuum states of the electron, having positive energies. We approximate here the continuum states as freeelectron states. Discussions concerning this approximation will be given in Chapter 9. Since the level energy is continuous we consider free states of electrons having speed v within the range dv. This upper "level" is regarded as the collection of states of free electrons paired to the core ion (in the ground state) in the ionization stage z. Then eq. (2.3) is rewritten as

where nz(l,v)dv and gz(l,v)dv denote, respectively, the "population" and the "statistical weight" of the upper "level", and A£" is the energy difference between this upper level and the lower level p, i.e. AE = mv /2 + xz-i(p)We now introduce the phase space, i.e. a six-dimensional space for the motion of a free electron: three dimensions for the spatial coordinate (x,y,z) and three dimensions for the momentum coordinate (px,py,pz). In the x-px plane, we define a cell Sx • Spx having area h. This is another quantum cell (remember Fig. l.ll(a)) and has the significance that all the states of motion, the corresponding points (x, PX) of which fall in this cell, are regarded as a single state. Similar arguments apply to the y—py and the z—p2 planes. Thus, the "number of states" deriving from the motion of the electrons is given as

where Ax, Aj, and Az are the spatial coordinate widths allocated to one of the free electrons and t±px, Apy, and Apz are similarly the momentum coordinate widths. The former widths make a volume A V allocated to the electron, which is equal to l/ne. Since we assume the electron motion to be isotropic we use the polar coordinate system

Equation (2.5) gives the number of states for the electron motion. The "statistical weight" of the "upper level" is thus given as

24

THERMODYNAMIC EQUILIBRIUM

or

where ge and gz(T) are the statistical weights originating from the inner structure of an electron and that of the ground-state ion, respectively. The former comes from the electron spin and is ge = 2, It is noted that eq. (2.5a) is also encountered in solid state physics as the density of states of electrons in the free-electron model. Equation (2.4) is transformed as

where we have used eq. (2.2). In many situations we are not interested in the "population ratio" as given by eq. (2.6). Rather, the quantity of interest would be the ratio of the "ion" density nz(l) and the "atom" density nz_i(p) in level p. The former quantity is obtained by integration of the "populations" over the speeds of the free electrons, i.e. nz(l) = fnz(l,v)dv. By using the normalization condition for the Maxwell distribution we obtain the Saha-Boltzmann distribution

or

where Z(p) is called the Saha-Boltzmann coefficient. It is interesting to note that Z(p) is expressed in terms of the thermal de Broglie wavelength (see eq. (2.1)),

Equation (2.7) or (2.7a) is called the Saha-Boltzmann distribution. Under certain conditions, thermodynamic equilibrium may be established in our plasma and the population of an excited level p is actually given by eq. (2.7) or (2.7a). If this is the case, we say that "level p is in local thermodynamic equilibrium (LTE) with respect to ion z." This situation is called partial LTE. In the case that the LTE population, eq. (2.7a), extends down to the ground state p=\, this situation is defined as complete local thermodynamic equilibrium (complete LTE). These problems will be treated later in Chapter 5.

BLACK-BODY RADIATION

25

In traditional plasma spectroscopy, the term Saha equilibrium has been, and still is, used to describe the density ratio of the "atoms" (z — 1) and the "ions" z, when complete LTE is assumed for this system. The "atom density" is the sum of all the "atomic" level populations,

The summation in the r.h.s. (right-hand side) of this equation is called the partition function, and denoted as Bz_i(T&). We define the "ion" density Nz in a similar manner by introducing the partition function of the ions Bz(Te), Then, the density ratio of the "atoms" and the "ions" is given as

It is readily seen that, because of the nature of the statistical weight, g(p) = 2p2 for the case of hydrogen atoms and hydrogen-like ions, the partition functions diverge. This difficulty comes partly from eq. (2.8) itself, i.e. the notion that all the populations in excited levels of atoms belong to the "atom." We will see in Chapters 4 and 5 that this understanding is rather unrealistic; excited atoms are strongly coupled to ions rather than to the ground-state atoms. The difficulty of the divergence itself will be resolved toward the end of Chapter 9. One important fact is noted here: in deriving eq. (2.6) we "filled" the density of states for free electrons, eq. (2.5a), with the Boltzmann distribution, eq. (2.4), with appropriate statistical weights incorporated. Equation (2.6) itself includes the Maxwell distribution function, eq. (2.2). This fact clearly indicates that the Maxwell distribution is nothing but the Boltzmann distribution extended over the free-electron states. The normalization factor in eq. (2.2) or eq. (2.2a) makes this point less obvious. This aspect will be further examined in Chapter 9. See eq. (9.19a). 2.2 Black-body radiation We consider an ensemble of ions having lower level p and upper level q, and the radiation field, the wavelength of which corresponds to the transition energy between these levels. The temporal development of the upper-level population n(q) is given by the rate equation

where /„ [Wm 2 sr : s] (sr means steradian or a unit solid angle) is the spectral intensity of the radiation field at the transition frequency v = E(p, q)/h. Figure 2.1 illustrates the situation of eq. (2.10). The quantity Iv is called the spectral radiance

26

THERMODYNAMIC EQUILIBRIUM

FIG 2.1 The emission-absorption processes of atoms in a radiation field. in radiometry. It should be noted that we have assumed that, in eq. (2.10), the radiation field is isotropic and has virtually a constant intensity over the line profile of the transition.* The first term represents excitation of the upper-level ions by absorption of photons, and the second and third terms denote deexcitation by spontaneous transition and by induced emission, respectively. A(q,p), B(p, q) and B(q,p) are called Einstein's A and B coefficients. We further suppose that this system is surrounded by "mirror" walls that reflect radiation and ions completely. After a sufficiently long time a stationary state is reached and the time derivative vanishes. Then we have

We note the interrelationships between the coefficients A and B*

On our above assumptions, we may expect that our system is in thermodynamic equilibrium. Then, the population ratio should be given by the Boltzmann distribution, eq. (2.3),

* Equation (2.10) cannot describe the atomic system in a radiation field that violates these conditions. This is the case for radiation fields encountered in many practical cases; an extreme example is the field produced by a laser beam. This field is highly directional, monochromatic, and sometimes polarized. In such cases we have to employ an alternative approach by introducing the concept of an absorption cross-section for the transition, as defined by eq. (3.9) later. t Equation (2.10) is sometimes written in terms of the spectral energy density, (4w/c)Iv, in place of the spectral intensity Iv. Then the second relationship of eq. (2.12) takes a different form.

BLACK-BODY RADIATION

27

Substitution of eqs. (2.12) and (2.13) into eq. (2.11) leads to the intensity of the radiation field,

As has been noted the radiation field is assumed isotropic, i.e. no angular dependence and unpolarized. We have derived eq. (2.14) for a particular frequency region corresponding to the transition p q. We may readily extend the above argument to other regions by introducing other transition frequencies, and finally, to obtain eq. (2.14) for the whole spectral range. Equation (2.14) is called Planck's distribution or the black-body radiation. Figure 2.2(a) shows examples of the black-body radiation for several temperatures. Several properties of the black-body radiation are discussed. It has a maximum intensity at a certain frequency, depending on the temperature. The frequency which gives the maximum is readily obtained from the derivative of eq. (2.14),

See Fig. 2.2(a). In the low-frequency region oihv^_kT, eq. (2.14) reduces to

This distribution is called the Rayleigh-Jeans law, and is of an entirely classical nature as is understood from the absence of h. This distribution diverges with v. At the other extremum, for hv ;$> kT, we obtain

The energy density of the black-body radiation contained in a unit volume is (^/c)Bv(T)dv, in units of [Jm~3], and the total energy density is

with

Equation (2.18) is called the Stefan-Boltzmann law.

FIG 2.2 Planck's distribution of the black-body radiation: (a) eq. (2.14); (b) eq. (2.14a). The region of visible light is indicated. Note the relationship of eqs. (2.15) and (2.15a), respectively, as indicated with the dotted lines.

BLACK-BODY RADIATION

29

It is sometimes convenient to rewrite eq. (2.14) in terms of wavelength A,

Figure 2.2(b) shows the distribution, eq. (2.14a), for several temperatures. The peak wavelength in this expression is given by

where T is measured in [K]. The relationship of eq. (2.15) or (2.15a) is called Wien's displacement law. We take as an example the surface of the Sun; its temperature is 5770 K. From eq. (2.15) we have j/ max ~3.36 x 1014 s^1, or A ~ 891 nm, in the near-infrared. From eq. (2.15a) we have Amax ~ 503 nm, a blue color. See Fig. 2.2(a) and (b), respectively.

3

ATOMIC PROCESSES In a plasma, atoms and ions undergo transitions between their quantum states through radiative and collisional processes. Among these processes, the most important are spontaneous radiative transitions and collisional transitions induced by electron impact (collisions). In the following, we review these processes. In doing so we emphasize two points: 1. The radiative transition probability and the collision cross-section are not unrelated nor independent. Rather, they share some common tendencies through a parameter called the absorption oscillator strength. 2. Various properties of high-lying states continue smoothly across the ionization limit to those of the low-energy continuum states. This fact is reflected in the atomic processes involving these states. Figure 3.1 shows schematically transitions which are included in our theory in subsequent chapters. These transitions are: spontaneous radiative transition excitation by electron impact deexcitation by electron impact radiative recombination ionization by electron impact three-body recombination In the above, we have assumed that levels p and q are of the atoms or ions (sometimes called "ions" for the purpose of simplicity) in the ionization stage (z — 1) and that z is the ground state of the ions in the next ionization stage, e in the initial state (left-hand side or l.h.s.) means the incident electron inducing the transition and in the final state (the right-hand side or r.h.s.) is the scattered electron(s). hv is a photon with frequency v emitted in the transition. The symbols shown above the arrows represent the rate constants for the transitions. A rate constant represents quantitatively the likelihood of that reaction taking place in a plasma. It is noted that A(q,p) is a probability and has units of [s^1], C(p,q), F(q,p), (3(p), and S(p) are called the rate coefficients and have units [m3 s^1], and

RADIATIVE TRANSITIONS

31

FIG 3.1 The transitions included in the rate equation of populations in succeeding chapters. a(p) is also called the rate coefficients and has units [m6 s^1]. On the right, the pair of arrows connected by the vertical thin line indicates that these two transitions are inverse processes to each other so that their rate coefficients are related internally by a thermodynamic relationship. The transitions connected by the thick lines have properties which are common in their natures, as will be discussed in the following. 3.1 Radiative transitions The probability of a spontaneous radiative transition q^p + hv has been introduced already in Section 2.2 as Einstein's A coefficient, which is simply called the transition probability. This is given in terms of the absorption oscillator strength fp>?,

The absorption oscillator strength is a measure of the ability of the atom in state p to absorb light in making the transition p + hv —> q. This is defined by

with the electric dipole matrix element, or the dipole moment

Here, ^>p and if}q are the wavefunctions and r is the position vector of the electron taking part in this transition, the optical electron. The radiative transition of this type is called the electric dipole transition, or the optically allowed transition. Other kinds of transitions, e.g. the electric quadrupole or magnetic dipole transitions, also exist. These optically forbidden transitions are usually quite weak: for

32

ATOMIC PROCESSES

example, electric quadrupole transitions have oscillator strengths (the definition of which is different from eq. (3.3)) smaller than those of dipole transitions by about 10~7. We neglect these optically forbidden transitions in the following discussions. We now take neutral hydrogen for the purpose of illustration. The wavefunction is expressed as tfjnim(t") = Rni(r)Yim(0,) with the radial part Rni(r) and the spherical harmonic Yim(0, ) for the angular part.* The angular part is further written as Yim(0,(f>) = (l/V2jr')Pf (cos6>)eim, where Pf(cosff) is the associated Legendre function. Figure 3.2(a) illustrates examples of Pf(cosO) for several small values of /'s and m's, and the factor em4> to be multiplied. If we take the ground state Is as the initial state, the oscillator strength has non-zero values only for the final states np. See Fig. 1.1 l(b) and Table 3.1. This selection rule stems from the integration of r with the two spherical harmonics over the angular coordinate. It is obvious from Fig. 3.2(a) that the s (1=0) and d (1=2) state wavefunctions are spatially even functions, and r is an odd function, so that the matrix element vanishes for the initial Is state and the final s or d states. For the f (/= 3) and still higher-/ final states, the reasoning of the vanishing matrix element is more involved. The selection rule that A/ cannot be larger than 1 may also be understood from the fact that the photon carrying away the energy difference of the initial and final states has unit angular momentum h. Figure 3.2(b) shows the radial wavefunction as the form rRB/(r) (in atomic units, in which e = m = h=\)} In the integration (3.3) for p= Is and q = np, apart from the integration over the angular coordinate, which is common to all the np states, the magnitude of the matrix element, or of the oscillator strength, is approximately given by the degree of "overlapping" of the radial wavefunctions of the initial and final states. As is easily seen in Fig. 3.2(b) the main contribution to the integral of the radial functions comes from the first peak of the np wavefunction. With an increase in n the amount of the overlap decreases, and the oscillator strength /is>Bp decreases. Table 3.1 shows several examples of the radial integrals of r (squared) and the * In this chapter until eq. (3.6), p or q is understood to stand for nl in the example of hydrogen-like ions. It is noted that, in the r.h.s. of eqs. (3.2) and (3.2a), we ignored the presence of m, or the degeneracy of the levels. The correct expression of eq. (3.2) is

where p stands for «'/'. Each of the above matrix elements corresponds to the transition from one of the cells in Fig. l.ll(a) to another. For example, for/ 2p 3d these transitions are between the three cells of n' = 2, /' = 1 and the five cells of n = 3 and / = 2. Here we neglect the presence of the electron spin. Therefore, within this framework, we have g(n'l') = 3. The expression ( q \ r \ p ) 2 ineq. (3.2) should be understood to be the averaged value of (nlm \ r n'l'm') 2 over the lower and upper levels, i.e. J5)i5)EL=-/EL=-f \(nlm\r\n'l'm')\2. Note that, in reality, each cell in Fig. l.ll(a) is doubly degenerate owing to the electron spin and this degeneracy should be taken into account both in the summations and the statistical weights. The resulting oscillator strength value is unchanged. t The radial wavefunction for an s state, Rns(r), tends to a finite value for r^O, while other wavefunctions, Rni(r) with /^ 0, tend to zero as can be seen from the straight line starting from zero for the former case and the finite curvature for the latter.

RADIATIVE TRANSITIONS

33

FIG 3.2 Several examples of the wavefunctions of atomic hydrogen, (a) The associated Legendre function Pf(cosff) for several / and m. Pficosff) multiplied by eim yields the spherical harmonics (apart from the normalization factor), Yim(0, <j)), which is the angular part of the atomic wavefunction. (b) The radial wavefunction of ns, np, and a few nl states in the form of rRni(r), for

34

ATOMIC PROCESSES

oscillator strengths. The above feature is clearly seen. It is worth noting that the asymptotic value of both the quantities for large n is proportional to n~3. It may be interesting to find that this factor has already appeared in eq. (1.5), the energy width allocated to a level having principal quantum number n. Equation (3.2a) suggests that, in the case of E(p, q) < 0, or when the final state lies below the initial state (p > q), the oscillator strength takes a negative value. In this case, eq. (3.2a) is called the emission oscillator strength and is defined (for E(q,p)i>r3dr) 2 for neutral hydrogen in atomic units [afc]. (Adopted from Bethe and Salpeter (1977).) Initial

Is

2s

Final

np

np

_

_

«=1 2 3 4 5 6 7 8 n = 9 to oo together Asymptotic Discrete spectrum Continuous spectrum Total

3s

2p «s

nd

1.67 27.00 0.88 0.15 0.052 0.025 0.014 0.009 0.025

_ 22.52 2.92 0.95 0.41 0.24 0.15 0.42

np _

3p «s

nd

np

0.3 9.2 22.5 162.0 101.2 101.2 6.0 57.0 1.7 0.9 8.8 0.23 0.33 3.0 0.08 0.16 1.4 0.03 0.09 0.8 0.02 0.22 2.0 0.05

«s

4f

4d

4p

4s

3d

np

nf

nd

«g

_ _ _ _ _ _ 0.09 2.9 1.66 0.15 104.7 1.7 57.0 6.0 29.9 104.7 540.0 540.0 432.0 432.0 252.0 252.0 9.3 197.8 2.75 11.0 72.6 21.2 121.9 1.3 26.9 0.32 19.3 2.9 3.2 11.9 0.08 1.4 7.7 0.5 8.6 1.4 5.7 3.2 0.2 3.9 0.04 0.6 0.8 2.1 0.3 6.9 0.07 5.9 4.3 1.0 1.8

_ 314.0 27.6 7.3 3.0 4.5

nf

np

nd

1.666 0.267 0.093 0.044 0.024 0.015 0.010 0.032

27.00 9.18 1.66 0.60 0.29 0.17 0.10 0.31

4.7/T3 2.151

44.0«~3 3.7«-3 58. 6«~3 169«"~ 3 28«~3 248«~3 5/T3 198/T3 445«~3 102«~3 655/T3 33«~3 687«~3 6«-3 39.30 29.820 27.62 202.56 179.18 174.54 125.88 122.85 642.7 598.7 591.7 503.50 496.0 359.95

0.849

2.70

0.180

2.38

3.000

42.00

30.00

30.00

0.9 162.0 29.9 5.1 1.9 0.9 0.5 1.4

4.44

0.82

5.46

0.12

3.15

5.3

1.3

8.3

0.50

8.0

0.05

207.00 180.00 180.00 126.00 126.00 648.0 600.0 600.0 504.00 504.0 360.0

393/T3 356.4 3.6

360.0

TABLE 3.1(b) Oscillator strengths for hydrogen. (Adopted and modified from Bethe and Salpeter (1977).) Initial

Is

2s

Final

«p

np

n=l 2 3 4 5 6 7 8 n=9 to oo

0.4162 0.0791 0.0290 0.0139 0.0078 0.0048 0.0032 0.0109

_

2p ns

nd

-0.139 0.4349 0.1028 0.0419 0.0216 0.0127 0.0081 0.0268

0.014 0.696 0.0031 0.122 0.0012 0.044 0.0006 0.022 0.0003 0.012 0.0002 0.008 0.0007 0.023

2

3s

n

np

3d

3p ns

nd

np

_ _ _ -0.104 -0.026 - -0.417 -0.041 -0.145 0.641 0.120 0.484 0.032 0.619 0.011 0.045 0.121 0.0022 0.007 0.139 0.022 0.052 0.003 0.056 0.0009 0.012 0.002 0.028 0.0004 0.027 0.008 0.016 0.001 0.017 0.0002 0.024 0.048 0.002 0.045 0.0007

nf

4s

n

np

4p ns

4d nd

np

1.6n~3 3.7n~3 O.ln~ 3 3.3n~3 0.5650 0.6489 -0.119 0.928

3.5n-3 0.769

6.2n~3 0.3n~3 6.1n~3 0.07n~3 4.4n~3 1.302 0.707 -0.121 0.904 -0.402

0.4350 0.3511

0.008 0.183

0.231

0.293

0.010 0.207

0.002

Total

1.000

-0.111 1.111

1.000

1.000

-0.111 1.111

-0.400

4

4f nf

nd

_ _ _ _ _ _ -0.010 -0.009 - -0.285 -0.009 -0.034 -0.073 - -0.727 -0.097 -0.161 -0.018 -0.371 1.016 0.841 0.156 0.150 0.545 0.053 0.610 0.028 0.890 0.009 0.053 0.056 0.138 0.012 0.006 0.187 0.0016 0.149 0.025 0.060 0.006 0.063 0.002 0.072 0.0005 0.027 0.015 0.016 0.033 0.003 0.033 0.001 0.037 0.0003 0.042 0.082 0.006 0.075 0.002 0.081 0.0006 0.037

Asymptotic Discrete spectrum Continuous spectrum

1.000

3

ng

_ -0.002 - -0.030 - -0.446 1.345 1.038 0.183 0.180 0.058 0.065 0.032 0.027 0.045 0.066

5.3n~3 0.840

9.3n~3 0.7n~3 0.752 -0.126

9.1n~3 0.3n~3 8.6n~3 0.05n~3 3.5n-3 1.658 0.912 -0.406 1.267 -0.715

0.098

0.160

0.248

0.015

0.199

1.400

1.000

1.000 -0.111

1.111

0.006 0.133 -0.400

1.400

n

6.8n~3 0.876

0.001

0.056

0.124

-0.714

1.714

1.000

RADIATIVE TRANSITIONS

37

we will see later (Appendix 4A), this situation is actually realized under many conditions of practical interest. From now on, for neutral hydrogen and hydrogen-like ions, we assume this situation actually to be the case. Then, we may bundle these individual m and / states into one level, which is designated only by its principal quantum number. Figure 1.6 represents the reduced energy-level diagram. The oscillator strength in this scheme is given in a similar manner to the expression in the first footnote in p. 32:

In the following we use p or q in place of n' or n to denote the principal quantum number specifying the level. In this convention the oscillator strength is expressed by a simple formula which is based on the classical Kramers formula

where gbb is the Gaunt factor, or the quantum correction factor. The subscript "bb" means the bound-bound transition. Here "bound" is identical to "negative energy" and denotes the discrete levels. A few examples of the Gaunt factors are shown in Fig. 3.3 for the case of p= 1; their magnitudes are of the order of 1. Examples of the oscillator strengths are shown in Fig. 3.4 for p= 1,2, , 15. Two important features are seen: 1. With an increase in q of the upper level, fPA tends to be proportional to q~3. We have already seen this feature in Table 3.1. See also eq. (1.5). This feature will lead to important consequences later. 2. For the transition to the adjacent higher-lying level, p—>(p + 1), the oscillator strength is the largest in this series (remember the arguments concerning Fig. 3.2(b)), and is well approximated byfp^p+1 ~ (/? + l)/5, or even by

as is seen in Fig. 3.4. The transition probability is expressed from eqs. (3.1) and (3.6a) as

where rq is the period of one revolution of the electron of the upper level ion, eq. (1.4), in the Bohr atom picture. It is interesting to note that this expression

38

ATOMIC PROCESSES

FIG 3.3 An example of the Gaunt factors for the bound-bound transitions (Is—np) and the bound-free (Is—/cp) transitions for hydrogen. The lower level is the ground state p = ls. consists of the fine structure constant, the ratio of the typical atomic energy to the electron rest-mass energy, the frequency of the classical orbit motion, and other quantities of the order of 1 or smaller. Figure 3.5 shows several examples of the transition probabilities. For an initial level g> 1, and for a low-lying final level p ( q. We may call this process photoexcitation. For various reasons (see Chapter 7) the absorption line is not monochromatic, rather it has a profile with a finite width. Figure 3.6 shows an example of the photoabsorption spectra of atoms; the initial state is the ground state of neutral helium.* The transitions shown in this figure are indicated with the dashed lines in Fig. 1.4. * Actually, this spectrum is the energy loss spectrum of high-energy electrons, 2.5 keV, passing through a dilute helium gas. The energy loss spectrum in this energy range is almost exactly equivalent to the photoabsorption spectrum of the same initial state, as can be understood in the discussion later in Section 3.3. In the present spectrum, however, the line profile is determined by the resolution of the measurement apparatus, 55 meV, including the energy spread of the electron beam. This is inconsistent with our assumption of the line profile in the text. However, this inconsistency leads to no difficulty in our discussion to follow. Therefore, we regard this spectrum as the photoabsorption spectrum, averaged over this energy width.

RADIATIVE TRANSITIONS

39

FIG 3.4 The absorption oscillator strength f p q , eq. (3.6a), for several transitions of hydrogen. The approximation of eq. (3.7) is shown.

We may express the characteristics of an absorption line in terms of the absorption cross-section apoo, i.e. £"=24.59 eV. See Fig. 1.4. Equation (3.9b) suggests that this almost constant value continues to still lower levels, which is seen to be actually the case in Fig. 3.6. This property leads to important conclusions later.

42

ATOMIC PROCESSES

3.2 Radiative recombination For the purpose of deriving the cross-section for radiative recombination we start with its inverse process, i.e. photoionization. Photoionization Figure 3.7 illustrates the photoionization process of an ion in state/? by absorbing a photon having frequency v. The final state is one of the continuum states having energy e. The photoionization cross-section is given as

where qe represents the continuum state. Figure 3.2(c) shows examples of the continuum wavefunctions for the es states.* It is seen in Fig. 3.2(b) that, starting from the low-energy discrete states, with an increase in energy, the number of nodes of the radial wavefunction increases. The overall "shape" of the wavefunction changes smoothly from the discrete states ns (E < 0) across the ionization limit (£"=0) to the continuum states es (E>0). For the transitions between the discrete levels, eqs. (3.1)-(3.3), if we take/>= Is as an example, we have seen that the dominant contribution to the matrix element, eq. (3.3), comes from the overlapping of the Is wavefunction with the first peak of an «p wavefunction. See Fig. 3.2(b). This is also true for the continuum ep wavefunctions, the "shape" of which can be imagined from the ns and es wavefunctions. This continuation property has significant consequences, which will be discussed later. The final states are continuously distributed in energy, and we consider photoionization into the final states within the energy width de centered at e. The cross-section is sometimes expressed in terms of the differential oscillator strength

with d;/ corresponding to de = h dv. Note that this expression is quite similar to eq. (3.9). * Two points are noted here. Since the upper level qc is in the continuum states the wavefunction q£) is normalized over a unit energy interval, so that (p r q£) 2 has units of [m2 J~']. Equation (3.10), like eq. (3.2), ignores the presence of degeneracy. The correct expression is

where the summations over [p] and [qe] are understood to be summations over all the magnetic sublevels of the lower level p and the upper level qe.

RADIATIVE RECOMBINATION

43

FIG 3.7 The schematic diagram for explanation of photoionization and radiative recombination. In the case of hydrogen-like ions the formula, eq. (3.6a), can be extended to that for photoionization; the "principal quantum number" of the final state is imaginary, so that q is replaced by i/c, with real K, and gbb by gw, the bound-free Gaunt factor. Here "free" means the levels in the continuum states with positive energy. See Fig. 3.7. (i^)3 is replaced by K3.

An example of the bound-free Gaunt factor is shown in Fig. 3.3 for p=\. This formula represents the absorption oscillator strength for the final states lying within width d/c = 1. Therefore, we have dfp,e

and

From eqs. (3.10a)-(3.12) we have

= fp,KdK

44

ATOMIC PROCESSES

In deriving the last line we have utilized the relation hv = z2R(p 2 + K 2). Figure 3.8 shows examples of photoionization cross-sections for several levels of neutral hydrogen, z= 1; the cross-section has a sharp threshold and above that it is given by eq. (3.13). It is noted here that eq. (3.13) is exactly the same as eq. (3.9b), the averaged cross-section for photoexcitation over the series lines, except for the Gaunt factor. As Fig. 3.3 shows, with an increase in the final state energy, this factor continues smoothly from the discrete states, g\,\,, across the series limit or the threshold for photoionization at (hv/z2K) = 1, to the continuum states, gbt. An important conclusion is thus reached: the magnitude of photoabsorption of series lines from a particular lower level continues smoothly across the series limit or the ionization limit to the photoionization continuum absorption from the same level. There is no break of absorption spectrum at the threshold of photoionization, as might be imagined from Fig. 3.8. This feature is clearly seen in Fig. 3.6; i.e. no break is seen at the energy of 24.59 eV, corresponding to the ionization limit. In this figure actual cross-section values of neutral helium are estimated from the ordinate value of about 0.07 (eV^1) to be 7 x 10~22 m2 from eqs. (3.9) and (3.10a). This value is close to the threshold photoionization cross-section of the Is hydrogen in Fig. 3.8.

FIG 3.8 Examples of the photoionization cross-sections from some of the lowlying levels of neutral hydrogen.

RADIATIVE RECOMBINATION

45

From eq. (3.12) we may define the integrated oscillator strength for photoionization,

where VQ = (z2R/hp2) is the threshold frequency for photoionization. This oscillator strength is nothing but the oscillator strength sum over the continuum states mentioned with regard to the sum rule, eq. (3.5). From eq. (3.13) together with eq. (3.10a) we may obtain

where (gb{) is the Gaunt factor averaged over the continuum states. Table 3.1b contains/p>c for/? with different / levels resolved. It may be interesting to note that, in eq. (3.14), the dominant contribution tofp>c comes from the differential oscillator strength (dfpf/dv), or the photoionization cross-section (eq. (3.10a)), in the energy region close to the ionization threshold. See Fig. 3.8 and eq. (3.13). Radiative recombination cross-section and rate coefficient In Fig. 3.7, an ion in level p in ionization stage (z — 1) may be photoionized by a photon with frequency i/, producing a pair of a ground state ion in ionization stage z and an electron having energy e. In a plasma, suppose there are nz_i(p) ions present per unit volume and they are photoionized by a radiation field, which is isotropic and has intensity Iv&v at v over the frequency width dz/. The number of photoionization events per unit time in unit volume (we call this quantity the photoionization flux, which has units of m^ 3 s^ 1 ) is given as n z-\(p)\^TfIv/hv\(Tptl,(y)d.v. The inverse process to photoionization is radiative recombination, in which a ground state ion in ionization stage z captures an electron having energy e to form an ion in level p in ionization stage (z — 1) by emitting a photon v. If we introduce the radiative recombination cross-section ffe,P(s) the number of events in the plasma, or the radiative recombination flux, is given as nz(l)nef(s)<je^p(s)vds* Here the energy width de corresponds to the

* A target has area <JE:f(e) for this reaction, and if we suppose that the motion of electrons is unidirectional with speed u, then the number of collisions in one second on this target is ne<jEtp(s)v. The number of targets per unit volume is nz(l). Thus, the number of collisions per unit volume per unit time inducing this reaction is nz(l)neij£ip(e)v. Even if the velocity distribution is isotropic, this expression is valid. This quantity is called the radiative recombination flux. Since the electron speed is distributed we reach the expression in the text. Note that this quantity is proportional to nz(l) and ne, the densities of the reacting agents.

46

ATOMIC PROCESSES

frequency width dz/. If we assume thermodynamic equilibrium for our plasma these processes should balance with each other so that these fluxes should be equal:

where we have included on the l.h.s. the exponential factor in order to account for the process of induced emission, just like in the case of transitions between discrete levels. See eq. (2.10) with eq. (2.13). The subscript E is understood to mean that this equation holds in thermodynamic equilibrium. Since we assume thermodynamic equilibrium the ionization ratio should be given by the Saha-Boltzmann equation, (2.7), and the radiation field is given by the Plancks' distribution, eq. (2.14). By substituting these equations and eq. (2.2) into eq. (3.16) we obtain

which is called Milne's formula. Thus the cross-section for radiative recombination to produce a hydrogen-like ion p is obtained from eq. (3.13):

where gz(l) = 1 has been used, since the bare ions including protons have no internal structure.* It is noted that the radiative recombination cross-section has no threshold energy, and it diverges toward the null energy. For low-energy electrons of e z2R/p2, it is proportional to p~3s~2. It may further be noted that, in these limiting cases, the cross-section is proportional to z2 and z4, respectively. These positive dependences on z are in sharp contrast to the negative dependence of the area of the electron orbit of the bound state; it is proportional to z~2 as seen from eq. (1.2). As noted already, for a beam of electrons having a speed v, the radiative recombination flux is nz(l)ne<je>p(s)v. The magnitude of this flux divided by nz(l)ne is called the radiative recombination rate coefficient. In the present case, the rate * Actually a nucleus like a proton or a deuteron could have an internal structure stemming from the existence of nuclear spin. However, the statistical weight due to this structure is common to an ion z and an atom (z — 1), so that it does not affect eq. (3.17).

RADIATIVE RECOMBINATION

47

coefficient is av. Likewise, for plasma electrons having energy distribution f(E)dE, the radiative recombination rate coefficient is given as

with the exponential integral

FIG 3.9 The radiative recombination rate coefficient j3(p) for several levels of neutral hydrogen. The arrows show the temperature at which kTe = x(p)The temperature dependence for low Te and for high Te are shown. See also Fig. 3.5(b).

48

ATOMIC PROCESSES

In deriving eq. (3.19a) we have assumed the Maxwellian distribution, eq. (2.2a), for/(e), and gbf = 1- The exponential integral is well approximated for a small or large argument by

where 7 = 0.5772 is Euler's constant. Figure 3.9 shows examples of the radiative recombination rate coefficients for neutral hydrogen calculated from eq. (3.19) without the assumption of gbf= 1. Corresponding to the above limiting cases of the cross-section the rate coefficient has the following dependences: for low temperatures of kTe z2R/p2, we have (3(p) <xp~2-5T~1-5, where we have approximated ln(l/jc) to 0.7(l/^)°'5 for (1/x) of the order of 10. 3.3 Collisional excitation and deexcitation Excitation cross-section We consider excitation of an ion (atom) by electron impact. For the initial and final states of the ion/? and q, respectively, the excitation process may be written as p + e^q + e, where e stands for the incident or scattered electron. See Fig. 3.1. For an incident electron, the likelihood of this process to take place depends on its energy, which must be higher than the excitation threshold, and is expressed quantitatively in terms of the excitation cross-section. Figure 3.10 shows several examples of measured or calculated excitation cross-sections for two transitions: (a) for 1:S + e—>2 1 P + e of helium-like iron, or 24 times ionized iron ion, and (b) for the corresponding transition of neutral helium (see Figs. 1.4 and 3.6). These correspond to the resonance lines. In Fig. 3.10(a), starting from the excitation threshold of 6.7 keV, several theoretically determined cross-sections are plotted with small symbols. Let the incident and scattered electrons have momentum hk0 and hka, respectively. The final state of the scattered electron is expressed as a spherical wave : [/(/> —> q; 9, 2 :P of neutral helium. Several examples of the results of the most sophisticated calculations and recent experiments are shown. See the text for details. (Quoted from Goto 2003; copyright 2003, with permission from Elsevier.)

50

ATOMIC PROCESSES

potential is retained. This method is called the Born approximation for a neutral target and the Coulomb-Born approximation for an ion target. In Fig. 3.10(a) the Coulomb-Born approximation gives the largest cross-section just above the excitation threshold. The incident electron can "kick out" one of the target electrons while being captured by the target. This process of replacement of the two electrons is called exchange. The cross-section calculated with exchange taken into account (Coulomb-Born-Oppenheimer approximation) is the next largest result. This method, however, sometimes gives incorrect results, and further improvements are needed. The result of one such method (the Coulomb-Born-OppenheimerOchkur-Rudge-Bely approximation: the reader doesn't need to be bothered by such nomenclature) is still smaller. Another modification is to use more accurate wavefunctions for incident and scattered electrons, as contrasted to the CoulombBorn approximation. This is called the distorted-wave approximation. Several results of this method, some of which include some further sophisticated modifications, are shown as a group of cross-sections having the smallest values. For a long time, experimental confirmation of the theoretical calculation was not obtained for highly ionized ions, because it was virtually impossible to produce a strong enough beam of such highly ionized ions to enable a crossed beam experiment to be performed. Recently, a device called an EBIT (electron beam ion trap) has been developed: highly ionized ions are created and held in a small space by a magnetic-electric trap with the help of a high-current electron beam, which excites the ions. From the observation of the emission radiation of the xuv (extreme ultraviolet) line (A = 0.185 nm in this example), the cross-section is determined for this highly ionized iron, as shown by the closed circle in Fig. 3.10(a). The electron beam both ionizes and sustains the ions in the trap, and the beam energy cannot be changed freely. So, the cross-section is obtained only for one energy. This data is compared with the results of the sophisticated theories. The agreement is not perfect, but considering the difficulty of this experiment, especially in obtaining the absolute value, we may regard this agreement as surprisingly good. Now we look at Fig. 3.10(b). This is for the transition of neutral helium corresponding to the ion transition we discussed above. Helium is the most thoroughly studied atomic species, and this transition is the resonance transition. This figure contains only very sophisticated calculations and recent experiments. One method of calculating a cross-section is the close-coupling method. In this method, the wavefunctions during the collision process are expanded in terms of the atomic eigenstate wavefunctions, and the coupled equations describing the collision process are solved numerically. The number of atomic states is as large as practically possible, e.g. 15 or 29. The convergent close-coupling method is a further extension, in which "all" the atomic states, including the continuum states, are virtually taken into account. The result of this calculation is shown in this figure with the closed circles. Another sophisticated method is the R-matrix theory, in which the atomic wavefunction is treated differently in the core region and the outer region, making it possible to increase the number of atomic states

COLLISIONAL EXCITATION AND DEEXCITATION

51

included. The result of a further sophistication, the R-matrix with pseudo-states, is shown with the dotted line in this figure. There are two types of experiments for determining the cross-section of neutral atoms. The first is, by hitting a dilute helium gas in a cell or a helium beam with the electron beam, we determine the number of excitation events by counting the number of photons emitted by the excited atoms. The diamonds in Fig. 3.10(b) give the result. Another technique is to count the transmitted electrons with the relevant energy loss, 21.2 eV in this example. Figure 3.6 is an example of this energy loss spectrum, where the incident electron energy is quite high, 2.5 keV, the high-energy end of Fig. 3.10(b). In the experiment determining the cross-section for 1 :S —> 2 1 P, the magnitude of the first peak in Fig. 3.6 is measured. The results of two experiments of this type are shown with the open triangles and the squares. Agreement among the data is good, except in the region immediately above the excitation threshold. As can be seen in Fig. 3.10(a) and (b), these cross-sections for the ion and the atom have rather similar energy dependences. But, the reader may have recognized an important difference. For the neutral atom, with the energy approaching the threshold, the cross-section value tends to diminish, while for the ion it tends to a finite limit. This is a quite universal tendency. This point is more explicitly shown in Fig. 3.11 (a): for excitation ls^2p of neutral hydrogen and of hydrogen-like ions, the cross-sections are shown which have been scaled against the nuclear charge z. The abscissa is in threshold units: u = E/E(ls, 2p), and u=\ means the energy at the excitation threshold. The excitation cross-section of neutral hydrogen starts from 0 and reaches a maximum at around u = 3 ~ 4. For z = 2, ionized helium (Figs. 1.5 and 1.6), the threshold value becomes finite, and for z = oo, which stands for ions with z^>2, the threshold value is even larger. These differences come from the difference of the "motion" of the electron incident on (and scattered from) a neutral atom or an ion: in the former case the electron does not feel the presence of the atom except when it is close to the atom. In contrast, the ion exerts a strong attractive Coulomb force, which is of quite long range. The electron is attracted and accelerated to the target ion even at a very long distance from the ion. Another typical feature of excitation cross-sections of ions is explicitly shown in Fig. 3.12; this is for the resonance transition of sodium-like argon. The CoulombBorn approximation gives the smooth curve. The curve with the rich structure is the result of the close-coupling calculation. This structure is produced by resonances, which will be discussed in Section 3.5. Although the cross-section of neutral atoms is not fully free from resonance structure, resonance is far more conspicuous in the cross-section of ions. The result of an experiment is also given by the crosses. This experiment is performed by a method called the merged beam method: an electron beam is merged with an ion beam, sometimes in a straight part of a ring accelerator, and the relative speed of these two beams is adjusted to change the excitation energy. In this example, the electron beam energy is 27.15 eV, and the ion energy is varied. Again, the number of electrons with relevant energy loss is counted.

FIG 3.11 Excitation cross-section for (a) 1 s —> 2p and (b) 1 s —> 2s transitions of neutral hydrogen z—l and hydrogen-like ions with nuclear charge z. The abscissa is the collision energy in threshold units. Near the excitation threshold, the scaled cross-section, zVM(w), strongly depends on z, while in high-energy regions the scaled cross-sections tend to be independent of z. (c) Excitation and ionization crosssections from the ground state of neutral hydrogen, z—l, and hydrogen-like ions. Excitation cross-section: the results of the Born or Coulomb-Born approximation: for z — 1, for z — 2, and for z — oo. : More accurate cross-section for 1 —> 2 of neutral hydrogen corresponding to (a) and (b). : approximation leading to eq. (3.29). Ionization cross-section: for z—l, for z — 2, for z — oo. (Quoted from Fujimoto 1979a, with permission from The Physical Society of Japan.)

COLLISIONAL EXCITATION AND DEEXCITATION

53

Until now, we have looked at excitation corresponding to optically allowed transitions. We now imagine a situation in which an electron with very high energy passes by a target atom or ion. This atom or ion feels a pulsed electric field. This pulse may have some similarities to a half-cycle of the light wave, the frequency of which coincides with that of the absorption line of the transition. Then, this atom or ion can absorb this light wave and thus be excited. We may therefore expect some correlation between the magnitude of the cross-section and the absorption oscillator strength. When the electron energy is higher than 5-10 times the excitation threshold, the cross-section can be approximated by

Energy (eV)

FIG 3.12 Excitation cross-section for transition (Is22s22p6)3s28 -> (ls22s22p6)3p2P of a sodium-like argon ion. The result of the Coulomb-Born approximation is shown with the smooth line. The close-coupling calculation gives a result which is rich in resonance structure. The result of an experiment is shown with crosses with uncertainty bars. (Quoted from AMDIS and from Badnell et al, 1991; copyright 1991, with permission from The American Physical Society.)

54

ATOMIC PROCESSES

where u denotes the energy of the incident electron in threshold units. This asymptotic cross-section value is called the Bethe limit. As eq. (3.24) explicitly shows the cross-section value is proportional to the absorption oscillator strength of the transition. We also note that the z scaling becomes valid in this energy range as seen in Fig. 3.11(a). Figure 3.11(c) shows several examples of the excitation cross-sections of neutral hydrogen and hydrogen-like ion from the ground state. The cross-sections to low-lying excited levels are taken from rather simple approximate calculations. Excitation cross-sections to highlying levels are scaled according to the oscillator strength. See Table 3.1(b) and Fig. 3.4. Excitation of optically forbidden transitions, for which fp,q = Q and therefore eq. (3.24) vanishes, is by no means negligible. An example is shown in Fig. 3.11(b). In this case the absolute value of the cross-section for ls^2s is rather small as compared with that of Is —> 2p, and with the increase in the energy a cross-section decreases more rapidly than that for the optically allowed transition. Note that eq. (3.24) has a logarithmic dependence on energy, which is another characteristic of the cross-section for optically allowed transitions. Another example is seen in Fig. 3.13 for neutral helium. Figure 3.13(a), which is for excitation 1 : S^2 :S,

FIG 3.13 (Continued)

COLLISIONAL EXCITATION AND DEEXCITATION

55

FIG 3.13 Examples of measured or calculated excitation cross-sections of neutral helium for optically forbidden transitions, (a) For transition 1 1 S^2 1 S. Results of theoretical calculations with various sophistications are given with the curves, and those of several experiments are shown with the points. (Quoted from AMDIS.) (b) 1 1 S^2 3 S. (Quoted from Fujimoto 1979b; copyright 1979, with permission from Elsevier.) includes the results of several experiments, which are the energy loss measurements, and several calculations. Even for this important transition (see Fig. 1.4), the agreement among the data is rather poor. The reader can imagine what the situation would be for the cross-section for some particular transition of lessstudied atoms or ions. Figure 3.13(b), for 1 : S^2 3S, is another example for neutral helium. This is for a transition with a change in multiplicity, i.e. singlet to triplet: see Fig. 1.4. This transition is made possible only by electron exchange,

56

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which we mentioned above. This process takes place at rather low energies. Thus, the cross-section decreases rapidly with the increase in energy. Note, however, that the cross-section value just above threshold is significant, or even large, as compared with the cross-section of the optically allowed transition, Fig. 3.10(b). As we have seen above, with an increase in the energy of the incident electron, the cross-section for optically forbidden transitions decays much faster than that for optically allowed, or electric dipole, transitions. At sufficiently high energy, cross-sections only for optically allowed transitions would survive. This is the reason why the energy loss spectrum of Fig. 3.6, which reflects excitation and ionization by electron impact at a fixed energy, 2.5 keV in this example, is virtually equivalent to the photoexcitation and photoionization spectrum, which is due to the electric dipole transitions.* Instead of the cross-section, the collision strength is sometimes used.

Klein-Rosseland relationship and deexcitation cross-section Deexcitation is the inverse process to excitation (see Fig. 3.1), and therefore the former process is related to the latter by the principle of detailed balance. Figure 3.14 shows schematically the excitation and deexcitation processes in the energy-level diagram. Let E be the energy of the incident electron before excitation and e be that of the scattered electron after excitation. The corresponding speed of the electrons far from the target are v and v', respectively. Therefore, E=mv /2 and e = mv'2/2. The total number of excitation events p —> q in a plasma per unit volume and unit time, the excitation flux, by electrons within energy width dE is given by

In the deexcitation process, E and e change their roles, and the corresponding deexcitation flux is given by

* From comparison of Fig. 3.13(a) with Fig. 3.10(b), the reader may doubt that the optically forbidden transition, l ' S — > 2 ' S , may not entirely be negligible in comparison with the optically allowed transition, 1 ! S—>2 ! P, at this energy of 2.5 keV. However, this energy loss spectrum is for electrons with very small scattering angles. Scattered electrons by optically forbidden transitions have relatively large scattering angles, so that in Fig. 3.6 the energy loss peak for the 1 'S —> 2 'S transition is found to be 1/200 of the optically allowed transition peak.

COLLISIONAL EXCITATION AND DEEXCITATION

57

FIG 3.14 Schematic diagram for explanation of the relationship between the excitation cross-section and the deexcitation cross-section. In thermodvnamic equilibrium both the fluxes should be ecmal:

where the electron energy distribution is given by the Maxwell distribution function, eq. (2.2a), and the population ratio n(q)/n(p) is given by the Boltzmann distribution, eq. (2.3). We note the relationship E=E(p,q) + e. Then, eq. (3.26) reduces to

We may call this relationship the Klein-Rosseland formula. In terms of the collision strength this relationship is expressed as

Excitation and deexcitation rate coefficients The excitation rate coefficient is obtained with an equation similar to eq. (3.19). The Maxwell distribution, eq. (2.2a), is assumed:*

* When E is measured in units of eV, kTe in eq. (2.2a) is also in eV. This is also the case in eq. (3.28) except for \fE in the integration. This factor comes from v, the speed of the incident electron, and should be in units of \fj. Thus, the resulting number should be multiplied by 4.0 x 10~10 (=1/1.6 x 10~19[C]) to obtain the rate coefficient value.

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As we have seen already, the excitation cross-section is a complicated function of energy, with a particular energy dependence for a particular transition, so that no general expression for the excitation rate coefficient is available. Even the Bethe limit, eq. (3.24), results in a rather complicated function. In the discussions in the following chapters, however, it is sometimes useful to make an order-of-magnitude estimate of various quantities. For these purposes, we employ a very crude approximation: \nu in eq. (3.24) is replaced by a constant unity. An example of this approximation is shown in Fig. 3.11(c) by the thick dashed line for the excitation cross-section ls^2p. Then eq. (3.28) reduces to

with

The deexcitation rate coefficient is likewise obtained from the deexcitation cross-section by

It is readily shown that, by use of the Klein-Rosseland relationship, the deexcitation rate coefficient is related to the excitation rate coefficient as

or

with eq. (2.7) for Z(p). In thermodynamic equilibrium, therefore, the principle of detailed balance actually holds,

and the Boltzmann distribution, eq. (2.3), is established:

IONIZATION AND THREE-BODY RECOMBINATION

59

In expressing eqs. (3.28) and (3.31) the effective collision strength is sometimes used:

Then, the deexcitation rate coefficient is expressed as

and the excitation rate coefficient is given by eq. (3.31). 3.4 lonization and three-body recombination lonization cross-section and rate coefficient As has been shown in Fig. 3.2, with an increase in the principal quantum number, or in energy, of discrete states, the "shape" of their wavefunctions changes gradually, and, with a further increase in energy, this change continues smoothly across the ionization limit to the continuum state wavefunctions. This observation suggests that the process of excitation, in which a negative-energy or discrete-state electron is produced, has features much in common with those of ionization, in which a positive-energy or continuum-state electron is produced. In other words, an ionization process may be regarded as a continuation of the excitation process. Figure 3.15 illustrates this point for the example of neutral hydrogen. For a particular incident energy, e.g. E=9R, cross-sections from the initial state p= 1 are calculated for excitation to final states q = 2,3,4, and 5 and for ionization. The negative abscissa is the energy of the final state q of the atom, or of the electron in the atom, for excitation, and the positive abscissa is the energy of the ejected electron for ionization. In the ordinate, for excitation, the cross-section values divided by (2/q3) are plotted. This factor | d(l/q2)/dq | is the energy width allocated to level q in units of R (see eq. (1.5)). Therefore, the plotted quantity is a cross-section value averaged over this energy width, or the cross-section value per unit energy interval (R), In this figure the energy R is regarded as a unit energy. Compare these cross-section values in Fig. 3.15 with those in Fig. 3.11.* For ionization, plotted in Fig. 3.15 is the "cross-section" a\,E'(E) of producing a positive-energy (£"') electron as the final state. This quantity is also the crosssection for unit energy width (R). The usual (conventional) ionization cross-section is given as an integration of this "cross-section" over the energy E'. It * For example, at £ = 9R=122 eV and for q = 2 in Fig. 3.15, the real cross-section value is 2.8 x (2/23)™o = 0.7™2,. This is consistent with the cross-section aiSi2s + v\s,if at u = 9_R/[(3/4)_R] = 12 in Fig. 3.11(a) and (b). The numerical value of 0.77ra02 is 6.1 x 10~21 m2, with 7ra02 = 8.79 x 10~21 m2; this is consistent with the (Is — 2s, 2p) cross-section at 122 eV in Fig. 3.11(c).

60

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FIG 3.15 Cross-sections for excitation and ionization, showing the continuation properties between these processes. Shown are the cross-section values divided by 2/n3 for excitation and the partial cross-section <Ji^E>(E) to produce a continuum electron having energy E' for ionization. See inset. (Reconstructed after McCaroll, 1957.) is seen that the "cross-section" values* are consistent with the ionization crosssection in Fig. 3.11(c). The first point to be noted in Fig. 3.15 is that, for excitation, with an increase of q, the "cross-section" values tend to a finite value. This indicates that the real cross-sections are approximately proportional to q~3 for very large q toward the ionization limit. This is consistent with eq. (3.24), because the oscillator strength fp>q tends to be proportional to q~3 for large q as seen in Table 3.1 and Fig. 3.4. The second point is that, with the increase in the energy of the final state of the target electron, the excitation "cross-section" continues smoothly to the ionization "cross-section". The third point is that from the comparison of the "cross-sections" for different incident electron energies E it is obvious that both * Since the energy dependence of the production cross-section in Fig. 3.15 is approximately proportional to (l+£'/R)~3 (see eq. (3.13) and Fig. 3.8, and remember that our incident electron energy is high so that the collision processes have features in common with the radiative processes, photoionization in this case) it is straightforward to obtain the conventional ionization cross-section from the threshold value of the cross-section. For E = 9R, for example, the threshold value of 1.3 (wao/R) leads to cr ljC (9.R) = 0.65mJo = 5.7 x 1CT21 m2 on the assumption of the exact minus third power dependence. See the ionization cross-section in Fig. 3.11(c) at E= 122 eV.

IONIZATION AND THREE-BODY RECOMBINATION

61

the excitation cross-section and the ionization cross-section, in conventional terms, have a similar dependence on energy, or they have similar "shapes" as functions of incident energy. We see this point in Fig. 3.11(c) for excitation and ionization from the ground state of neutral hydrogen and hydrogen-like ions. It is noted, however, that we confine our discussion here to the high incident energy region where the Born approximation is valid. For lower energies, the "shape" of the cross-sections could be appreciably different. It is worth noting that the "shape" of the cross-sections in Fig. 3.15 is almost exactly the same as eqs. (3.9b) and (3.13). We now remember the discussion concerning eq. (3.15) that, for the absorption oscillator strength to the continuum, the differential oscillator strength, eq. (3.10a), or the absorption cross-section, concentrates in the rather narrow energy region above the ionization threshold. See Fig. 3.8. In fact, Fig. 3.15 indicates that the produced continuum electrons are actually concentrated in the energy region just above threshold. Thus, for sufficiently high incident energy the ionization crosssection, in conventional terms, is expected to be proportional to fftC and have a similar energy dependence to that of the excitation cross-section, i.e. (\nu/u), where u is now understood to be the energy of the incident electron in units of the ionization potential. The above discussions suggest that for ionization we may employ a crosssection formula similar to that for excitation. We find, however, the following formula better approximates the actual cross-sections:t

where u is the energy E in threshold units, i.e. u = E/x(p). It may be interesting to note the following: in Fig. 3.11(c), for high-energy, the excitation cross-section <TI^(E) and the ionization cross-section a\>c(£) have similar magnitudes. This is related to the fact that the oscillator strengths are/i >2 = 0.416 and/i,c = 0.435 (see Table 3.1), and that the threshold energies are E(\, 2) = (3/4)x(l). The ionization rate coefficient S(p) may be obtained from a similar equation to eq. (3.28). We adopt the following formula corresponding to eq. (3.35):1"

*Excitation and ionization cross-sections in the vicinity of the ionization threshold In the above discussions of the excitation and ionization cross-sections, we have been mainly concerned with the region of high incident energy. We saw in Fig. 3.15 t These formulas are due to Dr E. Baronova.

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ATOMIC PROCESSES

the continuation from the excitation process across the ionization limit to ionization, the production of the continuum states. We now consider the opposite case, i.e. excitation and ionization in the energy region of incident electrons in the vicinity of the ionization threshold. From the discussions at the beginning of the preceding subsection, we may expect that a continuity feature also manifests itself in excitation cross-sections to high-lying states and the ionization cross-section in this low-energy region. As an example we take the ground state of neutral hydrogen for the initial state. It is known that excitation cross-sections to very high-lying levels have finite values at the excitation threshold. For simplicity, we approximate the cross-section to have a constant value

starting from the excitation threshold. We assume this approximation in the narrow energy region considered here. An example is given in Fig. 3.16 for cr\tp(E) f o r p > 5 with the dash-dotted line. The parameter value [p], where [p] denotes the principal quantum number of the core electron p. When n is small and close to [p], the above approximate discussion becomes inadequate. For example, in the case of n equal to [p] no distinction can be made between the core electron and the spectator electron. We have to treat these doubly excited states as they are. In such cases the frequencies of the stabilizing transition lines, or the satellite lines, are separated appreciably from the frequency of the parent line. Figure 3.18 shows an example of the lithium-like satellite lines associated with the parent helium-like line (1 : S—2 :P) of iron ions. All the satellite lines for n = 2 and « g 3 are seen.

FIG 3.18 Experimentally observed spectrum from highly ionized iron in a tokamak plasma. The lines labeled w, x, y, and z are the "parent" helium-like ion lines. Other lines are the lithium-like satellite lines and the beryllium-like satellite line (/3). (Quoted from Bitter et al, 1981; copyright 1981 with permission from The American Physical Society.)

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The population density of the doubly excited state is given from the balance of the processes of dielectronic capture, autoionization, and stabilizing transition

or

where we have used eq. (3.46). The effective rate coefficient for dielectronic recombination from \z into (1, K/')Z-I is given as

Figure 3.19(a) shows an example of the dielectronic recombination rate coefficients from the ground-state hydrogen-like boron ion to various K/' states of a helium-like ion. In this example, only the 2p state is included as the core electron/? in eq. (3.50). Figure 3.19(b) shows the total dielectronic recombination rate coefficient which is the sum of eq. (3.50) over all the nl' states. Figure 3.20 shows the total recombination rate coefficient of the hydrogen-like boron ion to the helium-like ion; the dashed curve is the sum of the rate coefficient for the radiative recombination as discussed in Section 3.2 (the sum of the rate coefficients similar to those in Fig. 3.9) and that for the dielectronic recombination as given in Fig. 3.19(b). For temperatures lower than 4 x 105 K, or the reduced temperature of T e /z 2 ~2.5 x 104 K, the former recombination is dominant, and for higher temperatures the latter becomes dominant. This is because, for dielectronic recombination to take place, the doubly excited states, (2p,«/') in this case, which lie rather high in energy, should be populated by energetic electrons. See Fig. 3.19(b). This figure also includes the effective recombination rate coefficient for finite electron densities. Under these conditions, we cannot separate the radiative recombination and the dielectronic recombination, as will be discussed in Chapter 5. For ions having more than two electrons, e.g. lithium-like ions with three electrons Is22s, doubly excited levels of beryllium-like ions Is 2 2p«/ lie just above the ground state Is 2s. Dielectronic recombination for these ions Is 2 2s + e^ Is 2 2p«/^ Is 2 2s«/+ hv in this example, can be quite substantial even at low temperatures. Resonance contribution to excitation cross-section As we have noted above, any of the excited levels of an ion is accompanied by series of doubly excited states. We now take an example of the hydrogen-like 3p level; this level forms the ionization limit of the doubly excited helium-like (3p, ri)

FIG 3.19 Dielectronic recombination rate coefficient, eq. (3.50), for a hydrogen-like boron ion. (a) Breakdown into each term; (b) the total rate coefficient. Several results of different calculations are given. (Quoted from Fujimoto et al., 1982.)

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FIG 3.20 The effective recombination rate coefficient from hydrogen-like to helium-like boron. In the limit of low density ( ) this is virtually the sum of the radiative recombination rate coefficient, similar to eq. (3.19) (see Fig. 3.9) for all levels p, and the dielectronic recombination rate coefficient, Fig. 3.19(b). For finite densities, the collisional-radiative (effective) recombination rate coefficient (see Chapter 5) is given. (Quoted from Fujimoto et al., 1982.) levels. Here we omit the angular momentum quantum number in designating the spectator electron. An ion in the ground state Is may dielectronically capture an electron to one of these doubly excited levels (3p,«). This doubly excited heliumlike ion could return to the ground Is state of the hydrogen-like ion by autoionization, but it could also autoionize to leave an electron in a singly excited hydrogen-like level, 2s for example. Thus the following series of processes can take place: (dielectronic capture) (autoionization). This series is nothing but an excitation process ls^2s. This additional process should be added to the direct excitation which was discussed in Section 3.3; or the

AUTOIONIZATION, DIELECTRONIC RECOMBINATION

71

FIG 3.21 The excitation cross-section of hydrogen-like neon (z=10) for the ls^2s transition. The underlying almost-constant cross-section corresponds to the curve in Fig. 3.11(b) for z—>oo just above the threshold. The energy range of this figure corresponds to u= 1 — 1.19 in Fig. 3.11(b). The contributions from the resonance, eq. (3.51), are expressed as sharp peaks. (Quoted from Aggarwal and Kingston, 1991; copyright 1991, with permission from The Royal Swedish Academy of Science.) excitation cross-section should have a contribution from this series of processes. This contribution is called the resonance contribution. The above picture based on the two-step mechanism is too simplistic, and a realistic treatment of this process should involve re-diagonalization of the doubly excited states and the underlying continuum states. As a result, the cross-section shows very complicated sharp structure, which is named resonance. Figure 3.21 shows an example of calculations of excitation cross-sections that include resonance contributions; this is for excitation ls^2s of hydrogen-like neon (z=10). The abscissa ranges from the excitation threshold (15R) to the threshold of the 1 s —> 3 / excitation, 8 8. 9R, and the ordinate is the collision strength which was introduced by eq. (3.25). The underlying flat near-horizontal dashed line corresponds to the cross-section given in Fig. 3.11(b): at the threshold the collision strength value of 0.0065 in Fig. 3.21 corresponds to zVij^l)/™}} = 0.43, which agrees very well with the cross-section for z —> oo in Fig. 3.1 l(b). The sharp resonances are due to the doubly excited intermediate states (3s,ri),(3p, ri), and (3d,ri)with « = 3 at around 19R, n = 4 at around 83-84J? and n = 5 at around 85.5R, and so on. Resonances due to (41,41') are also seen at around 88R. We have already seen in Fig. 3.12 an example of the resonance structures of excitation

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cross-sections. In these examples, the resonance contribution, when averaged over energy, is rather minor. In some other cases the contribution is very large, e.g. more than an order larger than the underlying cross-section. *3.6 Ion collisions Until this point, we have exhausted the major atomic processes, radiative transition and collisional transitions due to electron impact, which are important in plasma spectroscopy. In many practical situations, the spectroscopic properties of the ensemble of ions in a plasma are controlled by these processes, and they are enough to describe various phenomena. However, in some other situations, we have to include collision processes other than electron impact. We briefly survey ion collision processes in this section. Excitation-deexcitation Roughly speaking, excitation or deexcitation by electron impact is caused by the time-dependent electric field exerted on the target ion by the incident electron as noted in the paragraph in p. 53 leading to eq. (3.24). This is especially true at high energies. The incident ion also exerts an electric field on the target with the sign reversed, and could induce a transition in this target ion, too. In this context, the essential feature is the temporal variation of the field, which is directly related to the velocity of the incident particle. Thus the important parameter is the velocity rather than the energy as implicitly assumed in eq. (3.24). We recognize two points: 1. The ion mass is much larger than the electron mass; ions are some 103 or 104 times heavier. If an ion is to have the same velocity, or a similar effect, as an electron has on the target, its energy has to be larger than that of the electron by this factor. Figure 3.22 shows an example of cross-sections by ion collisions. This is for excitation of neutral helium: He(ls2 :S) + H + -> He(ls 2p :P) + H +. This is to be compared with Fig. 3.10(b); when the abscissa of the former figure is reduced by M/m = 1.66 x 1CT27/9.11 x 1CT31 = 1.82 x 103 (see Table 1.1), both crosssections are found to be nearly the same. This is especially the case for higher energies. In many cases, however, the ion temperature is nearly the same as or much lower than the electron temperature. These facts suggest that ion collisions are rather ineffective for inducing transitions like this example. 2. Even so, for transitions between closely lying levels, ion collisions could be effective. This can be understood from eq. (3.24); for electron impact, u, the collision energy normalized by the energy difference between the two levels, tends to be very large and the cross-section becomes small. For ions, in terms of velocity rather than energy, the effective u could be small. This is the case for transitions between different / levels with the same n in helium-like and other ions with simple energy structure. Figure 3.23 shows an example of the cross-sections for the 3s —> 3p transition of neutral hydrogen. For energies of practical interest, the crosssection for proton collision is much larger than that for electron impact. For higher

ION COLLISIONS

73

FIG 3.22 Excitation cross-section for the transition 1 1 S^2 1 P of neutral helium by proton collisions. This corresponds to Fig. 3.10(b) for electron impact. (Quoted from Ito et al., 1993, with permission from JAERI.)

FIG 3.23 Cross-section for transition 3s ^3p of neutral hydrogen by proton collisions and that by electron impact. (Quoted from Sawada, 1994.) energies the energy scaling by the factor 1.82x 103 is almost exact. In reality, however, for neutral hydrogen the levels are split into fine structure, and transitions between these fine-structure levels should be considered. The calculation shown in Fig. 3.23 is based on the Born approximation with the fine structure neglected.

74

ATOMIC PROCESSES

Charge exchange collisions

When an ion approaches an atom, it attracts the atomic electrons, and it may eventually capture one or more electrons from the atom after its collision with that atom. This process is called charge exchange or charge transfer. The thick solid curve with the open circles in Fig. 3.24 gives an example of the cross-sections for charge exchange; this is for one-electron capture He + H+ —> He+ + H. The captured electron may not be in the ground state. The dashed and dash-dotted curves in Fig. 3.24 show the cross-sections for production of excited-state hydrogen, 2p and 2s, respectively. In this case, the production of excited states is

FIG 3.24 Charge exchange cross-section for He + H+ —> He++ H. — o — o —: total cross-section; : charge exchange cross-section producing excited atoms H(2s). : the same for H(2p). —•—•—: resonant charge exchange cross-section H+ H + ^H + + H. (Rearranged from Ito et al, 1994, with permission from JAERI.)

ION COLLISIONS

75

rather minor. In the case of a multiply charged ion, the situation could be different. Figure 3.25 shows the cross-sections for O6+(ls2) + He^O 5+ (ls 2 «/) + He+. In (a) the total charge transfer cross-section is shown with x. The crosssection for producing nl= 3s is shown with •. for 3p with •. and 3d with ». The sum of these three cross-sections for n = 3 is given with A. Figure 3.25(b) shows the distribution among n at the collision energy of 60 keV: o for ns, n for «p, and o for nd. The symbol + is for n = 2 including the ground state 2s (the collision energy

FIG 3.25 Charge exchange cross-section for O6+(ls2) + He -> O 5+ (ls 2 n/) + He+. (a) Total charge transfer cross-section ( x ). Cross-section for producing nl= 3s («, o), 3p (•, n), and 3d (», o). The sum of these three cross-sections for n = 3 (A). The open symbols connected with the lines are for « = 4. (b) Distribution of the product ions among n at the collision energy of 60 keV. KS (o), «p (n), and nd (o). +: for n = 2 including the ground-state 2s (the collision energy is 9 keV). (Quoted from Watanabe, 1998.)

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ATOMIC PROCESSES

is 9 keV). In this example production of excited-state ions is dominant, and the produced states are quite selective. The above selective nature of charge exchange collisions is qualitatively explained by the "classical overbarrier model". In this model it is assumed that the optical electron is resonantly transferred from the target atom to the projectile ion when the "barrier", which is due to the superposition of the Coulomb potential of the atomic ion core and that of the projectile ion, becomes lower than the energy of the atomic electron. This occurs at the internuclear distance

where z is the charge of the projectile ion and %B is the ionization potential of the target atom electron. Here all the quantities are measured in atomic units (au). In the above example, z = 6 and XB = 24.5 eV/27.2 eV = 0.90 gives Rcv = 6.5. The favored energy of the final state of the ion is given by

In this example, x* = 1-66 au or 45.2 eV. On the assumption of the hydrogenic level structure the principal quantum number for this energy is «* = 3.3. The strong selectivity in Fig. 3.25 is thus explained. In the case of the example of Fig. 3.24, the above model gives the favored principal quantum number n* = 0.66. Thus, the dominant product is ground-state hydrogen. In some other cases the target atom may have the same ion core as the projectile ion. Charge exchange in such a case is called resonant charge exchange. An example of the cross-sections is shown in Fig. 3.24 with the dot-solid curve: H + + H ^ H + H + . The "shape" of the cross-section has two characteristic features. The cross-section values in the low-energy region are quite large and show little energy dependence. In higher-energy regions, the cross-section value decreases sharply with an increase in energy. The critical energy roughly corresponds to the relative speed of the colliding particles which is equal to the speed of the atomic electron in its classical orbit. In this example, this energy is given from eq. (1.3) and the proton mass in Table 1.1. This is 4.9 x 104 eV, which is consistent with the critical energy in Fig. 3.24.

Appendix 3A. Scaling properties of ions in isoelectronic sequence Hydrogen-like ions Various atomic parameters of hydrogen-like ions change according to the nuclear charge z. We call this the z scaling. Some of the scaling laws have already been introduced in the text. We summarize these properties in this appendix.

SCALING PROPERTIES OF IONS

77

In the following, we express the quantity for a hydrogen-like ion z in terms of the corresponding quantity for neutral hydrogen z = l ; the latter quantity has suffix "H". The energy of atomic levels is scaled as

This scaling applies to all energies, e.g. the ionization potential x(p), eq. (1.1). The frequency of a transition line has the same scaling, The oscillator strength is independent of z, as is understood from its definition, eq. (3.2), and the scaling of the atomic radius, eq. (1.2), together with eq. (3A.2), This is also consistent with the sum rule, eq. (3.5), which gives the number of electrons, i.e. 1. The transition probability, eq. (3.1), scales from eq. (3A.2) as

From eq. (2.12) the B coefficient scales according to

The photoionization cross-section is expressed as eq. (3.10a), and from eqs. (3A.2) and (3A.3) we have

From Milne's formula, eq. (3.17), we have

where the speed of the plasma electrons to be captured is measured according to the scaling

Equation (3A.6) is also seen in eq. (3.18). It is interesting to see that eq. (3A.7) is consistent with eq. (1.3), the electron speed in the Bohr orbits. From the expression of the Maxwell distribution, eq. (2.2), with eq. (3A.7), we have

where we have adopted the scaling for the electron temperature JjJ1 may be called the reduced electron temperature.

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With this scaling of temperature, the radiative recombination rate coefficient, eq. (3.19) or eq. (3.19a), is scaled as

The excitation cross-section, eq. (3.24), except for the energy region near threshold (see Fig. 3.11), is scaled as

where for the electron energy we have used the threshold units u, which is independent of z if we adopt the scaling for the energy of the incident electron,

in accordance with eqs. (3A.I) and (3A.7). It is noted that eq. (3A. 11) is not consistent with eq. (1.2). The collision strength as defined by eq. (3.25) is scaled as

With the scaling eq. (3A.9), the integration, eq. (3.28), for the excitation rate coefficient gives the scaling

The deexcitation rate coefficient and the ionization rate coefficient follow the same scaling. The Saha-Boltzmann coefficient, eq. (2.7), is scaled as

where the energy and the temperature are scaled according to eqs. (3A.I) and (3A.9), respectively. The three-body recombination rate coefficient is given from eq. (3.40) and scales as

*Non-hydro gen-like ions In Chapter 1, it has been pointed out that the energy level structure of ions having a small number of electrons can be regarded as a modification of that of hydrogenlike ions. For instance, the level energy is expressed in terms of the effective principal quantum number, eq. (1.7a). Other quantities are also scaled according to z, the effective core charge felt by the optical electron at a large distance from the core. Instead of "H" we use "1" to denote the starting point in this isoelectronic sequence, i.e. the neutral atom. The parameter values for z = 1 which give a good scaling for large z may be different from the actual values for the neutral

THREE-BODY RECOMBINATION "CROSS-SECTION"

79

atom. We designate a level with quantum numbers nl. The energy difference between the levels with different n, or the ionization potential of a level, follows approximately the scaling of the hydrogen-like ions, eq. (3A.1). A similar situation applies to various quantities for different n levels. The energy difference between different / in the same n is scaled as The oscillator strength then follows the scaling

and the transition probability scales as

The following scaling properties concerning the doubly excited levels, (p, nl), are valid for helium-like ions. As in Section 3.5,/> denotes the core electron and nl the spectator electron. The dielectronic capture rate coefficient, eq. (3.43), has the same scaling as that for the excitation rate coefficient, eq. (3A.14),

The autoionization probability, eq. (3.47a), is independent of z:

where the factor [(z — l)/z]2 has been neglected. The dielectronic recombination rate coefficient is given by eq. (3.50), so that it obeys rather complicated scalings

In the case of Aa > AT, the dielectronic recombination obeys the same scaling as the radiative recombination, eq. (3A.10). With an increase in z, Ar tends to be larger than Aa, and the dielectronic recombination becomes less important as compared with the radiative recombination. *Appendix 3B. Three-body recombination "cross-section" As noted in the text, the three-body recombination process involves an ion and two electrons, and the likelihood of this process to take place cannot be expressed by a cross-section which has been defined for two-particle reaction processes. In this appendix, we derive a rate coefficient in terms of a kind of cross-section. We consider ionization from level p of atom (z — 1) to ion z, and the inverse three-body recombination. We start with the Klein-Rosseland formula, eq. (3.27),

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for excitation and deexcitation cross-sections (Fig. 3.14). In the present case the upper level is denoted by u. The cross-section for deexcitation u —>/> is given by

We assume that the upper level consists of a group of levels, ut, having virtually the same energy and common excitation-deexcitation cross-sections. Then we have

We now assume that these upper levels belong to the continuum states, having energy E' as measured from the bottom of the continuum, or the ground state of ion z. Figure 3B.1 shows schematically the relationship of the energies. Let gu(E') be the density of states per unit energy interval. The Klein-Rosseland relationship extended to this situation is written as

where dE' is the energy width within which the group of the upper levels are contained. It is obvious from the derivation of this equation that the "cross-section" ff P,u(E') has the dimension of [m2!"1]. This is a cross-section for production of upper levels in a unit energy interval. Instead of the continuum states, we assume the free states. Then, the quantity gu(E') is nothing but the "statistical weight" as

FIG 3B.1 Schematic energy relationship for three-body recombination z + e(£") + e(e) -^pz-i + e(£).

THREE-BODY RECOMBINATION "CROSS-SECTION"

81

given by eq. (2.5a). Thus, the "deexcitation" cross-section is expressed as

where g(p) has been suffixed with z — 1 to make clear that level p is an atomic level, and ge = 2. As is seen in Fig. 3B.1 the energies are related to each other: E' + e = E~x(p)- We now consider "deexcitation", or more exactly, recombination; The number of target "atoms" is given by nzf(E')dE', where f(E') dE' is the energy distribution of the continuum electrons. These target atoms are acted upon by the electrons nj'(e) de with a cross-section that describes the likelihood of this reaction to take place. The number of recombination events in unit volume per unit time Mm^3s^1l is siven bv

We may call the above cross-section in the integrand the three-body recombination cross-section. By substituting eq. (3B.4) into this equation we have

where we have used v (e) = ^J2eJm, From eq. (2.2a) it is readily seen that the integrand is rewritten as

We now shift the origin of the energy from the ground state of ion z to the position of the atomic level p. Then, we have

It is noted that the last integral is the ordinary ionization cross-section, aptC(E), as pointed out in Section 3.4 with regard to Fig. 3.15. Equation (3B.7) then reduces to

where use has been made of eq. (2.7). It is readily seen that this equation is eq. (3.40), which we introduced from the thermodynamic equilibrium relationship.

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References The discussions in Section 3.5 are mainly based on: Beigman, I.L., Vainshtein, L.A., and Syunyaev, R.A. 1968 Sov. Phys.-Uspekhi 11,411. The tables and figures are taken or based on: Aggarwal, K.M. and Kingston, A.E. 1991 Phys. Scripta 44, 517. AMDIS (Data and Planning Information Center, National Institute for Fusion Science, Toki). Badnell, N.R., Pindzola, M.S., and Griffin, D.C. 1991 Phys. Rev. A 43, 2250. Bethe, H.A. and Salpeter, E.E. 1977 Quantum Mechanics of One- and Two-Electron Atoms (Plenum, New York; reprint of 1957). Bitter, M., von Goeler, S., Hill, K.W., Morton, R., Johnson, D.W., Roney, W., Sauthoff, N.R., Silver, E.H., and Stodiek, W. 1981 Phys. Rev. Letters 47, 921. Fisher, V.I., Ralchenko, Y.V., Bernshtam, V.A., Goldgirsh, A., Maron, Y., Vainshterin, L.A., Bray, I., and Golten, H. 1997 Phys. Rev. A 55, 329. Fujimoto, T. 1979a /. Phys. Soc. Japan 47, 265. Fujimoto, T. 1979b /. Quant. Spectrosc. Rad. Transfer 21, 439. Fujimoto, T. and McWhirter, R.W.P. 1990 Phys. Rev. A 42, 6588. Fujimoto, T., Kato, T., and Nakamura, Y. 1982IPPJ-AM-23 (Institute of Plasma Physics, Nagoya). Goto, M. 2003 /. Quant. Spectrosc. Radial. Transfer 76, 331. Ito, R., Tabata, T., Shirai, T., and Phaneuf, R.A. 1993 JAERI-M, 93-117 (Japan Atomic Energy Research Institute, Tokai-mura). Ito, R., Tabata, T., Shirai, T., and Phaneuf, R.A. 1994 JAERI-M, 94-005 (Japan Atomic Energy Research Institute, Tokai-mura). Liu, X.-J., Zhu, L.-F., Jiang, X.-M., Yuan, Z.-S., Cai, B., Chen, X.-J., and Xu, K.-Z. 2001 Rev. Sci. Instr. 72, 3357. McCarroll, R. 1957 Proc. Phys. Soc. (London) A70, 460. Sawada, K. 1994 Ph.D. thesis (Kyoto University). Watanabe, H. 1998 Ph.D. thesis (Kyoto University).

4

POPULATION DISTRIBUTION AND POPULATION KINETICS Suppose we observe a plasma with a spectrometer to resolve the light it emits into a spectrum (see Figs. 1.3,1.5, and 1.7 for example) and deal with one of the spectral lines which corresponds to the transition p —> q. We make two assumptions: 1. The plasma is optically thin, i.e. all the photons emitted by ions (atoms) in the plasma leave the plasma without being absorbed inside the plasma. 2. The plasma is isotropic, i.e. the emitted photons are unpolarized and their angular intensity distribution is isotropic. Then the observed line intensity $(p,q) (this quantity is called the radiant flux or the radiant power in radiometry, and has units of [W]) is given by the product of the upper-level population n(p) and the radiative transition probability A(p, q);

where V is the volume of the plasma which we observe and dfi is the solid angle subtended by our optics, e.g. when we use a condenser lens it determines dfL Since we assume A(p, q) to be known, $(p, q) is determined by n(p), apart from geometrical factors. Thus, the features of the spectrum like those in Figs. 1.3, 1.5, and 1.7, or the distribution of the line intensities over the spectrum, which is the central problem of plasma spectroscopy, reduces to the problem of the population and its distribution over various excited levels like those in Figs. 1.4 and 1.6. In this chapter we investigate how, in a plasma, the populations are formed in excited levels, and what are the general characteristics of the population distributions in relation with the nature of the plasma. 4.1 Collisional-radiative (CR) model Rate equation From now on we confine our consideration to hydrogen-like ions (and neutral hydrogen) with nuclear charge ze, except when otherwise stated, z = 1 means neutral hydrogen. We use the term ions to denote atoms as well as ions. (In this and following chapters, we sometimes use the terms atoms and ions for the purpose of distinguishing ions in two successive ionization stages.) We assume the statistical populations among the different angular momentum states within the level with the same principal quantum number. Its validity is examined in Appendix 4A. We thus adopt the simplified energy-level diagram like Fig. 1.6, where p or q represents the principal quantum number.

84

POPULATION DISTRIBUTION AND POPULATION KINETICS

A change in the population of a discrete level is brought about by spontaneous radiative transitions and transitions induced by electron impact as examined in the preceding chapter. See Fig. 3.1. We omit other transition processes, e.g. photoionization and transitions induced by ion collisions. This is because our objective in this chapter is to study the most fundamental features of the populations of ions which are immersed in a plasma. Then the temporal development of the population in level p in ionization stage (z—1), nz_i(p), is described by the rate equation

where we have assumed that the plasma is homogeneous and the spatial transport of the ions does not affect the population dynamics. We use the convention that when we are considering level p, summation over "q) is approximately proportional to p~3/p~4'5 or p1'5. Therefore we have Figure 4.5 compares this approximation with the results of the numerical calculation. For the low density, this approximation is good for high-lying levels that are in the corona phase. For low-lying levels including p = 2, the above various approximations become poor, and the populations deviate from eq. (4.24). Figure 1.10(a) includes the approximate population distribution, eq. (4.24). *Cascade contribution In the above discussion we may express the excitation rate coefficient as where CQ is a constant. See eq. (3.29). Then eq. (4.23) with eq. (4.13) reduces to The populating flux to level p by cascade is given as ^2q>p n\(q)A(q,p), which may be approximated by the integration,

We use eq. (3.8) to yield (gbb is assumed to be 1)

102 POPULATION DISTRIBUTION AND POPULATION KINETICS We define the cascading contribution as

The value of C is found for p = 4, 6, and 10 to be 0.187, 0.190, and 0.193, respectively. If we take into account the cascade contribution in eq. (4.23b) from the still higher-lying levels, (, would be slightly larger than 20% for these lowerlying levels; this is in accordance with Fig. 4.6(a). Transition from the corona phase to the saturation phase - Griem's boundary In Fig. 4.4, with an increase in «e, level 5, for example, makes a transition from the low-density region, the corona phase, to the high-density region, the saturation phase, at about « e = 1017-1018 m~3. It is seen in Fig. 4.6 that at about this «e a transition takes place both in the populating flux (a), and in the depopulating flux (b) as well. We will examine whether these simultaneous transitions are a mere coincidence or not. First we look at the depopulating flux. This transition is the change of the dominant terms in the second line of eq. (4.2) from the radiative decay to the collisional transitions. At this «e we have

We have seen that the dominant transition in the r.h.s. terms is C(p,p + l)«e. (See eqs. (4.6), (4.9), and (4.11); see also eq. (4.14).) Thus, we may conclude that the transition in the depopulating flux out of level p takes place when

holds. This boundary «e is given approximately from eqs. (4.7) and (4.13) as

With regard to the populating flux, Fig. 4.6(a) indicates that the dominant flux in the lower-density region is the direct excitation from the ground state as we have seen, in the high-density region it is the excitation from the adjacent lowerlying level, level 4 in the present example. Thus, at this transition the following equation holds:

We note that, at this «e, level (/>—!) is still in the corona phase (see Figs. 4.4 and 1.10(a)), and its population is given approximately by eq. (4.23);

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103

By substituting eqs. (4.7), (4.13), and (4.23a) into eq. (4.27a) we have

For large p the value of «e given by eq. (4.28) is very close to that given by eq. (4.26). Thus we may conclude that in the populating and depopulating fluxes, Fig. 4.6(a) and (b), both the transitions should take place at about the same ne, and that these transitions give the transition of the level from the corona phase to the saturation phase. For a given ne, eq. (4.25) or (4.26) may be interpreted as giving the boundary level between the lower-lying levels which are in the corona phase and the higherlying levels which are in the saturation phase. This boundary level is denoted asp®. See Fig. 1.10(a). The boundary ne or the boundary level pG which gives the above transition is called Griem's boundary. This nomenclature is offered in honor of the worker who proposed a critical «e in his discussion of LTE. This problem of the validity of LTE will be discussed in detail in Appendix 5C of the next chapter. His criterion is essentially the same as the present eq. (4.25) and thus eq. (4.26). By substituting appropriate values in G and H we obtain for Griem's boundary

Instead of this equation the following simple expression

or

gives even more appropriate numerical values. Figure 1.10(a) includes this boundary with the label "GRIEM" attached. In Fig. 4.5 the boundary level pG, as given by eq. (4.25), or given from the comparison of the total collisional depopulation rate and the radiative decay rate in the second line of eq. (4.2), is plotted with the dash-dotted line. Figure 4.8 shows the sketch of the dominant fluxes of electrons among the levels at ne = 1018 m~3, where Griem's boundary level is given by the dash-dotted line.

104 POPULATION DISTRIBUTION AND POPULATION KINETICS Saturation phase - ladder-like excitation-ionization Figure 4.6 shows that, for level 5, in the «e regions higher than Griem's boundary the dominant populating process is collisional excitation from level 4 and the dominant depopulating process is collisional excitation to level 6. Here we remember eq. (4.9), i.e. collisional excitation is more likely than deexcitation, and eq. (4.11), i.e. ionization is rather minor. The above feature is also seen in Fig. 4.8 for the levels lying above Griem's boundary. It is concluded that, for a level in the saturation phase, the dominant populating flux to this level is the collisional excitation from the adjacent lower-lying level and the dominant depopulating flux is the collisional excitation to the adjacent higher-lying level. The upward flux of population by stepwise excitation is thus established in eq. (4.2) with eq. (4.18): We name this mechanism of multistep excitation, ladder-like excitation. Since the chain of this excitation flux results in ionization, we may call it ladder-like excitation-ionization. Within the approximation of eq. (4.30), the magnitude of this flux is independent of p, then we have ni(p))oc/>~ 4 , or

FIG 4.8 Sketch similar to Fig. 4.7 except that « e = l x ! 0 l s m . The boundary level pG between the corona phase and the saturation phase, as given by eq. (4.25), lies between levels p = 3 and 4, as indicated by the dash-dotted line. The ladder-like excitation-ionization flux through a high-lying level, level p=lO, is shown with the filled arrow. (Quoted from Fujimoto, 1979b; with permission from The Physical Society of Japan.)

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105

We may call this distribution the minus sixth power distribution. Figure 4.5 compares this approximation with the results of the accurate numerical calculation. This simple approximation is surprisingly good. In Fig. 1.10(a), the upper half of the region higher than the boundary "GRIEM" shows the features of this saturation phase. In real plasmas in which the ionizing plasma component is dominant the temperature is high, and the boundary "BYRON" and the region lower than this boundary are absent, as will be discussed in the following subsection. With a further increase in «e no significant change is seen in Fig. 4.6(a) and (b), but the Ke-dependence of the population in Fig. 4.4 begins to show a new feature at about « e = 1021 m~3. We note that, with this increase in «e, Griem's boundary comes down, and that at « e ~10 21 m~ 3 it reaches the first excited level p = 2, Figure 4.9 shows a sketch of the dominant fluxes at « e = 1022 m~3. Except for the downward arrow 2 —> 1 all the arrows are collisional. Roughly speaking, the chain of the populating-depopulating fluxes of eq. (4.30) starts from the ground state and it controls the populations of all the excited levels. In this highest-density region, the ladder-like excitation-ionization mechanism controls the populations of all the excited levels, i.e. If we eliminate «e from this relationship it becomes independent of «e. This is the reason why all the excited level populations in Fig. 4.4 are independent of «e-

FIG 4.9 Sketch similar to Fig. 4.7 except that « e = 1 x 10 m . All the excited levels are in the saturation phase and the mechanism of the multistep ladder-like excitation-ionization, eq. (4.32), is established starting from the ground state. Only the downward arrow (2 —> 1) is the radiative transition. (Quoted from Fujimoto, 1979b; with permission from The Physical Society of Japan.)

106 POPULATION DISTRIBUTION AND POPULATION KINETICS The population distribution is given by eq. (4.31). Figure 4.5 shows that the population distribution of all the excited levels is given approximately by eq. (4.31). If we examine Fig. 4.6 and Fig. 4.9 in more detail, we find that the relationship (4.30) is in fact a rather poor approximation to the actual populating and depopulating fluxes. For example, eq. (4.30) accounts only for 60% of the total fluxes for level 5 and only 45% for level p = 20. Still, eq. (4.31) describes the population distribution in Fig. 4.5 rather well. This puzzling situation is resolved when we note that, if eq. (4.31) holds, other balance relations are valid:

and

for large/?. Equation (4.33) is derived rather straightforwardly from the relation (4.5) with eq. (4.7), for the distribution eq. (4.31). The proof of eq. (4.34) may be given on the basis of eqs. (3.6a) and (3.29). Of course, the exponential factor in the latter is neglected. It is seen in Figs. 4.6 and 4.9 that these relationships actually hold approximately. So far, we have assumed high temperatures so that the approximation (4.5) is valid in the above discussion. See Fig. 4.2. For excitation from the ground state, p = l, the exponential factor in eq. (3.29) cannot be neglected even on the present high-temperature assumption. Equation (4.7) and therefore eq. (4.31) cannot be applied to p = 1. If the temperature becomes low, this neglect of the exponential factor may not be justified even for excited levels. If the exponential factor is retained for excitation from excited levels,/? g 2, eq. (4.31) is no longer valid. Even in this case, so far as eq. (4.9) holds, eq. (4.30) should still be valid, but, in this case, the population distribution should include the exponential factor besides the p~6 factor. Instead of n(p)/g(p) we take ri(p);by doing so the exponential factor is absorbed in [Z(p)/Z(l)] as shown in eq. (4.20). Figure 4.10 gives a plot of r^(p) for several cases of neutral hydrogen as well as of hydrogen-like ions. This figure also shows the two lines representing (p~6 x 2) and (p~6/2). It is seen that in the range of ne/z7 g 1022 m~ 3 and Te/z2 g 3 x 104 K, the approximation

is valid within a factor of 2. We now consider the minus sixth power law, eq. (4.31), from another viewpoint. The density of atomic states in a unit energy interval is proportional to g(p)/AE(p)Ap=l, or to p5. See eq. (1.5). If we regard the population flux of the ladder-like excitation-ionization as the flux of electrons in the energy space, and if we require that the "speed" of this flow be finite, the population distribution should be n(p)/g(p)<xp~5. This is inconsistent with eq. (4.31) or (4.35). We have to think about the possibilities that something is wrong in our above discussions

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FIG 4.10 The population coefficient r\(p) against p. - - - - - - ; z = l , «e/z7 = 1023 m~3, and Te/z2 = 3.2 x 104 K; : z = l , « e /z 7 =10 23 m~3, and 2 5 7 r e /z =1.28x 10 K; : z = 2, « e /z =10 23 m~3, from top to bottom 2 3 4 Te/z = 8 x 10 K, 1.6 x 10 K, 3.2 x 104 K, and 5.12 x 105 K; : z = 2, « e /z 7 = 1021 m~3, and T e /z 2 = 5.12 x 105 K; : z = 2, « e /z 7 = 1019m-3, and Te/z2 = 5.12 x 105 K; : (p~6 x 2) and (p~6/2). (Quoted from Fujimoto and McWhirter, 1990; copyright 1990 with permission from The American Physical Society.) leading to these equations. One possibility is the approximation of C(p,p +1) based on the optically allowed transition, eq. (3.29) or eq. (4.7), C(p,p+ l)oc/> 4 . This approximation may not be valid for very large p. In our numerical calculation, the cross-sections for these transitions are based on the impact parameter method calculation which should take into account the optically forbidden transitions as well, but it is not certain whether they are very accurate for very large p. If, in the numerical calculation, we use the rate coefficients derived from the Monte Carlo calculation for classical electron orbits, we obtain a distribution n i(p)/g(p)°tp~5 -5 f°r levels lying higher than p~2Q, For lower levels which are important in practical situations thep~6 distribution is found to be valid. We may conclude that the present results are valid for excited levels of practical interest. Low-temperature case In practical situations we are unlikely to encounter the low-temperature case of ionizing plasma. However, for the purpose of keeping our theory transparent, we examine this case briefly.

108 POPULATION DISTRIBUTION AND POPULATION KINETICS

FIG 4.11 Population distributions r±(p) for several ne's. Neutral hydrogen with Te= 1 x 103 K. The boundary between eq. (4.9) and eq. (4.36) is given by the dotted line, and another boundary, pG, as given by eq. (4.25) is given by the dash-dotted lines. (Quoted from Fujimoto, 1979b; with permission from The Physical Society of Japan.) Table 4.1 (a) shows the population coefficients r^p) for T e = 1 x 103 K. The population distributions are shown in Fig. 4.11, which corresponds to Fig. 4.5. The reader may locate these plasmas in Fig. 1.2. In Fig. 4.11 the ordinate is the logarithm ofr^p) instead ofnl(p)/g(p), since the latter quantity has a too strong /i-dependence that is due to the exponential factor as discussed above. We remember that ri(p) = 1 means that level p is in thermodynamic equilibrium with the ground state, as has been noted near the end of the preceding section. As is seen in Table 4.1 (a) and in this figure, in the high-density limit, this equilibrium is actually established for the low-lying (p < 6) levels. In this limit the high-lying levels (p > 6) are still controlled by the ladder-like excitation-ionization flux, showing ri(p)<xp~6. The breakdown of the ladder-like excitation, eq. (4.30), for the low-lying levels results from the breakdown of eq. (4.9), which is due to the presence of the exponential factor in eqs. (3.29) and (3.31). Figure 4.2 shows the ratio C(p,p+\)/F(p,p— 1) against Te for several p's. In low temperatures the relationship opposite to eq. (4.9)

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109

holds for low-lying levels. The boundary between eq. (4.9) and eq. (4.36) is given in Fig. 4.2 for various p's as the ordinate value of 1 and by the dotted line in Fig. 4.11 for this particular temperature. The boundary "BYRON" in Fig. 1.10(a) is this boundary. A detailed discussion about this boundary will be given in Section 4.4. In this high-density limit the lower-lying levels are in the balance relation instead of eq. (4.30). This results in the thermal (Boltzmann) distribution between the levels/? and (p — 1), and therefore among all the levels 1

1019 m~3. The difference in the boundary «e values by one or two orders of magnitude between the two figures may be explained from

FIG 4.12 Population distribution of excited argon atoms in a positive column plasma. T e ~ 5 x 104 K. (a) Dependences on ne. This figure corresponds to Fig. 4.4 for neutral hydrogen, (b) Population distribution among the levels corresponding to Fig. 4.5. (Quoted from Tachibana and Fukuda, 1973; with permission from The Institute of Pure and Applied Physics.)

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the following two points: (1) we are comparing different atomic species, hydrogen and argon; (2) in the experiment, two of the four lowest-lying levels are metastable, and the resonance lines originating from the other two levels, which correspond to the Lyman a line in the case of hydrogen, are subjected to heavy radiation trapping, which will be dealt with later in Chapter 8. These facts result in a much reduced effective transition probability from the lowest-lying levels, which is not taken into account in the theory and tend to reduce Griem's boundary ne for the lowest excited level. Figure 4.12(b) shows the population distribution over the excited levels for « e = 1 x 1019 m~3, where all the levels are in the saturation phase. Since the levels have various energy values, we take the effective principal quantum number, eq. (1.1 a), as the abscissa. They follow the minus sixth power law except for the lowest-lying levels. Its deviation is in the opposite direction to those in Fig. 4.5. This inconsistency may be explained from the fact that the temperature of the experiment of Te = 5 x 104 K is not sufficiently high for the exponential factor to be neglected entirely. 4.3 Recombining plasma component - high-temperature case In this section we study the recombining plasma component of the excited-level populations. See Fig. 1.9. According to eq. (4.20) this is defined as

We further assume high temperature, i.e. Te/z2 is much higher than 1.5 x 104 K. This boundary temperature is different from that in Section 4.1. This boundary is relevant for the recombining plasma; as we will see later, in this range of temperature, collisional excitation from an excited level is more probable than collisional deexcitation from this level. See Fig. 4.2. We take neutral hydrogen in a plasma with Te= 1.28 x 105 K as an example for the purpose of illustration. The reader may remember the horizontal line drawn on Fig. 1.2 for the plasmas of Section 4.2. Table 4.1(b) gives the result of calculation of the population coefficient r0(p), A salient feature is that, for high densities, virtually all the levels have r0(p)= 1, indicating thermodynamic equilibrium populations, or LTE populations, as we have seen earlier. For lower densities the r0(p)'sdeviate from 1, but the degree of this deviation is rather small. Figure 4.13 shows the population of several levels as functions of ne, where the population per unit statistical weight, nQ(p)/g(p), has been further divided by the ion and electron densities, nzne. The population nQ(p) in this figure is regarded as n0(p)/z3for hydrogen-like ions with nuclear charge z. This is an approximation, but the degree of approximation is better than that for the ionizing plasma, and even exact in the low-density limit. In the following discussions of the recombining plasma component, we sometimes call 0(p)/g(p)nztie]simply the "population". This quantity is almost exactly [n

112 POPULATION DISTRIBUTION AND POPULATION KINETICS

FIG 4.13 The recombining plasma component of excited-level populations against «e [m^1]. The ordinate is n0(p) per unit statistical weight g(p) further divided by the ion and electron densities, nz and ne, respectively. Calculation is for neutral hydrogen with T e =1.28 x 105 K. (Quoted from Fujimoto, 1980a; with permission from The Physical Society of Japan.)

FIG 4.14 Population distribution of the recombining plasma component of the excited-level populations. The ordinate is the same as Fig. 4.13, and the abscissa is the logarithm of the principal quantum number of the level. Calculation is for neutral hydrogen with r e =1.28x!0 5 K. The Saha-Boltzmann population given by eq. (2.la) is shown with the solid line. The dash-dotted lines indicate the boundary between the CRC phase and the saturation (LTE) phase as given by eq. (4.25). (Quoted from Fujimoto, 1980a; with permission from The Physical Society of Japan.) proportional to r0(p) except for the small /^-dependent exponential factor in eq. (2.7) under this high-temperature condition. See also eq. (4.38). Figure 4.14 shows the population distributions for several electron densities. In Fig. 4.14 the population in LTE, or the Saha-Boltzmann population distribution given by eq. (2.la), i.e. r0(p) = 1, is shown with the solid line. As has been noted the actual populations

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in the high-density limit agree with this equilibrium population distribution. It should be noted that the range of the ordinate of these figures is very narrow. All the populations lie within a range of a factor of 1.5 for the whole of the range of ne and/?; this is actually seen in Table 4.1(b): the smallest r0(p) is 0.69. Therefore, we may conclude that, in the high-temperature case, the recombining plasma component is rather close to the Saha-Boltzmann value for low densities (for very high temperatures it is even larger than that - see Fig. 5.10 later; see also Appendix 5C) and tends to it at high densities. Figure 4.15 is similar to Fig. 4.6: the breakdown of the populating (a) and depopulating (b) fluxes for level p = 5. It should be noted that Fig. 4.15(b) is

FIG 4.15 Breakdown of the (a) populating and (b) depopulating fluxes concerning level p = 5 into individual fluxes. This condition corresponds to Figs. 4.13 and 4.14. Other explanation is almost the same as that for Fig. 4.6. Part (b) of this figure is identical to Fig. 4.6(b). (Quoted from Fujimoto, 1980a; with permission from The Physical Society of Japan.)

114 POPULATION DISTRIBUTION AND POPULATION KINETICS identical to Fig. 4.6(b), since the depopulating processes are common to the ionizing plasma component and the recombining plasma component. It is seen that, in the higher-density regions, higher than 1018 m~3, every depopulating flux in (b) is almost exactly balanced in (a) by the corresponding populating flux of its inverse process. In other words, the principle of detailed balance is actually realized. This is consistent with the fact that the populations are almost exactly equal to their Saha-Boltzmann values in these higher-density regions. In Fig. 4.15, we find two puzzling features: 1. For lower-density regions both the populating fluxes and the depopulating fluxes are radiative, i.e. the radiative recombination and the radiative cascade for the populating process in (a) and the radiative decay for the depopulating process in (b). No relationship like the principle of detailed balance is expected among the rate constants of these processes. Still the populations are very close to those given from thermodynamic equilibrium as we have seen. 2. The transition of the populating mechanism in Fig. 4.15(a) from the radiative processes in lower densities to the collisional processes in higher densities takes place at the density approximately equal to that in Fig. 4.15(b) for the transition of the depopulating mechanism, i.e. Griem's boundary given by eq. (4.25). We will investigate whether these features are a mere coincidence or they are a kind of necessary phenomenon stemming from a deeper basis. CRC phase Figure 4.16 is a sketch of the dominant fluxes of electrons in the energy-level diagram for « e = 1012 m~3. This figure together with Fig. 4.15 shows that, in low density, the populating fluxes to all the levels are the direct radiative recombination plus the cascade contributions from the higher-lying levels. The depopulating process is the radiative decay. We name this situation the capture radiative cascade (CRC) phase. The population is given from the above balance relation by

We approximate the populations of the higher-lying levels q to their SahaBoltzmann values, i.e. nQ(q) ~ Z(q)nzne (eq. (2.7a)). We rewrite eq. (4.39) as

The recombination rate coefficient (3(p) is given in Fig. 3.9 and approximately by eq. (3.19a). In the present high-temperature case, we adopt, for large p, the

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FIG 4.16 Sketch of the dominant populating and depopulating fluxes, or flows of electrons, in the energy-level diagram, corresponding to Figs. 4.13-4.15. «e = 1 x 1012m~3. Virtually all the levels are in the CRC phase. At right the total flux of recombination reaching the ground state is given by the downward arrow with the figure in it giving the collisional-radiative recombination rate coefficient. Other explanation is almost the same as that for Fig. 4.7. (Quoted from Fujimoto, 1980a; with permission from The Physical Society of Japan.) approximation for the exponential integral, eq. (3.21),

where Khas been given by eq. (4.12a). See also eq. (2.7). The second term in the numerator of eq. (4.39a) is expressed by use of eq. (3.8) with the approximation gbb = 1. By using a technique similar to that used in deriving eq. (4.12) we obtain the approximation

For the denominator we adopt the approximation

116 POPULATION DISTRIBUTION AND POPULATION KINETICS With the above approximations, eq. (4.39) yields

the near-Saha-Boltzmann population, or

It is thus concluded that the near-Saha-Boltzmann populations in the lowdensity regions in Table 4.1(b) or Fig. 4.14 are the result of the intricate relationships between the radiative recombination rate coefficient and the transition probabilities. It is further noted that the relative contributions in the numerator in eq. (4.39) from the radiative recombination, eq. (4.41), and the cascade, eq. (4.42), is approximately 2:1. This is in accordance with the accurate calculation shown in Fig. 4.15(a). Figure 1.10(b) gives a summary of the recombining plasma component. The left-side region indicates the CRC phase where the simplified population kinetics is depicted. The population distribution at high temperature is given in parentheses. Transition from the CRC phase to the saturation phase In Figs. 4.13 and 4.14, with an increase in «e, populations make a transition from the near-Saha-Boltzmann populations to almost exact Saha-Boltzmann populations. This transition is more clearly seen in Fig. 4.15 as transitions from the radiative processes to the collisional processes both in the populating and depopulating fluxes. The latter phase is called the saturation phase. Figure 4.17 shows a sketch of the dominant fluxes in the energy-level diagram at «e =1017 m~3. For levels lower than p = 5 the feature of the population kinetics is approximately the same as in Fig. 4.16, indicating that these levels are still in the low-density region, or in the CRC phase. Levels higher than p = 6 have entered into the highdensity region, or the saturation phase, so that the situation is completely different. In this example the boundary level which divides the low- and high-lying levels lies between p = 5 and 6. The transition from the CRC to saturation phases in the depopulating mechanism is the same as that for the ionizing plasma as given by eq. (4.25), or with the neglect of minor terms,

Among the populating fluxes the dominant processes at lower densities (or lower-lying levels) in the CRC phase are, as we have seen above, the direct radiative recombination and the cascade. At higher densities (higher-lying levels) in the saturation phase they are mainly the collisional deexcitation from the

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FIG 4.17 Sketch similar to Fig. 4.16 except that «e = 1 x 1017 m 3. The boundary level pG between the CRC phase and the saturation (LTE) phase, given by eq. (4.25), lies between p = 5 and 6, as indicated by the dash-dotted line. The downward recombination flux through a sufficiently high-lying level, level p = 10, is 1% of the net recombination flux as shown with the downward arrow. (Quoted from Fujimoto, 1980a; with permission from The Physical Society of Japan.) higher-lying levels. See Fig. 4.15(a). Thus the transition takes place when the magnitudes of these radiative and collisional fluxes become equal:

It is to be noted that the higher-lying levels q are already in LTE at «e at which this level p makes the transition from the CRC phase to the saturation phase, so that their population is given by n0(q) = Z(q)nzne. A similar procedure has already been employed in approximating eq. (4.39). We further note the principle of detailed balance, Z(q)F(q,p) = Z(p)C(p,q), eq. (3.31a). Then we rewrite the right-hand side of this equation (4.45) as

Equation (4.45) is then rewritten as

This equation is, within the approximations of eqs. (4.41), (4.42), and (4.12), identical with eq. (4.44), which was for the transition in the depopulating

118 POPULATION DISTRIBUTION AND POPULATION KINETICS mechanism. Thus it has been shown that, for both the populating and depopulating mechanisms, the transition takes place at almost the same electron density. The boundary is thus given by eq. (4.25) or (4.29a). Figure 1.10(b) includes this boundary, as labeled "GRIEM". Saturation (LTE) phase For higher densities the levels are in the saturation phase. At the beginning of this section we have already seen that, at high density, for an excited level, the populating fluxes from the higher-lying levels (deexcitation) and the continuum (three-body recombination) are almost exactly balanced by the respective inverse depopulating fluxes (excitation and ionization, respectively, see Fig. 4.15.). If the still higher-lying levels are in LTE and their populations are given by the SahaBoltzmann equation, eq. (2.la), the above balance relationship should result in the LTE population of this level. It is further noted from the discussion around eqs. (4.5)-(4.11) that the most dominant depopulating flux is the excitation to the adjacent higher-lying level, rather than ionization as is sometimes assumed. The most dominant populating flux is accordingly deexcitation from that adjacent level to this level, rather than three-body recombination, again as is sometimes assumed. Roughly speaking, therefore, the dominant flow pattern of the population is (p+ 1)—>/>—>(/>+ 1). This feature is seen in Fig. 4.15, and in Fig. 4.17 for levels/? > 6. Figure 4.18 shows a sketch of the dominant fluxes in the high-density limit. The flux has been further divided by nzn^, and its magnitude is given with

FIG 4.18 Sketch similar to Fig. 4.16 but for the high-density limit, where the magnitude of a flux has been divided by nznl and given by figures. All the excited levels are in the saturation (LTE) phase. (Quoted from Fujimoto, 1980a; with permission from The Physical Society of Japan.)

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the numbers. In this figure (and also in Fig. 4.15(a)) it is seen in fact that the direct contribution from the three-body recombination is substantially smaller than the deexcitation. Therefore, the dominant relationship of population balance is given by

and the resulting population ratio is the Boltzmann distribution, eq. (2.3) or eq. (3.32)

We may continue the string of this reasoning along p, (/>+!), (p+2),..., to reach very high-lying levels, denoted by, say, r. As has been discussed already, the atomic characteristics of these negative-energy discrete levels continue smoothly across the ionization limit to low-energy continuum states. Since we assume thermodynamic equilibrium for the "free" electrons, it would be natural to assume that the Maxwell distribution continues smoothly across the ionization limit to the negative-energy discrete levels. In other words, these levels are in thermodynamic equilibrium with the continuum, or they are in LTE, and their populations are given by the Saha-Boltzmann values, nQ(r) = Z(r)nzne, eq. (2.7a). See also the paragraph at the close of Section 2.1. If we now trace back the above reasoning down to level p, then we arrive at the conclusion that this level should be in LTE: Therefore, the saturation phase may be called the LTE phase. In Fig. 1.10(b), the upper region higher than the boundaries "GRIEM" and "BYRON" corresponds to this phase, and the relationship, eq. (4.46), is depicted schematically. The above discussion concerning the continuation of the Maxwell distribution of the "free" electrons to the negative-energy discrete levels may appear less convincing. This is partly due to our assumption of "free" electrons for the continuum state electrons, and partly to the ambiguity in treating very high-lying levels. The first assumption is obviously wrong for low-energy electrons, since the interaction of an electron with the ion cannot be neglected in comparison with its kinetic energy. Remember that the Coulomb force is strong and of long range. The second point is also problematic since we implicitly assume that the principal quantum number of levels can increase indefinitely. We then immediately encounter the difficulty that the total statistical weights of the levels, or the state density of the levels for a finite energy width, or the LTE populations, diverge. These points will be addressed and an adequate resolution will be introduced in Section 9.4. Then, the above arguments gain sound footings. At the close of Section 4.1 an important relationship was introduced:

120 POPULATION DISTRIBUTION AND POPULATION KINETICS which is valid in the limit of high density where all the radiative transitions can be neglected. Another approximate relationship was noted for high temperatures and high densities (Fig. 4.10): It is thus concluded that, in the limit of high density, we have It is easily seen that, for Te/z2^> 1.5 x 104 K, the deviation of r0(p) from 1 is very small except for, say, p = 2. See Fig. 4.3. This is another explanation of the LTE population, eq. (4.47). Equation (4.48) has been derived in the limit of high density. It should be noted, however, that this equation is also valid at much lower densities. As has been shown in Figs. 4.13 and 1.10(b), for example, an excited level is in the saturation phase for densities down to Griem's boundary, and r0(p) continues to take its high-density-limit value until that boundary. In Table 4.1(b), level p = 5, for example, has r0(5) = 1 — 5~6 = 1.00 from the high-density limit down to « e ~10 17 m~ 3 within a 10% deviation. This boundary is nothing but Griem's boundary as clearly seen in Fig. 1.10(b). 4.4 Recombining plasma component - low-temperature case In this section we examine the low-temperature case which is more important in practical situations. We assume the temperature to be low, i.e. Te/z2 -c 1.5 x 104 K. As an example we take neutral hydrogen with Te = 1 x 103 K. The reader can draw another horizontal straight line in Fig. 1.2. Table 4.1 (a) shows r0(p). Unlike the high-temperature case of Table 4.1(b), r0(p) is, in many cases, much smaller than 1. Only very high-lying levels have r0(p)~ 1 in high densities. Figure 4.19 shows the populations of several excited levels against ne, where, as before, the population per unit statistical weight nQ(p)/g(p) has been further divided by the ion and electron densities nzne. This quantity, [«o(/")/ g(p)nzne], is proportional to rQ(p); see eqs. (4.38) and (2.7). For an excited level, we may divide the range of ne into four regions according to the dependence of its population on ne: the low-density limit (for «e < 1012 m~ 3 in the case of level p = 5 taken as an example), the gradual increase in the population, or r0(p), with an increase in ne (10 12 < 7) never reach the Saha-Boltzmann values even in the limit of high electron density. We will examine below these features in detail. CRC phase Figure 4.21 shows a sketch of the dominant fluxes of electrons in the energy-level diagram in the low-density limit. The first point to note is the relative magnitudes of the fluxes of radiative recombination into various levels. In comparison with the high-temperature case, Fig. 4.16, the relative magnitudes are quite different: they have a much weaker dependence on the levels. This difference has already been noted at the close of Section 3.2 and in Fig. 3.9. For the recombination rate coefficient, eq. (3.19a), we adopt the approximation to the exponential integral, eq. (3.22), which is the opposite case to eq. (4.40), i.e.

This is valid for low-lying levels. Equation (3.19a) then reduces to

FIG 4.21 Sketch of the dominant fluxes of electrons in the energy-level diagram. Neutral hydrogen with T e = l x 103 K. The low-density limit, where all the levels are in the CRC phase. The recombination flux through a high-lying level, level p= 10 taken as an example, is shown with the solid arrow. (Quoted from Fujimoto, 1980b; with permission from The Physical Society of Japan.)

124 POPULATION DISTRIBUTION AND POPULATION KINETICS Therefore, /3(p) is proportional top~l and Te~°'5, and these dependences are quite different from the high-temperature case in which /3(p) otp~2'5Te~1'5. Figure 3.5(b) illustrates this point; at this low temperature for low-lying levels the p^1dependence is seen, while for levels />;$> 10, the ^-dependence is stronger. This weak /i-dependence is the reason why the higher-lying levels are more heavily populated in the low-temperature case than in the high-temperature case. Figure 4.22, corresponding to Fig. 4.15, shows the breakdown of the populating and depopulating fluxes for level 5. In the low-density limit, in comparison with the

FIG 4.22 Breakdown of the (a) populating and (b) depopulating fluxes concerning level/? = 5 into individual fluxes, corresponding to Figs. 4.19 and 4.20. The explanation is the same as for Fig. 4.15. (Quoted from Fujimoto, 1980b; with permission from The Physical Society of Japan.)

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high-temperature case, the cascade contribution is more important, about 50% of the total populating flux; this is a consequence of the heavier populations in the high-lying levels. The depopulating flux is the radiative decay. The population balance is given by eq. (4.39). It may be shown by a technique similar to the one leading to eq. (4.12) and eq. (4.42) that the ratio of the contributions from the direct radiative recombination, the first term on the r.h.s. of eq. (4.39), and that from the cascade, the second term, is about equal for any level, being independent of p. (In this derivation, however, the population distribution of higher-lying levels is assumed to be n0(q)/g(q)<xq, instead of eq. (4.51) below; this assumption is more accurate numerically as seen in Fig. 4.20(a).) Thus, for the purpose of deriving the population distribution among the levels we may neglect the cascade contribution in eq. (4.39) to reach the approximation

or

leading to the population inversion. In Fig. 4.20(a) this approximation is compared with the result of the numerical calculation. Figure 1.10(b) contains this distribution in the region of the CRC phase. With an increase in ne, very high-lying levels first enter into the saturation phase (Figs. 4.19 and 4.20). For collisional transitions concerning these levels, the energy differences of the important transitions (excitation and deexcitation to the adjacent levels; see the discussions around eqs. (4.5)-(4.11)) are much smaller than the electron temperature (multiplied by Boltzmann's constant). So, the situation is similar to the high-temperature case. Thus, the arguments in the preceding section concerning the transition of the population from the CRC phase to the saturation phase are valid; that is, Griem's boundary, eq. (4.25) or (4.29a), gives the transition, and the levels lying higher than this boundary are in the saturation phase and thus in LTE. This feature is actually seen in Fig. 4.20(a) and (b). Figure 1.10(b) includes the boundary "GRIEM" and the population kinetics in the upper part of the region of ne higher than this boundary. In the present case of low temperature, the Saha-Boltzmann populations, n(p)/g(p)nzne= Z(p)/g(p),for these levels are substantially higher than their populations in the CRC phase. As a result, with the increase in «e, the downward radiative cascading fluxes from these levels to lower-lying levels in the CRC phase increase substantially. This is the reason why, in Fig. 4.22(a), the relative contribution from the cascade increases with «e. The features described above may be understood in a different way: The lowering of Griem's boundary level with ne may be regarded as if it is the lowering of the ionization limit down to this level. The higher-lying levels are engulfed by the "continuum", and the threshold for "radiative recombination" is lowered to Griem's boundary level, resulting in an increase in the "radiative recombination

126 POPULATION DISTRIBUTION AND POPULATION KINETICS rate". Then, it is natural that, with the increase in «e, all the populations of the lowlying levels increase at almost the same degree; this is actually seen in Figs. 4.19 and 4.20 as parallel movements of the curves and the points, respectively, of the low-lying levels in the CRC phase; in Fig. 4.19 all the curves move almost in parallel for low densities, and in Fig. 4.20 the populations of low-lying levels move upward in parallel. The population inversion is conserved. It should be noted that the above argument has nothing to do with the lowering of the ionization potential, which will be discussed in Chapter 9, or with the merging of levels as discussed in Chapter 7. With a further increase in «e, the boundary between the CRC and saturation phases gradually comes down. See Fig. 1.10(b). Figure 4.19 and 4.20 show that, at « e ~ 1017-1018 m~3, level 6, and higher-lying levels have entered into the saturation phase. Figure 4.22(a) shows that, with this increase in «e, the contribution from the collisional deexcitation to level 5 begins to increase; in particular, that from the adjacent higher-lying level 6 is dominant. On the other hand, the dominant depopulating mechanism is still the radiative decay for « e < 3 x 1018 m~3. The presence of this intermediate density region is characteristic of the lowtemperature recombining plasma. This persistent cascading nature of the depopulating process is the origin of the nomenclature of the capture-radiativecascade phase. The population balance in this highest «e region of the CRC phase is given approximately by

Since the upper level (/>+!) has entered into the saturation phase, nQ(p+l) is almost proportional to nzne. Thus, nQ(p) is approximately proportional to nzn^. This explains the steep slope in Fig. 4.19 of nQ(p) in this highest-density region of the CRC phase. Saturation phase - Byron's boundary As seen in Fig. 4.22(b), with a further increase in «e, the depopulating process changes from radiative decay to collisional depopulation. This transition of the depopulating mechanism is given from eq. (4.25) and the boundary «e value is shown in Fig. 4.19. The boundary between the lower-lying levels in the CRC phase and the higher-lying levels in the saturation phase is shown in Fig. 4.20 with the dash-dotted line. We have already noted this boundary at the beginning of this section. This boundary is also shown schematically in Fig. 1.10(b). Note that the actual boundary «e is higher by about two orders of magnitude in this lowtemperature case. See eq. (4.59) later. Figure 4.19 shows that, with the above increase in «e, level 5, for example, enters into its high-density limit, or into the saturation phase at « e ~ 1019 m~3. Figure 4.23 is a sketch of the dominant fluxes of electrons in the energy-level

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FIG 4.23 Sketch similar to Fig. 4.21 except that «e = 1 x 1020m 3. The boundary pG given by eq. (4.25) is shown with the dash-dotted line and/?B given by eq. (4.55) with the dashed line. (Quoted from Fujimoto, 1980b; with permission from The Physical Society of Japan.) diagram for ne = 1 x 1020 m~3, where levels p > 4 are in the saturation phase. This figure and Fig. 4.22 for level 5 indicate that, in this phase, the dominant populating process is the collisional deexcitation from the adjacent higher-lying level 6. The dominant depopulating process is the collisional deexcitation to the adjacent lower-lying level 4. This feature still holds in the higher «e regions as seen in Fig. 4.22 and Fig. 4.24 for the high-density limit. This feature is contrasted to the high-temperature case in the preceding section, Figs. 4.15 and 4.18, where the dominant depopulating flux was excitation to the adjacent higher-lying level. This new feature is common to many lower-lying levels (Fig. 4.24). For these low-lying levels the exponential factor of the excitation rate coefficient, eq. (3.29), is no longer negligible, and eq. (4.36), F(p,p—1)> C(p,p+l), holds instead of eq. (4.9), F(p,p—T) < C(p,p+l), See Fig. 4.2. In other words, for these levels the energy separation between the adjacent levels (proportional to p~3, see eq. (1.5)) becomes quite significant for the electrons whose average energy is of the order of kTe. The population balance is therefore

Thus, the ladder-like deexcitation flow of electrons is established in the energylevel diagram (Fig. 4.24). It is noted that eq. (4.53) is nothing but eq. (4.33), and

128 POPULATION DISTRIBUTION AND POPULATION KINETICS

FIG 4.24 Sketch similar to Fig. 4.21 except that this is for the high-density limit. The magnitude of the effective recombination flux has been divided by nzr%, and shown with the figure in the arrow at right. (Quoted from Fujimoto, 1980b; with permission from The Physical Society of Japan.) therefore, we have a population distribution similar to eq. (4.31), the minus sixth power distribution,

for low-lying levels for which eq. (4.36) holds. Figure 4.20(a) shows that this approximation is good for high density and for low-lying levels, except for the lowest-lying levels for which some of the above approximations break down. Figure 1.10(b) depicts these features: in the density region higher than Griem's boundary and for low-lying levels, shown are eq. (4.53) for the population kinetics and eq. (4.54) for the population distribution. For higher-lying levels for which eq. (4.9) is valid instead of eq. (4.36), the situation is different. As is seen in Fig. 4.24 and expected from the discussion in the preceding sections, the dominant depopulating process from such a level p is the collisional excitation to the adjacent higher-lying level (/H-l). The balance of the population is approximately given by eq. (4.46), n0(p+l)F(p+l,p)ne= n0(p)C(p,p + l)«e, as was the case in the high-temperature case. Thus, eq. (4.47), n0(p) = Z(p)nzne, holds, and the discussion leading to this equation shows that all of the high-lying levels for which eq. (4.9) holds are in LTE. This is actually seen in Fig. 4.20 to be the case.

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For high enough density, the boundary level p between these high-lying levels and the low-lying levels is given from the boundary between eq. (4.9) and eq. (4.36):

Figure 4.2 shows C(p,p+l)/F(p,p—l) against Te for several levels. For the temperature of the present example, 103 K, this boundary lies between/? = 6 and 7 as shown in Figs. 4.20, 4.23, and 4.24. With the approximate rate coefficient, eq. (3.29) along with appropriate approximations like eq. (3.7), fp,p+\—p/4, and eq. (1.2), E(p,p+l)~2z2R/p3,together with the Taylor expansion of the

exponential factor and the approximation [(/>—1)//>]6~ 1 — (6/p), we arrive at an approximate expression for the principal quantum number of this boundary

The subscript B denotes "Byron et al.", since this expression was first derived by these authors. Byron's boundary, as expressed as the boundary temperature, is given in Fig. 4.2. It is seen that this approximation is good for large p, as is expected from the approximations. Figure 1.10(b) includes the boundary bearing the sign "BYRON", as determined by eq. (4.55) or (4.56). The Saha-Boltzmann population, eq. (4.47), for the high-lying levels as shown in Fig. 4.20(a) has negative slopes in this figure. We define for the abscissa

We differentiate the Saha-Boltzmann populations in this figure with respect to a:

We note that exp(o) is nothing but p, so that at Byron's boundary, pB, as given by eq. (4.56), the slope of the Saha-Boltzmann distribution is —6. This means that the Saha-Boltzmann populations in the higher-lying levels continue smoothly at Byron's boundary to the minus sixth power distribution for the low-lying levels, eq. (4.54). We can thus approximate the whole population distribution: for P >/"B, eq. (4.47) is valid, and for p B ~ 5.5, which are given in the figure with the dashdotted line and the dashed line, respectively. Also shown are the calculated populations by the collisional-radiative model for the recombining plasma component. This distribution illustrates well the characteristics of the low-temperature recombining plasma as discussed in the present section. For levels p>pv the population is given by the Saha-Boltzmann value, for pG Po and why these levels enter into the final stationary state at this time (see Fig. 4B.3). We thus conclude that ties is given by tTi(pG). Until now we have assumed that, in the time scale of tr\(p), the ground-state population «(1) is constant. In Appendix 5B we discuss the temporal development of n(l) and nz. Equation (3.35a) indicates that the ionization rate coefficient in the present example is of the order of 10~14 m 3 s^ 1 . The effective depletion rate coefficient of the ground state atoms is of similar magnitude. The ground-state

142 POPULATION DISTRIBUTION AND POPULATION KINETICS population is thus depleted with a time constant of 1CT4 s. Thus, our assumption of constant n(l) in the time scale of Figs. 4B.1 and 4B.2 is justified. Recombining plasma Protons ofnz=l m~ 3 are immersed in an electron gas or a plasma with Te = 0.1 eV and « e = 1018 m~3. The reader may locate this plasma on Fig. 1.2. At t = 0 they begin to be acted upon by the plasma and to recombine. Figures 4B.4 and 4B.5 show transient characteristics of the populations. The dashed lines in Fig. 4B.4 and the closed circles in Fig. 4B.5 show the final stationary-state values, which approximately correspond to the distribution in Fig. 4.20(a). In Fig. 4B.5, the Saha-Boltzmann, or LTE, populations, eq. (2.7a), are shown with the squares. Figure 4B.6 shows ttr(p) (crosses) determined from Fig. 4B.4 in a similar manner to the ionizing plasma case. This figure also includes the collisional and radiative depopulation rates and tr\(p). We now examine the transient populations. Figure 4B.7 shows the rate coefficients for radiative recombination and three-body recombination. It is noted that

FIG 4B.4 Temporal development of the excited-level populations for nz= 1 m 3, Te = 0.1 eV, and «e = 1018 m~3. The dashed lines show the steady-state values, and the dash-dotted lines show the result of eq. (4B.8). The assumption of constant nz is valid until ?~ 1CT2 s, the relaxation time of the ion density. (Quoted from Sawada and Fujimoto, 1994; copyright 1994, with permission from The American Physical Society.)

QUASI-STEADY-STATE

APPROXIMATION

143

FIG 4B.5 The population distribution for several values of t. •: the steady-state values, n: the LTE populations given by eq. (2.1 a). (Quoted from Sawada and Fujimoto, 1994; copyright 1994, with permission from The American Physical Society.) the former rate coefficients are almost the same as those in Fig. 3.5(b). For levels p>4 the latter dominates over the former, which in turn is dominant f o r p < 4 . In the early times of t< 1CP10 s the population distribution in Fig. 4B.5 directly reflects Fig. 4B.7, suggesting that the dominant populating process is the direct recombination, radiative, or three-body process. Remember that in Fig. 4B.5 the population has been divided by the statistical weight. The linear increase in the populations with time is seen in Fig. 4B.4. In this figure the dash-dotted lines show

It is interesting to note that population inversion is established for/? > 4 because of the dependence of a(p) <xp6. It is seen in Fig. 4B.4 that, with an increase in t, the population of levels/? > 4 deviates upward from eq. (4B.8), and they reach the final QSS values starting from very high-lying levels. The last point may be regarded as suggesting that ttr(p) is given by tri(p). However, ttr(p) is found to be appreciably longer than tri(p) as seen in Fig. 4B.6. The above two behaviors are explained from the fact that these high-lying levels, when their populations come close to the stationary-state or LTE values, are strongly coupled to each other by collisional transitions. See Figs. 4.22-4.24. As a result, a level, when its population is lower

144 POPULATION DISTRIBUTION AND POPULATION KINETICS

FIG 4B.6 A: the radiative decay rate; O: the collisional depopulation rate; and D: the total depopulation rate; or the relaxation time tri(p), eq. (4.4), which is referred to the r.h.s. ordinate. For Te = Q.l eV. x : the time at which the population comes close to the steady-state value. (Quoted from Sawada and Fujimoto, 1994; copyright 1994, with permission from The American Physical Society.) than the LTE value, receives additional populating flux from the neighboring higher-lying levels, or it is "pulled" upward toward its LTE value. This level, p, reaches its final value only when deexcitation flux from this level to the adjacent lower-lying level (p — 1) is balanced by the excitation flux from (p — 1) to p. In other words, when the population of a level is close to the LTE value, it is "pulled" downward by the lower-lying levels which are still far from LTE. These levels cannot be considered as independent levels. This is the reason of the appreciable deviation of ttr(p) from tr\(p). Among the lower-lying levels, but still higher than pG, there is Byron's boundary level p#. In the present example, />B lies between p = 6 and 7. See also Figs. 4.20-4.24. Between the levels lower than/> B (and still higher than/>0), collisional coupling becomes only downward, eqs. (4.36) and (4.53), and the population of a level is controlled by that of the adjacent higher-lying level. It might thus be assumed that ttT(p) for these levels, as in the ionizing plasma case, is given by ttT(pv). This turns out not to be the case. This is because lower levels have longer tri(p)'s and their populations lag behind n0(pB), In Fig. 4B.4 the slope of the population of level p = 2 first deviates downward from the linear relationship, eq. (4B.8) with a(p)ne replaced by (3(p), at

QUASI-STEADY-STATE

APPROXIMATION

145

FIG 4B.7 Recombination rate coefficients. O: radiative recombination, A: threebody recombination multiplied by « e =10 18 m~3. (Quoted from Sawada and Fujimoto, 1994; copyright 1994, with permission from The American Physical Society.) t~2x 10~9 s, which corresponds to tr\(2); see Fig. 4B.6. If the cascading contributions from levels/? > 2 were absent, the population would have reached a final value at this time. In fact, the cascading contributions are substantial and even dominant in later times. See Fig. 4.23. Depending on the time dependence of populations of the levels that contribute to the cascading population of level 2, its population increases with time. A similar, but less prominent, feature is seen with level 3. This is the reason why these lower-lying levels cannot reach their stationary-state values until Griem's boundary level po = 6, which has the largest tri(p), reaches its stationary-state value; see Fig. 4B.6. We thus again conclude that ?res is given by tA(pG). In Chapter 5 and Appendix 5A it turns out that the effective recombination rate coefficient for the present example is of the order of 10^16m3s^1, so that nz changes appreciably only in 10~ s. Thus, during the transient of the excited-level populations discussed above, nz is virtually constant. Validity condition of QSS We have seen that both for the ionizing plasma and the recombining plasma the overall response time of the excited-level populations, tres, is given by tr\(pG), the

146 POPULATION DISTRIBUTION AND POPULATION KINETICS

FIG 4B.8 The relaxation time of Griem's boundary level pG as calculated numerically are given by the lines. The response time tTes as determined from the temporal development of the populations, O: ionizing plasma of Te = 10 eV and x : recombining plasma of Te = 0.1 eV. +: recombining plasma of T e = 10 eV. (Quoted from Sawada and Fujimoto, 1994; copyright 1994, with permission from The American Physical Society.) longest relaxation time among the excited levels. Figure 4B.8 compares tTes, as determined from Figs. 4B.1 and 4B.4 and from similar calculations in the broad range of «e, and tr\(pG), as determined numerically. First we assume «e and Te are fixed, so that tTes is constant. We have shown that, under the condition of constant n(l) or nz, if the transient population of level pG, which determines the response time tres, reaches its stationary-state value, all the excited levels enter into the final stationary state. We now examine the case of ionizing plasma in which «(1) changes with time. This change may be due to its depletion by ionization or for some other reasons like spatial transport. The temporal development of the population of level pG is approximately expressed by

QUASI-STEADY-STATE APPROXIMATION

147

where n(i) and N(t) stand for n(pG) and n(l), respectively, at t. In the QSS approximation, the time derivative is set equal to zero, eq. (4.18), and the population is given as

We assume that the temporal change of N(f) is expressed in terms of a time constant T as

where NO is the initial value and T is positive for decreasing N(f) and negative for increasing N(f). Under the condition of t, \T\ ;$> tres, eqs. (4B.9)-(4B.11) lead to

This relation indicates that the parameter (ties/T) gives a measure of deviation of the CR population from its actual value. Thus the validity condition of QSS is that

In our example of Figs. 4B.1-4B.3, the change in «(1) is assumed to be only due to ionization. Even if we include the depletion of «(1) at a later time, the requirement (4B.13) is well satisfied, and QSS is valid. Next we examine the temporal change in nz for a recombining plasma. The dominant populating flux of level pG is deexcitation from the adjacent higher-lying level. We now take into account "the time lag" of this population. Then the population of pG is approximately given as

where N(f) is the ion density, in this case, at t, and Z(p) is the Saha-Boltzmann coefficient, eq. (2.7). In the QSS approximation, the population of po is expressed within this approximation as

If N(f) is again expressed by eq. (4B.11), then eqs. (4B.14) AND and (4B.15 lead to

for t, | T\ > tres. The validity condition of QSS is again given by eq. (4B.13).

148 POPULATION DISTRIBUTION AND POPULATION KINETICS When «e changes with time, we may start with eq. (4B.9) or eq. (4B.14). Instead of N(f) in eq. (4B.9), or N(T- tres) in eq. (4B.14) the factor ne now changes. We may proceed with our discussion in a similar way to the above cases. We then reach the same conclusion as for the previous cases. The change in pG or tres with the change in ne, which we have neglected so far, has little effect in the above reasoning. This is because n(pG) is almost linearly dependent on «e. The case of a temporal change in Te is not straightforward because the collision rates have nonlinear dependence on Te. First we examine an ionizing plasma and start with eq. (4B.9). Instead of the change in N(f), C(l,pG) changes this time. When we note that tres is very weakly dependent on Te (see Fig. 4B.8), we can proceed with our discussion in much the same way as for the previous cases. The validity of QSS would then be \ties[dC(l,pG)/dt]/C(l,pG)\ < 1. The excitation rate coefficient is approximately given by eq. (3.29). Except for very high temperature, the temperature dependence is mainly determined by the exponential factor, and the T,T1/2 factor may be neglected. Then the validity condition is written as

For a recombining plasma, except for the case of pG 10 are substantial at ne= 1018 m~3. We may express the increasing ionization rate coefficient as

154

IONIZATION AND RECOMBINATION OF PLASMA

FIG 5.2 Collisional-radiative ionization and recombination rate coefficients versus the reduced electron density, «e/z7, for hydrogen-like ions. The reduced electron temperature is Te/z2 = 6 x 104 K. All the quantities are scaled against the nuclear charge z. Approximations according to the formulas in the text, eqs. (5.10), (5.13), (5.18), and (5.24), are compared with the numerical calculation. The scaled ionization ratio [z4«z/n(l)j is also shown. The range of «e/z7 is divided into three regions: I, II, and III. (Quoted from Fujimoto, 1985, with permission from The Physical Society of Japan^) The discussions in the preceding chapter, especially that in p. 125-126, suggest that the lowering of Griem's boundary may be regarded, in effect, as lowering of the ionization limit. Those discussions were concerning a recombining plasma. Here, however, for the ionizing plasma, similar arguments can be made: i.e. the ionization limit comes down to Griem's boundary level. The ionization crosssection, eq. (3.35), may be modified, and if we return to the approximation of eqs. (3.29) and (3.35a), we have the approximation

COLLISIONAL-RADIATIVE IONIZATION

155

FIG 5.3 Comparison of approximate formulas and numerical calculation for the collisional-radiative ionization and recombination rate coefficients in the lowdensity limit and in the high-density limit. Sg^ and a~R are related by eq. (5.23). (Quoted from Fujimoto, 1985, with permission from The Physical Society of Japan.) with

Under the condition ofpo^> 1, the oscillator strength, eq. (3.6a), is approximated to/i^ ~ l.6p~3 (Table 3.1(b)) and/o is given in this approximation as/o — 0.8/>Q2. Since the excitation rate coefficients have a slightly different dependence on Te from that of S(l), we adopt an approximate expression starting from eq.

with pG given by eq. (4.25) or eq. (4.29b). In Fig. 5.2 we compare this approximation with the numerical calculation for Te/z2 = 6 x 104 K, where we have adopted eq. (4.29b) for pG_

156

IONIZATION AND RECOMBINATION OF PLASMA

With a further increase in ne, the boundary pG further comes down and reaches p = 2 at n£°, which is the upper boundary of region I. See Fig. 1.10(a). Note here that, since we assume high temperature, Byron's boundary does not appear in this discussion. This density is approximately given from eq. (4.29a) for pG = 2 as

This boundary is shown in Fig. 5.2. Regions II and III (ne > nf) At about this boundary nf, SCR approaches the high-density-limit value as seen in Figs. 5.1 and 5.2. (See also Fig. 4.4.) As we have seen in Section 4.2, in the highdensity regions of present interest, the multistep excitation-ionization ladder is established among all the excited levels starting from the ground state, as seen in Fig. 4.9. Virtually all the excitation fluxes leaving the ground state contribute to the net ionization flux. In these high-density regions, all the excited levels are in the saturation phase (Fig. 1.10(a)), and the interrelationship between the population coefficients r o(p) + ri(p)= 1> ecl- (4-21), holds. In eq. (5.3), if we neglect the radiative decay terms,* which is justified in the highest-density region as seen in the next section, we have

where we have used eq. (3.31a). The superscript oo means the high-density limit. When we remember that, at high temperature, r0(q) is approximated to (1 —p~6) (eq. (4.48)), or very close to 1, we have

This situation is actually seen in Fig. 4.9. Among the fluxes originating from the ground state the dominant flux is the excitation to the first excited level p = 2, and the following approximation is found to reproduce well the numerical calculation.

* In region II, A(q, 1) in eqs. (5.3) and (5.4) is neglected in comparison with, say, C(q, q + l)«e, that is, the radiative decay is neglected in the depopulating processes. However, A(q, 1) is still larger than the corresponding competing collisional rate F(q, l)«e. See Section 5.2 later.

CR RECOMBINATION - HIGH-TEMPERATURE

157

FIG 5.4 Collisional-radiative recombination rate coefficient aCR for neutral hydrogen. The boundary density «^°+ given by eq. (5.20) with p = 2 is shown with open circles, and that by eq. (5.25) with closed circles. (Quoted from Fujimoto, 1980a, with permission from The Physical Society of Japan.) This approximation is shown in Fig. 5.2 with the horizontal line and also in Fig. 5.3 as converted to a£?R (see later). This simple expression is good to within a factor of 2 in the temperature range of Fig. 5.3. In Fig. 5.2, in the density range beyond n£° the approximation, eq. (5.10), is extrapolated to connect to the highdensity limit value, eq. (5.13). 5.2 Collisional-radiative recombination - high-temperature case Figure 5.4 shows «CR, the collisional-radiative recombination rate coefficient for protons and electrons to form neutral hydrogen atoms in the ground state against «e for several temperatures. The reader may identify the parameter range of this figure in the ne-re plane of Fig. 1.2. Note that we are now treating the case of z=l. Figure 5.2 shows an example of «CR for fully stripped ions to form hydrogen-like ions. Figure 5.5 shows, for the high-temperature case of Te= 1.28 x 105 K treated in Section 4.3, the breakdown of the fluxes of the final steps of recombination reaching the ground state, or each term of eq. (5.4), normalized by the total recombination flux. The hatched areas show the contributions from the radiative transitions, i.e. direct radiative recombination, f3(l)nz «e (denoted by j3), and the spontaneous transitions from excited levels (the upper level is indicated by the numeral).

158

IONIZATION AND RECOMBINATION OF PLASMA

FIG 5.5 Breakdown of the collisional-radiative recombination flux into the individual fluxes of the processes reaching the ground state, normalized by the total collisional-radiative recombination flux. See eq. (5.4). The hatched areas show the radiative transitions and the blank areas the collisional transitions. On the right-end ordinate, the corresponding breakdown of the collisional-radiative ionization rate into the individual rates starting from the ground state (see eq. (5.3)) is shown for the high-density limit. Note that this breakdown is different from that shown in Fig. 5.1. For the relationship, see the footnote in p. 152. See also Fig. 4.9. Neutral hydrogen with Te = 1.28 x 105 K. (Quoted from Fujimoto, 1980a, with permission from The Physical Society of Japan.)

Region I (ne < n£°)

Figures 5.2-5.4 give the low-density limit «CR, which is common to hydrogen-like ions and neutral hydrogen. In this limit (Fig. 4.16) the effective recombination rate coefficient is given as

In the present high-temperature case, /3(p) is approximately proportional to p 2'5 for/? > 2 (Section 3.2; see Fig. 3.9 and also eq. (4.41)), and the summation of f3(p) over these excited levels gives a contribution comparable with (3(1) as actually seen in Fig. 5.5. Note that the cascading contribution as a whole is equal to S(/,>2)/3(/>). The argument of the exponential integral, eq. (3.19a), of /3 (1) in the temperature range of Figs. 5.2 and 5.3 is of the order of 1, and the Te dependence of /3(1) is neither T~1-5 (for extremely high temperature, eq. (3.21) or eq. (4.41)) nor T~°-5 (for low temperature, eq. (3.22) or eq. (4.50)) in these intermediate cases.

CR RECOMBINATION - HIGH-TEMPERATURE

159

See Fig. 3.9. We find that, in the range of Te shown in Fig. 5.3, the following approximation is rather good This approximation is shown in Figs. 5.2 and 5.3. Figure 5.5 shows that, with an increase in ne, the relative contribution from the cascade increases slightly. This is due to the slight increase in the "populations", [Ko(/0/g(p) n z n e], of the excited levels that have entered into the saturation (LTE) phase (Figs. 4.13, 4.14 and 1.10(b)), resulting in a slight increase in aCR in Fig. 5.4. This increase, however, is barely noticeable. This increase is more pronounced for lower temperatures in Fig. 5.4 and also in Fig. 5.2. On the assumption that the populations of the levels in the CRC phase are very low in comparison with the Saha-Boltzmann value in the saturation (LTE) phase (which is incorrect for the case of Te= 1.28 x 105 K and higher but is correct for lower temperatures) the effective rate coefficient may be expressed as

The idea of this approximation is that the levels lying higher than Griem's boundary are strongly collisionally coupled, directly or indirectly, with the continuum, and these levels are the origin of the cascading transitions to the lower-lying levels. This picture may be interpreted in another way: on p. 125-126 we regarded the lowering of Griem's boundary level as an effective lowering of the ionization limit. In the present context, the threshold energy of the radiative recombination is lowered to the energy of Griem's boundary, and the transitions nQ(p)A(p, q) for q<Pa

l, the direct recombination terms Y>pf3(p) for p>pG^>\ are extremely small as compared with, say, (3(1). From eq. (5.17), we have It is found that eq. (5.18) is a good approximation in the temperature range considered here rather than the low-temperature case.* An example for comparison is seen in Fig. 5.2. * At extremely high temperature, the populations in the CRC phase are even higher than the LTE values (see Fig. 5.10 later), and with an increase in «e they decrease to their LTE values. As a result, OCR decreases slightly. In such a case eq. (5.18) does not apply, of course.

160

IONIZATION AND RECOMBINATION OF PLASMA

With a further increase in ne, Griem's boundary comes down to reach the first excited level p = 2 at n£° as given by eq. (5.11). For higher densities all the excited levels have the LTE populations or their high-density-limit values. Region II (nf nf+) In this highest-density region all the terms of radiative transitions in the rate equation, eq. (4.2), or in eq. (5.4), are small in comparison with the terms of the corresponding competing collisional transitions, and they can be entirely neglected. As we have seen in Section 4.1, the interrelationship between the population coefficients r0(/>) + r1(/>) = 1, eq. (4.21), is valid. It is straightforward to see that eqs. (5.3) and (5.4) lead to another interrelationship:

162

IONIZATION AND RECOMBINATION OF PLASMA

Then we obtain from eq. (5.13)

This is the approximation which has been compared in Fig. 5.3. Figure 5.2 also shows the comparison of this approximation with the numerical calculation. Figure 5.6 shows (a~R/»e) calculated for neutral hydrogen. Equation (5.24) is also plotted on this figure. The small discrepancies may be attributed partly to the difference of the scaled excitation cross-sections near the threshold energies for ions which are considered here, and those for neutral atoms as seen in Fig. 3.11. Figure 5.5 shows the high-density limit of the breakdown of the final recombination fluxes. On the right-end ordinate, for the ionizing plasma, the breakdown of the initial steps, excitation and ionization, is shown. See eq. (5.3). This breakdown is different from that in Fig. 5.1. It is seen that the principle of detailed

FIG 5.6 The high-density-limit values of the collisional-radiative recombination rate coefficient divided by «e for neutral hydrogen. The approximation (5.24) is plotted for the high-temperature case of Te Si 1.5 x 104 K. For low-temperature the approximation eq. (5.27) is given. (Quoted from Fujimoto, 1980a, with permission from The Physical Society of Japan.)

CR RECOMBINATION

LOW-TEMPERATURE

163

balance is actually realized between the individual elementary processes in ionization and in recombination. 5.3 Collisional-radiative recombination - low-temperature case

In this section we examine the recombination process in the low-temperature case (T e /z 2 l, are in LTE, and they are so strongly coupled with each other and, therefore, with the continuum states that the downward fluxes into these levels are almost balanced by the inverse upward fluxes from these levels (Fig. 4.24 and also Figs. 4.15,4.17, and 4.18). In contrast to this, the downward fluxes among the levels lying below Byron's boundary, p < 6, are not balanced; this is because of the relationship F(p,p — 1) > C(p,p + 1), or eq. (4.36), for these levels. Thus, the multistep ladder-like deexcitation flux is established, which eventually reaches the ground state, resulting in recombination. In other words, the electrons which

CR RECOMBINATION - LOW-TEMPERATURE

165

originate from the continuum states and reach the levels in LTE (p>pv) can return to the continuum states easily, while those which have crossed Byron's boundary downward and left these high-lying levels can no longer return to these levels and thus to the continuum states. They simply flow down, finally reaching the ground state, completing the process of recombination. Now we look for the critical process that determines the magnitude of «CR, the effective rate of recombination, in this high-density limit. As we have seen the three-body recombination to the ground state never plays a dominant role. See in Fig. 5.7 that the contribution from a is virtually absent. As we have seen already, any collisional process involving the levels lying above Byron's boundary, p >p^, is almost balanced by the respective inverse process, so that none of them can determine the rate. A collisional deexcitation flux between levels lying lower than Byron's boundary, p (l)iB is given on the left-end ordinate. For hydrogen-like iron (z = 26) with T e /z 2 = 5.12 x 105 K. The horizontal dotted line is regarded as the upper bound of the approximate Saha-Boltzmann population, b(p)= 1+0.1. (Quoted from Fujimoto and McWhirter, 1990; copyright 1990, with permission from The American Physical Society.)

IONIZATION BALANCE

171

ordinate, and bi^(p) of excited levels are shown with the thick lines. The subscript IB means the ionization balance. The reduced density ne/z7 is given as a parameter. The power of ten is attached to each curve. If the reader tries to identify the parameter range of this figure in Fig. 1.2, not the scaled ones but the real ne and Te, he or she will find them extremely high. Also shown with the thin lines are the recombining plasma component, rQ(p). The horizontal dotted line indicates b(p) = 1 ± 0.1, i.e. the range of approximate Saha-Boltzmann populations. In this very high-temperature case, the recombining plasma component of the population of levels in the CRC phase, or in the case of «e/z7 = 1012m~3 levels/? lying lower than pG ~ 20, is close to the Saha-Boltzmann value, or even slightly larger than that. This is slightly different from Fig. 4.14 for Te/z2= 1.28 x 105 K, where the population was slightly smaller. Note that radiative recombination strictly follows the z scaling. These near Saha-Boltzmann populations of the high-temperature, lowdensity recombining plasma component have been already discussed in Section 4.3. The addition of the ionizing plasma component, ri(p)bi^(l), results in total populations substantially larger than the Saha-Boltzmann values as seen in Fig. 5.10. They are about one order larger, as shown below. We now consider the relative magnitude of the ionizing plasma component and the recombining plasma component in ionization balance plasma. See Fig. 1.9. In the low-density limit the ionization ratio is given by eq. (5.29). The ionizing plasma component is given approximately by eq. (4.23)

The recombining plasma component is given by eq. (4.39), and with the neglect of the cascading contribution (see Fig. 4.15(a)) we have

We now compare the magnitudes of these components for large p, say p of the order of 10. On the assumption of high temperature we finally arrive at

where (9(0.1) means "the order of 0.1". Here we have utilized the approximate relationships, eq. (4.41) or /?(/?) oc/?~2'5, eq. (4.23a) or C(l,/?)oc/?~3, and 5*(1)~ C(l, 2) (see the cross-sections in Fig. 3.1 l(c)). This last relationship comes from the oscillator strengths, /i,2—/i, c (Table 3.1(b)). See also Fig. 4.7. We have already seen an example of this conclusion in Fig. 5.10. Thus, we have come to an important conclusion: In the low-density limit, 1. the recombining plasma component is close to the Saha-Boltzmann equilibrium value, and 2. the ionizing plasma component is larger than that by about one order of magnitude.

172

IONIZATION AND RECOMBINATION OF PLASMA

These properties are independent of Te so long as the temperature is high. In particular, the second property is salient. In the case of neutral hydrogen, for example, at Te = 8 x 103 K, which is even outside the range of high temperature, a numerical calculation shows that the ionization ratio nz/n(l) is 3 x 10~5 and "oGOMGO is 0.1-0.7 for^ > 5. At Te= 1 x 106 K, the ionization ratio is 2 x 107, more than 10 orders of magnitude higher, yet nQ(p)/ni(p) is 0.2-1. A similar argument can be done for different ions. Suppose we fix the reduced temperature re/z2 at a certain value and change z. As we have seen in the previous subsection, for Te/z2 = 6 x 104 K the ionization ratio [nz/n(l)] changes by almost six orders of magnitude for the change of z from 1 to 26. Still, eq. (5.35) is valid for all the ions. Figure 5.10 is also an example of this statement. It is sometimes assumed that the excited level population in a plasma is given by eq. (5.33), with the neglect of the contribution from the recombining plasma component. The above arguments show that, so long as the plasma is in ionization balance, this assumption is approximately correct. With an increase in ne, it is seen in Fig. 5.10 that the recombining plasma component (ro(/>)) tends to the Saha-Boltzmann value; this is the problem which we have examined in Section 4.3. The total population, eq. (5.32a) or eq. (4.20), tends to its Saha-Boltzmann value, too, starting from high-lying levels. This feature is related to the problem of the establishment of LTE, which is important in practical plasma spectroscopy. This problem will be discussed in Appendix 5C. *Excited-level population - low-temperature case Figure 5.11 shows the population distribution of neutral hydrogen for Te = 4 x 103 K. The reader may identify the parameter range of this figure in Fig. 1.2. This temperature is, from the practical viewpoint, unrealistically low for a plasma in ionization balance. In this figure, however, an important feature is manifested which is characteristic of the low-temperature case. We note that, at low density, the total population, which is the sum of the ionizing plasma component and the recombining plasma component, of highlying levels, e.g. for/? ~ 10-20 at «e = 1012m~3 at which/? G ~ 20, is very close to the Saha-Boltzmann value. There appears to be no obvious basis for this to happen. We now examine this point. At these low temperatures, the rate coefficients for excitation and ionization from the ground state are determined by the cross-section values near the threshold for each process. We have established the interrelationship between these crosssections values for excitation to very high-lying levels and for ionization, eqs. (3.36)(3.38). As its consequence, we have an approximate relationship between the rate coefficients,

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FIG 5.11 Similar to Fig. 5.10 but for neutral hydrogen with Te = 4x 103K. Other explanations are similar to those for Fig. 5.10. The thick dotted curve shows the populations in slightly recombining plasma for ne= 1016m~3 with b(l) = 9 x 106 instead of bi#(l) = 1-65 x 107. This condition happens to give 6(3)= 1. (Quoted from Fujimoto and McWhirter, 1990; copyright 1990, with permission from The American Physical Society.) for large p. The ionizing plasma component of the population of level p in the corona phase is given by eq. (5.33). By using eqs. (5.36) and (5.29) we obtain

We now consider £/3(g). For lower-lying levels q we adopt the approximation (4.50) for (3(q), Then we have

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IONIZATION AND RECOMBINATION OF PLASMA

with

where q\im means the upper limit of q below which the approximation (4.50) is valid. This limit may be 5, 10, or 20, depending on Te. See the example of the /^-dependences of (3(p) in Fig. 3.5(b), where q\im is around 10. See also Fig. 3.9. For levels lying above q\im another approximation, eq. (4.41), is valid so that (3(q)<x q~2'5, and the summation converges rapidly. Therefore, S is of the order of 3. We are concerned with a higher-lying level p, which may be/>~ 10-20, where the radiative decay rate is approximated by eq. (4.12):

Here, \np = ft q ! dq is of the order of 3. By noting that the Saha-Boltzmann coefficient is written as

we conclude that ni(pf of eq. (5.37) is about (2/3)Z(p)nzne, or n(p)bIB(l')~2/3. In the same approximations as the above it may be shown that the recombining plasma component is not far from one-third of the Saha-Boltzmann value, i.e. n0(p)° ~(l/3)Z(p)nztie. This is the reason why the total population at low temperature and low density is close to the Saha-Boltzmann value. Excited-level population - high-density case In the limit of high density, or in region III defined in Section 5.2, the following interrelationship holds:

The ionization ratio is therefore given by the Saha-Boltzmann relationship

We remember

In this limit, the factor n(l)/[Z(l)nzne]

is unity. From another interrelationship,

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175

we have for all the excited levels. This equation, together with eq. (5.42), indicates that, in this high-density limit, we have the Saha-Boltzmann equilibrium (LTE) populations with respect to the ion density for all the levels including the ground state. This conclusion is independent of Te. This situation is nothing but the complete LTE as defined in Section 2.1. We now examine the above discussion in more detail. Figure 4.3 gives the highdensity-limit values of rQ(p) and r\(p) for severalp's against temperature. We have seen above that the r.h.s. of eq. (5.32) reduces to rQ(p) + ri(p), leading to the LTE populations for all p's. Therefore, Fig. 4.3, for a particular temperature, directly gives the ratio of the recombining plasma component and the ionizing plasma component of the total population for each level. As we have noted toward the end of Section 4.4 (p. 130-131), Byron's boundary/IB determines which of ro(p) or r\(p) is dominant in the total population; for levels lying above /?B the recombining plasma component is dominant, and for levels below, the ionizing plasma component is dominant. Figure 1.9 is a schematic illustration of this situation. Of course, all the total populations are LTE populations. Figure 1.7 is the spectrum of a hydrogen plasma virtually in the high-density limit (ne ~ 2 x 1023 m~3, higher than K^°+), and the emission line intensities as given from the LTE populations are shown with the dotted lines, where line broadening, which will be discussed in Chapter 7, and the effect of radiation reabsorption, which will be discussed in Chapter 8, are taken into account. As Fig. 4.3 indicates, for Tc=\ eV, except for level p = 2, which has a contribution from the ionizing plasma component of about 10%, the population of other levels virtually consists only of the recombining plasma component. Note in passing that the LTE populations for p 3. We note here an important characteristic of excited level populations in ionization balance plasma. We remember our discussions above and look at Figs. 5.10-5.12. For high temperature, in the low-density limit, where eq. (5.35) is valid the total populations are larger than the Saha-Boltzmann values by an order, or bi^(p) ~ O(10) in terms of eq. (5.32a). For low temperatures, the total populations tend to be close to the Saha-Boltzmann values even in low densities. See Fig. 5.11. An important conclusion is that, in ionization balance plasma excited level populations tend to be larger than the Saha-Boltzmann values in low densities. The only exceptions are that, for extremely low temperature and density, very high-lying levels can have slightly smaller populations than the Saha-Boltzmann values. See Fig. 5.11 with « e =10 12 m~3. With an increase in «e all the total populations come exactly to the Saha-Boltzmann values, or bi^(p) = 1. This problem will be further discussed in Appendix 5C. Ionizing plasma and recombining plasma So far in this chapter, we have assumed that ionization balance is established between the "atoms" n(l) and the "ions" nz. In many practical situations this assumption

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is not correct; rather the ionization ratio is far from the ionization balance. If the plasma is time dependent, or if the plasma is spatially inhomogeneous so that transport of the plasma particles affects the ionization-recombination, then ionization balance is not established. It should be noted that, in the above discussions of the excited level populations in a plasma in ionization balance, both the contributions from the ionizing plasma component and the recombining plasma component could be substantial, or even comparable in magnitude; i.e. at low density and high temperature the former contribution is larger than the latter by about an order, and at high density and rather low temperature both the components are necessary for the levels to have the LTE populations, eq. (5.43). Now, we come to an important conclusion: If the plasma is far from ionization balance, either of the components tends to predominate over the other, and we may neglect the other contribution entirely in the total population. We introduce here the concepts of an ionizing plasma and a recombining plasma: an ionizing plasma is defined by [nz/n(l)] [«z/«(l)]iB- In this case, nz is highly "overpopulated", so that the recombining plasma component will be much larger than the ionizing plasma component, which could be neglected entirely. Consider typical examples of actual plasmas. The first is a hydrogen (z= 1) positive-column plasma in a low-pressure discharge with Te = 5 x 104K, nz = « e = 1017 m~ 3 and «(!)= 1022 m~3. See Fig. 1.2. This is the condition which corresponds approximately to a helium-neon gas laser discharge. This plasma has nz/n(l)= 10~5, which is to be compared with [nz/n(l)]i# = 103. This overpopulation of «(1) by eight orders of magnitude is the result of the transport or diffusion of ions to the discharge cell wall, owing to the spatial inhomogeneity of the plasma. In order to take this effect into account the terms describing the spatial transport of the ions and atoms should be added to the r.h.s. of eq. (5.2). The recombination term aCRnzne is, in this case, smaller than the ion diffusion loss term by eight orders of magnitude. The ionization term balances with this diffusion loss term, in this case. Thus, this is a typical example of an ionizing plasma. Remember that the example cited in Section 4.2 (Fig. 4.12) was a positive-column plasma. In fact, Te of that plasma was 5 x 104 K and «e was 1017-1020 m~3. Similar situations are realized for hydrogen and impurity ions in a magnetically confined plasma, and in a plasma heated by a shock wave during the process of ionization relaxation. Figure 5.13 shows an example of an ionizing plasma, where [nz/n(l)] is smaller than [«2/K(l)]iB by two orders of magnitude. For the lower-lying levels than/? = 7, the ionizing plasma component is predominant, while for the higher-lying levels this is not the case. The reason why the recombining plasma component becomes dominant in this ionizing plasma is that, with an increase in/?, the ionizing plasma component decreases very rapidly, r^p) <xp~6, while the recombining plasma component, r0(p), stays almost constant (~ 1). It should be noted that the total

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IONIZATION AND RECOMBINATION OF PLASMA

FIG 5.13 Similar to Fig. 5.12 except that the plasma is slightly ionizing, nz/n(\) ~ KT2[«Z/«(1)]IB. re = 4 x 104 K, n(l)= 1 x 1019 m~3and ne = nz=lx 1020 m~3. Even in this ionizing plasma, the populations of the high-lying levels are dominated by the recombining plasma components. population for low-lying levels is larger than the Saha-Boltzmann values. This situation of b(p) > 1 for low-lying levels is the common feature for ionizing plasmas. An example of a recombining plasma is an afterglow plasma of neutral hydrogen, Te = 2 x 103 K, and nzn&= 1018 m~3. See Fig. 1.2. This condition gives [«Z/«(!)]IB ~ 1CT27, or K(!)IB ^ 1045 m~3, which is too large to be realistic. Since n(l) is "underpopulated" by more than 20 orders of magnitude, the ionizing plasma component is almost completely predominated over by the recombining plasma component. The examples cited in Section 4.4 (Fig. 4.25) were the flowing afterglow plasmas. There are many kinds of plasmas which fall in this class of recombining plasma. For instance, the plasma produced by illumination of intense laser light on a solid target sometimes shows this characteristic. A plasma surrounding a star which emits strong ultraviolet radiation, and being photoionized by it is another example. Highly ionized impurity ions diffusing from the central plasma to the outer region of a magnetically confined plasma are also an example of the recombining plasma. Divertor plasmas sometimes fall into this class.

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lonization flux, recombination flux, and emission-line intensity In Chapter 4 we examined various features of the populations of excited levels, and in this chapter we have studied the features of ionization and recombination of a plasma in the second sense. Both the ionizing plasma component, eq. (4.22), and the ionization flux, eq. (5.2), are proportional to the ground-state atom density, n(l). Both the recombining plasma component, eq. (4.38), and the recombination flux, eq. (5.2), are proportional to the ion density, n2. See also Fig. 1.9. These observations suggest that from the measurement of an excited level population, or of an emission line intensity, we can infer the magnitude of the flux of ionization or that of recombination, or even both. Figure 5.14 shows an example of the relationships between these quantities. Suppose we have neutral hydrogen atoms of «(1) = 1 m~ in a stationary plasma of « e = 1018 m~ 3 with varying temperatures. Shown are the ionization flux and the corresponding emitted photon numbers of the Balmera a(p = 2-3) transition per unit time and volume. It is seen that these two quantities have a similar dependence on temperature. Figure 5.15 shows the proportionality factor [5>cR«(l)«e/»i(3)^4(3,2)], or the number of ionization events per Balmer a photon emission. Figure 5.14 also shows similar quantities for a recombining plasma: for

FIG 5.14 The ionization flux and the Balmer a line intensity, or the number of photons emitted by excited atoms, originating from the ionizing plasma component; «(1) = 1 m~ 3 and «e = 1018 m~3. The recombination flux and the line intensity from the recombining plasma component; nz=\ m~ 3 and « e = 1018 m~3. For the situation where n(l) = nz=l m~ 3 is assumed, the sum of the line intensities is given with the solid curve. (Quoted from Goto et al., 2002; copyright 2002, with permission from The American Physical Society.)

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IONIZATION AND RECOMBINATION OF PLASMA

FIG 5.15 (a) Proportionality factor for the number of ionization events per Balmer a photon emitted, (b) Similar quantity for the recombining plasma. (Quoted from Goto et al., 2002; copyright 2002, with permission from The American Physical Society.) protons of KZ = 1 m 3 in a plasma of ne = 1018 m 3 the recombination flux and the corresponding emitted photon numbers are shown. Again both quantities have a similar temperature dependence. Figure 5.15 shows the proportionality factor [aCRnzne/n0(3i)A(3,2)], or the number of recombination events per Balmer a photon emission. It is interesting to note that, for recombination, the efficiency of producing photons per recombination event has weak dependences on Te and ne in the parameter range of this figure. We now return to a plasma in ionization balance in the present context. Figure 5.16 shows an example. The total number density of atoms and ions is assumed constant, i.e. The atom density

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FIG 5.16 A plasma in ionization balance, where n(l) + nz= 1 m 3 is assumed. The densities, n(l) and nz, are shown. Also shown are the magnitude of the flux of ionization-recombination and the number of the emitted Balmer a photons with its breakdown into the rates coming from the ionizing plasma component and the recombining plasma component. (Quoted from Goto et al., 2002; copyright 2002, with permission from The American Physical Society.) and the ion density against a change in temperature, as calculated from SCR and aCR (Fig. 5.8), are given by the thin solid lines. For low temperatures the atoms are dominant and for high temperatures the ions are dominant. In this figure, also shown are the flux of ionization, ScR_n(l)ne, or that of recombination, aCRnzne. Since the ionization balance is defined by eq. (5.28) or ScRn(l)ne = acRnzne, both fluxes are equal. It is to be noted that the magnitude of the ionizationrecombination fluxes have the maximum at about the optimum temperature at which «(1) = nz holds. The reason may not be obvious, but it can be understood from physical considerations; For low temperatures, the atom density is high, but the CR ionization rate coefficient is small. See Fig. 5.8. For high temperatures, the CR ionization rate coefficient is large, but the atom density is now low. A similar argument can be made concerning the recombination flux. At the beginning of this section we showed that, for a plasma in ionization balance, the relative magnitude of the ionizing plasma component is larger than the recombining plasma component by about an order of magnitude, eq. (5.35). The Balmer a line intensity as given by eq. (4.1), or the photon number, coming from the ionizing plasma component [Ki(3)J(3,2)] is given in this figure by the upper dashed line and the line intensity from the recombining plasma component [n0(3)A(3,2)] is given by the lower dashed line. The above statement is seen to hold, especially at high temperatures.

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IONIZATION AND RECOMBINATION OF PLASMA

The total intensity, i.e. the intensity we observe from this plasma, is given by the dash-dotted line. It is noted that the total line intensity moves approximately in parallel with the magnitude of the flux of ionization or recombination. Thus, the line intensity takes its maximum near the optimum temperature Teo. If we look at this figure more closely, we recognize that the actual temperature of the maximum emission intensity is slightly higher than the optimum temperature. The ionization-recombination flux takes the maximum at a still higher temperature. This problem is examined in Appendix 5D. We now consider a plasma out of ionization balance, or an ionizing plasma or a recombining plasma. In Fig. 5.14 suppose we fix «(1) = nz=\ m~3, and the temperature changes. The total intensity, eq. (4.20) or Fig. 1.9, is the sum of the intensities from the ionizing plasma component (the dashed curve) and from the recombining plasma component (the dashed curve), to yield the total intensity given by the thin solid line. The important point to note here is that the total intensity takes the minimum at about the optimum temperature. The situation at this particular temperature is rather close to the situation of the ionization balance plasma, Fig. 5.16; as mentioned above, near the optimum temperature, the intensity took the maximum. Note the consistency of the intensities of these minimum and maximum in these figures, respectively. In Fig. 5.14, for temperatures higher than the optimum temperature, the line intensity becomes high, indicating that the ionization flux is large because this plasma is an ionizing plasma. For temperatures lower than the optimum temperature, the line intensity is again high indicating that, this time, the recombination flux is large for this recombining plasma. We have now come to another important conclusion. When a plasma emits line radiation this is an indication that this plasma is undergoing ionization or recombination, or even both of them simultaneously in the case of ionization balance. A plasma out of ionization balance tends to emit intense radiation, indicating that this plasma is undergoing strong ionization or strong recombination. The line intensity is thus a measure of the magnitude of this ionization flux or the recombination flux. In other words, a plasma in ionization balance emits the weakest radiation among the plasmas in various ionization-recombination states. 5.5 Experimental illustration of transition from ionizing plasma to recombining plasma As has been noted in the preceding section, a plasma cannot attain ionization balance if it is spatially inhomogeneous or time dependent. The example of experimental observations of the ionizing plasma shown in Section 4.2 and that of the recombining plasma in Section 4.4 were both stationary. So the origin of the deviation from the ionization balance was that these plasmas are inhomogeneous and the spatial transport of the plasma particles was important in determining the ionization-recombination state of the plasma. In this section we take an example of a time-dependent plasma, a pulsed gas discharge. When a pulsed current (Fig. 5.17(a)) is drawn through a gas, helium in

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FIG 5.17 A discharge plasma is produced from a helium gas of 2torr filled in a discharge tube of 5 mm inner diameter, (a) Discharge current, (b) Emission line intensity of the neutral helium line Hel A587.6 nm (23P — 33D) measured from the side of the tube. See the dotted line in Fig. 1.4. The first peak is called peak A and the second peak B. (Quoted from Hirabayashi et al., 1988; copyright 1988, with permission from The American Physical Society.) this case, a plasma is produced, and the emission line intensity shows two peaks during the course of time as is seen in Fig. 5.17(b). This is a neutral helium line Hel A587.6 nm(23P - 33D). See the energy-level diagram of Fig. 1.4. We now call the former and latter peaks A and B, respectively. Peak A corresponds to the time during the current rise. Peak B appears just after the finish of the discharge current and the intensity decays rather slowly. From the discussions at the close of the previous section, we may suppose that peak A indicates that the "gas" is undergoing ionization during this period and its intensity is proportional to the ionization flux, and that peak B and the subsequent emission indicate the recombination flux. We observe the emission line spectra from these plasmas, as shown in Fig. 5.18 (a) and (b) for peak A and peak B, respectively, and deduce the excited level populations according to eq. (4.1). We plot the population distribution in a graph similar to Figs. 4.5 and 4.20(a). Figure 5.19(a) is the result. Peak A clearly shows the minus sixth power law, and the distribution at peak B looks similar to Fig. 4.20(a) at high densities. Figure 5.19(b) is another plot, the Boltzmann plot like Fig. 4.20(b), of the peak B populations. The minus sixth power population distribution is characteristic of the highdensity ionizing plasma for levels lying higher than Griem's boundary (Figs. 4.5 and 1.10(a)), or it also occurs to levels lying lower than Byron's boundary in the

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FIG 5.18 (a) Observed spectrum at peak A. (b) at peak B. The series lines of 23P — «3D and, in (b), the recombination continuum are seen. See also Chapter 6. (Quoted from Hirabayashi et al,, 1988; copyright 1988, with permission from The American Physical Society.) high-density, low-temperature recombining plasma (Figs. 4.20(a) and 1.10(b)). From the fact that (1) the minus sixth power distribution in Fig. 5.19(a) seems to extend to very high-lying levels, and (2) peak A appears when the plasma is in the build-up process from a gas during the current rise, we may conclude that the plasma at peak A comes from an ionizing plasma and the emission intensity indicates the magnitude of the ionization flux. From the facts that peak B appears in the period of plasma recombination and that the population distribution matches the low-temperature recombining plasma, we may conclude that the plasma at peak B is a recombining plasma and the emission intensity indicates the magnitude of the recombination flux.

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FIG 5.19 Excited-level population distribution for «3D levels for peak A and peak B (O). See Fig. 1.4. The calculated result is shown with +. (a) Plot similar to Figs. 4.5 and 4.20(a). (b) The Boltzmann plot of the population distribution for peak B. Note that the experimental populations extends to the continuum states. See Chapter 6. (Quoted from Hirabayashi et al., 1988; copyright 1988, with permission from The American Physical Society.)

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We now construct a collisional-radiative model for helium. The ground-state atom density is given from the filling pressure of the gas, i.e. n(l :S) = 6 x 1022 m~3. An excited-level population is given by eq. (4.20), and for peak A we take only the second term, the ionizing plasma component, and for B only the first term, the recombining plasma component. By fitting the calculated populations to experiment (see Fig. 5.19), we obtain Te and ne for these plasmas. The peak A plasma has Te of 4-5 x 104 K and ne ~ 1020 m~ 3 and the peak B plasma has Te = 5.1 x 103 K and ne= 1.25 x 1021 m~3. It is noted that the ion density is virtually equal to the electron density under the present condition. The reader may locate these plasmas on Fig. 1.2. We are concerned here with neutral helium, z = 1. We can thus explain the temporal development of the excited-level population in Fig. 5.17(b) as follows. During the current rise, Te is high so that energetic electrons ionize the gas; this means that the situation similar to Figs. 4.9 and 4.12 is actually realized. Therefore, the ionizing plasma component dominates over the recombining plasma component, and the population is high, indicating a large flux of ionization. Figure 5.20(a) shows the calculated total population distribution in this plasma including both components. The above conclusion is clearly seen. After the current takes the maximum the population begins to decrease. Until this time a plasma with a sufficient ionization ratio has been formed. The total number of ionization events during this initial stage, or the resulting ion density of this plasma at its peak, may be inferred from the area under the peak A. During the current decay, there is no need for the plasma electrons to ionize the gas further so that Te decreases to about 2 x 104 K, close to the optimum temperature. An ionization state close to the ionization balance is established, and the intensity shows a minimum, being consistent with Fig. 5.17(b). When the current ceases, there is no mechanism to sustain high Te, and it drops very rapidly. With this decrease in Te the recombining plasma component increases, and the ionizing plasma component becomes small by many orders of magnitude, as is seen in Fig. 5.20(b). Thus, the intensity is high, again suggesting a high recombination flux. From the area under peak B we may infer the total number of recombination events during this afterglow. If we adopt the proportionality factor for the hydrogen Balmer a line, Fig. 5.15, for our case of He (23P — 33D), the number of ionization events during peak A is almost equal to the number of recombination events in peak B. We could even deduce the time history of ne. Unfortunately, however, the intensity is measured only relatively (Fig. 5.17(b)), so that we cannot determine the absolute value of ne. It is noted that the situation realized in this experiment is similar to the situation of Fig. 5.14, where both the ground-state atoms and ions are present simultaneously and temperature changes over a wide range. In the present example, we start with a high temperature during the current rise leading to the strong emission intensity; during the current decay the temperature decreases giving the minimum of the intensity; and the further temperature decay after the finish of the current gives rise to the strong emission again. The reader will understand that this figure is quite universal, i.e. the condition of n(l) = nz= 1 m~ 3 is not a strong constraint in understanding real situations.

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FIG 5.20 Calculated distribution of the total population, eq. (4.20), for «3D levels together with the ionizing plasma component and the recombining plasma component resolved, (a) For peak A; (b) for peak B. (Calculation by M. Goto.)

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Appendix 5A. Establishment of the coUisional-radiative rate coefficients In Appendix 4B we examined the transient characteristics of the excited-level populations approaching the final stationary state. We worked out the conditions under which the QSS approximation is valid, or we can use the population coefficients, r0(p) and r^p). We likewise have to justify eqs. (5.1)-(5.5) for the description of ionization and recombination of our plasma in the second sense. We take the example of the ionizing plasma treated in Appendix 4B. The temporal development of excited-level populations has been shown in Fig. 4B.1. As has been noted, at early times the level populations are simply accumulating. This means that the ground state has a depopulating flux, while it has no returning populating flux. Figure 5A. 1 shows the temporal development of the net depletion rate coefficient from the ground state. It starts with the sum of the excitation rate coefficients to all the levels and the ionization rate coefficient. This corresponds to the first two terms of the r.h.s. of eq. (5.3). With the course of time the excited level populations develop, and they finally reach the stationary state. The returning flux to the ground state gradually increases, so that the net depletion flux decreases.

FIG 5A. 1 Temporal development of the effective rate coefficients for depletion of the ground-state population and that for production of ions, corresponding to Fig. 4B.1. Both the rate coefficients tend to SCR after all the excited levels come to QSS. (Quoted from Sawada and Fujimoto, 1994; copyright 1994, with permission from The American Physical Society.)

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189

This process corresponds to the development of the third negative term in the r.h.s. of eq. (5.3). At about t= l(T7s, or at tres, this transient state is over, and the net depletion rate coefficient tends to the collisional-radiative (CR) ionization rate coefficient, SCR. This figure also includes the net production rate coefficient of ions. It starts with the direct ionization rate coefficient. With the accumulation of the excitedlevel populations it also tends to the CR ionization rate coefficient. We remember that the time constant of depletion of the ground-state atoms is 1CT4 s in this example. So, the CR ionization rate coefficient is established much faster. We may draw two conclusions: 1. The CR ionization rate coefficient lies somewhere between the ionization rate coefficient and the sum of the rate coefficients for excitation and ionization, all from the ground state. The actual value of the rate coefficient depends on ne and Te of the plasma. 2. As in the case of the excited-level populations the response time ties gives the time for the validity of using SCR to describe the effective ionization rate of the plasma.

FIG 5A.2 Temporal development of the effective rate coefficients for depletion of ions and for production of the ground-state population, corresponding to Fig. 4B.4. Both the rate coefficients tend to aCR after all the excited levels come to QSS. (Quoted from Sawada and Fujimoto, 1994; copyright 1994, with permission from The American Physical Society.)

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Figure 5A.2 shows a similar plot for the recombining plasma treated in Appendix 4B. Corresponding population developments have been given in Fig. 4B.4. In the present figure, the effective depletion rate coefficient of the ions and the effective production rate coefficient of the ground-state atoms are shown. If we include recombination into very high-lying levels, at the start, the former quantity diverges, because, as is seen in Fig. 4B.7, the three-body recombination rate coefficient becomes very large for large p. The production rate coefficient of the ground-state atoms is the direct radiative recombination rate coefficient (see Fig. 4B.7). This corresponds to the first two terms in eq. (5.4). Note that, for the ground state, under our present condition the tree-body recombination rate is much smaller than the radiative recombination rate. With the course of time both the net rate coefficients tend to the collisional-radiative (CR) recombination rate coefficient. At about t = tres the CR recombination rate coefficient is established. Thus, in this case again, the response time for excited-level populations gives the validity of using aCR to express the effective recombination rate of the plasma. We again remember that the depletion time constant of the protons is 10~2 s in this example, much longer than the time for establishing the CR recombination rate coefficient. Appendix SB. Scaling law In Appendix 3A we have seen the scaling properties of atomic parameters of hydrogen-like ions against the nuclear charge z. The speed and energy of plasma electrons also scaled by the scaling of the atomic parameters. Therefore, Te scaled according to z2: Te/z2 is the reduced temperature. In Chapter 4, we introduced the rate equation and the collisional-radiative model, and investigated the populations of excited levels. In the present chapter we examined the processes of ionizationrecombination and the ionization balance of a plasma. In this appendix, we investigate the scaling law which enables us to scale various quantities which appear in the CR model for hydrogen-like ions with respect to those for neutral hydrogen. We also point out the limitation of the scaling laws. We want the rate equation, eq. (4.2), to become independent of z. We adopt the scaling laws of the rate coefficients as introduced in Appendix 3A into this equation. Then it is obvious that the electron density should scale as

Then, the time scales as

Unfortunately, this scaling is inconsistent with the scaling for the bound electron, eq. (1.4), i.e. z~2 scaling. It is readily seen that, if we adopt the above scaling laws, eq. (4.2) becomes independent of z except for the last line representing

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191

recombination. If we further require that the recombination follow the scaling, the ion density should scale according to

If we adopt this scaling the conservation of particles is violated in the recombination processes. The scalings of the collisional-radiative coefficients are obvious:

We consider the ionization balance, eq. (5.2):

or eq. (5.28). This leads to the same scaling as eq. (5B.3). This is the reason why we used the quantity [z4nz/n(l)] in the discussion of ionization balance. See eq. (5.30) and Fig. 5.2, for example. We now turn to recombination of the plasma, eq. (5.2) with eq. (5.5):

In this case the conservation of particles, nz, should be valid. Then, the time for recombination scales according to

being inconsistent with eq. (5B.2). These inconsistencies concerning recombination lead to an interesting exception of the scaling law for Te as discussed in Section 5.4 and shown in Fig. 5.9. We again note another limitation of our scaling law; as Fig. 3.11 shows, the cross-section values for excitation and ionization near the threshold do not follow a simple scaling law, so that our results, eqs. (5B.4), (5B.4a), (5B.5), and (5B.5a), become less valid for low temperatures. *Appendix 5C. Conditions for establishing local therniodynaniic equilibrium In Chapter 2, we introduced local therniodynaniic equilibrium, which is abbreviated to LTE. As we have seen in Chapter 4 the populations in high-lying

192

IONIZATION AND RECOMBINATION OF PLASMA

Rydberg levels are strongly coupled with the continuum state electrons, and they tend to be in thermodynamic equilibrium with the continuum electrons more easily. Partial LTE is the state that these levels are actually in thermodynamic equilibrium, or that these levels have Saha-Boltzmann populations. Low-lying levels, however, may depart from this equilibrium relation. There is thus the lower-bound level above which the higher-lying levels are in LTE. In the case that this lower-bound level comes down to the ground state, this situation is called complete LTE. We have considered this problem already in the text. In this appendix, we treat this problem further, with the aim of providing numerical expressions for conditions for establishing partial and complete LTEs. Partial LTE We start with the expression for the excited-level populations, eq. (5.32a),

where b(p) is the population normalized by its Saha-Boltzmann value. See eq. (5.32). We now define LTE for level p to be

Our problem is to find the lower boundary level which satisfies this definition in a particular plasma in the first sense, and to a certain degree, in the second sense. We have calculated rQ(p) and r\(p) for neutral hydrogen in Chapter 4, as given in Table 4.1, and examined the properties of the excited-level populations for a wide range of «e and Te. Similar calculations have been done for hydrogen-like ions, z = 2 or ionized helium, and z = 26 or 25 times ionized iron. We use the reduced electron temperature and the reduced electron density

respectively, in this appendix. We first consider recombining plasma. The definition of a recombining plasma in Section 5.4 is expressed as 6(l)R with TJ as a parameter as determined from the numerical calculations ofr0(p) for z =1,2, and 26. The "dip" near O«2-3 x 105 K is a rather exceptional situation for high temperature; r0(p) — 1 holdsevenfor very low«eas discussed in Section 4.3. Since this situation violates the spirit of LTE, we ignore this dip. In fact, for very low density, even our assumption of the statistical distribution among the different / levels is no longer valid (see Fig. 4A.2), so that this special situation should be taken with some care. The region in the left-bottom corner, i.e. />R cannot become small for low temperature, comes from Byron's boundary. In this figure are plotted the numerically fitted expressions for the conditions of LTE; the thin solid curve is

and the thin dotted line is

FIG 5C.1 The principal quantum number of the lower boundary level />R for establishment of partial LTE in recombining plasma. The thick curves are the result of numerical calculation. z=l; z = 2; z = 26. The eq. (5C.3); eq. (5C.4). (Quoted thin lines are numerical formulas. from Fujimoto and McWhirter, 1990; copyright 1990, with permission from The American Physical Society.)

194

IONIZATION AND RECOMBINATION OF PLASMA

FIG 5C.2 The principal quantum number of the boundary level />R in the 77 — O plane for recombining plasma. The region to the right of a curve is the region of density and temperature in which the higher-lying levels than the boundary level are in LTE. This figure is constructed from the three cases (z=l, 2, and 26) in Fig. 5C.1, and is therefore approximate. (Quoted from Fujimoto and McWhirter, 1990; copyright 1990, with permission from The American Physical Society.) Both conditions should be met simultaneously. It is noted that eq. (5C.3) is essentially the same as eq. (4.29) or even eq. (4.29b), and that eq. (5C.4) is the same as eq. (4.56). The small differences, e.g. instead of the 3We in the denominator of eq. (4.56), eq. (5C.4) adopts 2kT&, are partly due to the difference in the definitions of LTE; it is more stringent in this appendix, eq. (5C.2), than eq. (4.55) which gives Byron's boundary level. Figure 5C.2 is another plot of Fig. 5C.1: the region of temperature and density is shown in which levels p >/>R are in LTE with/> R as a parameter. In order for a level to be in LTE, the radiative transition processes concerning this level should be predominated over by the competing collisional transitions and can be neglected. Equation (5C.3), or eq. (4.25) with the equality sign replaced by an inequality sign, expresses this condition. If this condition is met the problem becomes the relationships among the collisional transition processes. We have seen that among the collisional transitions from a level, the dominant ones are excitation or deexcitation to the adjacent level, eq. (4.6). Figure 5C.3 is a schematic diagram

LOCAL THERMODYNAMIC EQUILIBRIUM

195

FIG 5C.3 Schematic diagram of the four possible cases of the dominant populating and depopulating processes of level p in which we are interested, under high-density conditions where the radiative transitions are neglected. (Quoted from Fujimoto and McWhirter, 1990; copyright 1990, with permission from The American Physical Society.) for the dominant collisional populating and depopulating processes concerning level p which is under consideration. This figure depicts the four possible schemes which can be realized in dense plasmas. For the recombining plasma, in which the population flux originates from the continuum state electrons, scheme (b) or (d) is possible. In order for the level p to be in LTE, scheme (d), eq. (4.36), should be excluded. Equation (5C.4) is the expression for this condition. We now come to the ionization balance plasma in which b(l) in eq. (5C.1) is given values &IB(!)- We investigated this problem in Section 5.4; for high temperature, starting from low density with b(p) ~ G = 345eao67/?7CU18.* Again the minor difference in the two expressions is due partly to the difference in the definitions, LTE on one hand and Griem's boundary level on the other. Figure 5C.5 is another plot of Fig. 5C.4. In this case, the lowering of/>rB for low G has been ignored, and approximate boundaries are determined from z = l , 2, and 26. If we compare Fig. 5C.4 (or Fig. 5C.5) with Fig. 5C.1 (or Fig. 5C.2) for a recombining plasma, we recognize a substantial difference: at high temperature the addition of the ionizing plasma component makes it difficult for levels to enter into LTE, while for low temperature the addition makes it easy for them. The first point is easily understood from Fig. 5.10. The recombining plasma component Professor Griem gives in his new book, which was mentioned in the references of Chapter 1, a similar criterion, />« 8436°' 059 /tf' ns . The numerical factor was about half in his book of 1964.

LOCAL THERMODYNAMIC EQUILIBRIUM

197

FIG 5C.5 The same as Fig. 5C.2, but for an ionization balance plasma. (Quoted from Fujimoto and McWhirter, 1990; copyright 1990, with permission from The American Physical Society.) alone is quite close to LTE, so that the additional population hinders the total population to be in LTE. In order for the population to return to the LTE values the ionizing plasma component should become sufficiently small. The lowtemperature case is not straightforward. We start with

This equation is rewritten as

In the high-density limit we have the factor [acR/ne]/[Z(l)Sci(]= 1. As Fig. 5.4 shows, with a decrease in «e from K^°+, [acR/«e] keeps its high-density limit value until «e reaches eq. (5.25). We now examine SCR and r\(p) for low-lying levels, especially p = 2. Figure 4.11 shows that, in the high-density limit, the levels p p so long as level (p—1) is in Boltzmann equilibrium with lower-lying levels. See Fig. 5C.3(c). At high density, starting from the high-density limit all the populations of excited levels are controlled by the population of the lower boundary level in Boltzmann equilibrium. This level is level 2 down to ne at which level 3 deviates from Boltzmann equilibrium with level 2. In this low-temperature case, in the high-density region the dominant contribution to ionization is the ladder-like excitation-ionization as suggested by the p~6 distributions in Fig. 4.11. These populations, and thus the ionization flux, are controlled by the population of the low-lying level p = 2 or 3. Thus, it is natural that SCR behaves much the same as the population r^p), especially ^(2). We expect that

for p = 2, 3 , . . . , pB, Numerical calculations actually confirm that the relationship (5C.10) is quite accurate. As seen in Fig. 4.3, for this low temperature, the LTE populations in the highdensity limit [n(p)/Z(p)nzne] = r0(p) + r\(p) = 1 consists mainly of the second term, the ionizing plasma component, for low-lying levels. With a decrease in ne, the decrease in ri(p) (Table 4.1(a)) is compensated in eq. (5C.6) almost exactly by the decrease in SCR according to eq. (5C.10), keeping the same value of the second term. The decrease in TQ(P) as seen in Fig. 4.19 does not affect much the population, because the first term is quite small for these low-lying levels. See Fig. 4.3.

LOCAL THERMODYNAMIC EQUILIBRIUM

199

For temperatures around the optimum temperature, the situation is different as we have noted concerning Fig. 5.12: for this low density, the populations are close to LTE, while, the relative contributions from the recombining plasma component and the ionizing plasma component is quite different from that in the high-density limit. We take level p = 2 for an example. In this temperature range, eq. (5C.10) is still valid in the density region close to nf. This is because ionization is controlled by the ladder-like excitation-ionization starting from/? = 2 at around this density. With a decrease in ne from the high-density limit, r0(2), the first term of eq. (5C.6), decreases substantially. At the same time r:(2) decreases, but this decrease is compensated by a decrease in SCR, as in the case of low temperature. In the present temperatures, however, (aCR/»e) increases (see Fig. 5.4), leading to an increase in the second term. This substantial increase compensates, or even overcompensates, the decrease in the first term. The resulting population tends to be higher than the LTE population for lower densities as has already been noted in the text. We now turn to the ionizing plasma, i.e. 6(1) > &IB(!)- It is clear that, if our plasma is purely ionizing, i.e. nz = 0, and lacks the first term of eq. (5C. 1), no LTE populations can exist. In many practical situations, we encounter plasmas which are ionizing, yet the first term is substantial to a certain degree. In these cases, in Figs. 5.10 and 5.11, for example, the populations would lie somewhere above the thick curves, or the second term tends to be larger than "it should be". Thus, the second term tends to make the level populations larger than the LTE values, preventing the levels from entering into LTE. This extreme situation is schematically illustrated by Fig. 5C.3(a): the population flux coming from the lower-lying level is too large, resulting in the ladder-like excitation-ionization. Under certain conditions, e.g. 6(1) is larger than £>IB(!) only by a small amount, some levels may be in scheme (b) and they can be in LTE. An example has been shown in Fig. 5.13: this plasma is slightly ionizing, i.e. 6(1) ~ lOO^iB(l), and levels p > 10 are in LTE. Thus, the condition of LTE includes, besides ne and Te, some constraint about the overall balance of ionization of the atoms or ions under consideration. The conditions for LTE in the present ionizing plasma reduce to r0(p) ~ 1 and

The first condition has been taken care of already, eqs. (5C.3) and (5C.4). In practical situations where LTE becomes an issue in this class of plasma, i.e. high density and temperature, these conditions are almost always met. Thus, the most crucial condition is eq. (5C.11). We now remember that, for high density and temperature, r\(p) is well approximated by eq. (4.35) or ri(p)=p~6. See Fig. 4.10. Within this approximation, eq. (5C.11) is rewritten as

This set of conditions for establishment of LTE involves several parameters. It would be illustrative to show the condition for certain particular cases. Table 5C. 1

200

IONIZATION AND RECOMBINATION OF PLASMA

TABLE 5C. 1 The critical level p\ for establishment of partial LTE in an ionizing plasma. Numbers in brackets denote powers of 10. 4 21 3 (a) 6 = 3.2x 10 K, 77 =10 mz=l z=2 z = 26 6IB(1) = 8.53[1] 6m(l) = 3.78[l] 6m(l) = 2.58[l]

eq. (5C.12)

10 104 106

2.6 6.4 13.2

3.2 6.8 14.8

b(l)

4 23 3 (b) e = 3.2x!0 K, 77 = 10 mz=l z =2 z = 26 6 m (l)=1.33[0] 6 m (l) = l-22[0] Ml) = l-71[0]

eq. (5C.12)

10 103 10s

2.2 4.9 10.1

2.2 4.7 10.0

b(l)

21 3 s (c) 6 = 5.12x 10 K, 77 =:10 mz=l z =2 z = 26 Ml) = l-49[2] 6 m (l)=1.35[2] Ml) = l-31[2]

eq. (5C.12)

10 10s 107

4.2 9.4 21.5

4.7 10.0 21.7

b(l)

5 23 3 (d) e = 5.12x!0 K, 77 =: 10 m" z=l z =2 z = 26 Ml) = 2.09[0] Ml) = 2.12[0] Ml) = 2.24[0]

b(l) 2

3

10 103 10s

2.0 4.4 9.8

2.9 7.2 15.1

2.2 5.3 11.1

4.3 9.7 22.5

2.0 4.5 10.1

3.2 7.3 16.2

2.3 5.3 11.6

4.3 9.8 22.6

2.0 4.5 10.1

eq. (5C.12) 22 4.7 10.0

5 19 3 (e) e = 2.56x!0 K, 77 =10 mz=l z =2 z = 26 Ml)=1.88[4] Ml) = 2.12[4] Ml) = l-82[4]

eq. (5C.12)

10 107 109

7.0 15.6 36a

10.0 21.7 46.8

b(l)

4 23 3 (0 O = 1.6x 10 K, 77 =10 mz=l z=2 z = 26 Ml) = l-60[0] 6 m (l)=l-25[0] 6 m (l) = l-15[0]

eq. (5C.12)

10 103 10s

2.7 6.0 12.2

2.2 4.7 10.0

b(l) s

1

Extrapolated.

7.2 16.5 39.0

2.6 6.5 13.7

7.3 17.0 41.0

2.7 6.5 14.5

LOCAL THERMODYNAMIC EQUILIBRIUM

201

gives several examples of the lower boundary level p\. The boundary determined from the numerical calculations for z= 1, 2, and 26 are compared with eq. (5C.12) Complete LTE In the text we examined the dependences of SCR and «CR on «e, and concluded that nf+ gives the region in which these rate coefficients take their high-densitylimit values. Thus, nf+ gives the lower boundary of complete LTE, perhaps except for a small numerical factor as discussed below. This boundary was given from eqs. (5.19) and (5.20), the comparison between the radiative and collisional transitions terminating on the ground state. This boundary density is larger than Griem's boundary for/> 0 = 2. In fact, in the discussions after eq. (5.21a), we have seen that ri^+ almost coincides numerically with Griem's boundary, eq. (4.29a) extended to/>Q= 1- Figure 5C.6 shows the lower boundary ?ys for complete LTE as determined from numerical CR model calculations for z= 1, 2, and 26. A substantial difference for different z is seen. Our numerical expression includes z

FIG 5C.6 Boundary electron density for establishment of complete LTE for an ionization balance plasma. The thick curves show the results of numerical calculation. : z = 1; : z = 2; : z = 26. The thin curves are : eq. (5C.13) with eq. (5C.14); — .. — . . — : 10,4(2, I)/ ^(2.1); — . — .—: 10/3(l)/a(l). (Quoted from Fujimoto and McWhirter, 1990; copyright 1990, with permission from The American Physical Society.)

202

IONIZATION AND RECOMBINATION OF PLASMA

with

Figure 5C.6 compares the above formula with the result of the numerical calculations. In this figure the boundaries determined by eq. (5.19) and eq. (5.20) with p = 2 are given, where a factor 10 has been multiplied to be consistent with the present definition of LTE, eq. (5C.2). The plasma density of Fig. 1.7 is barely outside of the above criterion for complete LTE. In this case, however, the plasma is optically thick toward Lyman lines which terminate on the ground state (see Fig. 1.6), and the effective A coefficients are substantially reduced in eq. (5.20). Therefore, the above criterion should be relaxed substantially. Boltzmann equilibrium A short remark is given here on the condition for the establishment of the Boltzmann distribution, eq. (2.3), of an excited-level population with respect to the ground-state atom density. The case of an ionization balance plasma has been examined already. We now consider an ionizing plasma. The population kinetic scheme should be case (c) in Fig. 5C.3, with the chain of this relationship continuing down to the ground state. We have seen an example in the close of Section 4.2, Fig. 4.11. In this example, in the limit of high density the Boltzmann distribution with the ground state is established up to p = 5. We remember that this upper boundary for the levels comes from the difference between the scheme (c) and scheme (a) in Fig. 5C.3. Thus, for this distribution to be realized, the density should be high and the temperature should be low, so that r^p)^ 1. When we remember that the condition for complete LTE comes from eq. (5.20), the same condition for density, eq. (5C.13), applies to the present problem. For temperature, the left-bottom corner of Fig. 5C.1 gives an approximate region for this distribution. More quantitatively, the upper bound given by the thin dotted line is reduced by a factor of 2, i.e. p < 5 for O = 103 K and p = 2 for O = 104 K. This factor of 2 again partly comes from the present definition of LTE. If we examine these parameters it is concluded that an ionizing plasma having a high density, eq. (5C. 13) and low temperature as given above is too extreme to be practically possible. Appendix 5D. Optimum temperature, emission maximum, and flux maximum For an ionization balance plasma we introduced the optimum temperature, i.e. the temperature at which the ionization ratio [nz/n(l)] is unity. As Fig. 5.16 shows the emission line intensity is strong at around this temperature (1.35eV). But if we look at this figure more closely, we realize that the temperature at which the intensity takes a maximum is slightly higher in this example of neutral hydrogen. Furthermore, the ionization-recombination flux takes a maximum at still higher

OPTIMUM TEMPERATURE

203

temperature (1.7 eV). At this latter temperature [nz/n(lj\ is 11.5, much larger than 1. We consider this problem now. As in Fig. 5.16, we pose a constraint,

In the ionization balance plasma the ionization ratio is given by eq. (5.28):

First, we consider the ionization-recombination flux, ScRn(l)ne = acRnzne, and look for a temperature at which this flux takes a maximum:

It is straightforward to rewrite eq. (5D.3), under the constraint of eqs. (5D.1) and (5D.2), as

Figure 5.8 shows SCR and «CR as functions of Te. For low density, SCR is well approximated by S(l), eq. (5.6), which may be given by eq. (3.35a). It is straightforward to show that the slope of S(l) against Te is given by R/kTe, with the neglect of the small Te dependence of G. Figure 5.16 shows that the flux maximum takes place at Te= 1.7eV. The slope at this temperature should be 13.6/1.7 = 8. However, the approximation eq. (3.35a) is too crude for the present quantitative discussion. See Fig. 3.11(c): eq. (3.35a) corresponds to the dashed straight line. Actually, the slope of SCR is about 10-12 as seen in Fig. 5.8. The slope of «CR is about —1. See also eq. (5.15) and Fig. 5.3. Thus eq. (5D.4) gives the ionization ratio [nz/n(l)] of 10-12. This is consistent with the value 11.5 as we have seen in Fig. 5.16. As Fig. 5.15 suggests the emission efficiency of ionization, or the number of photons emitted per ionization event, has a substantial temperature dependence at low temperature: the efficiency is higher at lower temperature. The emission efficiency of recombination has a small but opposite temperature dependence. These facts make the peak of the emission line intensity shift slightly to a lower temperature than the peak of the ionization-recombination fluxes. We have now to consider the problem of whether the emission maximum can be lower than the optimum temperature or not. For this purpose, we consider, at the optimum temperature Teo at which nz = «(1), whether the emission line intensity has a positive slope or a negative slope. Under the same constraint, eq. (5D.1), we consider the temperature derivative of [C(l, 3)n(l)]. We assume that, in the vicinity of Teo ~ 1.35 eV, the excitation rate coefficient is proportional to (Te/Teo)^ and the

204

IONIZATION AND RECOMBINATION OF PLASMA

ionization rate coefficient is proportional to (Te/Teo)a, Then, it is rather straightforward to show that if 2/3/ (a + 1) is larger than 1, the slope of the line intensity is positive. From the above argument and from Fig. 3.1 l(c), it is obvious that both the slopes, a and j3, of the rate coefficients have similar magnitudes of around 15. Thus, the line intensity has a positive slope at Teo so that it should take a maximum at a temperature higher than the optimum temperature. Thus, we are able to understand the slight differences among the optimum temperature, the emission maximum, and the flux maximum. We have examined above the example of neutral hydrogen. As we noted in Section 5.4 (Fig. 5.9), the optimum temperature, Teo/z2, becomes high for high-z ions. For z = 26, hydrogen-like iron for example, Fig. 5.9 gives Teo/z2 ~ 2 x 105 K. In Fig. 5.8, at this reduced temperature the slopes of SCR and «CR are approximately 1 and —1, respectively. Equation (5D.4) suggests that, in this case, the optimum temperature and the maximum flux temperature coincide with each other. We further note that j3 for the excitation rate coefficient is around 1 and a is already noted above. Thus, the slope of the emission intensity is about null. Then, the emission maximum also coincides with the optimum temperature. References

The discussions of this chapter is based on: Fujimoto, T. 1979a /. Phy. Soc. Japan 47, 265. Fujimoto, T. 1979b /. Phy. Soc. Japan 47, 273. Fujimoto, T. 1980a /. Phy. Soc. Japan 49, 1561. Fujimoto, T. 1980b /. Phy. Soc. Japan 49, 1569. Fujimoto, T. 1985 /. Phy. Soc. Japan 54, 2905. Fujimoto, T. and McWhirter R.W.P. 1990 Phys. Rev. A 42, 6588. Hirabayashi, A., Nambu, Y., Hasuo, M., and Fujimoto, T. 1988 Phys. Rev. A 37,77. Appendix 5A is based on: Sawada, K. and Fujimoto, T. 1994 Phys. Rev. E 49, 5565. Appendix 5C is based on: Fujimoto, T. and McWhirter R.W.P. 1990 Phys. Rev. A 42, 6588. Appendix 5D is based on: Goto, M., Sawada, K., and Fujimoto, T. 2002 Phys. Plasmas 9, 4316.

6

CONTINUUM RADIATION Figure 1.3, which was a spectrum of neutral helium in a recombining plasma, shows continuum radiation, underlying the series lines, or in the shorterwavelength region, extending from the series lines. Figure 1.7 also shows a prominent continuum. In this chapter we investigate the characteristics of the continuum radiation. 6.1 Recombination continuum We consider the radiative recombination process as schematically depicted in Fig. 6.1, which is essentially the same as Fig. 3.7; an electron having energy e is captured by an ion in its ground state "1" in ionization stage z, and an ion (atom) in levelp is formed in ionization stage (z—1). A photon is emitted which carries the released energy away. Since, in a plasma, the energy of the electrons are distributed over the continuum from zero to high energy, the energies or the frequencies of the emitted photons are distributed from the threshold to high energy (frequency). Thus, this spectrum is continuous, extending from the threshold to higher frequencies. We call this continuum radiation the recombination continuum, In Section 3.2, we introduced the radiative recombination cross-section, crej,(e), and its explicit expression was given by eq. (3.18) for the case in which a fully stripped ion captures an electron to produce a hydrogen-like ion after recombination. The radiated power of the recombination continuum in

FIG 6.1 Radiative recombination of an electron having energy e with the groundstate ion z to form an "atom" in level p. A spontaneous transition q^>p is also shown. Level q is allocated the energy width /zAz/.

206

CONTINUUM RADIATION

FIG 6.2 (a) Schematic illustration of the recombination continuum and the accompanying line emissions. The intensity of the line q—>p, fP^(i/)di/, is replaced by P^(v)/^v. (b) Boltzmann plot of the discrete-level populations and its extension to the continuum electrons. The three points in negative energy correspond to the three emission lines in (a) and the point in positive energy corresponds to the recombination continuum in (a). frequency width dz/ from a plasma having ion density nz(l) and electron density «e is given by

where/(e)de is the energy distribution function of the continuum electrons and the energy width de is equal to hdv. See Figs. 6.1 and 6.2(a). In the literature, instead of the radiative recombination cross-section, the photoionization cross-section is given as standard data. By using Milne's formula, eq. (3.17), we rewrite eq. (6.1) in terms of the photoionization cross-section, where the Maxwell distribution,

CONTINUATION TO SERIES LINES

207

eq. (2.3a), is assumed for/(e)de:

The recombination continuum radiation for recombination of a fully stripped ion z to form a hydrogen-like ion (z— 1) in level p is explicitly given from eq. (6.2) and the photoionization cross-section, eq. (3.13):

with

In deriving the above equation (6.3) we have used the relation hv = e + z2R/p2, See Figs. 6.1 and 6.2(a). Thus, on the assumption that the Gaunt factor is unity, the slope of the recombination continuum spectrum gives Te. 6.2 Continuation to series lines The radiated power of a transition line