Spectral Line Broadening by Plasmas

Spectral Line Broadening
by Plasmas

HANS R. GRIEM

Department of Physics and Astronomy
University of Maryland
College Park, Maryland

ACADEMIC PRESS New York and London 1974

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Library of Congress Cataloging in Publication Data

Griem, Hans R
Spectral line broadening by plasmas.

(Pure and applied physics, v. )

Includes bibliographical references.

1. Plasma spectroscopy. I. Title. II. Series.

QC718.5.S6G74 543'.085 73-5300

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Preface

Many problems have been solved by the very active experimental and
theoretical research on Stark broadening that began in the 1950's, although
without question a number of the more difficult problems have not yet
yielded even to rather high-powered approaches. Hopefully, the nature of
these problems will become clear to the reader of the appropriate sections
of this book in spite of the fact that a "minimum theory" approach was
generally preferred. However, not only the formal aspects of theoretical
work had to be somewhat abbreviated, but also the routine functions of
experiments designed principally to measure certain Stark broadening
parameters had to be neglected to a large extent in order to gain space for
the discussion of critical experiments. Such experiments have contributed
more than their share to our present understanding of the subject and will
probably continue to offer serious challenges to the theoreticians.

References (close to five hundred) are numbered throughout the text to
avoid repetition. If several papers are listed under one number, they are
usually distinguished by a), b), c), etc., in the text, unless an entire group
of papers is referred to. No value-judgement should be attached to the
ordering or the multiplicity of papers under one reference number. As a
matter of fact, a large number in such a group may well mean that the
subject of these papers is particularly interesting and the research on it
unusually active.

Future work in this area will be much facilitated by the establishment
of a data center for spectral line shapes and shifts at the United States
National Bureau of Standards. The two main objectives of the center are:
(1) the collection and cataloging of all literature relevant to the broadening

ix

X PREFACE

and shift of atomic spectral lines; and (2) the preparation and publishing
of bibliographies and critical reviews of various topics in atomic line
broadening. Its first publication is a "Bibliography on Atomic Line Shapes
and Shifts" by J. R. Fuhr, W. L. Wiese, and L. J. Roszman (NBS Spec.
Pubi. 366), U.S. Government Printing Office, Washington, D. C , 1972. A
supplement to this publication was issued in January 1974.

Acknowledgments

This monograph was begun while the author was a Guggenheim fellow
at the Culham Laboratory in England during 1968-1969, and the manu-
script was completed in the course of a one year's stay at the European
Space Research Institute at Frascati, Italy. The hospitality of spec-
troscopists, plasma-, and astrophysicists in both laboratories has made
much of the tedious work possible that would have stretched out over an
even longer period otherwise. However, the major part of the manuscript
was written in the two intervening years 1969-1971 at the University of
Maryland, a preliminary version serving as the basis for a special lecture
course in the spring of 1970. The students of this class and other colleagues
and students of the University of Maryland have given so many critical
comments and suggestions that individual acknowledgments are out of
the question.

Equally valuable have been numerous discussions with scientific col-
leagues all over the world, who offered their criticisms of the 1970 Maryland
lecture notes, answered questions regarding their own work, or contributed
unpublished results to this first attempt at a comprehensive review of the
Stark broadening of atomic and ionic spectral lines. Again I have to plead
for the understanding of these readers if I do not acknowledge their con-
tributions individually.

Almost all of the draft manuscripts and the entire final manuscript were
typed patiently and critically by Mrs. Mary Ann Ferg. For this I thank her
with all the unnamed scientific colleagues who contributed so much to
this work.

List of Symbols

a (Integral) width function Fo Holtsmark (normal) field
Bohr radius strength
do (Differential) width function,
Transition probability, Fourier $ Bates and Damgaard factor
A transform of field strength dis- g Gaunt factor, Statistical
tribution function, Ion broad-
AM ening parameter weight, Two-particle correla-
b Asymmetry tion function
Time derivative of reduced G Green's function
B field, (Integral) shift function
Magnetic field strength, (Dif- Giß) Chandrasekhar function
c ferential) shift function,
C Parameter for dynamical h Profile parameter
corrections fi Planck's constant divided by
C(s) Velocity of light
Stark effect coefficient, Phase 2?r
'3 > CA shift parameter
Ci, Autocorrelation function H Holtsmark function, Hamil-
tonian
d Interaction constants
D Stark coefficient Ha Balmer a line, etc.
e Dipole (subscript), Stark shift 3C Effective impact broadening
En Dipole operator
Ei Electron charge Hamiltonian
E„ Ionization energy of hydrogen i Initial state (subscript)
Atomic energy levels I Intensity
f Ionization energy or series limit / ( « Chandrasekhar function
Scattering amplitude, Velocity
F distribution function, Final h Bessel function
F,F>? l , f t state (subscript), (Collision) Im Imaginary part
frequency, Oscillator strength
Electric field strength 3 Angular momentum quantum
Relaxation theory functions number

j(x) Reduced line shape

J,S Total angular momentum

quantum number
k Wave number, Momentum,

Boltzmann constant, Trans-
formed field variable
K Wave number

KM Modified Bessel functions of

the second kind

xu

LIST OF SYMBOLS Xlll

I Reduced line shape, (Orbital) S Spin quantum number
angular momentum quantum t(s, 0) Schrödinger evolution operator
number, Thickness t Time
T Transition matrix, Kinetic
L,£ Orbital angular momentum temperature
quantum number Tr Trace
u(s, 0) Heisenberg evolution operator
« « ) Line shape U Interaction Hamiltonian
L« Lyman a line, etc. Electric field energy density of
uQ plasma waves
£(ω) Relaxation operator
m Magnetic quantum number, V Velocity
Electron mass Ve Electron velocity
Wir Radiator mass Vi Ion velocity
mp Perturber mass V Volume
m' Reduced mass w Stark (half) half-width
M Ion mass, Magnetic quantum W Field strength distribution
number function
X Reduced wavelength, Carte-
max Maximum (of) sian coordinate, Correction
min Minimum (of) function, Dimensionless vari-
9fïl Total magnetic quantum able
Xa Coordinate (operator)
number y Coordinate
n, n » , ri/ Principal quantum number Y Spherical harmonic, Dimen-
sionless variable
n Integer, Total number of per- z Dimensionless variable, Co-
turbers ordinate
z Nuclear (or core) charge of
Π\ , 7l2 Parabolic quantum numbers radiator
N- Electron density z» Perturber charge
NP Perturber density
P Power, Probability, Projection a Scattering angle, Fine struc-
operator ture constant, Index for 1, 2,
P , Paschen a line, etc. and 3, (Holtsmark) reduced
Pn Configurational partition func- wavelength
tion ß Reduced field Strength
q Quadrupole (subscript) y Damping constant, Euler's
Q Perturber coordinates
constant
Q(r) Configuration space distribu- Gamma function
tion function r Reduced frequency separation,
Kronecker symbol, Dirac's
r Distance, Position δ
ri Position vector of perturbing
delta function, New variable
ion for hyperbolic path functions
rP Mean ion-ion radius (separa- Δ Difference

tion) Ad Dipole operator in line space
R Reactance matrix, Debye Δ(/3) Correction to Holtsmark func-
tion
shielding parameter
Re Real part (of) Δω Frequency separation from un-
rms Root mean square perturbed line

s Time variable
S Spectral density, Spin quan-

tum number, S matrix, Line
strength
£+.- Satellite intensities
θ ( « Kogan function

XIV LIST OF SYMBOLS

e Dielectric constant, Kinetic φ Bates and Damgaard correc-
energy, Dimensionless param- tion function
eter, Eccentricity
Φ Phase shift, Polar angle, Im-
η Coulomb parameter, Coulomb pact broadening operator
phase, Decrement, Imaginary
part of phase shift χ Ionization energy
χ' Screening function
Θ Polar angle, Broadening opera- φ' Two-particle correlation func-
tor in "line" space
tion
λ Wavelength, Azimuth angle ψ( y) Generalized phase shift correla-
Ä de Broglie wavelength (divided
tion function
by27r) ω Frequency
μ Summation index cos Mean Stark splitting
v Summation index coF Field fluctuation frequency
£ Parameter in hyperbolic classi- ω0 Doppier width
ω»/ Unperturbed frequency of
cal path theory
P Charge density, Impact param- spectral line
coo Unperturbed frequency of
eter
PD Debye radius spectral line
pi Statistical (density) operator ωρ (Electron) plasma frequency
ωα, Separation of unperturbed
σ Cross section, Transition in-
tegral energy levels
Ω Frequency of plasma waves,
©(<£) Relative line strength
©(2fTZ) Multiplet strength Angle, Solid angle, Potential
energy
T Duration of collision, Relaxa-
tion time, Dimensionless time
variable

CHAPTER I

Introduction

Effects of electric fields from electrons and ions (both acting as point
charges) on spectral line shapes can be important over a wide range of
plasma parameters, especially of charged particle densities. At one extreme
of the density range are so-called H II regions (N « 103 cm-3) emitting
radio-frequency radiation due to transitions between highly excited states
(principal quantum numbers n « 102) of atomic hydrogen; at the other are
stellar interiors (N « 1026 cm-3) in which some radiative energy transport
may be provided by Stark-broadened resonance lines of highly ionized
atoms (such as 25-times ionized iron, Fe XXVI) in the X-ray region of
the electromagnetic spectrum. In between are laboratory plasmas with
densities of N « 1013 cm-3 (rf discharges) toiV ~ 1019 cm-3 (laser-produced
plasmas, etc.) at the extremes, and with spectral lines from those of neutral
atoms mostly in the visible part of the spectrum to those of multiply ionized
light or medium atoms in the vacuum ultraviolet region.

The temperature range of both astronomical objects and laboratory
plasmas is smaller in comparison—say, from T « 2 · 103 K in some dis-
charge sources and, perhaps, certain H II regions to T « 2 · 107 K in
both stellar interiors and very high temperature laboratory sources. Other
plasma parameters, namely those describing the spectrum of plasma waves
and details of the electron and ion velocity distribution functions, tend to
be of minor influence in regard to line shapes, unless deviations from
thermodynamic equilibrium are large. These additional parameters, besides

1

2 I. INTRODUCTION

electron and ion number densities and kinetic temperatures, are of course
superfluous in case of complete thermodynamic equilibrium.

The very wide range of densities, temperatures, wavelengths, and ionic
charges would seem to discourage hopes for a unified and practical, as
opposed to formal, theoretical treatment of the subject. However, for any
particular spectral line, the relevant range of plasma conditions for Stark
broadening to be important is fortunately much smaller. The density range
of interest may typically be estimated from the inequality

COD < w < | ω,·/ — co»'/' \j

which involves Doppler width COD (almost always much larger than the
natural width), Stark width w (approximately proportional to Np} with p
ranging from f to 1 in actual cases), and unperturbed frequencies ω,/ and
co»'/' of the line in question and a neighboring line. Even for widely spaced
lines, the ratio of maximum and minimum densities is therefore only of
order (c/v)1,2p, c being the velocity of light and v the mean velocity of the
emitting or absorbing systems relative to the observer. For all astronomical
and laboratory conditions mentioned above, this ratio stays below
iVmax/iVmin « 104 according to this consideration and is much smaller than
that in most cases—say, Nma*/Nmin « 102 to 103 for any given line. For all
plasmas but those showing extreme deviations from equilibrium, the tem-
perature range is restricted by

10-2χ <kT < χ,

X being the ionization energy of the radiating atom or ion, whose relative
abundance would be vanishingly small at other temperatures.

Also, the frequency separation Δω relative to the unperturbed line fre-
quency coo tends to vary by no more than a factor about 102 (corresponding
to variations in relative intensities by factors of 104-105) over directly
observable or otherwise important (e.g., for radiative transfer) portions of
the line profile. At relatively high densities, this factor is still smaller be-
cause of the overlap with neighboring lines, and it is thus fair to say that
for a particular line, the three main variables (N, T, Δω) usually vary only
by about a factor of IO2. A description of the line profile in terms of one or
two rather extreme approximations to a more general, but less practical,
theory thus becomes a much more likely proposition.

Whether or not the same approximations will be useful for a large class of
lines depends on the relative magnitudes of, say, all the characteristic
frequencies entering the general problem. Of these frequencies, the angular
frequency corresponding to the wavelength λ of the line, namely co = 2rc/\,
and that corresponding to a quantum of kinetic perturber energy, namely
cok ~ kT/h, tend to be comparable and much larger than most other char-

I. INTRODUCTION 3

acteristic frequencies. If we multiply cok by Z/n2, i.e., divide by a typical
radial matrix element (in atomic units), we obtain the usually small
Weisskopf-Unsöld frequency cow. Other usually small frequencies are the
following: Stark (half) half-width w (and shift d); Doppler width COD ;
difference Δω of actual (co) and unperturbed frequency of the radiation
(coo = co» — co/ ΞΞ co»/) ; characteristic frequency v/p for a single collision
occurring with relative velocity v and at impact parameter p; Keppler or
Coulomb frequency œh = (Zie2/mpz)112 in the case of an electron (mass m)
perturbing a positive ion of charge Z» ; and electron plasma frequency
cop = (4:wNe2/m)112. (Other characteristic frequencies of the plasma can
normally be ignored except, e.g., in cases of unusually high magnetic fields
or strong instabilities.) Then there are the frequencies corresponding to
energy level separations of the unperturbed atom or ion, say, ωα> and ω/f .
These may be "large" or "small" in the above sense, so that there are two
or three types of large frequencies and (at least) six distinct small fre-
quencies in the general problem. Fortunately, it transpires that the detailed
ordering of the large frequencies (including cotl', etc.) is not too important
in the general theory, nor is that of the small frequencies w(d), Δω, and coA .
The essential remaining ordering is therefore only between, say, Δω, v/p,
and cop . Moreover, since the plasma frequency is always below the char-
acteristic frequency for single, binary collisions, there remain only three
orderings of general physical interest:

cop < v/p < I Δω |, cop < | Δω | < v/p, and | Δω | < cop < v/p.

As discussed in the following survey of theoretical work (Chapter II), the
parameter cop tends to be of lesser importance (exceptions are cooperative
effects as covered in Sections II.2a and II.5), and the cases | Δω | <K v/p
and | Δω | » v/p are seen to correspond to the two most extreme, and com-
plementary, approximations to the general theory, namely the impact
approximation (Section II.3) and quasi-static approximation (Section
II.2). For | Δω | « v/p, the theory is naturally much more complicated,
and intermediate approximations (Section II.4) are thus of necessity
usually rather restricted in regard to the permissible range for other char-
acteristic frequencies.

The survey of the theory is followed by a review of Stark broadening
experiments (Chapter III), with special emphasis on experiments capable
of checking the accuracy or validity limits of the various approximations.
Finally, applications (Chapter IV) in laboratory plasma physics and
astronomy are taken up, which often require considerable extrapolation
beyond parameter ranges where the validity of a given version of the
theory is either obvious or verified by laboratory experiments. Appendices
contain tables of calculated Stark profiles or broadening parameters.

CHAPTER II

Theory

Stark broadening, i.e., pressure broadening caused by charged perturbers,
has been the main or exclusive concern of many reviews (see following
paragraph, preface, and Chapter III) of pressure broadening since the
publication in 1958 of the papers by Baranger [1-3] and by Kolb and the
present author [ 4 ] . Much of the subsequent theoretical work has been
based on these papers, which constituted necessary generalizations of
Anderson's [ 5 ] impact theory from 1949 for types of lines actually occurring
in plasmas. Anderson had shown that in the impact approximation
(| Δω | <3C v/p) the line broadening problem reduces to a calculation of the
scattering matrix giving the connection between radiator states before and
after a collision. However, he assumed that the unperturbed radiator states
were either single or completely degenerate, and that the perturbers followed
classical paths which were independent of the internal state of the radiator.
The classical path approximation, which is entirely satisfactory in most
applications, was removed by Baranger [1, 3], who also gave the generaliza-
tion [2, 3 ] of Anderson's theory for radiator states that were neither single
nor completely degenerate. This generalization was derived independently,
using the classical path approximation, in the other basic paper [4], which
also stated the quantum-mechanical equivalent of the results.

Because of the repeated theoretical reviews [6-10] concerned with
modern line broadening theory, its foundations will be dealt with rather
briefly in the following section, and the derivations of the impact approxi-
mation and the quasi-static theory first introduced by Holtsmark [11] in

4

I L I . GENERAL FORMULATION 5

Sections II.3 and II.2. The remaining sections in Chapter II then either
describe details of actual calculations (Sections II.2a-3f) based on these
two extreme versions of the general theory or discuss some intermediate
approximations (namely Anderson and Talman's general phase integral
method [12] in Section II.4a; Smith and Hooper's relaxation theory [13],
which derives from earlier investigations of Fano [14], in Section II.4b;
Kogan's [15] method for correcting the quasi-static approximation for
hydrogen lines in Section II.4c; Baranger's one-electron approximation [6]
in Section II.4d) and characteristic plasma effects (Debye shielding of
binary electron collisions in Section II.5a; shifts of ion lines from non-
vanishing time-averaged interactions in Section II.5b; and plasma reso-
nances mostly near forbidden Stark components in Section II.5c). Finally,
in Section II.6, the question of simultaneous Stark and Zeeman effects and
other influences of magnetic fields on line profiles is discussed following the
work of Nguyen-Hoe, Deutsch, Drawin, Herman, and collaborators.
(Literature references can be found in that section.)

Returning to the impact approximation calculations, it is worth noting
that these are closely related to the problem of electron-atom (or ion)
cross-section calculations. This point has been especially emphasized by
Van Regemorter [9] and Sahal-Brechot and their collaborators (see Section
II.3d). However, we should keep in mind that for line broadening calcula-
tions, one mainly needs total cross sections of excited atoms or ions, while
most other atomic collision problems are concerned with inelastic cross
sections of systems in their ground states.

ILL GENERAL FORMULATION

Stark broadening tends to be important only for lines originating from
allowed electric dipole transitions. When Stark widths are large, collision-
induced transition rates are large as well, i.e., relative populations of
neighboring levels are nearly statistical and intensities of so-called forbidden
lines therefore negligible. (These forbidden lines, corresponding to electric
quadrupole, magnetic dipole, etc., transitions must not be confused with
"forbidden components" arising from the breakdown of parity selection
rules in electric fields, or with "plasma satellites" corresponding to two-
quantum transitions; see Sections II.3cß and II.5c, respectively.) A suitable
point of departure is therefore the formula for the spectral power Ρ(ω) of
spontaneous electric dipole emission from a single quantum-mechanical
system,

P ( « ) = (4<oV/3c*) Σ *(« - ω?,) |</ | xa | i)\* p,·. (1)

if"

6 II. THEORY

(Similar formulas hold of course for induced emission or absorption.)
Here e and c are the usual constants, the (f\xa\i) are the matrix elements
of the components of the position vector (measured from the nucleus) of the
radiating electron (or of the sum of such vectors if several electrons are
involved), and p» gives the probability of finding the system in the initial
state i. The δ function takes care of energy conservation according to the
Bohr frequency condition,

*«?, = E* - EA (2)

where 2?ts and Efs are the initial and final energies of the stationary states
of the complete quantum-mechanical system consisting of radiators and
perturbers.

In most applications, it is not necessary to consider the corresponding
formulas for absorption and induced emission separately, as they are
normally related to each other by Kirchhofes law, at least as far as their
relative line shapes are concerned. Also, radiation from different radiators
adds incoherently in situations where Stark broadening is important (which
tends not to be the case in gas lasers), and the factor ω4 in Eq. (1) is usually
almost constant over the frequency range of a line. It is thus customary to
introduce a normalized line shape by

L(«) = Z ' * ( « - « ? / ) K / l * l i > l f * , (3)
ifct

assuming the ( / | xa \ i) to be constrained by

ΣΊ</Ι*«ΙΟΙ2Ρι· = ι. (4)

i/a

The primed sums extend only over states contributing to a given line, and
the system now consists of one radiator but in principle all perturbers.
(Radiators or absorbers and perturbers are assumed to be different here,
which may not be a valid assumption, e.g., for He II resonance lines
broadened by singly charged helium ions. However, usually it is sufficient
to consider one radiator or absorber only, their number being so much
smaller than that of the perturbers.) While Eq. (3) can be used directly for
"quasi-static" perturbers (Section II.2), the Fourier transform C(s) of the
line shape Ζ/(ω) is more amenable to theoretical analysis in case of "fast"
perturbers, i.e.,

exp(—ίωδ) L(o>) άω (5)

— oo

= Σ'βχρ(-ί«!,β)Κ/ι*.ιοι»Λ.
i/c

I I . 2 . QUASI-STATIC APPROXIMATION 7

Because it satisfies C( —s) = [C(s)]*, only values for s > 0 are required,
and the line shape can be calculated back from

L(CU) = ( 1 / T ) Re ί βχρ(ίωβ) C(s) ds. (6)
•'o

The above expression for C(s) can be written as a trace, which facilitates

the transition to more convenient representations. From Eqs. (2) and (5)

follows

C(s) = Σ ' (i \*a\f) exp[(t/Ä) JE,V] </ | Xa | i) e x p [ - (i/Ä) Α,Ββ]Ρ<

i/a

= Σ Tria?« *t (s, 0) ζα * (s, 0) p], (7)

a

Tr' standing for Σϊ/ a n ( i I / ) a n ( i I 0 being eigenstates of the Hamiltonian

H. Further, t(s9 0), the time development operator for the complete system,
is given by

t(s,0) = er<*·* (8)

where H is the total Hamiltonian (including interactions between radiator

and perturbers but excluding interactions with the radiation field). Finally,
Jf(s, 0) denotes the Hermitian conjugate of t(s9 0), i.e., the transpose of
t( — s, 0), and p is the density (statistical) operator given, in the case of

thermodynamic equilibrium, by

p = e-*'**. (9)

It can usually be represented by diagonal matrices and assumed to be
independent of time also in other representations, making it possible to
write Eq. (7) as

C(s) = Σ Τ φ β ( 0 ) χ α ( 8 ) ρ ( 0 ) ] . (10)

a

This version suggests calling C(s) the autocorrelation function of the light
amplitude [6]. Note also that any additional terms which might be intro-
duced by using the trace notation will normally not be "picked" by the
Fourier integral in Eq. (6) as contributing to the line in question, and that
there are now intermediate states %' ?* i and / ' ^ /.

II.2. QUASI-STATIC APPROXIMATION

If particles are moving sufficiently slowly so that, e.g., the frequencies
characterizing the actual time-dependence of the perturbing electric field
F(t) produced in the vicinity of the radiator during the interaction arjB

8 I I . THEORY

much smaller than the resulting Stark shifts Aw,-/(F) = uif(F) — ω»/(0),

i.e., if

\F(t)/F(t) \«\Au>if(F)\, (11)

errors incurred by neglecting relative motions will be negligible. The line
profile then consists of an average over perturber configurations of profiles
calculated for fixed perturber configurations, and the theoretical problem
splits into three distinct parts : the determination of the relative statistical
weight of a given configuration (distribution functions, Section II.2a) ;
solution of the time-independent Schrödinger equation for the complete
system in a given configuration (by perturbation theory, Section II.2b);
and the evaluation of atomic matrix elements for the unperturbed radiator
(Section II.2c). (The latter are required for other line broadening cal-
culations as well.)

In simple cases, perturbation theory yields

Aœif(F) = CifF», (12)

with n = 1 and 2 for linear and quadratic Stark effects, respectively.

(There are of course some intermediate cases; see Sections II.2b and

II.3cß). Also, relatively large fields tend to be produced by just one nearby
perturber (charge Zpe, distance from the radiator r ) . With F « Zpe/r2 and
Eq. (12), the usual [16, 17] validity criterion for the quasi-static approxi-

mation follows now from Eq. (11) to

| Δω | » (y2/Zpe)-/^-1)C7/1/(2n-1), (13)

using (F/F) ~ v/r, where v is the (relative) velocity of the perturber. For
linear Stark effect, a suitably averaged value of the Stark coefficient (see
also Section II.2b) is approximately

Gif « (3h/2emZ) (m2 - nf2), (14)

where rti and Uf are the principal quantum numbers of initial and final
states, and Z = 1 for hydrogen, Z = 2 for ionized helium, etc. With
v2 « 3kT/m' (where m! is the reduced perturber-radiator mass), the above
validity criterion thus becomes

| Δω | » 2ZmkT/hZpm,(ni2 - nf2). (15)

When the right-hand side of Eq. (15) approaches the half-width w of the
line, which, in the quasi-static approximation and using F « 8ZpeiVp/3 (see
the following section), is estimated by

w « (12ZpÄ/Zm) (m2 - nf2)NÎ/3, (16)

(actual widths being smaller for low series members), then | F/F \ « v/r is

I I . 2 . QUASI-STATIC APPROXIMATION 9

an overestimate for the characteristic frequency of the perturbing field
[15]. Two or more perturbers are now involved simultaneously, leading to
a considerable smoothing of the field fluctuations (Section II.4c), and the
quasi-static approximation remains valid also in the line core, i.e., well
within the half-intensity points separated by about 2w. By requiring w to be
about equal to or larger than the right-hand side of Eq. (15), one thus
obtains as the criterion for the validity of the quasi-static approximation
throughout the profile of a line subject to linear Stark effect

Np > (kT/6m')W(Zm/Zph)*(ni2 - n,2)"3. (17)

[In the terminology of Section II.4c, this corresponds to (coF/ws)2 < 10.
Note also that the criterion for ions may be relaxed somewhat in the
presence of significant electron impact broadening, and that (COF/ÎOS)2 < 10
rather than < 1 arises, because mean field strengths are substantially
larger than F0 in Eq. (36).]

To see how useful the quasi-static approximation for the entire profile is
in practice, this lower limit for the perturber density must be compared
with two other critical densities. First, the Stark width w should also be
larger than the thermal Doppler width

_ (vr_\1/2 = /fcry/2 zv /j_ _ j _ \

\mTc2/ ■* \mTc2/ 2hao\nf2 n?) '

*· 5 £ » (S)'" 0? - hT <-* - " "·amT being the mass of the radiating species, i.e.,Λ_ ! (Ι9)

Second, this Stark width should certainly stay below the separation be-
tween two adjacent lines of a given series (Inglis-Teller [18] limit) for the
lines to remain reasonably distinct. This condition leads to the upper limit

The useful range for the quasi-static approximation for hydrogen lines
broadened by electrons or protons (using their reduced mass and a tem-
perature of 104 K) according to these considerations is indicated in Fig. 1.
The following conclusions are evident :

A. Lyman series:

1. The approximation is valid and useful for protons as perturbers over
density ranges corresponding to factors ~ 2 · 105 (La) to ^ 2 0 (L20).

10 I I . THEORY

2. For electrons as perturbers (having a reduced mass smaller than that
of protons by a factor ~ 9 2 0 ) , the quasi-static approximation is valid, at
best, only very near the Inglis-Teller limit or on the line wings (at this
temperature).

3. There is a small density rainge for L« (factor ~ 3 ) where Stark
broadening may already dominate but the quasi-static approximation is
not yet valid even for protons.

B. Balmer series:

1. The approximation is valid and useful for protons over density
ranges corresponding to factors between ~ 1 0 5 (H^) to ^ 1 5 0 (H2o).

2. Again, the approximation may be valid for electrons over entire
profiles only near the Inglis-Teller limit.

3. The quasi-static approximation is not yet valid, but Stark profiles are
already required a factor ^ 1 5 for Ha and ~ 2 for Ήβ below its limit of
validity for protons. (Note that this interesting region is actually at some-
what higher densities, because the Stark widths of these lines are really
smaller by factors of about 6 and 2 than assumed here.)

C. Paschen Series:

1. The approximation is valid and useful for protons over density ranges
spanning factors between ~ 5 · 104 (Ργ) and ~ 5 0 0 (P20).

2. The approximation may apply also to electrons over entire profiles
when the density is near the Inglis-Teller limit.

3. There is a range below the validity limit for protons covering factors
~ 3 0 (Pa), ~ 6 (P/3), and ~ 3 (P7) where Stark profiles are required. (Its
actual location should be somewhat higher than indicated in Fig. 1.)

For the subsequent series, (1) the validity range for protons as perturbers
shrinks further for low series members, the maximum range corresponds to
still higher members, and the range increases for higher members; (2) the
approximation is practically never useful for electrons over entire profiles;
(3) there is an increasing requirement for other than quasi-static Stark
profiles for lower series members broadened by protons, both in regard to
the number of lines and the density range.

The above conclusions are easily generalized to other temperatures by
multiplying the densities by appropriate factors according to Eqs. (17),
(19), or (20). Assuming T « 104Z2 K, the same curves may be used for
He II and Li III broadened by protons, if the densities are multiplied
by Z9/2 « 23 or «150. (The reduced perturber mass is approximately
proportional to Z for Z < 3, and mx varies approximately as Z2 from H to

I I . 2 . QUASI-STATIC APPROXIMATION 11

FIG 1. Validity and useful electron density ranges of the quasi-static approximation
over entire profiles of hydrogen Lyman, Balmer, and Paschen series lines at a tempera-
ture of 104 K for perturbing protons (hatched area) and electrons (cross hatched area).
The various curves correspond to the Inglis-Teller limit beyond which adjacent members
of a series are no longer distinct, the lower limit for a quasi-static treatment of broadening
by electrons, the analogous limit for protons, and the density where Doppler and Stark
widths are about equal. Temperature scaling and adaptation to other ions as perturbers
or hydrogenic ions as radiators are discussed in the text. Note also that the density scale
for the Lyman lines is shifted upward by one decade relative to those for the Balmer and
Paschen lines.

Li.) For higher Z and other perturbers, the appropriate scaling must be
inferred directly from Eqs. (17), (19), and (20).

The question naturally arises whether (1) correlations between charged
particles, (2) quantum-mechanical effects, or (3) deviations from linear
Stark effect might invalidate one or the other of the present conclusions
regarding the validity and utility of the quasi-static approximation. The
answers can be given in rather general terms:

1. Correlations (discussed in more detail in the following section), e.g.,
between hydrogenic ions as emitters and perturbing ions, will cause rela-
tively minor modifications as long as the mean Coulomb interaction energy

12 I I . THEORY

\_~(Z — l)Zpe2iVp/3] is below the thermal energy kT> i.e., when ion densi-

ties fulfill

Nv < LkT/(Z - \)ZpéJ. (21)

(For perturber-perturber correlations to be not overly important, a
similar relation with Z — 1 replaced by Zp must be fulfilled.) Comparison
with the Inglis-Teller limit, i.e., Eq. (20), shows that such correlation
effects are generally small or at least not dominant, La lines of high Z ions
being the most important exceptions, especially when broadened by mul-
tiply charged ions. [See Preist [19] for situations where collective effects
are important for high series members, but note that the discussion pre-

ceding Eq. (20) of this work is misleading.]

2. Quantum mechanically, the implicit assumption of exactly localized
point charges must of course be relaxed to requiring only a relatively small
range Ar for the perturber separation, so as to keep the uncertainty in
perturber momentum (velocity) relatively small. Because of Heisenberg^

uncertainty relation (mp Av Ar « h), this about amounts to saying that
the relative angular momenta of the perturbers with respect to the emitter,

Zp « mprv/h, must be large (Ar/r « h/mpr Av ~ h/mprv). Again using
r « JV~1/3 and v « (kT/mp)112, this requirement can also be expressed in
terms of an upper limit for the density, namely,

NP < (mpkT/V)w, (22)

which turns out to be less restrictive than the Inglis-Teller limit even for
electrons as perturbers. (Some La lines are possible exceptions.) The
predominance of large angular momenta further implies that classical theory
is normally sufficient to evaluate correlation effects. (See, however, Section
II.5b, and note that Av ~ v is permissible above.)

3. Deviations from linear Stark effect can occur in three ways: (a) The
lines may be so broad as to invalidate the implicit assumption of negligible
interactions between levels of different principal quantum numbers, which
actually do give rise to a quadratic correction term (see Section II.2b).
However, this effect is important only when adjacent series members begin
to merge and is thus of no interest in the present context, (b) In addition
to monopole-dipole interactions, which are responsible for linear Stark
effect in the case of hydrogen and hydrogenic ions, one may have to con-
sider higher multipole interactions. The latter will be relatively small only
when the Bohr radii of excited states, rn = n2a0/Z, remain smaller than
typical perturber separations, r ~ N~1/3, i.e., when

iVp < Z*/nW. (23)

Also this limit is well above the Inglis-Teller limit in practically all cases.

I I . 2 . QUASI-STATIC APPROXIMATION 13

(c) Deviations from linear Stark effect are naturally also encountered
when the widths of nonhydrogenic lines are not much larger than separa-
tions of unperturbed levels of a given principal quantum number. However,
there is no need to rediscuss the question of the validity of the quasi-static
approximation for the entire profiles of this large class of lines. First of all,
their Stark widths cannot significantly exceed those of the corresponding
hydrogen or hydrogenic ion lines; as for the latter, the quasi-static approxi-
mation will therefore almost never apply over the entire profile for electrons
as perturbers. Second, in regard to ions as perturbers, it is sufficient to
argue that the latter will either be as effective as in the hydrogenic case, so
that the discussion summarized in Fig. 1 can be used, or (more commonly)
much less effective, in which case electron broadening would dominate
except on the far wings of the lines, where corrections can easily be made
(see Section II.3f). A fourth cause for deviations from linear Stark effect,
fine structure, can usually be dismissed (see Section IV.6).

After this broad discussion of the validity and utility of the quasi-static
approximation (which is supplemented by Section II.4c in regard to inter-
mediate situations where it becomes valid only beyond some point on the
line wings), its formal derivation can be stated rather quickly. The quanti-
ties oify , xa , and pi in Eq. (3) now depend only on the position coordinates
of the perturbers, called Q collectively; cofy is equal to the corresponding
quantity œif for the radiator (perturber energies being assumed to be
independent of the internal state of the radiator), and xa refers only to the
atomic electrons (all other charges being at rest and therefore not radi-
ating) . Assuming that the part of the interaction between radiators and
perturbers which does depend on the internal state of the radiator is small
compared to thermal energies (i.e., Δω « kT/h), the statistical factor can
be replaced by pi&P(Q) dQ, where p*a now describes only the probability of
finding the radiator ("atom") in a given initial state, while P(Q) dQ is the
probability of finding the perturber coordinates in Q, Q + dQ. (This is not
an essential approximation and can be removed if necessary.) In this
manner, Eq. (3) becomes

£.(«) = Σ ' I ά\ω - «<,(<?)] \(f\Xa(Q) | i)\2 Pi*P(Q) dQ

ifa JQ

= Σ " l</ I Xa(Q') I i)\2 Pf P(Q') I dQ/dœ |<wr , (24)

ifa

with Qr corresponding to zeros in the argument of the δ function and the

double prime serving as a reminder that the sum is only over atomic states

contributing to the line in question and being capable of zero argument in

the δ function.

14 II. THEORY

As discussed above, most applications of the quasi-static approximation
to entire line profiles are those involving linear Stark effect, i.e.,

ω = o>if = cot/(0) + CifF(Q). (25)

Neglecting all asymmetries (see Section III.9), the quasi-static profiles
then follow from Eqs. (24) and (25) and introducing a distribution func-
tion for the field strength W(F) dF = P(Q) dQ as

£.(«) = Σ " l</ I x* I Ol2 P<a Wm | df | Λ (26)
i/a

with Ff determined by Eq. (25) and with the understanding that ( / | and
| i) here are Stark effect ("parabolic") wave functions, and that the p»a

are now simply statistical weight factors.

In almost all other cases, the so-called nearest neighbor approximation is

sufficient, because one is then mainly interested in relatively large fre-

quency separations from the unperturbed line. If we therefore write co,·/ =
o)if(r), etc., and replace P(Q) dQ by 4πΛΓρΓ2 dr (the probability of finding a

single perturber in r, r + dr), the quasi-static approximation, Eq. (24),

is replaced by the asymptotic relation

L.(«) ~ 4ΤΓΛΓΡ Σ " I </ I *"(r0 I <>lf Pi* exp ί 1 i \dr'
i/a L (27)

Here the Boltzmann factor involving the ratio of Coulomb interaction

energy between charged radiators and perturbers was added for complete-

ness, and some possible sources of asymmetry were again allowed for

[except for those already omitted in Eq. (24)]. Quantum-mechanical

considerations [20] suggest that the classical Coulomb interaction energy
Zp(Z — l)e2/r' should not be used here for electrons (Zp = —1) when it
exceeds ~Z2e2/a0 in magnitude. The reason is of course that the perturbing
electron cannot be localized any better than within Ar' ~ a0 in the vicinity
of a proton (see also Weise [21]). The nearest neighbor approximation as

such is justified if the field strength distribution function deviates only

slightly from its asymptotic value, i.e., if the correction terms in Eq. (39)

of the next section are small.

II.2a. Field Strength Distribution Functions

As indicated in the preceding section, Coulomb interaction energies both
between various perturbers and between perturbers and charged radiators
tend to be smaller than thermal energies under conditions for which lines
involving upper states of different principal quantum numbers remain

I I . 2 . QUASI-STATIC APPROXIMATION 15

isolated. It is therefore appropriate to begin the discussion of distribution
functions with that derived by Holtsmark [11], who assumed all particles to
be statistically independent. The probability of finding the field vector F
in F, F + dF is in general

TTo(F) = / · · - / « ( F - t F y i p i n ^ , . . . , ^ * ! * , , . . . , * · , ,

(28)

where the (Coulomb) field produced by the jth particle is

Fy = -Zpery/r/, (29)

and where P is the probability of having particle 1 in r i , ri + drL, etc.
(The r / s are measured from the center of the radiator, where we want to

know the field.) For statistically independent particles, the probability P
is simply V~n in terms of the actual volume V of the system, i.e.,

Wo(F) = V-* f · · · / * ( F - t ^ f t i i f t î , . . . , * . . (30)
Easier to calculate is the Fourier transform

A ( k ) = f exp(ik - F)Wo(F) dF

= V~n ' · · I exp ( ik · Σ ^j ) ^3ri ^3r2 y · · · > d*rn (31)

(The quantity k is not a wave vector, in contrast to normal usage.) For

isotropie plasmas (strongly magnetized plasmas are a possible exception;

see Section II.6), A ( k ) obviously depends only on | k | = k, as Wo(F)

depends only on | F | = F} so that the desired distribution function can be
calculated from

W(F) = 4irFWo(F) = ( 2 / T ) F Γ kA(k) sin(kF) dk. (32)

[Note that in the integration over angles in k space, / exp ( — zk · F) A ( k) dzk
reduces to

2π j A(k) k2 dk j exp(-ikF cos Θ) sin Θde

= (4T/F) f kA(k) sm(kF) dk,

16 I I . THEORY

and that a factor {2τ)~ζ arises from the inverse transform.] As might be
expected for independent particles, the multiple integral in Eq. (31) is
actually the nth power of a three-dimensional integral,

A(k) = r ^ l e x p ^ k . F O d V x T

-[v {**£*]

4-F'W;0-s-¥)£l· <«>

where the second line results by integration over polar angles in ri space

[analogous to the integration discussed following Eq. (32)], the third line
by separating 4π / r2 dr = V, and the last line by introducing Y = | Zpek/r2 |
as a dimensionless variable. The Y integral, when integrated by parts,

reduces to

/;0-=¥)£-*/"£?"-*<*>-■ (34)

Substitution into Eq. (33) and taking the limit n —> <» with n/V = Np =
constant then leads to

A(k) = exp[-(/cF0)3/2] s exp(-x3/2), (35)

where F0 is the Holtsmark (normal) field strength

Fo = 2TT(TV)2/3 I Zpe | Νψ « 2.603 | Ζρβ | Νψ, (36)

which, by the way, deviates only very slightly from the field strength

produced by a perturber located at one mean ion-ion radius rp defined by
(4ir/3)fpWp = 1. (The numerical factor in this case would be (4π/3)2/3 «
2.599. Also, for singly charged perturbers, rp is often called rQ in the litera-
ture.)

The above result suggests introducing a reduced field strength β through

β = F/Fo (37)

whose distribution, according to Eqs. (32), (35), and (37) in the Holts-

I I . 2 . QUASI-STATIC APPROXIMATION 17

mark limit with x = kF0, and normalized to / * H{ß) dß = 1, is

H(ß) = FoW(F) = (2/π)β Γ exp(-xW) sin(ßx)xdx. (38)
•'ο

This integral has been evaluated numerically by various authors (see below
for tabulations). Particularly useful is an asymptotic expansion for large ß
obtained by expanding the exponential in Eq. (38) and integrating term
by term,

H(ß) ~ (1.496//35'2)[1 + (5.107//33'2) + (14.43//33) + · · · ] · (39)

[The general expression is

It indicates clearly the deviations from the nearest neighbor approxi-
mation, which almost exactly corresponds to the leading term, for, say,
β < 10. This approximation is thus seen to be as accurate as 10% or better
when the field strengths are at least as large as those produced by one
perturber within about one quarter of the mean ion-ion radius mentioned
above.

With this in mind, the nearest neighbor approximation may now be used
for a first estimate of correlation effects, using, in analogy to Eq. (27),

WC(F) ~ 47riVpr2 | dr/dF | e x p [ - Z p ( Z - l)e2/(WcT)] (40)

and, instead of Eq. (29), a Debye-screened field, namely the derivative of
the Debye-Hückel potential

F = (ZPe/r2)[l + (r/pD)] exp(-r/pD), (41)

pu being a suitably chosen screening radius (see below). Expansion of the
exponential in Eq. (41), etc., with*0 = F/F0 « rp2/^2, then yields

„ / / Λ 1.50 Γ , 5.1 5 / r p \ 2 , 1 ΓΖΡ(1 - Z)e2 1

(42)

when the first correction term from Eq. (39) is included. [Note that the
leading term does not exactly agree with the asymptotic expansion of
H(ß) in Eq. (39).] In the case of fields produced by ions, one may argue
that these are essentially shielded by electrons only, i.e., choose

PD = (kT/ÏTNe2)"2, (43)

18 I I . THEORY

N being the electron density. A somewhat better approximation, at least in
the case of equal ion and electron temperatures, is probably obtained b y
allowing approximately for (perturbing) ion-ion correlations as well by
using, instead of PD ,

PD' = lkT/4re2(N + ZpWp)]1'2. (44)

For the case Zp = 1 (i.e., Np = N and rp = n ) , one thus estimates from
Eqs. (42-44)

Both perturber-perturber correlations and perturber-radiator correlations

(given by the third term and the exponential, respectively) therefore

depend on the parameter ri/pD , whose cube is essentially the usual plasma
parameter (inverse of the number of electrons in a Debye cube). The ex-

plicit expression for η/ρη is

R = r i / P D = WWifWH/kTyi* « 2.2(fNW/kT)1**, (46)

i.e., its dependence on temperature and, especially, density is rather weak,
numerical values ranging only from ~ 0 . 1 to ~ 1 in typical line broadening
applications.

For a more systematic and general treatment of correlations between
charged particles, it is necessary to modify both the probability
P ( r i , r 2 , . . . , rn) of a given perturbing ion configuration and the expres-
sion for the field produced by a single ion. This was first done by Baranger
and Mozer [22], who used Eq. (41) with the electron Debye radius from
Eq. (43) for the field but also modified the configuration probabilities by
writing

P ( n , r2, . . . , rn)

= F-[l + jΣ<kί ( r , , r*)] exp Lί ^ 1 ^ Σj λrojexp (\ - ^PDΥ/ΙJ , (47)

hi

where g = g(\ ry — r* |) = g(rjk) is the two-particle correlation function,
and where the exponential accounts for perturber-radiator correlations.
(Earlier attempts [23,24] to obtain complete distribution functions
corrected for correlations omitted the g correction and instead used a
smaller screening radius in the expression for the field.) Debye-Hückel
theory (linearized) was used for the two-particle correlation function in

I I . 2 . QUASI-STATIC APPROXIMATION 19

Eq. (47), namely

g(rjfrk) = e x p [ - (e2/rjkkT) exp(-ry*/pD')] - 1

« - (e*/rjkkT) exp( -ry*/pD'), (48)

and it was argued that the screening radius appropriate for the ion-ion
correlation terms (all ions still assumed to be singly charged) was as given
by Eq. (44), i.e.,

AD' = PD/V^. (49)

Neglecting first the exponential factor in Eq. (47), the original authors [22]
were able to show that Eq. (35) for the Fourier transform of the field-
strength distribution was now to be replaced by

A(k,R) = exp{-xU*Zx'(Rx1!2) - ^(Äs1'2)]}. (50)

The function χ' originates from the replacement of Coulomb fields by
screened fields, i.e., χ'(0) = 1, while the function ψ incorporates the two-
particle correlations between perturbing ions interacting with each other
via the effective (Debye-Hückel) potential. It therefore tends to ψ' = 0
for large values of its argument, i.e., complete screening of the effective
interactions.

Both χ' and ψ' must be evaluated by numerical integration (and subse-
quently also the inverse Fourier transform). Mozer and Baranger's results,
as discussed by Pfennig and Trefftz [25], suffered some from numerical
errors in ψ' which the latter authors corrected (for Z = 1), thus obtaining
a more accurate set of distribution functions (for neutral radiators only)
for 0 < β < 10, 0 < R < 0.8 in the Baranger-Mozer approximation, but
without linearizing as in Eq. (48).

Mozer and Baranger [22b] had calculated distribution functions for
singly charged radiators, Z = 2, as well, by essentially allowing the ex-
ponential factor in Eq. (47) to act only on the first term ("1") in brackets,
consistent with their neglect of explicit three-body correlations. However,
these results also were affected by numerical inaccuracies in ψ'. Hooper
[26] then reexamined the Baranger-Mozer approximation theoretically,
starting with

P(r!,r2,. . .,rn)

= F/xp(-è) = F e xp[- iê gae xp (-ë)] ' (51)
n

where Pn is the configurational partition function (see below for the ensuing

20 II. THEORY

normalization problem). His interaction potentials are screened by elec-
trons only, and the summation begins with i = 0 if the radiator (at r0) is
singly charged or with i = 1 if the radiator is neutral. Instead of proceeding
immediately with the cluster expansion, whose first two terms would be
analogous to the first bracket in Eq. (47), the interaction energy is re-
written

Ω = Ωο + Σ («Vr<w) exp[-û:(ro»/pD)] (52)

with a real and positive, but otherwise arbitrary for the time being. (Note
that Ωο depends in a complicated way on all r,*.) The object is to optimize
the accuracy of the calculations for a given computational effort. (In con-
trast, the Baranger-Mozer method is based on a more intuitive choice of
the screening parameter pD' = on/V2 in the effective interaction.)

The necessary integrations are considerably more complicated, but these
difficulties could be overcome by the use of collective coordinates. Fortu-
nately, it turned out that the results were insensitive to the actual value of a
over a wide range, which suggests that errors from both termination of the
cluster expansion and the integration method are very small. Hooper [26]
verified this by Monte Carlo calculations based essentially on Eqs. (28),
(41), (43), and (51) for R = 0.8 and charged radiators; subsequently
Bailey also verified the result (see Fig. 2). The accuracy is further sub-

0.6

0.5

0.4
H(/3)

0.3

0.2

0.1
~0 1.0 2.0 3.0 4.0

ß
FIG. 2. Comparison of Monte Carlo calculations (by the late R. E. Bailey of the Uni-
versity of Florida) of the ionfieldstrength distribution at a charged point with the theory
of Hooper [26b]. The value of the Debye shielding parameter is R = ΤΊ/PD = 0.8.

I I . 2 . QUASI-STATIC APPROXIMATION 21

stantiated by the excellent agreement [within 1% except for R = 0.6,
ß = 0.1 (2%) and for R = 0.8, ß = 0.1 (5%), ß = 0.2 (4%), £ = 0.3
(3%), 0 = 0.4 (2%)] with the Pfennig-Trefftz results [25] for neutral
perturbers. Errors exceeding ~ 1 % in Hooper's tables [26b] for the ion
field ("low frequency component") distribution functions for Zp = 1,
Z = 1 or 2, thus appear rather unlikely. Beyond their range (ß < 10),
tables (for neutrals and ions) and asymptotic relations (for ions) analogous
(and almost equal) to Eq. (45) are available [27]. O'Brien and Hooper
[28] generalized Hooper's calculations to mixtures containing also doubly
charged ions, using F0 for electrons, but no detailed calculations seem to
exist for other multiply charged ions. For them, Eq. (42) with PD replaced
by PD and rp given by

rp = [(4π/3)ΛΓρ]-1/3 (53)

should therefore be particularly useful if one wants to estimate deviations
from the Holtsmark theory in such cases. [Note also that contributions to
W(F) from various types of perturbers are additive in the asymptotic
regime and that definitions of F0 may differ.]

This section has been devoted almost entirely to fields produced by ions,
as these tend to be the only perturbers whose effects can be treated in the
quasi-static approximation (see the preceding section). In most of the
papers on corrections to the Holtsmark theory mentioned above, dis-
tributions of the fields produced by electrons ("high frequency com-
ponents") were presented as well, Hooper's results [26a] again probably
being the most accurate ones. (For electrons, the fields are of course not
screened at all, while the effective interaction energies in the Baranger-
Mozer approximation now just allow for screening by electrons, ions in both
cases simply serving as a neutralizing, uniform background.)

When electrons also act quasi statically, we will require the distribution
of the total field from electrons and ions. In the Holtsmark limit, this is
obviously obtained by using N + Ni = 2iV in Eqs. (36), etc., if we once
more assume all ions to be singly charged. Generally, the distributions
Wo(F) for the two species must be convoluted. (Note that the fields add
vectorially.) In view of Eq. (32), this constitutes no difficulty in principle.
Alternatively, one may retain the total (Coulomb) field produced by
particles of both species and then perform a cluster expansion. This has
been attempted by Weise [21a], using Kirkwood's superposition principle
to estimate three-body correlations, which were included to reach about
the same degree of accuracy as that of the Baranger-Mozer approximation
(where the use of electron-screened ion-produced fields and of a uniform,

22 I I . THEORY

neutralizing ion-background in the electron calculations speeds up the con-
vergence of the cluster expansion and allows for the different time scales
implied in these cases). However, as his distribution functions deviate even
less from the corresponding Holtsmark functions than the electron-field
distributions obtained from the Baranger-Mozer method [22a] or from
Hooper's theory [26a], correlation effects are evidently underestimated in
Weise's results. (The probable reason [29] for this is the omission of one of
the terms arising from the correlation function.) In his first paper, Weise
further discussed (besides the distribution of the micropotential) the theo-
retical problem encountered in normalization, as the classical configura-
tional partition function diverges in the case of point charges of different
signs. He argues that a quantum-mechanical calculation (see also Engel-
mann [20]) would deviate from the classical calculations for r < α0,
causing convergence without changing the distribution functions in any
substantial manner as long as N<&a^3 « 1025cm~3. In a second paper
Weise [21b] considers relativistic effects and magnetic microfields.

Explicitly or implicitly, all of the above calculations were based on the
assumption that particles much beyond a Debye radius from the radiator
would not contribute significantly to the field, i.e., long-range cooperative
fields corresponding to plasma waves were ignored. Actually, such fields do
exist even in thermodynamic equilibrium, when, e.g., the electric-field
energy density in longitudinal plasma waves follows from the classical
equipartition theorem (half of the wave energy being in the field, the other
in kinetic energy) to

FW2/8TT = \1CT(2TC)-* / K*dK~ fcT/24^pD3, (54)

where (1/2π2) Κ2 dK is the usual density of modes with wave vectors of
magnitude between K and K + dK. The upper limit of the integral,
-Kmax « 1/PD , is chosen both because waves with K > l / p D are heavily
Landau damped, and because correspondingly short wavelength density
fluctuations are already accounted for in the usual (Holtsmark, etc.) cal-
culations. The latter yield mean ("particle") field strengths of

Fp « 8.8eN** (55)

(ignoring correlations and electron contributions), and the ratio of rms
wave fields and mean particle fields, with Eqs. (43), (46), (54), and (55),
for a thermal plasma containing no multiply charged ions, is

(FV)W/TP « W/WV.2) (eW^/feT)1'4 = 0.17/Ü1'2. (56)

I I . 2 . QUASI-STATIC APPROXIMATION 23

(A Holtsmark rms field cannot be defined, unless one modifies the small r
contribution.) For the experimentally relevant values of Ä « 0.1 to 1,
wavefieldsthus turn out to be considerably smaller than the particle fields
under equilibrium conditions. Moreover, the waves in question are then
mostly electron waves, ion waves being still more heavily damped in the
case of equal ion and electron temperatures, and the frequencies associated
with the wave fields are accordingly slightly above the Langmuir plasma
frequency,
ωρ = (4*#e2/™)1/2, (57)

i.e., often too high as to permit the application of the quasi-static approxi-
mation. It is therefore natural to combine the wave fields with the "high-
frequency" fields from electrons, which has been done more rigorously by
Ecker and Fischer [30a] and also by Dalenoort [30b]. A more important
effect of wave fields, however, is the generation of "sidebands" to certain
lines, rather than a broadening in the usual sense (see Section II.5c).

The distribution function for a wave field component has a Gaussian
character, as proposed by Hunger and Larenz [31a, b, c] on the basis of
an investigation of the limiting process n —> <*>,n/V = N in the derivation
of the particle field distribution and of computer experiments. The total
field distribution would then follow from a convolution of the particle and
wave field distributions. (The width of the latter is much overestimated by
these authors, because of a number of errors [29, 3 Id]. Also their singling
out [31c] a particular experiment [32] as being about the only one in a
range where particle field contributions dominate is not justified, R being
about 0.3 in this case, which is right in the middle of the usual range for
this parameter.)

The above is not to say that wavefieldscan never be important in regard
to profiles of allowed lines, since an enhancement of the wave energies by a
factor of about 102 above the thermal equilibrium level will already make
wave and particle fields comparable to each other. For plasmas having
substantially higher electron than ion temperatures, ion waves may then
be strongly excited as well. Their frequencies will usually be small enough
for the quasi-static approximation to be valid. (Allowance must be made,
however, for Doppler shifts of the plasma frequency in interstreaming
plasmas.) Then hydrogen or ionized helium lines may develop a near-
Gaussian profile instead of the usual Holtsmark shape (~Δω~5/2) on the
far wings. This effect requires still larger enhancements of wave energies,
say by factors of about 103 or greater, because rms wave fields should now
be much larger than mean particle fields in order to dominate over a sub-
stantial portion of the (convoluted) distribution function for the total

24 I I . THEORY

field. Calculations [33] of field strength distribution functions for turbulent
plasmas necessarily involve rather special assumptions, but should help
considerably in correlating spectroscopic measurements (Sections III. 10
and IV. 3) with plasma theory provided the dynamical response of the
radiating atoms to the field is either quasi-static or can be handled by
perturbation theory. (The final result of Kim and Wilhelm [33d], based on
perturbation theory, is invalid because the latter is insufficient in the
quasi-static regime, | Δω | > ωρ.)

II.2b. Perturbation Theory

To the extent that all perturbers (including ions with bound electrons)
act as point charges having relative velocities with respect to the radiator
well below the velocity of light, the interaction Hamiltonian describing the
effects of a given perturber configuration on the radiator is generally well
represented by the multipole expansion of the electrostatic interaction for
I r» I ^ I ra | beginning with the monopole-dipole term, namely

« -era - F. (58)

Here ra is the position vector operator for the radiator electron (s) and r»
the position vector of the ith perturber (of charge Zve), all measured from
the nucleus of the radiator. The approximation in terms of the total field
strength F, which is produced by the perturbers at the radiator's nucleus, is
good when higher multipole terms describing quadrupole-monopole inter-
actions, etc., are negligible. Also, the omission in this expansion of a leading
monopole-monopole term (for charged radiators) is usually justified,
because it does not depend on the internal state of the radiator as long as
the perturbers are well outside the range of the radiator wave functions.
Effects of this term on initial and final radiator states therefore tend to
cancel. (See, however, Section II.5b.)

Effects connected with internal structure of perturbing ions possessing
one or more bound electrons are generally small, because they are of the
order of neutral gas pressure broadening. However, portions of the line
profile well beyond the half-width, e.g., of ionized helium absorption lines
broadened by singly charged helium ions are possible exceptions to this rule.

Since the perturber coordinates are assumed fixed, the remainder of this
section is mainly a recapitulation of standard (time-independent) quantum-
mechanical perturbation theory, wrhich is presented mainly to clarify
terminology and to discuss some subtle points. In the case of one-electron

II.2. QUASI-STATIC APPROXIMATION 25

systems (hydrogen, ionized helium, etc.), there is mostly linear Stark effect
(neglecting quadrupole interactions, etc., for the time being), i.e., the
various Stark components of a level are shifted by amounts [34, 35]

Aœ(F) = friri! - n2) (eaQ/hZ)F - n«(aQz/hZ*)F* + · · · (59)

(in terms of principal quantum number n and parabolic quantum numbers
U\, n2), where the second term gives the (approximate) correction from
quadratic Stark effect. If the latter is important, quadrupole interactions

^-~m^^-^mm^ <«>must not be neglected either. They give rise (by themselves) to shifts
in terms of n, orbital angular momentum quantum numbers Z, and the

projection quantum number m (determined with respect to the direction

of r p ) . Clearly, simple addition of Eqs. (59) and (60) is not quite enough,
I no longer being a good quantum number on account of the Stark effect.

Moreover, when quadratic Stark effects and quadrupole interactions are

to be included in the level shifts, it also becomes necessary to use corrected

wave functions for the matrix elements of xa in Eq. (26) rather than the
usual parabolic wave functions. All these considerations are perhaps best

demonstrated by a specific example.

Consider hydrogen Le , theoretically about the simplest case, perturbed
by a single proton at r = (0, 0, z). In the w, l, m representation, the non-

vanishing perturbation energy matrix elements (within the n = 2 sub-

space) are then

(21 ± 1 | U I 21 ± 1) = 6(eW/r3), (61a)

(210 | V | 210) = - 1 2 ( e W / r 3 ) = t/q , (61b)

(210 | U | 200) = eiAio/r1) = Ud . (61c)

For m = zfcl, corresponding to the unshifted Stark components, no special
difficulty arises, i.e., the frequency of these components is shifted by

Δω0 = 6 (/tao/mr 3 )[ l - (101/8) (a0/rp) + · · · ] , (62)
p

where the leading term is due to first order quadrupole interactions ac-
cording to Eq. (61a), and the correction is due to quadratic Stark effect,
using the exact formula [34, 35] rather than the second term in Eq. (59),
and allowing for a (small) shift in the ground state. For m = 0, the de-
generacy makes it necessary to construct first linear combinations of the
(210) and (200) wave functions which diagonalize the perturbation energy

26 II. THEORY
matrix inclusive of the quadrupole terms [36, 37]. They are

±;ΙΒ'-£+···)|210>> (63">

with the energy level shifts due to linear Stark effect and first order quad-
rupole interactions of

AE+,_ = =b[£/d2 + (it/Q)2]1/2 - *tfq (64)
« ±3(e2a0/r2) - 6(eW/r3) ± 6(e2a03/r4) + · · · ,

if the various expressions are expanded in powers of a0/r. (Note that the
general formula for the quadrupole term given by Sholin [37] has sign

errors in regard to the dependence on parabolic quantum numbers.) Clearly,

ΔΕ+ and | 2+), for example, correspond to energies and wave functions of

the upper level of that component of La which is shifted to higher fre-
quencies, but to really obtain wave functions to order (ao/r)2 and energies
to order (a0/r)A, it is necessary to consider quadratic Stark effect as well.
(For higher states, first order octopole terms would also contribute [37].)

Since inclusion of second order dipole interactions with states of different

principal quantum numbers gives rise only to terms of the highest order in

a0/r considered here, it is sufficient to write the two wave functions in
question as linear combinations of the two functions given by Eq. (63a)
and unperturbed wrave functions of hydrogen for n τ* 2, and to use only

the first term from Eq. (64) and dipole interactions in the ensuing energy

matrix. Its approximate diagonalization leads to the following corrections

toEq. (63a),

A|2+-)ai^:^(|200):F|210))

v , 1 < 2 1 0 | E / | n 0 0 ) | 2 - |(200 | U\ wl0)|2 + 1(2101 U\n20)\i

*^ TP TP

n IL n — Uli

T à Σ' Έ~=γ2 (| η00> <η0° 'υ '210>

± | ηΐθ) <η10 | U \ 200) + | η20) (η20 | U \ 2 1 0 » , (63b)

I I . 2 . QUASI-STATIC APPROXIMATION 27

where the En are the unperturbed energies of the hydrogen levels, and the
[/-matrix elements are those of —z multiplied with e2/r2. The first term of
this correction is characteristic of the degeneracy. Together with the
correction terms in Eq. (63a), it was omitted by Griem [38], whereas the
entire degeneracy is properly accounted for by Nguyen-Hoe et al. [36] and
Sholin [37]. For the ground state, the corrected wave function is simply

I 1> « I 100) - Σ ' 11/(En -E{)1\ nlO) (nlO | U | 100). (63c)

n

Rather than the wave functions, one actually needs the relative intensi-
ties / + , / _ , 70 of the various Stark components, which are proportional to
Σ « \(f I x« I Ol2· Using Eqs. (63a-c) and summing over principal quantum
numbers of discrete and continuum states in a standard manner, these
relative dipole strengths are then

/+ = i [ l - 2(oo/r) + 16(a0/r)2 + · · · ] , (65a)
/_ = i [ l + 2(ao/r) - 16(a0/r)2 + · · · ] , (65b)
/o = f [ l + ((ao/r)3)]. (65c)

The intensity of the unshifted component is so insensitive to the per-
turbation because | 21 ± 1 ) is actually an exact wave function of the sum
of unperturbed Hamiltonian and An = 0 interactions, and because first
order dipole interactions with states of different principal quantum number
do not contribute. (The ground-state only "mixes" with np levels which
cannot combine with 2p for dipole radiation, and the upper state with ns
and nd, neither combining with Is.)

An expression for the energy or frequency corrections for the two shifted
components is obtained by combining Eq. (64) and the exact formula
[34,35] for quadratic Stark effect (of the line rather than the level),
namely

Δω+ = 3(Ä/ror*)[l - 2(a„/r) - (101/4) (a0/r)2 + · · · ] , (66a)

Δω_ = -3(Ä/mr*)[l + 2(a0/r) + (117/4) (a0/r)2 + · · · ] · (66b)

Inspection of Eqs. (65a)-(66b) reveals that the various corrections to the

leading terms remain below or near ~ 1 0 % only when the perturber-atom

separation obeys r > 20α0, corresponding to wavelength separations from
the unperturbed La line of Δλ < ± 2 5 A. Note also that near these limits
corrections for quadrupole interactions and dipole interactions with states

of different principal quantum number to the usual linear Stark effect

approximation are quite similar in magnitude. For the unshifted com-

ponent, Eq. (62) would suggest requiring r > 100a0. However, higher-

28 II. THEORY

order correction terms probably have smaller coefficients, and rp > 20a0
may again be permissible, or Δλ < + 2 A in this case. [To reach
Δλ » —15 A, r « 5a0 would be required by Eq. (62), which is therefore
not likely to happen.] Since this component is actually much more affected
by electron impact broadening, the probably general conclusion from this
example is thus that the simplest form of perturbation theory suffices to
obtain quasi-static profiles with errors from this source below ~ 1 0 % , as
long as perturber-atom separations are larger than about five times the
mean radius of the excited state, i.e., 5n2a0/Z. As discussed earlier, in
Section II.2, this requirement does not usually impose any additional
limitations. To be more specific, r « 20a0 yields Δλ « 25 A for La but
already ~ 5 0 A for Lß, leaving less than half of the separation of the un-
perturbed lines. This would invalidate our implicit assumption of well-
separated levels with different principal quantum numbers.

It remains to discuss the relative importance of dipole and quadrupole
interactions when energy changes are considerably smaller than inherent
level spacings, as in the case of quadratic Stark effect. Then dipole inter-
actions contribute only in second order, yielding level shifts

i ^ (t 1 Uä 1 ϊ) <t* 1 ud 1 i)
Acod « - ZJ ™™
n if Hi — bji>

Γ 2Z2ffH |Ί1η6&αο2 rf_

~ Vn\Ev -E%) ^ J Z ' r o r * ' K}

where the contribution from levels i' of the same principal quantum number

but different orbital quantum number (V rather than Z) is estimated very
roughly by using one effective matrix element (I \ Ud \ V) ~ n2a0e2/2r2, and
where Eq. (59) is used for the contributions from n' ^ n. (En is the

ionization energy of hydrogen.) Typical first order shifts from quadrupole

interactions, for comparison, according to Eq. (60), are

! Δωα | « (n*/Z>) (h/m) (a0/r*), (68)

i.e., the two shifts are comparable when the (singly charged) perturber is
at a distance

Γ 2Z 2 £ H , ,1η2α ο ,AQ.m

rd« ~ U *r - EÙ +l\-¥·

Normally, the factor in square brackets is large—say, about 10—and for
r > 5n2a0/Z (as required for the multipole expansion to be reasonably
accurate), quadrupole shifts and quadratic Stark effects are therefore

typically of similar magnitude even for low Z. Still, because the quad-

I I . 2 . QUASI-STATIC APPROXIMATION 29

rupole effect vanishes on the average over magnetic quantum numbers, it
is not very important in practice (see also Section II.3f).

Finally, the transition between linear and quadratic Stark effects must
be considered, which occurs when dipole interactions cause level shifts
comparable to separations between relevant unperturbed levels. In the
general case of, say, n states coupled through dipole interactions, this
requires the diagonalization of nth order matrices. However, often only
two levels coupled by dipole interactions are so close as to require this exact
treatment, which then amounts to the solution of the matrix eigenvalue
equations

/E1-Ei' U12 \/CiA (70)

) 1-0

Here Ei and E2 are the unperturbed energies, E/ (i = 1, 2) the shifted
levels, Un = U2\ the dipole interaction energy matrix elements (there
being no diagonal elements for unperturbed states of definite parity), and
Cu and C2,· the expansion coefficients of the perturbed wave functions in
terms of the unperturbed wave functions. By requiring the determinant of
the energy matrix to vanish, the perturbed levels are obtained, namely,
assuming Ei > E2,

[ U^k)J ·1 + / 2Γ7 VI1/2 (71)

The two resultant systems of linear equations lead to the (properly nor-
malized) expansion coefficients

C u = C w = [1 + * - \ l + *)mjn ' (72a)

C 1 2 - - C 2 1 - [ [! - (2!_ + e»)*/»]/V2 2 , (72b)
(73)
i+ e (i + e2)1/2]1/ (74)
(75)
where the parameter e is defined as
***2Uli/(El-Et).

In the limit t2 « 1, Eq. (71) gives

&JEi,t<*±U\t/(E1-Et),
the quadratic Stark effect formula, and in the limit «2 : » 1

ΔΑι.1 « ±h(E1 - Et) ±Ua,

30 I I . THEORY

the usual linear Stark effect displaced by one half of the inherent level
splitting. Suitable expansions give corresponding results for the coeffi-
cients Cu , etc., which can be useful, e.g., in the calculations of intensities
of forbidden components (see Sections II.3c0 and IV.4b) and also in the
evaluation of further shifts due to interactions with more remote states
from second order perturbation theory.

11.2c. Atomic Matrix Elements

Except for one-electron systems, even the unperturbed wave functions
describing the radiator states in the absence of perturbers are only approxi-
mately known. The preceding discussion of the quasi-static approximation,
and that to follow in the next section of the impact approximation, make
clear that the most important requirement on such approximate wave
functions is to yield accurate matrix elements for the dipole operator. There-
fore, calculations or measurements of oscillator strengths, or the various
Einstein coefficients, for electric dipole transitions provide essential data
also for line broadening calculations. The relevant formula for the square
of the absolute value of the coordinate (rather than dipole) operator matrix
element, summed over all components of r and magnetic substates of total
angular momentum J', in terms of absorption oscillator strength fj>j or
spontaneous transition probability (per unit time) AJJ> , is

£|<^™ = CT·"'—1(A)* , 2J' + 1 KAJJ.

2J + 1 EK '

(76a)

where a on the right-hand side is the fine structure constant, a « 1/137.
This expression actually gives the average over magnetic substates of J ,
which is appropriate for most applications, and is of course valid only for
Ej' > Ej . Otherwise, one must use the analogous relation

i{J |r<|J>l = E7^E7.ao ^T\fjJ'= AE7^E7J a029 hAj.j

EK

(76b)

or, in both cases,

Σ \(J'\ ri I / ) | 2 = [ao2/(2J + 1)]S, (77)

S being the so-called line strength (in atomic units), which is symmetric
in J' and J.

While critically compiled tables [39, 40] of oscillator strengths, etc., for
the twenty lightest elements (except for transitions between highly excited

I I . 2 . QUASI-STATIC APPROXIMATION 31

states) now exist, corresponding data for heavier elements are still scarce
and not systematically analyzed. For such elements, and for line strengths
of transitions between highly excited states of light elements, direct cal-
culations remain necessary. These are usually based on the assumption of
pure LS coupling, i.e., on

S = ©(TO) ©(£) σ2, (78)

where @(2(TC) and ©(£) are the well-known multiplet and relative line
strengths, respectively, which are widely tabulated [41]. Only σ, the
transition integral, depends on the radial wave functions, i t is therefore the
most difficult quantity to calculate. However, especially for the transitions
between highly excited states, the relatively simple Coulomb approximation
of Bates and Damgaard [42] will often be sufficient. Then one has

* « - ! , I - 1; wi*, 0 = 2 I ~2~ V 4P _ i ) I ^71*-1 ' Ul ' Z )

= (1/Z)ff(n,*, 1)φ(η!1ι > ni*, l). (79)

The correction function φ (which is unity when the effective quantum

numbers n*_x and nf are equal integers) is tabulated [43] for I < 4, and
the effective quantum numbers are readily obtained from empirical energies

[44] E of the corresponding terms and their series limits E^ through

η* = Ζ[£Η/(#οο-£;)]1/2. (80)

[In case the term in question belongs to a series converging on an excited
state of the resulting ion, it would be better to add its excitation energy to
the usual ionization energy. Also, if I > 4 enters, one may normally use
Eq. (79) with φ = 1 for An = 0 and hydrogen values [34] divided by Z2
for An 9* 0.]

Occasionally, the LS coupling approximation will have to be replaced by
an intermediate coupling calculation, pure pair coupling not being realized
in many cases. Also, the radial integrals, in particular when low-lying ex-
cited states or even the ground states are involved, sometimes demand more
realistic wave functions than those provided by the Coulomb approximation
(say, Hartree-Fock wave functions accounting, perhaps, for configuration
interactions). Obviously, no general and accurate formula can be expected
to represent the results of such calculations or measurements. However,
application of Coulomb and LS coupling approximations usually yields
surprisingly accurate results, especially for the widths of lines broadened by
electron impacts, because in line broadening sums are taken over inter-
acting states «/'. These sums tend to be more accurate than the individual

32 I I . THEORY

contributions, a fortunate circumstance which is connected with the
existence of the sum rules.

Other matrix elements occasionally required in correction terms are those
of the quadrupole moment operator, i.e., in the case of one-electron systems
and for diagonal elements,

(nlm |r2(fcos20 - i ) | nlm)

This formula should be sufficiently accurate for two- and more-electron
systems as well, when n is replaced by the effective quantum number
according to Eq. (80). Off-diagonal elements {ài = ± 2 ) tend to be of
little practical importance for nonhydrogenic systems, because then levels
with Al = ± 2 are normally widely separated. (They can also be obtained
from an extension [43] of the Bates-Damgaard method.)

Returning to radial matrix elements of the dipole operator, another
powerful method [45, 46] should be mentioned which is based on extra-
polations along isoelectronic sequences. Using this method, a considerable
number of matrix elements, especially between low-lying states of various
ions, can be determined with fair accuracy and little effort. Very useful in
this connection are of course the bibliographies on atomic transition
probabilities [47].

Line broadening calculations must depend not only on atomic matrix
elements but also on a reliable system of atomic energy values and classifica-
tions. Except for one-electron systems, empirical energy values [44] or
ionization energies [48] are considered preferable, although present data
may be insufficient when large angular momentum states act as perturbing
levels. As to the classifications, serious doubts affecting line broadening
predictions probably occur mainly in cases of intermediate coupling, which
had not been recognized as such in the term analysis. Comparison between
measured and calculated line profiles thus might occasionally serve as an
auxiliary tool in unraveling some of these classifications. Here the reader
should be aware of the revised multiplet tables and atomic energy levels
[49] and the corresponding bibliographies [50].

ILS. IMPACT APPROXIMATION

When the duration of effective perturbations is short compared to time
scales relevant for the Fourier integral which represents the line shapes
according to Eq. (6), details of the actual time-dependence of perturbed

I I . 3 . IMPACT APPROXIMATION 33

radiator wave functions are of little consequence, and only net changes in
the wave functions caused by such effective collisions matter. Because of
the general properties of Fourier integrals, the time scale As of interest here
is determined by the smaller of 1/| Δω | and 1/w, the inverses of angular
frequency separation or half-width. [For times larger than | Δω |_1, the
integrand in Eq. (6) oscillates rapidly; for times larger than url, its mag-
nitude decreases exponentially for the cases considered here. Note also that
in the case of lines with forbidden components, Δω may have to be counted
from the originally forbidden line.] The actual duration of a collision char-
acterized by velocity v and impact parameter p may therefore be neglected
in comparison to As when v/p^> max(| Δω |, w). This condition for the
validity of the so-called impact approximation is the exact corollary to that
for the quasi-static approximation (Section II.2). These approximations to
the general theory of pressure broadening therefore complement each other,
and both become questionable when v/p « max(| Δω |, w). In this transi-
tion regime between their domains of validity, it is necessary to employ
more appropriate ("intermediate") approximations, as discussed in
Section II.4.

In view of the above remarks and after the detailed discussion in Section
II.2, there is no need for as broad a discussion of the validity domain for the
impact approximation. For hydrogen, ionized helium, etc., the impact
approximation becomes valid below the curves (for electrons as perturbers)
in Fig. 1 which are determined by Eq. (17), at least for portions of the
profiles near or within the half-intensity points. Far out on the line wings,
the corollary of Eq. (15) must hold true for the impact approximation to be
valid in the case of these lines. From the discussion in Section II.2, it thus
follows that the broadening by electrons almost always falls within the
impact domain, distant wings of lines subject to linear Stark effect being an
important exception. However, the broadening of such lines by protons,
not to mention heavier (i.e., slower) ions, can be described by the impact
approximation only for early members of hydrogen (and hydrogenic) line
series, in particular when also the lower state is highly excited (n-a lines,
etc.; see Section IV.5).

In case of two- and more-electron systems, the impact approximation is
practically always applicable to broadening by electrons (the far wings of
diffuse neutral helium lines constituting a noteworthy example to the
contrary) and in the line cores occasionally also to the broadening by ions.
However, here it would not be realistic to discuss the validity of the impact
approximation for ions as perturbers entirely separate from the effects
caused by electrons. The latter naturally dominate the time variations of
the perturbing fields, and those of the ion-produced fields can thus be

34 I I . THEORY

ignored as long as they occur over time scales larger than l/we, where we is
the (angular frequency) half-width caused by electron impacts. Accord-

ingly, the quasi-static approximation for ions might remain reasonably valid

throughout the profile if the following inequality [51a] is fulfilled (see

also Section IV.4c, however) :

werp/v = (wjv) ΟΛττΛΓρ)1'3 > 1. (82)

(In this case, v is a mean value of the relative velocity between radiators
and perturbing ions.) Conversely, the impact approximation can often be
used for perturbing ions when this inequality is reversed, but must be
replaced by the quasi-static approximation on the far wings of the lines as
soon as one has | Δω | > v/p. For other than one-electron systems as
radiators, broadening by ions is generally less important than broadening
by electrons, so that relatively crude approximations usually suffice to
evaluate corresponding corrections (see Section II.3f) to electron impact
profiles. Unless specifically stated otherwise, the perturbers in the sub-
sequent discussion of the formal and practical aspects of the impact
approximation will therefore be taken as electrons.

According to Baranger [1], the formal derivation of the impact approxi-
mation most conveniently begins with Eqs. (5) and (6). Writing the former
in terms of the stationary states and energy levels (instead of frequencies
oty) of the complete system (radiator plus perturbers) and introducing [3]
time-evolution operators through Eq. (8), i.e., in our representation,

t(s, 0) = e x p [ - (i/h)E*s~], (83a)

*+(s,0) = exp[+(i'//i)#ss], (83b)

the correlation function C(s) becomes, since p is diagonal for stationary
states and as anticipated in Eq. (7),

C(s) = Σ <t I «. I /></1 tHs, 0) | /></1 *. | i) (i | t(s, 0) | i) (i\p\ i).
Ha
(84)

If we include "intermediate" states f 9* f and %' τέ iy this is valid in any
representation, especially in one corresponding essentially to products of
perturber and radiator wave functions. Assuming that the density operator
remains diagonal in such new representation and that all initial states of the
radiator contributing to a given line are equally likely, the factor p may be
dropped or, rather, be replaced by an average over (unperturbed) per-
turber states. The use of slightly corrected product wave functions and the
assumptions regarding the density matrix are all justified when line widths

I I . 3 . IMPACT APPROXIMATION 35

and shifts are much smaller than kT/h, as is usually the neglect of changes
in perturber, e.g., wave-packet [ 6 ] states. For the latter reason, the
t operators from now on will be assumed to affect radiator states only.
(However, we must keep in mind for later that actually perturber states do
change as well.) With these provisions and assuming that the xa act only on
radiator electrons, i.e., neglecting so-called perturber radiation (see Section
II.4d), the correlation function can be written as

C(s) = Σ ' [<t | xa | / ) </ I ft(*f o) |/'> </' I xa I i'> <i' 1t(s, 0) I <>]aT

= Σ ' << \Xa\f) </' I Xa I i') [ < / I <*(·, 0) \f) <i' I t(8, 0) I Z)]av

- Σ' <«r i Ad i »r·» <«r i M*, o) i in)

= Tr[Ad^av(s,0)]. (85)

Now i,f, i', f are quantum numbers characterizing (unperturbed) states of
the radiator only, over which (in addition to a = 1, 2, 3) the sums are
taken. (The prime indicates that one needs to sum only over states con-
tributing to the line in question, and "av" stands for the average over
perturber states.) The next-to-last line serves to define [2, 3, 6 ] the two
operators Ad and Θ in "line space/ ' in which a radiator wave function of one
of the upper states of the line is associated in a direct product with the
complex conjugate of a wave function of the lower state. (This corresponds
to the Liouville operator formalism first used by Fano [14] in line broad-
ening theory.) The sum over a is included in Ad .

In the special case that perturbations of the lower states ( / , / ' ) are
negligible, i.e., that

</ ! tHs, 0) | / ' ) = e x p [ ( t / Ä ) M à». , (86)

the result is much simpler, in particular when Ef is chosen as the origin of
the energy scale. Then Eq. (85) reduces to

C(s) = Σ ' {ιΊ^Ι/Χ/Ι^Ιί'ΧίΊΜβ,Ο) | i ) S T r D M « , 0 ) , (87)

which is entirely analogous to the "two-state" formula, Eq. (85), except
that the trace now only involves upper states (i, i') rather than "doubled"
states. (Note, however, that the D operator not only implies a sum over
components a but also over degenerate sublevels/' of the lower state.) In
view of this analogy, it is sufficient to continue the development of the
impact approximation with Eq. (87) and to generalize the result at the end.

To affect a further separation of quantities related to radiator and per-
turbers, respectively, it is more convenient to consider the interaction
representation evolution operator, whose matrix elements with respect to

36 I I . THEORY

the unperturbed time-independent radiator wave functions correspond to
the usual Cmn(t) coefficients in time-dependent perturbation theory, i.e.,

w(e,0) = exp[(i/A)ffe]f(e,0), (88)

where H is the unperturbed Hamiltonian for the radiator. To arrive at the
average time dependence of this operator, one may write

Auay(s, 0) = u&y(s + Asy 0) — uav(s, 0) = {[u(s + As,s) — l]u(s,0) }ftv .
(89)

The impact approximation [2, 4, 51a] now consists of treating the two
factors inside the braces as statistically independent but still taking the
average matrix element of the first factor to be small in magnitude. This
factor has the perturbation expansion (Dyson series)

u(s + As, s) - 1 = (ih)'1 f dx U'(x)

dx / dy U'(x) U'(y) + · · · , (90)

which is the iterated solution of the time-dependent Schrödinger equation
in the interaction representation,

ihu(s,0) = expl(i/h)Hs'] U e x p [ - (i//i)/fs>(s, 0) = U'(s)u(s, 0),

(91)

U being the usual interaction Hamiltonian which describes the effects of
the perturbers on the radiator [see Eq. (58)].

Strictly speaking, U is of course a sum of contributions from various
perturbers. However, the corresponding entanglement in the multiple
integrals of Eq. (90) does not really occur if the impact approximation is
valid [1-4, 51a]: there may be either one "strong" collision in the time
interval As in Eq. (89), so that concurrent weak collisions can be ignored,
or several "weak" collisions overlapping in time, for which the first few
terms in Eq. (90) are sufficient. Except for fourth and higher (even) order
terms, any entanglement disappears then on the average over various
(statistically independent) perturbers, and it is thus actually sufficient to
calculate [u(s + As, s) — 1] for single perturbers, adding their con-
tributions afterwards. (When electron-electron correlations are allowed for,
a shielding correction appears in the second order term; see Section II.5a.)
Moreover, it is necessary for the usual impact approximation (in contrast

I I . 3 . IMPACT APPROXIMATION 37

to extensions [6, 52] of this approximation as discussed in Sections II.4b
and II.4d) to assume that the duration of a single collision is much shorter
than As. In this manner, one obtains (now counting x, y, etc., from time s)

[ti(e + As, s) - l ] a v « Σ / i Δβ exp[(t/ft)ffe] | (ih)"1 f " dx U/(x)

+ W"2 ί+"ώ f dy U/(x) U/(y) + · · ·]

•'—oo ·'—oo J

X exp[-(i/Ä)ffs]

= βχρ[(ί/Α)Η«][Σ/,(5, - l)]exp[-(t/Ä)H«]Ae

i

= exp[(t7&)i7s] φ exp[— (i/h)Hs] As

= (ζ'/ί)-1 exp[(i/Ä)Äe] 5C e x p [ ~ (*'/Α)ΙΓβ] As,

(92)

where j stands for all parameters of a given type of collision which occurs
with frequency / , · , and where Sy is the corresponding scattering (S)
matrix. The next-to-last line gives the definition of the collision broadening
operator φ of Kolb and the author [ 4 ] , the last that of the effective (non-
Hermitian) perturbation Hamiltonian 3C introduced by Baranger [1-3].

In terms of this effective (time-independent) Hamiltonian, the impact
approximation equivalent of Eq. (89) is

ifiùnv = exp[(t'/Ä)iTs] 3C exp[— (i/h)Hs'] uAy , (93)

which corresponds to

ifd%y = ( # + 3C)*av, (94)

Eqs. (93) and (94) being Schrödinger equations for the averaged evolution
operators in the Heisenberg (interaction) and Schrödinger representations,
respectively. In the impact approximation, the line broadening can there-
fore be described by the addition of a constant but non-Hermitian Hamil-
tonian to the unperturbed Hamiltonian of the radiator. The effective
Hamiltonian is expressible according to Eq. (92) in terms of the S matrices
Sa and S/j describing the scattering of the perturbing electron on upper
and lower levels, respectively, if one now generalizes to the two-state case
using the analogy between Eqs. (85) and (87). Observing further that the
operator Θ can also be interpreted [ 2 ] as the product of {ι' \ t(s, 0) \i) with

38 II. THEORY

( / ' I t(s, 0) I / ) * , this expression is found to be

oc = ΟΦ = -in Σ/,(ΐ - Α,·Λ*,) = -ih ΣΜΤ« + T*fj - τ«τ*η),
3ì
(95)

where the second version is written in terms of the transition (T) matrix,

T = 1 - S. (96)

(Note that the i-f product term usually appears only for elastic scattering,
i.e., only if perturber energies change by much less than the inverse collision
time. Otherwise, there would be no overlap between the two scattered
waves [3, 6]). Either way, the principal remaining problem in the cal-
culation of 5C or φ is the determination of the elastic (or nearly elastic)
scattering of electrons by radiators in states i and /. Practical solutions to
this problem will be discussed in the following sections.

Resuming the formal derivation of impact profiles, the integral of Eq.
(94),

M s , 0) = e x p [ - (i/h) (H + 3C)s], (97)

is substituted into Eq. (87), or the equivalent expression for 0, in which
H is replaced by Hi — Hf*, into Eq. (85). (The subscripts of H are to
indicate that this operator generates unperturbed energies of either the
upper or the lower groups of levels.) Finally, the Fourier integral in Eq.
(6) is performed, resulting with H* = H in the profile

L(co) = - ( Ä / T ) ImTr(Ad[fco - (Hi - H, + OC)]"1}

= - ( l / τ ) Re Tr{Ad[t« - (i/h) (Η{ - Hf) + φ]"1}. (98)

When 3C (or ψ) can be represented by diagonal matrices (in line space),
these profiles are therefore Lorentzians with widths and shifts given by

w = - (1/Ä) Im<«r I 3C I r » = - R e « 2 / * I Φ I if*)), (99)

d = (1/Ä) Re«2/* | 3C | in) = - I m < « r I Φ I if*)) · (100)

(Note that ψ has occasionally been used [4, 53] only under rather special
conditions, and that there have been numerically inconsequential errors
in regard to complex conjugate signs, etc., associated with it. Also, using,
as we do, half of the so-called full width between half-intensity points is
obviously more convenient than to use, say, the damping constant y = 2w.)

Clearly, the electron impact profiles described by Eq. (98) thus con-
stitute a generalization of Lorentz [54] profiles to situations where various

I I . 3 . IMPACT APPROXIMATION 39

lines corresponding to transitions between distinct sublevels of upper and
lower states overlap, which is a very common situation in plasmas. More-
over, Eq. (95) gives the correspondence between line broadening param-
eters and quantities giving the net change of state from single collisions, a
connection [5] whose existence in the impact limit was surmised in the
first sentence of this section. In this limit, line broadening and usual
electron-atom (or ion) scattering theory are thus closely related, as was
first discussed in detail by Baranger [1-3] (see also Section II.3e). The
impact approximation is expected to be valid as long as line widths (and
shifts) remain much smaller than perturber energies (divided by h) and
the duration of effective collisions is smaller than the times of interest in
the Fourier integral describing the line shape (see the opening paragraph
of the present section). Fulfillment of the former requirement not only
eliminates difficulties with the perturber density matrix but also ensures
that the lower quantum-mechanical limit h/kT for the duration of collisions
stays well below the relevant time scales for the Fourier integral. All of
these conditions also appear when the line broadening theory is formulated
[55] in terms of the density matrix rather than the autocorrelation func-
tion.

A logical continuation of the above developments would be a discussion
of quantum-mechanical methods for the calculation of JC or φ, i.e., of S or
T matrices. Instead, we will follow a more pragmatic approach by treating
the perturbing electrons as point particles moving along prescribed trajec-
tories. This approach has been very successful for neutral atom and ionized
helium lines, but is of course questionable when, e.g., in the partial wave
treatment of the scattering problem, low i-values predominate. This cer-
tainly happens for resonance lines of some singly charged ions, and it is
for this practical reason that the fully quantum-mechanical theory of
electron impact broadening will be resumed only in Section II.3e.

II.3a. Hydrogen Lines

Lines from one-electron systems are rather strongly affected by a quasi-
static (and linear) Stark effect from ion-produced fields (see Section II.2) ;
i.e., they consist really of groups of Stark components displaced more or less
from the unperturbed line positions. The "mixing" of these components by
electron collisions is then described by the impact broadening operators
3C and φ, which were introduced in the preceding section. To account for
both electron impact and quasi-static ion broadening, Eq. (98) must
therefore be averaged over ion-produced fields F whose distribution is
given (see Section II.2a) by W(F). Assuming linear Stark effect for the
energy levels and neglecting any other dependences on the ion field, the

40 I I . THEORY

appropriate equation for Stark profiles of hydrogen, etc., lines is thus [53]

L(co) = - ( l / π ) Re Tr Γ dF W{F) {Ad[i Δω - iCF + φ]-1}, (101)
•'ο

where Δω is the frequency separation from the unperturbed line, and C is
an operator whose matrix elements (in terms of parabolic wave functions)
[56] are the differences between the linear Stark coefficients for the upper
and lower levels [see Eq. (59) ] .

Equation (101) has served as the basis of most realistic calculations
[53, 57-62] of Stark profiles for one-electron systems. (See also Section
II.4b, however.) Its validity rests not only on the two fundamental ap-
proximations for ions and electrons, respectively, but also on the accuracy
of Ad , C, and φ-matrix elements calculated using zero-field (parabolic)
wave functions and energies. (Transformation coefficients from spherical to
parabolic coordinates are available [56].) In the case of Ad and C, higher
order corrections are not expected to be important (see Section II.2b),
except that they will cause small asymmetries (~10%) in the profiles
(see Section III.9) at high densities where lines involving upper states of
different principal quantum numbers begin to merge. This leaves φ as the
most likely source of errors (in addition to those stemming from the two
basic approximations) in actual calculations, since W(F) values should
normally be accurate to within a few percent unless there is strong tur-
bulence and except near F = 0.

Almost all calculations of φ rely on the classical path approximation for
perturbing electrons (straight for hydrogen or hyperbolic for ionized
helium). The frequency of collisions occurring within impact parameter
and velocity intervals p, p + dp, and v, v + dv is then simply

df = 2rNf(v) pv dp dv, (102)

which replaces / , in Eq. (95) and where f(v) is the velocity (magnitude
only) distribution normalized to

Γ f{v) dv = 1.

•'o

Unless stated otherwise, it will be taken as Maxwellian,

f(v) = (2/T)1/2(m/kTy/2v2exp(-mv*/2kT), (103)

and we shall assume isotropy in velocity space. It still remains to perform
an average in Eq. (95) over angles associated with the vectors ρ and v
entering, e.g., the straight classical path expression for the radius vector of

I I . 3 . IMPACT APPROXIMATION 41

the perturber relative to the nucleus of the radiator, namely

r ( 0 = 9 + vt, (104)

counting time from the point of closest approach. Inserting Eq. (104) into
Eq. (58) for the first two multipole terms of the interaction energy and
integration over t results in

nJ-oo mpv I \ «op / P L \ a0p /

+ \ a0v / \ a 0 / \)

mpv {2-αc0 os0 + ρ \ao/

X I 2 (cos2 Θ - y + cos2 θ' - l- 1 . (105)

Clearly, this vanishes on the average over the angles Θ and Θ' (between ρ,
ra and v, ra) for isotropie plasmas, so that for hydrogen and other neutral
atoms there is no first order term in the "one-" or "two-state" equivalents
of Eq. (92) for φ (except, perhaps, for the strongly magnetized plasmas
discussed in Section II.6).

To obtain the second order terms in Eq. (92), it is sufficient to employ

I " 0 0 dx f dyf(x) f(y) = h \ / + Q ° / ( z ) dx\ , (106)

—oo —oo L —oo -1

assuming U'(t) = U(t), i.e., neglecting the quasi-static Stark splitting
between the various sublevels and inelastic collisions. In this manner

φ « -2TN fdvvfiv) f dpp(h/mpvy Σ {(2/3ao2)[(rt | i") · <t" | r<)

- 2r,· · x, + (r, | / " ) · (f" | r , ) ] + (2/15a0V) (r,·211"> <t" | r,s)}
(107)

follows from Eqs. (92), (95), (102), and (105), and from, e.g., (cos20),v = \,
(cos2 Θ cos2 0') av — I T , and (cos4 0) av = z> for diagonal elements. [[The
relations for (cos2 Θ) a v , etc., can be obtained from an integration over
angles for g and v, observing that the two vectors are orthogonal. Also,
the rt and 17 correspond to ra above, but operate only in the i and / sub-

42 I I . THEORY

spaces.] In this expression, quadrupole terms for the lower states and a
quadrupole upper-lower state product term were omitted because they are
a correction anyway. (First order dipole-quadrupole product terms vanish
on the directional average.) Also, second order dipole terms for the lower
states and the (negative) product of first order dipole terms were added, as
indicated by the appearance of SiSf* in the two-state formula for ψ and 3C,
Eq. (95). The sums over intermediate states only involve states of the same
principal quantum numbers as those of states i and/, respectively, and the
second order dipole terms are written so that they are valid for off-diagonal
«/»-matrix elements as well (in regard to other than principal quantum
numbers).

There are two more correction terms to Eq. (107) which may have to be
considered. First of all, fourth order dipole terms in the Dyson series can
be estimated to be of the order

{l/hy{i | Γ dxL f l dx2f2 dxnf* dx, Ufa) U(x2) Ufa) Ufa) \ i)w

•'—oo · ' — oo •'—oo ^—oo

« (2/15a04) (h/mpvy \(i | r< | i')\* « (2/15a„*) (h/mPvY{i | r,·21 i)\

(108)

using the generalization of Eq. (106), Eq. (105), (cos40)av = i , and the
indicated approximations for the combination of r» matrix elements. (This
estimate is not strictly valid also because the potential operators do not
commute within the i subspace of fixed principal quantum number [62a,
63b]). Comparison with the second order quadrupole term in Eq. (107)
shows that the latter is a factor of about kT/E-a. times the estimated fourth
order dipole term, which is therefore numerically more important (at least
in the case of hydrogen). The other correction is for second order dipole
terms with intermediate states k ?* i, i.e., accounts for inelastic collisions.
This effect is limited by adiabaticity to p < v/W , as discussed in more
detail in Section II.3ca.

Leaving out for the time being all but the second and fourth order dipole
terms which operate within the i subspace, estimating a typical matrix
element of i\ · r< (summed over intermediate states) by n4a02/Z2 and one
of rt2 by %n4a02/Z2, and ignoring the velocity average, one would have

(i | φ | i) « -2irNv f päp {h/mpv)2H{n*/Z2) - f (h/mpv)2(n*/Z*)']

« - {±irßv)N(h/m)2{n*/Z2)[_l + ln(pmax/pmin) - * ] . (109)
Here p ^ n = n2h/Zmv was chosen as a lower cutoff for the p integral to