Spectral Line Broadening by Plasmas

I I . 3 . IMPACT APPROXIMATION 93

(See Barnes and Peach [108a] for generalizations of Eqs. (207) and
(211).) A major practical difficulty in applying this method to line
broadening problems lies in the necessity for considerable extrapolation and
interpolation when using the R matrices originally obtained for fixed
energies, mostly in the inelastic region, of the total (radiator plus electron)
system, rather than for fixed energies of the perturbing electron.

The results of these calculations for the line widths are in fair but not
perfect agreement with experiment (see Section III.7) and also with some
of the semiclassical results, and indicate that fine-structure effects are
indeed small, as surmised earlier. The situation for the shifts is much less
satisfactory [108b], as is that for the widths of the Be II resonance lines
(see also Griem [64c]).

Similar calculations have been made [64a, 67] for hydrogen and ionized
helium resonance lines, ignoring any modifications for the degeneracy in
J (or L). They are useful for the assessment of the broadening by inelastic
collisions, for the determination of the "strong collision" term, and for the
calculation of higher multipole interactions. Quantum-mechanical calcula-
tions for multiply charged ions of astrophysical interest are not yet avail-
able, but will be anticipated in Section IV.6 for estimates of widths and
wing shape parameters.

II.3f. Corrections for Ion Broadening of Isolated Lines

For lines other than those from one-electron systems (Sections II.3a
and b) or lines showing forbidden components (Section II.3c0), ions tend
to be much less important as perturbing particles than electrons, and even
for these special cases ion broadening is not entirely dominant. There could
be one exception to this rule, namely the case of some so-called high
principal quantum number n-a lines [111] (transitions from n to n + 1),
etc., from one-electron systems, for which the impact approximation holds
for ions as well (see Section II.3) but broadening by inelastic electron
collisions is not yet dominant (see also Section IV.5). Some isolated lines
from highly charged ions might be another exception, but as discussed
below, this also is not very likely to happen. Isolated lines in general,
therefore, are broadened primarily by electron impacts, and relatively
crude corrections are sufficient to allow for ion effects. The first question
then is which of the two extreme versions of the theory to use in estimating
these corrections, quasi-static or impact approximations.

To answer this question in a practical way, one should always keep in
mind the overriding influence of electrons, especially in the line cores, and
use as a sufficient condition for the quasi-static approximation

HIT ^ max[w, | Δω(τ-) | ] . (212)

94 I I . THEORY

In other words, the frequency corresponding to the duration of a given ion
collision only has to stay below the electron impact width or the instan-
taneous frequency shift caused by the ion in question. Assuming for the
time being that the quasi-static approximation is indeed valid throughout
the line profile, one can argue that typical values of r are of the order iV~1/3
and thus obtain the simple (and usually sufficient) criterion for the validity
of the quasi-static approximation in the present context from Eq. (82),
namely

w / W O £ 1, (213)

where the near equality sign was justified numerically [51a] by comparisons
with calculations based on a suitable intermediate approximation (see
Section II.4a). Omitting logarithmic factors in the dipole contribution to
the electron impact width estimated analogously to, e.g., Eq. (175) and
neglecting lower state broadening and higher multipole contributions, Eq.
(213) can also be written as

vevp \mz) AT1,'1 ~ b kT \mz) NlJ* ~ h ^

where n is the (effective) principal quantum number of the upper state of

the line, and Z = 1 for neutral atoms, Z = 2 for singly ionized radiators,

etc. Even for protons as perturbing ions, with thermal velocity vp and mass
rap , the right-hand side of Eq. (214) tends to be above unity in most
laboratory experiments, at least for Z = 1 or 2, while the inverse situation

is typical for stellar atmospheres.

However, it would be premature to conclude from the latter observation

that the impact approximation should be used to estimate the ion broad-

ening of isolated lines in stellar atmospheres, because what is really needed

here are asymptotic expressions for the far wings of the lines. The relevant

criterion for the validity of the quasi-static approximation in these cases

is therefore

vjr < | A«(r) I « nWe2/hZ*r*, (215)

assuming, in order to be on the safe side for most lines of interest, quadratic
Stark effect (see also below) in the "hydrogenic" approximation, Eq. (59),
and singly charged ions as perturbers. Expressed solely in terms of fre-
quency separations from the line center, this criterion is

| Δω | > (hZ%A/nWe2)m « (Z4m/n6Äao2)1/3(fcr/mp)2/3. (216)

It is at least marginally fulfilled on the "damping" wings of most lines
showing any significant Stark broadening in stellar atmospheres, so that

I I . 3 . IMPACT APPROXIMATION 95

dynamical corrections to the quasi-static approximation (see Section IV.4c)
remain small. Only in the rare cases in which both criteria, Eqs. (213) and
(216), are violated, must ion broadening, and especially also shifts of
isolated lines, be estimated from the impact approximation. This is a
relatively simple matter, as the adiabatic approximation (vp/p « ω»'»·,
etc.) tends to remain valid. For neutral atoms, the well-known phase shift
theory of Lindholm [80] and Foley [81] can therefore be used [see Eq.
(263)]; for charged perturbers, an equivalent approximation [101] based
on hyperbolic rather than straight classical paths can be used, provided
always that quadratic Stark effect dominates. (Roberts and Davis [101]
actually deal with attractive hyperbolas, but the required changes of
signare trivial.)

Before turning to the quasi-static calculation of ion broadening through
quadratic Stark (second order dipole) effects, it is worthwhile to consider
broadening of isolated lines solely through first-order quadrupole inter-
actions. According to Eq. (60), the latter give rise to shifts

| Δω(τ·) | « (n4ao2e2/Z2Är3)Zp (217)

from Zp-times charged perturbers or, using (47r/3)rWp « 1, to (half)
half-widths

Wi « (4τ/3)Νρ(ηΑήα0/Ζ2τη)Ζρ . (218a)

[Note that an estimate based on the impact approximation and Eq. (138)
yields a very similar result, except for the numerical factor and inclusion
of lower-state interactions. See Eq. (218b), which follows Eq. (229a).]
Comparison with Eq. (175) shows the quadrupole broadening by ions to be
less effective than electron impact broadening by a factor of about ma^Jh,
which is indeed small at typical temperatures for low ionization stages,
but rarely much below, say, 0.1. In cases of negligible quadratic Stark
effects from ion-produced fields, a correction of this order added to the
electron impact widths may thus be called for. However, because the first-
order quadrupole splitting is symmetrical, no such correction should be
applied to the shifts.

When quadratic Stark effects are more important than the above quad-
rupole interactions, it is not proper to simply add their contributions to the
line width as estimated separately. This would ignore the fact that in
expressions for instantaneous level shifts caused by a charged perturber,
which account both for second-order dipole interactions, i.e., quadratic
Stark effects (~r~4), and for first-order quadrupole interactions (~r~z),
the coefficient of the r~4 term is only weakly dependent on the magnetic
quantum number while that of the r~3 term depends strongly on this

96 I I . THEORY

quantum number, to the extent of being zero on the average over corre-
sponding orientations. To assess remaining quadrupole effects on the far
wings, corresponding to small r, consider therefore

M r ) « -(CVr4) + (C,/r») (219)

for a given perturber-radiator orientation. In the nearest neighbor approxi-
mation and treating the quadrupole term as a small correction, the asymp-
totic contribution to the line shape for this particular orientation is then

IM dr
- 4TTÌVP7·2 d Δω

TNPC\/A 1 - c4 Δω +/ÇA2 |C4_ 1/2 (220)
Δω
32 \CJ |Δω

After averaging over magnetic quantum numbers, the first correction term
will become very small under the present assumptions. The second correc-
tion term can be estimated using coefficients for quadratic Stark and
first order quadrupole effects, as in Eqs. (59) and (60). This gives a relative
correction to the wing intensity produced by ions of no more than

AI/I « A(C3/C4)21 C4/Aœ 1/2 (Z2/20nZp) | En/h Δω 11/2 (221)

in most cases (since most lines actually have larger Stark effects) for

I > 1, or Al/I = 0 for I = 0. Before deciding whether or not such correc-

tion is negligible, one should of course verify that quadratic Stark effects

from ions are important compared to electron impact broadening. The
latter contributes Ie(u) ~ w/π Δω2 on the line wings, so that the ratio of
the two contributions is about

/ι(ω)//β(ω) « TrWpC^4 Αω^/w (Ä Δω/£Ή)1/4, (222)
« 0.7(kT/EB)^(Npnl/iZ9J2/NZ)

again using Eqs. (59) and (175) and ignoring slowly varying factors. For
situations where the ion contribution estimated in this manner is more
than, say, 20% of the total wing intensity, quadrupole corrections ac-
cording to Eq. (221) account for less than or about 10% of this total
intensity, provided we have kT/En < 0.1.

If the ion broadening is at least as important as that, it is thus rather safe
to write the total Stark profile of an isolated neutral atom line as a con-
volution of electron impact profiles with quasi-static, quadratic Stark effect

I I . 3 . IMPACT APPROXIMATION 97

profiles for the ion broadening,

I ( « ) = (to/x) f [W{F) d F ] / [ > 2 + (Δω - d + CF2)2], (223)

C = C4/e being a mean value over magnetic quantum numbers. Such
profiles have been calculated [7, 51a] [in terms of reduced wavelengths or
frequencies x = (Δω — d)/w~\ and depend on two dimensionless param-
eters, namely, for Zv = 1 and Np = N,

A = (CF<?/wy<*

3 _|3w> W I w (2i* + 1)««· \ *' k * >

£v (2mIa/x+(^l),«IS/)/(A/1I ral,r7VlJl f 4'

and

β = η/ΡΌ = 61/3π1/6(β2/Α:Τ)1/2ΛΓ1/6. (225)

The first parameter, which has been called a previously, is a measure of
the relative importance of ion broadening, the second of Debye shielding
and ion-ion correlations. Appendix IV.b contains such j (x) profiles, and it
turns out [7, 112] that widths and shifts of the corresponding 7 (co) profiles
are well represented by

wtotai « w + 1.75A(1 - 0.75Ä)w (226)

dtotai « d ± 2.00A(1 - 0.75R)w (227)

for A < 0.5, R < 0.8. (The sign in the shift equation is equal to that of the
low-velocity limit for d.) Furthermore, on the line wings, we have the
asymptotic expansion [51a]

7(co) ~ (w/w Δω2)[1 + (3ΤΓ/4)Α I Αω/w |1/4] (228)

for Δω of the same sign as the quadratic Stark effect, and only the leading
term otherwise.

In view of what has been said above, these formulas are usually applicable
only for A > 0.05, because at lower A, quadrupole interactions should be
included. An upper limit A « 0.5 is imposed by the transition to linear
Stark effect in typical cases, leaving a useful fange of 0.05 < A < 0.5 for
Eqs. (223)-(228), which fortunately accommodates most isolated lines
emitted by neutrals. (Note that A scales as iV1/4.) For typical ion lines,
A values would be too small for these relations to be appropriate, especially

98 I I . THEORY

after the latter have been corrected for the radiator-perturber Coulomb
interactions. In the asymptotic region, these will lead to a reduction in the
ion contribution by a Boltzmann factor

L — ^ τ — J ^ L — * r — I e r i Je x pΓ (Z - 1)Ζρβ2Ί e x p Γ (Z - l)Zpe2 I Δω 11/41

Γ (Z - 1) Zpe2 ΜΙΓΝΛ1/Ζ I Δω 11/41

^eXPL *Γ V U 7 IVI J'

(229a)

now assuming Zp-times charged perturbers. This factor tends to be well
below unity when ion broadening is at all important on the far wings of
such lines.

Still, to estimate corrections to the half-width, Coulomb effects can
normally be neglected, and the correction can either be obtained from Eq.
(226) or by adding a "quadrupole" ion impact width

Wi « 2πΛ^ρ[(ηί2 - n/2)2^ao/Z2m]Zp , (218b)

whichever is larger. [Equation (218b) follows from Eq. (138) with some
simple approximations for matrix elements, etc., and may, because of its
independence of velocity, also apply when the quasi-static result approxi-
mated by Eq. (218a) would seem more appropriate.] The third possibility,
namely ion impact broadening and shift through second order dipole inter-
actions (quadratic Stark effects), is not of much practical interest for
isolated lines, as it usually stays below Eq. (218b), indicating that quad-
rupole interactions involving diagonal matrix elements are typically stronger
than dipole interactions with nondegenerate states. Clearly, lines with
perturbing levels Al = ± 1 unusually close to upper or lower levels can be
exceptions to this rule, but would more often than not fall in the quasi-static
regime in regard to their broadening by ions. If not, they can be treated by
a suitable modification for the signs of perturber and radiator charges of the
adiabatic approximation for electrons on hyperbolic paths [101].

A final remark is in order for highly charged radiators. For them, Eqs.
(218a) and (218b) would suggest an ion-produced width exceeding the
electron impact width at higher electron temperatures, say, kTe > 100 eV.
(Note that higher multipole interactions are now more important also for
the electron broadening.) Then a careful calculation along the lines of
Sections II.3ca and II.3d is indicated, but Eq. (218b) may nevertheless
give a first estimate, if the impact approximation holds and provided ions
cannot produce many real transitions between Al = ± 1 levels or substan-
tial phase shifts from "virtual" transitions between such levels, nor are kept

I I . 4 . INTERMEDIATE APPROXIMATIONS 99

apart by their Coulomb interactions. (Note that relevant impact param-
eters tend to stay well below excited state Bohr radii, so that no significant
errors are expected from our use of the multipole expansion.) For these lines,
however, one is often interested in large frequency separations and densi-
ties, where the quasi-static approximation is more appropriate and quad-
rupole interactions and quadratic Stark effects could be comparable.
Assuming the former interactions to dominate, we then estimate for the
Boltzmann reduction factor analogous to Eq. (229a)

r (z - ΐ)ζρβη r .(z-i)(ZpZ)"j?H h Δω 1/3Ί
exp rk—T J « ex^p L—2 nmkT En

(229b)

which is usually rather small for the lines in question. Also impact broad-
ening would be reduced by Coulomb interactions, roughly by a factor
corresponding to [ · · · ] * / 2 in Eq. (123) but without [68] the π, because
the 0 integration goes via θ = 0 for repulsive orbits.

II.4. INTERMEDIATE APPROXIMATIONS

Although most actual situations are covered reasonably well by one of
the two extreme approximations to the general theory of pressure broad-
ening, these quasi-static and impact approximations are not always
sufficient when a high degree of accuracy is desired. Moreover, a satisfactory
estimate of all theoretical errors is not possible in the absence of a theory
valid in a regime intermediate to those where the usual extreme approxi-
mations are appropriate. Third, by working out a more general theory
spanning this intermediate range, more insight might be gained into the
meaning of various subsidiary approximations made in practical cal-
culations.

While there are thus enough and weighty reasons for developing inter-
mediate approximations in the above sense, it should not be overlooked
that hopes for a truly unified and practical theory are dim indeed. As
enumerated in Chapter I, there are far too many parameters in the general
problem to make such an ambitious program realistic. This is not to say
that calculations based on a more general theory are not possible for some
otherwise relatively simple cases, e.g., for hydrogen lines (see Section
II.4b), without additional approximations. However, intermediate approxi-
mations generally involve severe assumptions and idealizations in other
respects and should therefore be regarded as model calculations. Most of
the subsequent sections deal with such models, which were designed to

100 I I . THEORY

elucidate one or another specific detail of the theory, rather than to produce
results that could be compared directly with experiments or be used in the
various applications of line broadening theory. (The relaxation theory
[52, 113] as applied to hydrogen lines is an exception.)

Before discussing any of the intermediate approximations in detail, one
may want to speculate about the minimum requirements for such models to
be at all realistic. First of all, they must consider the perturbing field as
some stochastic function of time, the field being the key quantity in view
of the dominance of dipole interactions. As emphasized by Frisch and
Brissaud [114], this stochastic function must resemble the actual statistical
field fluctuations at least in two respects in order for the model to reduce
properly to the two extreme limits. Namely, the function used for the model
calculations instead of the real perturbing field must have the same auto-
correlation function (see Section II.5a) to give the impact limit and the
same field strength distribution function to give the quasi-static limit. This
still leaves considerable freedom in the choice of the theoretical model,
allowing on one side simplifications for otherwise very difficult calculations
of, e.g., lines with forbidden components, but also introducing ambiguities.
In practice, these may not be any more serious than the limitations imposed
by the failure of, e.g., the so-called phase shift limit (Section II.4a) and
the relaxation theory (Section II.4b) to contain the entire field strength
distribution function, or of the theory due to Kogan [15] to encompass
the autocorrelation function. However, these limitations can be translated
into validity criteria, while the ambiguity of mathematical models seems
more difficult to assess. Be this as it may, it is well to remember at this
point that, in some sense, all "practical" theories going beyond pure
Lorentz or Holtsmark profiles are based on models, notably also those
involving Eq. (101) or its relaxation theory generalization. But then,
invention and study of models has always been a powerful method in
theoretical physics.

II.4a. The Phase Integral Method

An approximate solution of the time-dependent Schrödinger equation
for the evolution operator of the radiator (see Section II.3), which must
not be confused with the transition matrix of Section II.3e, namely

iht = (H + U)t, (230)
is (for diagonal elements) the adiabatic approximation

(i\11 i> = exp [- ΐ (Ed +Σ[Ο WE^EÎ *)] ' (231)

I I . 4 . INTERMEDIATE APPROXIMATIONS 101

provided the diagonal matrix elements of the interaction Hamiltonian U
vanish and frequencies characterizing its time variation are much smaller
than | Ei — EV \/h. (This requirement of course excludes hydrogen and
similar lines.) Also, the off-diagonal matrix elements of U must be small
compared with the inherent level splittings. Equation (231) can be verified
by substitution into the components of Eq. (230),

ih(i \i\i) = Ei{i \t \i) + Σ <i | U | ï) {%' | t | t>, (232)

i'

ih{i' | 11 i) = Ev{i' 11 \i) + Σ <*'l U I *"> <*'" I * I *>» (233)

i"

for diagonal and off-diagonal ^-matrix elements, respectively, and the
off-diagonal matrix elements are seen to be approximated by

(234)

They therefore remain negligibly small under the above assumptions.
To make further progress [12, 51a], one now splits U(s) into contribu-

tions Uj(s — Sj) from individual collisions occurring at Sj and neglects all
cross terms / 9e j in the product required here, i.e., assumes

I(i \U(8)\ i'W = Σ I<*· I Uj(s - Sy) 11'>|*. (235)

3

This assumption of "scalar additivity" is of course not really justified in
case of Stark broadening, because we then have

(i | U(s - Sj) | i') = -e(i | r | %') - Fy (236)

in terms of the field strength produced by the j t h perturber. Strictly
speaking, these field strengths should therefore be added vectorially to
obtain (i \ U(s) \ i'). (The reader will note that we forsake at this point
the reduction to the exact quasi-static limit.)

However, the resulting cross terms do not contribute to the average of
(i | t | i) over perturber configurations in two limiting cases [51a]. For
small s, the probability of having two or more collisions overlapping in
time is vanishingly small, and there is thus no problem with cross terms.
This is obvious also for another reason: this situation corresponds to the
nearest-neighbor, quasi-static approximation. In the other extreme, namely
for times much exceeding the duration of individual collisions, the time
dependence of the average of (i \ t \ i) can be discussed along the lines of
Section II.3, where the corresponding quantity in the interaction repre-

102 I I . THEORY

sentation was considered. The differential equation corresponding to Eq.
(94) is, in the present case,

Ìh(Ì | t | i)av

-s£W-i(*-+çrra*)H.

X (i | t | î ) a v

= {E'+*2:/'h(-iǣ"^^*)-1]}<'"|i>-

(237)

if As can be chosen much larger than the duration of a collision but much
smaller than time intervals between collisions. (The index j now designates
all parameters of a given kind of collision occuring with frequency fj.)
Cross terms f 9^ j clearly do not arise in this case, and were the relevant
s values in the Fourier integral for the line shape indeed much larger than
durations of individual collisions, one would obtain Lorentz profiles char-
acterized by the adiabatic approximation to the impact broadening, namely

(i | 3C|i') = ih(i | φ | ΐ ' )

(238)

This is the so-called phase shift limit of Lindholm [80] and Foley [81],
which, for monopole-dipole interactions (quadratic Stark effect), has been
worked out both for neutral [51a] [see Eq. (263)] and charged radiators
[101].

Returning to the question of scalar additivity, no substantial errors from
this assumption are therefore expected if the impact approximation remains
valid sufficiently far into the wings so that the transition to the quasi-static
approximation occurs under conditions where only nearest neighbors need
be considered. Should this transition region be too close to the line center,
one may argue that since scalar additivity does not cause any errors in the
impact limit, actual errors from this source should be smaller than the
difference between quasi-static profiles calculated with and without this
assumption. However, this theory is exactly valid and really useful only if

I I . 4 . INTERMEDIATE APPROXIMATIONS 103

the transition occurs on the line wings, where it then agrees with Holstein's
correction [17b] to the quasi-static limit.

To treat intermediate situations within the adiabatic and scalar addi-
tivity assumptions we now follow Anderson and Talman [12], and write
the following, using Eq. (231) and taking all perturbers to be equivalent:

exp[(z'/ft)i£i$] (i\t\ i ) a v

ifr / i r f'\(i\Ui\i')W\\

» { È e x p [ - t P y ( s - a/) ] } = Lxpl-iP^s - 8l) ] } "

\j=l ) av \ / av

= [1 + {exp[-iPi(e - * ) ] - l}av>
—> exp[n{exp[—iPi(s — Si)] — l } a v ]

= exp 2TNV I dsi I pi dpi {exp[—iPi(s — Si)] — l } a v . (239)

(The probability of finding a perturber moving along a straight path whose
time of closest approach and impact parameter are in Si, Si + dsi and
Pi j Pi + dpi is given by the ratio of the volume 2πν dsi pi dpi to the total
volume V.) The various collisions are therefore assumed to be statistically
independent, i.e., relevant impact parameters pi are assumed to be much
smaller than the Debye radius PD . Also, Pi was assumed to be small on the
average in taking the limit n = NV —> °o. (There still remains an average
over perturber angles or radiator magnetic quantum numbers.) Since the
volume over which perturbers are reasonably independent of each other is
of order V = 47TPD3/3, phase shifts caused by perturbers with pi « PD must
accordingly be vanishingly small. Usually, this additional requirement is
met. (Remember that hydrogen lines had to be excluded from the present
discussions.) Otherwise, the perturbing ions should be replaced by
"dressed" particles which are not correlated (see Section II.5a). Often, the
requirement that instantaneous perturbations be smaller than inherent
level spacings is more restrictive, i.e., using Eq. (236) with F « e/p2, that

(e2/p2) \(i I r | i')\ « (nW/mZp2) < | Ei - Ε{· \ (240)

or

p > [(rc2/Z) ( 2 E H / | Ei - Ei. |)]1/2a0 (241)

104 I I . THEORY

for the second-order perturbation theory used here to be valid. Since p must,
however, at the same time be smaller than p « 22?HGO | EÌ — E{> |_1, where
quadrupole interactions would become important, Eq. (239) is useful only
when the relevant impact parameters fulfill

\Z | Ei - E* \) α0 Ί Ei - Ev \ ' K]

always assuming that there are no substantial cancellations in the actual
sums over intermediate states i'.

Instead of averaging exp[—t'Pi(s — Si)] in Eq. (239) over angles, which
could be accomplished analogously to the average in the case of electron
impact broadening (see Section II.3ca), it is customary to employ the
average of Pi(s — Si) in the exponential, i.e., using Eqs. (236) and (239),

{Ρΐ(β - Si)}av

ha0 ^ max(tf,Z<Q / I r I V f*
= 3^ Ç (2I, + υ {Ei _ ^
^ | -1 ιή ]o i n* - *) i2 *

Ξρί' ^s = _Ç_ Γ _! A* - vsA Plv(s - si) T
Pi2 + v*(s-Sly\Q
Λ [Pi2 + *>2(* - *ι)2]2 2Ρι3ί>Γ V Pi /
(243)

for singly charged perturbers on straight classical paths. This treatment of
the angular average is correct for U = 0 under all circumstances, neglecting
spin as usual, but should not cause very large errors in any case. (The
impact and quasi-static limits turn out to involve C2/3 and C4/3 respectively;
i.e., the linear average of C is a good compromise.)

Returning to Eq. (239), we therefore have to calculate

u&v(s) = exp[m(i/h)Eis'] (i\t\ i ) a v

2rNv I dsi I pi dpi

x h (-*f.w+û -,)?)-']}■ (244)

I I . 4 . INTERMEDIATE APPROXIMATIONS 105

which is seen to depend only on a single parameter, namely, for our - ^ τ - 4
polarization forces,

h = NC/v, (245)

if we substitute [12b]

pr = {C/vY'h, (246)

VSl = (C/v)mz, (247)

vs' = (C/v)l,3u, (248)

vs = (C/v)l/3y. (249)

In terms of the new variables, we have

Mav(s) = e x p [ - A *(?/)], (250)

Hv) = ~2π C dZ /." "dT [βΧΡ (-*' f. [r« + ( Î - *)>]') - '] '

(251)

where wav is essentially the correlation function C(s) used in Section II.1.
According to Eq. (6), a line profile in terms of reduced frequencies

Ò = Aw(C/v*)m (252)

can then be obtained from

lk(5) = ( 1 / T ) Re Γ e x p [ % - λ φ(y)] dy = WC)1'* L ( w ) . (253)

[Note that u&y fulfills the usual condition u&v( — s) = w*v(s)·]
The key quantity therefore is the function ψ(ν), for which analytic

approximations will now be derived both for small and large y. [Corre-

sponding expressions for Van der Waals interactions were given by

Anderson and Talman [12b]. Their results should be more accurate than

those of Lindholm [80b], who idealized the behavior of the inner integral

in Eq. (251).] To proceed for small y, the variables of integration r and z
(cylindrical coordinates) in Eq. (251) are changed to R(l — x2)1/2 and Rx}
respectively (spherical coordinates with x being the cosine of the polar

angle). This yields

ψ(ν) = 2ττ Γ R2 dR f dx [1 - e x p ( - t ï ) ] (254)

106 II. THEORY

with

ir_ . du

θ Rß44JΛ0 [π1 -- (2ux/R) + (M2/ß2)]2

~ « ; ü « - «' -"r1 /""Γ1 + 4 +ux (12x, 2)u2 + Ί
1 / rc 12a:2 — 2 \
(255)
-*{v+2R*+-ür-*+-·)'
The next step is to replace Ä2 dR by a differential form in 0, namely

-«-^-werhi'^-è)^···]}
-werhi-^x-î)·^—]}
- - 5* [* -χθ1v'4+ ( T " 0 *1/y/2 + ' ' ' ]de-

(256)

This can be integrated over x, i.e.,

/ dx ß2 dß = - Ι ^ [l + 1 Ö'V2 + · · · ] dô, (257)

and ψ(ΐ/) becomes the single integral

φ(ν) = 7Γ ΓάθΙ(ν"*/θ·"<) + Ä(2/9/4/Ö3/4) + · · · ] [ 1 - e x p ( - # ) ] .
(258)
•'ο

Taking the limit 0 —» <χ> is correct to this order in y because, going back to
Eq. (254), the exponential averages out for large Θ and therefore small
R < R', so that the contribution to }p(y) from small R is of order Rn. Now
R' must be somewhat larger than y for the above expansions to be valid,
and the error committed by extending the 0 integration into the R < R'
region is thus seen to be some multiple of y3, making it pointless to continue
the expansion in Eq. (258) to higher orders. The remaining integrals are

I I . 4 . INTERMEDIATE APPROXIMATIONS 107

of the form

Γ (d0/0l+<)[l - e x p ( - Ä ) ] = ri'/(g T(g) sin wg), (259)
•'ο
and the small y analytic representation is therefore

,(„) = MMym + ***** + [(,3)]. (260)

ΨΚυ> 3 Γ(3/4) 8in(3r/4) 6 Γ(1/4) sin(7r/4) Τ LKU >Δ Κ '

The behavior of tp(y) for large y may be obtained from Eq. (251) after
the substitution u — z = u' in the exponent,

HV) = - 2 . / _ + ; * [ r dr [exp ( - < f_~° j ^ f ^ ) - l] . (261)

Now the bracket is vanishingly small for practically all z values except
those fulfilling 0 < z < y (giving J i " dz « y)i whereas for almost all of
the latter, the integral in the exponent may as well be taken over the
interval — <*> < u' < <*>. A good approximation is therefore

♦<„> « - * , f· r * [«p ( - /;; ^^) -1]+c+(Q)

_ihrjr,*[i-ep(-i<)]+c+(Q)

2-2/3^8/3^2/3^ + ^ 3ö4 y «"+ //A\ (262)

Γ(2/3) sin(27r/3) \ -,

using the transformation 7r/2r3 —» 0 and Eq. (259) to evaluate the integral.
(The constant term C in the large y expansion was obtained from Anderson
and Talman's general expression [12b] by the application of L'HospitaPs
rule.)

The two analytic approximations to the function yp{y) are shown in Fig.
10, and it is seen that they approach the curves obtained by numerical
integration very closely. These curves were then used to calculate reduced
profiles from Eq. (253), which are shown in Fig. 11 for various values of
the parameter h. As expected, for Λ <3C 1 (necessitating large y), these
profiles approach simple dispersion profiles of width and shift [51a] (in
the δ scale)

Wi + id5 = 2"2/3π8/3ζ2/3Λ/[Γ(2/3) ΒΪη(2ιτ/3)], (263)

108 I I . THEORY

FIG. 10. Correlation function ψ(ρ) for quadratic Stark effect from the Anderson-
Talman [12] theory. Also shown are large y and small y (real part only) analytic approxi-
mations which lead to impact and quasi-static (wings only) limits. For the imaginary
part, our numerical results practically coincide with the small y limit.

which follow from Eq. (262) and correspond to Lindholm [80] and Foley's
[81] result, i.e., to Eq. (238). However, for large h (small y), we do not
quite recover the quasi-static profile (except on the stronger of the wings),
because the assumption of scalar additivity is then not justified (see the
discussion earlier in this section). The proper quasi-static profiles in the
δ scale are, rather,

Z8(<5) = ^(2Μ2/Ψ/2)~1 H(ô1/2/2M2/s)} (264)

where Η(β) is the Holtsmark distribution as discussed in Section II.2a.
Such a profile is also shown in Fig. 11 (for h = 1). For δ > 2, it agrees
reasonably well with Ζ(δ), which in turn, for ò > 1, almost coincides with
the quasi-static profile (not shown) as calculated from the assumption of
scalar additivity, i.e., using the first term in Eq. (260) only. Fortunately,
all these profiles are therefore not too different from each other for h « 1,
which suggests [51a] that errors incurred from the above assumption are
usually small under conditions requiring consideration of the time de-
pendence of ion-produced fields.

It is not difficult to extend the treatment discussed in this section to
perturbations of the lower level of the line by simply taking the difference
of the appropriate h parameters. However, ions broadening ion lines would

I I . 4 . INTERMEDIATE APPROXIMATIONS 109

FIG. 11. Reduced line profiles 1(δ) for quadratic Stark effect from the phase integral
method of Anderson and Talman [12]. These profiles are labeled by the value of the
parameter h (see text). Also shown are the impact (Lorentz) profile for h = ^ and the
quasistatic (Holtsmark) profile for h = 1.

require a different calculation because of their hyperbolic paths, and the
various auxiliary approximations made above must also be kept in mind.
Especially restrictive is the condition imposed by Eq. (242), which of
course applies to all quadratic Stark effects. (See also Section II.3f, how-
ever.)

To summarize, the method discussed here may be useful for the cal-
culation of ion broadening of isolated lines in intermediate situations, but
normally cannot be applied to electrons, because for them the adiabatic
approximation tends to be invalid. Since ion broadening of such lines is not
very important, the phase integral method has actually not yet had many
applications in Stark broadening, but it may find some use when distant
line wings are important (see Section IV.4c). In this case, it agrees with
Holstein's dynamical correction [17b], coefficients of which are tabulated
in Appendix IV.a.

In some respects, almost the opposite can be said of the intermediate
approximation to be discussed in the following section. This "relaxation
theory" is particularly suited for the electron broadening even of over-
lapping lines in intermediate situations. Moreover, since electron broad-

110 I I . THEORY

ening of hydrogen and similar lines is important in the transition regime,
there are numerous applications. However, because it does not contain the
entire field strength distribution function, it cannot fully describe the
transition from impact to quasi-static broadening either.

II.4b. Relaxation Theory

Line broadening through collisions may be considered a relaxation
phenomenon [13, 14, 52, 113] in the sense that the radiators are, after their
initial excitation, trying to equilibrate with a heat bath consisting of the
perturbers. The asymptotic (long time) approach to equilibrium as de-
scribed by the autocorrelation function introduced in Section II. 1 is of
course always governed by a simple exponential of the time, corresponding
to the impact approximation (see Section II.3), also called impact theory in
the present context. Such exponential behavior certainly prevails after a
long time has elapsed compared to the duration of any single collision, and,
in the case of overlapping weak collisions, also after the passing of a suitable
correlation time such as the inverse of the electron plasma frequency
[6, 7, 51a]. Howrever, before reaching this asymptotic time dependence, the
correlation function reflects much more detail of the radiator's evolution
during individual collisions. Such a situation wTas discussed in the preceding
section for the case of quadratic Stark effect, and the relaxation theory
[13, 52, 113] will be shown below to be in some respects a generalization of
the phase-integral method [12], at least as far as actual calculations are
concerned. (In regard to the formal developments, the newer theory is
much less restricted. See also papers by Dalenoort [115a] and Zaidi [116]
and especially Bezzerides [117], Dufty, and Lee [118].)

Although a completely quantum-mechanical formulation is possible
[13, 113d], we shall follow here the classical path version of the relaxation
theory [52]. This is no serious limitation, because truly quantum-mechani-
cal effects are significant, if at all, only for isolated ion lines (see Section
II.3e), for wrhose electron broadening the impact approximation is almost
always sufficient. (Another effect sensitive to quantum-mechanical correc-
tions, namely the plasma polarization shift discussed in Section II.5b, is
usually eliminated at a rather early stage from both impact and relaxation
classical path theories.) For simplicity, we will discuss the "one-state" case,
i.e., correlation functions as in Eq. (87) or, using also Eq. (88),

C(s) = T r D e x p [ - ( t 7 Ä ) Ä e ] t i a v ( e , 0 ) = Tr Z> *av(s, 0). (265)

(The "two-state" generalization is analogous to that in the impact theory,
but, as discussed at the end of Section II.3a and by Griem [64b, c], there is
some ambiguity and lack of "unification" in this case.) The main task

I I . 4 . INTERMEDIATE APPROXIMATIONS 111

therefore is to calculate

uav(s, 0) = f Q(r) u(r; s, 0) dr = f P ( r , s) dr, (266)

where Q(r) is the normalized distribution for the 6-n dimensional vector r
consisting of initial coordinates and velocities of all n perturbers, and
u(r; s, 0) is the radiator's evolution operator for a given perturber con-
figuration r. This operator is the solution of Eq. (91) for the initial condi-
tion u(r; 0, 0) = 1, with U = U(r, s) being the interaction Hamiltonian.
The quantity

F(r,s) = Q(r)u(r;s,0) (267)

accordingly obeys

ih (d/ds) P ( r , s) = [/'(r, s) P ( r , *), (268)

and the problem is to transform this dynamic equation into a relation

for u&v(s, 0) = / P ( r , s) dr.
To accomplish this transformation, and to eliminate irrelevant portions

of the total information, a projection operator [52, 119] is introduced

through

P/(r) = Q(r) j V ( r ' ) dr', (269)

and P ( r , s) is written as

F ( r , s ) = P F ( r , s ) + (1 - P) P ( r , s) s ^ r , « ) + F 2 ( r , e ) . (270)
[Voslamber [113] uses, instead of the projection operator formalism, a

BBGKY hierarchy method, which leads to the same result for u&v(sy 0),
etc.] Because of

Λ(Γ,β) = P F ( r , s ) = Q(r) JF(T',8) dr' = Q(r) u a v ( s , 0 ) , (271)

the function F\(r,s) then contains all required information. Dynamic
equations for the new functions are derived by applying P and (1 — P)
respectively, to Eq. (268),

ih (d/ds) F1(ry s) =P t/'(r, s) [ ^ ( r , s) + P2(r, s ) ] , (272)

ih (d/ds) F2(r, s) = (1 - P) U'(r, s) [ * Ί ( Γ , S) + P2(r, e ) ] . (273)
The second of these equations has the formal solution

P2(r, s) = - (i/ft) f G(r, β, *') (1 - P ) U'(r, s') i \ ( r , e') de' (274)
•'ο

112 I I . THEORY

in terms of the Green's function

G(r; s, s') = exp Γ - (i/h) f (1 - P) E/'(r, s) d e l , (275)

in which the usual time ordering is understood. [This- solution can be
verified by substitution into Eq. (273) since

G(r;s,s) = 1,

(d/ds) (?(r; s, β') = (1/ih) (1 - P) U'(r, s) G(r; s, β'),

Λ(Γ,Ο) =0,

the last relation following from F(r, 0) = Q(r) according to Eq. (267) and
Fi(r, 0) = Q(r) according to Eq. (271).] Substitution of Eq. (274) into
Eq. (272) then yields as dynamic equation for the relevant function
Fi{r,8)

ih (d/ds) FI(T, S) - P £/'(r, s) Fx(r, s)

- (i/h) f PU'(r,s)G(r]8,sf) (1 - P)

X C7,(r,s,)ί7l(r,s,)ds^ (276)

Using Fi(r, s) = Q(r) u&v(s, 0) from Eq. (271) and J Q(r) dr = 1, this
can be integrated over r, resulting in the desired equation for u&v(s, 0),
namely

ih (d/ds) u&v(s, 0) = {[/'(r, s)}
av ^ a v (e,0)

- (i/h) f {U'(r,s)G(r;s,s') (1 - P)

X U'(r,8')}„u„(8',0)d8'. (277)

[Note that

f Pg(r) Q(r) dr = f Q(v) dr f g(r') Q(r') dr' = \g(v)}av .]

At this point, the average perturbation is assumed to vanish, i.e., the
plasma polarization shift from "initial correlations" is neglected. The first
term on the right-hand side and the P operator under the integral thus
disappear.

[P U'(r, sf) Q(r) = Q(r) f C/'(r', O Q(r') dr' = Q(r) {£/'(r,s') Jav = 0.]

I I . 4 . INTERMEDIATE APPROXIMATIONS 113

Furthermore, because of the stationary property, namely

{U'(8) U'(S') - - - t/'(e<»>)}av
= 6χρ[(ί/Α)#β<»>]{Ι/'(β - «<»>) U'(8' - s(n)) · · · Γ7'(0)}.ν
X expl-(i/h)Hs^2

and the relation

G(r; 8, s') -> expl(i/h)Hs'~] (?(r; s - s', 0) e x p [ - (Z"/Ä) ίϊβ'], (278)

which can be proven by expansion and use of the stationary property,
the integral term also can be simplified. The dynamic equation for

*av(s, 0) = exp(—(i/h)Hs) u&v(s, 0)
in this way becomes

ih (d/ds) tav(s, 0) = H *av(s, 0) - (i/h) f exp[-(i'/Ä) H (s - s')]
• •'ο

{U'(r,8-8')G(r]8-8\0)U'(r,0)}„Uy(8',0)d8'.

(279)

Instead of solving for £av(s, 0) and then taking the Fourier transform
required to calculate the line profile from Eqs. (6) and (87), it is possible

to obtain the Fourier transform of t&v(s, 0), i.e.,

F (ω) = ί exp(icos) tav(s, 0) ds, (280)
•'o

directly by taking the Fourier transform of Eq. (279). Because of £av(0, 0)
= 1 and t&v( °°, 0) = 0, the transformed equation is

-ih + hœF(o>) = HF(œ) + £(co)F(co)

= HF(œ) - (i/h) Γβχρ[(ί/Α)(Αω - # ) s ]

. {U'(r,s)G(r;s,0)U'(r,0)}&vdsF(u>). (281)

[To evaluate the last term, s' is replaced by a new variable s — s' in all

but the £av(s', 0) factor, cos is written as cos — cos' + cos', and the integration
over s — s' is performed first using the convolution theorem.] According

to Eqs. (6), (87), (280), and (281), the line shape is finally

L(co) = - (h/π) Im Tr D[hù> - H - J^co)]"1, (282)

always using the lower (unperturbed!) level as the origin of the frequency
scale.

114 I I . THEORY

Comparison with Eq. (98) now shows that the relaxation theory is
formally the same as the impact theory, but replaces Baranger's [1-3]
3C or Kolb and Griem's [4] ιήφ, which were constant operators, by the
frequency dependent operator £(ω), namely

£(«) = -(i/Ä) Γβχρ[(ζ7Α)(Αω-#)δ]
•'ο

• j t / ' ( r , s) exp| - (i/Ä) / V ( r , s') ds'|î7'(r, 0 ) | ds (283)

from Eqs. (275) and (281). [The P operator in the Green's function can
also be omitted because of the assumed vanishing of {f/'(r, s) }av .] Only
for frequency separations Δω from the line center (or from forbidden com-
ponents) such that |Δωτ| <<C 1, where r is the correlation time for the per-
turbations, can the first exponential factor in the integrand be omitted.
Then we have

£(«) « f "(d/ds) j e x p i - (i/K) / V ( r , β') de'ltf'(r, 0)1 ds

= j e x p | - ( i / Ä ) / " V ( r , s) de |J7'(r, 0)1

= lim (ih/T) { e x p | - ( i / Ä ) [ t/'(r, e) del - l i . (284)

T-*oo IL •'o J ) av

The second line again follows from {U'(r, s)}&v = 0; the third line can be

verified by expanding the exponentials, using stationarity and JV {U'(r, s) ·

U'(r, 0)} av ds —> / " {· · ·} av ds, etc. If we now assume perturbations to be

caused by statistically independent (quasi-) particles j colliding with the

radiator at well-separated times and with frequency / , · , we may also write

£(«) - ih Σ/y [exp(-(i/Ä) / + V ds) - l] = ih Σ/iOSy - Di

(285)

noting that the integral from — oo to 0 vanishes for collisions occurring at
positive times and replacing / Q(r) dr by Σ / J ' ^ · AS expected, this agrees
with the one-state version of Eq. (92) for the impact theory operator 0C.
(Any overlap in time of different collisions would cause errors only in terms
of fourth and higher orders in U/f i.e., "weak" collisions may occur simul-
taneously without invalidating the impact theory [6, 7, 51, 53].)

I I . 4 . INTERMEDIATE APPROXIMATIONS 115

Since the very general Eq. (283) is not amenable to direct calculations,
it has been necessary to make (except for the "completed collision" as-
sumption) all the assumptions leading from Eq. (284) to Eq. (285) also
in the relaxation theory, in which case Eq. (283) reduces to the "impact
approximation" version [52, 113] of the relaxation theory ("unified line
shape approximation"). In other words, { }av is replaced by n{ }i, i.e.,
the average for a single perturber times their total number, or

£(ω) « -^fex^(hœ-H)s^

. j l Y ( r i , s) β χ ρ Γ - -h f'win , s') de'] Win , 0 ) | ds
= n exp - (ήω — H)s\

• £ | β χ ρ Γ - If'ui'in , s') ds'] [ / / ( n , 0)1 ds

= - %- (A« - H) f~exp\\ (&* - H)s]

o

• | β χ ρ Γ - If'ui'in , s') ds'] W ( n , 0 ) 1 ds

= η(ήω — H) I exp - (fiw — H)s

•is{A-if_,u^>s')ds']lds

= -n(hw - H) - l- (fio, - H)* / " β χ ρ Γ - - (fio, - H)s\

. | e x p r - ^ y V i ( n , e ' ) d e ' j J ds

in (δω - HY /°°exp| ~ (fiw - H)s\{ui(n ;s,0) - l}i ds.
(286)
~h

116 I I . THEORY

(This "binary approximation" is also used in "exact" solutions [63c,
113c].) Here Ui(ri ; $, 0) is the evolution operator describing the effects of a
single collision, and the third line follows on integration by parts, using

e x p i - (i/h) flJSin , s) del = 0 and { W ( n , 0) U = 0.

(In evaluating matrix elements of <£, e.g., (a \ £ \ a'), (a' \ H \ a') should
be used only in the second factor, hw — H.) As in the preceding section,
the single particle average may be written in terms of impact parameters
pi and times of closest approach s\, leading to

n K ( n ; s , 0 ) - l}i = (n/V) f 2*PldPl j v dsl {· · - U = NF™(s)f

(287)

wrhere the remaining average is over velocities and angles (or radiator
orientations). The final formula is therefore

£ ( ω ) = - (i/h)N(ho> - HY Γ exp[(t/ft) («ω - ff) e] F(l)(s) de, (288)

which is valid as long as the average perturbation vanishes and "strong"
collisions do not overlap in time, meaning that, e.g., U\ — 1 and u* — 1
must not simultaneously deviate from zero by more than some small
fraction of unity.

However, Eq. (288) is not subject to the additional condition | Δω r | « 1
required in the impact theory (r « co"1), i.e., it remains valid beyond
Δω « ωρ , the electron plasma frequency, provided this portion of the
profile is not influenced by overlapping strong collisions. But for a collision
to be strong, matrix elements of the interaction Uj multiplied writh the
duration (~PJ/V}) of a given collision must be comparable to or larger
than h, which is equivalent to saying that their effects could at least be
estimated from the quasi-static approximation (see Section II.2). Accord-
ingly, Eq. (288) should certainly be used only if the transition to the quasi-
static broadening occurs well on the wings of the line, where the nearest
neighbor approximation [6] discussed in Section II.4d is valid as well.
(This is practically always true for electrons as perturbers, but not for ions
causing linear Stark effects. The latter case will therefore be discussed
separately in the following section.) Except for a normalization factor, the
relaxation theory should deviate from the impact theory only near or
beyond ωρ , and no large differences should occur between calculations
based on Eq. (288), on the one hand, or on the impact theory as modified

I I . 4 . INTERMEDIATE APPROXIMATIONS 117

by the Lewis cutoff [65], which follows [6, 13, 116] from the relaxation
theory in the one-electron, weak collision approximation. The essential
advantage of the much more involved relaxation theory calculations is the
preservation of normalized profiles, while the impact theory with the Lewis
cutoff (which allows for the actual duration of collisions) violates the
normalization, although only by small amounts as long as ωρ falls well
beyond the half-width. A further advantage is the smooth, but perhaps
numerically not always quite correct (see also below), transition to the
nearest-neighbor, quasi-static result if U\ is calculated to all orders in the
Dyson series [63e, 113c]. (In a previous estimate [120a], this transition
was achieved by simply adding a quasi-static contribution from low velocity
electrons; see also Section IV.4a.) The important point is that calculations
according to Eqs. (282) and (288) are valid and, at least potentially also
worthwhile, if the transition to a quasi-static profile behavior occurs well
outside the half-width and ωρ is near or even within the half-width. The
latter condition implies for hydrogen lines whose lower state broadening
is negligible and whose upper states have principal quantum number n,

a0Nm > 0.1/n4 ; (289)

the former implies

(aoN^y « O.likT/Em*), (290)

using Eq. (16) and the corollary of Eq. (17), respectively. These conditions
are compatible for practically all N and T, and Eq. (290) is met for many
experimental conditions. Still, differences between the two theories remain
small, because the effective impact parameter cutoff [59a, 66] of the
impact theory over much of the line profile is then determined by the quasi-
static splitting from the ion field, i.e., is of order pmax ~ v/w or a factor of
about kT/n2hw larger than the minimum impact parameter pmin estimated
by Eq. (111). This factor has to be large, as can be seen from Eqs. (16)
and (290), which yield

Pmax/Pmin « (kT/24Enn4) (a0iV1/3)-2 » 1, (291)

if the present version [52, 113] of the relaxation theory is valid. Therefore,
most impact theory profiles (see Appendix I ) , which depend only loga-
rithmically on Pmax/pmin , do not differ very much from those of the relaxa-
tion theory [52, 113b, 113c, 121], which effectively replaces the half-width
in the cutoff by some combination of the plasma frequency (also of order w)
and the frequency separations from the various Stark components. Oc-
currence of any substantial deviations indicates either that neither of the
two theories is really valid or that the calculations differ in details, e.g., in

118 I I . THEORY

Debye shielding corrections or in regard to quadrupole interactions. The
only exceptions, besides the core of Ηα , are the line wings, where use of some
frequency-independent cutoff [59, 61] leads to an overestimate of the
electron broadening beyond ωρ . Asymptotic formulas are then preferable
and indeed agree here better with the relaxation theory and measurements.
(The differences for Ha indicate [64b, c] that the "two-state" generalization
[121d, f] of the relaxation theory needs revision.)

The above discussion shows that not much improvement over impact
theories employing suitable cutoffs can be expected for the electron broad-
ening of hydrogen lines, etc., unless overlapping strong collisions are treated
properly. Such treatment would seem necessary for electron broadening of
hydrogen lines whenever Eq. (290) is violated, which tends to be the case
for the highest members of the various hydrogen line series still observable
at a particular density. In other words, for them the transition to quasi-
static electron broadening happens near the half-width, and a theory [15]
extending the quasi-static rather than the impact approximation is called
for (see the following section). Such treatment gives [120b], on the line
wings, a negative correction to the quasi-static result, while that from the
relaxation theory [121] is positive there, perhaps because the condition in
Eq. (290) was not sufficiently met or because of "nonadiabatic" effects.

For details of calculations of the function F(1)(s) in Eq. (287) and (288)
and for resulting profiles, the reader is referred to the literature [52, 113,
121] and reminded that these calculations are rather similar to those dis-
cussed in Section II.3a for the implementation of the impact theory.
(References to some other theoretical approaches are given at the end of
the following section.) There is but one essential difference: In the relaxa-
tion theory, we deal with time integrals over binary collisions taken between
finite limits; in the impact theory, these limits are extended over the entire
duration of such collisions, and the breakdown of this assumption of
completed collisions must be taken care of by a cutoff in the integral over
impact parameters. Finally, while the above applications of the relaxation
theory have been to the electron broadening of hydrogen lines (differences
from the impact theory should be similar for ionized helium lines, etc.), the
theory may of course also be used, e.g., for the ion broadening of neutral
atom lines through quadratic Stark effect. Using Eq. (288) and the quanti-
ties introduced in the preceding section, the profile is then seen to be given
by a "dispersion" profile with frequency-dependent width and shift
parameters (in the ò scale)

w(6) + i d(6) = -0% Γ exp(tty) Hy) dy. (292)

I I . 4 . INTERMEDIATE APPROXIMATIONS 119

FIG. 12 Frequency-dependent width and shift parameters in units of the parameter
h (see text) as functions of the reduced frequency separation δ from the unperturbed
position of a line broadened by quadratic Stark effect (after the relaxation theory ver-
sion of the phase integral method). Also shown are some analytical approximations (see
text).

Compared with Eq. (253), this is a considerable simplification. It can easily
be shown to reduce to the impact theory result for \f/(y) ^ y, the large y
approximation to \//(y) as given by Eq. (262). For large δ, w(ô) and d(ò)
both turn out to be equal to π2 | δ \l,Ah for ô's having the same sign as C,
corresponding to the nearest-neighbor quasi-static approximation, and
vanishingly small otherwise. These asymptotic results, from the first two
terms in Eqs. (260) and (262), respectively, together with intermediate
values obtained numerically, are shown in Fig. 12. Another, future, applica-
tion would be to the broadening of early members of the hydrogen line
series by protons at low densities.

II.4c. Dynamical Corrections to the Holtsmark Theory

In terms of the characteristic frequencies ωρ , v/p} and Δω discussed in
Chapter I, the relaxation theory may be said to have the most merit (for
hydrogen lines) in the case of the ordering ωρ < Δω < v/p, if p is taken as
the Weisskopf radius or strong-collision impact parameter p « hrft/mv, and
if in addition v/p » w is fulfilled. As one is generally interested in Δω > w,
violation of the condition v/p ^> w therefore implies Δω > v/p, suggesting
the quasi-static treatment for a first approximation in situations where the

120 I I . THEORY

implemented version [52, 113] of the relaxation theory breaks down
because of the predominance of overlapping strong collisions. However, in
view of the near equality of frequency separations Δω from the line center
and inverses v/p of the duration of collisions, it becomes necessary to allow
for the actual time dependence of the perturbing electric field as a dynamical
correction.

As long as this correction is small, it is sufficient [15,17b, 122] to evaluate
the general expressions for the line shape, namely Eqs. (6) and (10), by
employing "adiabatic" wave functions as a basis, i.e., for linear Stark effect
parabolic wave functions with the z axis in the direction of the instantaneous
field. Concerning the solution of the Schrödinger equation, Eq. (230), it is
also clear that only diagonal matrix elements of U need be considered, as
long as the initial field F(0) is not likely to change much in the times of
interest (t < \ Αω | _ 1 ) . This becomes obvious if the field is expanded into a
Taylor series, because off-diagonal ^/-matrix elements are then seen to
involve only derivatives of F(t), while the diagonal elements also contain
F(0). To evaluate corrections to the quasi-static approximation, it is thus
sufficient to use in analogy to Eq. (231) of the phase integral method for
shifted Stark components

(i 111 i) = exp I - (i/h) (EÌS + f (i\U\ i) ds\\, (293)

and to ignore off-diagonal ^-matrix elements. (Unshifted components
cannot be treated by this adiabatic approximation.)

Although there is actually an additional time dependence in the correla-
tion function from the rotation of the z axis for the basis functions, this
complication can again be ignored in the present context. Such effects
would occur on the time scale for significant field variations and may there-
fore be neglected with those corresponding to off-diagonal matrix elements,
as long as substantial phase changes explicitly accounted for by Eq. (293)
occur on a somewhat shorter time scale. Using these approximations and
Eqs. (6), (84), and (293), the line shape of a single Stark component char-
acterized by a linear Stark constant C can be written as

L(Aco) = (1/TT) Re f jexp i (àœs - C f F dt\\\ ds, (294)

where the average is over all fields F(t). At this point, a step is taken which
makes it impossible in practice to recover the correct autocorrelation func-
tion of the electric field. Namely, F(t) is expanded into a Taylor series, as

I I . 4 . INTERMEDIATE APPROXIMATIONS 121

is the ensuing correction factor of expp(Aco — CF(0)s], yielding
expUÎAws - C f Fdtjl

= βχρ[ζ(Δω - CF<°>)«] I 1 - iCFM 2* - iCF<»6J - iCF™24**

(CF<«)2- +

i d2
1 + 2- CFM dAto2 h 6- CF& ά—Αω* 2—4 CF™ d■ Δω4

- J (C7™>)2 - ^ - + - - · 1 expp(A« - CF<w)e], (295)
8 α Δω4 J

where Fin) denotes the nth derivative of the function F(t) taken at t = 0.
Significant phase changes are seen to occur in T^ « (TF/CF)1/2 ~
(TF/ACO)1/2, Τγ = TF(F) being a characteristic time for field fluctuations.
However, for the Fourier integral to be accurate, s must be allowed to reach
Δω-1 without invalidating the expansion. This gives (CF/TF)1/2 Δω_1 ~
(Aù)TF)~1/2 « 1 or, with the first condition, indeed ΤΦ<£Τ¥ , as long as
dynamical corrections are small, thus justifying the neglect of rotation and

off-diagonal matrix elements.

Integration over s transforms the remaining exponential factor in Eq.
(295) essentially into δ(Αω — CF(0)), which, on substitution into Eq. (294)
and integration over F(0), gives back the Holtsmark (quasi-static) pro-

file L8(Aco) or, with the correction terms, and only keeping the real part,

Ζ,(Δω) = L.(A«) + Ci (d*/dAo>*) 4*(/™>)2{^(2)}av

- i C (d*/d Δω4) 4TT(F(°>)2{ ( i ™ ) 2 } a v + · · · ] , (296)

the remaining averages to be taken at fixed Fw = Αω/C. [Since the F(1),
etc., do depend on F(0), the differential operators must be moved to the left
[123, 124] before performing the average over Fi0), and the factors
47r(F(0))2 arise from the integration over the directions of the vector F(0).]

It is convenient to introduce dimensionless variables through

β = Δω/cos = Αω/CFo = F/F0, (297)

122 I I . THEORY

with Fo (the Holtsmark normal field strength) defined by Eq. (36), and

T = ωρ* = l(2kT/mp)(F0/Zpe)J/2t « (2kT/mpy/2(t/r1)) (298)

so that the ß profile l(ß) = Ζ,(Δω) άω/dß becomes

l(ß) = Η{β) + (ωρ/ωΒ)2[* (d»/<W 4π/32{/3(2)}αν (299)
- i ( d 4 / ^ ) 47rß2{(^(1))2)av] + · · ·

in terms of derivatives with respect to reduced field strengths ß = F/F0,
averaged at fixed ß. [Note that for a single Stark component, the reduced
Holtsmark profile simply equals the Holtsmark distribution H{ß) of the
reduced field strengths as given by Eq. (38).] Kogan's approach [15] is
thus seen to give a power series expansion in COF/COS , the ratio of typical
fluctuation frequencies and Stark displacements, higher terms of which
could be obtained similarly to the second-order term derived here. However,
if these terms are important, deviations from adiabaticity and also rota-
tional effects are no longer negligible.

The remaining task is to evaluate {0(2)}av and {(0(1))2}av, the differ-
entiations being with respect to the dimensionless time variable r. Closely
related quantities were first obtained for the gravitational case [125],
assuming as throughout this section that individual perturbers act in-
dependently, i.e., that kT is much larger than mean interaction energies
between perturbers. In preparation of this evaluation, the first derivative
of the (reduced) field amplitude is written as

0<i> = 5 . £(1)0-1, (300)

which immediately leads to (301)
0(2) = g . cwß-1 + (0(ΐ))20-ι - (g . ^Yßr\

If a (stationary!) coordinate system with the x axis pointing in the direc-
tion of (5 (0) is now introduced, there result

0(D = ß?\ (302)

0(2) = ß™ + [ ( O 2 + (ßWlß-1. (303)

Furthermore, the ensemble average of ß^2) can be shown to vanish, while
those of (ß™)2 and (ß™)2 are equal to each other. To prove the former
statement, consider the contribution from a single particle ßx ~ x(t) r~3,
with x(t) = x + us, etc., and u, v, w being constants. The second derivative

I I . 4 . INTERMEDIATE APPROXIMATIONS 123

is then proportional to [ßu(xu + yv + zw)r2 + 3x(u2 + v2 + w2)r2 —
lòx(xu + yv + zw)2]r~7. Because of (u2) = (v2) = (w2) and (uv) =
(uw) = (wz) = (vii;) = 0, this vanishes for all spherically symmetrical
velocity distributions. Accordingly, Eq. (299) becomes

- ^ 4 ^ { ( ^ υ ) 2 Ι » ν ] + · · ·, (304)

i.e., we actually need only "fluctuation moments" involving first deriva-
tives, both at right angles to and in the direction of the instantaneous field.

To calculate these fluctuation moments, Chandrasekhar and von
Neumann [125] analyzed the joint distribution function for (5 and (5(1) = b,
namely

T7(ff, b) = (2ΤΓ)"6 jj e x p [ - * ( 0 · 5 + d - b)]A(j>, d) d*pd*a (305)
with A (9, d) in generalization of Eq. (33) given by

A(9} d) = lim \(1/Vr) / ϊ / β χ ρ [ ζ ( ρ - fc + d - ^ ) ] Λ · Λ > Τ . (306)

(The quantities 9 and d of course have nothing to do with impact param-
eters or cross sections.) Here (3i and bi are the (reduced) field strength and
its derivative from a single perturber,

ffi = r I r I"3, (307)
bx = v I r |-3 - 3r(r · v) | r |"5, (308)

x and v being single particle coordinate and velocity vectors multiplied
with (2π)1/2(4ΛΓρ/15)1/3 and (mp/2kT)l/2, respectively. Also, 1/Vr =
3/47rrfTlax « 0.238/n gives the probability of finding the perturber in any unit
volume of r space, while / is the corresponding probability in v space,

/ = ir-3/2exp(-|v|2), (309)

for a Maxwellian velocity distribution.
Fortunately, it is not necessary to calculate A (9, d), the Fourier trans-

form of W(§, b ) , in all generality. For example, for the (second) moment

124 I I . THEORY

of b in the field direction, we only require

uo2}.v = yV(5,b)6,«d%

= (2*·)-« JJf exp[-t(j» · ff + * · b>] A(p, d) Ò,2 d»6 d»p dV

- - ( 2 » ) - · / T e x p ( - t » · 5) «"(»,) ί(σ,) «(σ.) A(p, d) dVd'p

= - (2ir)"s j " exp(-tp · J) [(aU (», *)/βσ.*]ι.ι_» dsp, (310)

i.e., the second derivative of A (g, d) with roebsptaeicntstoforσχ{a(t/S|"d))*| }=»v0,,axnbdetinhge
the direction of Ç. An analogous relation
remaining problem therefore is to evaluate A(p, d) for small values of
I d |. Because of F71 JffcPr dh = 1, Eq. (306) may be rewritten as

A ( M ) = Hm [ l - (1/Vr) / / U - exp[t(» · fc + * · bi)J} / Λ · Λ>Τ
n—»00 L J

= exp ([-15/4(2π)3/2] j j {1 - « p [ i ( p · fc+ d · b i ) ] } / * r c f t \

(311)
noting that the density in r space is n/Vr = 15/4(2ττ)3/2. The exponential
of i(ô · bi) is then developed up to second order so that the desired deriva-
tive can be taken, namely

(312)

because the integral with bix vanishes. The second factor corresponds to the
Holtsmark case, Α(ρ,0) = exp(-p3/2) from Eq. (35). Using Eqs. (307)
and (308), again leaving out terms whose velocity average vanishes
(vxVy ,etc.) and employing vx2 = $v* and, from Eq. (309), v2 = / v2fdzv = f,

I I . 4 . INTERMEDIATE APPROXIMATIONS 125

the second derivative can be reduced to

& A, M 1 5 r/, , J A /.»·ΛΛ· ,,

( . g - r\ d*r

X exp

15
8(2τ)1/2

(313)

where the last line follows by integration over polar angles defined with
respect to 9. (Strictly speaking, we deal here with relative velocities, whose
distribution is not usually isotropie. But for electrons as perturbers, such
isotropy holds to a very good approximation. See below for perturbing
ions.) In terms of a new variable y = p/r2 and using

co 4 /"*

/ Vmin-*0 2Γ5/2 sin y dy = 2 ^ 2 - °- '/ymin ym sin y dy

l/ml.-tO 2rn _ 4 ArV*J ' ' l i m/i2.

pl/2 3 W ' (314)

the r integration can also be performed, yielding

£ A < » , 4) |,,u, = - ^ e x p ( - p - ) ( p - + 4 ^ P - )

= - g exp(-p^) ( l - ^ ) ρ-»\ (315)

(Note that the term involving rmax does not contribute after differentiation.)
The fluctuation moment from Eq. (310) therefore is

{wn» - dsb / 0 - 7)exp(-^ ·5 - >'/2) ? · (316)

If we now introduce polar angles with respect to (3, the desired fluctuation

126 II. THEORY

moment in the field direction becomes cos Θ - p3/2) d cos Θ p1/2 dp
{ ( O ' U = —45 jf (1 - cos2 Θ) exp(-ipß

45 / / ( 1 + -2 ^ j exp( -ipß cos Θ - p3/2) d cos 0 p1/2 dp

64ττ2

/45 Κ · I) ^-<-<"·»■»*

32π2

; /3~3/2 y/n (sin x - x cos x) e x p [ - (x/ß)m~] x—5/2 , (317)

where x = pß was introduced as a new variable after differentiating with
respect to ß. For the fluctuation moment at right angles to the field,
1 — cos2 Θ must be replaced, e.g., by 1 — sin2 Θ cos2 Φ or £(1 + cos2 Θ) on
the average over the azimuth Φ, i.e., there results

<*">·.--&/(.-1£)^-Ρ<-^*

45 f00
= β~3/2 / (x2 sin x + x cos x — sin x)

X e x p [ - ( z / / 3 ) ^ ] —dx . (318)

It is convenient at this point, still following Chandrasekhar and von
Neumann [125]], to define two new functions, in addition to the related
Holtsmark function from Eq. (38) and x —* χ/β, namely

G(ß) = (2/ir) Γ expl-(x/ß)3n]sinx (dx/xm), (319)

I(ß) = (2/») f e x p [ - ( x / ^ ) 3 / 2 ] ( s i n a ; - a ; c o s a ; ) (dx/x*'*), (320)
so that Eqs. (304) and (317)-(320) lead to [15]

(321)

I I . 4 . INTERMEDIATE APPROXIMATIONS 127

Because of the differentail equation for Ι(β), namely

ß (dl/dß) + f/ = G, (322)

the correction term can be simplified somewhat, and the relative deviation
from the Holtsmark result finally becomes

l(ß) - H(ß) \ ω 8 / H(β) 64 dß'iP LW WJ'
H{ß) \o>s/ Η(β) (323)

Here Δ(/3) is related to the function Six) introduced by Kogan [15]
through

Δ(/3) = 2*iWSiß) « 2.65(0), (324)

and using

(?~(2/π)1/2,

7~(î)(2/»)w,
and

#(0) ~ (γ)(2/»)1Λ/8-»Λ(1 + 5.107/r3/2 + · · · ) ,

there follows for large ßiß > 10) (325)
Mß)/Hiß) ~ -(5/64(8)[1 - (5.1O7/03/2) + · · · ] ·

For small /3 one estimates, from the small j8 expansions of H, G, and 7,

Aiß)/Hiß) « -(1/12/S»)r(i) = -0.223/18». (326)

The entire behavior of this ratio (times ß2) can be seen in Fig. 13, which
was partially drawn from Kogan's results. To estimate errors from using
the quasi-static approximation for the electron broadening of hydrogen
lines, Fig. 13 or the asymptotic result of Eq. (325), which, by the way,
agrees with Holstein's dynamical correction [17b], together with

/2Y kT
9 · 2.6 EK {aoNl«[n(m - nt) - ri (m' - n / ) ] } " 2
\cos/

kT (327)
— [αοΛΓ1/3(η2-η'2)]-2
EH

from Eqs. (59), (297), and (298), will generally be sufficient. (Here n,

n\, and n2 are the principal and the two parabolic quantum numbers for

128 II. THEORY

1.0

0.8

0.6

OX

0.2

0

-0.2

-0.4

-0.6

-0.8

-1.0

-1.2

-1.4

'"0 1 2 3 4 5 6 7 8 9 10
ß

FIG. 13. Characteristic functions A(ß) (after Kogan [15]) and Δ'(/3), multiplied with
β2/Η(β), for dynamical corrections to the Holtsmark profile Η(β) of a single linear
Stark effect component.

the upper state, the primed quantities those for the lower state.) Of course
the errors have to be small for the present approach to be valid, a situation
not likely to happen in the case of perturbing electrons, except on the far
wings where agreement with the relaxation theory should obtain unless
there is substantial Debye shielding, or near the Inglis-Teller limit. As
mentioned in the preceding section, there is actually still no such agreement
with the relaxation theory, dynamical corrections in the asymptotic regime
being in opposite directions [120b] even for lines that have no unshifted
component.

For ions as perturbers, (COF/COS)2 is smaller by the electron-to-ion mass
ratio, and errors from the quasi-static approximation estimated as above
will usually be very small. However, applying corresponding corrections is
not really justified, because anisotropies in (relative) velocity distributions
were neglected up to this point. Corresponding effects in the gravitational
case had again been evaluated by Chandrasekhar and von Neumann

I I . 4 . INTERMEDIATE APPROXIMATIONS 129

[125], who obtained in this way the dynamical friction between a particular
moving star and a system of randomly (in phase space) distributed field
stars. For isotropie distributions of both field and test star velocities, the
average of ß™ over both velocity distributions still vanishes, as can be seen
by a simple extension of the argument given following Eq. (303), with the
velocity there replaced by the difference between the two velocities.

However, the results for the averages of (0£1})2 and (0<1})2 + (ßll))2 are
different, and Eq. (323) must be replaced by [126]

Η(β) ~ W p LV + rnj Η(β) + mr H(ß) J (328)

with the function

Δ'(0) = (25/256π) (d*/dß*) [β-*Η™ + ψβ-'Η - 2//<1>] (329)

in terms of the Holtsmark function from Eq. (38), its integral

and derivative #<-i> = f H{ß') dß',
•'ο

H™ = dH/dß.

Also, (COF/WS)P is to be calculated using the actual perturber mass mp in
Eq. (298), i.e., multiplying Eq. (327) by the electron-to-ion mass ratio

m/mp , and mT is the radiator mass. (All kinetic temperatures are assumed
to be the same. If not, the mass ratios must be divided by the corresponding

temperature ratios.) For electrons as perturbers, mp/mT—>0, Eq. (328)
reduces to Eq. (323), as it should. The correction factor (1 + mJmT)
accompanying A(ß) originates from using relative instead of just perturber

velocities in Eq. (308), while the term characterized by Δ'(0) comes from

the linear term in the expansion of expp(d · b i ) ] following Eq. (311). Its

integral over perturber velocities and impact parameters is proportional

to the radiator velocity, and the second derivative of Λ(ρ, d) involves the

square of the resulting expression, whose average over radiator velocities

therefore is not zero.
The function Α,(β)/Η(β), again multiplied with β2, is also shown in Fig.

13. For large ßy is has the asymptotic expansion

Α'{β)/Η{β) ~ -0.514β-5/2(1 + 41.7/3"3/2 + · · · ) , (330)

and for small β, it behaves as

Α'(β)/Η(β) « O.299/02. (331)

130 II. THEORY

Figure 13 indicates that when radiator and perturber masses (and tem-
peratures) are similar, the two terms in Eq. (328) are comparable for all
but rather large ß values. In other words, corrections to the Holtsmark
theory from the source discussed in this section are often somewhat larger
than was suggested by the original work of Kogan. Finally, to make
corresponding estimates for ionized helium lines, etc., n2 — n'2 in Eq.
(327) should be divided by Z = 2, etc., while in the case of multiply
charged perturbers, one should multiply by the number of these charges
and interpret N as Np , the ion density. The reader will notice that the
present theory breaks down in the line center, has to be worked out for
each Stark component separately, and does not incorporate correlations
between particles. All this may explain why it has found so few applica-
tions. However, some recent criticism [127a] of the original theory [15]
can be refuted, because it was partially based on the use of perturbation
theory for (i\t\ i') instead of Eq. (293), and is therefore not valid in or
near the quasi-static regime. (Note that Lisitsa and Kogan [127b, c]
essentially confirm the early results [15, 17b, 120b], although deviations
from adiabaticity seem very important for La with its strong unshifted
Stark component.)

The theories discussed in this and the two preceding sections are unique
in the sense that they do yield quantitative estimates of the deviations from
the two extreme approximations to the general theory, albeit only in more
or less special cases. There have been other papers [117,118] by Bezzerides,
Dufty, and Lee on more general treatments which, as yet, have failed to
yield numerical results or to really separate effects from instantaneous
fields and field fluctuations (which are not independent of each other), but
should be consulted together with the literature [52, 64b, 113-116, 121] by
readers interested in remaining theoretical difficulties in this area.

II.4d. The One-Electron Approximation

If the impact approximation is valid somewhat beyond the half-intensity
points, the duration of a typical collision must obey r < url> and the effects
from different perturbers are additive in the effective perturbation Hamil-
tonian 3C (see Section II.3) for times less than about tir1. Therefore, con-
tributions to the profile well beyond these points, corresponding to times
shorter than url in the Fourier integral describing the line shape, and
accordingly also shorter than the interval between effective collisions,
must obviously come from perturbers acting individually, because the
intensity is then proportional to 3C. In other words, we may calculate these
parts of the profile by considering one electron (perturber) at a time,
multiplying in the end with the total number of electrons; under the above

I I . 4 . INTERMEDIATE APPROXIMATIONS 131

assumption, there will then always be a region of overlap between the
one-electron (Δω»ι#) and the impact approximation (Δω<£τ~ι). (The
reader will notice the similarity in this regard to the relaxation theory
calculations discussed in Section II.4b.)

The one-electron approximation as first discussed by Baranger [6] thus
has a wide range of applicability. However, its main virtue lies in its basic
simplicity, which allows a direct attack on the basic formula for the line
shape, namely Eq. (3), for dipole radiation. Writing this formula in the
one-perturber approximation and reducing it then to the more familiar
semiclassical or quantum-mechanical versions, e.g., of the impact approxi-
mation, is of considerable help in elucidating the nature of these approxi-
mations. Furthermore, in this process an entirely new phenomenon pointed
out by Burgess [128] (the interference between line and so-called perturber
radiation, i.e., bremsstrahlung) can be discussed for the first time.

Without much loss of generality, we may assume that interactions (other
than those causing pure Coulomb scattering) between perturbers and the
radiator in the lower state / of the line in question are negligible.
(Generalization to the "two-state" case wOuld be possible, in analogy to
the derivations in Section II.3.) The lower state of the total system is
accordingly described by the product (neglecting exchange) of plane wave
(or free Coulomb wave) functions | k) with the radiator wave function
| /). However, the initial states of the total system are built upon initial
radiator states %' and perturber states &J"t·» , i.e.,

| i'V) = Σ I i") I Κ-Λ (332)

| kfi"i') describing a (scattering) state for an electron of initial momentum
k' interacting with the radiator in state i' and leaving it in state i" at
momentum k. The general formula, Eq. (3), may now be written with Ω
designating the direction of k relative to k' and remembering that xa
stands for the total dipole operator:

£(«) ~ Σ Σ / «(«?· - £/s - M I Σ <*ί·v I k) «" i *. i/>
i',a J i"

+ <*/,' I *· I *> </1 /)l2 Or (dn/dk') dkf (dÛ/4r). (333)

(Proper normalization of the \ k) takes care of the k integral.) The dipole
moment of the whole system is seen to arise from two processes: In one of
them, the perturber interacts with the radiator, causing a transition from
state i' to state z", which is followed by the emission of the photon. In the
other process, the perturber emits the photon while interacting with the
radiator in state/. The latter process is usually neglected in line broadening

132 I I . THEORY

theory by using xa = xTa instead of xa = xra + xpa, xra and xpa being
radiator and perturber dipole operators, respectively. (The subscripts r and
p are of course superfluous in the above equation.)

Factoring the density matrix into an "atomic" factor g^ and a "per-
turber" factor (with the Boltzmann factor omitted for the time being) is
certainly in the spirit of the one-electron approximation, there being, most
of the time, no perturber-radiator interactions (except for the pure Coulomb
interactions in the case of ionized radiators, which are explicitly allowed
for by the use of Coulomb rather than plane waves in these cases). The
factor giving the density of perturber states is, as usual (i.e., neglecting
spin until later),

dn/dk' = {V/2ir2)k'2dk\ (334)

where V —> iV-1 is the volume available for each electron. At this point, we
recognize that Eq. (333) implies, e.g., the use of F~1/2exp(2k · x) rather
than the more conventional (2π)~3/2 exp(ik · x) perturber wave functions,
so that an additional factor (27r)6F~2 must be applied before making contact
with scattering theory. (See any standard text on quantum mechanics.)
With these factors and integrating over some small range of kf values, the
profile simplifies to

Ζ,(ω) « (2τ)*(ηι/ή)Νν Σ / I Σ <*<'"<' I *> <*" I *« l/>

i',a %"

+ <*;<Ί»«Ι*></Ι/>Ι,^Λ. (335)

There is also the energy conservation relation (from the δ function)

fua = Ei> + e' - Ef - e = ήωίΊ + e' - e (336)

in terms of radiator energies (Ei> and Ef) and initial and final kinetic
energies of the perturber (e' = h2k'2/2m and e = h2k2/2m).

The perturber functions obey the system of coupled Schrödinger equa-
tions used in "close coupling" calculations (see Section II.3e),

[ £ ψ + e' + E* - E<.. +{Z ~ 1 ) e 2 ] i vri.) = Σ t W " I *:■··<·>,

(337)

Ui"i>>> being the interaction potential {%" \ U | %'") (minus the Z — 1
monopole term) appropriate for a given pair of radiator states. The corre-
sponding equation for \k) has U = 0 in the "one-state" approximation.
Multiplication of Eq. (337) with (k \ and of the equation for (k \ with

I I . 4 . INTERMEDIATE APPROXIMATIONS 133

! &,'"<') an(i subtraction then results for k ^ k' in
(k | *;»,.> = (#,- - Ei» + ί' - ί)-1 Σ (k I £W<< I &,·<<<,·<>

= Α-»(ω - ω^ν)"1 Σ <* I Ui"i>" I Κ-Λ (338)

where Eq. (336) was used in the second expression. If | k' | and | k | obey

W(k2 - k'2)/2m « Ei - Ev = # W - fc'2)/2m,

the matrix elements may be replaced by scattering amplitudes [1, 3, 129].
With p as the range of the potential, this remains (for | Ak/k | < 1,
Ak ΞΞ k — ko) a valid procedure if

p | Ak | « mp | 2?i' — JE»/' + Λ(ω — ω,·'/) |/Ä2Ä; = p | ω — ω»"/ |/ϋ <3C l,

which, for a line centered at ω»"/ , is just the usual condition for the validity
of the impact approximation on the line wings and therefore says that
energy of the perturber-radiator system should be approximately con-
served. Within this approximation, Eq. (338) becomes, in terms of scat-
tering amplitudes (not to be confused with oscillator strengths),

(k | Αψν> « [-δ/(2τΓ)ν](ω - «rv)-l/<"i'(0), (339)

and the nearest neighbor, impact approximation line shape can be written as

L(co) « A-Nk> Σ f" | a ; a | / ) f " | X a | / ) * [f*..,WUMQ)dug,

- 4 * ^ ' Re Σ ^ 1 ^ 1 ^ /" <*;,. | xa | *>·/*»«· (0) <» ?,',

a.i'.i" « - «<"/ J

(340)

omitting a third frequency independent (continuum) term and using
To make contact with earlier discussions, we will now consider the case

i" = i>» = i ("isolated line"), for which the first term in Eq. (340) yields

Ζ,(ω) « ΙΝν'/2τ(ω - ω„)2] |(i \ xa | />|2 Σ *«'ΰ<> , (341)
i'

the σ,·»' standing for the various cross sections. [J |/»'<(Ω) \2 dû = σ»',·.]
Using detailed balancing [see Eq. (143)], we can replace σα^υ' by
ai'iCiV. Normalizing J^a |(i I x<* \f)\29i = 1 as usual, the impact approxi-

134 I I . THEORY

mation width defined by L(œ) ~ (w/v)(<a — cot/)~2 is thus seen to be
[6, 129]

Wi = $Nv Σ m , (342)

in agreement with Eq. (210) for the "one-state" case. [Actually, Eq. (143)
would give an additional factor (ν/ν') in Eq. (342), which can be omitted
if the implied Maxwell average is taken over v rather than v'.]

The second term in Eq. (340) gives something new, namely, as men-
tioned above, the interference effect [128] between line and continuum
(perturber) radiation. To assess this effect, the interaction Hamiltonian
will now be approximated by the dipole term according to Eq. (58), and
the perturber wave functions by free Coulomb functions, i.e., | fcJ",·')—>
| kf) bi"i> . The scattering amplitudes then become, in this Coulomb-Born-
Bethe approximation,

U>v « [(27r)2me2//i2] Σ <i" I xß \ i') (k \ xß/r* \ V). (343)

ß

Also, using the relationship between matrix elements of acceleration and
force [τηω2Χβ—-> — e2(Z — \)xß/rz~\, the dipole matrix element for the
perturber taken between Coulomb wave functions can be written as

(k | xa | kU) « ~ie2(Z - l)/m<o2] (k \ xß/r* \ k') òfi. . (344)

[Conservation of energy, i.e., Eq. (336), confirms e' — e = hœ for %' = / . ]
Substitution of these estimates into Eq. (340) yields, for the asymmetry
of the profile of an isolated ion line from this particular mechanism,

L(CÜ,·/ + Δω) - L(ù>if — Δω) h Δω
Α(ω> = 77 1 Λ Ν . Ί ( Γ Τ ~ 2 ( Ζ " Χ) 2"
L{o)if + Δω) + L(o)i/ — Αω) mœif

Re Σ <* I *« I/) ( / I Xß 11) I (k I xjr* \ k') (kf \ xß/r* \ k) dQgf
a,ß J

X

Σ \(i\xa\ f)\2W I Xß | i)(i | Xy | ϊ) f(k' | Xß/r* I k)(k I xy/r* \ k') dû Qi.

a,ß,y,i'

Z\(i\x°\f)\29(ki,k/)gf

W - 1) *H _! . (345)
3/./ ω,/ Σ | (î'| xa | i')\2 g(ki, k'r) gt>

I I . 4 . INTERMEDIATE APPROXIMATIONS 135

The last version follows with £)« \(i \ x<* | / ) | 2 = î(e2a0/hœ)fif, where fif
now stands for the absorption oscillator strength of the line or the average
absorption oscillator strength of the multiplet. The Kramers-Gaunt factors
are defined by

g(ki, K) = Wyftkik't Σ 11<*.· I **/*1 K>)\2 d% (346)
aJ

assuming (27r)~3/2 exp(ik · r) for plane wave states. (The integrals for
a j* β vanish, and | Αψ) of course corresponds to | kf) above, and | A»)
to | *>.)

In the special case that only i' = / contributes, Eq. (345) simplifies to

Α(ω) « v2[4(Z - 1)/3/ν](Δω/«</), (347)

if a factor of v2 is added to account for spin. [If spin is included, radiator
matrix elements in Eq. (340) are reduced by a factor of V2, but the per-
turber dipole matrix elements remain the same.] This result of Burgess
[128], who used radial matrix elements instead of oscillator strengths, is to
be compared with other sources of asymmetry, e.g., that stemming (for
emission) from the factor ω4 in Eq. (1), which gives rise to A (co) «
4Δω/ω;/ . Then there are the ion effects as discussed in Section II.3f, which
often counteract the above asymmetries, but may be more or less suppressed
by the Coulomb repulsion. Also, if interacting states i' ^ / are important,
Eq. (347) is obviously an overestimate of the line-continuum interference
effect. As a matter of fact, in this case, Eq. (345) would not be correct
because it was derived here, but not in the original paper [128], assuming
that perturbations of the lower state of the line were negligible. This
internal inconsistency leads to errors except in the special case assumed for
Eq. (347). In general, we must consider perturbations of the state / as
well, whose effects on the line shape are simply additive for isolated lines
in the dipole approximation for the interaction Hamiltonian, there being
no upper-lower state dipole interference terms for isolated lines (see Section
II.3ca). This generalization of Eq. (347), again neglecting fine structure
except for the gross correction factor v2, is

A(«)

^ 4(Z - 1) Δω hAU 1 r \ lj)* g(kt, h/) [(21, + l ) ' 1 + {21, + 1 ) - ' ]

3/./ or Σ U'Mh-\r\li»)tg<Jc,k'){2li.. + l)-1 '

136 IL THEORY

where the (Zt·' | r \ U») « fn(n2 — 12ί^^)1/2α0/Ζ are reduced radial matrix
elements, and Z,-'ti" is the larger of U* and U» . (This approximate expression
for the radial matrix elements is exact for ionized helium, etc., when i' and

i" correspond to the same principal quantum number. In general, it must

be multiplied by a correction factor φ calculated, e.g., according to the

Bates-Damgaard [42, 43] method.) Equation (348) can be derived by

adding to Eq. (340) an analogous relation with / replaced by i} and by
summing or averaging over magnetic quantum numbers as in Section

II.3CÛ!. [Because of the summations over magnetic quantum numbers, the

Qi, even times exp(—2?,-/fc!T), are all the same in the validity range of the

Gaunt factor approximation.] Finally, it should be remembered that the

electron momenta in the "broadening" Gaunt factor g(k, k') are related by

t" = i

k2 = k'2 - (2m/h) (ω - ω ν =F ω ^ ) , (349)

and that some allowance for deviations from the Coulomb-Born-Bethe
approximation can be made by using effective [103] (semiempirical [102])
rather than actual Gaunt factors (see also Sections II.3d and IV.4c).

Of greater practical interest than the asymmetry of an isolated ion line is
its width, for which Eqs. (340), (343), and (346) lead, in analogy to the
derivation of Eqs. (341) and (342), to

» ^ 4 ^ (—Y Σ ^τΐττ ν< Ir I ^2 *<*' - *··'>> (35°)
33/2ι> \maj if 2U + 1
if we again introduce reduced radial matrix elements and imply a Maxwell
average over v. An analogous result holds for the ground state (i—»/),
and comparison with Eqs. (151) and (161) reveals that the Gaunt factor
corresponds to V3Ü(Z)/T in the classical path approximation, or rather to
the same expression for the "hyperbolic" case with a functions as given in
Table IIIc. However, its argument is frequency-dependent, as required by
the relaxation theory [52, 96a, 113, 121] (see Section II.4b). In other
words, we must remember that /ψ = mv'/h corresponds to the initial
perturber wave number, while the relation for ki, from Eq. (349), is

fc.2 = fcji _ (2m/h) (ω - œif - ω<'<), (351)

i.e., the perturber has to make up for the energy differences between levels
%' and i and between hœ andfaunf.

I I . 4 . INTERMEDIATE APPROXIMATIONS 137

The Gaunt factor depends mainly on the variable

Z-l/1 1\ - e 2Z- lη{/ω - vi'/)
Ç = cio \Ik—{> - k- i/ I « nv mv2—

= -η[Α(ω ~ ωίΊ)/πιυ2~\. (352)

It is close to g(£) = 1 for | ξ | > 1. For small | £ |, there holds

g(t) « (V3/T) ln(2/7 I ί I) = (vS/τ) ln(1.123/| £ |), (353)

always assuming η = (Z — l)e2/hv > 1. As discussed by Baranger [6],
?7 > 1 implies that there are no large quantum corrections to the classical
approximation. This follows because the relative angular momentum
quantum number of the perturber is approximately

I « mpv/h = V[mv2p/(Z - l)e2], (354)

i.e., I > η for all impact parameters for which Coulomb scattering angles
are small. But for large Coulomb scattering angles (eccentricity e « 1),
the weak-collision dipole contributions are small anyway, at least in the
classical path approximation, so that η > 1 indeed is a sufficient condition
for the validity of the classical limit for weak dipole interactions, provided,
in addition, that the two momenta ki and k^ are not too different, i.e.,
h I ω — coi'/ \/mv2 <£ 1. Otherwise, use of Eq. (343), the Coulomb-Born
approximation for dipole interactions, may not be justified. (There could
be large deviations from Coulomb functions for the perturber wave func-
tions, especially for Z = 2 ions.) For h \ ω — ωίΊ \/mv2 > 1, Eq. (350) and
the preceding estimates for the effects of perturber radiation must thus be
viewed with caution, and it is doubtful whether use of g(ki, k^) instead
of g (| £ |) would lead to any real improvement here. This view is supported
by fully quantum-mechanical calculations [107, 108] (see Section II.3e),
which tend to yield effective Gaunt factors g « 0.2 instead of g ~ 1 for
such conditions, as do numerous Z = 2 experiments (see Section III.7).

There are two more precautions to take before using the Coulomb-Born
approximation with dipole interactions only. As it does not conserve proba-
bility, we must make sure that the "Coulomb cutoff" [6, 57a] correspond-
ing to Z « 77 is more effective than that to be imposed to avoid serious viola-
tions of unitarity, namely I ~ n2/Z (see, e.g., Section II.3d). While for
η > n2/Z use of lowest order perturbation theory is always sufficient, for
η < n2/Z a cutoff may have to be introduced near I = n2/Z in a partial
wave expansion of the Born approximation scattering amplitudes, and

138 II. THEORY

lower I partial wave contributions may have to be estimated either from a
saturation argument or, better, from higher-order perturbation theory.
Second, there is the problem of higher multipole interactions. For I > η > 1,
their relative contribution is of the order [ ( Z — 1)η2/ΖηΓ^. This is indeed
small for all I > n2/Z, so that higher multipole contributions are not im-
portant here except in the lower partial wave contributions to be calculated
from higher-order perturbation theory. For I < η, no simple estimate for
higher multipole contributions seems to exist. However, estimates for
ionized helium (Section II.3b) and the fully quantum-mechanical calcula-
tions mentioned above suggest that they may cause the widths of a few
lines to increase by as much as a factor of about 2. A similar situation prob-
ably prevails for higher than singly charged ions, for which, in the limit of
large Z, the Coulomb-Born approximation with the entire interaction
Hamiltonian becomes valid.

As already indicated, Eq. (350) does not quite correspond to the impact
approximation because the Gaunt factor still depends on the actual photon
energy through Eq. (351). Only if we replace ω by coty do we obtain the
(Coulomb-Born-Bethe) impact approximation, while keeping the fre-
quency dependence is analogous to the second order relaxation theory
[52, 96a, 113,121] (Section II.4b) result for the line wings or to the "Lewis
cutoff" [65]. The decrease of the Gaunt factor toward the line wings is
mainly due to a reduction in higher partial wave contributions, so that here
use of higher-order perturbation theory for low partial waves becomes more
important. It is for this reason that the transition to the "quasi-static"
wings cannot be effected within the Born approximation. Therefore Eq.
(350), i.e., L(œ) ~ wt/7r(Aw)2, should not be used near or beyond the fre-
quency separation where the quasi-static approximation would be appropri-
ate (see Section II.2). For isolated lines (subject to quadratic Stark effect),
this is usually not a serious limitation to the usefulness of Eq. (350) or its
equivalents.

Besides such generalization of the impact approximation for isolated lines,
there is another interesting application of the one-electron approximation,
namely that to the overlapping-line problem [ 6 ] (see also Section II.3cß).
From the first term in Eq. (340), again using the (dipole) Coulomb-Born
approximation and summing over magnetic quantum numbers, follows

x(«in/)-w"iri/)«irio(^ir|f) (355)
(ω — o)if) (ω — ω<"/)

I I . 4 . INTERMEDIATE APPROXIMATIONS 139

if we again invoke detailed balancing, etc., normalize with respect to the
unperturbed i —> / transition, and neglect the lower state perturbations.
This formula is particularly useful in the "windows" between lines, which
may be important in opacity problems with weak continua. However, for
frequencies ω = ω»'/ (corresponding to a forbidden component), the argu-
ment ξ of the Gaunt factor vanishes according to Eq. (352), leading to a
logarithmic divergence in Eq. (353). Comparison with classical path calcu-
lations as in Section II.3CQ: reveals that this divergence is due to large im-
pact parameters, at which Debye shielding should have been taken into
account. As will become clear in Section II.5a, this can be done approxi-
mately by never letting the argument of the Gaunt factor decrease below
£s ~ ηήωρ/πιν2) where cop is the electron plasma frequency. Away from such
forbidden lines, g « 1 may often be a reasonable estimate for multiply
ionized radiators, although all the caveats mentioned in connection with
Eq. (350) are not to be overlooked here either.

Between lines i —> / and i" —> /, the i, i" interference term in Eq. (355)
is always negative. Therefore, the intensity in this region is less than would
be expected from a superposition of isolated line (dispersion) profiles.
Near, say, the i —>/line, this correction is of order (ω — ω,·/)/ω,·",· times a
factor involving the radial matrix elements, which is typically well below
unity. Still, this interference effect between different lines may be just as
important as the interference between line and perturber radiation dis-
cussed earlier in this section.

Almost all the above discussions could have been made in the framework
of the classical path approximation. A practical reason for preferring a
quantum-mechanical discussion is the availability of results from Coulomb
excitation theory; e.g., the functions/Ei of Alder et al. [74a, b ] are related
to the Gaunt factors through

fmivi, Vk) = exp(-27T | { |) (2V/35/2) g(Vi, Vk), (356)

with ξ = Vi — η/c and, e.g., 77; = (Z — l)/a0k. Analogous relations can be
derived [74c] for higher multipole interactions, but are more complicated
than Eq. (356). Finally, in order to use, say, Eq. (355) for neutral atom
lines, one simply replaces the Gaunt factors by V3 α(ζ)/π or by (vS/π)
ln(pmax/pmin) for isolated or overlapping lines, respectively, with the further
understanding that z « n?h \ «,·,»·/ \/2kT is calculated from max(| ω —
ω»/ — coi'»· |, ωρ) instead of | ω»·',· |.

Before closing this section, which has covered a rather rich but probably
incomplete class of phenomena to be understood in terms of the one-electron
approximation, the reader is reminded that most of the results given here
are somewhat qualitative in nature.

140 I I . THEORY

II.5. CORRELATION EFFECTS

Except for (electron) Debye shielding and ion-ion correlation effects on
the "low-frequency" ion field strength distribution functions discussed in
Section II.2a, most theoretical developments described in the preceding
sections explicitly or implicitly involved the notion of statistically inde-
pendent perturbers (and radiators). As pointed out in Section II.3a, this
led to a logarithmic divergence in the original impact theory calculations
[53] for the electron broadening of hydrogen lines at large impact param-
eters. It was recognized already at that time that actually electron-electron
correlations would reduce the contributions of distant collisions, and a cut-
off in the integral over impact parameters was therefore introduced at the
(electron) Debye radius as given by Eq. (43). A similar divergence occurs
when the observed frequency corresponds to a forbidden line (see the pre-
ceding section), and a somewhat weaker shielding effect must of course be
expected for isolated lines as well.

Besides this Debye shielding (Section II.5a) in the electron impact
broadening mainly of hydrogen, ionized helium, etc., lines and the effects
on ion field distribution functions mentioned above, at least twO more cor-
relation or collective effects have been discussed in the literature: a "plasma
polarization" shift (Section IL5b) of ion lines first proposed for the
X4686 A line of ionized helium [130] as due to the difference between
radiator-electron and radiator-ion correlations, and the effect of supra-
thermal plasma oscillations on lines with forbidden components predicted
by Baranger and Mozer [131] to give rise to "plasma satellites" (Section
II.5c). Such suprathermal plasma oscillations may also lead to substantial
changes in field strength distribution functions, as briefly discussed at the
end of Section II.2a, and to corresponding changes in line profiles should
the quasi-static approximation be valid, i.e., if the lines extend well beyond
the relevant plasma frequencies. Again, the effect has been observed [132]
(see also Section III. 10), but in this case the theoretical problems are
mostly in the theory of field strength distribution functions for turbulent
plasmas, a subject more appropriate for advanced texts on plasma kinetic
theory. Some of the related atomic physics problems were first discussed by
BlochinzewT [133a], who considered the effects of an rf electric field on
hydrogen lines, while the very first observation of related plasma effects
was reported by Lifshits et al. [133b] and a theoretical discussion of the
simultaneous effects of static and rf fields given by Cohn et al. [133c].

II.Sa. Debye Shielding

In the calculation of electron broadening using either the impact theory,
i.e., Eqs. (92) and (98), or the relaxation theory, i.e., Eqs. (282) and (288)

I I . 5 . CORRELATION EFFECTS 141

with Eq. (90), we encountered terms of first, second, etc., order in the inter-
action Hamiltonian U for a collision with a single electron, which was as-
sumed to be statistically independent of the rest of the electrons. The first
order term had been seen to vanish on the average over directions (still
neglecting the plasma polarization shift), while higher than second order
terms were recognized as important only for electrons coming as close as
Pmin ~ h(n2 — n'2)/mv in the case of hydrogen [Eq. ( I l l ) ] or closer than
that for any other radiating system. Such values of pmin are usually very
much smaller than mean electron-electron separations r\ « (4iriV/3)~1/3,
and it is thus very unlikely that two or more electrons would simultan-
eously be as close as that to the radiator. Some distant collisions may occur
simultaneously with such strong collisions, but should then be negligible,
leaving the second order term in U as the only term that might be sig-
nificantly affected by correlations, i.e., for which the assumption of having
statistically independent perturbers may have to be removed [51a].

Such removal is most expeditiously done by returning to Eq. (283) of
the general (classical perturber path) relaxation theory. (Again, this
generalization can also be performed using BBGKY hierarchy techniques
[134].) A matrix element of <£(ω), to second order in the interaction
Hamiltonian accounting for all electrons, is then

(i | £(«) | %') = - (t/Ä) Σ fexp[i(« - a*v)G

. {<t | t/(r, t) | i") <t" | *7(r, 0) | i ' > U Ä, (357)

if ω is now the actual frequency. As in Section II.4b, the average is over all
initial electron positions and velocities. Since correlation effects are ex-
pected only for distant collisions, not only classical paths but also the dipole
approximation for U from Eq. (58) can be used. Assuming isotropy of the
perturbing fields, one thus obtains

(i | £(«) | %') = - (teV3A) Σ <* I r | t"> · <t" | r | %')

t"

• f exp[i(« - o>i»f)t] {F(0 · F(0) | a v * , (358)

and the key quantity is seen to be the Fourier transform of the autocorrela-
tion function of the electric field [13, 52]. (For the line wings, this was
shown already by Lewis [65] and Baranger [6].) As discussed in Section
II.4d, the usual impact approximation amounts to using, e.g., ω = œif ,
so that for hydrogen lines, etc., the exponential factor is unity. For this
particular case, the first discussions [51a, 65] of Debye shielding going

142 II. THEORY
beyond a simple cutoff procedure [53] involved a two-step average: First,
the position and velocity of one particular electron, say at time t = 0, is
fixed, and the average is over the motion of all other electrons. The question
here is what to expect for the total average electric field at time t. It was
argued that in the vicinity of the first electron, this should be close to its
(statically) Debye-shielded field, because velocities somewhat below the
mean thermal speed are most important for the broadening of hydrogen
lines. Second, one must average over the paths of the particular electron,
which produces a Coulomb (unshielded) field. After averaging over its
initial position and velocity as in Eq. (239) by assuming straight classical
paths, one finally multiplies with the total number of electrons. This
amounts to saying that all electrons are equivalent and that the total initial
field is just the sum of Coulomb fields. The p integral ensuing from the
average for the "first" electron converges at large impact parameters or, if
the results are written as proportional to ln(pmax/pmin), the equivalent cut-
off is found [51a] at pmax « 1.1 PD . [Here PD is the Debye radius account-
ing for electrons only as given by Eq. (43).]

The above result has been criticized [135a] because actually electron-
electron interactions do also cause some curvature in the perturbing elec-
tron paths in the sense that an electron already present in the vicinity of
the radiator will deflect others and thus make them less effective. As a
matter of fact, the calculation (see also below) including this curvature
effect shows that both fields in Eq. (358) should be shielded, leading to an
equivalent cutoff at pmax « 0.6 PD . However, this reduction in the equiva-
lent cutoff by almost a factor of 2 was challenged [135b] on physical
grounds because it ignores the electron-ion interactions: Since (for neutral
radiators) an ion is just as likely close to the atom as an electron, the cor-
responding curvature correction may about cancel the electron-electron
curvature effect. In any event, there is the ambiguity associated with the
separation of electron and ion effects in Eq. (101) and its equivalents.

For a more formal discussion of correlation effects, it is advantageous to
make contact with plasma kinetic theory. Instead of the Fourier transform
of the autocorrelation function for electric fields, consider now the spectral
density, e.g., for density fluctuations, by writing

retàwt (F(j) . F(0) }av dt = 8e2 f S+CK, i Δω) dK (359)
in Rosenbluth and Rostoker's [136] notation, except for our use of K for
the wave number in the spatial Fourier transforms of density, etc. [A
factor (4ve/K)2 arises from the change from density to field fluctuations,
and we still assume isotropy.] Because of its importance for light scattering