Spectral Line Broadening by Plasmas

I I . 3 . IMPACT APPROXIMATION 43

avoid serious violations of the unitarity condition for the S matrix, and a
"strong-collision" term (corresponding to "f ") was introduced on the basis
of the observation that the ÜT-matrix element (1 — S) will oscillate around
unity for p < pmm . ( 7 = 1 would actually give " i , " but T could be as
large as 2, the mean of these possibilities leading to "f.") The fourth-order
term is represented by —"■§■," combining with the strong-collision term to
a "strong-collision constant" J. To allow for still higher order (positive)
terms, this will here be doubled rather arbitrarily, in which case it agrees
better with more detailed calculations or exact solutions [17a, 38, 62-64a]
for the line shape operator (see also Voslamber [113c] below). With this
in mind and introducing again lower-state broadening and second-order
quadrupole terms (using the same estimates for the atomic matrix elements
as in the fourth-order dipole term but multiplying by a factor of 2 to ac-
count for different intermediate states), φ can now be written in terms of
parabolic wave functions as

Φ = ~ ^ 3Ν ( —\ma) 0/2 Jf-f(vv) ΓL (ifΣf Γ , Κ ' ) · (i"\ri-2zizf

/// Y± Pmin O ÛH /

+ Σ r, | * " > · < * " I *« In —
kff^iff Pmin

+ Σ r, | *">·<*" I r/ l n - ^ - 1 , (110)

kffytfff PminJ

with the lower p cutoff generalized to

Pmin ~ (tt;2 - nf2)h/Zmv. (Ill)

(The "strong collision constant" depends only weakly on pmin, and for

most of the line profile only ZiZf appears, because xtx/ or 2/ti// correspond to

inelastic scattering unless the quasi-static splitting of the various levels is

smaller than their electron impact widths [64b, c].) The last two terms in

Eq. (110) allow for interactions with states having principal quantum

numbers different from i or /, respectively. Therefore, e.g., pk% must be

chosen as

Phi ~ v/au (112)

in accordance with the well-known condition for collision-induced transi-
tions.

44 I I . THEORY

Most difficult, generally, is the choice of the upper p cutoff pmax (or
rather, of pmax/pmin) for the usually dominating second order dipole terms
involving intermediate states of the same principal quantum number as
initial or final states of the line. [Were these terms not the largest, the
estimates for the various corrections would not be valid either—in particu-
lar, the (incorrect) assumption that these corrections are the same for all
φ-matrix elements. Some questions related to this assumption, which is not
especially critical (see, e.g., Voslamber [113c] below), have been discussed
in great detail in the literature [62, 63]. Also, simply adding inelastic terms
as in Eq. (110) may well be an overestimate of the effects of inelastic
collisions.] Three distinct physical effects not yet included in the above
treatment of electron broadening of hydrogen lines are related to this
choice, i.e., if properly included, they would have produced convergent
results also for arbitrarily large p values. These effects are related to electron
correlations or Debye shielding [53] (see also Section II.5a), to the finite
duration of actual collisions [65] (see also Section II.4b), and to the
removal of the degeneracy by quasi-static Stark effect [66], which in-
validates the approximation U'(t) « U(t) in Eqs. (107), etc. In any event,
the influence of collisions outside pmax « ν/ω0 would be much reduced,
with the critical frequency being the largest of the electron plasma fre-
quency ωρ , the frequency separation | Δω | from a given Stark component,
or the mean separation between Stark components, depending on which
effect dominates [59a]. The latter two frequencies are of course com-
parable in practice, except on the far wings of the line, while in the first
case one has pmax ~ PD , with pD being the Debye radius accounting for
electron-electron correlations only as given by Eq. (43). Depending on the
velocity of the perturbing electron, pmax may thus be estimated either by
this shielding radius or by ν/ω0, with ω0 being of the order of w in Eq. (16)
or larger.

Now the velocity integral in Eq. (110) can be performed, leading, except
for numerically insignificant differences in correction terms, cutoffs, etc., to
the expression for φ which was used in improved calculations [59a, b ] for
hydrogen. (These profiles are tabulated in Appendix La. Dipole terms in-
volving states k" ^ i, / , were, however, omitted. Their inclusion [59c] for
low Balmer lines appears not to improve the agreement with experiment
except, perhaps, in that they smooth the central portions of the profiles;
see Section III.2b.) In the integral, velocities for which pmin > pmax must
of course be ignored. The corresponding electrons would more properly be
treated as quasi-static perturbers, but as long as their fraction is small, it
may just as well be included into an error estimate. Further (relative)
errors are of the order of the correction terms or correspond to, say, a factor

II.3. IMPACT APPROXIMATION 45

of about 3 uncertainty in pmax/pmin. However, as shown in Fig. 3, such
estimated theoretical errors are below or near 10% for conditions under
which neighboring lines, say, in the Balmer series are well isolated. (See
Fig. 1 for the Inglis-Teller limit.) Errors from the quasi-static approxima-
tion, etc., for ions are usually negligible if we ignore small asymmetries.
(Corrections from actual motions of ions and atoms are discussed in Section
II.4c, following Kogan [15].) Also, the use of the classical path approxi-
mation for perturbing electrons is well justified in case of hydrogen, because,
according to Eq. ( I l l ) , the relevant impact parameters exceed the
DeBroglie wavelength of the colliding electrons by a factor well above
(n,2 — η/2)/2π, and because relative changes in perturber energies are
vanishingly small in almost all cases.

For La , the classical path approximation can be verified by fully quan-
tum-mechanical calculations [64a, 67]. And indeed, the semiclassical
results discussed in the present section do agree in the relevant energy
range and within the errors estimated above with widths obtained [64a]
from close-coupling calculations of Ormonde and coworkers, in which
n == 3 states were included. The other calculation [67], without n = 3
states, was only for one energy (~0.5 eV). It here confirms the conclusions
of Griem [64a], assuming that the quantity given is Ω/(2Ϊ + 1). However,

ΙΟΟΟΟ K
|L 20000 K H

0Κ)13 IO14 IO0 IO16 IO17 1 0 * KP
N (cm"3)

FIG. 3. Estimated theoretical errors in the Stark profiles of the first four Balmer lines
from approximations in the electron broadening calculations (after Kepple and Griem
[59a]). Note that actual errors may be larger by a factor of about 2, e.g., in the core of
the He line, because here quasi-static broadening by ions is not too important.

46 II. THEORY

errors from our use of the impact approximation are additional (see Sec-
tions II.4b and IV.4a), as are those from ambiguities arising in the course
of correcting the ion (F) and electron (φ) broadening for electron (Debye
shielding) and ion (quasi-static level splitting) effects, which were originally
clearly separated in Eq. (101). Even this original separation was intuitive
rather than theoretically derived (see also the papers cited at the end of
Section II.4c). Finally, we have the uncertainty connected with the
transition from only a ZiZf upper-lower state interference term to a full
r» · T/ term in going from the line core to the line wings [64b, c]. (This transi-
tion must not be confused with the transition from impact to relaxation
theory discussed in Section II.4b.)

II.3b. Hydrogenic Ion Lines

For ionized helium and higher members of the one-electron sequence, not
only must appropriate changes be made in radial matrix elements and
straight classical paths be replaced by hyperbolic trajectories, but the
validity of the classical path approximation and the use of the multipole
expansion for the interaction energy should be reexamined as well.

With these two approximations the integral of Eq. (58) becomes, for
singly charged perturbers,

l/_+>di = -^T)r"-[v(00) -v(-°°)]

-ür°J-T^)T~dt' (113)

if one observes [57a] that the "perturber factor" in the dipole term can be
expressed through the classical equation of motion (for electrons) as

- A ( < ) / | r ( « ) |3 = m v ( 0 / ( Z - l ) . (114)

The cosine of the angle 0" between vectors ra and r(t) in the quadrupole
term may be written in terms of azimuthal (λ) and polar (φ) angles asso-

ciated with ra, defined with respect to the orbital plane and the line be-
tween the asymptotes as the Z axis (see Fig. 4), and the polar angle (0)

of r (0 in this coordinate system, namely

cos 0" = sin λ sin φ sin 0 + cos φ cos 0. (115)

The remaining integral in Eq. (113) can be transformed into an integral

II.3. IMPACT APPROXIMATION 47

FIG. 4. Schematic of an electron [radius vector r(t)] collision with a positive ion (repre-
sented by the radius vector r» of the perturbed electron). The angles and Z axis are as
defined in the text, the origin is at the nucleus of the perturbed ion with the X axis in
the orbital plane and at right angles to the Z axis, and the Y axis is orthogonal to both.

over 0, using the expression for (attractive) hyperbolic orbits,

1/r = (l/p)(l - ecoso), (116)
where e and p are defined by

c = {1 + [mpvy{Z - 1) W = [1 + (LA)2]1'2, (117)
p = p(e2-l)1/2, (118)

and (from the conservation of angular momentum) (119)
pvdt = r2dd.

[In the second version of Eq. (117), the quantities L and η are the relative
angular momentum L = mpv/h and the Coulomb parameter η =
(Z — l)e2/fe>.] The integral in the quadrupole term thus becomes

f [(cos2 β" - \)/r3] dt = (l/ppv) [(sin2 λ sin2 φ - è) / (1 - « cos 0) d0

— 00

+ (cos2 φ - sin2 λ sin2 φ) J (1 - € cos Θ) cos2 0 etó], (120)

omitting a term linear in sin Θ which vanishes on integration, and replacing
sin2 Θ by (1 — cos2 Θ). The limits of the Θ integrals correspond to cos Θ = 1/e

48 II. THEORY

(via 0 = π), and the final result follows with / cos2 0 άθ = \ sin 0 cos 0 + \B
and / cos8 0 dd = sin 0 — $ sin3 0 to

£/™- 2mvra 1) cos Λ, . α , 3eVa2 <, ,(s.inβ2 % .. - J)
Ä(Z - 0' sin -2 + fztvpp sin1 φ
λ

X (e2 - 1)1/2 + T - cos-1 ( - ) + (cos2 φ - sin2 λ sin2 φ)

* Kl· a)<"-»'» + H — (;)]}■ <121>

where now the dipole term ra · [v(<*>) — v(—«>)] from Eq. (113) is
expressed through the scattering angle a, initial (final) velocity v, mag-

nitude ra of r „ , and angle 0' between the vectors ra and v(oo) — v( — » ) .
For Coulomb scattering, the following equation holds:

sin (a/2) = 1/6. (122)

Therefore, using Eqs. (106), (117), (118), and (121) and integrating over
the angles 0', λ, and φ gives the following expression for the angular average
of the second-order term in the Dyson series:

[(1/Ä*) / dt f dt'U(t) U(t')~\

if we again account only for interactions between the degenerate sublevels
of the same principal quantum number ( £/' —► U). Comparison with Eq.
(107) reveals that the factors [ · · ·]<ι a n ( i [' · *]q represent the correc-
tions for hyperbolic (rather than straight) classical paths in dipole [57a]
and quadrupole terms [68], respectively. While the dipole term is always
reduced, although usually only slightly, the quadrupole term is enhanced
substantially for all e < 10 (see Fig. 5). For larger € values, we have

Ε···],-»1+(3τ/4*), (124)

and [69] for e « 1 (small velocities), (125)
[ · · -]q-*3irV4(e2 - 1) = | | > ( Z - l)t?/mpifij.

I I . 3 . IMPACT APPROXIMATION 49

FIG. 5. Enhancement of second order "diagonal" quadrupole contributions to the
classical path ^-matrix element due to Coulomb interactions between electron and radi-
ating ion as a function of the eccentricity parameter €. Also shown is a simple interpolation
described in the text.

Also shown in Fig. 5 is the sum of Eq. (125) and 1, the leading term in Eq.
(124). This sum represents the actual quadrupole correction factor within
about 20%.

Using this approximation and estimating strong collision (i.e., higher
order dipole) terms as in the hydrogen case, Eq. (123) finally leads, in
parabolic coordinates as in Eq. (110), to

φ - - TN (er)' f τ / w f(Σr< i »""> *{i" ' " - ***
ό \mao/ J v L iff

+ Σ r/ I / " ) ' < / " I r,) (-. + In ^ ) + ^ Σ rt \ i") « " | r,2
fit \4 «min/ DÛT ilt

/3*»(Z - 1)2Z4 Z*mv* \

X \16(n<2 - η,ψ 4(n,·2 - n / ) 2 B H /

+ ( Σ ',· I *"> ' <*" I r.) ^j|f« + ( Σ '/ I *"> ' <fc" I */) JSeA>
ktf^iff Vó kff*f>f Vö J

(126)

if inelastic dipole terms are added according to the effective Gaunt factor
approximation (see Section IV.4c). Actually there are also lower state and

50 II. THEORY

upper-lower state interference contributions to the quadrupole terms, as
discussed in the next section. These may very summarily be accounted for
by writing the quadrupole matrix element as a fraction of the combination
of dipole matrix elements, i.e., by replacing the velocity average of
[£ + ln(€max/€min)] as follows:

\ 4 €min/av 4 \ €min/av 2 [ 4(tt»2 - Π/2) J 4 EK

using the same relative values for the matrix elements as for hydrogen
(except for the Z~l scaling). The third term in Eq. (127) describes the
increase of quadrupole interactions due to Coulomb effects, while the last
term corresponds to straight classical path quadrupole interactions. It is
clear from their derivation that they scale as p^t and p~£ , respectively,
so that the preliminary choice of pmm according to Eq. ( I l l ) must be
reexamined to avoid serious overestimates of the quadrupole corrections,
especially for the lines analogous to Le and Ha .

Often, e « 1 is relevant in these cases near pmin , and in the extreme that
quadrupole interactions should dominate near pm'in , Eqs. (123) and (125)
would yield a strong collision impact parameter

Pq « ΙΤΛ/5(Ζ - l)Z/2(nt - n/)]l'3pmin . (128)

For the ionized helium "La" and "Ha" lines, this gives pq « 1.33pmin or
pq « l.lpmin , suggesting revised cutoff impact parameters ρ^η « 1.4pmin
or p^in « 1.2pmin for these lines when both dipole and quadrupole inter-

actions are considered. The last term in Eq. (127) must therefore be divided

by factors^2 a n d ~ 1.4 for these two lines, and the third term by similar

factors, as its decrease may be compensated some by an increase in the

strong collision constant (J) from quadrupole interactions. (Note, how-

ever, that for € « 1 dipole interactions do not give any strong collision

term.)

With these corrections and after performing the velocity average with

€max chosen analogously to pmax in the case of hydrogen, Eqs. (126) and
(127), or numerically equivalent expressions, have been used in extensive

calculations of ionized helium Stark profiles [61]. (Tables of these profiles

are given in Appendix I.b.) Estimated errors in these profiles stay below

about 20%, in spite of the rather crude estimates of higher multipole inter-

actions (which account for less than 20% of the electron broadening in

most cases). To further improve the accuracy of electron broadening cal-

culations, in particular for "Le," a fully quantum-mechanical theory
[64a, 67] (see also Section II.3e) may be necessary, because the strong

collision contribution here corresponds to relative angular momentum

I I . 3 . IMPACT APPROXIMATION 51

quantum numbers L < 2. For higher members of the isoelectronic series,
both quantum-mechanical corrections and higher multipole interactions
should become even more important. Ignoring the former and taking note
of the fact that the multipole expansion terminates with the quadrupole
term if restricted to the n = 2 subspace, we find that Eq. (128) would
suggest for the width of La , in the high Z and € « 1 limit,

w « (9T/Z*v)N(h/myi(TTT/5/6)Z(Z - 1)]2/3 « (9*/Z™v)N(h/m)\

(129)

However, this formula should not be used for Z values violating

Z 2 / [ ( Z - 1)Z]2/3 < (Z*EH/kT), (130)

because otherwise, pmin falls inside the n = 2 Bohr orbit. [Then the implicit
assumption in Eq. (58) that the perturbing electron is farther out than the
bound electron fails.] This condition is reasonably fulfilled for £ < 30,
but Eq. (128) implies orbital angular momentum quantum numbers
L « 3/Z1/3 for the perturbing electrons, so that fully quantum-mechanical
calculations using the entire electrostatic interaction would definitely be
required for, say, Fe XXVI. In lieu of such calculations, more circumspect
estimates for electron impact widths of multiply ionized systems are dis-
cussed in Section IV.6. They suggest that Eq. (129) is already an over-
estimate for Z « 4, rendering it rather useless.

II.3c. Two- and More-Electron Atoms

II.3ca. Isolated lines

For one-electron systems, both diagonal and off-diagonal matrix elements
of the electron broadening operators φ or 3C were required. However, when
levels of different angular momentum are not degenerate, the matrix in-
version implied by Eq. (98) can be performed approximately by expanding
[ 7 ] essentially in terms of ((if* | 3C | i'f'*))l(Hi - # / ) - Hf - F / ) ] " 1 ,
etc. If 3C matrix elements are much smaller than frequency splittings
between lines i-f and z'-/', first- and higher-order terms in this expansion
can be neglected, leaving us with the single term i\ f = i, f if we ignore the
magnetic substates of i a n d / for the time being. This situation corresponds
to that discussed in Section II.3 following Eq. (98), i.e., the line has a
Lorentz (dispersion) profile with width and shift given by Eqs. (99) and
(100).

In reality, there always exists some rotational degeneracy, at least for
one of the states involved, and the question arises how to perform the
inversion in magnetic quantum number Mi and Mf subspaces associated

52 II. THEORY

with initial («/,·) and final (Jf) angular momenta. This question may be

approached [2, 3, 5] with the help of the Wigner-Eckart theorem [70],

because X and φ are spherically symmetric (in "line space") in the absence

of applied external fields and of anisotropies in the perturber distributions.

The Wigner-Eckart theorem implies that the matrix elements of the dipole

moment are proportional to Clebsch-Gordan coefficients, which are essen-

tially integrals over products of three spherical harmonics. When x =h iy

and z are used instead of the usual Cartesian components, the factors

corresponding to the dipole moments are the spherical harmonics Υίμ (or,
in the case of the second factor, y*M) with μ = ± 1 and 0, and the Ad
operator in Eqs. (85) and (98) may accordingly be represented by [ 2 ]

«Jlfiilf,* | Ad | Mi'Mj'))

(i\r\f) 2 ; Μ/μΜ/)
2Ji + 1 Σ C(Jf\Ji ; MvM<)C(JAJi

Ih

= (i\r\f)^(-l)2Jf+M<+Mi'\
\M,

(131)

Here (i\r\f) is a reduced radial matrix element which is often also called
the radial integral. The second version of Eq. (131) is written in terms of
3-j symbols [70] in place of the Clebsch-Gordan coefficients. (See Cooper
[55] for the case of other multipole transitions and Tsao and Curnutte [71]
for a more general application of angular momentum algebra to line
broadening problems.) Actually, any spherically symmetric operator such
as 5C, ψ, or Hitf and functions thereof—in particular [ · · · ] _ 1 ί η Eq. (98)—
can be written [ 2 ] in terms of reduced matrix elements and 3-j symbols
according to Eq» (131), since the d-j symbols fully describe the rotational
transform of the wave functions. After substitution of such expressions
(with μ replaced, say, by v in case of [ · · · ] _ 1 ) , Eq. (98) therefore becomes

L(w) = - ( * / » ) \(i\r\f)\nm(if\Z

/Jf 1 /, \/Jf

χΣ

\M/ μ -Μ//\Μ/

= - (h/w) | « | r | / ) | 2 Im((if ! [ · · · ] \if))-\ (132)

I I . 3 . IMPACT APPROXIMATION 53

where the second version follows on summing over all magnetic quantum
numbers, including μ and v, from the orthogonality relations of 3-j symbols
[70]. (The phase factors always combine to +1.)
The impact approximation profiles of lines arising from levels that are
degenerate only in magnetic quantum numbers are thus also of the simple
Lorentz-Weisskopf type with widths and shifts given by Eqs. (99) and
(100), except that the matrix elements there are to be replaced by reduced
matrix elements of 3C or φ. These reduced matrix elements are readily
obtained by multiplying both sides of the equivalents of Eq. (131) with
the factor accompanying | (i \ r \ f) |2 and summing over all magnetic
quantum numbers, again using the above orthogonality relations. This
procedure leads to

IJS 1 Ji \/Jf 1 Ji \

(«flœlff)) = Κ - Ι Ρ ^ Ί
\M, μ -Mi/Xu/ μ -M//
X {{MM,* I OC I Mm*/)). (133)

When 3C or φ can be written as the sum of two operators affecting only
initial and final states, respectively, this simplifies considerably. Then we
have either Mi = M/ or Mf = Af/, and the orthogonality relation for
3-j symbols allows one to write, e.g.,

(i | 3d | 0 = [1/(2Λ + 1)] Σ {Mi | Ki I Mi), (134)

Mi

which is just the average over magnetic quantum numbers one would have
expected from the outset. In view of this average, it is sufficient [5] to do
actual calculations, say, in the classical path approximation, only for one
especially convenient perturber orientation, e.g., with velocities in the
x direction and impact parameters in the y direction ("collision axes"),
and then to rotate the atom. Alternatively [4, 51a], the average over
magnetic quantum numbers may be replaced by the average over perturber
orientations ("angular average"). Then the diagonal matrix elements do
not depend on M, and the others vanish.

In most calculations, one now uses perturbation theory, i.e., uses the
Dyson series given in Eq. (90) and takes the classical path approximation
together with the multipole expansion of the perturbation Hamiltonian in
Eq. (58). As for hydrogen lines, JC is then usually dominated by second
order (Dyson series) dipole-monopole (multipole expansion) terms also
for the "isolated" lines considered here. But up to these terms, 3C is now
indeed a sum of terms as given by Eq. (134), because the upper-lower state
interference term vanishes for allowed dipole transitions on account of
selection rules. Furthermore, all first order Dyson terms and products of,

54 I I . THEORY

say, first order dipole and quadrupole terms vanish on the average over
orientations. Therefore, second order quadrupole terms as given by Eq.
(134) and an upper-lower state first order quadrupole interference term
constitute the next correction to the leading dipole-dipole terms calculated
originally [51a]. This interference term involves products of first order
quadrupole contributions as given by the second term in Eq. (105),
namely

l Γ UM * . «a t^Jt. is. Ά (L'Y" ir„ + Y,_t)t (I35)
n J_o0 mp2v a 2 mp2v a02 \ 1 5 /
0

when g and v are used as x and z (collision) axes, and when the components

of the atomic electron position vector are called x, y} and z. Ignoring spin,
the matrix elements of the spherical harmonics are [70]

(I'm! | r2±21 lm)

Ti/2 fl 2 ly )(
= (-1)* [j-(2l' + l)(2!+l)T(
L 4 x J \ 0 0 0 / \ - m ' ± 2 ml

(136)

and the required average of the cross term [72], according to Eqs. (133),
(135), and (136), is

- \t ((tf I /_+" K,i(0 dt £* UifV) dt | ify

-l(^)(''(li),l'')W(s),W*+"*+»

/h 2 Ιλ/lf 2 ΙΛ II, 1 U \/l, 1 li \

\0 0 0 / \ 0 0 0 / \mf μ -mi/\m/ μ -m//
I li 2 h \ I lf 2 l, \
\—m,' 2 mi) \—m/
2 m,J

/h 2 l\/lf 2 l,\ ii 2 i<) (137)
\ 0 0 0 / \ 0 0 0> lt 1 l/J

I I . 3 . IMPACT APPROXIMATION 55

[Note that the reduced matrix elements in Cooper and Oertel [72] already
contain, e.g., the factors

(U 2 l\
(2U + 1)

\0 0 0/

and that the product of the remaining 3-J symbols is negative for
| U — h I = 1 which, in turn, is the only case of present interest.] The
sum in the second version of this expression is over all magnetic quan-
tum numbers, and the factor 2 compensates for taking only products of
Y22. (Mixed terms Y22 Y2-2, etc., do not contribute because the sum of the
magnetic quantum numbers in all 3-j symbols must be zero.) To obtain
the final result, the magnetic quantum number 2 is replaced by another
summation index, which must be corrected for by dividing by 2 · 2 + 1 = 5,
and the sum is then seen to be a Q-j symbol. All phase factors can be ignored
in this derivation, which leads to a special case (/»·—> U, etc.) of a more
general formula [68] where spin is taken into account.

An analogous derivation of the second order quadrupole terms, using in
addition Eq. (106), yields [68, 72]

hdf u^dtf w)*' |t)

_125_{W±V)Z(Vw \\(!)I '\a\oU/ )I,</ *+»Ç'1'\,01 0 0/ (138)

What is actually needed, except for Z/ = U , is a corresponding expression
with U(t) replaced by U'(t) both in Eq. (138) and in the corresponding
expressions for the dipole contributions, e.g., in

^ ( i | / " Ua{t)dtf Ua{t')dt' \i\

/ Ir I V A " 1 l%\2

+Ι
0
)Σ(^1-.Ι'·)'*' »( , J= 2(— V
3 \mpvt
.KAYW^UY-ei™
3 \mpv/ w, \ |a0| / 2li + 1 (139)

[This expression follows by specializing the (hydrogen) results in Eqs.

(105) and (107) to %' = i, use of 3-j symbols as in Eq. (131), and by

summing over all magnetic quantum numbers with the help of the appro-

56 I I . THEORY

priate orthogonality relation. The second version is then based on an
algebraic relation [70] for the remaining 3-j symbol.] Comparison of the
(dipole interaction) double time integrals over U(t) and U'(t), i.e., of
integrals with or without exponential factors exp(za>/"i0 involving the
angular frequency separation ωνι between levels I" and I, shows that Eq.
(139) must be multiplied by [51a]

A(z)+iB(z)=-J^ dx j _ M dx> exp[>(z - x ' ) ] ( 1 + xi)m{l + x,i)m

(140)

to account for the inherent level splittings. [Were the angular average not
taken, (1 + x) (1 + x') would occur instead of 1 + xx'.~] The quantity
z, namely

z = ωι»φ/ν, (141)

is a measure of the deviation from hydrogen-like behavior. The magnitude
of this dimensionless variable tends to stay below z « 1 for significantly
contributing perturbing levels (Z"), impact parameters (p), and electron
velocities (v).

The real part A{z) of the double integral over x = vt/p, etc., can be
written [51a] in terms of modified Bessel functions of the second kind,

A(z) =z*lK1*(\z\) + # o 2 ( | * | ) ] . (142)

It naturally also occurs in semiclassical (impact parameter method)
calculations [73] of inelastic cross sections for electron-impact induced
transitions, the contribution to the line width w involving A (z) being one
half the corresponding inelastic collisional rate (see also Sections II.3e
and II.4d). Since impact parameter method results for cross sections are
improved [73, 74a] if symmetrized to fulfill reciprocity relations between
cross sections (as functions of electron energy) for inverse processes

aitivi) = {gk/gdivk/viYak^Vk), (143)

where Vi (vk) is the initial (final) velocity of the electron causing the
transition i —► k and where the g are statistical weight factors, sym-
metrization might also improve semiclassical line broadening results [68,
72]. However, this procedure is by no means unique, and it is also not at
all clear that A ( | z | ) should be set to zero for electron energies below
threshold energies for inelastic processes [75]. Fortunately, these questions
are not very important for neutral atoms, because for them thermal elec-
tron energies are usually larger than such threshold energies for transitions
essential to the broadening. (The situation is different for positive ion lines,
especially those involving low-lying levels.)

I I . 3 . IMPACT APPROXIMATION 57

Since A(z) and B(z) are real and imaginary parts of the same complex
function, B(z) may be obtained [51a] from A (z) as a Cauchy principal
value (P) integral (Hilbert transform), making use of the fact that A(z)
is an even function,

T J-n Z — Z T J0 Z2 — Z2

= irz2tKo(z) h{z) - Kx{z) h(z)l (144)

(The closed expression in terms of Bessel functions was obtained by
Klarsfeld [76a], using contour integration.) For numerical convenience,
the integral has been transformed in various ways [51a, 72, 77a], but all
results as shown in Fig. 6, together with those for A (z) and the corre-
sponding quadrupole functions (see below), are in agreement with each
other. While A (z) vanishes exponentially at large | z \ according to

A{z) ~ 7 r | * | e x p ( - | 2 | z ) , (145)

the function B(z) decreases much more slowly in this "adiabatic" limit,
namely

B(z) ~T/4Z. (146)

Higher-order terms in this expansion in inverse powers of z have also been
calculated [72, 77b]. They are useful in a z range where direct use of Eq.

Ι.Θ 11 11 1 1 1 11

1.6

1.4 UlrULL

1.2 - QUADRUPOLE
1.0

v B0.8
'">\ -»«»
y^^~ ~ ^/ \ \\
0.4 Bvs.0.6

0.2 ^

1. . i ~7T~—

12 345

Z

FIG. 6. Characteristic functions of the variable z = ωρ/ν for straight classical path
diagonal ^-matrix elements (after Cooper and Oertel [72], except that the quadrupole
functions Aa and Bq were multiplied by i).

58 I I . THEORY

(144) is hampered by large cancellations. Also, as expected from the
hydrogen results, A (z) and B(z) approach one and zero, respectively,
for z = 0.

Corresponding functions [72, 76a, 77a] for the second order quadrupole
terms are

Aq(z) + iB^z) = - / dx dx' exp[*z(:r — a/)]

4 •'-oo '-oo

2(l+*2)(l+s'2) -3(s-a;')2

(1 + x2)5/2(l + z'2)5/2 *l ;

(Note that, e.g , Cooper and Oertel [72] defined functions A4(z) and £*(z)
without the f factor.) Again, the real part is familiar from the impact

parameter method for (quadrupole transition) cross section calculations

[71, 78] and can also be expressed in terms of modified Bessel functions

of the second kind,

A^z) = (*V4)[K22(| z I) + 4JW(| z |) + 3tfo2(| z | ) ] . (148)
Klarsfeld [76a] has an analytic result for the corresponding B function,

£ q = z2[£(z) + b(z) - irffiW/iOO] (149)

in our notation. These quadrupole functions are compared in Fig. 6 with
the corresponding dipole functions, with which they coincide only at z = 0.
Asymptotic relations for large | z \ are

Aq(z) ~ 7 r | z | 3 e x p ( - 2 | z | ) , (150a)

£q(z) ~9ττ/16ζ. (150b)

(The higher-order terms given by Cooper and Oertel [72] for J5q disagree
with Klarsfeld's [76a] general results.) Deriving characteristic functions
for higher multipole interactions is of course possible, but is of little value
unless the Dyson series is first taken to fourth and higher orders. Since
(V | r 11)2 « (I | r2 \ I) « 2 n W for typical matrix elements, it is clear from
Eqs. (138) and (139) that the multipole expansion parameter is
(I | r2 | l)/p2 « 2n4(a0/p)2, while according to Eq. (109) that for the Dyson
series is (n2h/mpv)2 = η4(λ/ρ)2. However, the square of the thermal
DeBroglie wavelength (divided by 2ir) is typically

λ2 = (h/mv)2 = h2/mkT = (2EH/kT)a02f

so that the "Dyson" parameter is larger by a factor En/kT than the multi-
pole expansion parameter, as has already been noted following Eq. (108)
for hydrogen. For neutral atoms, quadrupole interactions can thus be

II.3. IMPACT APPROXIMATION 59

important only when splittings between levels connected by dipole inter-
actions (Al = zbl) are unusually large. Since levels with Al = ±2 will
normally be still farther apart, causing Aq(z) to be even smaller than
A (z), it is more than sufficient in practically all cases to allow for the
diagonal {Al = 0) quadrupole terms [68] in addition to the dipole terms.

Before summarizing the quasi-classical method of calculations for neutral
atom lines, it might be appropriate to discuss the applicability of (dis-
persion) relations like Eq. (144) to entire widths and shifts [79], which,
according to Eqs. (99) and (100), are real and imaginary parts of the same
complex function. The question is whether this function of the electron
velocity is analytic, which is not at all obvious, e.g., in the presence of
resonances in elastic cross sections below inelastic thresholds. In general^
the dispersion relation would thus contain bound state terms in addition to
the usual integral. Only in a consistent quasi-classical calculation in which
thresholds, symmetrization, and conservation of energy are ignored would
dispersion relations without such terms be justified. (They are then useful
to check the internal consistency of cutoff procedures, etc.) Since bound
states (of the complete radiator plus perturber system) are more likely to
occur in the case of positively charged rather than neutral radiators, even
greater caution in the application of such dispersion relations is indicated
for positive ion lines.

Using Eqs. (92), (99), (100), (137)-(140), and (147), the "weak colli-
sion" widths and shifts in the classical path approximation for isolated
neutral atom lines, whose terms are well described by a one-electron model,
can be written as

w' + id' = 2πΝν Γ pdp\l (—Y Σ fo ILI ! max(I'jj, U,f)
2U,f + 1
Uf)

X [A«',) + <*(,&)] + U^) Σ («, I^YI hj)

( 15 \mp2v/ ^ \ | \a0/ \ /
V- 2 l- \
+ämimm\(u\'>)
/li 2 1%\ /I/ 2 lf\ 1 li 2 Ιχ J

(151)

60 I I . THEORY

(Note again that the upper-lower state term is negative for | U — lf \ = 1.
Also, in the case of equivalent electrons, allowance for l-l coupling should

be made.) We have postponed the velocity average and defined, e.g.,

zi SE l(Ei - Εν)/*ϋ1ρ, (152)

*/' Ξ l(Ef> - Ef)/h)]p. (153)

[This assumes that the (odd) B (z) functions are positive for z > 0.] A

lower impact parameter cutoff pmin is necessary to avoid serious errors at
small impact parameters, where the Dyson series and perhaps also the

multipole expansion should be carried further. Even then, such extended
relation would become invalid below p « η2α0, because it was assumed in
the multipole expansion in Eq. (58) that the perturbing electron was out-

side the atomic electron, and in any event below p « λ = h/mv) because
here the classical path assumption would certainly break down. However,
usually pmm in Eq. (151) has to be chosen larger than both n2a0 (which at
the same time ensures that higher multipole terms are small) and λ, since

higher order Dyson terms are important already at larger impact param-
eters p « n2h/mv, at least for small \z\. That z's corresponding to such

p values are indeed small (\z\ < 1) follows from

| « | « (AE/fiv)p ~ (n2 AE/3kT) « (2EK/3nkT), (154)

the splitting AE between relevant levels almost always being much less

than the distance of adjacent principal quantum number terms of hydrogen.

Several procedures have been proposed [51a, 68, 69, 72] to determine

Pmin and to estimate contributions from collisions with p < pmin . The
simplest of these uses pmin values corresponding to

ID.· · ] | « ι (155)

in order to avoid violations of unitarity conditions for the S matrix, where

[. . . ] = i — S/*Si is the integrand in Eq. (151), except for the factor p.

The collisions within pmin (the "Weisskopf radius") are then assumed to
contribute essentially only to the width as fully disruptive collisions would,

namely by a "strong collision" term

wB = irNvp2min . (156)

Such a procedure can be modified slightly [51a] (without affecting the
high velocity results) by introducing small corrections, which ensure that
the second-order dipole contributions in the large | z | (adiabatic) limit
agree with the so-called phase shift approximation [80, 81] as given by

I I . 3 . IMPACT APPROXIMATION 61

Eq. (238). (This approximation ignores inelastic and, perhaps more
seriously [9], superelastic collisions, but otherwise treats the Dyson series
exactly. It is of course also correct only to the extent that higher multipole
interactions are negligible.) The hope is that having both high- and low-
velocity limits right will result in satisfactory results for intermediate
velocities (| z \ « 1) as well. Other slight modifications may ensure com-
patibility with dispersion relations [79] and can thus be used to estimate
errors from the approximate treatment of strong collisions, typically ± 15%
of the width for the shift and somewhat less than that for the width.

Another approach, theoretically more straightforward but computa-
tionally more involved, is based on the analogy between Dyson series and
Taylor series expansion of ex. Ignoring time ordering, etc., it is then natural
to assume [82, 83]

w + id = 2ΊΓΝΌ [pdpll - e x p ( - [ · · ·])]> (157)

and to use pmin « 0 (or, say, λ) as the lower limit. This procedure is com-
patible with both unitarity conditions and dispersion relations, but does
not necessarily yield more accurate results than the numerically simpler
method discussed above.

A related method is to separate contributions from elastic and inelastic
collisions by beginning with an extended phase-shift approximation,

w + id = 2irNv ] pdp[l - exp(-27j - 2ίφ)], (158)

η(ρ} ν) and φ(ρ, ν) being real and 2(η + ΐφ) to be identified again with the
large bracket from Eq. (151). Sometimes, it is now assumed [84] that

either η or φ is zero for a given collision and that η is always small, i.e.,

w « 2rNv [ p dp [2η + 2 sin2 φ], (159)

d « 2rNv I pdp [sin 2φ]. (160)

This has been further simplified by replacing the sine functions by their
arguments, which still leaves elastic terms in the width, even though only
the second order dipole term of the Dyson series is actually calculated.
This inconsistency and the arbitrary assumption that either η or φ vanishes,
which is manifestly incorrect for | z | « 1 (see Fig. 6), are obvious short-
comings of this procedure, while correct separation [2, 3 ] of inelastic terms

62 I I . THEORY

could be of some advantage. They are obviously related to excitation and
deexcitation rates, and experience gained in atomic collision theory could
be used as a guide in the choice of cutoff impact parameters and strong-
collision terms.

Similar to the extended phase shift approximation is an approach [68,
69, 85, 86] based essentially on Eq. (95) or its equivalent [ 3 ] in terms of
inelastic cross sections and differences of elastic scattering amplitudes for
scattering on atoms in the upper or lower states of the line (see also Sections
II.3e and II.4d). The semiclassical approximation for the inelastic cross
sections [73], which corresponds to the so-called unitarized Born approxi-
mation, then yields the dipole contribution to the width as given by our
Eq. (151) plus a strong-collision term. For the purely elastic dipole terms
the second Born approximation for scattering on a polarization potential
could be shown [87] to give the adiabatic or phase-shift (large z) limit
[80, 81] referred to above, while elastic quadrupole terms were taken as
in Eq. (151). With suitable cutoffs and symmetrization, this procedure
[68, 69, 85, 86] yields widths that are continuous functions of the incident
electron energy, as are the corresponding expressions for the shifts. How-
ever, the use of the second Born approximation for elastic cross sections
and of only the first Born approximation for inelastic cross sections (corre-
sponding to fourth and second order Dyson series terms, respectively)
seems inconsistent. Moreover, this procedure is computationally rather
involved.

Most numerical results [7, 51a, 72] have been obtained with the pro-
cedure first described. A priori, it, as all the others, will only be reliable
when strong-collision contributions are small, say <20%. (Actually, a
width contribution of about 50% from strong collisions is more typical for
neutral atom lines.) Very useful in such calculations are functions [72, 88]
(see Fig. 7) such as

a(z) = Γ A(z) (dz/z) = \z\Ko(\z\)K1(\z\)) (161a)
Jz

aq(z) = f A^z) (dz/z*) = iJ^fl z |) [2 | z | K0(\ z |) + Κ,(\ z | ) ] ,
(162a)
JZ

which enter through the p integral in Eq. (151). (Originally [51a], a (z)
and b(z) had been calculated by numerical integration, but, as pointed out
later [72, 88], with errors as large as 20%. Fortunately, corresponding
errors in widths and shifts [7, 51a] were much smaller.) The integrals for

I I . 3 . IMPACT APPROXIMATION 63

FIG. 7. Characteristic functions of the variable z = ωρ/ν for the straight classical path
weak collision contributions to the widths and shifts of isolated lines from dipole and
quadrupole interactions (after Cooper and Oertel [72] and Klarsfeld [76a], except that aq
and ba were multiplied by $).

the b functions are [76a]

b(z) = £ T - nKo(z)Ii(z)9 (161b)
6q(s) = b{z) - ^πΚ1(ζ)Ι1(ζ). (162b)

They are also shown in Fig. 7, and their values and values of the other
characteristic functions for the electron impact broadening are collected in
Table I. Table II contains values of 3-j and 6-j symbols required for the
quadrupole contributions, while the reader is referred to the literature
[42, 43] for convenient and usually adequate calculations of radial matrix
elements. [See also Section II.2c, and note that our (I \ r2 \ I) is as in Eq.
(81) without the last factor, while (V | r | I) is as in Eq. (79) multiplied
with (4P — 1)1/2.] The only other data necessary for calculations of widths
and shifts are excitation energies, which should be appropriately averaged
over quantum numbers J associated with given I for the outer electron.
Only when that is considered too crude an approximation must additional
vector coupling coefficients be inserted [68] into the formulas given in this
section. Still, quantum-mechanical effects remain obscure in such more
elaborate classical path calculations, which are not found [89] to be of any
superior accuracy for neutral atom lines broadened by electron impacts,
even though most of the earlier calculations [ 7 ] neglected lower-state

TABLE I

C H A R A C T E R I S T I C " D I P O L E " F U N C T I O N S A(Z) A N D B(Z) F O R R E A L A N D I M A G I N A R Y P A R T S O F T H E S T R A I G H T C L A S S I C A L P A T H S M A T R I X
A N D C O R R E S P O N D I N G F U N C T I O N S a(z) A N D b(z) F O R T H E W I D T H S A N D S H I F T S O F I S O L A T E D N E U T R A L A T O M L I N E S 0

z A{z) a(z) B(z) b(z) AQ(z) aq(*) BQ(z) bq(z)

0 1.000 00 0 1.571 1.000 00 0 0.7854
0.1 1.030 1.005
0.2 1.035 2.392 6.094 · IO"2 1.533 1.021 50.95 4.380· IO"4 0.7579
0.4 0.9622 1.074 13.08
0.6 0.8287 1.674 0.1621 1.460 1.128 3.360 4.575- io-* 0.7063
0.8 0.6799 1.161
1.0 0.5396 0.9738 0.3588 1.285 1.155 1.457 3.902. 10"2 0.5851
1.2 0.4181 1.113 0.761
1.4 0.3181 0.6078 0.4981 1.111 1.040 0.435 0.1171 0.4692
1.6 0.2387 0.944 0.261
0.3897 0.5759 0.9558 0.1610 0.2303 0.3699
0.1014
0.2534 0.6059 0.8233 0.3606 0.2890

0.1661 0.6031 0.7127 0.4897 0.2248

0.1094 0.5801 0.6212 0.6041 0.1747

7.237 · 1Ò"2 0.5459 0.5459 0.6957 0.1359

1.8 0.1771 4.797 · IO"2 0.5068 0.4838 0.837 6.467- IO"2 0.7614 0.1060
2.0 0.0830
2.5 0.1302 3.186 · IO"2 0.4669 0.4325 0.726 4.166 · IO"2 0.8020 4.632- IO"2
3.0 2.705 - IO"2
3.5 — — 0.3760 0.3384 — — 0.8140 1.665. IO"2
4.0 1.079- IO"2
5.0 2.537 · IO"2 4.185 · IO"3 0.3055 0.2764 0.2806 4.992- IO"3 0.7483 5.215 - IO"3
6.0 2.911 - IO"3
7.0 — — 0.2536 0.2334 — — 0.6554 1.797. IO"3
8.0 1.189 - IO"3
9.0 4.486 . IO"3 5.572- IO"4 0.2158 0.2022 0.0832 6.353 · IO"4 0.5638 8.289 - IO"4
10.0 6.012. IO"4
7.496· IO"4 7.465 - IO"5 0.1663 0.1598 2.102 - 10~2 8.282- IO"5 0.4215

— — 0.1358 0.1342 — — 0.3313

— — 0.1151 0.1131 — — 0.2736

— — 0.1001 9.878- IO'2 — — 0.2344

— — 8.859 -IO"2 8.769 -IO"2 — — 0.2077

— — 7.947 -IO"2 7.884 -IO"2 — — 0.1828

β From Griem et al. [5la], Cooper and Oertel [72], and Klarsfeld [76a]. In the last four columns are the equivalent "quadrupole"
functions [from Cooper and Oertel [72J and Klarsfeld [76a] but multiplied by a factor i to achieve Aq(0) = 1]. For all functions,
values at larger z than tabulated here can be estimated from asymptotic formulas, i.e., Eqs. (145), (146), (150a), and (150b) or their
integrals over dz/z for the dipole functions and over dz/z3 for the quadrupole functions.

66 I I . THEORY

TABLE II

NUMERICAL VALUES OF SOME 3-j AND 6-j SYMBOLS FOR ELECTRON IMPACT BROADENING
CALCULATIONS

(a) 3-j symbols:

[o o o j

For k = 1 and | U — li> \ — 1, we have

m β I" max(*»,^) ] 1 / 2

L(2^ + i)(2^ + i)J

For k = 2 and | î< — Z<» | = 0, we have

Ix

0 0.0
1 2/\/3Ö_
2 -2/V70
3 2/VÎ05
4 -W\/385

(b) 6-j symbols:

x= \ \, L = max(ï», Î/), | Z,· - 1/ | = 1

[h 1 I/J

If Z< or If = 0, then x = 0.
If Z,· and Î/ > 1, then

, , [tf^ - 1) X ( 4 ^ - 9)]^2
1 1 L X (4L* - 1)

broadening and quadrupole interactions. (Lower-state dipole interactions
were accounted for in the original calculations [51a] for neutral helium
lines.) Calculated Stark broadening parameters from the first procedure
described above can be found in Appendix IV.a.

II-3cß. Overlapping lines.

When electron impact widths, calculated on the assumption that the line
in question is isolated in the sense of the preceding section, are comparable
to the splitting between energy levels involved, higher-order terms in the
expansion mentioned at the beginning of that section must be included.
Frequently, the "unperturbed" levels in such cases are substantially

I I . 3 . IMPACT APPROXIMATION 67

influenced by quasi-static Stark effects caused by ions, and besides the usual
(quadratic, linear, or intermediate) Stark shifts, so-called forbidden com-
ponents occur, which are related to but not the same as the "plasma
satellites" caused by intense oscillating fields (see Section II.5c) from
suprathermal fluctuations. In the limit that electron impact widths are
actually much larger than inherent atomic level splittings, and that there is
no longer a meaningful distinction between allowed and forbidden com-
ponents (linear Stark effect), one would then come back to the situation
met in case of one-electron systems (see Sections II.3a and b ) . The present
section is thus mainly a discussion of situations intermediate between
hydrogen lines, etc., and isolated lines being well separated from each
other (except for rotational degeneracies). Such a delineation of the concept
of overlapping lines implies that this section cannot be concerned with
the problems encountered when lines arising from upper states with
adjacent principal quantum numbers begin to overlap. While there are
similarities, the latter problem is much more difficult to solve and, as a
matter of fact, has not yet been amenable to a solution which is both useful
in practice and theoretically satisfying. (The one-electron approximation
[ 6 ] for the "window spectrum" discussed in Section II.4d is a possible
exception.) In other words, not much has been said that substantially
improves the well-known Inglis-Teller [18] treatment (see Section II.2).

The starting point of detailed calculations [90-93] (see also below) in
the present category is Eq. (101), except that the linear, ion-produced
Stark shift is replaced by the "exact" shift ACO(JF), i.e.,

L(o>) = - (1/ΤΓ) Re Tr Γ dF W{F) {Adp Δω - i Aco(F) + φ]-1}.
•'ο

(163)

As Aœ(F) is not a spherically symmetric quantity, rotational invariance
arguments cannot be used here to reduce the order of the matrices to be
inverted. However, when the ion field direction is chosen as the z axis, then
Aœ(F) has no matrix elements between states with different magnetic
quantum numbers, nor has the electron broadening operator φ, unless its
symmetry is destroyed, e.g., by magnetic field effects or anisotropies in the
electron velocity distribution. Usually, the calculation of Eq. (163) can
thus proceed for fixed magnetic quantum numbers, over which the results
are then simply averaged at the end. [Strictly speaking, this diagonality
in magnetic quantum numbers only holds when electron impact broadening
of the lower levels of the line can be ignored, but experience gained with
hydrogen and ionized helium [59] suggests that φ-matrix elements corre-
sponding to changes in magnetic quantum numbers are negligible over

68 I I . THEORY

most of the line profile [64b, c]. See also the discussion, e.g., of XiXf following
Eq. (111).]

In general, the quantities Ad and φ in Eq. (163) do depend on the ion
field, the former because "Stark" eigenfunctions will be used in place of the
unperturbed (nlm) eigenfunctions, the latter both for this reason and
because field-dependent (rather than unperturbed) energies may have to
be employed in calculating the arguments of the A and B functions from
Eq. ( 141 ). With such Stark eigenfunctions, the usual diagonality of φ in the
i-quantum number no longer holds, and it becomes necessary to calculate
φ-matrix elements also for Al = 1, etc., because of corresponding admixtures
to the unperturbed wave functions (in the dipole approximation for the
perturbation by ions). In this case, the double integrals in the Dyson series
for φ involve exponentials with different frequencies; e.g., z(x — x') in Eq.
(140) is to be replaced by zx — z'x', and A and B become functions of both
z and z'. These new (dipole) A functions are straightforward generaliza-
tions of Eq. (142), namely [77, 91]

A(z,z') = \zz'\Kl{\z\)Kl{\z'\) +zz'Ko(\z\)Ko(\z'\), (164a)

while B(z, z') is given [76a] (for z > z' > 0) by

B(z, ζ') = τζζ'ΙΚο(ζ) h{z') - Kx{z) h(z')J (164b)
For z/z' < 0, one has [76a] B(z, zf) = 0. The relations [77]

A(z, ζ') = Α(-ζ,-ζ') (165)

B(z,z') = -B(-z, -z') (166)
A(z,z') = A{zf,z) (167)

B(zfz') =B(z',z) (168)

finally characterize the symmetry properties of these new functions.
The signs of z and z' are opposite when the intermediate (perturbing)

level i" lies between the two levels i and i' connected by φ, and are the same
otherwise; i.e., z and z' are to be calculated from

z = ±l(Ei - E^/hv^p, (169)

z' = ± [ ( J ^ - Ei..)/fà>lp, (170)

for broadening of upper and lower levels, respectively. Since for small
absolute values of z} zf the A and B functions approach the hydrogen values
A = 1, B = 0, the smallest relevant values of p will often be well estimated

by Eq. ( I l l ) , and therefore those of, say, z by

Zmin « W - t t / 2 ) (AE/2kT), (171)

I I . 3 . IMPACT APPROXIMATION 69

AE being a typical level separation between levels connected by dipole
transitions. For diffuse helium lines (2P-nD), which provide outstanding
examples of overlapping lines (with levels nF, etc.), this quantity is less
than or about 10~2 eV in most cases, and here we usually have | zm-m | < 0.1.
The hydrogenic (A = 1, B = 0) approximation is then good to better
than 10%, at least for the widths. This estimate can be improved upon by
considering the functions a (z, z') and b(z, z') which arise after integration
over impact parameters. Assuming again that z > z' > 0, a{z>z') is a
generalization [91] of Eq. (161), namely

a{z,z') = / (dy/y)A(y1z'y/z)

Jz

= [»'/(« + 2')][#o(| z I) Ki(| *' I) + *o(| z' I) κ,{\ z | ) i

(172a)

with the corresponding b function [76a]

b{z, z') = τ[ζζ'/{ζ + *')X*iOO h{z') - Ko(z) h(z'n (172b)

(Characteristic "nondiagonal" functions for the case of quadrupole inter-
actions are derived by Deutsch and Klarsfeld [76b].) For small absolute
values of z and z', the width function a (z, zf) is well represented (to 5%
or better for | z |, | ζ' \ < 0.2) by

a(z,z')~ln(l/z>), (173)

z> being the larger of the two absolute values. This is just as expected from

the hydrogen result a = ln(pmax/pmin), with pmax « hv/AE determined
from considerations of adiabaticity, if pmin and z> are chosen according to
Eqs. ( I l l ) and (171), respectively, and using the larger of the two level

splittings AE involved here.

For the shift functions [91], on the other hand, we have

\b(z,z')\<*/2, (174)

so that they are definitely less than 10% of the width functions only for
z> < 10~6. [The small z-limit of ò(z, ζ') is [76a] ò(z, pz) -> irp/(l + p), in
generalization of [51a] b(z) —> π/2.] This is often much too strong a
requirement for neglecting the 6 functions, e.g., because contributions to φ
from them tend to cancel. For nF, etc., levels, one actually finds z> < 10~2 ;
i.e., for n = 4 (where no cancellation is possible because of the absence of
a 4G level), the shift function is about 30% of the width function. Still,
the resultant shift of the forbidden component 2P-4F will hardly be
significant for the n = 4 triplet line. First of all, the forbidden component

70 I I . THEORY

depends on mixing by ion fields of 4 3D and 4 3F, so that the effective φ for
the forbidden component is some linear combination of the φ'β calculated
for pure 4 3D and 4 3F levels (see below), the former having a negligible
electron impact shift because of the near cancellation of the 4 3P and 4 3F
contributions [see Eq. (175) ] . Second, broadening by ions of the forbidden
component is comparable to that by electrons, and the shift in question is
thus not likely to exceed, say, 5% of the full half-width [90] of the 2 3P-4 3F
feature. For higher members of the triplet series [92], errors from omitting
the b functions are even smaller (in the vicinity of 2 3P-n 3D, F, etc.), but
this approximation would not be sufficient for the singlet series. Here n XD
lies below both n lP and n *F, i.e., is "red shifted" by both. However, the
level splittings are so small that the high temperature approximation
b « TT/2 is applicable (to < 5 % ) , and both a and ò functions are thus so
weakly dependent on their arguments that even here there is no real need
to use the a(z9 zf) and b(z> z') functions. Moreover, Debye shielding is quite
important, leading to a reduction especially of the shift (see Section II.5a).
Except for the necessary replacement for ò = π/2, a good approximation
for all I > 2, n > 4 neutral helium levels is therefore the so-called high-
temperature limit [51a]

Γ1 / 2kT \ E,-Ew >1

in which, in addition, dipole matrix elements of hydrogen and a strong
collision term ("f") equivalent to assuming an average value near 1 for
the matrix element of 1 — S at small impact parameters were employed.
(Quadrupole corrections are again of order kT/En when compared to the
strong collisions term and can thus usually be neglected, as can the Δ η ^ Ο
perturbations and those of the lower state.)

To proceed further, φ must be transformed to the Stark representation,
with the transformation coefficients (a\lm) and Stark shifts to be deter-
mined from the generalization of Eq. (70), namely the system of linear,
homogeneous equations (with a = V for F = 0)

Σ {LEi - Eam(F)-]òn + eFzvl(m)) (a \ Im) = 0. (176)

I I . 3 . IMPACT APPROXIMATION 71

(Remember that the ion field is taken to be in the z direction.) Here

Eam(F) = hü)am(F) are the shifted levels in the ion field which are needed
in Eq. (163), Ei is the unperturbed (empirical) energy, and the z-matrix

elements [34] for V = l db 1 are in the hydrogen approximation

*"■'<"> - * » L ( 2 T + ! ) ( » + ! ) J "* (177)

(All other An = 0 elements are zero, and Z> is the larger of V, I.) The new
φ-matrix elements {a 9* a corresponding to inelastic collisions) are then

4>a>a{m) = Σ <*' I &*> <« I lm) Φιι , (178)

i

and the matrix elements of the operator Ad (or rather D, since we make the
"one-state" approximation throughout this section) are similarly

<«'l D | a) = Σ (<"'ß* I A | aß*)) = [1/(2Z + 1 ) ] <<*' | Zm> (a | Zm),

(179)

assuming the expansion coefficients to be normalized ( Σ « |(α | lm)\2 = 1)
and Z here to correspond to the allowed component.

Equation (163) withEqs. (175)-(179), summed over magnetic quantum
numbers (m = ± 2 , ± 1 , 0) of the n 3D level and averaged over the fine-
structure components of the 2 3P level (with minor refinements for φοο and
φιι, neglecting Im Φ22, etc., and approximately considering Debye shielding
or finite duration of the collisions) have been used to calculate detailed
profiles of the [90] n = 4 (4471 A) and [92] n = 5 (4026 A) triplet lines.
Calculations not based on the high-temperature limit but rather on the new
α(ζ,ζ') and b(z, z') functions discussed above have been made [91] for
n = 4 singlet (4922 A) and triplet lines, the latter agreeing usually to
within 10% with the other calculation [90], as was to be expected. Analo-
gous to the work of Barnard et al. [91] is a calculation [93] for 2 xS-3 Ψ
(5016 A), with 2 ^-3 XD being the forbidden component. Results also
exist (see Ya'akobi et aL [233] and Greig et al. [234] below) for the
21P-3o1D (6678 A) line, with the forbidden 2 1 P - 3 1 P component near
6632 A. (In the work of Ya'akobi et al. [233], the unshifted m = 2 com-
ponent was omitted.) Compared to isolated lines but also hydrogen lines,
all these profiles naturally show a much richer and density-dependent
structure. There is accordingly no simple scaling with electron density like
the approximate linear and two-thirds power laws in the two limiting cases,
which are only approached at low or high electron densities, respectively.
The same of course holds true for the 5D-6G transition of cesium [94, 95]
(see also Wu and Shaw [228] below).

72 I I . THEORY

Before closing this section, some cautionary remarks must be made
regarding the validity of Eq. (163) for overlapping lines. In regard to the
validity of the impact approximation, Eqs. (92) and following are justified
for off-diagonal elements only if we have, e.g., As | ω,·,·' | « 1. But off-
diagonal elements of the φ operator are required only when the inherent
level splitting is of the order of the width of the line, so that for essentially
nonoverlapping (ωα> ^> ωρ) collisions, this provides no additional restric-
tion—a fact already pointed out by Baranger [ 6 ] . If collisions overlap in
time (coil* < cop), however, As must be chosen as large as ω~λ (see Section
II.5a) for the collisions within the time interval As to be statistically
independent of those occurring earlier. This value of As tends to be large
enough to invalidate the impact approximation in cases where Debye
shielding is at all important, and it then makes little sense to use the
a(z, z') functions with cutoffs, say, at the Debye radius. As discussed by
Voslamber [96a], one should rather drop the impact approximation and
use an extension of the one-electron approximation [6] (see Section II.4d),
but numerically this would make very little difference [96b, c] near the
allowed and forbidden components.

Also, the quasi-static approximation for ions, which are actually respon-
sible for the generation of forbidden components rather than the electrons
which essentially only broaden them, can be questionable. Since this diffi-
culty might be most serious at low densities (see, however, Section III.5)
as encountered in stellar atmospheres, its discussion and that of asymptotic
formulas are deferred to Section IV.4b.

Forbidden components are naturally not restricted to neutral helium,
other good examples being neutral lithium and cesium (see Section III.5).

II.3d. Two- and More-Electron Ions

The principal difference between the broadening of neutral-atom and
positive-ion lines lies in the presence of the long range Coulomb interactions
between radiators and charged perturbers [69]. Since these long range
interactions are independent of the internal state of the radiator, they of
course cannot be the direct cause of widths and shifts. ("Penetrating"
monopole interactions, however, must not be overlooked here; see Section
II.5b.) Nevertheless, using the classical path picture, it is clear that, com-
pared with the straight classical path result, there will be an enhancement
in the instantaneous perturbation from the Coulomb attraction but a
reduction in the duration of the collision from the acceleration, always con-
sidering positive ions perturbed by electrons. As evaluated in Section II.3b,
the net result for helium ions, etc., is a reduction of the dipole and an

I I . 3 . IMPACT APPROXIMATION 73

increase of the quadrupole contributions to the second-order terms in the
Dyson series.

In Tegard to the usually dominant dipole contribution, a more detailed
analysis [57a] indicated, for He II and at typical electron velocities, a
breakdown of the second-order perturbation theory before this reduction
becomes significant for decreasing impact parameters, and it was thus very
tempting to neglect such Coulomb effects also for other ion lines [7]. This
generalization turned out to be premature, because it was based on argu-
ments ignoring the actual level splittings ω,ν between Al = ± 1 levels
involved in the second order theory, which would only have been justified
for

P < I ν/ωα* \. (180)

Corresponding values of the hyperbolic (eccentricity) parameter €, ac-
cording to Eq. (117), are

Γ , / m i M 2 ] 1 ' 2 Γ , /2kT\2SkTf2 /ιοιλ

for singly ionized systems. For the stronger lines from singly charged ions,
one usually has | ήωα* | « kT, and kT « EK , i.e., emax ~ 1. Therefore, the
above inequality tends to be inconsistent with the assumption of small
Coulomb effects (ey> 1), and a more careful examination of the situation
for isolated ion lines with | ήωα> | « kT becomes necessary.

Before embarking on the detailed discussion of semiclassical calculations
appropriate for singly ionized | ήωα> | « kT lines, it is worth noting that for
isolated ion lines from highly excited states, ω»ν is often small enough for
Eq. (180) to be consistent with small Coulomb effects. Since broadening
of the lower level and also quadrupole contributions tend to be small, a
fair approximation for the widths of such lines is in analogy to the high-
temperature limit for atomic lines as given by Eq. (175), using the ex-
pression for ionized helium

£ ( 1 — lAmin) + ln(€max/Cmin) « 1 + ln(€max/€min)

[see Eq. (126)] instead of è + ln(pmftX/pmin) :

Η(^)·^{<<+^-<<->φ'"('£>]

+ ï„,_P][;+h(<iu)]}. (182)

(More involved strong-collision terms would not yield any real improve-

74 I I . THEORY

ment.) For large arguments of the logarithms, this agrees with the one-

electron approximation of Baranger [ 6 ] based on Coulomb excitation

theory (see Section II.4d). Otherwise, the smaller of the two results is to be

preferred, i.e., (€max/€min) from Eq. (184) below should be replaced by
| ξ I"1 from Eq. (186) if this gives a smaller width. The €U±i in Eq. (182)
correspond to emax calculated from Eq. (181) using the cou±i level splittings,
while €min is to be estimated, e.g., so that unitarity of the S matrix is not
seriously violated. In analogy to Eq. ( I l l ) and using Eq. (123), this

condition results in

6min « (n2/2) (hv/e2) « (n2/2)(3kT/2EKy<2, (183)

as long as the corresponding p values, etc., are larger than n2a0, which is
the usual situation for singly charged ions. (Otherwise, penetrating orbits

are important, so that the multipole expansion must be modified.) In the

limit of large emax , these relations reduce to the neutral atom high-tempera-
ture limit, Eq. (175), except that n2 is replaced by n2/2, etc., from the

Z dependence of the dipole matrix elements. It is clear from the above dis-

cussion that Eq. (182) is hardly valid unless emax is indeed large, and we
therefore write

€U±1 4Α:Γ 2kT /SkT\1/2\ <184)
« min " T i l ΐ ' Ϊ Ί \(^r) l·
n2 I ηωι,ι±ι \ | ηωι,ι±ι | \2EH/ )

using Eqs. (181) and (183) and insisting on €min > 1. Of course, higher

multipole contributions may again be estimated by adding terms like those

discussed for ionized helium to the logarithms [see Eq. (127) and also

below].

To treat cases where both Coulomb effects and finite level splittings are

important, i.e., where €min is small but €max not very large, the double time

integrals appearing in Eq. (92) for the classical path approximation must

be evaluated for hyperbolic orbits. The real part of these (dipole) integrals

is characterized by a function A (£, e) which is a generalization [68, 97] of

the A (z) function for neutral atoms, namely

A(fc e) = (e2 - l)£2exp(,r£) [\ Κ'«&) I2 + [(*2 - D A 2 ] I X « ( « | 2 } ,

(185)

where K^(x) and Kf^(x) are a modified Bessel function of imaginary order
and its derivative. [Our function Α(ξ, e) differs by a factor (e2 — l)/c2
from its original definition [97], which also involved changes in other fac-
tors in the width expression, while the present definition does not.] The

I I . 3 . IMPACT APPROXIMATION 75

new dimensionless parameter £, for singly ionized radiators, is

ξ = (e2/hv)(hü>ii>/mv2) = η(ήωα>/την2), (186)

with the second factor often thought to be necessarily small for the un-
perturbed hyperbolic path to give a reasonable approximation to the actual
perturber path. For the lines of particular interest here, this is not always
the case, and some improvement has been sought [68, 69, 84] by replacing
the initial electron energy by some mean value of initial and final (\mv2 +
hœn>) energies, assuming implicitly that the inelastic transition has indeed
happened. Such modifications need not lead to better results, however, nor
do the closely related symmetrization procedures used to ensure consistency
with the reciprocity relations [Eq. (143)] for cross sections of inverse
inelastic processes. The point is that the quantities of primary interest for
these lines are elastic scattering amplitudes, for which the original (hyper-
bolic) classical path approximation may remain good even for | ήωα> | «
kT as long as inelastic transition probabilities are small, i.e., certainly as
long as second order perturbation theory remains valid for the majority of
the collisions. For the same reason, it may not at all be advantageous to
split the classical analog of Eq. (99) by means of the optical theorem (see
the following section) into elastic and inelastic contributions (proportional
to the A functions), setting the latter to zero below relevant inelastic
thresholds. Unless one is careful with the actual choice of cutoff parameters,
etc., such a procedure would introduce discontinuities into the line width
considered as a function of electron energy. This is rather unphysical and
would disagree in almost all cases with theoretical threshold laws also to be
discussed in the next section.

In short, for the weak collision contribution, the analog of Eq. (151)
with suitable replacements for the characteristic functions A (z) and B(z)
probably constitutes the best quasiclassical approximation. Figures 8a and
b show some A(£, e) and !?(£, e) [or, rather, A(£, δ) and Β(ξ, δ) with
δ = (e — 1)£], i.e., the hyperbolic path dipole functions which, in the limit
ξ —> 0, e —> oo, reduce to the straight classical path functions A (z), B(z)
with £e —► z. (The new variable δ is more convenient for interpolations.)
Corresponding quadrupole functions can be evaluated [74] as well, but it
is almost certainly sufficient to account only for the V — I quadrupole
terms, for which the ionized helium results are applicable in view of the
negligible level splitting (£ = 0). The £q($, e) function is then identically
zero, while Eq. (123) gives (see Fig. 5)

76 II. THEORY

2.00 A(e,8)
1.50

1.00

i / / Λ ^ \/ i=io.o\Y \ \ \
/ / A\ \0.50 L'/
/ \
f = ioo.\ \ \

s.p.'

%: 1.0 100 100.

(a)

Λ i \1.00 r\ I \ Βί*·δ)

0.50 sT%\i\ \

0.00 V ' f;i.o / «ywoo ^ = - ^ ^

></ / / /
' ^ / J\ \ ///

-0.50
o.i rz—>*.*' 10.0 KX).

(b)

FIG. 8. Characteristic "dipole" functions of the variables £ and δ — (e — 1)£ for hyper-
bolic classical path ^-matrix elements (after Sahal-Brechot [68], Feautrier [98], and Klars-
feld [99]: (a) real part; (b) imaginary part. Also indicated are the straight classical path
(s.p.) limits.

I I . 3 . IMPACT APPROXIMATION 77

The upper-lower-state quadrupole interference term must also be multi-
plied with this function.

The principal difficulty again arises in the determination of a suitable
minimum impact parameter pmin (—>emin) in the analog of Eq. (151), and
in estimating the strong-collision term. All procedures discussed in Section
II.3ca have been tried for isolated ion lines, but usually with much less
accurate results than for neutral atom lines. The reason is mainly that
strong collisions tend to be relatively more important, to the extent of being
dominant for many lines. Any real improvement of such classical path cal-
culations would thus have to come from higher than second-order solutions
of the time-dependent Schrödinger equation for the radiator-perturber
system. However, even such a major effort may prove futile, since in gen-
eral the corresponding fourth- and higher-order terms in the Dyson series
would mostly involve rather small angular momenta of the perturber,
making any quasi-classical calculation very uncertain. (Such higher-order
calculation has been attempted by Roberts [100] for inelastic contribu-
tions to the broadening of some selected lines.)

There is perhaps one exception to this, namely the low velocity limit for
impact widths and shifts. Very slow traversais will certainly not be able to
cause further excitation, but are also hardly capable of inducing deexcita-
tion through superelastic collisions, leaving elastic collisions as most ef-
fective. In Eqs. (158)-(160), the quantity η (not be to confused with the
Coulomb parameter) can now be neglected and the real phase shift φ
calculated in the adiabatic approximation ( | ξ | —> <» ), as far as dipole
interactions are concerned. The A (£, e) functions are then negligible, and
for small e the function Β(ξ, e) has the asymptotic expansion [68]

Β(ξ, €) ~ 3TT/2£(€2 - 1)8/2 = 3^4/2uit->mVp3, (188)

again multiplied by (c2 — l)/e2. [Taking the e —» 1 limit is not correct
mathematically, since the adiabatic condition that the relative rate of
change of the dipole interaction energy be much smaller than ω»ν is not
fulfilled at the two (finite) points of strongly curved hyperbolas where the
interaction vanishes, i.e., where (radiator) dipole and (perturber) field are
at right angles to each other. However, higher-order (Dyson series) terms
then of course dominate, and use of the above limit does indeed agree with
the corresponding limit of the phase shift approximation [101].] Compari-
son of Eqs. (151) and, for small ψ, of Eq. (158) therefore yields, e.g.,

vp\mpv/ \mpv/ u>, \ max(Z/', U)
= C(v)/p* s k/{pv)\ \a0\ / (2lu + 1)ωΐϊ"

(189)

78 II. THEORY

However, the angular average used here is appropriate only for the second
order Dyson term. If this difficulty can be ignored, Eqs. (158) and (189)
suggest, for the low-velocity limit (elastic collisions only),

Wo + ido = 2wNv f Pdp{l- exp[-t(C/p5)]}

= 2wNvC2/^ i°°(dx/xl+2/5)ll - e x p ( - i x ) ]
Jo

= 2TNVC2/ÒITTÌ2/*/2 sin (2ττ/5) Γ ( 2 / 5 ) ]

« 2irNv | C |2/5(0.60 ± i0.44) = 2wN(k2/5/v) (0.60 ± 0.44), (190)

a result which corresponds to an elastic cross section first derived by Bre-
chot and Van Regemorter [69a]. In analogy to the first (dipole) term in
Eq. (151), this may be written

, . 7 4x / ÄV ΛΤ ^ / „ I r I , V m a x ( V , W , .r N , _ N
3t> \ m / " \ I oo I / 2U + 1

with ac and òc obeying

ac + Ä. = I (0.60 ± Λ.44) [ftr Σif -*L A , I J! I z Y Î^^^-^>1T2/5

2 L Awn// \ I «o I / 2li + 1 J

L if \ I ao I / 2i,· + 1 J

Numerical calculations for actual cases demonstrate this expression to
be rather insensitive to the detailed properties of the ion in question, and
for strong lines of singly charged ions a suitable mean value turns out to be

ac + ibc ~ 0.4 ± z'0.3. (193)

The real part of this agrees rather well with an extension [102] of the
effective Gaunt factor (g) approximation [103] for inelastic collisions to
the predominantly elastic (low energy) range. [See Eqs. (458) and (459),
and note that, e.g., ac corresponds to ng/VS ~ 0.35 for an effective Gaunt
factor of 0.2. Compare also with the one-electron approximation of Baranger
[6] in Section II.4d.] The semiempirical Gaunt factor approximation, in
turn, agrees with measured widths of singly charged ions to within an
average factor of 1.5, a fact which was well documented by Griem [102]
(see also Section III.7), notwithstanding some repeated criticism [86, 104].

I I . 3 . IMPACT APPROXIMATION 79

Interestingly enough, most of the widths calculated essentially according to
Sahal-Brechot [68] for some ultraviolet lines [104] (C II, N II, Si II)
and visible lines [105] (C II, Mg II, Si II, S II) also agree fairly well with
the effective Gaunt factor approximation. The only clear exceptions are
the first five Si II multiplets of Sahal-Brechot and Segre [104] whose
calculated damping constants seem improbably low and too different
from each other. For higher ions (C III, C IV, N III, N V, O VI, Si III,
Si IV, S III, S IV, S V I ) , however, the detailed calculations [104] do
yield widths that correspond, especially for lines from 2s-2p and 3s-3p
transitions in Li- or Na-like ions, to larger effective Gaunt factors than had
been adopted by Griem [102]. The increase is by about a factor of 2 at
threshold (see also Section IV.6), which would be consistent with Gaunt
factors from measured excitation rate coefficients [106]. Nevertheless,
since the Gaunt factor approximation represents measured widths of
singly charged ions within an average factor of 1.5, incorporation of ac and
òc into the a and ò functions (see below), which arise from the integration
over impact parameters of the weak collision terms analogous to Eq. (151),
should result in a rather accurate description of the impact broadening and
shift at all electron energies for these ions.

As for the choice of the minimum impact parameter pmin (or €min), one
might argue that the value used for the high temperature approximation,
namely Eq. (183) foremin > 1 with the additional requirement pmin > ^n2a0
(neglecting here the curvature), i.e.,

Pmin « max{l(n2h/2mv)2 - ( θ 2 / ^ 2 ) 2 ] 1 / 2 , |η2α0} (194)

could be applied throughout, because it is certainly appropriate at high
perturber energies and because the weak-collision terms are in any case
negligible at low energies. When broadening of both upper and lower states
is important, a proper treatment of the strong collision contributions to the
width would necessarily contain some interference term also in the dipole
approximation. This term and a "direct" lower state term may be accounted
for by subtracting a lower state contribution from the sums in Eqs. (189)
and (192), as in other treatments [68, 69, 83, 84] of elastic collisions based
directly on the difference of φ» and φ / . However, while these treatments
appear to be more or less in line with Baranger's quantum-mechanical
expression for the width [ 3 ] (see the next section) or for that matter also
with his corresponding general semiclassical formula [2], they more likely
than not still lead to an overestimate of this interference term. This has
been demonstrated by fully quantum-mechanical calculations for the
Mg II resonance lines [107]. In this case, quadratic Stark coefficients
(~k) for upper and lower states almost cancel, but still the interference

80 I I . THEORY

Ο.Ι 1.0 IO 100

(b) 8

FIG. 9. Characteristic functions of the variables £ and δ - (e — 1)£ for the hyperbolic
classical path weak (dipole) collision contributions to the widths and shifts of isolated
lines (after Sahal-Brechot [68], Feautrier [98], and Klarsfeld [99]: (a) width; (b) shift.
Also indicated are the straight classical path (s.p.) limits.

term was found to amount to only about 25% of the width. The rather
approximate consideration of the angular average in the quasi-classical
estimates of higher order Dyson terms mentioned above may in part be
responsible for this discrepancy, but the complete disregard for the very

I I . 3 . IMPACT APPROXIMATION 81

important resonances in elastic cross sections in the semiclassical calcula-

tions (except for Roberts [100]) is probably a major cause for the large

errors in the interference term, perhaps together with the improper use of

the multipole expansion also for penetrating orbits. (See also Griem [64c].)

In conclusion, a quasi-classical calculation based on Eq. (151) with the

appropriate substitution of hyperbolic-path characteristic functions and

with ac or òc from Eq. (192) allowed for along with the analogs of the
straight path a and b functions, which arise after integration over impact

parameters, is probably just as accurate as more elaborate calculations, or

more so. (The sign of bc here of course is that of the corresponding weak-
collision term.) The new a function can be expressed [98] in terms of

modified Bessel functions, namely

a = F2(£, €min) = βΧρ(τξ) £€min Ki^€min) I K'i}t(&min) |, (195)

which is another generalization of Eq. (161). Its values as functions of
δ = (e — 1)£ and ξ are listed in Table III (see also Fig. 9) together with
those of the 6 function, which were first obtained by numerical integration
[68] and for which, as for Β(ξ, e), more convenient integral representations
and expansions are now available [99]. For the (elastic) quadrupole con-
tribution to the width a curvature correction factor

lim 2z2 a^z) -+ 1 + (3π2/8) {h/mpminvY{9miJa,Y (196)

z-*0

follows, in analogy to Eq. (126) in sufficient approximation, assuming
singly charged ions. Numerical calculations show that quadrupole inter-
actions are not very important even for resonance lines.

Appendix V contains a table of Stark broadening parameters computed
as discussed above, except that strong-collision terms, etc., were adjusted
to also give agreement with certain straight classical path limits. For
multiply charged ions, the reader is referred to Section IV.6.

CM rg IM CM oj tn m en *ι <*ι *■ y y ar 9- in

82

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IIIIII II
(S O)
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ί ί I* I I I I* I
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III I II I
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r^. nw4av- i(\ij ^io\ t«cr.(^\ i ai »*,c oa i aa cσ hN aM rO- cier ra- «(\-·! f^c oce σa cveve^ rσi H(vit ivr . ev tiir·a ai i aC £ % o C5
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îÎ III I I o2

^51 53 SÌ £S nia
tf +s-. , ,
|
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I I . 3 . IMPACT APPROXIMATION 87

II.3e. Quantum-Mechanical Calculations of Electron Impact
Broadening

The calculations discussed heretofore were based on the classical path
approximation, either in its pure form (straight or hyperbolic unperturbed
trajectories with initial and final velocities being the same) or modified by
symmetrization in regard to initial and (postulated rather than calculated)
final velocities, and by imposing threshold conditions on the characteristic
A functions. While there are a number of physical arguments [75] (see also
the preceding section) against these modifications as "improvements,"
fully quantum-mechanical calculations appeared necessary for settling
these questions and for assessing the accuracy of the classical path approxi-
mation as described in Sections II.3a-d. This is not to say that there cannot
be any real improvements within the classical path approximation, such
as the replacement of the impact approximation by more generally valid
procedures (see Sections II.4a-d), the evaluation of higher order "Dyson"
terms, or the inclusion of higher multipole interactions with proper allowance
for penetrating orbits. Many such improvements have been made, espe-
cially in the case of hydrogen La [38, 62, 63], but in regard to the two last-
mentioned improvements there is again considerable doubt whether they
are substantial in the sense of leading to better agreement with measure-
ments or fully quantum-mechanical calculations [64a, 67] (see also the
end of this section).

Besides using fully quantum-mechanical calculations to verify the
accuracy of the much simpler classical path approximation, there will
certainly also be situations where a quantum-mechanical calculation is
essential to obtain even a first estimate of the line profiles. For high mem-
bers, e.g., of the one-electron (hydrogen, ionized helium, etc.) sequence,
Eq. ( I l l ) for the minimum impact parameter suggests that angular
momentum quantum numbers for the perturbing electrons as small as

I « (nt·2 - n?)/Z (197)

are important, although higher multipole interactions will increase this
estimate some (see the end of Section II.3b). However, for corresponding
two- and more-electron systems, relevant I values are smaller, and quantum-
mechanical calculations are thus definitely required for the electron
broadening of some strong lines from multiply ionized systems. Stark
broadening of such lines has not yet been investigated experimentally, but
their wings may be important, e.g., for the opacity of stellar interiors (see
Section IV. 6). We should also mention in this connection that with respect
to the relative importance of quantum-mechanical effects when going from
neutral to charged radiators, the situation in line broadening may be some-

88 I I . THEORY

what different from that for excitation cross sections. The reason for this
difference would of course be connected with the need for total cross
sections in line broadening calculations, at least for cases where only one of
the states is significantly perturbed.

A third reason for doing fully quantum-mechanical calculations might
be a search for possible fine-structure effects. These were purposely neglected
in the preceding sections, because use of actual energy levels rather than
those averaged over fine-structure components and insertion of the appro-
priate vector coupling coefficients [68, 86] into the expressions for dipole
matrix elements, etc., to split the total contribution from some I —» V
transition into components could not possibly give a full account of spin-
related effects. To do this, it is obviously necessary to also include ex-
change, which cannot be done naturally in semiclassical calculations. Again,
carrying a classical path calculation beyond its natural domain by trying
to account for parts of the fine-structure effect may not be an actual
improvement. One might imagine spectra not obeying LS coupling as an
exception to this rule, but experimental evidence, e.g., for Ar I (see
Section III.6) points to the contrary, the line widths (in frequency units)
being very nearly the same (within about 10%) for analogous transitions.

The formal quantum-mechanical theory of electron impact broadening
was given by Baranger [1-3] and reviewed very briefly in Section II.3.
According to Eqs. (95), (99), (100), and (133), this theory allows us to
express the width and shift of an isolated line (which does not extend so far
as to overlap with lines from adjacent levels with different J values, or for
which φ-matrix elements involving changes in J are much smaller than
those that do not) in terms of transition (Γ) matrix elements as

WAV iJ< 1 Ji \iJf 1 Ji\
„ -Mt'l\Mt μ -MJ
w + id = - ( - ) N Σ (_i)JW«'4V/(

^W W

X liiJiMi'jm | T | JiMijm) δ(Μ/, Mf)

+ {JfM/jm | T | JfMfjm)* δ(Μ/, Μ , ) ]

- (JiM/jm' | T | JiMijm) (JfM/jmf | T \ JfMfjm)*}. (198)

The sum is over all magnetic quantum numbers, including μ = —1,0,

and 1, and over initial and, in principle, final total angular momentum

quantum numbers j and / of the colliding electron. Also, the frequency of

collisions of a given type was written as

/=* (x/aO(Ä/«)W, (199)

I I . 3 . IMPACT APPROXIMATION 89

assuming the electron | jm) wave functions to be normalized over all space
and deferring the velocity average until later. The Kronecker symbols
δ(Μ/, Mf) and δ(Μ/, Mi) are remnants of the two-state representation
used in Section II.3. They come about physically because the same electron
is scattered by upper and lower states, and because the linear terms corre-
spond to scattering in which one of these states, and therefore also that of
the electron, remains unchanged. In these terms, m! = m was accordingly
employed. (Note that f = j always holds for elastic scattering on systems
which are not degenerate with respect to J.)

The two terms linear in T reduce on account of the orthogonality rela-
tions [70] for the 3-j symbols, e.g.,

Mf>\Mf μ -Mi'/\Mf μ -MJ ^-t"1

to the average over magnetic quantum numbers of the diagonal matrix
elements. (That only M/ = Mi or M/ = Mf contributes would already
have followed from m' = m.) To simplify the remaining interference term,
more explicit use of the conservation of total angular momentum g = Jitf + j
and one of its components 9ΊΤ = Mitf + m = M'if + w! must be made
by transforming from the JMjm to the JjgÎiïl representation with the
coefficients

/J j S\
(JMjm | #TC> = ( - 1 ) ^ - ^ ( 2 5 + 1)1/2 - (201)

\M m -9TC/

After some algebra, the summations over all magnetic quantum numbers
can be performed, leading to

w + t-d-f (-ΥΛΓΣ 1 XΣ ( 2 & + 1 ) Ϊ Ά

2v \m/ j 2Ji + l

+<^TÏÇ(2*'+1)T^

Ui et i ) 2 (202)

- Σ (2& + i) (2& + i) ΓΑΠΙ,
3is' [Jf J< j)

(In the original derivations [86, 107, 108a], the simplification f = j was
not explicitly made, although it would not have entailed any additional

90 I I . THEORY

approximation in these applications.) Here the Tjg are the diagonal T-matrix
elements in the new representation, which are independent of 3U because
of the rotational invariance of the total Hamiltonian, and {: : :} is the
6-j symbol [70]. Actually, total orbital (<£) and spin (S) angular momenta
are conserved separately, if only electrostatic interactions are considered,
making it advantageous to transform from the (l\)jJ$5ïl to the (LI) £(S%) S
representation. The old T-matrix elements can then be written as

LV£ Ll £

Tsg= (2j+l)(2J+l) Σ {£>*,...} S i S 2 S rp£$

[J 3 3) [J j â)
(203)

in terms of 9-j symbols [70] and the newT T-matrix elements. Also,
{<£,§,...} is defined by

{ £ , § , . . . } = (2JB + 1 ) ( 2 S + 1 ) . . . , (204)

and V(l) is the perturbing electron's orbital angular momentum after
(before) the collision, and L is that of the radiator whose spin is S. Or-
thogonality relations again permit some simplifications in the linear terms,
and the final result is

w + «i = ^ ( - Υ Λ Γ Σ T (2£{ + 1) (2S,· + 1) £iSj
i&(2Lt+l){2Si+l)
"

+&gt/:»^'.)(,y),-w+i)w+i)

(Si âf i ) 2

>M/*i*/M/ Li l £i [j, Jt j)
Li V £, Lf V £f Lf l &/

X òi 2 &i S/ \ S/ Sf \ Sf

[Ji i ai) [Ji 3 êi) [Jf 3 âf) [Jf 3 S/)

X Tf? (Tff) (205)

I I . 3 . IMPACT APPROXIMATION 91

Equation (205) is the basis of quantum-mechanical calculations for
isolated lines. It can be considerably simplified [107] when one of the states
involved has L = 0, because the corresponding 9-j symbol is then propor-
tional to a 6-j symbol. Also, if the T matrices are independent of the total
spin S = S ± £, i.e., when exchange effects are negligible, one may sum
over S, <0, and j , obtaining, on account of the orthogonality relations for
9-j symbols, etc.,

υ \m/ i L &i 2Li + 1 &l 2Lf + 1

k, (2Lt+l)(2L/+l) T"{Tll) J" (206)

The upper-lower (i-f) state interference term here involves additional
approximations [£»■ « <£/, Tfj « (2L + l ) " 1 ] £ £ Tf J ; it should there-
fore be replaced by the corresponding term in Eq. (205) if small I partial
waves are important.

To avoid difficulties with long range Coulomb effects, it is necessary in
practice to use T matrices from which pure Coulomb scattering has been
eliminated by writing

Tft = expptor + m)l Tft + òvi[l - exp(2z77z)]. (207)

(This relation was stated incorrectly by Bely and Griem [107], but the
actual calculations of line widths were not affected by this.) As always, the
total orbital angular momentum ranges from <£= | ί — L\ to £ = I + L,
and the Coulomb phase shifts are defined as usual,

m = arg T(l + 1 - ie2/hv). (208)

The δι>ι term and the phase factor cancel [108a] on substitution into Eqs.
(205) and (206), which therefore remain valid also if the f matrices are
used instead of T. (Coulomb scattering per se naturally contributes nothing
to the line broadening or shift.) For width calculations, the optical theorem
is very useful [3], namely

Re Ff? = J Σ I Tft |* = 1 + ReCexp(2i„) (fff - 1 ) ] + h Σ I Tf, \\

VV

(209)

On substitution of this relation into the linear terms and of Eq. (207) into
the product term of the width expression, the first two terms on the right-

92 I I . THEORY

hand side of Eq. (209) cancel, leaving

w . L (h-\N v ΓΙ y (2^ + D(2st-+i) M
2. W £ L2 Ä (2L, + 1) (2S< + 1) '

+ 2 S (2L/+1)(2S/+1) ' T / f ' - mterferenCe termJ (210)

= *Μ(σ< + σ,) - - · · ,

the interference term being as in Eq. (205), but with f instead of T. The
two cross sections in the second version of Eq. (210) are total cross sections
(for inelastic plus non-Coulomb elastic scattering of electrons on radiators
in upper and lower states of the line), not merely inelastic cross sections as
in Baranger's original formulation [3]. [His result can be recovered by
combining the elastic contributions to the first two terms in the first version
of Eq. (210) with the interference term.]

Writing the width expression in terms of total cross sections is advan-
tageous for actual calculations, because these cross sections are continuous
when electron energies cross inelastic thresholds, provided one averages
over elastic resonances below such thresholds. In this sense, the inter-
ference term also is normally continuous because only one of the T factors
is oscillatory in any given below-threshold energy range [108, 109]. Only if
thresholds for upper and lower states coincide, would both factors oscillate
synchronously; resonances would then give a net contribution, causing a
step in the line width (averaged over resonances) at these coincident
thresholds. However, as the interference term seems to be small in such
cases [107] and to change only by some fraction of itself across thresholds,
these discontinuities are not likely to exceed, say, 10% of the width. They
are therefore next to impossible to observe, since the average over velocities
would smooth them considerably. (The general properties of electron-
impact-produced line widths as a function of electron energy are not only
important for quantum-mechanical calculations but may also allow one to
choose between various semiclassical methods as discussed in Sections
II.3caandII.3d.)

The f matrices used in quantum-mechanical calculations [107, 108]
mainly of Mg II and Ca II resonance lines (see Sanchez et al. [273b]
below for similar calculations of the Be II resonance lines) were taken
from "close coupling" solutions [110] to the scattering problem which are
given in terms of the (real and symmetric) R matrices, from which T
follows through

T = 2R/(i + R). (211)