Spectral Line Broadening by Plasmas

I I I . 5 . LINES WITH FORBIDDEN COMPONENTS 193

shallower, and the forbidden components somewhat weaker but broader
(see Fig. 23). The original paper contains entire families of profiles for both
lines, which show very nicely the evolution of these complicated line shapes
with density. The quantitative comparisons were somewhat tentative,
because no consistent density values could be determined. However, these
difficulties have since been traced [230b] to non-LTE effects, and most of
the conclusions are well confirmed by an independent and simultaneous
pulsed arc (mostly end-on) experiment [231] at higher density and tem-
perature [N = (3-10) · 1016 cm~3, kT « 4 eV]. As a matter of fact, the
deviations from theory are more pronounced and clearly outside the experi-
mental errors, which were estimated to about 10%. (The suggestion that
the attraction of the peaks is caused by interactions with n = 5, etc., levels
will probably not stand up to quantitative examination analogous to that
made by Gieske and Griem [92], because similar shifts should then also
occur for H/j ; and the correction for doubly charged perturbing ions, while
most likely negligible, would certainly work in the opposite direction.
Also, errors in the Debye shielding corrections, as discussed in Section II.5a,
seem unlikely causes of this deviation.) Another interesting indication from
this experiment [231] is that of a satisfactory agreement in the vicinity of
the 2P-4P components, especially near 2 3P-4 3P at X4517 A.

The latter observation is substantiated by Abel inverted measurements
[232] on a helium (spark) plasma [N = (3-5) · 1016 cm"3, kT « 3 eV]
generated by a repetitive CO2 laser, as is the reduction of the 4D-4F
separation and the filling-in of the minimum between these two com-
ponents. The same apparatus and method were used [233] to measure the
profiles of the He I, X6678 A (2 Ψ-3 *D) line with X6632 (2 Ψ-3 Ψ) as

100

0.1

°U60 U70 Xj-J] U80

FIG. 23. Comparison of measured (Birkeland et al. [230a], N » 1.0 · 10ιβ cm"8, T «
2.0 · IO4 Κ) and calculated (Ref. [90], dashed curve; Ref. [91], solid curve) profiles of the
He I, X4471 A (2 «P-4 3D, 3F) line.

194 I I I . EXPERIMENTS

forbidden component. These profiles were compared with calculations
according to Griem [90], evidently not allowing for the unshifted Stark
component of this line. While agreement for the allowed line was good, the
relative peak intensity of the forbidden component was found to be smaller
than calculated, and its far wing to be depressed by a factor of about 2.
Furthermore, on the wing of the allowed line toward the forbidden com-
ponent, the measured intensity was higher than calculated. The same line
[along with the He I, X4471 A (2 3P-4 3D, 3F) line] has also been meas-
ured in a high-pressure T-tube experiment [234] at higher densities and
lower temperatures [iV = (2-6) · 1017 cm-3, kT « 2 eV]. Compared to
calculations allowing for the unshifted Stark component, no significant
discrepancies were observed, and one would expect the same to happen to
the discrepancies reported by Ya'akobi et al. [233], if comparison were
made to similar calculations.

Deviations as found in the experiments [230-233] from theory based on
the quasi-static approximation for ions and the impact approximation for
electrons were first reported by Burgess and Cairns [235a] for the X4471 A
line and then [235b] also for the X4922 A line. They made measurements
in the afterglow of a z pinch at intermediate electron densities and tem-
peratures [_N = 3-10) · 1014 cm-3, kT < 1 eV], where the term inter-
mediate here refers to the regime between that of Vidal [32] (N « 3 · 1013
cm~3, kT « 0.16 eV) and all other experiments. Because of the difficulties
encountered with density gradients in this light source at earlier times
[221], several checks were made to ensure that the rather large deviations
(a factor of about 1.5 in the relative intensity of the forbidden compo-
nents) were not due to such density gradients or to optical depth correc-
tions near the peak of the allowed line X4471 A (see also refs. [235c, d ] ) .

Burgess suggested [236] immediately that these deviations were due to a
breakdown of the quasi-static approximation for ions near the forbidden
component. This is very reasonable, because rather large fields and there-
fore close collisions are required to excite forbidden transitions, say, col-
lisions with impact parameters p, which therefore correspond to a frequency
uncertainty (dynamical width) of ω, « v{/p. As discussed in more detail
in Section IV.4b, because of the astrophysical importance of these lines,
the corresponding additional broadening can be estimated [237a] (see also
[237b] and especially [237c, d ] which, however, give dynamical correc-
tions for the allowed line opposite to what would be expected from Section
II.4a) more quantitatively from an extension of Baranger's one-electron
approximation [ 6 ] (see Section II.4d). For lack of a complete theory, these
"dynamical" profiles should then be convolved, in the vicinity of the for-
bidden components, with their profiles calculated from the usual approxi-

I I I . 5 . LINES WITH FORBIDDEN COMPONENTS 195

mations. At least in principle, such convolution corrects for all the devia-
tions discussed above, and for some cases this correction is almost quantita-
tive. It is, however, somewhat smaller than the deviations observed by
Burgess and Cairns [235a, b ] , but then a recent experiment [235f] gives
no deviations under similar plasma conditions.

The most important parameter for the dynamical broadening should be
the ratio r of unperturbed separation ωη of the two components of the line
and the dynamical width ω, as calculated, e.g., from Eq. (446) below. As
summarized in Table VI, there is indeed a good correlation between r and
the degree of deviation from calculations not corrected for dynamical
broadening. In contrast to some expectations [236], the correlation with
density is rather weak, although the exact nature of disagreement (separa-
tions of the peaks, peak intensities, intensities between the components,
etc.) may well also depend on density. While further work is needed to clear
up this point, it seems safe to conclude that with dynamical corrections,
agreement of about 20% obtains near forbidden components (and about
10% near allowed components). For lines from other elements prone to

TABLE VI

CORRELATION OF THE DEVIATIONS BETWEEN MEASURED PROFILES OF NEUTRAL HELIUM
LINES WITH FORBIDDEN COMPONENTS AND CALCULATED PROFILES"

r Peak Density
Minimum Separation Transition [cm-3] Reference
2.0 Extreme
2.0 Small Extreme — 2 3P-4 3D, 3F 3 <• 101β [221]
2.5 — Large Large 2P-4D, F
3.5 Small Large Small 2P-4D, F 4<• 101β [2311
3.5 Large Small None 2P-4D, F
6.0 Small Small None 2P-4D, F 5 ■. 1 0 « [232]
7.5 None Small None 1 <• 1016 [230]
9 None None None 2 Φ-3 lD, Ψ
10 None None 2 Φ-3 *D, Ψ 1 «» 101δ [235a, b]
10 None — None 2 !S-3 Ψ, TO 4 «• 101β [233]
12 None None None 2 »Ρ-5 3D, *F 3 ■• IO17 [234]
38 None None None 2 3P-4 3D, 3P 1 . 4 .. IO17 [229]
None None 2 3P-4 3D, 3F 3 ■• IO13 [32]
None 23P-103D,3P 4 .• 10ιβ [231]
5 ■> IO16 [232]
3 ■• IO13 [32]

° Based on the quasi-static approximation for perturbing ions, r is the ratio of
separation of unperturbed levels and "dynamical" width. Listed are deviations of the
peak intensities of the forbidden components relative to the allowed line, deviations of
the intensity minima between these two features, and deviations of the separations of the
intensity maxima. The scale is necessarily somewhat subjective.

196 I I I . EXPERIMENTS

give forbidden components, such as lithium [238a, b ] or cesium [94, 95,
228], the situation should be rather similar. However, one recent experi-
ment [235d] not included in Table VI shows more substantial deviations
from theory based on the quasi-static assumption for ions than other
experiments at similar r values (r « 4), possibly pointing toward a need
for a more drastic revision of the theory or, more likely, to experimental
difficulties. As to the theoretical proposals [237c, d ] made to account for
these more substantial deviations, it should be pointed out that they in-
volve [235d] an ion impact parameter cutoff [111] that would a priori
seem much too small for the lines in question here. Moreover, it is not
likely that the separation [118, 237d] of effects connected with the in-
stantaneous ion fields and effects from field fluctuations is indeed correct.
(In evaluating the latter, Dufty and Lee average over all perturber coordi-
nates, without attention to the fact that for a fixed microfield we can no
longer assume equal a priori probabilities in coordinate space.)

III.6. ISOLATED NEUTRAL ATOM LINES

The number of measurements of profiles (or rather, in most cases, simply
widths and, to a lesser extent, also shifts of Stark broadened lines) having
almost Lorentzian shapes is so large that a complete review is nearly im-
possible. The conclusions reached in earlier reviews [6; 7, Chapter 15-3;
189] had been that calculated widths [7, 51a], mostly due to electron im-
pacts, had an accuracy of about 20% for the majority of the lines, and that
shifts of the maxima were typically predicted to within about 30% of the
half-widths. Indications were that some of the deviations were due to
uncertainties in radial matrix elements, obtained using the Bates-Dam-
gaard method [42], and sometimes also in atomic energy levels. Specifically,
there was some deterioration in the agreement obtained when going, say,
from helium to argon, i.e., from a simple system with nearly hydrogenic
wave functions to a complex system which, moreover, is not too well
described by the LS coupling approximation used in the calculations.
However, after averaging the ratio of measured to calculated line widths
over a number of lines, deviations tended to reduce to 10-15% regardless
of the atom under study, which suggested that all the basic approximations
and procedures in the calculations [7, 51a] (see also Appendix IV.a) were
well justified for neutral atom lines.

As will be seen below, this average agreement comparable to experi-
mental errors in width and density measurements is not to be construed
as ruling out substantial discrepancies in individual cases by as much as a
factor of about 2, nor does it necessarily imply as good agreement for entire

I I I . 6 . ISOLATED NEUTRAL ATOM LINES 197

profiles, especially on the far wings of the lines. (The latter are of particular
interest in astrophysical applications, because the cores of isolated lines
are here often dominated by Doppler broadening.) Also, there have been
many fewer shift than width measurements. In later work, the emphasis
should therefore be on entire profile (or wing shape) measurements and on
accurate shift determinations, in addition to high precision or survey type
width measurements (to find strongly deviating lines).

In many respects, helium is the most suitable element for such studies,
in spite of the relatively high temperatures required for sufficient excitation
and the large electron densities if local thermal equilibrium considerations
are to be used in diagnosing the plasma. As a matter of fact, the good agree-
ment mentioned in the preceding section for the allowed components of
the helium lines discussed there can be taken as direct evidence for the ac-
curacy of the theory for isolated lines, which in turn may be considered as
the low density limit of overlapping lines. The often rather small depend-
ence of helium energy levels on orbital angular momentum is, however,
somewhat unusual, so that the isolated line approximation must be viewed
with caution at higher densities. A case in point involves the X5016 A
(2 xS-3 Ψ) and X3889 A (2 3S-3 3P) He I lines investigated by Greig and
co-workers [229, 239a] in a helium-hydrogen T-tube plasma (1016 cm"3 <
N < 6 · 1017 cm-3, kT ~ 2 eV). This experiment also constitutes an ex-
ample of difficulties in the analysis of broad lines, especially of the H^ line,
which served as electron density standard. In particular, the red wing of
H,? is perturbed by the almost equally broad X4922 A (2 Ψ - 4 XD) He I
line. A first [239a] (graphical) analysis of the data had therefore resulted
in about 10% higher densities at N > 1017 cm-3 than computer-fitted
theoretical profiles to the blue wing of H^ only. A further difference was
the use of logarithmic versus linear intensity scales in the two analyses, the
latter probably more realistic in view of a more or less constant error in
intensity measurements. While a 10% change in the electron density is
barely outside experimental error limits, the change in the dependence of
electron density on discharge conditions is probably significant, especially
because there are indications from other experiments [201, 206] that com-
parison with the blue wing of Hß yields slightly higher N values at high
densities than would be obtained from the entire ΈΙβ profile (see also Section
III.9). Be this as it may, the new analysis [229] yields widths and shifts
more closely linear in density than was thought first [239a] with remaining
deviations from linearity being consistent except, perhaps, for the shift of
X5016 A, with estimated Debye shielding corrections to the impact theory
calculations. Probably this last discrepancy could be reduced further, if
the shift were calculated according to Section II.5a.

198 III. EXPERIMENTS

The measured profiles of the X3889 A and X5016 A lines could be fitted
satisfactorily to j(x) profiles [see Eq. (223) and Appendix IV.b], using
calculated ion broadening parameters A) etc. Such a comparison is given
in Fig. 24. It shows that ion broadening is mostly through (quasi-static)
quadratic Stark effects. Average ratios between measured and calculated
widths are near unity for both lines, suggesting that most of the calculations
[7, 51a, 72] gave about 10% accuracy for the widths of these lines. How-
ever, the most involved classical path calculations [86] fall short by about
20-30% for the triplet and singlet lines, respectively. Some of the refine-
ments to the original classical path theory [51a] thus appear to actually
lead to a deterioration in the accuracy, as had been surmised in Section
II.3ca. (The widths of Dyne and O'Mara [51b] fall in between.)

Although the shot-to-shot scanning method is not too well suited for
shift measurements, it seems clear that the X5016 A line shift still increases
with density more slowly than would be consistent with the Debye-shielded
impact theory results. This defect of the theory may be due to a breakdown
of the isolated line approximation, as evidenced by the occurrence of a
forbidden 2 lS-S *D component with about 10% relative intensity, which
has already been discussed in the preceding section. In regard to absolute
values of the shifts at low densities, agreement with the calculations of
Griem [7] is not too good, calculated values being smaller than was meas-

0.3 T i i i i r

j(x)

0.2

0.1

0 JIIIL
-2-10 1 2 3 45

X

FIG. 24. Comparison of measured (Greig and Jones [229], N = 1.7 · 1017 cm"8, T =
2.3 · 104 K) and calculated (Griem [112] or Appendix IV.b) reduced profiles of the He I,
Λ5016 A (2 *S-3 Φ) line.

I I I . 6 . ISOLATED NEUTRAL ATOM LINES 199

ured by about 20% of the half-widths. Better agreement was obtained with
respect to the original calculations [51a], which allowed for more interact-
ing levels and also for perturbations of the lower state of the line, as do the
calculations in Appendix IV.a.

Of considerable interest for other line broadening experiments is the
establishment of the width of the X3889 A line as a density standard which,
provided the calculated widths of Griem [7] are increased by about 5%
or those of Cooper and Oertel [72] (and Appendix IV.a) decreased by
about 5%, should be almost as accurate as H/j. (If Griem et al. [51a] is
used, there is no correction.) The errors quoted by Greig and Jones [229]
and Greig et al. [239a] were standard deviations of the individual meas-
urements as defined, e.g., by Young [240]. Mean values of the correction
factors to calculated widths are probably even more accurate than these
standard deviations might suggest. Compared to density determinations
based on Hß profile calculations ( ^ ± 7 % ) , additional uncertainties after
the above corrections have been made may therefore be as small as 2%.
Such accuracies can of course be achieved only if the corrections for broad-
ening by ions are applied according to Eqs. (226) and (227). They are well
supported by comparisons and computations referred to by Jenkins and
Burgess [221] but in direct conflict with a pulsed arc experiment [239b],
which must have suffered, e.g., from thermal instabilities [64b].

In the category of survey experiments, work on lithium [241], carbon
[89, 242], nitrogen [243], oxygen [88, 244], fluorine [245a], neon [89,
246], sodium [247], silicon [89], sulfur [89, 248], chlorine [89, 245b],
and argon [197a, 207, 249, 250] lines must be mentioned. (We postpone
the discussion of measurements on cesium and omit measurements pub-
lished prior to 1965 or measurements that could not be compared with
realistic calculations.) Many of these experiments were performed with
stabilized arcs, either using Ή.β as density standard [207, 242, 243, 248,
250] or calculating electron densities from measured argon line intensities
[197a, 249a]. As discussed by Bues and co-workers [197a], use of Έίβ profiles
yielded about 25% smaller electron densities than such line intensities
(see also Griem [64b]). After correcting the measured argon line widths
of Bues et al. [197a] and Schulz et al. [249a] so as to refer them to ΈΙβ as
well, almost all argon lines widths are consistent with each other and with
theory [7, 112] to about ±20%. (Evans et al. [249c] is the sole exception.
Also, the phase shift theory [80, 81], i.e., Eq. (263), gives widths that are
too small by a factor as much as about 2 for most argon lines.) Measured
and calculated shifts of argon lines, however, scatter rather widely [197a,
207], although agreement tends to be better for relatively large shift
values. Of four neon lines [246], the widths of two agreed with calculations

200 I I I . EXPERIMENTS

[7], while those of two others were smaller by a factor as much as about 2.
(Agreement with calculations presented in Appendix IV.a would not have
been any better.) In these cases, the phase shift theory was found [246]
to be more accurate.

Other arc experiments confirm a ± 2 0 % accuracy of width calculations
[ 7 ] for carbon [242], nitrogen [243], and sulfur [248] lines. The latter
work also gives two shift comparisons, both rather favorable. For two of the
nitrogen lines, widths could only be compared with the phase shift theory,
which in these cases agrees within about 10% with the measurements. This
is not at all surprising in view of the large separation between the upper
level of the lines (3s' 2D) and the nearest perturbing level (3p' 2D), which
is 1.3 eV and thus sufficiently large for the adiabatic approximation, i.e.,
Eqs. (145) and (146), to be applicable. Calculations along the lines of
Griem [ 7 ] and Griem et al. [51a] or Section II.3ca would thus also give a
result close to the phase shift limit.

The plasma flame protruding from a pulsed lithium hydride capillary
discharge served for measurements of four lithium lines [241], again relative
to H/3 profiles. All four lines had Lorentz profiles, and the widths of three
of them could be compared with calculations [ 7 ] . Two of them gave about
10% agreement, one showed almost twice the calculated width. The fact
that the shift of this line (2s 2S-3p 2P) is in much better agreement with
these calculations suggests that its profile may have been affected by ab-
sorption in the source. Opposing such an explanation, however, is the ob-
servation of a Lorentzian profile also for this line, so that some additional
line broadening mechanism might be considered to be responsible for the
discrepancy. Estimates of both resonance and Van der Waals broadening
rule out this explanation, so that the discrepancy remains unresolved.

The Na I, X5682-5688 A and X4978-4982 A lines were measured by
Oettinger and Cooper [247] in a sodium-helium discharge, using the He I,
X3889 A line as a density standard and employing a rapidly scanning Fabry-
Perot interferometer [184]. The widths of the X5682-5688 A lines (3p 2 P -
4d 2D) were found to be consistent with theory [7], but the X4978-4982 Â
lines (3p 2P-5d 2D) were narrower than calculated by a factor of 3. Possible
causes for this discrepancy are Debye shielding and, more likely, break-
down of the isolated-line approximation [247, 251].

Miller and Bengtson [89] performed an extensive study of Stark widths
and shifts of carbon, oxygen, neon, silicon, (one) phosphorus, sulphur,
chlorine, and (one) argon lines in a shock tube filled with suitable gas
mixtures, neon being the carrier gas. The electron density was determined
both from Ή.β profiles and from pressure-temperature data, with an average
consistency of ± 3 % in the density range (3-12) · 1016 cm"3. Agreement

I I I . 6 . ISOLATED NEUTRAL ATOM LINES 201

between measured and calculated widths (from Griem [7] except for
chlorine for which a similar calculation of Roberts [252] with new per-
turbing levels was used) was within 20% in almost all cases, with an aver-
age agreement much better than that for the 14 lines allowing direct com-
parison with the calculations of Griem [ 7 ] . However, even the revised
chlorine calculations [252], as do those presented here, give widths for two
lines that are too small by as much as a factor of 2, an observation which is
contradicted by the results of a z-pinch experiment [245b]. The shift
calculations of Griem [7] were found to agree with measurements also to
within ± 2 0 % , except for the C I, X5380 A line, where some important
perturbing levels had been omitted in the original calculations.

Miller and Bengtson [89] further noticed that refinements in the calcu-
lations according to Cooper and Oertel [72] actually lead to a deteriora-
tion in the agreement with measurements for heavier elements, and pointed
out that broadening by "strong" collisions is very important for the chlo-
rine lines. Furthermore, they found reasonable agreement with a simple
semiempirical formula for the widths analogous to that first proposed for
isolated ion lines [102], if the effective Gaunt factors determined for
neutral helium lines are increased by a factor of 1.5. The importance of
strong collisions has also been emphasized by Assous [244] for some infra-
red oxygen lines measured (and calculated) by him in a plasma produced
by induction heating. All but two of the seven lines have widths that com-
pare well with those calculated according to Section II.3ca (see Appendix
IV.a), as does the large shift of one of the lines.

For fluorine lines measured in a z-pinch experiment [245a], for which
strong collisions are very important, measured widths are smaller than
calculated values (Appendix IV.a) by factors of 2 or more, while measured
shift-to-width ratios tend to be twice as large as calculated. Absolute values
of the shifts are therefore in fair agreement with theory, which might sug-
gest that measured widths were overcorrected for Doppler and instru-
mental broadening or affected by density gradients. (Note that for these
lines, calculated shifts were revised considerably when lower state contri-
butions were included, etc.)

In none of the above measurements of isolated lines could profiles be
measured over much of an intensity range, nor were any of them concerned
with resonance lines. Since the "damping wings" of isolated lines are of
great interest, e.g., in astrophysical applications, and because for resonance
lines the relatively large level splittings and strong deviations from hydro-
gen wave functions might cause especially large errors in the usual electron
impact broadening calculations, measurements of helium [253] and argon
and neon [192] resonance lines emitted from optically thick layers fill a

202 I I I . EXPERIMENTS

substantial gap in the experimental verification of classical path calcula-
tions. The measurements were done on T-tube plasmas involving suitable
admixtures of hydrogen and were analyzed according to Eq. (413), the
portions of the theoretical line shape L(co) actually checked being below
the peak of L(œ) by four or more orders of magnitude. In most cases, re-
sults were consistent with an estimated 20% accuracy of the profile calcu-
lations [7, 51a], but tended to be somewhat below the calculated values.
Quasi-static ion broadening according to Eq. (228) remained a small cor-
rection even that far out on the optically thin line shape, being equivalent
to a small shift. As a by-product, measured wing shapes [192] of hydrogen
L« and Lß were found to agree with theoretical ratios [59] to well within
10%. (The L/3 line was used as a density standard by Moo-Young et al.
[192].)

The most extensive and systematic study of isolated line shifts was made
by Majkowski and Donohue [227] in an rf excited cesium plasma [iV =
(0.5-3) · 1014 cm-3, kT « 0.25 eV), with reference wavelengths provided
by a low density microwave discharge viewed simultaneously through es-
sentially the same optical system. On the basis of three preceding and inde-
pendent experiments, including those of Stone and Agnew [226], the widths
of cesium lines were taken as about ± 1 0 % density standards. This high
accuracy of, mostly, electron impact broadening calculations, which has
recently been verified [254] for high members of the fundamental series,
is of great interest in respect to the electron impact broadening of hydrogen
lines. (The latter is quite similar, but is always accompanied by a compar-
able ion effect.) Agreement between measured and calculated [7, 112]
shifts was reasonable for intermediate members of the series studied (S, D,
and F ) , while the first members of these series all gave about 30% larger
shift values than calculated and the high members of the diffuse (D) and
fundamental (F) series had about 20% smaller shifts (see Fig. 25). Some
of the latter deviations may be due to the transition from quadratic to
linear Stark effect in the ion broadening, a conclusion supported by the
smaller asymmetry of the measured profiles, and to Debye shielding (see
Section II.5a). Still, after correction for these effects, there would remain
an excess shift for the first members of the diffuse and fundamental series,
not to mention three of the four sharp series lines shown in Fig. 25. This
may be due to a breakdown of the classical path approximation, as es-
pecially for the line 6P-9S thermal electron energies are no longer larger
than separations between perturbed and perturbing levels. (Note, how-
ever, that except for the sharp series, deviations always amount to less than
20% of the half-widths, although use of corrected 6 functions [72] would
accentuate the deviations for low series members.)

I I I . 7 . ISOLATED LINES FROM SINGLY CHARGED IONS 203

i 1 1 1 1 1r

)6I I I I I I I I 1 L
k 5 6 7 δ 9 10 11 12 13
Principal Quantum Number n

FIG. 25. Ratio of measured (Majkowski and Donohue [227], N « 1014 cm"3, T «
3 · 103 K) and calculated ([7] or [112]) shifts of cesium lines of the fundamental (F),
diffuse (D) and sharp (S) series as function of the principal quantum number of the
upper states.

The principal conclusion of experimental studies on isolated neutral atom
lines is therefore an agreement within about 20% or less with calculated
[7, 51a, 112] profiles (see Section II.3ca and Appendix IV) in the over-
whelming number of cases, provided ion broadening (Section II.3f) and
Debye shielding (Section II.5a) are allowed for. Furthermore, some re-
finements [68, 72, 86] of the original classical path calculations do not give
any better agreement, and occasionally a reasonable first estimate of line
widths may be obtained from the (adiabatic) phase shift theory [80, 81].
However, as demonstrated by Kusch and Oberschelp for zinc [255] and
cadmium [256], very large errors might then be incurred from calculated
values of the quadratic Stark effect coefficient. It will thus often be safer
to use the semiempirical method [89, 102] for such width estimates (see
also Section IV.4c and Miller et al. [257a], and Cowley [257b]) in cases
where calculations as presented in Appendix IV.a are not possible, or where
the high-temperature limit given by Eq. (175) is not applicable.

III.7. ISOLATED LINES FROM SINGLY CHARGED IONS

Quantitative experiments on isolated ion lines did not exist when, e.g.,
Wiese's review [189] was written, the first two papers [207, 258] on ex-
periments in this category appearing only in 1965. One of these papers
[207] contained results for the width and shift of the Ar II X4806 A

204 I I I . EXPERIMENTS

line which exceeded calculated values [7] by about a factor of 3. The other
paper [258] was concerned with a number of N II lines whose widths
(and in most cases also shifts) were found to agree quite reasonably with
the straight classical path calculations [ 7 ] then available, except for one
line which might have been overlapping with the Si III, X4552.65 A line
from wall impurities. However, one year later Jalufka et al. [259] confirmed
the factor of about 3 larger width of the Ar II, X4806 A line and found
similar factors for the widths of three other lines of ionized argon. As in
the N II experiment [258], a T-tube was used for these Ar II width
measurements, and again a neutral helium line was used as density stand-
ard. (The first Ar II measurement [207] was done in an arc with H^ as
density reference.) A major difference between the N II and Ar II T-
tube measurements was that in the latter case radial homogeneity in
density and temperature could be verified, and subsequent N II measure-
ments [260, 261] indeed gave widths larger by factors of 2-3 than the first
N II profile measurements. Control experiments of Jalufka and Craig
[261] also showed that density inhomogeneities can lead to erroneous re-
sults by such factors, so that the ratios of widths (shifts) and electron
densities of Day and Griem [258] should probably all be multiplied by
about 2.

Results of four additional experiments [248, 262, 263a, 264a] concerned
with C II, S II, Ca II, and Ar II lines were available by 1968, all
supporting the above evidence for excess line widths. These experimental
data and those discussed in the preceding paragraph (after correcting those
of Day and Griem [258]) and the widths of some Si II lines served as an
empirical basis for simple formulas [102] [Eqs. (458) and (459)] which
represented measured widths within an average factor of 1.5, deviations by
factors of 2 being rare exceptions. Besides radial matrix elements, etc., only
one dimensionless parameter appeared in this semiempirical formulation,
namely the ratio of thermal electron energies and characteristic unper-
turbed level splittings. For large values of this ratio, inelastic weak col-
lisions dominate, while for small values the reasonable agreement with the
semiempirical formula was interpreted as evidence for the predominance of
elastic collisions, whose effects cannot be described by second order per-
turbation theory. Other important effects for such ion lines accounted for
by Griem [102], but not in the (straight) classical path calculations of
Griem [ 7 ] , are lowerstate broadening and curvature effects. As discussed
in Section II.3d, these effects were also included in more deductive classical
path calculations [68, 83-86]. However, although these calculations are
much more involved, they do not at all agree any better with measurements
than the semiempirical formula in most cases.

I I I . 7 . ISOLATED LINES FROM SINGLY CHARGED IONS 205

Almost simultaneously with the survey of experimental data given by
Griem [102], a measurement of sixteen C II line widths was reported
[265]. These measurements were relative to H^ , and the light source was a
pulsed discharge tube [266] designed for operation in a wide density and
temperature range. Analysis of these data and other more recent results on
Si II [267a, b ] , Ar II [268, 269a], and Ba II [269b, 270] according to
Griem [102] supports the above conclusions, i.e., for ratios of mean per-
turber energies \kT to characteristic level splittings AE between about 1
and 100, the measured and calculated widths certainly agree on the average
to within a factor of 1.5, provided that the contributions from various per-
turbing levels are treated individually for %kT/AE > 10 as in Eq. (458).
Actually, the agreement is within 10% for four of the five Si II lines
measured in a T-tube, and within 20% for almost all of the about twenty
Ar II lines measured in a pulsed arc [268a], a plasma jet from a dc arc
[268b], and a T-tube [269a]. Lines of the Ar II multiplet 31 are the most
flagrant exceptions, being narrower than calculated by a factor of about
1.5. However, larger deviations in the opposite direction reported [271a, b ]
for some Ca II and Sr II lines are more likely than not due to experi-
mental errors, as is a similar deviation [269b] for the Be II resonance
line. Finally, a trend for the ratio of measured and calculated widths of
Cl II lines [271c] to decrease with increasing excitation energy of the
upper level is not real, in the sense that, in the calculation of atomic matrix
elements for these levels according to the Bates-Damgaard method [42,
43], effective quantum numbers were calculated using always the ioniza-
tion energy of Cl II rather than its sum with the excitation energy of the
Cl III level to which the Cl II level series in question converges. (See
also Section II.2c.) After correcting these errors, deviations between meas-
ured and calculated widths are as usual [271d].

To confirm that elastic collisions are really accounted for by the semi-
empirical method, extension of the data to \kT / AE <<C 1 was called for, a
condition which is attainable only for resonance lines of elements having
sufficiently large ionization energies. An experiment analogous to that for
neutral argon and neon lines [192] (see the preceding section) was there-
fore performed [272] for C II and N II resonance multiplets in a high
pressure T-tube filled mostly with helium, and using the He I, X3889 A
line as a density standard. Under the prevailing optically thick conditions,
fine structure was not resolved; theoretical profiles were therefore super-
imposed using appropriate oscillator strengths and unperturbed wave-
lengths. The best fit to the measured profiles corresponded to 1.0-1.5 times
the semiempirical width for C II and N II, respectively. Judging from
these data, there is therefore no substantial deterioration of the accuracy

206 I I I . EXPERIMENTS

even at $kT/AE « 0.3, where most of the electron impact broadening must
be due to elastic collisions. (Superelastic collisions were estimated [272]
to give a contribution of a few per cent, while inelastic collisions ought to be
entirely negligible.) In contrast to the neutral atom resonance lines, no
correction was necessary for ion broadening. The most important compet-
ing line broadening mechanism was probably Van der Waals broadening by
helium atoms, but even this should cause no significant corrections.

Of other resonance lines, those of the magnesium and calcium ions are of
particular interest, because the first fully quantum-mechanical calcula-
tions [107, 108] were for these ions. (Analogous calculations have since
been made for the beryllium ion [273b].) The Ca II, X3934 A (4 2S-
4 2P) line has been measured in arc-plasma jets [262, 274], in T-tubes
[264a, 273a], in a parallel-rail shock tube [275], and in arcs [263b, 264b]
using argon, helium, hydrogen, and C02 in various mixtures as carrier gases
for the small Ca impurity required to avoid self-absorption. Electron
densities were near N = 1017 cm-3 in all these experiments, and because of
the small Stark widths, corrections had to be applied for Doppler and
instrumental broadening and, at least in the work of Jones and co-workers
[273a] also a correction of about 5% for Van der Waals and ion broaden-
ing. As shown in Fig. 26, there is rather reasonable agreement between
measured and calculated temperature dependence of the width, if one ex-
cludes the two low points at low temperatures. However, even the quan-
tum-mechanical width values [108a] seem too high, not to mention the
curve labeled "semiclassical," which was obtained according to Section
II.3d and Appendix V. (Agreement with a semiclassical calculation re-
ported by Chapelle and Sahal-Brechot [274] would have been better, but
these procedures [68, 86] work less well than ours for the vast majority of
isolated lines from atoms or ions.) The Mg II, X2803 A (3 2S-3 2P) line
was measured in three of these experiments [263b, 274, 273a], measured
widths agreeing fairly well with semiclassical calculations, but, except for
[263b], being a factor of 1.5 larger than the fully quantum-mechanical
results [107, 108b]. This defect may or may not be due to the use of only
three states (3S, 3P, 3D) in the underlying close coupling calculations [110]
of the R matrices. For the Ca II, X8542 A (3 2D-4 2P) line, the deviation
of measured [274] and calculated [108b] width is even larger, but in the
opposite direction. Substantial disagreement with theory exists for the
Be II, X3131 A (2 2S-2 2P) line [273b], perhaps because of the large
interference term [64c]. (As mentioned before, the measured width from
Platisa et al. [269b] is most likely much too large. See also Hadziomerspahic
et al. [271e] and note that agreement with semiclassical calculations is
satisfactory.)

I I I . 7 . ISOLATED LINES FROM SINGLY CHARGED IONS 207

0.15

w
[Â]010

005

10000 J L
20000 30000

τ[·κ]

FIG. 26. Comparison (updated from Jones et al. [273a]) of measured widths of Ca II
resonance lines (normalized to N = 1.0 · 1017 cm"3; A, Ref. [274]; B, Ref. [273a]; E,
Ref. [262]; F, Ref. [264a], G, Ref. [275], H, Ref. [263b], and I, Ref. [264b]) with quantum-
mechanical (QM, Ref. [108a]) and semiclassical (SC, Appendix V) calculations.
Measurements by Kusch and Pritschow [271b] and Puric et al. [271a] give w » 0.5 and
0.6 Â at T « 2 - IO4 and 1 · IO4 K (see also the introduction to Table VII).

Returning to Fig. 26, we observe that at least some measured widths for
the Ca II resonance line may have errors of about 20%. That such errors
are typical for ion line measurements can also be inferred from Table VII,
where measured widths from the various experiments discussed in this
section are collected and compared with semiclassical calculations (Ap-
pendix V). On the average, deviations between the various data for a given
multiplet and temperature are also about 20%, so that presently it only
makes sense to investigate the reasons for the few cases of discrepancies of,
say, about 30% or more. As for neutral atom lines [189], the average ratio
of measured and calculated widths is close to unity, namely 1.05 =b 0.03.
For each individual line, the probable error of the calculated width comes
to about 25%, i.e., perhaps half that of the much simpler semiempirical
method [102]. Also, ion broadening and other broadening mechanisms were
evidently not important in most of these experiments, and the shape of the
Stark profiles, where reported in detail, was always very close to a Lorentz-
ian curve. For many of the multiplets, the measured widths are averages
over several lines, whose widths, however, almost never seem to differ by
much more than 10%. (An independent analysis by Jones [273c] essen-
tially confirms these conclusions and those discussed in the following
paragraphs.)

208 III. EXPERIMENTS

TABLE VII

COMPARISON OF MEASURED (wm) AND CALCULATED (WC) WIDTHS OF ISOLATED LINES
FROM SINGLY CHARGED IONS°

Spectrum Multiplet T[eV] wm [Â] WC[Â] Wm/Wc Reference

Bell 1 2.0 0.035 0.042 0.85 f273b]
CII 1.5 1.59 [272]
Nil ÎUV 1.0 2.7 · 10~3 1.7 · 10"8 1.72 [265]
15UV 1.0 0.91 0.75 [265]
Mg II 4 2.5 0.41 0.53 1.09 [264a]
4 1.0 0.51 0.55 1.22 [265]
6 2.5 1.10 0.47 0.80 [264a]
7 1.0 0.74 0.90 0.93 [265]
8 2.5 2.01 0.93 1.43 [264a]
8 1.5 2.58 2.17 1.80 [272]
1.5 1.8 · 10~8 1.80 0.88 [258]
1UV 1.5 0.14 0.88 [260b]
5 2.0 0.14 1.0 · HT3 1.18 [260a]
5 2.0 0.20 0.16 1.00 [261]
5 1.5 0.17 0.16 1.29 [258]
5 2.0 0.27 0.17 1.43 [261]
18 1.5 0.30 0.17 0.82 [258]
18 2.0 0.20 0.21 1.21 [261]
29 1.5 0.29 0.21 1.53 [258]
29 2.0 0.58 0.24 1.16 [260a]
30 2.0 0.44 0.24 1.32 [261]
30 1.0 0.50 0.38 1.20 [274]
30 1.5 0.38 1.00 [273a]
1.5 0.06 0.38 0.55 [263b]
1UV 1.0 0.045 0.81 [274]
1UV 0.025 0.05
1UV 1.05 0.045
4 0.045
1.31

α The measured widths (from the references stated) are normalized to an electron den-
sity of N = 1017 cm-3, as are the calculated electron impact widths (from Appendix V).
The experimental data are averaged over different lines of the same multiplet, if data for
more than one such line are contained in a given reference. The C I I multiplet 13UV
of Kusch [265] is omitted because of possible overlap of its components, and in accord-
ance with Jalufka and Craig [261], the measured N I I widths of Day and Griem [258]
are multiplied by a factor of 2. Since it disagrees with three independent measurements
[268b, 269a, 263a] by a factor of about 4, the Ar II multiplet 10 value of Jalufka et al.
[259] was discarded, as were the Be II data of Platisa et al. [269b] and most Ca II
data of Puric et al. [271a] and Kusch and Pritschow [271b]. (The widths from these two
experiments for the Ca II multiplet 1 are larger than those of Yamamoto [262], Roberts
and Eckerle [264] and others [273-275] by almost an order of magnitude, probably be-
cause of self-absorption [27le]). Results of Puric et al. [271a] for Ca II multiplets 3 and
4 seem more reliable and are therefore included, while the Ca II multiplet 2 value of
Chapelle and Sahal-Brechot [274] was rejected as it gives a smaller damping constant
than that of multiplet 1.

o O cc

P ^SOiOìOOOìQOìWH CO W H ^ ί Μ Ο Ο OS rf* tO ►-*

^ ^ · £»■ rf^ O O O

HHMtOHMMHHM MbOHHtOHHtOlOHHMHtOtOHPMHHtOtO H- H-* H- H * H* H- h-» H-t-»00

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210 I I I . EXPERIMENTS

In regard to shifts of isolated ion lines, the situation is less satisfactory.
Measurements as reviewed by Griem [102] seemed to indicate that calcu-
lated electron impact shifts (from the semiempirical widths and using a
dispersion relation [79]) were insufficient to explain the data, leaving room
for a plasma polarization shift (see Section II.5b). However, subsequent
measurements [277a] of 25 Ar II lines were found to be consistent with
semiclassical calculations according to Sahal-Brechot [68], although the
measured line peaks tended to be at somewhat shorter wavelengths than
calculated, especially when the shifts attributed to ions are omitted.
(These amounted to about 50% of the electron effect and were estimated
from the impact approximation, because durations of ion collisions were
shorter than times between ion collisions. Actually, this may not have been
the relevant criterion in this case, because one is interested in frequencies
from the line center somewhat larger than the ion-produced width.) A
particularly serious disagreement exists for the Ca II resonance lines
where most measured shifts [262, 263b] are very small, whereas the quan-
tum-mechanical theory [108a] gives substantial blue shifts, albeit with
large errors [108b]. Adding a plasma polarization shift may or may not
make matters worse. A difficult experimental problem is posed by the often
small shift-to-width ratios in the usually transient plasma. This has been
overcome in a z-pinch experiment [278] on the strong Ar II, X4806 A
line by the use of a Fizeau fringe interferometer [279].

Results of the experiments mentioned in the preceding paragraph, of the
few earlier experiments [207, 258, 260a, 276, 268a, 271b], which gave line
shift data as well, and of Puric and Konjevic [277b] are summarized in
Table VIII and compared with our semiclassical results for the shifts
(Appendix V) caused by electron impacts. There is clearly a large scatter
between measured (dm) and calculated (dc) shifts, but the experimentally
and theoretically more significant ratio (dm — dc)/w is not all that large
in most cases, its mean being 0.2 ± 0.1. Other shift mechanisms, e.g., those
from ion broadening or from the plasma polarization effect, thus seem on
the average not to be important. However, it should not be overlooked that
details such as the strong temperature dependence found in the work of
Chowdhury [278] probably cannot be explained solely on the basis of elec-
tron impacts, and that the predictive power of theory is much lower for the
shift of any given ion line than for its width. Especially uncertain is the
situation in regard to the shifts of the Mg II and Ca II resonance lines.
Omitting these data, the mean value of (dm — dc)/w comes to 0.03 =b 0.03,
i.e., to an agreement similar to that for the widths. This is as expected
theoretically [79], provided both widths and shifts are caused mostly by
electron impacts.

I I I . 7 . ISOLATED LINES FROM SINGLY CHARGED IONS 211

TABLE VIII

COMPARISON OF MEASURED (dm) AND CALCULATED (dc) SHIFTS OF ISOLATED LINES FROM
SINGLY CHARGED NITROGEN, ARGON, AND ALKALINE METAL IONS0

Spectrum Multiplet T [eV] dm [Â] dc [Â] (dm — de)/w Reference
Bell
Nil 1 1.5 -0.03 -0.04 +0.25 [277b]
Mg II
Aril 5 1.5 +0.15 +0.16 -0.07 [258]
18 1.5 +0.35 +0.22 +0.48 [258]
Call 29 1.5 +0.15 +0.20 -0.26 [258]
30 1.5 +0.55 +0.31 +0.43 [258]
30 2.0 +0.80 +0.37 +0.98 [260a]

1UV 1.5 +0.015 -0.03 + 1.80 [263b]
2UV 1.5 +0.08 +0.14 -0.42 [277b]

1 1.0 +0.06 +0.10 -0.28 [277a]
2 1.0 +0.10 +0.09 +0.06 [277a]
6 1.0 -0.09 -0.09 [277a]
6 1.0 -0.07 -0.09 0.00 [276]
6 2.0 -0.05 -0.02 +0.07 [278]
6 2.5 -0.03 -0.01 -0.16 [278]
6 2.5 -0.08 -0.01 -0.10 [268a]
6 3.0 -0.02 +0.01 -0.35 [278]
6 3.5 -0.01 +0.02 -0.15 [278]
7 1.0 -0.05 -0.08 -0.15 [277a]
7 1.0 -0.03 -0.08 +0.20 [276]
7 2.5 -0.05 -0.01 +0.38 [268a]
10 1.0 +0.09 -0.22 [277a]
14 1.0 0.00 -0.18 -0.70 [277a]
14 1.0 -0.08 -0.18 +0.41 [276]
14 2.5 -0.04 -0.14 +0.58 [268a]
-0.08 +0.25
1 1.0 -0.13 [262]
1 1.5 -0.02 -0.12 +2.00 [263b]
1 1.5 -0.02 -0.12 + 1.10 [277b]
1 2.5 -0.09 -0.10 +0.30 [263b]
3 1.5 -0.025 +0.34 +0.75 [277b]
3 1.5 +0.18 +0.34 -0.25 [271b]
4 1.5 +0.37 +0.25 +0.06 [277b]
+0.18 -0.27

a The measured shifts (from the references stated, with those from Day and Griem
[258] again multiplied by a factor 2) are normalized to an electron density of N = 1017
cm"8 (assuming an effective average density equal to this nominal value in the case of
Blandin et al. [277a]), as are the calculated electron impact shifts (from Appendix V).
The experimental data are averaged over different lines of a given multiplet if data for
more than one such line are contained in a given reference. Where available, the measured
widths (from Table VII) have been used to calculate the ratios of deviations between the
two shift values and the width; otherwise, widths have been estimated from similar ex-
periments. Data [277c] for Cl II are not included because calculations for this spectrum
are hampered by the scarcity of atomic energy level data. See, e.g., [277a] for measured
variations of shifts for different lines of a multiplet.

212 I I I . EXPERIMENTS

III.8. ISOLATED LINES FROM MULTIPLY CHARGED IONS

Very little is known experimentally or, for that matter theoretically,
about the Stark broadening of lines from more than singly charged two-
and more-electron systems. One practical reason for this is the smallness
of Stark broadening for the stronger lines, i.e., those from low-lying excited
states, under most experimental conditions. Theoretically, some of the
experience gained in going from calculations for neutral atoms to singly'
ionized systems suggests that for strong lines from multiply ionized atoms,
more than the first nonvanishing terms in the various expansions should be
considered. Moreover, one suspects that the classical path approximation
could be rather unreliable. Fully quantum-mechanical calculations of the
electron impact broadening may thus be required, as well as measurements
at extremely high densities (N > 1019 cm"3), as obtainable in laser pro-
duced plasmas [280, 281], low-inductance sparks [282, 283], or plasma
focus devices [284].

One important reason for the study of charged particle effects on lines
in third and higher spectra is the possibility of errors in wavelength meas-
urements and therefore in deduced atomic energy levels. In view of the
possibility of a plasma polarization shift (see Section IL5b) and, in some
cases, of ion impact broadening, the usual argument that shifts should be
much smaller than widths is not necessarily correct. It would be much
safer to measure wavelengths of such lines at various electron densities but
essentially the same temperature, and then to extrapolate to zero density.
This was the principal conclusion of the first systematic experimental study
[285], for some Sn IV, Al III, Si III, and Si IV lines, of such shifts
(and widths) in a spark source. Using a theta pinch at different initial
pressures, substantial shifts were then found [286] in the wavelengths of
the 4d-5f transitions of N V and C IV. These were inconsistent with
quasi-static Stark effects, at least relative to the 4f-5g transitions, but the
observed shifts can perhaps be explained [287] by ion impact broadening
in the adiabatic (phase shift) limit.

While the lines for which Stark effects were suspected by Mazing and
Vrublevskaya [285] and Hallin [286] all come from rather highly excited
states, sufficiently high electron densities (iV « 5 · 1020 cm-3) were reached
in a laser-produced plasma [280] so that Stark broadening of lower lying
transitions in 0 VI or of intermediate transitions in an even higher ion,
K IX, could be clearly established. In other words, the profiles were
Lorentzian rather than Gaussian and relative values of the widths could
not reasonably be explained by any other line broadening mechanism.
However, as in experiments discussed earlier in this section, no truly inde-

I I I . 8 . ISOLATED LINES FROM MULTIPLY CHARGED IONS 213

pendent determination of the electron density was made. Instead, it was
argued that in view of the small inherent level splitting, straight classical
path calculations analogous to those of Griem et al. [51a] are sufficient
for some of the lines, and that ion broadening is not too important except
for one line showing almost linear Stark effect. Excluding this line, electron
densities determined from measured widths with the aid of such calcula-
tions appeared to be consistent within a factor of about 2. At least for
lithium-like ion lines studied in this experiment and that of Hallin [286],
relatively simple estimates of electron impact broadening may therefore
be assumed to reach a similar accuracy. This contention is substantiated
by the fair consistency of the measured widths [280] of the O VI and
K IX lines with the effective Gaunt factor approximation [102], which
is also within a factor of 2 at N = 5 · 1020 cm-3. Interestingly enough, a
reanalysis of the data of Mazing and Vrublevskaya [285] in terms of this
simple semiempirical formula leads to the same conclusion for lines from
somewhat more complicated spectra. The question therefore arises whether
the effective Gaunt factor approximation is indeed only reliable to within
such a factor for multiply ionized systems, or whether some of the devia-
tions are due to experimental errors.

This question is probably resolved by the results of a measurement
[288] of some C III and C IV lines in a small theta pinch operated in
helium with a small admixture of methane. In this case, the density could
be determined to N « 4 · 1017 cm"3 from the profile of the He II, X3203 A
line (see Section III.3), and the temperature was about 5 eV. As can be
seen from the comparison in Table IX, agreement with the effective Gaunt
factor approximation is indeed less satisfactory than for second spectra.
Except for the C I V resonance line, agreement is rather better when com-
parison is made with straight classical path calculations according to Griem
et al. [51a]. This is not surprising, because for most lines, the splitting be-
tween perturbed and perturbing levels is much smaller than kT. For the
C IV resonance line, the situation is opposite, so that its width is much
better represented by Eq. (190), i.e., by a suitable version of the phase
shift (adiabatic) limit [69a, 104] of the impact theory. Also noteworthy is
the good overall agreement with estimates from Section IV.6.

No doubt considerable work is still required in this category to reduce
probable errors in line widths below, say, 40%. Experimentally, we need
for this well-calibrated Stark widths of prominent Unes, say, of C III,
X2297 A (2s 2p Φ-2ρ2 XD), as secondary density standards, because ionized
helium lines are not always suitable and because density determinations by
other methods are often very difficult in the light sources of interest here.
However, an early measurement [265] of this line is most likely in error,

214 I I I . EXPERIMENTS

TABLE IX

MEASURED F U L L WIDTHS Δλ OF C III AND C I V L I N E S COMPARED WITH CALCULATED
F U L L W I D T H S 2wa

λ Transition Δλ 2wa 2wb 2we A\/2wc

5696 3p ψ-3ά ιΌ 1.9 0.8 1.0 2.0 1.0 c m
4647 3s 3S-3p 8P 0.95 0.8 1.4 0.7 cccc mmmm
4326 3s Φ-3ρ *D 2.1 0.55 0.9 1.3 1.6
4187 4f >F-5g *G 4.1 0.6 3.6 5.3 0.8
3609 4p 3P-5d 3D 6.2 3.8 7.5 7.3
2163 3d 'Ό-Λΐ Ψ 7.6 0.9
1532 3p ^ ^ d lD 0.46 0.47
5802 3s 2S-3p 2P 0.43 0.39 0.49 0.60 0.8 cc mm
5812 0.44 1.35 0.49 0.8
1584 2s 2S-2p 2P 1.6 0.84
1551 1.7 0.9 CIV

0.024 0.0064 — 0.016 1.5 CIV

° Measured full widths Δλ at N = 4 · 1017 cm"3 and T = 6 · 104 K from Bogen [288];
calculated full widths 2w from various approximations: (a) according to Griem [102]
(semiempirical method with effective Gaunt factor g = 0.2 at threshold); (b) according
to Griem et al. [51a] (straight classical path method); (c) according to Eq. (526) of Sec-
tion IV.6. Wavelengths and line widths are in angstroms and the (extrapolated) value of
Sahal-Brechot and Segre [104] is 2w = 0.018 Â for the last two lines.

the measured width being larger than the widths of similar C II lines in
the same plasma. Finally, we must not overlook the possibility of substan-
tial ion broadening, e.g., in the case of lines which are especially sensitive to
Stark broadening. If this broadening is in the quasi-static domain, it will
manifest itself through deviations from a Lorentzian shape. Otherwise,
theoretical estimates according to Section II. 3f may be necessary to esti-
mate corrections to the normally dominating electron impact broadening.

III.9. ASYMMETRIES AND SHIFTS OF HYDROGEN LINES
To a very good approximation, Stark profiles of lines from one-electron

systems are symmetrical about the unperturbed line center, in contrast to
lines from two- and more-electron systems, for which quasi-static quadratic
Stark effects tend to cause asymmetries whenever broadening by ions is at
all important (see Section II.3f ). However, at high densities or far from the
line center, hydrogen, etc., lines also show deviations from symmetry,
and are shifted. These effects are normally ignored by comparing only

I I I . 9 . ASYMMETRIES AND SHIFTS OF HYDROGEN LINES 215

symmetrized measured profiles with theory (using, e.g., Appendix I.a),
but do have some intrinsic interest in that they are a measure of a number
of higher-order effects left out in most calculations. Moreover, the shifts
are of astrophysical interest in connection with relativistic red shifts, as
discussed by Wiese and Kelleher [289]. The most complete theoretical
estimates of such effects have been made [36-38] for L a , and there is
semiquantitative agreement with experiments [190a, 191] even if only the
quasi-static quadrupole effect is considered following Sholin [37]. (See
also Section II.2b, and note that the ion broadening of the unshifted com-
ponent was probably overestimated by Griem [38].) This is true for the
far wings, measured asymmetries [191] nearer to line center being in favor
of the blue wing.

Boldt and Cooper [190a] had measured the absorption coefficient of an
argon plasma containing a small hydrogen admixture. In addition to the
quadrupole correction to the quasi-static broadening, their asymmetry is
therefore influenced by a factor proportional to the actual frequency, which
is omitted in the usual expressions for the line shape, by quadratic Stark
effects, and by the associated ion-field dependence of the oscillator strengths,
not to mention the transformation from frequency to wavelength incre-
ments. However, if the electron (impact) broadening is assumed to be
entirely symmetrical, then first-order quadrupole effects dominate, giving,
according to Eq. (487), relative corrections of ±(2Δλ/λ)1/2 to red or blue
wings of the quasi-static profiles. (These corrections [36, 37] are twice as
large as the quadrupole corrections estimated by Griem [38], where
quadrupole effects on the wave functions were ignored.) At the largest
wavelength separations (Δλ = ± 1 3 A) obtained in the experiment, the
asymmetry was equivalent to a red shift of over 0.5 Â, which should be
compared with an "optically thin" (full) Stark width of about 0.05 A.
Corresponding emission profiles would be expected theoretically [36-38]
to have a somewhat larger asymmetry also favoring the far red wing, e.g.,
at points where the intensity is more than five orders of magnitude below
the peak of the calculated Stark profile, by perhaps 15% over the mean of
the two wing intensities. (We note in passing that electron impact shifts
are negligible, even if An ?* 0 interactions are accounted for [62d].)

The only other line for which both experimental and a fair amount of
theoretical data are available is Ή.β , which has long been known [290] to
give more emission near the peak on the blue than near that on the red
side of the profile, the difference between the heights of the two intensity
maxima being about 5% at N « 1017 cm"3. Moreover, at least two authors
[201, 206] have come to the conclusion that the entire blue side is favored
at such densities, although some ambiguity exists as to the choice of a

216 I I I . EXPERIMENTS

reference wavelength. As a matter of fact, by not insisting on the wave-
length of the central dip as reference, extremely symmetrical profiles can be
obtained [203a] out to about ±300 A, with asymmetries probably not ex-
ceeding 15%. However, in contrast to L« , these smaller asymmetries are
probably still in favor of the blue wing in the measurements of Vujnovic
[206] and Wende [201], and some indications [206] are that a similar
situation exists for higher Balmer lines. While the above measurements
were performed either at relatively high electron density (N < 3 · 1017
cm-3) in a hydrogen plasma [206] or at moderate density (N « 7 · 1016
cm"3) in an argon plasma [201] with a small admixture of hydrogen, a
probably more accurate experiment [198, 291] has been performed in a
hydrogen arc at N < 9 · 1016 cm"3 which gives, in contrast, a 15% asym-
metry in favor of the far red wing (see Fig. 17). In this case, all data were
referred to the unperturbed line center, and careful corrections were made
for continuum and wings of other lines.

Besides asymmetries from trivial sources, such as the ω4 and Boltzmann
factors or scale transformations, three mechanisms have been considered
to explain the observed asymmetries of Hp . These are the quadratic Stark
effect and the field-dependence of oscillator strengths [290], quadrupole,
etc., interactions [37, 292], and the so-called dissolution effect [206] first
discussed theoretically by Lanczos [293]. The latter effect corresponds to
the spontaneous ionization by tunneling, when an atom is placed into an
external field. It clearly weakens the red wing first as the field increases,
which was shown experimentally by Gebauer and Rausch von Trauben-
berg [294]. In the absence of calculations accounting for all three mecha-
nisms and the "trivial" sources, it is not yet possible to single out any of
them as being most important. However, from the apparent change in sign
at high densities of the asymmetry in going from L« to Ή.β , etc., one can
tentatively conclude that the dissolution effect as proposed by Vujnovic
[206, 295] is indeed important for the latter lines, and the numerical esti-
mates [290, 292] of the other effects suggest that they are not negligible
either. (Note that the dissolution effect has a rather sharp onset and that
the required microfields in the case of La would be much higher than those
reached in Boldt and Cooper's experiment [190].) On the other hand, from
the fact that at moderate densities the red asymmetry of Hp (minus that
from trivial sources) is observed [291] to be only about half that expected
from higher multipole interactions [37], we may infer that these effects
are then indeed important, but that the other causes of asymmetry would
have to be considered as well in a realistic theory.

Similarly inconclusive is the situation with respect to the relative heights
of the two maxima of Kß . Here both calculations of higher multipole ef-

I I I . 9 . ASYMMETRIES AND SHIFTS OF HYDROGEN LINES 217

fects [37, 292] and the earlier estimates of quadratic Stark effects [290],
etc., agree quite well with measurements [198, 201, 290, 291], and it seems
clear that a combination of all proposed mechanisms would give asym-
metries larger than observed. This, like some of the remaining discrepancies
on the line wings, might point out the need for a consideration of asym-
metries in the electron broadening, which is usually assumed to be sym-
metrical.

Shifts of otherwise almost symmetrical profiles have been observed
[289, 290] especially for Ηγ , approaching 1 A at N = 1017 cm"3. They
and corresponding shifts [198, 289] of Ha and H^ are all to the red, i.e.,
they are in qualitative disagreement with what would be expected from the
theory [37] of higher multipole interactions. However, at least for HT ,
they are consistent [290] with the displacement of the unshifted com-
ponent by ion impacts calculated according to Lindholm [80] or Foley
[81], i.e., using the adiabatic approximation. This may indicate that a
theory of these shifts must not be based on the quasi-static approximation.
We conclude this discussion by mentioning that Wiese et al. [198, 291]
observed asymmetries similar to those on Η# for Ηγ , and especially for
Ha as well, which would seem to be in contradiction to the results of earlier
work [206]. Perhaps this is due to difficulties with the corrections for other
lines and for the continuum, or to differences in the plasma conditions.

Up to this point, we have been concerned with asymmetries and shifts
of La and the early members of the Balmer series. Although the question
of asymmetries of higher members of the Balmer and Paschen lines for
conditions in the experiments of Schlüter and coworkers and of Vidal (see
Section III.2c) has evidently not been investigated in great detail, these
lines appear to be symmetrical to a high degree over an intensity range of
three decades, and it was perhaps taken for granted that they were not
shifted either. However, a special experimental investigation [296] re-
vealed measurable shifts with the definite pattern of relatively large shifts
for lines having an "unshifted" Stark component, and no shifts for the
alternate members of the Balmer series. As with the various effects dis-
cussed above, there is presently no detailed theoretical understanding of
these shifts, but there appears to be an intimate relation to quadratic Stark
effects.

Almost nothing can be said about the symmetry or asymmetry of io-
nized helium lines, because none of the experiments discussed in Section
III.3 possessed sufficient accuracy over a large intensity range to detect
asymmetries as small as those found for hydrogen lines. The tendency has
been to blame any deviations from symmetry on impurity lines. Depending
on the conditions of excitation, there is also the possibility of satellite lines

218 III. EXPERIMENTS

arising from doubly excited states of neutral helium [149c, 150a]. These
satellite lines have slightly longer wavelengths than the parent lines, but
form Rydberg series converging on the position of the parent line. It is
therefore often sufficient to know the displacement of the first member of
the Rydberg series, e.g., from the 1/Z expansion method [297], and then to
assume a l/nz dependence for the displacements of the higher series mem-
bers. The latter may be broadened appreciably and may have played a
role in the simulation of line shifts (see Section II.5b).

III.10. EFFECTS OF SUPRATHERMAL FIELD FLUCTUATIONS

The influence of electric fields from plasma waves rather than from indi-
vidual charged particles on line profiles has only recently become the sub-
ject of experimental studies. This is not to say that there have not been any
such cooperative effects, besides Debye shielding, in experiments discussed
heretofore. For example, as mentioned already in Section III.2a, the filling-
in of the central minimum of ΤΆβ found experimentally may, for one-
dimensional turbulence, well be due to the presence of a near Gaussian
distribution of wave produced fields with rms amplitudes exceeding those
given by the equipartition theorem [see Eq. (56) ] . In that case, the total
"low frequency" field strength distribution would be the convolution of the
particle-produced field distribution discussed in Section II.2a with the
wave-field distribution, the latter usually being important only at rela-
tively small field strengths. However, for strong wave excitation or turbu-
lence, the collective contribution may dominate most of the convoluted
distribution so that shapes, e.g., of hydrogen lines could be very different
from line profiles calculated according to Section II.3a. Provided the rele-
vant plasma frequencies are all smaller than the resultant line widths, the
quasi-static approximation would be applicable. Since electron impact
broadening should be negligible under such circumstances, the line profile
would mainly reflect the distribution function of the collective fields.

Following some early work [133b] on ϋβ , these general ideas were first
verified by a measurement [132] of the He II, X4686 A and X3203 A
lines in a high voltage, low density theta pinch. The lines exhibited near-
Gaussian profiles with enhanced wings. Their widths, however, were not
at all consistent with Doppler broadening alone, the X3203 A line being
broader than the X4686 A line, which is less sensitive to Stark effect. On
the other hand, by considering both Doppler and "turbulent" Stark
broadening, the line widths could be explained by reasonable values of ion
temperature and rms field strength. Similar effects have since been seen in

I I I . 10. EFFECTS OF SUPRATHERMAL FIELD FLUCTUATIONS 219

toroidal high ß and in gun-produced hydrogen plasmas [298, 299], i.e., the
method seems well established for probing amplitude distributions of turbu-
lent electric fields. All of these experiments should be compared with a
measurement [300] of the half-widths of H^ and Η γ in a toroidal "Toko-
mak" device with turbulent heating currents. In this case, the profiles were
not at all Gaussian near the peak of the turbulence and decidedly wider
than ordinary Stark profiles would have been at the independently meas-
ured electron density.

Using an image intensifier and a Fabry-Perot interferometer, Zagorod-
nikov et al. [301a, b ] and Zavoiskii et al. [301c] were able to obtain "streak"
pictures of the H/j and Ha profiles produced in and behind the "collision-
less" shock front in a low density theta pinch device and in a linear (mirror
machine) turbulent heating experiment. In the latter and behind the shock
front, the H/? profile exhibited distinct sidebands whose separation from
line center was interpreted as a measure of the mean field. The authors
pointed out, presumably for fully developed three-dimensional turbulence,
that the wave fields would have a Rayleigh function distribution, and that a
convolution of these with the Holtsmark distribution for the particle-
produced fields would result in a shift of the most probable total field
strength to larger values. Assuming the quasistatic approximation to be
valid for all these fields, lines such as Rß or H$ would then indeed have more
pronounced and displaced intensity maxima than the ordinary Stark pro-
files. In the shock front, however, the Ή.β profile was single-peaked, though
substantially broadened. This was interpreted [160, 301a] as due to the
nonadiabatic action of high frequency turbulent fields, which would give a
Lorentz profile whose width obeys essentially Eq. (358). This approach is
of course justified only in cases for which this "collisional" width is much
smaller than the relevant plasma frequencies, a condition which was only
marginally met in the experiment even for turbulence at or near the elec-
tron plasma frequency. As an alternative explanation, one should therefore
consider the quasistatic action of lower frequency one-dimensional turbu-
lence in the shock front. In that case, the appearance of shifted peaks with
increasing separation from the shock front would be analogous to the shift
of the most probable speed from zero in a one-dimensional Maxwellian to
the peak of a three-dimensional Maxwellian. A similar situation prevails for
a beam-plasma experiment [302] on Ha .

By making side- and end-on measurements, say, on a high-power theta
pinch discharge or by using polarizers, one can actually determine any
preferred direction of the turbulent electric field. Such measurements were
made by Berezin et al. on a linear high frequency discharge [303] and
an rf current discharge [304], while the techniques described by Zagorod-

220 I I I . EXPERIMENTS

nikov et al. [301a, b ] and Zavoiskii et al. [301c] were employed for the
analysis of other turbulent heating devices [305]. In addition to displaced
peaks such as those discussed above, there has also been some evidence
[303, 305, 306] for satellites on hydrogen lines more or less analogous
to those of Baranger and Mozer [131] on neutral helium lines. While
theoretical models for such satellites can be constructed [133c], their
interpretation is still so insecure in general that they are deserving of much
more work. Therefore, before going on to the theoretically more transparent
case of neutral helium, the reader must be referred to the present and future
literature on this question and be made aware of the existence of competing
effects [307] to the "turbulent" Stark broadening of hydrogen and ionized
helium lines if field strengths are small.

When instantaneous Stark shifts are smaller than characteristic fre-
quencies of the plasma, which is often the case for two- (or more-) electron
systems as radiators, we may expect to see the "plasma satellites' ' pre-
dicted by Baranger and Mozer [131]. As discussed by these authors (see
also Section II.5c), two such satellites should appear near wavelength
positions symmetrical to unperturbed positions of forbidden components,
the "near" satellite (disposed toward the allowed line) being somewhat
stronger than the "far" satellite, and their relative intensities with respect
to the allowed line being proportional to the electrical energy density in the
plasma waves.

The first observation [155] of (far) plasma satellites was made in a
theta-pinch helium plasma, which evidently sustained wave energies three
to four orders of magnitude above thermal levels. To guard against coinci-
dences with impurity lines, etc., both singlet and triplet X4922 A and
X4471 A (2P-4D, F) lines were scanned from shot to shot, and the location
of the satellites was found to correspond to densities of 3 · 1013 and 4 · 1013
cm~3, respectively, if electron plasma waves were assumed responsible.
The X3889 A (2 3S-3 3P) line profile was measured as a control, in case
the satellites should have been simulated by Doppler shifts from mass
motions. Had the quadratic Stark shifts of the allowed lines caused by the
wave fields been taken into account [see Eq. (397)], only slightly smaller
frequencies or densities would have resulted, especially at later times in
the theta-pinch implosion. This is not to say that such shifts were the only
complication. Actually, they may even have been masked by other higher
order effects, which were discussed following Eq. (397) in Section II.5c.

The identification of the plasma waves in this early experiment was
somewhat uncertain, but there is no way to explain the satellites in terms
of forbidden components caused by particle-produced fields. For the meas-
ured relative intensities, this would have required [90, 91] N « 5 · 1014

I I I . 10. EFFECTS OF SUPRATHERMAL FIELD FLUCTUATIONS 221

cm-3 or, for the shifts of the satellite maxima, even N « 1015 cm-3, densities
that are entirely inconsistent with the absence of any appreciable Stark
broadening of the allowed line and other independent estimates of the
density. Still, one major difficulty remained for the interpretation of these
observations in terms of plasma satellites, namely, the failure to detect
near satellites stronger than the clearly observed far satellites. As already
indicated, this deviation from second order theory may have been due to
higher order effects, although some overlap of an incipient near satellite
with the allowed line cannot be ruled out either.

Both near and far satellites for a plasma frequency larger than the in-
herent level splitting, so that the satellites would also be on different sides
of the allowed line, were reported ([308] one year later for the He I,
X4471 A (2 3P-4 3D, 3F) line emitted during the afterglow phase of a laser-
produced plasma in 10 atm helium gas. From the separation of the satellites
and assuming electron plasma waves, or from the width of the He I,
X5876 A (2 3P-3 3D) line, there followed electron densities of about 1016
cm-3 or more, i.e., densities probably more than two orders of magnitude
higher than in the first experiment [155]. These high densities of course
imply very strong particle-produced fields, which by themselves would
cause a forbidden component having at least half the strength of the al-
lowed line [90, 91] and being positioned between allowed line and far
satellite. Since no trace of this forbidden component was evident on the
observed line profile, there is thus a severe inconsistency in the interpreta-
tion of these data in terms of plasma satellites. As a matter of fact, rather
similar experiments [232, 233] can be interpreted (see Section III.5)
entirely on the basis of particle-produced fields if the raw profile data are
Abel-inverted to remove the effects of spatial gradients. A much more
plausible explanation of the profile data of Baravian et al. [308] is therefore
in terms of a spatially inhomogeneous plasma with a dense core (N >
1017 cm"3) emitting a wide double-peaked ' 'thermal· ' Stark profile simulat-
ing the satellites. The more tenuous surrounding layers (N < 1015 cm-3)
would then be responsible for what was originally interpreted as the allowed
line, with some self-reversal and a relatively strong red wing lifting up the
"near satellite."

Also continuation [156] of the first measurements [155] on a much
higher voltage and faster theta pinch yielded no near satellites, but a more
rigorous extension of the perturbation theory suggested that this was not
inconsistent with the presence of rather strong (i.e., about 20% or less of
the allowed line) far satellites. The density in this case was about 7 · 1013
cm-3, small enough to neglect forbidden components caused by particle-
produced fields, but too high for the measured satellite wavelengths to be

222 I I I . EXPERIMENTS

consistent with electron plasma oscillations. Ion plasma oscillations stand-
ing in the moving magnetic structure and thus presenting a Doppler-shifted
wave to the neutral atoms, which do not participate in this motion, were
therefore thought to be responsible for the satellites. Should correction for
quadratic Stark shifts have been necessary, i.e., the effective frequency
have been smaller, the wavelength of the ion waves would have been larger
in proportion. An alternative explanation [157] in terms of fourth-order
satellites leads to inconsistencies, as discussed in Section II.5c and demon-
strated in Fig. 14. Therefore, the most probable situation here seems to be a
coincidence of near and far satellites within the spectral resolution of these
measurements. This was verified by further measurements [309] on the
same apparatus as function of time and magnetic field orientations, which
clearly showed a separation of the single-peaked structure into two Bar-
anger-Mozer satellites (Fig. 27). Both far and near satellites were also
seen in two experiments permitting better spectral resolution, one [310]
involving an expanding gun-produced plasma as in the work of Ben-Yosef
and Rubin [299], the other [158] a plasma rendered unstable by an elec-
tron beam. The gun experiment was particularly interesting in that it
operated at high density (N « 1016 cm-3) and therefore large plasma fre-
quencies. This necessitated the use of lines which are rather insensitive to
Stark effects, namely the He I, X6678 A (2 Ψ - 3 Ψ, Ί ) ) and He I,
X5016 A (2 lS-3 Ψ, XD) lines, but caused a spread in the plasma wave spec-
trum beyond the limits of resolution. The satellites each exhibited a distinct
double structure, and profiles of near and far satellites, whose relative in-
tensities were consistent with second order theory, were mirror images of
each other, as expected. From the relative intensities with respect to the
allowed line followed turbulent electric field energy densities two orders of
magnitude above thermal. The satellite positions revealed that electron
plasma waves of wavelengths considerably larger than the Debye length
were responsible, since, according to the warm plasma dispersion relation,
wavelengths near the Debye limit would have resulted in too high fre-
quencies. The satellite lines were found to be polarized, which indicates
[153] that the turbulent field was not isotropie. The reader may also wonder
whether at these high densities there should not have been a normal for-
bidden component. However, calculations [233b, 93] predict relative in-
tensities of less than or about 1% in these cases, while the satellites had
about 10% of the allowed line intensity.

In the electron beam experiment of Cooper and Hess [158], on the other
hand, plasma densities had been much lower, namely, less than about 1014
cm"3, but the satellites, in this case of the He I, X4922 A (2 Ψ - 4 *D, Ψ)
line, wrere further split by the Zeeman effect in the 7 kG magnetic field

III. 10. EFFECTS OF SUPRATHERMAL FIELD FLUCTUATIONS 223

>-
>

4918 4919 4920 4921 4922 4923 4924

FIG. 27. Measured profiles (Davis [309]) of the He I, X4922 Â (2 Φ-4 Φ, *F) line
at two times in a turbulent plasma showing first two merged second-order plasma
satellites and then their separation. The vertical arrows indicate the actual positions of
allowed and forbidden components, the adjacent vertical lines show the corresponding
unperturbed positions.

employed, and well-resolved with a Fabry-Perot interferometer. The meas-
ured profiles of the satellites and the forbidden component for both π and
σ polarizations are shown in Fig. 28, the "fine structure" of the σ profile
being a characteristic of the Zeeman effect. However, perhaps against
naive expectations, these profiles are consistent with isotropie electric
fields, combined Stark and Zeeman effects for other fields being still more
complicated. These features of the Stark effect in the presence of a strong
magnetic field were further exploited in an experiment [159] on turbulent
heating. In this case, almost the entire splitting was due to Zeeman effects,
and from the polarization pattern the electric field was found to be of rela-
tively low frequency and to be mostly at right angles to the magnetic field
with random azimuths.

224 III. EXPERIMENTS

II Γ~Γ Γ
"P" polarization IX

Forbidden
line

Resolution:-HI«*- flII

"S" polarization

-2J.0 I -1.L0
Δλ(Α)

FIG. 28. Measured profiles (Cooper and Hess [158]) of plasma satellites and forbidden
component associated with the He I, X4922 Â (2 Φ-4 *D) line from an electron beam
experiment for both π ("P") and σ ("S") polarization. The allowed line is not shown.

The existence of three satellite pairs corresponding to electron plasma
oscillations and oscillations near 0.8ωρ and 0.2ωρ was reported [311] for a
turbulent 2-pinch experiment. In this case, magnetic fields were relatively
small (3 kG), so that at the quoted density of 5 · 1014 cm-3 Zeeman effects
should have been negligible, which might otherwise have simulated the
existence of several distinct frequencies. This experiment is further of
particular interest, because it demonstrates the transition from particle-
produced forbidden components at high densities and low turbulence
levels to plasma satellites at lower densities and higher turbulence levels.
Another turbulent heating experiment [312] is distinguished by the need
to correct for deviations from statistical populations of the two upper levels,
and a conical z-pinch gun experiment [313] by the combined use of plasma
satellites and "turbulent" Stark broadening.

Meanwhile, additional satellite measurements have been reported
[305, 314a, 314b] and there has been a short review [315] of these difficult
but rewarding measurements and their interpretation. Some of the remain-

I I I . 10. EFFECTS OF SUPRATHERMAL FIELD FLUCTUATIONS 225

ing questions and problems will be taken up once more in Section IV.3, in
hopes of giving the reader some guidelines for the choice of suitable lines,
etc. As with forbidden components, plasma satellites are naturally expected
not only in neutral helium but, e.g., also in lithium. Such satellites have in-
deed been observed [238a], interestingly enough in a plasma where, be-
cause of the small number of particles in the Debye sphere, even thermal
fluctuations were sufficient to excite them (see also Section II.2a).

CHAPTER IV

Applications

A scientific discipline as narrow as the subject of this book would hardly
attract so much attention and hard work, were the results not useful in a
wider field. And fortunately, there is no dearth of applications for our re-
sults, with the physical parameters of the objects of such applications span-
ning an impressive range as indicated in Chapter I. A listing of these ob-
jects with some of their physical characteristics would probably have been
quite sufficient for the specialist. However, for the many users of Stark
broadening methods whose principal interest is in other disciplines, a more
detailed guide to the relevant parts of our specialty will be helpful. The
following sections are meant to lead into a few areas where broadening of
spectral lines by charged particles or collective electric fields is known to be
important or interesting for a variety of reasons.

There is necessarily some redundancy in what follows. On the other hand,
the occasional reader may want to start with the most appropriate section
here and then work his way back to the corresponding portions of the pre-
ceding chapters or to the tabulated material appended to this book.

IV.l. DENSITY MEASUREMENTS
In the past two decades, a major stimulus for quantitative research into
the Stark broadening of spectral lines has been the need for accurate de-

226

I V . 1. DENSITY MEASUREMENTS 227

terminations of physical parameters characterizing laboratory plasmas
generated for many different purposes. The electron or ion density is ob-
viously the most accessible of all plasma parameters to Stark broadening
methods, because Stark profiles depend primarily on the electron density.
(Plasmas with strong suprathermal field fluctuations are an important
exception; see Sections II.5c, III. 10, and IV.3.) Furthermore, if the degree
of ionization exceeds a few per cent, other pressure broadening mechanisms
(see, e.g., the literature [7,8,10]) are almost always negligible, as is natural
line broadening under conditions where the technique to be discussed here
is at all practical. This leaves Doppler broadening as the most likely com-
peting mechanism, which must be considered along with apparatus broad-
ening and the effects of suprathermal field fluctuations, before measured
profiles are interpreted in terms of electron densities.

There are three main procedures for separating Doppler and Stark
broadening. In the simplest case, when kinetic temperatures of the radiators
are well known from independent considerations, one simply calculates the
thermal Doppler broadening in order to decide whether it can or cannot
contribute significantly to the observed broadening, after correcting for
apparatus broadening, etc. If a significant contribution is then suspected,
calculated Stark profiles should be convolved with these Doppler profiles
before comparison is made. For purely impact-broadened isolated lines, this
leads to the well-known Voigt profiles (which are conveniently tabulated
[316-318]), provided the two broadening mechanisms are statistically
independent. For broadening by ions, which in turn may control the
motion of the radiators by gas-kinetic collisions, this may not be the case,
and a more general theory [187, 188, 319, 320] could be necessary. Fortu-
nately, this complication does not arise when Stark broadening by ions,
if at all important, is well described by the quasi-static approximation, as
in most situations of interest here.

If kinetic temperatures are not sufficiently well known, it is best to
measure profiles of at least two lines of greatly different sensitivity to Stark
effect. Again assuming Doppler and Stark broadening to be independent,
we can use the width of the least sensitive line as an upper limit for the
Doppler broadening, which can be improved upon once an approximate
value for the electron density has been established. However, if only one
line can be measured and the kinetic temperature is not known, it is neces-
sary to fit convoluted Doppler and Stark profiles to the data and vary, e.g.,
the ratio of Lorentz (impact) and Gaussian (Doppler) widths to obtain
the kinetic temperature and electron density from the optimum fit. Unless
reliable profile data over about two decades in intensity are available, this
procedure is rather inaccurate in comparison to the two-line method.

228 I V . APPLICATIONS

Here and in most of the remainder of this section, use of line widths or
even entire shapes is implied. If shifts of intensity maxima are used instead,
and if the profiles are symmetrical, there is of course no correction for
thermal Doppler broadening. However, this Stark shift method usually
reaches the accuracies possible with widths or shapes only after cross cali-
bration as, e.g., in the cesium experiment of Majkowski and Donohue
[227]. Sometimes even more subtle aspects of line profiles, such as asym-
metries or the distant-wing shapes, are proposed as density monitors, but
they will generally lead to much larger errors than width measurements.
The Inglis-Teller method [18] is probably of intermediate accuracy, if the
criterion discussed in Section II.2 is suitably sharpened [321] and if the
density scale is calibrated empirically for a reasonable range of electron
temperatures. This has not yet been done, nor are there sufficiently accurate
calculations for conditions where lines with upper levels of different princi-
pal quantum numbers merge.

In view of these facts, and since actual experimental procedures are
naturally analogous to those described in Section III.l, it is sufficient to
conclude this section by a listing of suitable "standard" lines, i.e., lines
whose Stark widths or profiles are sufficiently well known for this, perhaps,
most demanding application. The first place in such list should be taken
by the Ή.β line of hydrogen or the corresponding deuterium line, Ώ$ . Ac-
cording to the experimental evidence discussed in Section III.2a, calculated
profiles (see Appendix I.a) of this line have an accuracy commensurate
with the best experimental accuracies yet attained, except for regions near
the central minimum and beyond the 10% relative intensity (of the aver-
aged peaks) points, and also neglecting asymmetries, i.e., averaging over
"blue" and "red" halves of the measured profile. Therefore, after Abel
inversion (see Section I I I . l ) , etc., as necessary, measured profiles, when
fitted to the calculated profiles, should yield electron densities with errors
corresponding almost entirely to the scatter in the experimental data. In
practice, one would determine rms deviations between measured and calcu-
lated profiles for a number of electron densities and thus find the most
probable value and error of the density. In these comparisons, the deter-
mination of the continuum level, which must be subtracted from the
"line" signals, may pose a difficult problem. The usual procedure is to treat
this level, and for broad lines also its slope versus wavelength, as a free
parameter in the best fit procedure, if it exceeds, say, 1% of the peak in-
tensity. Subsequently, the line-to-continuum intensity ratio can often be
used to determine the temperature [7], taking proper account of the actual
plasma composition. Another free parameter may be a constant factor ap-
plied to all "line" signals, unless calculated line-to-continuum ratios are

I V . 1. DENSITY MEASUREMENTS 229

assumed. The problem of overlapping lines is more difficult. To a first
approximation, contributions from neighboring lines can be estimated from
relative total line intensities and asymptotic line shapes (see Section IV.4a
and Appendix II) of the adjacent hydrogen lines, although interference
terms as discussed in Section II.4d should not be overlooked either. In
practice, it may be sufficient to treat these contributions from other lines by
replacing the essentially wavelength-independent continuum term by a
sum of a constant and a term linear in Δλ. For more accurate procedures,
the reader is referred, e.g., to Wiese et al. [198].

H/s is useful as a density standard for electron densities from about 1015
cm"3 to 3 · 1017 cm"3. Using H 7 at lower densities is of no particular ad-
vantage, although it can be useful up to N « 1017 cm-3, e.g., in cases where
H/3 is perturbed by impurity lines. (Remember from Section III.2b that
half-width comparisons for H7 were just as favorable as for Η^ , but not
those of entire profiles.) However, uge of H5 enables measurements down to
N « io14 cm-3 and of Ha (both also tabulated in Appendix I.a) up to
N « 1019 cm""3, provided radiative transfer effects are small or can be
allowed for also in the latter case. In contrast to H#, systematic errors in
electron densities from the profiles of these lines may not be negligible, as
remaining discrepancies between experiment and theory correspond to
about 15% uncertainty in the electron density. In view of this, half-width
(Δλι/2) comparisons will often be sufficient and also much simpler, espe-
cially for the lines with no unshifted Stark components. The electron
density is then readily calculated from the formula, valid also for quarter
and eight widths,

N = 10"(Δλ1/η)3/2 Σ CnmDogioiAXi/n)]-1. (414)

Calculated coefficients for H/s and Ha are collected in Appendix Hl.a.
Extension of the method to N « 1013 cm"3 is possible with the aid of higher
series members (Section III.2c), but calculated line profiles [59, 121c, 215]
are available only for one particular condition. Fortunately, simple asymp-
totic formulas [120] (see Section IV.4a and Appendix II) are reasonably
accurate [215] beyond the half-intensity points. By fitting them to area-
normalized measured profiles, electron densities can thus probably be
determined to ± 2 0 % . (See Chester and Bengtson [205] for more recent
results on high series members at intermediate densities.)

While the difficulties of vacuum ultraviolet spectroscopy hardly warrant
the use of Lyman lines, nor those of infrared spectroscopy the use of Paschen,
etc., lines, a very worthwhile extension of the temperature range (up to

230 I V . APPLICATIONS

kT « 10 eV) is possible with some He II lines. (Calculated data for such
lines can be obtained from Appendices I.b, II, and IILb.) Especially
valuable in this context are the He II, X4686 A and X3203 A lines. As
should be clear from the discussion in Section III.3, not too much reliance
should be placed on portions of the profiles much beyond the half-widths,
but the latter may well be as accurate as those of most hydrogen lines,
suggesting systematic errors in the electron density of about ± 1 5 % if only
one line is used, and probably slightly less than that if an average of values
obtained from both lines is taken. Theoretical data for some vacuum ultra-
violet He II lines, also given in Appendices I.b and IILb, might be useful
at extremely high densities, where the two visible lines become too broad.

Perhaps somewhat surprisingly, widths of many neutral helium and
heavier neutral atom lines are just as reliable density standards as most
hydrogen and ionized helium lines, especially the width of the He I,
X3889 A line, which comes close to H/3 in attainable accuracy (see Section
III.6). One reason is the almost linear dependence of isolated line widths
and shifts on electron density, versus the nearly iV2/3 dependence for lines
subject to linear Stark effect. The other is the overcompensation of the
apparent simplicity of the one-electron case by a number of complicating
factors such as Debye shielding and breakdown of the impact approxi-
mation for the electron broadening of these systems, not to mention wave-
field and dynamical effects in the ion broadening.

Appendix IV.a contains calculated electron impact and quasi-static ion
broadening parameters for many lines from the elements He through Ca,
arranged analogously to the tabulations [39, 40] of oscillator strengths,
and for cesium, which provides so many laboratory plasmas. From these
parameters, scaled with density as indicated in Appendix IV.a, and from
Eqs. (226) and (227), combined Stark (half) half-widths and shifts are
easily calculated as a function of electron density and, in principle, also
temperature. Fortunately, the temperature dependence is often so weak
that this causes no particular difficulty. One thus obtains curves of half-
width versus electron density of 10-20% accuracy in almost all cases.
Analogous curves for line shifts will be of inferior accuracy except, perhaps,
when shifts are about as large as the (half) half-widths. Also, systematic
errors in electron densities from measured widths can probably be reduced
to 10% or less if an average over several neutral atom lines is used [189]
and if Debye shielding corrections are small.

Appendix V gives calculated (semiclassical) electron impact width and
shift data for singly ionized atoms (Li II to C a l l ) . The widths are
believed to be accurate to within an average factor of 1.25 (see Section
III.7). Again, averaging over electron densities obtained from widths of

I V . 2 . TEMPERATURE MEASUREMENTS 231

various lines will reduce systematic errors from uncertainties in the line
broadening calculations, in this case, say, to less than 20%. Making correc-
tions for broadening by ions according to Section II.3f is hardly worth-
while here, because these corrections would typically amount to about
10%. Use of shifts instead of widths is even less recommended for these ion
lines because of the large uncertainties in calculated electron impact shifts,
the plasma polarization shift (Section II.5b), and the often relatively large
contributions by ions for which neither of the two extreme approximations
may be applicable.

In regard to high temperature plasmas emitting lines from multiply
ionized atoms, not enough data (see Sections III.4 and 8) are available to
make much of an assessment of their utility for electron density measure-
ments. Note, however, that suitable lines in all but extremely dense plasmas
will involve rather highly excited states, so that classical path calculations
of their widths may be sufficient [288] for the present application. Such
calculations would be entirely analogous to those described in Sections
II.3b and d. For lines involving lower-lying excited states, theoretical
estimates as given in Section IV.6 or by the effective Gaunt factor method
[102] (see also Section IV.4c) would be preferable.

IV.2. TEMPERATURE MEASUREMENTS

Inspection of isolated neutral atom line shift-to-width ratios shows that
this quantity is rather sensitive to electron temperature. This observation
led Burgess and Cooper [322] to propose measurements of the temperature
from precise determinations of the shift-to-width ratio, e.g., with a Fizeau
interferometer [279]. The advantage of this method is its independence
from the usual excitation and ionization equilibrium assumptions required
for most other spectroscopic temperature measurements. However, its
application has been hampered by the scarcity of reliable d/w values,
especially for ion lines. (Note again that ion broadening contributions to d,
and perhaps also plasma polarization shifts, can be quite important.)

Much wider use has been made of an entirely different method which,
however, is restricted to plasmas where the ionization equilibrium, i.e., the
electron density as function of temperature, etc., can be calculated. Match-
ing these electron densities with those obtained from Stark broadening
according to the preceding section, a value for the electron temperature
may thus be derived which can be of great precision ( ^ ± 2 % ) in partially
(~10%) ionized plasmas. Stabilized arc and gas-dynamic shock tube
plasmas are particularly amenable to this method.

232 I V . APPLICATIONS

IV.3. AMPLITUDES AND SPECTRA OF PLASMA WAVES

In the past, Stark broadening methods have had most of their laboratory
applications in equilibrium plasmas, the latter term often being interpreted
to mean local and instantaneous thermal equilibrium not only in regard to
translational degrees of freedom but also with respect to excitation and
ionization. It must therefore be emphasized that these methods may apply
to nonequilibrium plasmas as well. (The second temperature measurement
method, unless suitably modified, is an exception.) As a matter of fact, the
present section is concerned with methods almost exclusively suited for the
investigation of strongly excited collective modes in a plasma, i.e., modes
whose energy far exceeds the equipartition value, say, at the kinetic electron
temperature.

Nonequilibrium laboratory plasmas in this sense are, perhaps unfortu-
nately, rather commonplace, and many experimental methods for their
analysis are well established [186]. Still, the two Stark broadening methods
based on theoretical considerations of Baranger and Mozer [131] and of
Blochinzew [133a], respectively, add substantially to this arsenal of
plasma "probes," especially because they are noninterfering, as are all
other spectroscopic methods.

Both methods come into their own if electric fields from plasma waves
exceed particle-produced fields, the latter being the only ones allowed for
in most Stark broadening calculations. Ideally, a theory of Stark broad-
ening by wave fields would enable one to calculate the radiator's response to
fields of arbitrary frequencies and amplitudes, or actually, to superpositions
of such fields. This general approach is not very practical, and one should
always endeavor to find lines for which this response can, at least in prin-
ciple, be obtained from one of the two extreme versions of such a general
theory. In one case, the wave fields must be strong enough to cause in-
stantaneous linear Stark shifts in hydrogen, ionized helium, or other
"hydrogenic" lines which exceed all characteristic frequencies of the plasma
waves or the turbulence. Then the quasi-static approximation is applicable
(Section II.2), and the line profile essentially traces the distribution func-
tion of wave field amplitudes. This case is one of the limits of Blochinzew's
[133a] theory, and the only analytical difficulty in deducing distribution
functions from profiles dominated by this effect comes from the necessity
to allow for the presence, in a given line, of several components char-
acterized by different Stark constants and relative intensities. (These
quantities can be found in the work of Underhill and Waddell [323].) A
trial and error procedure will probably be best, in which one assumes a
distribution function, calculates the profiles, and repeats the calculation

I V . 3 . AMPLITUDES AND SPECTRA OF PLASMA WAVES 233

until satisfactory agreement with measurements is achieved. In cases where
the profiles depend on polarization, one further needs to make suitable
assumptions on the anisotropy of the electric fields and calculate profiles
for σ and π polarizations separately. (Such calculations can, of course, also
be made for other than hydrogen, etc., lines, especially for neutral helium
lines with forbidden components.)

If the shape of the distribution function is not as important as an estimate,
e.g., of the rms wave field strength, this cumbersome procedure can be
circumvented by using approximate averages [132] for the linear Stark
coefficients corresponding to Eq. (14). Usually, controls against other line
broadening effects are more important than improvements to this pro-
cedure. Since "turbulent field" Stark profiles may well resemble Gaussians
(see also Section III. 10), great care must be taken to eliminate Doppler
effects. This is again best accomplished by measuring at least two lines with
substantially different ratios of Stark coefficients and Doppler broadening.
Also, reliable estimates of an upper limit for the electron density are
required so that ordinary Stark broadening from particle-produced fields
can be evaluated, e.g., from Appendix I or, more crudely, from Eq. (16).
Only if this Stark broadening turns out to be negligible can reliable deter-
minations of the wave fields be made. In this connection, the reader should
be aware of the fact that the mean particle-produced field is larger than the
Holtsmark field strength in Eq. (36) by a factor of about 3, unless Debye
shielding, etc., are very strong (see Section II.2a). Other competing effects
should be considered as well, e.g., Zeeman effects and fine structure [307].

Much more detailed information on the spectrum of plasma waves can
be obtained when instantaneous Stark shifts are small compared to both
relevant atomic frequency differences and plasma frequencies. This was
the case Baranger and Mozer [131] had in mind in their paper on light as a
plasma probe, and which is discussed in some detail in Sections II.5c and
III. 10. Ideally, the profiles of the ensuing plasma satellites, with intensities
measured relative to the corresponding allowed lines, directly reflect the
frequency spectrum of plasma waves. (The frequencies are of course as seen
by the atoms, i.e., might be Doppler-shifted in case of atom-plasma
streaming.) However, higher order effects as enumerated in Section II.5c,
magnetic field effects [158, 159] (see also below) and, especially, other line
broadening mechanisms often complicate matters.

For example, in plasmas of low electron density but not very low kinetic
atom temperatures, Doppler broadening may be important. While it can
then be rather hopeless to measure the spectrum of the waves, one might
still be able to determine their total electric energy density. This quantity
can, at least in principle, also be obtained from the shift of the allowed line,

234 I V . APPLICATIONS

using Eq. (391) or its generalization for less hydrogenic systems. In the
case of ion waves, i.e., low frequency oscillations, even the resolution of
satellites and ordinary forbidden components is often not possible by purely
spectroscopic means. It is then essential to make sure that the forbidden
components from particle-produced fields as calculated, e.g., in the litera-
ture [90-95] (see also Section II.3cß) are indeed much weaker than the
features, i.e., the unresolved plasma satellites, actually seen. To estimate
effects of particle (ion)-produced fields, the following formula (see also
Section IV.4b) for the relative intensity of a forbidden component is
usually sufficient, namely

*-««[jj^O-M] h 3/2 (415)

meo»·',» N

in terms of q u a n t u m numbers n, I for t h e upper state of t h e allowed line
and n, V for t h a t of t h e forbidden component, which is displaced b y a n
angular frequency cot',t· (see Appendix V I ) from t h e allowed line. Also, l>
is t h e larger of I, V. Another precaution in this a n d other cases is t o measure
t h e profiles of a t least t w o lines with parameters sufficiently different t h a t
Doppler effects from mass motions corresponding, say, to two drifting
Maxwellian distributions can be excluded, not to mention coincidences with
impurity lines.

If measured profiles are substantially broader than estimated Doppler
widths or profiles measured from other lines with smaller Stark effects, it
may still not be possible to interpret the measured satellite profile directly
in terms of a spectrum of plasma oscillations, unless t h e satellite profile is
significantly broader t h a n t h a t of t h e allowed line, a n d this additional
broadening is not trivially caused by electron density variations along the
line of sight. T h e former condition arises from t h e fact t h a t calculated
widths of allowed a n d forbidden components from particle-produced fields
tend to be quite similar and that this broadening is always present, while
t h e second condition comes from t h e square-root dependence of b o t h elec-
tron and ion plasma frequencies on electron density. (Naturally, the second
condition does n o t apply if gyro frequencies in a homogeneous magnetic
field are involved.) Some allowance for t h e usual S t a r k broadening can
probably be made b y treating t h e satellite profile as a convolution of t h e
allowed line profile and plasma wave spectrum, although there is no sound
theoretical basis for this procedure. Density gradient effects can sometimes
be eliminated by an Abel inversion (for cylindrical plasmas) or accounted
for by superimposing suitable calculated profiles (see also below).

Atomic parameters for helium lines prone to show plasma satellites can
be obtained from Eq. (393) and Appendix VI. (Probably, satellites could

I V . 3 . AMPLITUDES AND SPECTRA OF PLASMA WAVES 235

also be found in other neutral atoms or nonhydrogenic ions, but no sys-
tematic search has been made as yet.) A number of considerations enter into
the choice of lines for a given experiment, such as freedom from inter-
ference by impurities, convenient separation of allowed and forbidden
components, and suitable sensitivity to the Stark effect as determined by
the ratio of the relevant matrix element and atomic frequency splitting.
Depending on the width of the plasma spectrum, the sensitivity must not
exceed a value beyond which higher-order corrections to Baranger and
Mozer's theory are important, such as quadratic Stark shifts from wave
fields as estimated by Eq. (397), etc. Higher-order effects are particularly
important in the presence of a second perturbing level, say 5G in the case
of the 2P-5D, F transitions [154], and perturbation theory then may not
be of much use. Only in exceptional cases wOuld it be worthwhile to analyze,
e.g., fourth-order satellites (see Section II.5c) accompanying the allowed
line rather than second-order satellites of lines with smaller sensitivity to
the Stark effect.

No discussion of the plasma satellite method would be complete without
mentioning magnetic field and polarization effects on the profiles. (Anisot-
ropies of the radiation pattern are small [153], even if plasma fields exist
essentially only in one direction.) Without magnetic fields, plasma satellites
are only weakly polarized—about 15% in extreme cases [153]. However,
even for electron gyro frequencies, and therefore Zeeman shifts, sub-
stantially below the plasma frequency, profiles of satellites depend rather
characteristically on the polarization, and a comparison with calculated
Zeeman patterns of the satellites can be used [158, 159] to determine
whether the plasma waves were predominantly longitudinal (with respect
to the magnetic field), perpendicular with random E-field directions,
perpendicular with left- or right-hand circular polarization, or more or less
completely random as far as the electric fields are concerned. As a matter
of fact, such a determination of the dominant polarization of the plasma
waves may be essential [158] before deducing plasma field frequencies and
amplitudes, because otherwise errors of the order of the gyro frequency
could not be avoided.

This dominant polarization is easily determined in the case of linear Stark
effect (see the literature [303-305]). Finally, in this case and for "low
frequency" (unresolved) satellites, one should not overlook the possibility
of superimposed Stark effects from macroscopic fields, which must of course
be substantially smaller than the fields corresponding to the observed
effects so as not to invalidate the various interpretations discussed here.
Also, before using measured satellite intensities for determining electrical
energy densities of plasma waves, one should make sure that the two

236 I V . APPLICATIONS

relevant upper levels have statistical populations, or else correct for
deviations from this situation.

In closing, a powerful method [324] must be mentioned which combines
the advantages of light scattering and emission spectroscopy. Basically,
it involves "scanning" allowed and forbidden lines, etc., by a tunable laser.
Observing, say, at right angles to the laser beam, one thus has the spatial
resolution of usual light scattering experiments [186], and the scattered
spectrum of course contains the same information on plasma waves, etc.,
as do the emission profiles. Last but not least, there is a significant increase
in sensitivity.

IV.4. STELLAR ATMOSPHERES

In stellar atmospheres, electron densities and the degree of ionization are
often considerably below those in laboratory light sources used for Stark
broadening measurements. Therefore, Doppler effects and pressure broad-
ening caused by neutral perturbers tend to be more important, and this
complicates matters. On the other hand, the lower electron densities also
lead to some simplifications. Other things being equal, the usual impact
approximation for the electron broadening is now even more appropriate,
and the broadening by ions is still less important than in typical laboratory
situations, hydrogen and similar lines being notable exceptions. Also, so-
called asymptotic formulas are almost always sufficient, because as-
tronomers are not often interested in portions of the line profiles within or
near the Stark widths. For convolution, e.g., with Doppler profiles, Stark
profiles can be put together from asymptotic profiles on the wings joined by
an almost arbitrary curve near the line center, such that area normalization
is preserved. Such schematic Stark profiles are particularly useful in opacity
calculations (see Section IV.6).

IV.4a. Hydrogen and Ionized Helium Lines

For lines subject to linear Stark effect, it is convenient to express Stark
profiles in terms of reduced wavelengths or, rather, wavelength differences
. a, namely

a = | λ - λ0 |/Fo = | Δλ |/2.6eiV2/3, (416)

where F0 is the Holtsmark field strength as defined by Eq. (36) and to be
calculated from electronic charge e and electron density N. Asymptotic

wing profiles, assuming singly charged perturbing ions and the corrections

I V . 4 . STELLAR ATMOSPHERES 237

according to Eqs. (426) and (431) to be small, are practically always well
approximated by

Sn'n(a)

+/:.-f-^/;-?)]·

a formula which combines features of several previous wing formulas
[38, 59, 120]. Normalized asymptotic line shapes, say, in wavelength
units, follow then from /(Δλ) = S (a) \ da/d Δλ | = S(a)/Fo, and Cn>n
is of course the usual constant characterizing asymptotic Holtsmark profiles
which account only for broadening by statistically independent "quasi-
static" ions. Numerical values of these constants for some early series
members can be found in Appendix II, while values for other lines, except
for low series members, can be calculated [325] to within about ± 5 % from

_ 6.1 - 10-* ^10'
Zi5/2 [! __ (n*/nt) J A

for angstrom wavelength and electrostatic cgs field strength units. The
Z scaling with Z = 1 for hydrogen, Z = 2 for ionized helium, etc., in this
expression is exact, and n(n') is the principal quantum number of the
upper (lower) level of the line.

The square bracket in Eq. (417), i.e., the correction factor to the asymp-
totic Holtsmark result, will be discussed next. Beginning with the first line,
the first term added to the " 1 " in the first factor accounts for deviations
from the asymptotic ion field strength distribution function, and the term
associated with the " 1 " in the second factor on this line accounts for the
"quasi-static" contribution from slow electrons. The constants Dn>n can
be shown by comparison with entire Holtsmark profiles [323] to corre-
spond to

6 · 10~6 n3n'6 (419)
D»'» ~ ~^Γ [1 - (n*/n*)J* ~ 10Cn'n '

238 IV. APPLICATIONS

and the parameters Yn>n obey

(n* - n")fi | A« | (n« - n'^hcFp \ a |

r - mr = äür 10 ' (420)

both expressions being in the same units as used for Eq. (418). Obviously,
Dn>n/az/2 must always remain relatively small, but no such restriction is
implied for the quasi-static electron broadening term.

The second line in Eq. (417) describes the contribution of electron impact
broadening. (For Ha , P« , etc., one might expect larger contributions from
detailed calculations [53, 57, 326] of the atomic matrix elements occurring
in Eq. (110), but — 2{xiXf + ytyf) must now be included.) The first factor
on this line essentially allows [38] for deviations from the asymptotic
impact or Lorentz profile, which should stay relatively small for the electron
impact broadening correction to the asymptotic Holtsmark profile to be
valid. The two terms \ exp( — Yn>n) and i(kT/En) exp( — 7n'n) in the last
factor, which account for "strong" collisions and higher multipole inter-
actions (see Sections II.3a and b), constitute interpolations between
various estimates, but may well be in error by a factor of about 2. In view
of this, explicit inclusion of inelastic collisions is hardly worthwhile, except,
perhaps for ionized helium, etc., lines (see Section IV.6), nor should the
various estimates be trusted for kT > EK . In the case of ionized helium,
one might also consider an additional term corresponding to the third term
in Eq. (127), which again seems insignificant in the present context.

The negative term on the first line of Eq. (417) stands [38] for Debye
shielding of the perturbing ions with an approximate numerical factor. It
reduces the quasi-static broadening below the Holtsmark result, but must
be small for the correction to be meaningful (see Section II.2a). Similarly,
the negative term on the third line corrects for Debye screening in the
electron impact broadening, with 72 estimated [59, 120] by

1 / Δ ω \ 2 τη(Αω)2 2ττ2 /4ττ \1/3 m&

the last expression again for angstrom wavelength units and electrostatic

cgs field strength units. (Use of y2/4 probably gives more accurate results.
See below.) For this correction to be relatively minor, 72 has to be larger
than Yn>n. From Eqs. (420) and (421) and from the requirement
| Δα> | > 2wj since asymptotic formulas would certainly not be of much use

for | Δω | < 2w> the corresponding requirement with w from Eq. (16) is

Yn'n/Y* < ( T / 6 ) {*W*/kT) < 1. (422a)

I V . 4 . STELLAR ATMOSPHERES 239

A little consideration shows that the above condition is always fulfilled

when the shielding correction to the quasi-static contribution is small.
Specifically, assuming a > (2w/F0) \2/2TC, one obtains for the latter
correction the condition

5(e2N1/s/kT) ((%*/<*) < 0.5 (e2Nm/kT) < 0.3, (422b)

say, for errors of about 10% or less from this source.
One validity condition for Eq. (417) is therefore

l(4w/Z)NJ*(*/kT) = eVnfcT < 1, (423)

ri being the mean separation of perturbing ions as defined by Eq. (53).
While Eq. (423) usually does not impose any serious restriction, we must
not lose sight of the fact that the underlying theory of Debye screening
(see also Sections II.2a and II.5a) is, strictly speaking, valid only if
e2/nkT is much smaller than unity. Even such a sharpened criterion tends
to be fulfilled in most laboratory experiments, including those at low
densities [212] and temperatures (see also Section III.2c), and in stellar
atmospheres. More restrictive is the condition imposed by the requirement
that the corrections to the leading terms in the asymptotic expansion of the
entire line shape be small. Assuming that the fractional corrections in the
asymptotic expansion should not exceed, say, 0.3 for a 10% accuracy, one
thus estimates from Eqs. (418) and (419) that Eq. (417) should be
employed only for a's obeying

, , . 7 - IO"4 TIV4 ,AnA.

' « ' * Z* 1 - ( η « / η ' Γ (424)

The right-hand side is a factor of about 2 larger than the (half) half-width
corresponding to Eq. (16), as surmised above.

That entire calculated Stark profiles for the relatively low electron
densities of interest here can indeed be approximated to within about 10%
beyond these a values is borne out by comparison with corresponding
profile calculations [59] (see also Appendix I.a) for hydrogen lines which,
in turn, agree with laboratory experiments to within about 20% or less
(see Sections III.2a-c). At higher densities, larger deviations do occur,
so that Eq. (417) should then be used only well outside of | Δω | « 2w;
however, its accuracy at densities prevailing in stellar atmospheres is well
supported by recent relaxation theory [52, 113, 121] calculations, as dis-
cussed after the next paragraph, provided the parameter Y2 is divided by a
factor of about 4. This corresponds to halving the usual Debye impact
parameter cutoff, which can also be justified theoretically [135] for fre-
quencies well above the ion plasma frequency (see Section II.5a).

240 I V . APPLICATIONS

Before making analogous comparisons for ionized helium lines, allowance
may have to be made for another physical effect not yet included in Eq.
(417), namely the radiator-perturber correlations caused by the Coulomb
interactions between charged radiators and perturbers. This effect gives
rise to the exponential (Boltzmann) factor in Eq. (40) for the asymptotic
field strength distribution function in the vicinit}' of a charged radiator.
Assuming the nearest neighbor approximation to be reasonably valid, the
radiator-perturber separation r occuring in this formula may be estimated
from the corresponding reduced field strength, namely

ß = F/F, « A\/CFo = a/C « a(fCn,n)"2/3, (425)

the last step following from the observation that Cn'n is 1.5 times the f
power of the linear Stark coefficient C in the wavelength scale and averaged

over all Stark components. Using also Eqs. (45) and (46), the exponential

factor can thus be written

for singly charged perturbing positive ions and "quasi-static" electrons,
respectively. The "ion" factor replaces the " 1 " in the last factor on the
first line of Eq. (417) ; the "electron" factor multiplies the integral in this
factor. With this modification, Eq. (417) agrees with the calculations [61]
(see also Appendix I.b) of ionized helium lines for a values fulfilling Eq.
(424), and again for relatively low densities, to within about 15%. Coulomb
corrections analogous to those described by the above exponential factors
to the electron impact broadening are generally negligible [57, 61]. (See
also Section II.3b.)

It is of considerable interest to compare Eq. (417) with relaxation theory
calculations [52, 121] (see also Section II.4b) of hydrogen Unes, in which
the interpolation between quasi-static and impact approximations for the
broadening by electrons is relegated to a deeper layer of the theory, at
great expense in calculational effort. Such a comparison [120b] shows
typical agreements to within about 5% in the asymptotic range as defined
by Eq. (424), if the reduced Debye cutoff is used. This is close to the
accuracy of any of the various calculational procedures. Moreover, because
of the large number of correction terms in the factor multiplying the
asymptotic Holtsmark result and the often opposing wavelength- or
«-dependence of these terms, the square bracket in Eq. (417) exhibits a
weaker a-dependence than one might think, resulting in a better fit to an
a~5/2 law of the Stark profile than expected. Also this is in good agreement

I V . 4 . STELLAR ATMOSPHERES 241

with the appropriate relaxation theory calculations [121c] and the corre-
sponding measurements [212] (see also Section III.2c), but is evidently of
no deeper physical significance. Agreement in respect to such extended
a-5/2 dependence is not so good when comparison is made with calculations
[215] involving the original impact approximation, though perhaps still
within mutual tolerances. These deviations can be traced to the replace-
ment of Yn'n in one of the exponential integrals in Eq. (417) by a constant
corresponding to an a value about equal to the half-width of the Stark
profile, i.e., to the only approximate inclusion of the "Lewis" [65] correc-
tion (see Section II.3a) in the impact broadening calculations [59, 61, 215].

Comparison of Eq. (417) with an often used empirical procedure [216]
for the inclusion of electron broadening is less satisfactory [120b], If
expressed in terms of the dimensionless frequency Yn'n , this procedure
usually gives a sharper transition from the simple Holtsmark result to
about twice this relative intensity, when Yn>n increases from Yn*n « 1 to
Yn'n > 1. Except in the case of strong Debye shielding this is inconsistent
with all of the calculations but of course not with the underlying experi-
mental data [212] (Section III.2c), perhaps in part because the transition
in question was somewhat obscured by the other corrections to the asymp-
totic Holtsmark formula under the conditions of the experiments. For these
reasons, but mainly because of the inclusion of the temperature dependence,
Eq. (417) with F2 divided by 4 in most cases, or the equivalent calculations
of entire line profiles should be given preference over the empirical pro-
cedure.

While Eq. (417) is thus more likely than not correct to within ± 1 0 %
provided the conditions (423) and (424) are well fulfilled (and that the
factors given by Eq. (426) are inserted in the case of charged radiators and
that we have kT < 2?H), there still remains the question as to the accuracy
of the Holtsmark theory [11] of ion broadening, the basic theory used as a
reference in all these considerations. Of the various approximations required
for its derivation (see Sections II. 1 and 2) from the more general expres-
sions for spectral line shapes, only two seem subject to reasonable doubts
for the lines considered in this section, namely the dipole approximation to
the radiator-perturber interaction Hamiltonian and the quasi-static
approximation. The discussion of the former can be disposed of very
quickly, because any deviations from it would primarily cause asymmetries
in the profiles, which are known to be small (see Section III.9) even when
neighboring lines almost merge, a situation that would be rather incon-
sistent with the use of asymptotic wing formulas.

To estimate deviations from the quasi-static approximation, Kogan's
treatment [15] (see Section II.4c) is appropriate. It gives, according to

242 I V . APPLICATIONS

Eqs. (327) and (328) but using an averaged Stark coefficient appropriate
for the line wings, a relative correction to the Holtsmark result £»·(«) of

ASM 1 m kT / n* - n'>\~> Γ/ m p \ A(j9) mp A'Q9)1
Si(a) 2mpEH\° Z ) L\ rrtr) H{ß) ^ mr H(ß) J '

(427)

with β to be estimated, e.g., from Eq. (425) and with the functions
Α(β)/Η(β) and Α'(β)/Η(β) as discussed in Section II.4c. (The masses m,
rap , and mT are those of the electron, perturber, and radiator, respectively.)
The largest deviations generally occur for small β values, so that for pur-
poses of an error estimate we may use β ~ 14. (Note that smaller β values
are of no interest here, because corrections to asymptotic formulas per se
would be too large.) With Eqs. (325) and (330) we therefore obtain

Δ&(«) . 1 m kT ( Λ 7 1 / , η 2 - η , 2 \ - 2

Stia) ~ 2 mp EK \ Z)

X Γθ.0050 (l + — ) + 0.0015 — ] , (428)
L \ mr/ rrtri

the actual corrections being negative in the asymptotic regime. For

kT < EK , this error tends to be small except for high principal quantum
number n-a, n-ßf etc., lines, which will be discussed separately in Section
IV.5, and for some early members of the Lyman, Balmer, and Paschen

series at low densities. However, Doppler broadening is then usually

important even in the near asymptotic region of the Stark profiles, and we

may use a larger value of ß in Eq. (427). Essentially, this allows us to

multiply Eq. (428) by the ratio of, say, 2w, with w from Eq. (16), and the

Doppler width from Eq. (18), i.e., by a factor

2w/œO « 9 · 103(nV2/Z3) (mTEK/mkT)1/2 a0W2/3. (429)

Provided that this factor is below unity, the error estimate (428) may then
be replaced by

^ 25 / m r \ 1 / 2 (m m\( ηη' V /fcT\1/2 „οΛΧ

Si(a)

if we omit the much faster decreasing term stemming from A'(ß) in Eq.
(427). Even this error may not always be negligible, e.g., for the solar H„