Spectral Line Broadening by Plasmas

I V . 4 . STELLAR ATMOSPHERES 243

line (<30%) and should therefore, at least to a major extent, be avoided
by subtracting

| ASi(a)/Si(a) | « (5/64Fn,n) (m/mp) (1 + mJmT) (431)

from the first factor on the first line of the square bracket in Eq. (417).
This estimate follows from Eqs. (297), (298), (325), (328), (418), (420),
and (425), except that we divided by 2 to allow more properly for the
strongly shifted components (see also Griem [120b]). The A'(ß)/H(ß)
term is again neglected.

We finally observe that such a negative correction to the asymptotic
Holtsmark result, which is only important for Yn>n < 10~3, may occa-
sionally more than compensate for the electron impact broadening correc-
tion. In some cases, the empirical procedure [216] could thus be more
accurate than Eq. (417) without the correction expressed by Eq. (431).
However, the physical reason is exactly the opposite of the one often
assumed: the broadening by ions was overestimated, not that by electrons.
(Note that none of the detailed profile calculations [52, 59, 61, 113, 121,
215] make allowance for this last correction.)

IV.4b. Lines With Forbidden Components

o

It has long been recognized that forbidden helium lines such as X4470.0 A
(2 3P-4 3F) arise in stellar atmospheres from a breakdown of the usual
dipole selection rules in the presence of a perturbing electric field, e.g., the
ion microfield. If the quasi-static approximation can indeed be applied for
calculating the effect of this field on, say, the 2 3P-4 3D, 3F composite
profile, and if the impact approximation is valid for the evaluation of
electron broadening, then procedures as discussed in Section II.3c/3 can be
used to obtain entire Stark profiles. Such calculations [90-93, 96c, 233,
234] have been made for a number of helium lines and their forbidden
components, and the asymptotic formulas as given by Griem [90] and
Gieske and Griem [92] are of particular interest for astronomers. (The
prescription [51a] mentioned by Barnard et al, [91] is not suitable, because
it does not account for the transition to linear Stark effect.)

Especially simple are the upper state principal quantum number n = 4
lines. For them, we may often ignore all but the 4D and 4F levels, in which
case the transition from quadratic to linear Stark effect can be handled
analytically. (See also Section II.2b.) Furthermore, each magnetic sub-
state can be treated separately, and fine structure of the upper levels and
Stark effects of the lower levels are negligible. With these approximations

244 I V . APPLICATIONS

and procedures, one obtains [90] for the X4471.48 Â (2 3P-4 3D, 3F) line

I(x1) ~ ? Γ ( « / ' ) , 0.81 · 10-'W 1
~ 9 U>2 + (Δλ)2 | Δλ |7/41 Δλ + 1.44 H

1 Γ (w/v) 0.81 · lQ-'W

+ 9 [w2 + (Δλ - 0.20)2 | Δλ - 0.20 \Vi | Δλ + 1.24 \M

(432)

a formula valid on the opposing wings of the Stark components corre-
sponding to the two distinct fine-structure levels of the lower state. Between
the Stark components, only the first (electron broadening) term of the
above expression should be used or, rather, the Stark profile

7 ( λ ) ~ W w* + (Δλ)« + ( ô ï ) v? + (Δλ - 0.2)' · ( 4 3 3 )

The (half) half-width from impact broadening is [90] (434)
w « h · 10-16#[Â]

with fc3 « 1.51, 1.34, 1.14, 1.03, and 0.94 for temperatures of T = 0.5,
1,2,3, and 4 · 104 K. Shifts from electron impacts are negligible at densities
for which asymptotic formulas are useful, say, N < 3 · 1014 cm"3.
Entirely analogous formulas hold for the n = 4 singlet (X4921.93 A)
line, except that in this case there is of course no fine structure and that
w = ki · 10~16 N [A] and the constant in the second (ion broadening)
term must be multiplied by factors of about 1.50 and 1.35, respectively, to
allow for their scaling with wavelengths, etc. The wavelength separation
Δλ is naturally to be measured from the unperturbed wavelength of the
allowed line at 4921.93 A, and instead of 1.44 A for the zero-field separation
of the Stark components we now have 1.34 A.

The reader will notice that Eq. (432) reduces, in the limit of much larger
wavelength separations than the zero-field separations of the two com-
ponents, essentially to the leading terms in the formulas for hydrogen lines
of the preceding section, where the ion broadening term was written first.
However, the wavelength dependence of the limits of the first exponential
integral is now ignored, and the second exponential integral which took
care of Debye shielding is omitted. All these simplifications seem well
justified in most astrophysical applications of Eqs. (432) and (433), namely
at N < 3 · 1014 cm-3 and outside the Doppler widths of the two com-
ponents, but still well within Δλ values corresponding to the zero-field
separations from the 23P-43P (X4517.4Ä) or 21P-41P (X4910.8Â)

I V . 4 . STELLAR ATMOSPHERES 245

components. The latter restriction, i.e., | Δλ | < 25 A for the triplet and
| Δλ | < 5 A for the singlet line, must be imposed because ion field inter-
actions with 4P levels were neglected in Eq. (432). (The detailed profile
calculations [90, 91, 96c] are not subject to this limitation.) For wave-
lengths just beyond the 2 3P-4 3P line positions, another asymptotic formula
can be derived [90], namely, now neglecting fine structure,

1.05-10-19iV , 1.62' 10-16iNT (485)
h ( λ ) ~ (λ - 4517.4)3/4 + (λ - 4471.5)" '

The first term accounts for 4D-4P interactions caused by ion microfields
and the second term for the combined ion and electron effect on the 4D-4F
group, estimated by doubling the appropriate limit of the ion broadening
term in Eq. (432). The latter procedure seems reasonable, because the
parameter Yn>n of the preceding section has values of about 0.1 or more at
these wavelengths, so that electron and ion broadening of the 4D-4F group
should indeed be comparable [120]. The corresponding singlet formula is

/ / / λ \ « 1.6 * IO-18 AT 2.2 - IQ-16 AT (}
1 ( } ~ (4910.8 - λ)3/4 (4921.9 - λ)5/2 '

because Yn'n is also near 0.1 here so that the quasi-static approximation
becomes valid. Neither of these formulas should be used when the wave-
length separation from the 2P-4P zero-field positions approaches a mag-
nitude corresponding to the unperturbed 4P-4D separations, i.e., about
36 A or 10 A, respectively. (Beyond, a —§■ power law might be very close
to the truth.) This condition arises because quadratic Stark effect was
assumed for the 4P levels in the derivation of Eqs. (435) and (436). [See
also Eq. (451).]

For electron densities below N « 3 · 1Ö14 cm-3, the asymptotic formulas
per se are expected to be valid to within a Doppler width from the various
profile maxima. It will thus be sufficient to connect the solutions of Eqs.
(432) and (433) near the allowed component in any reasonable way that
preserves area normalization of the entire profile. However, to carry through
such normalization in the presence of the forbidden components, their
relative profile area must first be estimated. This can be done either by
comparison with the detailed profile calculations [90, 91] or by a per-
turbation theory calculation based on Eq. (72b) of Section II.2b. [Direct
integration of, e.g., the first terms in Eqs. (435) and (436) is of no use
because of their limited range of validity.] The relative (integrated) in-
tensity in question is then given for small e by

| Cn |2 « €*/4 = IU12/(E1 - E2)J, (437)

246 I V . APPLICATIONS

the index 1 designating the upper level of the allowed line, the other that
of the forbidden component. For the interaction energy, we now substitute
the dipole approximation from Eq. (58) and replace the ion microfield in
this expression by the nearest neighbor approximation, F = en | n |~3.
This leads to

/''MA-'c-'-(<**ïrbL· (438)

the average being over directions and magnitudes of r i . The directional
average is entirely analogous to that leading to Eq. (139) in the impact
theory, and using radial matrix elements of the hydrogen atom, the relative
profile area becomes

J 4 21 + 1 \mcoi2/ ViVav

where n is the principal or effective quantum number of the upper state,
I the orbital quantum number corresponding to the allowed component, and
Z> the larger of this quantum number and that corresponding to the for-
bidden component. Also, ωί2 is the zero-field separation in angular fre-
quencies of the two levels (see Appendix VI).

The remaining average is over the magnitudes of r i . It leads to the
divergent integral

(l/n4)av = j (4wNi*/f*) dr « 4irN/rmin, (440)

with the minimum nearest ion separation from the perturbed atom to be

determined such that the assumption of quadratic Stark effect remains

reasonably valid. This is equivalent to saying that substitution of rm-m into
Eq. (439) should yield a result close to unity, say, about J. With this

choice, we finally obtain

Γ 07 / f, \ 2-13/4

/

7,(X) dX « 2"W [^fj nHn> - W ( — ) J , (441)

i.e., about 1.6 · 10"16 N for 2 3P-4 3F and 2.4 · 10~16 N for 2 Ψ - 4 *F, relative
intensities which are in about 20% agreement with the detailed calculations
[90,91] for the 2P-4F components at all densities below about 3 · 1014
cm"3. An implicit assumption at this point is that the wing of the allowed
line as given by electron impact broadening is subtracted from the total
intensity near the forbidden component, before 7/ is determined from the

I V . 4 . STELLAR ATMOSPHERES 247

entire Stark profiles. [Note, however, that the temperature dependence
implied by Eq. (41) of Griem [90] is not borne out by these estimates.]
Application of Eq. (441) to the 2P-4P components yields about
1.45 · 10~18iV and 1.6 · 10~17iV for the relative intensities of the triplet
and singlet components, again in 20% agreement with at least one of the
detailed calculations [90], if a background is now subtracted which
corresponds to the second (2P-4D and 4F) terms in Eqs. (435) and (436).
Not surprisingly, this agreement extends to higher densities than N =
3 · 1014 cm"3, as does the validity of Eqs. (435) and (436) in the appro-
priate wavelength range.

Returning to the construction of Stark profiles by piecing together the
various asymptotic formulas, one should therefore multiply Eqs. (432) and
(433) or their equivalents for the singlet line by factors of (1-1.6 · 10~16 N)
and (1 — 2.4 · 10~16iV), respectively, to compensate for the intensity of
the major forbidden components. These modified formulas can then be
used to calculate Stark profiles for all wavelengths except those beyond the
2P-4P components, provided the ion broadening term in Eq. (432) or its
equivalent is omitted within, say, one Doppler width from the allowed line.
It is not necessary to introduce a cutoff near the forbidden component,
because the integrated intensity is convergent with respect to the limit at
the zero-field wavelength, and because the unphysical profile spike at the
zero-field position of the forbidden component is washed out after con-
volution with Doppler profiles, etc. At and somewhat beyond the 2P-4P
components, the Stark profiles must be continued according to Eq. (435)
or (436), but there is no need for another area renormalization, and the
profile spike is again of no real consequence.

After convolution with the appropriate Doppler profile, line shapes
constructed as suggested here agree with the corresponding results of the
numerical profile calculations [90, 91, 96c] to within about 20% at most
wavelengths, if we discount a small repulsion of the allowed line from the
various forbidden components of the order of the product of the zero-field
splitting with the integrated intensity of the forbidden component. [See
Eq. (397).] However, as first emphasized by Burgess [236], there is
another important contribution to the broadening in the vicinity of a for-
bidden component, which is caused by deviations from the quasi-static
approximation for the ion broadening, and may well be responsible for some
of the difficulties in the interpretation [327, 328] of stellar profiles. As dis-
cussed in Section III.5, such effects are also evident in several laboratory
experiments conducted at higher electron densities than those envisaged
here. To estimate this "dynamic" broadening, we turn [237a] to the "one-
electron" approximation [6] of Section II.4d, which gives, according to the

248 I V . APPLICATIONS

remarks at the end of that section,

Ζ,,(ω) « —Vi (\m- ^Δ-ω) / 2^17+- T1n 2 ( n 2 - *>2)α(Ι Δ ω - ω» I AninAi) (442)

for the profile of a forbidden component associated with an allowed neutral
atom line. Here a (z) is the characteristic width function given by Eq.
(161), Vi a typical relative velocity of the perturbing ions, and pmìn a
minimum impact parameter still to be estimated. All other symbols are as
used earlier in this section, except that we have returned to angular fre-
quencies.

Since a(z) decreases exponentially for | z \ > 1, the forbidden com-
ponent is concentrated near Δω « ωΐ2 in the limit of small velocities, and
we have in this limit for its integrated intensity

L,(«) άω « — ( ) — J - n2(/i2 - l>2) / a{z) dz
Pmin \mun/ 21+1 J^

= TTW / h V 3l> nu\n2 2 - 7l2>.2). (AM.
I ) —— (443)
4pmin χτηωη/ 21+1 quasi-static

Comparison with the corresponding (i.e., nearest neighbor)

result in Eq. (441) yields

U T Ì »·<»'- <>·>]'"Pmin2l«v 1/2 (444)
4
a«,,

&C012

and our conjecture [237a] is that this estimate can be used for all reason-
able ion velocities. (With different cutoffs [237c, d ] , the total intensity of
forbidden components also might depend on ion dynamics. There is as yet
no theory of this effect, which ought to be extremely weak because pmln
depends at most linearly on the reciprocal relative perturber velocity.) If
we further assume the dynamic ion broadening to be statistically in-
dependent of other broadening mechanisms, especially Doppler broadening,
and take it to be substantially less than the inherent level splitting, then
combined Doppler and dynamic ion broadening would produce a normalized
line shape

L.M . t £ ï p . ( I " ' - - I ) exp f- ("-^)']U (445)

for the forbidden component. Here COD is the Doppler width and cot the

I V . 4 . STELLAR ATMOSPHERES 249

''dynamical'' width

ω. = IL « fey*

■ΐΦΓ-ΙΑϊ^-"] 1/4 00i2ü>L 1/2 (446)

The latter estimate follows from Eq. (444), OJL stands for the angular
frequency of the Lyman limit of hydrogen, and a is the fine-structure
constant, 1/137. Also, the ratio of radiator mass mT and reduced perturber
mass ra' = (m"1 + m~^)~l comes in, because we should of course use some
mean value for the relative velocity of the perturbing ions.

Switching back to wavelength units (A), we therefore have

Ct ωΌ Α\Ό a\m') \_2l + I { >]\ Δλΐ2 1/2 (447)

912

for the ratio of the two widths, i.e., Xi « 3(mr/m')1/2 for both 2P-4F
components, a?< « 14(mr/ra')1/2 for 2 3 P - 4 3 P , and x{ « 7(mr/m')1/2 for
2 Ψ-4 Ψ. Accordingly, the dynamic broadening can be quite important,

but to assess its influence quantitatively one should consider profiles, say,

in the x = Δλ'/Δλχ scale with Δλ' measured from the zero-field position

of the forbidden component in question, i.e.,

X(x, Xi) = Ζ/(ω) | dœ/dx \

= (4χί/π5/2) ί " o d » ' I ) e x p [ - ( x ~ α : , ) 2 ^ 2 ] d α : , . (448)

Such profiles have been calculated numerically (see Appendix VII) by
excluding the interval | xf \ < 10~3, which, because of a(\ x |) —♦ In | x |_I,
should contribute less than 1% for the xx values considered here. These
profiles, after transforming to the wavelength scale, may finally be used to
estimate the combined effect of dynamic and Doppler broadening by con-
volving them near forbidden components with the second terms in the
square brackets of Eq. (432) or the first terms in Eqs. (435) and (436).
It then turns out that the 2P-4F components are more affected than the
2P-4P components by the additional broadening. (Because of the common
origin of dynamic and quasi-static ion broadening, there is actually no
theoretical foundation for this convolution procedure. The method of
Lee [237c, d ] may seem less arbitrary but gives similar results for these
lines, except for an erroneously strong wing of the allowed lines. See also

250 I V . APPLICATIONS

Section III.5.) This convolution should always become negligible as one
goes from the zero-field position of the forbidden component, say, half way
toward the allowed line, and in the vicinity of the latter, one should con-
volve only with an ordinary Doppler profile.

In case the reader wonders why we did not consider electron contribu-
tions to forbidden components, we might point out that for them the re-
duced mass in Eq. (447) is essentially the electron mass, so that we have
Xi « 250 for the 2P-4F components. Dynamic broadening would therefore
smooth the profile beyond recognition, so that it could not be distinguished
from the wing of the allowed line. However, we should not overlook
that all estimates in the present section ignore collective effects, which
can give rise to "satellites" to the forbidden components, especially in
the case of nonequilibrium plasmas (see Sections II.5c and III. 10). In
extreme cases, these plasma satellites may obscure the forbidden components
and result in erroneously high density determinations.

The procedures discussed here can naturally also be used for forbidden
components other than those associated with the He I, 2P-4D lines. For
example, detailed calculations for the 2 3P-5 3P, D, F, G lines are available
[92] (without the dynamical correction), and have been used to test asymp-
totic formulas for extensions to densities below 1015 cm-3 and to other
members of the diffuse series. Especially for higher series members, much
of the ion broadening is now via linear Stark effect, so that the formulas of
the preceding section often constitute satisfactory approximations. How-
ever, the 4P-nP components and any forbidden components such as
2S-3S, 2S-3D, 2P-3P, 2S-4S, 2S-4D, or 2P-4P usually remain well iso-
lated. At least in the vicinity of their zero-field positions and away from
the corresponding allowed lines, formulas analogous to the first terms in
Eqs. (435) and (436) are therefore quite useful.

Such formulas may be obtained by comparing the integrands in Eqs.
(439) and (440). This leads to

7 / h \2 1 \ dr \
7,(λ) « 3TN — J - n2(n2 - l>2) (
Ί -L (449)
21 + 1 χτηωη/ r2 \ d\ \

where r is still to be related to λ by an estimate of the quadratic Stark
effect. According to Section II.2b, the latter is equal to le2(E2 — 2?i) in
energy units, so that Eqs. (438) and (439) and transformation to wave-

lengths give

λ _ λ2 « (λ2 _ λ ι ) - - ^ - η*{η* - 1>η (—Y -4. (450)

I V . 4 . STELLAR ATMOSPHERES 251

With Eq. (449) the desired formula is thus

T^ *Nr3 TNW*mcy*>? χ2 Γ3 l> Λ12(Λ1 ~2yT)' 4 '
7/(λ)
~ μΤ=^ϊ ~ l χ2 - λι π χ - |- Li 2ΤΤΊ J

(451)

where the zero-field splitting is expressed through the wavelength dif-
ference. (If angstrom units are used instead of centimeters, the right-hand
side must be multiplied by a factor of 10~12.) Again, this formula is valid
only for relatively weak forbidden components and | λ — λ2 | < | λ2 — λι |,
and a suitable background must be added to allow for the usually electron-
produced wing of the allowed line, not to mention the Doppler and ion
dynamics corrections. Also, the reader should not be too surprised that
Eq. (451) gives, e.g., about 10% higher intensities than Eq. (436). This
difference can be traced to the approximate averaging procedure over mag-
netic substates used in deriving Eq. (451). Finally, the use of hydrogen
wave functions also for 2S-nS components would give a similar overesti-
mate. This is easily rectified by multiplying with suitable correction factors
as calculated, e.g., from the Bates-Damgaard [42, 43] procedure. [These
correction factors would correspond to the f power of the function <p(n*i_i,
ni*,l) defined by Eq. (79).]

With the Bates-Damgaard correction, the procedures discussed above
should be reasonably accurate for forbidden components to lines other than
helium neutral atom lines as well, provided the background from the elec-
tron impact broadening of the allowed line is calculated as discussed in the
following section, the component in question stays relatively weak, and
there is no other forbidden component of comparable strength associated
with the same allowed line. (See Section IV.6 for ion lines.)

IV.4c. Isolated Lines

Lines whose Stark widths are much smaller than zero-field separations
between interacting levels, e.g., the 4S, P, D, and F levels of He I dis-
cussed in the preceding section, are called isolated. Thus the 2 3S-4 3P
line is isolated at all densities below, say, N = 3 · 1016 cm""3, because its
width is then much smaller than the separations of the principal perturbing
levels (4 3S and 4 3D) from the upper state. However, the 2 3P-4 3D line is
isolated only below N « 1015 cm-3, so that the interacting level 4 3F is
about ten or more Stark widths from the upper state. Most strong helium
lines, except those of the two diffuse series, can thus be considered isolated
in stellar atmospheres. Moreover, for heavier atoms, not to mention heavier

252 I V . APPLICATIONS

ions, zero-field splittings tend to be larger than in the neutral helium
spectrum, so that an overwhelming number of strong lines in stellar spectra
not from hydrogen or ionized helium falls into the present category. Weak
lines, on the other hand, often involve highly excited states, whose separa-
tions from perturbing levels may not substantially exceed their relatively
large Stark widths. For them, the procedures described below must there-
fore be used with care and, if necessary, supplemented by estimates
analogous to those presented in the two preceding sections. Fortunately,
this is not as difficult as it might appear on first sight, because the corre-
sponding wave functions are usually well approximated by the wave func-
tions of hydrogen.

For isolated lines, most of the Stark broadening is caused by electron
impacts, which give rise to a dispersion or Lorentz profile

J.00 = - [ ( Δ λ - ί ί λ ) 2 + ^λ2]-1

where w\ and d\ correspond to the widths and shifts in Appendices IV.a
and V. (The tabulated values were calculated for N = 1016 and 1017 cm-3,
respectively, and should be scaled linearly with the ratio of actual density
to this nominal density.) The second version of Eq. (452) is usually suf-
ficient, because Doppler widths tend to be larger than both w and d, but
the linear correction term in d/Αλ must not be overlooked. For neutral
atom lines not listed in the tables, w and d values can often be estimated
with almost the same accuracy (| Aw/w | « | Ad/w | < 0.2) from the high
temperature approximation [51a] as given by Eq. (175), if the value of
the energy splittings between interacting levels of the same principal quan-
tum number n or, in case Debye shielding is important, the "plasmon"
energy ήωρ = h(4:TNe2/m)1/2 is, say, less than 0.1 kT/n2. (Even simpler
though somewhat cruder estimates mainly for lines with larger level
separations will be discussed at the end of this section.) Equation (175)
can also be used for ion lines, e.g., from singly charged ions, provided
Φαα = w + id is divided by 4 and the argument of the logarithm by 2, but
remains larger than, say, about 5.

Ion broadening of neutral atom lines is normally well described by the
quasi-static quadratic Stark effect approximation, which leads [51a], ac-
cording to Eq. (228), to

Ii(A\) « |Awx3/4/l Δλ |7/4 (453)

IV.4. STELLAR ATMOSPHERES 253

on the wing toward which the line is shifted by quadratic Stark effect, i.e.,
for Δλ corresponding to a low temperature limit of A\/d\ > 0, and to no
contribution on the opposing wing. The parameter A given in Section
AlV.a for ΛΓ = 1016 cm-3 must be scaled with iV1/4, and values of this ion
broadening parameter for lines not listed can be calculated from Eq. (224).
(This A was called a originally [7, 51a], which might be confused with the
reduced wavelength a — A\/F0 for lines subject to linear Stark effect.)

Several checks ought to be made before simply adding Eq. (453) to
Eq. (452) to obtain combined electron and ion Stark profiles of isolated
neutral atom lines. One consists of calculating x = (Δλ — d\)/w\ and veri-
fying that even near the Doppler width this quantity corresponds to
j(x) values in Section AlV.b which are well represented by the analog of
Eq. (228). Another is to make sure that A exceeds about 0.05, because
otherwise multipole interactions higher than monopole-dipole should have
been included which, according to Section II.3f, increase the damping
constant from electron impacts by typically 10%. But A should remain
smaller than about 0.5, because beyond this limit, transition to linear Stark
effect would occur in typical cases. If this happens, the procedures of the
preceding section must be employed. Lastly, the validity of the quasi-
static approximation must be investigated for the relevant values of Δλ.
As shown in Section II.4a, this approximation requires δ = Δω((?/#»4)1/3 >
1, with C being proportional to the quadratic Stark coefficient.

For a more quantitative estimate, we note that Eqs. (260) and (292)
yield for the frequency-dependent "widths" and "shifts" of the relaxation
theory as derived near the end of Section II.4b

w(6) = Λττ2δ1/4[1 - (15/128)δ-3/2], (454a)

d(i) « Αττ2δ1/4[1 + (15/128)δ"3/2], (454b)

On the far wings, the lh(ò) profiles are accordingly (455)
lh(d) « w(ô)/irô2 « (TTÄ/57/4)[1 - (15/128δ3/2)],

if we omit corrections as in Eq. (452). The leading term corresponds to
Eq. (453), and the correction to it from (relative) ion-atom motions is
therefore [17b]

Ah _ _ 15 _ _ 15^2 _ _ 15(Fp/e) I Δω 11/2 / νΛ2

li ~ 128δ3/2 - ~ 128C1/2 | Δω |3/2 " " 128A2/3 | w \ \Αω) '

(456)

where we have used Eq. (224) in the last step of this reduction. (Note
that C corresponds to Ce). Since we are interested only in frequency separa-

254 I V . APPLICATIONS

tions larger than the (1/e) Doppler width Δω0 « ν(ω/ο, we may finally
write

Δ7, = 45(Fo/e) · 1Q-16 /ΔλοΥ'2 (mA /λ_ Υ I Δλρ 13/2 _ _ m, -B

7, ~ 256A2/3 \wx ) \m') \2π) | Δλ | ~ ro'| Δλ |3/2 '

(457)

with di2 = 3 kT/m' and if angstrom units are used throughout. [The factor
mT/mf must be applied for the same reason as in Eq. (446) of the preceding
section.] A little consideration (F0 ~ NmyA ~ N1/4, w ~ N) shows that
this dynamic correction is actually independent of electron (ion) density,
so that the coefficient B as defined here is only a (linear) function of tem-
perature. Values of these coefficients can also be found in Appendix IV.a.
They are typically well below 0.1 [A3/2], and this correction is negligible
for most lines and wavelength separations, even after multiplication with the
appropriate atom-to- (reduced) perturber mass ratio. Whether or not it is
responsible for a discrepancy between measured [329] and calculated A
values of the He I, X4713 A line is an open question.

For ion lines, such B coefficients would often be closer to 1, and the quasi-
static approximation is no longer so useful for a quantitative evaluation of
their broadening by other ions, except at rather large | Δλ |. An important
qualitative conclusion, on the other hand, that may be drawn from Eqs.
(224), (452), and (457) for most ion lines and Δλ values is that broadening
of isolated ion lines by ions can usually be ignored in view of the relatively
large uncertainties in calculations or measurements of the dominant Stark
broadening mechanism, namely the electron impact broadening. As a
matter of fact, these uncertainties are still so large that as comprehensive a
tabulation of width and shift parameters as for neutral lines might seem
premature. We will therefore conclude this section with the description of a
simple semiempirical procedure capable of yielding calculated widths of
very similar, if not superior, accuracy to that achieved even by some of the
most involved calculations (see Section II.3d), namely, to within an aver-
age factor of 1.5. This is to be compared with the estimated accuracy of
our width data in Appendix V, which show an average agreement of ± 2 5 %
with measured widths (see Section III.7) of lines from singly charged ions.

The semiempirical procedure [102] for the calculation of electron impact
widths explicitly accounts only for monopole-dipole (perturber-radiator)
interactions from "weak" collisions. From Eqs. (151) and (210), it then
follows that the width is given by half the product of the density, the
velocity, and the sum of total (but non-Coulomb) cross sections for upper
and lower states, i.e., essentially by a total rate coefficient. One now as-
sumes that also the inelastic contributions not involving any change in

I V . 4 . STELLAR ATMOSPHERES 255

principal quantum number, which usually dominate in the inelastic terms
of Baranger's general formula [ 3 ] [see below Eq. (210)], are well repre-
sented by Van Regemorter's effective Gaunt factor approximation [103].
The average of these rate coefficients over the Maxwell distribution leads
to an integral starting at the threshold velocity for the particular inelastic
process. (Superelastic rates tend to be small.) By beginning the integration
at zero velocity instead, the semiempirical method attempts to account for
(non-Coulomb) elastic scattering at the same time. Such elastic scattering
is very important for the broadening of many strong ion lines, because for
them splittings between relevant perturbing levels tend to be of order kT
or larger.

With these approximations, the (angular frequency) width of a line,
i.e., one half its damping constant, becomes

+ Σΐ(/Ί>/α,Ι/)Ν(2|^Ε)|)], (458)

if furthermore we replace the arguments of the slowly varying Gaunt fac-
tors by suitable thermal averages. [Note that g(x) corresponds to b(e/W)
in the astrophysical literature [41], and that its numerical values for singly
charged ions are g(x) = 0.20 for x < 2 and g(x) = 0.24, 0.33, 0.56, 0.98,
and 1.33 for x = 3, 5, 10, 30, and 100.] The sums are over all levels i'(f)
which combine with the upper (lower) state of the line according to electric
dipole selection rules. They can usually be calculated to sufficient accuracy
with the help of Eqs. (77)-(80). (Note that the £ a | {J' \ra\J) |2 corre-
spond to | (i' | r | i) |2.) Frequently, the effective Gaunt factors for the
important contributions to the sum are practically all the same, and we
can sum over i' and / ' to obtain

o / x \ 3 / 2 ha, /EH\1/2 Γ y . , · / · . .v / 3*Γ \

+ </k2/ao2|/),(|^)], (459)

where AEi and AEf should be chosen close to the smallest values of | E? —
Ei I and | Ef> — Es |, respectively. The remaining matrix elements can be
estimated using effective quantum numbers from

(i | r2/a021 i) « H T K / Z ) 2 ^ · 2 + 1 - 3U(U + D ] , (460)

256 I V . APPLICATIONS

with rti to be determined from Eq. (80). Furthermore, U is the orbital
quantum number of the valence electron, and Z = 2, 3, etc., for singly,
doubly, etc., charged ions.

Experimental indications [102] (see also Section III.7) for singly charged
ions are that Eq. (459) may be used for all kT/AEi < 3. For say, 3 <
kT/AE < 50, Eq. (458) is of superior accuracy, while for kT/AE{ > 50,
the high temperature approximation, i.e., essentially Eq. (175) with
Wi = — Re φιαί and divided by Z2, should be preferable. (Coulomb effects,
which are allowed for by the effective Gaunt factors, are then quite small.)
There is some evidence [100, 263b] that the effective Gaunt factors under-
estimate the actual energy (temperature) dependence, but the correspond-
ing errors are hardly significant in the present context. Not much is known
about the accuracy of Eqs. (458) and (459) for multiply charged ions, i.e.,
for Z > 3. (See also Section III.8.) The effective Gaunt factor method as
such then almost certainly deteriorates in accuracy when applied to the
evaluation of specific cross sections, because higher multipole interactions
become increasingly important. However, the total cross sections may still
be of reasonable accuracy, especially if a threshold value g = 0.4 or larger
is used [330, 331] for 2S-2P and 3S-3P transitions, and are at least suffi-
ciently accurate so that Eqs. (458) and (459) estimate the broadening of
multiply ionized systems, say, to within a factor of 2. [Note that for multi-
ply charged ions, the impact approximation may be valid for broadening
by ions as well, whose contribution should then, e.g., be estimated analo-
gously to Roberts and Davis [101]. See also Sections II.3f and IV.6.]
For further and probably slightly more accurate estimates for the electron
impact broadening of multiply charged ions, the reader is referred to Section
IV.6, in particular to Eq. (526).

Equations (458) or (459) have been applied with fair success [89, 257]
also to neutral atom lines, with the effective Gaunt factor now rising
smoothly from near 0 at kT/AEi ~ 0 to join the Gaunt factor for positive
ions discussed above near kT/AEi « 2. In this case, the major use should
be only for estimates of Stark broadening relative to other line broadening
mechanisms, or for lines for which sufficient atomic data are not available
to implement the methods of Section II.3ca.

With the help of a dispersion (Kramers-Kronig) relation [79] the semi-
empirical method [102] could be made to also yield estimates of electron
impact shifts of ion lines to which plasma polarization shifts (see Section
II.5b) may have to be added. In general, these shifts are rather small, and
therefore not necessarily much larger than ion-produced shifts. One might
estimate the latter, e.g., analogously to Roberts and Davis [101] in cases
where the impact approximation is valid also for ions, at least near the line

I V . 5 . RADIO-FREQUENCY LINES 257

center. None of these shift estimates are very reliable, but fortunately all
shifts seem to be small enough to be negligible in most applications. For
neutral atoms, shifts are larger but can usually be calculated from Eq.
(227) and Section AlV.a with much better accuracy than for ion lines. In
stellar atmospheres, wre have R <K 1 in almost all cases. Debye shielding is
therefore normally negligible in the ion contribution, as had been implicitly
assumed in this and the preceding section also for the electron contributions.
(See also Section II.5a and the introduction to Section AlVa.)

IV.5. RADIO-FREQUENCY LINES

When some radio-frequency lines mainly from galactic H II regions
were first identified as n —► n + 1 transitions of hydrogen with n values
ranging from n « 90 to 170, their profiles were surprisingly narrow under
conditions where estimates such as those leading to the Inglis-Teller cri-
terion [18] (see also Section II.2) would seem to rule out sharp lines.
This problem was resolved [111, 332] by pointing out that the quasi-static
approximation is invalid for these lines under most conditions, so that the
broadening by ions is much reduced below its "Holtsmark" value, even if
for the latter the relatively small difference between averaged Stark
coefficients of the two principal quantum number levels is taken. Since
impact broadening of hydrogen lines is approximately proportional to the
inverse of relative perturber velocities (see Section II.3a), broadening by
ions might still be expected to dominate. Its contribution was therefore
calculated by Griem [111] by starting, in effect, with the dipole-monopole
interaction terms in Eq. (107), or by writing, e.g., for z polarization,

WÌ~^N(3—VYÌ \maj J[ ^p ΣVm, ( | < n + l ï + l m | r | n + l I W > | »

- 2<n + 1 V + 1 ra'l r | n + 11 + 1 m) · (nlm | r | nl'm!)

+ | (nl'm' | r | nlm) |2). (461)

[Note that in contrast to the situation for the electron impact broadening
of lines broadened at the same time by the quasi-static action of ions,
the XiXf and ytyf contributions, which were omitted in Eqs. (110), etc.,
are now also due to elastic collisions.] In this relation, which essentially
corresponds to Eq. (19) of Griem [111], except that we did not yet sum
or, rather, average over orbital quantum numbers I of the lower states,
use is made of the facts that radiative transitions with opposing changes in
principal and orbital quantum numbers and collision-induced transitions

258 I V . APPLICATIONS

caused by ions involving changes in principal quantum number are neglig-
ible. Also, the ratio of (n I — 1 m' \ z \ n + 1 Vm!) and (nlm \ z \ n + 11 +
1 m) is assumed to be close to one for m' = m, m ± 1, which is consistent
with the further reduction of the matrix elements by recurrence relations
and approximate radial integrals. Such reduction [111] leads to

' 3vi \ZmJ J p \ 4 L2 I (21 + 1) (21 + 3) J (462)

or, after averaging over I with suitable partial oscillator strengths and as-
suming statistical populations of the (I, m) sublevels, to

-ΙΜ0ΜΗΚΙ»)Μ;Ξ> <-»

(Here e is the base of the natural logarithm.)
The divergence of the integral over impact parameters at small p stems

from the use of perturbation theory in Section II.3a, and in analogy to
Eq. ( I l l ) we therefore estimate

Pmin ~ nh/ZmVi. (464)

Convergence at large p could always be obtained by using the relaxation
theory (Section II.4b) instead of the much simpler impact theory. Its
major modification can, however, be allowed for by a cutoff at

PÎnax ~ Vi/\ Δω | « Vi/Αωΐ) « c/ω = λ/2ττ. (465)

The physical meaning of this is that the duration of any contributing col-
lision p/vi should not exceed the times | Δω |_1 relevant in the Fourier inte-
gral representing the general line shape. With the further observation that
we are now interested only in frequency separations Δω from the unper-
turbed line of the order of a Doppler width, the last versions of this esti-
mate for pmax follow. With ω « 2Z2En/hnz and adding a ' 'strong collision"
term, the average width from ion impact broadening is finally

-^>(0{ϊ»·[ΜΚ1»)Μΐ+Κ^)} <->

where a « xiy is again the fine-structure constant. When averaged over
relative ion velocities, this becomes Eq. (31) of Griem [111]. (The other
original paper [332] on this subject had no estimate of the ion broadening.)

For electrons as perturbers, the neglect of collision-induced transitions
between levels of different principal quantum numbers would not be justi-
fied. As a matter of fact, in a generalized version of Eq. (461) with the sum

I V . 5 . RADIO-FREQUENCY LINES 259

also extended over one of the principal quantum numbers in the two posi-
tive contributions, terms with n' J£ n + 1 or n then completely dominate
for almost all Z's, while the terms diagonal in principal quantum number are
very nearly compensated for by the negative contribution. One of the two
almost equal "inelastic" contributions is therefore

Σ | (n'l'm' | r | nlm) |2 = (nlm \ r2 \ nlm) — X) | {nVmr | r | nlm) \2

= (nW/2Z2) (on2 + 1 — 3Z2 — 31)

- ( 9n 2 a 2 / 4 Z 2) (n2 - I2 - I - 1)
0

= (n2a02/2Z2) (\n2 + fI2 + !? + ¥■)

« (7iW/4Z2) (n2 + 3l2), (467)

where the second line follows from exact matrix elements for hydrogen. The
electron impact width, in analogy to Eq. (462), is then

we « (±T/3ve)NWZm)2 f (dp/p) (n2/2) (n2 + 3P) ln(pemax/pemin), (468)

or, after averaging over statistical equilibrium populations of the sublevels,

we « (±T/3ve)N(h/Zm)2 1.43n4 ln(pemax/p^n). (469)

The numerical coefficient in this expression is about 15% larger than in
the previous paper [111] because of an improved accuracy in the average
over Ϊ, and a factor of 1.5 smaller than in the work of Minaeva et al. [332].
(Note that the latter comparison is not very meaningful, because in this
work only the terms diagonal in n were considered, while the negative
interference and the actually dominating An τ* 0 terms were neglected.)

For the minimum impact parameter for electrons one estimates, in
analogy to Eq. (464),

Pemin « n2h/Zmve, (470)

which naturally agrees with the corresponding estimate of Section II.3a
for lines whose lower levels are not perturbed, because the upper and lower
state contributions are additive for inelastic collisions, and because n' = n
and nr j* n matrix elements are comparable in magnitude. Equation (470)
also agrees with the corresponding estimate of Minaeva et al. [332] where
Pmin is only 8% larger, but for p^ax the differences are very large. In Griem
[111], it had been argued that the cutoff should come at pem&x « νβ/ω, the
"adiabatic" cutoff for the dominant An = ± 1 inelastic collisions, while

260 I V . APPLICATIONS

in the other work the Debye radius was employed, which is much larger
and therefore irrelevant. (This incorrect choice was predicated on the
neglect of interference and inelastic terms.) We therefore adopt

Pmax « vc/a> « (hve/e2) (n*/Z2)a0 (471)
and finally obtain for the averaged width from electron impacts

we « (Aw/Svc)N(h/Zm)2 1.43n4[i + \n{mv2n/2EilZ)~], (472)

again adding a "strong-collision" term.

We are now in a position to compare the averaged widths caused by elec-
tron and ion collisions. The arguments of the logarithms in Eqs. (466) and
(472) are in the ratio

Pmax Pmin _ UViE^ ViÇn _ / _ m \ 1 / 2 ^ U

Pmin Pmax a2cmve2 2ve2 \ r a y v~e °2 '

which is usually rather large, say, of order 103. (mf is the reduced radiator-

ion mass.) Still, the argument in Eq. (472) tends to exceed 10, so that we

may use a value of about 2.5 for the ratio of the factors in square brackets.

With the further approximation ln(fn) « 4.5, the width ratio then be-

comes

Wi/we « (10/n2) (m'/m)1/2, (474)

always assuming equal temperatures for all perturbers and radiators. At
least on the average over Z, broadening by ions is thus entirely negligible, a
conclusion that ought to be substantiated by verifying that ion collisions
are not very effective in inducing Δη ^ 0 transitions. Furthermore, al-
though Wi should be multiplied by factors 22, 32, etc., for η-β, n-y,
etc., lines (accompanied by a reduction in the argument of the logarithms
bjr similar factors), ion broadening of those lines will usually be negligible
just the same.

We may thus return to Eqs. (468) and (472) for the electron impact
width to compare it with the major broadening mechanism for radio-
frequency lines, namely Doppler broadening. Since our expressions for the
electron broadening apply equally well for An = 2 or 3 transitions under
most circumstances, the ratio of the collision (half) half-width and the
Doppler (half) 1/e width is

ΔωΒ ~ 3 Vir m \kT/ \kTO) 2Z4 An |_2 \EKz)J
(475)

I V . 5. RADIO-FREQUENCY LINES 261

for individual I components of n-a, η-β, etc. lines, or, averaged over I,

we 4 n hca<? / m γ / 2 / mr V'2 Ar 1.43n7 f l , . / kTn\] ,_

if we use (vj^av = (2m/wkT)l/2 and introduce an effective Doppler tem-

perature for the radiators of mass mT. (In the "individual" formula, some
numerically insignificant terms were dropped, and the same estimate was

used for the logarithm as in the "average" formula.) It follows from Eq.

(476) that the electron density at which Stark and Doppler broadening

might be expected to be comparable is

Λ7 « 5 · 1014[(mH/mr)7TD]1/2(Z4 An/n1), (477)

if temperatures are measured in degrees Kelvin and the density per cubic
centimeter. [This is a generalization of Eq. (42) of Griem [111] to An 9e 1
lines and to ionized helium (Z = 2, mT = 4mn) and other hydrogenic ion
lines.] For, e.g., the 109-a line of hydrogen and T « ΤΌ « IO4 K, this rela-
tion suggests N' « 5 · 104 cm-3, an estimate which seems inconsistent with
the failure of various attempts [333-336] to detect collisional broadening
corresponding to the theory presented in this section.

On the other hand, broadening solely by inelastic collisions ought to re-
sult in an irreducible width, provided the cross sections for An 5* 0 transi-
tions corresponding to Eq. (468), etc., are indeed correct. Comparison
with Eq. (210) shows that we have effectively employed

or, averaging with (Z2)»v = \n2 and assuming only An = ± 1 transitions to
be important and equally probable,

5χαο2η4 / £ Η \ Γ1 , . (n m»f/2\\ (479)

*~* - ~W Uva) h + in{z -ΒΓ)\·

This cross section, in the relevant energy range (kT < 4 eV), is about a
factor of 1.3 larger than some earlier results [337] from the semiclassical
dipole approximation, which also differ by a similar factor from the more
refined calculations of Percival and Richards (see below). The accuracy of
our coefficient for the factor in square brackets can be verified by compari-
son with Born approximation calculations [338, 339] if we substitute
Pmin ~ n2a0/Z for Eq. (470), because in the high energy limit multipole
interactions higher than the dipole type and penetrating orbits should have

262 I V . APPLICATIONS

been accounted for. Instead of (479), we then obtain for the n —» n + 1
dipole contribution

~wU^;L H*i*ryJ'l+0"n-n+l (48/A0Qn).
5 π α 0 ν / EH \ Γ (n* m».'/2\]

where a factor J arises from squaring the argument of the logarithm.
This is indeed consistent writh extrapolated results, e.g., of Omidvar and
Khateeb [339b], but is a factor of about 1.6 larger than the corresponding
values of Saraph [337], in the limit of large n. Further support for the ac-
curacy of Eq. (479) is provided by two semiclassical calculations [340,
341], which agree in our parameter range with Eq. (479) to within factors
of 1.3 and 1.1, or even factors of about 1.1 and 0.9 if allowance [338b,
339b] is made for the fact that Eq. (479) also includes n —» n + 2, etc.,
transitions. For large ratios of incident to transition energy, another semi-
classical calculation [342a] also agrees very well with our estimate, and the
theoretical question [343] arises why the semiclassical dipole approxima-
tion comes so close, e.g., to the much more circumspect theory [341] of
Percival and Richards. Be this as it may be, the latter authors' cross sec-
tions and therefore also corresponding calculated line widths [344] do
confirm the first estimates [111] and would be preferable, if at all, only
when the argument of the logarithm in Eq. (476) approaches 1 from above.
(Were An > 2 transitions and higher multipole contributions more im-
portant, the theoretical situation would be different [343].)

Any disagreement between observations and predicted collision broad-
ening is thus not likely to be due to errors in the total inelastic cross sections
used here or by Griem [111]. However, the mean value of we is appropriate
only on the far wings of the Stark profile, which then indeed is given by

L(«) « ^/ττ(Δω)2. (481)

To describe the entire Stark profile due to electron impact broadening, one
should average the individual dispersion profiles according to

L( ì ~ 4 Γ MDlew"dl ^'

W ~ 7 r ( e 2 + l ) . / 0 [ w . ( I ) ? + (Δα,)2'

where le2l,n [which integrates to (e2 + l ) / 4 ] is the weight function ob-
tained by multiplying statistical weights and approximate oscillator
strengths, and where i ~ I + | is treated as a continuous variable. If we
now introduce a new frequency variable,

Y = Aœ/we, (483)

I V . 5 . RADIO-FREQUENCY LINES 263

the Stark profile of any high principal quantum number radio frequency
line of hydrogen, etc., from inelastic electron collisions is, from Eqs. (468)
and (472),

7(F) = L(«) do) 26.4 J0 (0.352 + 0.264a:2) xe* dx, (484)
dY (0.352 + 0.264X2)2 + F

with x = 21/n. This function is shown in Fig. (29) together with the
corresponding dispersion profile. The (half) half-width of I(y) is Y = 0.84.
In other words, the entire Stark profiles are narrower than w by only 16%
according to this simple calculation. (More involved calculations [344b]
suggest that the actual effect is about half of this estimate, which ignores
the I dependence of the logarithmic factors and any residual broadening
from elastic collisions.)

Also these numerical improvements in the average therefore cannot
explain why collision broadening of these lines has proved so difficult to
observe. One might even consider deviations from statistical populations,
i.e., overpopulations of low Z-states, in which case the lines might be
narrower by a factor of about 3. However, such deviations seem out of the
question at densities where collision broadening should be important. More
likely explanations can probably be found in instrumental or source
geometry and structure effects. Such suggestions are supported by some
evidence for collision broadening from systematic observations [334b,
335] of η-οί, -ß, etc., lines at almost the same frequency and from the

1 11

03
A

I(y)

0.2

0.1

0.0 0.5 2.0 2.5

FIG. 29. Comparison between the profile of a large principal quantum number n-a
line calculated using Z-dependent cross sections (solid line) with a Lorentz profile corre-
sponding to an averaged cross section (broken line).

264 I V . APPLICATIONS

failure [345] to detect the Η109ο; line in the NGC 7027 planetary nebula,
perhaps because the line was Stark-broadened and its central intensity thus
substantially depressed [345, 346]. This explanation, while not unique,
could be consistent with the observation [347] in NGC 7027 of the H85a
line, whose Stark width should be smaller by a factor of 2.5, but appears
less likely in the light of more recent observations [348]. It has further been
demonstrated theoretically [349] that plausible density distributions in
the source and maser action driven by the continuum in the outer, more
tenuous, layers can give profiles and total intensities in agreement with
observations, in particular if one notes that broad but weak damping wings
may have been subtracted out with the base line correction [350]. Another
suggestion [170], that the Gaussian component may be enhanced by
Lorentz effects (see also Section II.6), can be ruled out for all practical
purposes. The idea here was that the moving atoms see an electric field
2?L = vxB/c in a magnetic field B. This causes, according to Eq. (14), a
linear Stark effect of

AÜJS ~ (3h/2emZ) (n? - n?) vJB/c « (Sh/emZ) n An v±B/c.

The ratio of Acos and the Doppler width ACOD is therefore AWS/ACOD ~
(2a02/e) Bn4/Zs « 10~7 BnA/Z3, where B is in gauss. For all rira, -ßy etc.,
lines yet observed, i.e., for n < 250, the magnetic field would thus have to
approach a milligauss for this effect to be important.

We finally note that the widths estimated in this section can also be used
for radio-frequency lines of two- and more-electron atoms or ions, provided
they turn out to be larger than the spread of levels with different orbital
angular momentum quantum numbers. An extension to lower n values,
i.e., to microwave or even infrared frequencies, is also possible, provided
elastic electron collisions are then included analogously to the treatment of
elastic ion collisions by Griem [111], and broadening by ions is allowed for
either in this manner or by a quasi-static calculation, whichever is appro-
priate. In the latter case, which seems to correspond to the situation
encountered in a laboratory measurement of the 12-a: line [351], the com-
plication discussed following Eq. ( I l l ) may arise for the elastic electron
contribution. (This complication is not allowed for in any of the calcula-
tions discussed in the present section.)

IV.6. OPACITY CALCULATIONS

An implicit assumption in practically all of the preceding discussions was
that the Stark-broadened profiles of individual lines were more or less

I V . 6 . OPACITY CALCULATIONS 265

directly observable, while for calculations of radiative transport through
line radiation only certain averages [171-173] over profiles of many lines
are required. Especially important for these averages are the wings of the
lines, so that asymptotic formulas as discussed in Sections IV.4a-c are
usually sufficient for hydrogen, ionized helium, and neutral or singly
ionized atom lines in general. But line contributions to opacities, e.g., in
stellar interiors, are often from highly ionized systems, for which Stark
broadening is negligible under almost any laboratory condition. We there-
fore depend heavily on theory for the prediction of the corresponding line
shapes, and are in this respect faced with a situation similar to that for the
radio-frequency lines discussed in the preceding section. In that case,
densities are too low to allow laboratory verifications of the Stark broad-
ening calculations, while in stellar interiors they are somewhat high for
direct simulation.

Before going into discussions of estimates for multiply ionized systems,
one limitation on the application of Eq. (417) to the very important La
line of hydrogen should be mentioned, which was not pointed out in Section
IV.4a because it usually imposes no serious restriction in applications to
stellar atmospheres. However, in some radiative transfer problems, it
might be tempting to extend the Δλ~5/2 wings far toward the visible part
of the spectrum. For such large Δλ, both electrons and ions would act
quasi-statically and individually, and there is thus a unique radiator-
perturber separation for each Δλ value, namely, according to Eq. (66b),

I 3h |1/2 /Sh\1/2 = 2α0 2λ 1/2 (485)
T ~ I m Αω I ~ ( \πΐωο/) Δλ Δλ

But the magnitude of the correction terms, e.g., in Eqs. (65b) and (66b),

also suggests that the linear Stark effect (first order dipole) approximation

is reasonably valid only for rp > 10a0, so that we have the additional
requirement

I Δλ/λ | < 0.1 (486)

for the applicability of Eq. (417) to the wings of the La line. Corresponding
restrictions for higher members of the series are not of much practical

importance because of the merging of neighboring lines (see Section II.2),

and other than resonance lines tend to give relatively small contributions to

the radiative energy transport in stellar interiors.

In view of this unique position of La , we will now show how Eq. (417)
might be corrected for Δλ/λ approaching 0.1. For absorption, the line shape

is proportional to the actual frequency and the square of the dipole matrix

266 I V . APPLICATIONS

element, i.e., we have, from Eqs. (65b) and (66b),

I<->-ÌS['+»7-KT),+---H"III

-2fKi+i)[i+27-i6©']ihs7^(?)1
,^(i)%,M-K^)H?-<)']

4-Μ(?)ΐΗ?+<*)Ί

where all first-order (quadrupole) terms in a0/r were dropped in the last
step, because they in fact have opposite signs for electrons and ions as
quasi-static perturbers. (See Section III.9 for a discussion of these terms
in situations where only ions act quasi-statically.) With

- Δ ω = (2irc/Xo2) Δλ (1 - Δλ/λ0 + · · · ) ,

Eq. (485), and multiplying by 2 to allow for electrons and ions, the nor-
malized profile of the absorption coefficient in the wavelength scale becomes

Z.(X) ~2L(») | - | - 2 [ L ( — ) - - ^ - ) - J - ( l - 2 - )

^4«/ ΔλΛ /2irc AX\

~ λο2 \ 2XoJ V λο2 /

(The second term in the square brackets on the first line comes from the
correction term in the Δω to Δλ transformation.) Except for the last factor,
this is twice the usual Holtsmark result. The fractional correction to this,

I V . 6 . OPACITY CALCULATIONS 267

mainly from quadratic Stark effect, is thus about 1.7 Δλ/λ0. It is therefore
probably still reasonably accurate when Δλ/λ approaches 0.1, except for
satellites [352]. For Δλ/λ > 0.1, a realistic theory would treat the hy-
drogen-electron system in terms of the H~ ion [353] with allowance also for
the 2 *P resonance, and the hydrogen-proton (or other ion) system as a
molecular ion [354, 355]. In other words, we would have to leave the realm
of line broadening theory.

We now return to the principal subject of this section, namely the
estimate of wing shapes for resonance lines from multiply ionized atoms. It
is natural to first stay with La lines also for this purpose. As discussed in
Section II.2, for them ion broadening is usually well described by the
quasi-static approximation. Even for protons as perturbers and assuming
kT « Z2 eV, Eq. (17) yields only N > 2 · 1015 Zm cm~3 as a criterion for
the validity of this approximation over the entire Stark profile. This is often
fulfilled in situations where line opacity is important, so that we have, in
straightforward generalization of Eq. (487), but now neglecting asym-
metries, for the ion contribution to the asymptotic line shape of any La line,

Li (co) « (xV3/| Δω |5/2) (h/mZy1* £ Ζψ Νρ . (489)

P

Here Zp is the net charge of a given perturbing ion species, and Z the
nuclear charge of the radiating one-electron ion. To obtain corresponding

formulas for Lß, L7 , etc., one would have to multiply by factors 8.7, 23,
etc., and in order to allow for dynamical corrections according to Sections

II.4c and IV.4a, multiply Eq. (489) by a factor

(Li + ΔLi)/Li « 1 - 3 · IO"5 ZkT/h \ Αω | (490)

for protons as perturbers. (This estimate neglects Coulomb interactions,
but uses the actual Stark coefficient for La . For the β and y lines, the
correction term is smaller by factors of about 2 and 5.) While this correction
must not be too large to be meaningful, no such restriction is imposed on
the Boltzmann factors which may have to be inserted in the sum over
perturbing ions in order to allow for radiator-perturber correlations. In
analogy to Eq. (426), these factors are

eXPLΓ~ Zp(Zrk- Tl)e»Γ1 [-2Γ\Γn*(ΒZΓZv)hA<Am —(Z -WP~^ lH·] , ,mQ n)

with the " 6 " to be replaced by about 12 or 24 for L/j or Lr . They tend to
introduce effective cutoffs in the "ion" profiles before the various asym-

268 I V . APPLICATIONS

metry corrections discussed above for La of hydrogen would become too
important.

In addition to Coulomb repulsion between radiating and perturbing ions,

fine structure of the unperturbed energy levels also might lead to a reduc-

tion of wing intensities below those given by Eq. (489) or its equivalents.
The relative value of the heretofore neglected splitting between the 2 2Pi/2
and 2 2P3/2 levels is

(«i/t - ωι/2)/ωο « Z2a2/S « 1.78 · 10"5 Z2, (492)

and 2 2P3/2 will exhibit linear Stark effect only for Δω/ωο values exceeding
this ratio substantially [34, 204]. For some of the very heavy ions this may
not yet be the case, and one would then have to treat separately the
12Si/2-2 2Pi/2 and 12Si/2-2 2P3/2 transitions, whose absorption oscillator
strengths are in the ratio 1:2. Only for the former transition would Eq.
(489) apply, except that it should be divided by a factor 33/4 to account for
a reduction in the linear Stark coefficient [34]. For the latter transition,
quasi-static broadening would now be through quadratic Stark effect and
thus usually be negligible. Even in the transition regime, where 2 2P3/2
gives rise to a forbidden component, 12Si/2-2 2Si/2, this would seem to be
true in the present context, because this component is very close to the
12Si/2-2 2Pi/2 line. (Lamb shifts are probably always negligible in applica-
tions to opacity problems, but ion-quadrupole effects as discussed in
Section II.3f may be important at high Z.)

Before turning to the discussion of broadening by electrons, wre should
note that the products of asymptotic ion-produced line shapes and absorp-
tion oscillator strengths are approximately the same for all lines of the
Lyman series of a given one-electron ion. However, the frequency separa-
tions from the line center beyond which Eq. (489) or corresponding rela-
tions for higher series members are valid are about twice the width as
estimated in Eq. (16). Within these separations, which scale with the
square of the principal quantum number, absorption coefficients therefore
decrease very rapidly along the series. To estimate the peak height of the
ion-produced Stark profiles, one might use, e.g., Eq. (489) outside some
frequency separation Δωι, and assume a constant line shape equal to the
asymptotic value at Δωί for smaller separations. Fixing Δωι such that area
normalization is maintained, i.e., i for each shifted Stark component of
L« , we find in this manner for this line

Acoi = (lOwVS)^ (h/mZ) ( Σ Zy2 iVp)2/3 (493)

P

I V . 6 . OPACITY CALCULATIONS 269

and for | Δω | < Δω»

L(co) « 1/(10 Awi). (494)

As to be expected, Eq. (493) is essentially the same as Eq. (16), except that
it is smaller by a factor 2.5 and allows for several perturbing ion species.
For higher series members (n = 3, etc.), the analogous relations are

Δωι « 4(n2 - 1) (h/mZ) ( Σ ^ #P)2/3, (495)
(496)
p

L(«) « 3/(10 Δωι),

and the asymptotic line shape corresponding to these estimates is

L ( c 0 ) , S Ï O T ^ r = 5 | Δ ω ρ \^Ζ) Ç V ^ P . (497)

which agrees well with Eq. (418) for the coefficient in the asymptotic
expression for the reduced Holtsmark profiles (after transforming to
the a scale).

To compute entire Stark profiles, it will normally be sufficient to con-
volve such schematic "ion" profiles with electron impact profiles, adding
in the case of La an unshifted "electron" (Lorentz) profile having two
thirds of the total profile area. For L^ there is no such unshifted Stark
component, and for L7 its share of the total profile area is 26.6%, so that
the corresponding correction is here not significant. The next line again has
no unshifted component, and that of L€ contributes only 17.2%, etc. In
the asymptotic regime, the "electron" profiles are dispersion profiles,
albeit often with frequency-dependent damping constants y = 2w, which
can at least be estimated according to Section II.3b. The dipole inter-
action, weak collision term from Eq. (126) with appropriate matrix ele-
ments for nP levels gives

w « (STN/Ό) {h/mZy n2(n2 - 3) ln(emax/€min). (498)

The parameter e of the classical hyperbolic orbits for the perturbing elec-
trons of course always obeys e > 1, but is further restricted by the unitarity
requirement to

e > (hv/e*)[n/{Z - l ) Z ] ( f n 2 - f)1/2. (499)

This condition follows from Eqs. (117) and (123), if we postulate that the
required matrix element of Eq. (123) does not exceed 1. Another restriction
must be imposed such that the distance of closest approach remains larger
than about n2a0/Z, because otherwise the dipole approximation would

270 I V . APPLICATIONS

break down. With Eqs. (116)-(118), this leads to

6 > 1 + [mv2/2(Z - 1 ) Z # H > 2 « *min , (500)

because it typically turns out to be larger than Eq. (499) and is therefore

the determining condition for emin .
To estimate emax , it is safe to assume e ^> 1, i.e., according to Eq. (117),

€ « mpv2/(Z - l)e2, (501)

and to assess the upper limit for p more or less as discussed in Section II.3a
following Eq. (112). Since we are especially interested in frequencies beyond
those corresponding to the mean quasi-static Stark splittings, we first need
to consider Debye [53] or Lewis [65] cutoffs (see also Sections II.5a and
II.4b). Usually, the former would apply for frequency separations some-
what beyond Δαη according to Eq. (493) or (495), while the latter cutoff
should be used on the far wings. However, especially for large Z, fine struc-
ture may impose an "adiabatic" cutoff near p = v/AœF , and in the line
cores we have another adiabatic cutoff near p = ν/Αωί. All four cutoffs are
summarized by

Pmax « V/0)C (502)

with

coc = max{| Δω |, cop = (4irNe2/m)l/2)

coF « 2Z2a2co0/3n, Δωί « 4n2(h/mZ) ( Σ Ζψ iVp)2/3}, (503)

so that the argument of the logarithm in Eq. (498) for the electron colli-
sion width becomes

«max mv* Γ mv2n2
~^~ ~ ( Z - l ) e V L 2(Z-1)ZEH

(kT)*/2 Γ kTn2 I-1

~ (Z - l)EU2ho>c L (Z - 1)ZEK\ ' (}

[For j = §, the fine-structure splitting œF ought to be replaced by the
Lamb shift, wL « |α3Ζ4(#Η//ίη3) 1η(η3/α2Ζ2), but these two critical fre-
quencies are mainly of academic interest because they tend to be dominated
by | Δω | or Δω».] Together with (l/v)&v = (2m/irkT)ll2i this estimate
allows us to calculate the usually major contribution to the electron impact

broadening of Lyman lines from Eq. (498). A little consideration then

shows that the ratio of asymptotic "electron" profiles

■?ε(ω) ~ ΐϋ/π(Δω)2 (505)

I V . 6 . OPACITY CALCULATIONS 271

and asymptotic "ion" profiles from Eq. (489) or (497), at | Δω | = Δω\,
is estimated by

7e/7i « 3 (n2/Z) (EH/kT) ™ αοΛ™ m (emax/€min), (506)

which in many cases is not much below 1. To estimate the asymptotic
ratio at larger separations from the line center, one could simply multiply
with | Δω/Δω, |1/2 until this modified ratio approaches 1, and beyond use the
quasi-static approximation for electrons as well. If necessary, the appro-
priate correction for Coulomb attraction can be made from Eq. (491),
except for a change in sign.

If the argument of the logarithm in Eq. (498) is large and Z is small or
moderate, the estimates for broadening of hydrogenic ion lines presented
above will be fairly accurate. Otherwise, a number of corrections, mainly
for higher multipole interactions and for inelastic collisions, should be made.
Beginning with the former, their effect can probably not exceed that of a
"strong-collision" term w' ~ ττρ^νΝ, with ps being an impact parameter
close to that for which the perihelion is about equal to the excited state
Bohr radius. Such value of ps corresponds to €min as estimated by Eq. (500),
so that we obtain, using Eq. (117),

<t i$ TTvNao2(n2/Z)2t(4:(Z - l)ZEH/n2mv2) + 1]. (507)

However, this formula must not be used when the incident kinetic energy
\mv2 is below the binding energy of the bound electron, because in this case
the perturbing electron might suffer a substantial perturbation in a close
collision. It may be better to omit the "focusing" term in Eq. (507) and to
compare such reduced estimate with the effects of strong collisions through
dipole interactions only,

<l % {TrN/v)(h/m)2(nA/Z2) = wvNa02(n2/Z)2(2EH/mv2). (508)

[According to Eq. (123), one should actually multiply with a factor
1 — €~2, but this correction is not important in the present context.]
Simply adding the two widths, contributions from higher multipole inter-
actions and "strong" collisions are thus estimated by

where the higher multipole term was corrected to allow for the quantum-
mechanical limit, i.e., p > h/mv rather than pmin « n2h/Zmv, for the mini-
mum impact parameter. (See also Section II.3b, and remember that the

bracketed factor should not exceed, say, 2 for these estimates to be valid.)

272 I V . APPLICATIONS

Inelastic collisions causing dipole transitions may be assumed to con-
tribute according to the modified [102, 103] Bethe-Born approximation
(see Sections II.3d and IV.4c). Lumping all An ^ 0 dipole transitions and
estimating the sum of their matrix elements as in the preceding section, the
corresponding contribution follows from Eqs. (350) and (467) to

Win « {*N/v){h/mZ)2n*[l + (17/7I2)](TT/33/2)0, (510)

where g is near 0.2 for energies up to about three times threshold energies
[103]. By using the same value also below threshold one presumably
accounts [102] for contributions of An τ^ 0 intermediate states to elastic
scattering, and for high electron energies the discussion in Section II.4d
suggests replacing ng/VS by ln(emax/€min) with o>c chosen to correspond to
the n and n + 1 level separations, i.e., coc « 2Z2En/n3. (The n to n + 1
transitions dominate in the An τ^ 0 contribution.) A convenient inter-
polation formula is therefore

τΝ/ή\2η*/ 17V f (nhv/e2)* Γ mv2n2 I"1!

(511)

(Although the Gaunt factor used here may be somewhat small [330, 331]
for, e.g., 2P-3D, etc. collision-induced transitions, it seems a better choice
nevertheless because some inelastic contributions are already contained
in the strong-collision term.) Addition of Eqs. (498), (509), and (511)
yields for the collision width of Lyman lines from electron impacts

x In \(z - i)z^Wc [ + (z- i)zsj /

+ U + T - r \ 1 - 4 + U J JZ^Z^+ÌZ-DZEJ ))

(512)

I V . 6 . OPACITY CALCULATIONS 273

if we interpolate the quadrupole term according to the remark following
Eq. (509) and use an approximate Maxwell average.

The corresponding total cross sections for the riP states of the one-
electron ions are

(3w4 - 9n2) In j - Γΐ + En*

ΕΆ(Ζ - 1)Z.

E ({ E Z*V 1
ΕΗ \ ΕΗ η 4 /

(Ϊ^)'"Μ^)'?[' En2
ΕΆ{Ζ - 1)Ζ\

(513)

with E — \mvi and η = (Z — l)e2/hv, while I is the maximum orbital
angular momentum quantum number used in the divergent sum over
partial waves for the perturbing electron. This estimated cross section
should, if possible, be compared with measurements or with quantum-
mechanical calculations in order to check its accuracy. Such a comparison
with one of the calculations [64a] for La of He+ suggests an accuracy of
about 20% for E/Z2EH < 0.5, provided our extrapolation along the iso-
electronic sequence and to larger principal quantum numbers is correct.
The other calculation [67] results in substantially smaller widths, typically
by about 40%, if we assume that Fig. 2 of that paper actually gives
Ω/(2Ζ + 1). This defect is almost certainly due to the omission even of
n = 3 states in the latter calculation.

It is a relatively simple matter to generalize Eq. (513) for nP levels of,
say, helium-, lithium-, or beryllium-like ions. For them, the nS and nD
contributions must obviously be separated, and œc will now usually corre-
spond to the various inherent level separations œw . However, use of
hydrogen matrix elements, perhaps with effective quantum numbers etc.,
and carry-over of the above estimates for strong collision and An 9e 0
contributions should not cause any significant errors. Probably the Gaunt
factors in the An = 0 terms are larger and depend on Z, say, as g =
0.9 - 1.1/Z at threshold [330, 331]. [This is rather different from the
Z dependence predicted by Eq. (190) for elastic contributions, which,
however, may be invalidated anyway because the relevant collisions are
too close.] Interpolating again in the arguments of the logarithms, the
proposed total cross section against electron impacts for the nP states of

274 I V . APPLICATIONS
these ions is then

E / E ZV1

(514)

where the interpolation with the effective Gaunt factor method in the
Δ η ^ Ο contribution was done as for hydrogenic ions (g = 0.2), and where
ξιι>, in generalization of Eq. (186), is given by

«ir = (Z - l)e21 ωιν \/m* = η(ή \ ωιν \/mv*). (515)

As always, Z is the charge acting on the bound electron, while a distant
perturbing electron "sees" a charge Z — 1.

From Eq. (514), electron impact (half) half-widths can be obtained by
multiplying with $Nv and by averaging over the electron velocity dis-
tribution, and a further generalization of our formula for total cross sections
to lines whose upper levels have I = 0, 1, 2. . . . , etc., is obtained by the
following replacements in Eq. (514) :

(2n4 - 8n2) ->3(Z + l)n2[n2 - (I + 1)2]/(2Z + 1),

(n4 - n2) ->3Zn2(n2 - Z2)/(2Z + 1),

(rc4 + 17n2)/3 -> n2(n2 + 3Z2 + 31 + l l ) / 3 ,

not to mention the obvious changes in the coc values. With these total
cross sections, one can estimate the electron broadening [see Eq. (526)
below] of almost any "isolated" ion line of importance, if necessary by
adding lower-state widths, using effective quantum numbers and Bates-
Damgaard factors [42, 43], and by chosing o>c « | Δω | on the far wings.
(One should also check the validity of the impact approximation.)

It is interesting to note that our simple analytical estimates reproduce
more than half of the semiclassical results of Sahal-Brechot and Segre [104]

I V . 6 . OPACITY CALCULATIONS 275

for C III and IV, N III, Si III and IV, and S III, IV and VI to within a
factor of 2. Exceptions are most Si III multiplets, the last multiplets of
N III and Si IV, and the S III multiplet, whose widths are smaller by
factors of about 2 or more in the detailed semiclassical calculation, while the
low-temperature value for the width of the O VI resonance line is larger by
a similar factor. An interpretation of these deviations as errors mostly in
Sahal-Brechot and Segre [104] is suggested by comparisons between lines
of the same and different spectra, or between widths of a given line at
different temperatures. However, widths from Eq. (526) may be relatively
too large at low temperatures, because the possible cancellation of elastic
contributions to the widths was ignored. (This cancellation is not as strong
as for the high n-a lines of hydrogen and, at most, applies only to the
nonlogarithmic terms.) The strong temperature dependence of some of the
widths of lines from higher ionization stages in [104], on the other hand, is
almost certainly caused by the neglect of a condition analogous to Eq.
(500). On balance, Eq. (526) therefore appears to be more accurate, and
we will assign a ± 5 0 % probable error to our estimates for the lines in
question, judging also from the experience with the analogous formula for
one-electron ions, Eq. (512), and from measurements of collisional rate
coefficients [106]. This error estimate is further supported by a very
satisfactory agreement with a measurement [288] of C III and C IV lines
(see Table I X ) , but for singly charged ions the methods discussed in
Sections II.3d and IV.4c are preferable. (See Appendix V for numerical
results.)

Occasionally, impact broadening by protons or other ions may have to be
considered. Especially for multiply charged radiators, second-order per-
turbation theory allowing for dipole interactions only might provide a
good estimate for their contribution to the width, because the Coulomb
repulsion tends to suppress higher-order terms in both multipole and
Dyson series. The repulsive analog of Eq. (350) may thus be appropriate,
which, for £ > 1 and η » 1 (the usual situation) for singly charged ions
("protons") becomes

^ ~ wrk (à)n2{(Z+1} ^ -(*+1)2] -p(-2^-)

+ Z[n2 - P] e x p ( - 2 i r f M - i ) } , (516)

if we consider only An = 0 contributions and use hydrogen matrix ele-
ments. The £'s here correspond to those in Eq. (515), except that we now
have to use proton mass and velocity, and the exponential factors con-

276 I V . APPLICATIONS

stitute a suitable approximation [6] to the Coulomb excitation functions
[74] fm(vi, vu), multiplied with 36/2/32ir3. The Maxwell average of Eq.
(516), if approximated by the saddle point method, then leads to

wp » (2»Wp/2l + 1) (i)ie(Än/mZ)*(mp/*T) {(I + 1)Γ>2 - (I + 1)*>,#ι+,

X βχρ(-|«ΐρβ»ι.Μ.ι/*Τ) + l\jP - P>i.i_1exp(-frop»îi,_1/*T)}

(517)

with

Όι,ν = [6π(Ζ - l)e21 mv \ hT/mf]1* (518)

and the provision that the exponential factors should be smaller than
βχρ(-2ττ) « 2 · 10"3.

For intermediate ionization stages or higher series members, second-order
perturbation theory as used for Eq. (517) would provide an overestimate,
because the unitarity condition was ignored. In such situations, the analog
of Eq. (190) might seem more appropriate. Use of this relation presumes
that the broadening is due to elastic rather than inelastic collisions, that
we have £ 2> 1, and that the classical paths are strongly curved hyperbolas
for the relevant impact parameters. All but the last of these assumptions
may be fulfilled for the lines of interest, whose proton impact broadening
is not covered by the preceding estimate. Therefore, the straight classical
path version [80, 81] of the phase shift (adiabatic) approximation, namely,
in analogy to Eq. (263),

wp « 2.8πΝρ I — ) I /exl , ^N—— J — [n2 — (I + 1)2J

IE 12/3
+ —^(n2-*2) , (519)
ηωι,ι-ι J

is usually preferable whenever it gives a smaller width than Eq. (517)

due to of proton collisions. Both of these estimates could easily be gen-

eralized to allow for multiply charged ions and for perturbations of the

lower states, paying due attention to signs. Also, one of the proton collision

widths estimated here, i.e., the more appropriate of the two, is normally

smaller than the electron-produced width, provided the impact approxi-

mation was indeed valid for protons. This point can be checked by com-
paring the duration of an effective collision, pp/vp , with | Δω |_1, where pp
can be estimated from wp « πΝνρρρ2. It often turns out that the impact
approximation for protons breaks down within the Doppler width, so that

I V . 6 . OPACITY CALCULATIONS 277

any significant ion broadening would have to arise from quasi-static per-
turbations which are, as mentioned before, almost always negligible com-
pared with electron impact broadening in the case of quadratic Stark effect.
However, quadrupole interactions must not be overlooked. According to
Eqs. (218a) and (218b), they can cause substantial corrections, although
these equations may result in overestimates for high-Z lines. (See the end
of Section II.3f.) Nevertheless, ion-produced quadrupole interactions are
frequently more important than dipole interactions for the broadening of
high Z isolated ion lines, although they will seldom contribute more than
electron collisions.

Besides contributions to the line opacity from hydrogenic ion and
"isolated" ion lines, one should also consider contributions from additional
processes as described by the one-electron approximation of Baranger [ 6 ]
or its generalization by Burgess [128]. As discussed in Section II.4d, these
processes primarily introduce asymmetries to be estimated from Eqs. (355)
and (348), respectively, which as such do not seem very important in our
present applications. Moreover, the estimate in Eq. (348) rests on the
validity of second order perturbation theory and the dipole approximation
for the electron broadening, i.e., implies that the logarithmic terms in Eq.
(514) or its generalizations dominate. This is evidently not the normal
situation, and the estimated asymmetries should therefore be multiplied,
say, by the square root of total cross sections calculated without or with the
additional, nonlogarithmic, terms. However, in cases of very weak back-
ground continua (or scattering), even such asymmetries should not be
ignored, although they will only rarely lead to a truly vanishing total
absorption coefficient.

Another modification of the isolated line approximation is probably of
greater consequence, namely that for forbidden components. As in the case
of neutral atom lines, such features are mostly produced by ion collisions.
We first estimate their contribution to the normalized line shape of the
parent line in analogy to Eq. (442), replacing the a function by the repul-
sive analog (multiplied with TT/V3) of the Gaunt factor, whose argument,
according to Eq. (352) and the discussion following Eq. (355), should now
obey, say,

ξρ « [ ( Z - 1)βν/Μ;ρ][ΜΔω'2 + ωρ2) 1/2/mpi>p2] (520)

in the case of protons, where Δω' is the frequency separation measured

from the forbidden line, and ωρ' is the ion plasma frequency, ωρ' =
(4*Ne2/mp)1/2. Near the unperturbed position of a forbidden (ΔΖ = 0, ± 2 )
line, £p is typically much below unity, and also the repulsive Gaunt factor

278 I V . APPLICATIONS

is then well represented [74] by Eq. (353). The desired contribution to the
line shape is therefore

in situations where the Coulomb cutoff [ 6 ] dominates. Here Z> is equal to
the orbital angular momentum of the upper state for either the allowed line
(I) or the forbidden line (Γ), whichever is larger.

Such forbidden component will reach out to frequency separations from

its unperturbed position giving £p « 1, i.e., its "dynamic" width is esti-
mated by

w' « [yp/(Z - l)e2>pvp2, (522)

which is typically much larger than Doppler or electron impact broadening,
so that convolution of Eq. (521) with other line shapes is normally not
required. Before applying Eq. (521), one should, however, ascertain that
w' stays well below the unperturbed level separations ω π , because other-
wise, the forbidden component should not have been treated separately
from the wing of the allowed line. (This condition tends to be violated for
perturbing electrons.) Also, our use of Gaunt factors implies that per-
turbation theory and dipole approximation remain valid even for strongly
curved hyperbolas, which may not actually be the case. Corresponding
corrections would reduce the argument of the logarithm in Eq. (521) and
the estimate for w' by a factor emin , the minimum value of e according to
Eq. (117) that is consistent with these approximations. Comparison with
Eqs. (442) and (444) suggests that €min , now for perturbers of charge
Zp , may be estimated from

/ kT/E* Γ 3U* (»22 - I1" f EK f*\ (523)

«... « max jl, {Z _ w , L(â+ljâi V)\ If—] ) .

if second-order perturbation theory breaks down first. In case the product

of ί and €min is no longer small, further improvements could be achieved by
replacing 1η(1.1/ξ €min) with the function α(ξ, €min) defined in Eq. (195)
and tabulated in Table IIIc, but multiplied by exp(—2π£).

We also observe that the product of Eqs. (521) and (522) is essentially
proportional to mpvp2 « SkT and the first power of the perturber charge, if
appropriate factors Zp are inserted, namely Zp in Eq. (520) and Zp2 in Eq.
(521). It is therefore clear that forbidden components will be particularly

important in stellar interiors. Accordingly, we generalize our estimate as

I V . 6 . OPACITY CALCULATIONS 279

follows:

«"»-?'.£)"©?(=;;)'

[^Ί<» -! 1>η exp(-2ir|p)a(?P , «mi„) +π(Δwω)5 (524)

averaging approximately over the velocity distributions of relative veloci-
ties, i.e., using the reduced perturber-radiator mass m'. In Eq. (524) the
electron-produced wing of the allowed line was added and £p replaced by

ÎP « (Z - l)Z„(En/kTy^h(Aw" + cop2)i/V2fcT]. (525)

The ion plasma frequency, for the usual mixtures, can still be calculated
as before to sufficient accuracy, and the electron impact width of the
allowed line may be estimated, as discussed in the paragraph following Eq.
v„lô),irom

w 3*N /2m\1/i /_Ä_V l+l 1O2 - (i + i)2]
\*kT/ \mz) '' 21 +
nXln<!5--V^4z
+ {7lU [l + kTn1
En(Z - 1)Z

+I (n2-i2)ln^5 ^Vz + 1, kTn? /
21+1
EH(Z -i)z]

+ 3 3 Eu \ EH + n4/ + - (n2 + 3P + 31 + 11)

ΧΜ-(ΙΓΑ[ 1 + kTn2 1-n (526)
En((ZZ-1)Z\ J

using effective quantum numbers and Bates-Damgaard factors [42, 43],
and adding lower state contributions, if necessary. (Upper-lower state
interference terms are probably small for most of the lines of interest here.)

Comparison of Eqs. (524) and (526) shows that the relative amplitude
of the forbidden component with respect to the background from the
allowed line wing is essentially given by the square root of the ion-to-
electron mass ratio multiplied with the ratio of the square-bracketed

280 I V . APPLICATIONS

factors in the two equations. The latter ratio tends to be small, and the
relative amplitude is typically near unity, decreasing with temperature for
high-Z ions. However, the width of the forbidden component increases
with temperature. As a matter of fact, the ratio of wf from Eq. (522) and
unperturbed splittings ωι>ι increases along isoelectronic sequences. For-
bidden components thus actually become more important along such
sequences in this sense, until, well beyond the unperturbed position of
the forbidden component, and for corresponding frequencies on the other
side of the allowed line, a hydrogenic ion treatment using appropriate
linear Stark effect coefficients would be more realistic.

To summarize this chapter, we may say that except for the special
mechanisms [6,128] mentioned in the paragraph following Eq. (519), three
types of ion line Stark profiles are important in opacity calculations. The
first type, besides hydrogen L a , consists mainly of hydrogenic ion lines,
for which a first approximation for the broadening by ions can be obtained
as discussed above in connection with Eq. (489) and Eqs. (493)-(497).
To these schematic profiles one then either adds on the wings Le(œ) =
W/T(AÙ))2 with w from Eq. (512), but not allowing this contribution to
exceed the ion contribution and renormalizing by lowering the flat top of
the ion profiles, or convolves the two Stark profiles. Unshifted components
may be accounted for as discussed in the paragraph following Eq. (497).
At every step, one should examine the various modifications to the quasi-
static "ion" profiles discussed above and correct for them, if necessary.

"Isolated" lines form the second class, consisting of lines which are much
narrower than separations between relevant unperturbed levels. Their
Stark profiles are Lorentzians with (half) half-widths according to Eq.
(526), perhaps with small corrections from ion impact broadening according
to Eqs. (218b), (517), or (519), generalized for perturber charges Zp 5^1
if necessary, etc. Otherwise the nearest-neighbor quasi-static approximation
(see Section II.2) is usually more appropriate for the broadening of such
lines by ions. The third class, namely lines with forbidden components, is
intermediate to the two extreme types also with respect to the relative
importance of ion and electron broadening. Since profile calculations for
such lines have just been discussed, suffice it to say that these approximate
profiles should certainly be used only as long as forbidden components are
much weaker than the allowed line. Otherwise, use of hydrogenic ion line
shapes throughout would be better.

We conclude by reminding the reader once more of the detailed in-
formation available for neutral atom lines, which, for hydrogen lines, is
discussed in Sections II.2a, II.3a, III.2, IV.4a, and Appendices I.a, II
and Hl.a; for isolated lines, in Sections II.3ca, II.3f, III.6, IV.4c, and

I V . 6 . OPACITY CALCULATIONS 281

Appendix IV; and for lines with forbidden components, in Sections II.3c0,
III.5, IV.4b, and Appendix VII. Only slightly fewer particulars are known
about lines from singly ionized atoms, and Sections II.3b, II.3d, III.3,

111.7, IV.4a, IV.4c, and Appendices I.b, II, Ill.b, and V may be consulted
in connection with such lines. Lines from multiply ionized systems, on the
other hand, were essentially discussed only in the present section and, with
respect to a very few laboratory experiments, also in Sections III.4 and

111.8. Much more experimental and theoretical work is required to reach
for them the 20% or better accuracy (except, perhaps, for forbidden com-
ponents subject to dynamical ion broadening) attained in Stark profile
calculations for neutral atom lines, or the typical agreements of about 25%
(excepting shifts) achieved for lines from singly charged ions.

APPENDIX I

Stark Profiles for Hydrogen and
Ionized Helium Lines

This appendix lists Stark profiles S (a) for hydrogen [59b] (Section
Al.a) and ionized helium [61b] (Section Al.b) lines from transitions
between levels with principal quantum numbers NA and NB at various
temperatures and densities. The densities are characterized by the decimal
logarithm of the number (N) of electrons per cubic centimeter, and the
reduced wavelength separation (a) from the line center is in angstroms per
cgsfieldstrength unit. Some entries in the original tables have been omitted
here, either because they were for special conditions or because a was in a
range beyond characteristic values corresponding to the electron plasma
frequency by significant amounts. Beyond the a range of the present
tabulation, extrapolation with the help of an asymptotic formula, Eq.
(417) of Section IV.4a, will usually be possible without significant increase
in the theoretical error. Alternatively, the results [121f] of the relaxation
theory can be used, which are also available for T = 2 500 K. (Note,
however, that these calculations have large errors for the line core of H«
and smaller errors for that of Κβ , etc. [64b].) The latter results are avail-
able both with and without the inclusion of Doppler broadening and can
thus be used to estimate the influence of this effect, which is not accounted
for in the following tables because it is usually negligible.

282

Al.a. HYDROGEN 283

Al.a. HYDROGENJ

T = 5,000 K NA = 1 NB = 2

a\ LOG(N) !9 *
1.997E 03
05·.00E-06 2.860E 03 1.967E 03 1.645E 03 1.994E 03
1·00Ε-05 2.848E 03 1.963E 03 1.643E 03 1.985E 03
4.00E-05 2.813E 03 1.953E 03 1·637Ε 03 1.819E 03
7.00E-05 2·257Ε 03 1.766E 0J 1.537E 03 1.544E 03
1.00E-04 1.583E 03 1.463E 03 1.358E 03 1.264E 03
2.00E-04 1.096E 03 1.166E 03 1.156E 03 6.633Ε 02
4.00E-04 4.353Ε 02 5.717E 02 6.503Ε 02 2.704Ε 02
6.00E-04 1.995E 02 2.519E 02 2.860Ε 02 1.180E 02
8.00E-04 1.493E 02 1.633E 02 1.648E 02 5.311E 01
l.OOE-03 1.181E 02 1.170E 02 1.064E 02 4.256Ε 01
1·50Ε-03 9·193Ε 01 8.739Ε 01 7.776Ε 01 6.367Ε 01
2.00E-03 5.093Ε 01 4.867Ε 01 5.046Ε 01 3.208Ε 01
2.50E-03 2.540Ε 01 2.563Ε 01 2.737Ε 01 1.652E 01
3·00Ε-03 1.520E 01 1.558E 01 1.613E 01 8·157Ε 00
9.928Ε 00 1.018E 01 1.002E 01

Τ « 10,000 Κ ΝΑ=1 ΝΒ = 2
α\ LOG(N)
18 19 t

0. 3.365Ε 03 2.148E 03 1·576Ε 03 1.466E 03
5.00Ε-06 3.344Ε 03 2.143E 03 1.574E 03 1.465E 03
1.00E-05 3.286Ε 03 2.128E 03 1.569E 03 1.461E 03
4.00Ε-05 2.435Ε 03 1.877E 03 1.475E 03 1.392E 03
1.562E 03 1.496E 03 1.305E 03 1.263E 03
7.00Ε-05 1.018E 03 1.147E 03 1.113E 03 1.109E 03
1.00E-04 3.727Ε 02 5.224Ε 02 6.253Ε 02 6.702Ε 02
2.00Ε-04 1.732E 02 2·284Ε 02 2.797Ε 02 3·009Ε 02
4.00Ε-04 1.398E 02 1.583E 0> 1.743E 02 1.717E 02
6.00Ε-04 1·180Ε 02 1.213E 02 1·226Ε 02 1·107Ε 02
8.00Ε-04 9.584Ε 01 9.401E ο: 9·129Ε 01 8.081E 01
1.00E-03 5.595Ε 01 5.207Ε 5.062Ε 01 5.188E 01
1.50E-03 2.712E 01 Οι 2.765Ε 01 2.890Ε 01
2.00Ε-03 1.590E 01 2·702Ε 01 1.698E 01 1.711E 01
2.50Ε-03 1.028E 01 1.638E 01 1.120E 01 1.070E 01
3·00Ε-03 1.078E 01

Τ = 20,000 Κ ΝΑ = 1 WB = 2
17
α\ LOG(N) 16 18 19 Ä 20 #

0. 4.045Ε 03 2.443Ε 03 1.634E 03 1.291E 03 1·308Ε 03
5·00Ε-06 4.009Ε 03 2.436Ε 03 1.632E 03 1.290E 03 1·307Ε 03
1.00E-05 3.907Ε 03 2.414E Oli 1.626E 03 1.287E 03 1.305E 03
4.00Ε-05 2.589Ε 03 2.045Ε 03 1.516E 03 1.238E 03 1.258E 03
7.00Ε-05 1·498Ε 03 1.538E 03 1.321E 03 1.142E 03 1.166E 03
1.00E-04 9.186E 02 1.121E 03 1.107E 03 1.023E 03 1.050E 03
2.00Ε-04 3.139E 02 4.678Ε 02 5.947Ε 02 6.522Ε 02 6.807Ε 02
4.00Ε-04 1.494E 02 2.028Ε 0 2 2.626Ε 02 3.072Ε 02 3·161Ε 02
6.00Ε-04 1·295Ε 02 1.489E 02 1.710E 02 1.875E 02 1.798E 02
8.00Ε-04 1·158Ε 02 1.208E 0?. 1·267Ε 02 1.298E 02 1.160E 02
1.00E-03 9.788Ε 01 9.731E Ox 9.738Ε 01 9.622Ε 01 8.460Ε 01
1.50E-03 6.075Ε 01 5.662Ε 01 5.381E 01 5.306Ε 01 5.351E 01
2.00Ε-03 2·861Ε 01 2·862Ε 01 2.916E 01 2.995Ε 01 3.059Ε 01
1.639E 01 1.704E 01 1.793E 01 1.860E 01 1.825E 01
2·50Ε-03 1.045E 01 1.110E 01 1.192E 01 1.240E 01 1.149E 01
3.00Ε-03

t The asterisks indicate that field strength distribution functions had to be extrapo-
lated with respect to the Debye shielding parameter.

284 APPENDIX I. STARK PROFILES

30,000 K

Λ LOG(N) 18 19 20

0. 4.529ε 03 2.664Ε 03 1.707ε 03 1.255ε 03 1.131E 03
5·00ε-06 4.479ε 03 2.655ε 03 1.705ε 03 1.254ε 03 1.131E 03
ι.οοε-05 4.336ε 03 2.626ε 03 1.698ε 03 1.252E 03 1.129ε 03
4.00ε-05 2·649ε 03 2.157ε 03 1.569ε 03 1.205ε 03 1.098ε 03
7.00ε-05 1.43βε 03 1.555ε 03 1.347ε 03 ι . ι ΐ 4 ε 03 1.036E 03
ι.οοε-04 8.54ΐε 02 1.096ε 0 ) ι.ιιιε 03 ι.οοοε 03 9.543ε 02
2.00Ε-04 2·826Ε 02 4.350ε 02 5.746ε 02 6.413ε 02 6.637ε 02
4·00ε-04 1.37ΐε 02 1.882E 02 2.502ε 02 3.043ε 02 3.296ε 02
6·00ε-04 1.238E 02 1.42βε 02 1.664ε 02 1.89ΐε 02 1.969ε 02
8.00Ε-04 1.141E 02 1.194ε 02 1.266E 02 1.340ε 02 1.328E 02
9.857ε 01 9.839ε Oi 9.923ε 01 5ι..ο5ο07βεε 02 9.726ε 01
ι.οοε-03 6.317ε 01 5.934ε 01 5.609ε 01 01 5.47ΐε 01
1.50ε-03 2.933ε 01 2.946ε 01 3.012ε 01 3.114ε 01 3.168E 01
2·00ε-03 1.658E 01 1.733ε 01 1.839ε 01 1.947ε 01 1.96βε 01
2.50ε-03 1.050ε 01 ι . ι ΐ 9 ε 01 1.217ε 01 1.309ε 01 1.304ε 01
3.00ε-03

Τ - 40,000 Κ ΝΑ = : ΝΒ = 2

οΛ LOG(N) 16 17 18 19 20 *
1.247ε 03
0. 4.911E 03 2.841E 03 1.772ε 03 1.246ε 03 1.047ε 03
5·00ε-06 4.847Ε 03 2.829ε 03 1.769ε 03 1.244E 03 1.047ε 03
ι·οοε-θ5 4.666Ε 03 2.794ε 01 1.76ΐε 03 1.197ε 03 1.045E 03
4·00ε-05 2.672ε 03 2.236ε 03 1.616E 03 1.106ε 03 1·02ΐε 03
7.00ε-05 1.388E 03 1.56ΐε 03 1.370ε 03 9.920ε 02 9.703ε 02
ι.οοε-04 8.074ε 02 1.074ε 03 ι.ιΐ4ε 03 6.343ε 02 9.026ε 02
2·00ε-04 2.621E 02 4.120ε 02 5.595ε 02 3.006ε 02 6.49βε 02
4·00ε-04 1.292E 02 1.783ε 02 2.411E 02 1.885ε 02 3.350ε 02
6.00Ε-04 1.199ε 02 1.384E 02 1.626ε 02 1.353ε 02 2.043ε 02
8.00Ε-04 1.129ε 02 1.182ε 02 1.25βε 02 1.027ε 02 1.401E 02
9.893ε 01 9.892ε 01 ι.οοιε 02 5.662ε 01 1.033ε 02
ι.οοε-03 6.465Ε 01 6.121E 01 5.776ε 01 3.203ε 01 5.66ΐε 01
1.50ε-03 2.97βε 01 2.999ε 01 3.07βε 01 2.003ε 01 3.295ε 01
2.00ε-03 1.667E 01 1.749ε 0or1 1.866E 01 1.348E 01 2.07βε 01
2.50ε-03 1.053ε 01 1.122ε 1.229ε 01 ι.40ΐε οι
3.00ε-03

Τ = 5 , 0 0 0 Κ ΝΑ = 1 ΝΒ = 3

α\ LOG(N) 15 8.495ε 01 17 * 18 *
2·00ε-04 1.062E 02
4·00ε-04 5·βοοε οι 1.45ΐε 02 1.105E 02 1.412ε 02
5·00ε-04 ι.6ΐβε oj 1.269E 02 1.615E 02
6·00ε-04 8.533ε 01 1.744ε 02 1.554ε 02 1.886E 02
7.00ε-04 1.373ε 02 1.822E 02 1.662ε 02 1.942ε 02
8·00ε-04 1.614ε 02 1.854ε 02 1.733ε 02 1.93βε 02
9·00ε-04 1.802ε 02 1.848E 02 1.764ε 02 1.889E 02
1·92βε 02 1.812ε 02 1.76ΐε 02 Ι.θΐΐε 02
ι.οοε-03 1.990ε 02 1.756ε 02 1.732ε 02 1·72ΐε 02
ι.ιοε-03 2.000Ε 02 1.688E 02 1.684ε 02 1.631E 02
1.20ε-03 1.968E 02 1.534ε 02 1.624ε 02 1.546ε 02
1.40ε-03 1.910ε 02 1.355ε 02 1.556ε 02 1.469ε 02
1·60ε-03 1.83βε 02 1.170ε 02 1.410ε 02 1.330ε 02
1.80ε-03 1.696ε 02 ι.οιιε 02 1.254ε 02 1.187ε 02
2.00ε-03 1.466ε 02 7·19ΐε 01 1.09βε 02 1.035ε 02
2.50ε-03 1.244E 02 5.306ε 01 9.570ε 01 8.953ε 01
3.00ε-03 1.05βε 02 2.90βε 01 6.913ε 01 6.433ε 01
4.00ε-03 7.38ΐε 01 1.787ε 01 5.136ε 01 4.836ε 01
5.00Ε-03 5.52ΐε 01 7.194ε 00 2.956ε 01 2.812ε 01
7.50ε-03 2.794ε 01 3.752ε 00 1.863ε 01 1.748ε 01
1.656ε 01 7.765ε 00
ι.οοε-02 6.310ε 00
3.185ε 00

T « 10,000 K NA = 1 NB = 3
16
a\ LOG(W) 15 17

0. 4.495ε 01 6·86ΐε 01 9.172E 01 1.128E 02
2·00ε-04 7.103ε 01 8·94βε 01 1.067ε 02 1.254E 02
4.00ε-04 1.239ε 02 1.302ε 02 1.355ε 02 1·48ΐε 02
5.00ε-04 1.501E 02 1.495ε 0J 1.484ε 02 1.570E 02
6.00E-04 1.717ε 02 1·65ΐε 02 1.586ε 02 1.630E 02
7.00ε-04 1.873ε 02 1.762E 02 1.654E 02 1.660E 02
8.00Ε-04 1.964ε 02 1.825ε 02 1.688E 02 1.661E 02
1.99βε 02 1.846ε 02 1.693E 02 1.640E 02
9ιι·.·οι0οο0εεε---0θ0343 1·987ε 02 1.833E 02 1.673E 02 1.602E 02
1.946ε 02 1.794ε 02 1.635ε 02
1.20ε-03 1.889ε 02 1.74ΐε 02 1.584ε 02 1.553E 02
1·40ε-03 1.789ε 02 1.610E 02 1.457ε 02 1.495ε 02
1.60ε-03 1.574ε 02 1.432ε 02 1.308E 02 1.366E 02
ι.βοε-03 1.301E 02 1.233E 02 1.153ε 02 1.227E 02
2.00ε-03 1.101E 02 1.063E 02 1.011E 02 1.086E 02
2·50ε-03 7.678Ε 01 7.567ε 01 7.364Ε 01 9.567Ε 01
3.00ε-03 5.877ε 01 5.623ε 01 5.488Ε 01 7.022ε 01
4.00ε-03 2.849Ε 01 2.998Ε 01 3.127E 01 5.264Ε 01
5.00Ε-03 1.657E 01 1.808E 01 1.955ε 01 3.094Ε 01
7.50Ε-03 6.100E 00 7.062Ε 00 8.059ε 00 1.971E 01
3.027Ε 00 3.615E 00
ι.οοε-02

Τ - 20,000 Κ ΝΑ = 1

a\L0G(N) 15 16 17 18 1.137E 02
0. 3.512ε 01 5.566Ε 01 7.747Ε 01 9.551E 01 1.232E 02
2.00ε-04 6.017E 01 7.650Ε 01 9.226Ε 01 1.058E 02 1.409E 02
4.00Ε-04 1.135ε 02 1.190E 02 1.223E 02 1.266E 02 1.481E 02
5.00Ε-04 1.413E 02 1.403E 02 1.370E 02 1.364E 02 1.532E 02
6.00Ε-04 1.651E 02 1.585E 02 1.494E 02 1.444E 02 1.558E 02
7.00Ε-04 1.830E 02 1.721E 02 1.586E 02 1.501E 02 1.563E 02
8.00Ε-04 1.944E 02 1.809E 02 1.646E 02 1.534E 02 1.548E 02
9.00Ε-04 1.999ε 02 1.852E 01 1.674E 02 1.545E 02 1.519E 02
1.00E-03 2.005Ε 02 1.856E 02 1.675ε 02 1.536E 02 1.479ε 02
1.10E-03 1.977E 02 1.833E 02 1.654E 02 1.512ε 02 1.431E 02
1.20E-03 1.932E 02 1.793ε 0<ί 1.617E 02 1.475ε 02 1.320E 02
1.40E-03 1.881E 02 1.691E 02 1.510ε 02 1.375ε 02 1.197ε 02
1.60ε-03 1.655E 02 1.511E 02 1.366E 02 1.254E 02 1.07ΐε 02
1.80E-03 1.346E 02 1.289E 02 1.207E 02 1.125E 02 9.530ε 01
2.00ε-03 1.133ε 02 1.106E 02 1.059E 02 1.002ε 02 7.116ε 01
2.50Ε-03 7·890ε 01 7.85βε 0. 7.736Ε 01 7.490ε 01 5.393ε 01
3.00Ε-03 6.223Ε 01 5.907ε 01 5.785ε 01 5.666Ε 01 3.236ε 01
4.00Ε-03 2.875Ε 01 3.047ε 01 3·240ε 01 3.345ε 01 2.086Ε 01
5.00Ε-03 1.645ε 01 1.801E-01 1.995ε 01 2.132E 01
7.50Ε-03 5.866Ε 00 6.812E 00 8.016E 00
1.00E-02 2.871E 00 3.427Ε 00

30,000 Κ ΝΑ = 1

a\LOG(N) 16 17 18 19

0. 3.054Ε 01 4.939ε 01 7.05ΐε 01 8.809Ε 01 1.026E 02
2·00ε-04 5.510E 01 7.023ε 01 8.545ε 01 9.782Ε 01 1.100E 02
4·00ε-04 1.086E 02 1.136E 02 1.165ε 02 1.183E 02 1.252E 02
5·00ε-04 1.370E 02 1.360E 02 1.32ΐε 02 1.285E 02 1.324ε 02
6·00ε-04 1.618E 02 1.554E 02 1.456ε 02 1.372E 02 1.38ΐε 02
7.00ε-04 1.809E 02 1.704ε 02 1·56ΐε 02 1.439ε 02 1.42ΐε 02
8.00ε-04 1.935ε 02 1.804ε 0? 1.632ε 02 1.483ε 02 1.443ε 02
2.000ε 02 1.857ε 02 1.67ΐε 02 1.506ε 02 1·44βε 02
9ι..ο0ο0εε--00 34 2.014E 02 1.87ΐε 0? 1.682ε 02 1.509ε 02 1.438ε 02
1.993E 02 1·855ε 02 1.670ε 02 1.495ε 02 1.415ε 02
1.10E-03 1.953ε 02 1.822E 02 1.641E 02 1.468E 02 1.383ε 02
1.20E-03 1.932ε 02 1.739ε 02 1.545ε 02 1.383ε 02 1.296ε 02
1.40ε-03 1.696ε 02 1.556ε 02 1.403ε 02 1.270ε 02 1.19ΐε 02
1.60E-03 1.367ε 02 1.31βε 02 1.23βε 02 1.146E 02 1.080E 02
1.80ε-03 1·147ε 02 1.127ε 0J 1.085E 02 1.025ε 02 9.715ε 01
2.00ε-03 7.98ΐε 01 7.99ΐε 01 7.918E 01 7.715ε 01 7.403ε 01
2.50ε-03 6.425ε 01 6.061E 01 5.933Ε 01 5.860ε 01 5.676ε 01
3.00ε-03 2.882Ε 01 3.060ε 01 3.281E 01 3.455ε 01 3.452ε 01
4.00ε-03 1.637ε 01 1.789ε 01 2.193ε 01 2.243ε 01
5.00Ε-03 5.730ε 00 6.646Ε 00 2.οοοε οι
7.50ε-03 2.78βε 00 3.313E 00
7.900ε 00
ι.οοε-02

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T - 20,000 Κ HA = 1 ΝΒ = 4

ey.OG(N) 14 15 16 17 18
0. 1.494ε 02 1.144E 02 1.037E 02
2·00ε-04 3.688Ε 02 2.230ε 02 1.392E 02 1.109ε 02 1.016E 02
4.00ε-04 2.214ε 02 1·842ε 02 1.174E 02 1.022ε 02 9.628Ε 01
6.00E-04 1.093ε 02 1.261E 02 9.700ε 01 9.216E 01 8.963Ε 01
7.048Ε 01 9.052ε 01 8·280ε 01 8.334Ε 01 8.328Ε 01
8ι..ο0ο0εε--00 34 5.815ε 01 7.332ε 01 7.427ε 01 7.66βε 01 7.799ε 01
5.587ε 01 6.613E 01 6.964ε 01 7.202ε 01 7.387ε 01
1·20ε-03 5.763ε 01 6.400Ε 01 6.735ε 01 6·β88ε 01 7.072ε 01
1.40ε-03 6.082Ε 01 6.423Ε 01 6.626ε 01 6·67ΐε 01 6.823ε 01
1.60ε-03 6·422ε 01 6.530ε 01 6·565ε 01 6.50βε 01 6.613ε 01
1.80ε-03 6.688Ε 01 6.631E 01 6.509ε 01 6.369ε 01 6.423ε 01
2.00ε-03 6.815ε 01 6.681E 01 6.282Ε 01 6.019ε 01 5.959ε 01
2·50ε-03 6·804ε 01 6.577ε 01 5.913E 01 5.601E 01 5.464ε 01
3.00ε-03 6.485ε 01 6.247ε 01 4.803ε 01 4.567ε 01 4.400Ε 01
4·00ε-03 5.189ε 01 5.039ε 01 3.606Ε 01 3.524ε 01 3.410ε 01
5.00ε-03 3.737ε 01 3.677ε 01 1.682E 01 1.787ε 01 1.787ε 01
7.50ε-03 1.430ε 01 1.546ε 01 8.880Ε 00 9.977ε 00
6.75βε 00 7.683ε 00
ι.οοε-02 2.345ε 00 2.828Ε 00
1.164E 00
1.50ε-02
2.00ε-02

Τ - 30,000 Κ ΝΑ = 1 ΝΒ = 4

o\L06(N) 14 15 16 17 18
0. 2.453ε 02 1.589ε 02 9.823Ε 01
2.00ε-04 4.159ε 02 1.937ε 02 1.459ε 02 1.160E 02 9.640Ε 01
4.00ε-04 2.23ΐε 02 1.245ε 02 1.195ε 02 1.121E 02 9.168E 01
6·00ε-04 1.022ε 02 8.650ε 01 9.617ε 01 1.026ε 02 8·566ε 01
6.506ε 01 6.952ε 01 8.082ε 01 9.164ε 01 7.976ε 01
8ι..ο0ο0εε--00 34 5.43ΐε 01 6.302Ε 01 7.206ε 01 8.219ε 01 7.473ε 01
5.306ε 01 6.158E 01 6.760ε 01 7.517ε 01 7.075ε 01
1.20ε-03 5.55ΐε 01 6.24ΐε Οχ 6.564ε 01 7.03βε 01 6.77ΐε 01
1.40ε-03 5.922ε 01 6.402Ε 01 6.492Ε 01 6.725ε 01 6.536Ε 01
1·60ε-03 6·546ε 01 6.466Ε 01 6·520ε 01 6.345ε 01
1.80ε-03 6.3ΐοε οι 6.629ε 01 6.443Ε 01 6.375ε 01 6.180ε 01
2.00ε-03 6.587Ε 01 6.283Ε 01 6.258ε 01 5.795ε 01
2.50ε-03 6.618E 01 6.305Ε 0* 5.969Ε 01 5.969ε 01 5.38ΐε 01
3.00ε-03 6.765ε 01 5.126ε 01 4.897ε 01 5.609ε 01 4.435ε 01
4·00ε-03 6.807ε 01 3.739ε 01 3.674ε 01 4.637ε 01 3.496ε 01
5.00ε-03 6.524ε 01 1.537ε 01 1.683E 01 3.597ε 01 1.870E 01
7.50Ε-03 5.244Ε 01 7.525ε 00 8.747Ε 00 1.817ε 01
3.780ε 01 2.722ε 00· 1.004ε οι
ι.οοε-02 1.42ΐε 01
6.637Ε 00
1.50ε-02 2.267ε 00
2.00ε-02 1.125E 00

Τ - 40,000 Κ ΝΑ = 1 ΝΒ = 4

0V L0G(N) 14 15 16 17 18
0. 1.180E 02 9.553Ε 01
2·00ε-04 4.532ε 02 2·63ΐε 02 1.665E 02 1.137E 02 9.382Ε 01
4.00ε-04 2.223Ε 02 2.001E 02 1.511E 02 1.033E 02 8.936Ε 01
6·00ε-04 9.694ε 01 1.226E Οί 1.208E 02 9.157E 01 8.361E 01
8·00ε-04 6·140ε 01 8.344Ε 01 9.536ε 01 β·15βε 01 7.790ε 01
1.00E-03 5·ιβοε 01 6.684Ε 01 7.92βε 01 7.426ε 01 7.29βε 01
1.20E-03 5.126E 01 6·092ε Οι 7.045Ε 01 6.937ε 01 6.906ε 01
1.40ε-03 5.417ε 01 5.997ε 01 6.618E 01 6.625ε 01 6.605Ε 01
1·60Ε-0ΐ 5.822Ε 01 6.123ε 01 6.447ε 01 6.426Ε 01 6.375ε 01
1.80E-03 6.240ε 01 6.320ε 01 6.403Ε 01 6.296Ε 01 6.193E 01
2.00Ε-03 6.57βε 01 6.493ε 01 6.403Ε 01 6.194E 01 6.039Ε 01
2.50ε-03 6.735ε 01 6.597ε 0* 6.402ε 01 5.944ε 01 5.694ε 01
3.00ε-03 6.8ΐοε 01 6.595ε 01 6.287ε 01 5.620ε 01 5.326ε 01
4.00ε-03 6·54βε 01 6.344Ε 01 4.686Ε 01 4.452ε 01
5.00ε-03 5.276ε 01 5.182ε 01 6.οιοε οι 3.644ε 01 3.546ε 01
7.50ε-03 3.804Ε 01 3.77βε 01 1.831E 01 1.919ε 01
1.00E-02 1.414E 01 1.529ε 01 4.960ε 01 1.005E 01
1.50E-02 6.554ε 00 7.414ε 00 3.71βε 01
2.00ε-02 2.216ε 00 2.65ΐε 00 1.680E 01
1.102E 00 8.637Ε 00

287

288 APPENDIX I. STARK PROFILES

T = 5,000 K NA = 1 NB = 5

cALOG(N) 14 15 16 17
4·978Ε 01 5.546Ε 01
01·.00E-03 4.237E 01 4.738Ε 01 5.376Ε 01 5·849Ε 01
2.00E-03 6.048E 01 5.552Ε 01 5.371E 01 5.660Ε 01
3·ΟΟΕ-03 6.489E 01 5.733Ε 01 4.803Ε 01 4.954Ε 01
4.00E-03 5·403Ε 01 5.037Ε 01 4.141E 01 4.166E 01
5·ΟΟΕ-03 4.496Ε 01 4.293Ε 01 3.530Ε 01 3.465Ε 01
6·00Ε-03 3.772Ε 01 3.646Ε 01 2.993Ε 01 2.886Ε 01
7.00E-03 3.203Ε 01 3.093Ε 01 2.522Ε 01 2.407Ε 01
8.00E-03 2.667Ε 01 2.599Ε 01 2·115Ε 01 2.006Ε 01
9.00E-03 2.193E 01 2.164E 01 1.771E 01 1.674E 01
l.OOE-02 1.791E 01 1.795E 01 1.484E 01 1.400E 01
1.20E-02 1.432E 01 1.482E 01 1.061E 01 9·983Ε 00
1.40E-02 9.558Ε 00 U033E 01 7.845Ε 00
1.70E-02 6.730Ε 00 7.488Ε 00
4.267Ε 00 4.914E 00

Τ = 10,000 Κ ΝΑ = 1 ΝΒ = 5
15
o\LOG(N) 14 16 17
4.640Ε 01 4.866Ε 01
0. 3·769Ε 01 4.419E 01 5.123E 01 5.150E 01
1.00E-03 6.091E 01 5.522Ε 01 5.254Ε 01 5.133E 01
2.00Ε-03 6.777Ε 01 5·889Ε 01 4.746Ε 01 4.645Ε 01
3·00Ε-03 5.488Ε 01 5.124E 01 4.125E 01 4.046Ε 01
4.00Ε-03 4.538Ε 01 4.344Ε 01 3.556Ε 01 3.478Ε 01
5·00Ε-03 3.815E 01 3.699Ε 01 3.053Ε 01 2.971E 01
6.00Ε-03 3.281E 01 3.165E 01 2·600Ε 01 2.522Ε 01
7.00Ε-03 2·738Ε 01 2.674Ε 01 2.195E 01 2.131E 01
8·00Ε-03 2.255Ε 01 2.229Ε 01 1.845E 01 1.797E 01
9.00Ε-03 1.835E 01 1.844E 01. 1.548E 01
1.00E-02 1·444Ε 01 1.510E 01 1.104E 01 1.517E αι
1.20E-02 9·448Ε 00 1.036E 01 8.127E 00
1.40E-02 6.561E 00 7.415E 00 1.098E 01
1.70E-02 4.087Ε 00 4·788Ε 00

Τ - 20,000 Κ Ν Α = 1 ΝΒ=5

a\L0G(N) 14 15 16 17
3.305Ε 01 4.098Ε 01 4.413E 01 4.470Ε 01
0. 6.171E 01 5.566Ε 01 5.039Ε 01 4.754Ε 01
1.00E-03 7.113E 01 6.135E 01 5.304Ε 01 4.836Ε 01
2·00Ε-03 5.558Ε 01 5.237Ε 01 4.789Ε 01 4·460Ε 01
3.00Ε-03 4.574Ε 01 4.400Ε 01 4·154Ε 01 3·949Ε 01
4.00Ε-03 3.848Ε 01 3.749Ε 01 3.595Ε 01 3.454Ε 01
5.00Ε-03 3.353Ε 01 3·233Ε 01 3·109Ε 01 3.001E 01
6.00Ε-03 2.794Ε 01 2.741E 01 2-664Ε 01 2.587Ε 01
7.00Ε-03 2.306Ε 01 2·283Ε 01 2.256Ε 01 2·213Ε 01
8.00Ε-03 1.871E 01 1.882E 01 1.897E 01 1.884E 01
9.00Ε-03 1.445E 01 1.523E 01 1.585E 01 1·602Ε 01
1·00Ε-02 9.276Ε 00 1.025E 01 1.120E 01 1.168E 01
1.20E-02 6.363Ε 00 7.235Ε 00 8.170E 00
1.40E-02 3.898Ε 00 4.592Ε 00
1.70E-02

A I . a . HYDROGEN 289

T = 30,000 Κ ΝΑ = 1 ΝΒ = 5

α \ LOG(N ) 14 15 16 17
3.043Ε 01 4.323Ε 01
0. 6.227Ε 01 3.903Ε 01 4.301E 01 4.623Ε 01
l.OOE-03 7·320Ε 01 5.611E 01 5.036Ε 01 4.751E 01
2·00Ε-03 5.588Ε 01 6.305Ε 01 5.384Ε 01 4.409Ε 01
3·00Ε-03 4.591E 01 5.303Ε 01 4.842Ε 01 3.920Ε 01
4.00Ε-03 3.863Ε 01 4.431E 01 4.184E 01 3.446Ε 01
5.00Ε-03 3.392Ε 01 3·774Ε 01 3.621E 01 3.014E 01
6.00Ε-03 2.819E 01 3.272Ε 01 3.142E 01 2.614E 01
7.00Ε-03 2.332Ε 01 2·776Ε 01 2.700Ε 01 2.248Ε 01
8.00Ε-03 1.887E 01 2.311E 01 2·287Ε 01 1.921E 01
9.00Ε-03 1.442E 01 1.900E 01 1·920Ε 01 1.636E 01
1.00E-02 9.163E 00 1.525E 01 1·599Ε 01 1.195E 01
1.20E-02 6.245Ε 00 1.014E ο00ι 1.121E 01
1.40E-02 3.794Ε 00 7·105Ε 8.121E 00
1.70E-02 4·466Ε 00

Τ = 40,000 Κ ΝΑ = 1 ΝΒ = 5

α\ L O G ( H ) 14 15 16 17
4.223Ε 01 4·244Ε 01
0· 2.864Ε 01 3.762Ε 01 5.044Ε 01 4.560Ε 01
1.00E-03 6·269Ε 01 5.648Ε 01 5·454Ε 01 4.718E 01
2.00Ε-03 7.467Ε 01 6.432Ε 01 4.885Ε 01 4.391E 01
3.00Ε-03 5.604Ε 01 5.347Ε 01 4.206Ε 01 3.910E 01
4.00Ε-03 4.602Ε 01 4.451E 01 3.640Ε 01 3·446Ε 01
5.00Ε-03 3.872Ε 01 3·791Ε Οι 3.166E 01 3·024Ε 01
6·00Ε-03 3.417E 01 3.299Ε 01 2.723Ε 01 2.632Ε 01
7.00Ε-03 2.834Ε 01 2.798Ε 01 2.307Ε 01 2·269Ε 01
8.00Ε-03 2.3Α7Ε 01 2.329Ε 01 1.934E 01 1.942E 01
9.00Ε-03 1.896E 01 1.911E 01 1.606E 01 1.656E 01
1.00E-02 1.440E 01 1.524E 0* W119E 01 1.209E 01
1.20E-02 9.082Ε 00 1.006E 01 8.064Ε 00
1.40E-02 6.165E 00 7.009Ε 00
1.70E-02 3.725Ε 00 4.376Ε 00

Τ » 5,000 Κ ΝΑ = 2 ΝΒ = 3 17
15 16 1.487E 01
a^OG(N) 1.481E 01
0. 2.304Ε 01 1.754E 01 1.463E 01
1.00E-03 2.267Ε 01 1.742E 01 1.435E 01
2.00Ε-03 2·164Ε 01 1·708Ε 01 1.355E 01
3.00Ε-03 2.018E 01 1.655E Οχ 1.233E 01
5.00Ε-03 1.693E 01 1·514Ε 01 1.106E 01
7.50Ε-03 1.352E 01 1.319E 01 8.809Ε 00
1.00E-02 1.112E 01 1.141E 01 7.085Ε 00
1.50E-02 8.264Ε 00 8·711Ε 00 5.792Ε 00
2.00Ε-02 6.583Ε 00 6.906Ε 0J 4.807Ε 00
2.50Ε-02 5.384Ε 00 5.634Ε 00 3.437Ε 00
3.00Ε-02 4.485Ε 00 4.687Ε 00 2·558Ε 00
4·00Ε-02 3·257Ε 00 3.374Ε 00 1.372E 00
5.00Ε-02 2·452Ε 00 2.519E 00 8.094E-01
7.50Ε-02 1·289Ε 00 1·328Ε 00 3.563E-01
W00E-01 6.983Ε--01 7.553Ε--01
1.50E-01 2.756E-01 3.183E--01
2.00E-01 1·405Ε--01 1.695E-•01
2.50E-01 8.352Ε-•02
3.00E-01 5.495Ε-02

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14 15 2.677Ε 00 3.046Ε 00
\α L ü G ( N ) 2.82ΐε 00 3.158E 00
0. 1.14ΘΕ 00 1.705ε 00 2.253ε 00 3.168ε 00 3.423Ε 00
5 . οοε- •03 1·645Ε 00 3.566ε 00 3.715ε 00
2.754ε 00 2.066ε 00 2.485ε 00 3.904ε 00 3.947ε 00
1 . ,ΟΟΕ- ■02 3.939ε 00 2.894ε 00 3.033ε 00 4.130ε 00 4.082ε 00
1. 50Ε- •02 4.872ε 00 3.805ε 00 3.65ΐε 00 4.237ε 00 4.119ε 00
5.420ε 00 4.544ε 00 4.ΐ7ΐε 00 4.237ε 00 4·070ε 00
2 . ΟΟΕ- •02 5.61βε 00 4.150ε 00 3.955ε 00
5.613ε 00 4.000Ε 00 3.793ε 00
2 . 50Ε- •02 5.554ε 00 5.οιιε οο 4.519ε 00 3.807Ε 00 3.60ΐε 00
5.415E 00 3.589ε 00 3.392ε 00
3 . ΟΟΕ- •02 4.863Ε 00 5.216ε 00 4.688Ε 00 3.36ΐε 00 3.178ε 00
4.38ΐε 00 5.233ε 00 4.708Ε 00 2.908ε 00 2.757ε 00
3 . 50Ε- •02 4.017ε 00 5.138ε 00 4.615ε 00 2·490ε 00 2.372ε 00
3.395ε 00 4.926ε 00 4.433ε 00 2.125ε 00 2.037ε 00
4· ΟΟΕ- •02 2.689ε 00 4.576ε 00 4.186ε 00 1.815ε 00 1.751E 00
45 ·. 2.24βε 00 4.204ε 0 ) 3.909ε 00 1.332ε 00 1.304E 00
50Ε- •02 1.802ε 00 3.870ε 00 3.630Ε 00 9.927ε-01 9.875ε-01
ΟΟΕ- •02 1.215ε 00 3.252ε 00 3.089Ε 00 6 . 6 8 6 Ε ·- 0 1 6.790ε-01
8.137ε-01 2.664ε 00 2.596ε 00 4.738ε--01 4.892ε-01
65 ·. 50Ε- •02 4.915ε-01 2.18 IE 00 2.919ε--01
ΟΟΕ- •02 3.21βε-01 2.2ΐοε οο 1.835E 00 2.834Ε 00
1.804E-01 2.520Ε 00 2.931E 00
87 ·. ΟΟΕ- •02 1.136E-01 2.666Ε 00 3.168E 00
ΟΟΕ- •02 7.677ε-02 3.021E 00 3.445Ε 00
Κ ΝΑ = 2 3.437Ε 00 3.682Ε 00
9 . ΟΟΕ- •02 1·020Ε 00 3.802Ε 00 3.842Ε 00
1. οοε- •01 1.531E 00 4.059Ε 00 3.916E 00
1 . 2οε- •01 2·672ε 00 1.827ε 00 1.305ε 00 4.197E 00 3.909Ε 00
1 . 4οε- •01 3.89ΐε 00 1.259ε 00 9.404ε-01 4.225Ε 00 3.835Ε 00
1 . 7οε- •01 4.860ε 00 8.707ε-01 6.103ε-01 4.163E 00 3.711E 00
5 . 4 3 U 00 5.432ε-0. 4.2ΐοε-οι 4.031E 00 3.552Ε 00
2 . οοε- ■01 5.639ε 00 3.645ε-01 3.850Ε 00 3.372Ε 00
2 . »οε- 01 5.647ε 00 2.105E-01 2.514ε-01 3.640Ε 00 3.179E 00
3 . οοε- •01 5.615ε 00 1.348E-01 1·646Ε-01 3.416E 00 2.790Ε 00
3 . 5οε- •01 5.537ε 00 2.965Ε 00 2.422Ε 00
4.919ε 00 9.310ε-02 2.544Ε 00 2.095Ε 00
4.420ε 00 2·173ε 00 1.810E 00
Τ - 30,,000 4.055Ε 00 ΝΒ = H 2.092Ε 00 1.857E 00 1.359E 00
3.443ε 00 1.547ε 00 2.337ε 00 1.360E 00 1.033E 00
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2.279ε 00 2.792ε 00 3.577Ε 00 6.757E-01 5.134E-01
5«,οοε- . 0 3 1.811E 00 3.750ε 00 4.141E 00 4.763E-01
1«► ο ο ε--0 2 1.217ε 00 4.535ε 00 4.527Ε 00 2.914E-01
1.>5οε·-02 8.112ε-01 5.035ε 00 4.725Ε 00
4.863ε-01 5.260ε 00 4.769Ε 00
2.00Ε-02 3.163E-01 5.295ε 00 4.695Ε 00
2.50Ε--02 1.761E-01 5.225ε ΟΟ 4.522ε 00
3<► ο ο ε- 0· 2 1.104E-01 5.03ΐε 00 4.272ε 00
3.50Ε--02 7.398Ε-02 4.660ε 00 3.989Ε 00
4.00Ε--02 4.270ε 00 3.705Ε 00
4 . ► 50Ε- 0 2 3.932ε 00 3.154ε 00
5<► 0 0 Ε- 0· 2 3.31ΐε 0) 2.645Ε 00
5«► 5 ο ε- ·0 2 2.700ε 00 2.219E 00
2.240ε 00 1.864ε 00
6..οοε-02 1.845ε 00 1.318ε 00
7,,οοε- -02 1.265ε 00 9.415ε-01
8..οοε- -02 8.670ε-01 6.04βε-01
9«► ο ο ε- ·0 2 5.356ε-01 4.141E-01
1,»οοε«- 0 1 3.56βε-01 2.449E-01
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1.►4οε-οι ι.3θοε-οι
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2 , ► οοε-«0 1 8.940ε-02
2<» 5 ο ε -- 0 1
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9.391E-01 2·24ΐε 00 2.57ΐε 00 2.804Ε 00
0• 1.459ε 00 1 · 4 4 6 ε 00 2.844Ε 00 2.935Ε 00 3.029Ε 00
5 • οοε«- 0 3 2.622ε 00 1.835ε 00 3.535ε 00 3.366Ε 00 3.297Ε 00
1 ·οοε·-02 3.862ε 00 2 . 7 2 9 ε 0Ü 4.129E 00 3.749Ε 00 3.537Ε 00
1 ·5οε·-02 4.854ε 00 3.719ε 00 4.540ε 00 4.028Ε 00 3.709Ε 00
5.440ε 00 4.533ε 00 4.757ε 00 4.186E 00 3.801E 00
2 ·οοε·-02 5.652Ε 00 5.054ε 0J 4.815ε 00 4.231E 00 3.817E 00
2 ·5οε·-02 5.668ε 00 5.292ε 0 ) 4.753ε 00 4.183E 00 3.766Ε 00
3 .οοε -02 5.653ε 00 5.337ε 00 4·586ε 00 4.061E 00 3.664ε 00
5.624ε 00 5.285ε 00 4.333Ε 00 3.886Ε 00 3.524Ε 00
3 .5οε -02 4.952ε 00 5.107ε 00 4.044Ε 00 3.679Ε 00 3.359Ε 00
4 • ο ο ε -- 0 2 4.444ε 00 4.716ε 00 3.755ε 00 3.457Ε 00 3.180E 00
4 ·5οε -02 4.076ε 00 4.312ε 00 3.196ε 00 3.005ε 00 2.808Ε 00
5 ·οοε·-02 3.474ε 00 3.972ε 00 2.676Ε 00 2.579Ε 00 2.452ε 00
2.724ε 00 3.349ε 00 2.242ε 00 2.204Ε 00 2.129ε 00
5 • 50Ε-- 0 2 2.301E 00 2.722ε 00 1.882E 00 1.846E 00
1.816ε 00 2.260ε 00 ι.ββιε οο 1.376ε 00 1.391E 00
6 .ΟΟΕ - 0 2 1.217ε 00 1.855ε 00 1.018E 00 1.060E 00
7 • 00Ε-- 0 2 8.093ε-01 1.268ε 00 1.325ε 00 6.779E-01 7.316ε-01
8 .00Ε·-02 4.82βε-01 8.637ε--01 9.404ε-01 4.758E-01 5.270E-01
9 • ο ο ε -- 0 2 3.126ε-01 5.30ΐε--01 5.999ε-01 2.896ε-01
1 ·οοε·-01 1.734ε-01 3.514ε·-01 4.086ε-01
1 • 20Ε-- 0 1 1.085E-01 1.999ε·-01 2.40ΐε-01
1 ·40Ε·-01 7.21θε-02 1.26βε--01 1.556E-01
1 • 70Ε-- 0 1 8 . 6 9 4 ε - -Οί
2 .ΟΟΕ - 0 1
2 ·5οε -01
3 • οοε-01
3 • 50Ε-01