Spectral Line Broadening by Plasmas

I I . 5 . CORRELATION EFFECTS 143

by plasmas, the function S+ has been investigated widely using, e.g.,
Rostoker's dressed test particle concept [137]. The principal result in the
present context is the emergence of two contributions [136] to S+, one
from test electrons and their shielding clouds, the other from the shielding
electrons of test ions. In equilibrium plasmas and for Δω intermediate to
ion and electron (ωρ) plasma frequencies, the for us most important real
part of S+ is [136]

Re S+(X, i Δω) =\S(K, Δω)

-ïfè)"h®T

with ΚΌ = (PD)-1. The term involving the ion-to-electron mass ratio is
due to the ion-cloud electrons, while the leading term corresponds to the
expression of Chappell et al. [135a], if the same assumptions regarding
equilibrium and frequency range are also made there.

Insertion of Eqs. (359) and (360) into Eq. (358) and integration from
K = 0 to Km&x (corresponding to ρ~·η in the impact theory) results in

Im <t | £ (co) 11'> « - | ( — ) - N Σ <f I r \i" ) · <*" | r 11' >

for lfmax » KN. [See Eq. (368) for an estimate of Re(i | £(ω) | i'), which
is usually both smaller and less important than lm(i \ £(ω) | i').~\ The char-

acteristic function for the ion cloud electrons depends on the dimensionless

variable

X«» = HM/m)Z(o - ω ^ ν ) / ω ρ ] 2 (362)

and can be expressed as

Λ . ) - 1 [ < 1 + . ) ( * / ν - * ) - ΐ ] - ί ( ΐ - ί + . . . ) (363,

The " — J" term in Eq. (361) represents the double shielding correction
proposed by Chappell et al. [135a], which, however, should often be ob-

144 I I . THEORY

scured in practice by the "ion cloud" term involving the ion-to-electron
mass ratio, frequencies approaching the electron plasma frequency being
an exception. To evaluate S(K, Δω) in this neighborhood, Eq. (360) must
be replaced by [135a, 136]

\ S(K, Δω) « (N/K)(Tfn/2kT)1/2 \ e(K, Δω) |"2 βχρΙ-%(ΑωΚΌ/ωρΚ)2],

(364)

where c is the usual wave number and frequency dependent dielectric con-
stant of a plasma [138], which had been replaced by its low-frequency ap-
proximation 1 + (ΚΌ/Κ)2 farther above. Numerical calculations [139]
show that this low frequency approximation may actually be used right
up to Δω = cop , beyond which the numerical results rather abruptly join
those obtained from Eq. (364) and the high-frequency approximation
€ = 1 (no shielding), namely

lm{i | £(«) | i>) « - | ( — ) - ΛΓ Σ (i I r | i") · <t" | r | i>)

' ln(l "*'"<*) (365)

for large values of the argument of the logarithm. This is the familiar result
of Lewis [65], of the relaxation theory (neglecting electron-electron cor-
relations [13a]), and of the one-electron approximation [6]. [If the argu-
ment of the logarithms in Eq. (361) or (365) is not large, these should be
replaced by differences of a(z) functions (Section II.3ca) with z « | ω —
Ui"f | Κη/ωρ Km&x and z = | ω — ω,·»/ |/ωρ , respectively, provided this
gives smaller widths.]

While knowledge of Im £(ω) is indeed sufficient on the line wings,
evaluation of entire profiles according to Eq. (282) also requires the real
part of <£(ω), which of course has no counterpart in the impact theory
calculations [53, 57-61] for hydrogen lines, etc. (In calculations [62, 63]
that also consider penetrating orbits, there is a small real part and accord-
ingly also a shift, which, however, has nothing to do with correlations be-
tween perturbers.) A simple approximation [136, 140] for the electron
cloud contribution, approximately corresponding to Eq. (360), is then

—«"■>-i(rJ[-(f)T-(-1)HS*

~i(S)""[-(f)T*

I I . 5 . CORRELATION EFFECTS 145

with

x = ΑωΚΌ/ωρΚ = (Aco/Z)(m//cT)1/2, (367)

whose magnitude has to stay below unity for the linear approximation to
be reasonably valid. Substitution into Eqs. (358) and (359) and integration
over K yields

Re<* | JB(«) | i') « I y y e-N Σ <*' I r | i") · <t" | r | i'>

Since the x dependence of Eq. (366) is actually as x~l for large magnitudes
of this parameter, the square bracketed expression should, for large
I ω — oli"/ |/ωρ , be replaced by some decreasing function of the frequency
ratio and, in any event, not be allowed to exceed the high temperature
limit for the b function, i.e., we must certainly insist on

IE- · · ] | < τ / 2 . (369)

Comparison with Eq. (361) thus reveals that the real part of the £-matrix
elements is smaller in magnitude than the imaginary part by a factor of
about \n(KmAX/KO) for relevant frequencies close to ωρ . Since line profiles
involve Re £(ω) only to second and higher order, errors committed by
omitting Re <£(ω) in calculations of hydrogen, etc., lines should accordingly
be only of order [}n(KmSLX/KO)lr2 if line widths approach the plasma
frequency (and less than that for smaller widths). For widths larger than
ωρ , errors of order [ln(œpKm&x/KO | ω — ωνΊ | ) ] " 2 or less are suggested
from this source. These might be of practical interest, e.g., for higher mem-
bers of the Balmer series, which could well suffer some additional broaden-
ing from Re £(ω).

The reader will note that discussing the Debye shielding problem in
terms of plasma kinetic theory rather than using simple cutoffs has not
yielded corrections that are large compared, e.g., to those stemming from
the approximate treatment of close collisions in most calculations. The same
can be said of other more or less formal treatments [115. 116,134, 141-143]
of this problem. (Burgess [143], however, obtains the conclusion that for
small velocities, both fields should be shielded in width calculations, but
just one in shift calculations, assuming ions could indeed be ignored.)
Moreover, as long as the ion broadening is treated separately by using the
quasi-static approximation, there remains some ambiguity or model de-
pendence in all these treatments, but the plasma kinetic theory approach

146 I I . THEORY

will turn out to be more useful in the discussion of plasma satellites from
suprathermal plasma oscillations in Section II.5c and, perhaps, other non-
equilibrium situations [142].

Most discussions in this section have been given for hydrogen atoms as
radiators, although some of the results obtained apply to isolated neutral
atom lines as well. The principal difference here is the need for reasonably
accurate values of Re £(ω) for predictions of line shifts. In this case
(Δω « 0), it is suggested that Eq. (368) be used for small | ωα" |/ωρ .
For larger frequency ratios, the square bracket in this expression should be
replaced by the b(z) function (Section II.3ca) with z ~ | ω,·,·" | KT>/œpKm&x ,
but subtracting from it b(zp) with zp ~ | ω»,·" |/ωρ to make some allowance
for Debye shielding, and provided that this procedure gives a smaller re-
sult. At a frequency ratio of one and for large Km&JC/KO , the latter pro-
cedure yields about 0.75 instead of [ · · · ] ~ 1.25 from Eq. (368), which is
obviously an overestimate in this, perhaps, worst case. We may thus specu-
late that errors from such schematic treatment of Debye shielding in calcu-
lations of line shifts are usually not very serious. Finally, a separate dis-
cussion for ion lines appears superfluous, because Debye shielding is nor-
mally not noticeable for impact parameters small enough for Coulomb ef-
fects to be significant.

II.5b. Plasma Polarization Shift

The formulation of the line shape operator £(ω) in the preceding section
revealed charge density (or electric field) fluctuations as the primary source
of broadening, shifts, etc., at least under conditions where the impact
theory or its extension given by the "impact approximation" version of the
relaxation theory are valid (Sections II.3 and II.4b). An explicit or implicit
assumption, however, was that the average perturbing charge density in
the vicinity of the radiator was zero. Then the average interaction energy
between perturbers and radiator vanishes, because all other than mono-
pole-monopole terms in the multipole expansion of the interaction Hamil-
tonian average to zero for spherically symmetrical perturber distributions.
In the approximations involving time-dependent perturbation theory it is
now consistent to assume the radiator to be not perturbed initially, while
in the quasi-static approximation or time-independent theory, we may in-
deed begin with the dipole-monopole term in the radiator-perturber inter-
action Hamiltonian. The reader will recall that all these procedures were
adopted in the preceding sections, but that the fulfillment of the condition
of zero average perturber charge density was never verified.

Lest there be a misunderstanding, it must be emphasized that inclusion
of monopole-monopole interactions, e.g., in fully quantum-mechanical

I I . 5 . CORRELATION EFFECTS 147

calculations of electron impact broadening (Section II.3e) or in semiclassical
calculations [62, 63] employing the entire electrostatic interaction Hamil-
tonian, may not be fully equivalent to allowing for the effect under dis-
cussion here. First of all, these calculations do assume [in going from Eq.
(94) to (97) ] the radiators to be unperturbed initially, which is inconsistent
with the existence of a nonvanishing average interaction. In particular, the
sharp resonances in the elastic scattering regime may correspond to such
large collision times as to invalidate the impact approximation. Also, while
the first order term in the Dyson series for the impact broadening operator
[Eq. (92) ] by itself may be interpreted as the electron contribution to the
(monopole-monopole) average interaction, higher-order Dyson terms quite
often lead to a substantial reduction of the total impact-produced shifts
below the first-order result, because the collisions contributing to the
average of the first-order term are usually "strong."

As in the discussion of Debye shielding effects, a suitable point of de-
parture for an actual evaluation of the effects from nonzero average inter-
actions would be the relaxation theory, in this case in the form of Eq.
(277). However, instead of assuming (t/'(r, t) )av = (U'(r, 0) )av = 0, we
now try to allow for matrix elements of this quantity by writing

i*.y(*, 0) = e x p [ - (i/h) (U(r, 0) >.v*Kv(f, 0), (370)

assuming a diagonal representation. In the resulting equation for ur&v,
there is then only the integral term on the right-hand side with u&v(t', 0)
replaced by urw{t\ 0) exp[(z'/7i) (t/(r, 0))av(* — tf)~\. The exponential
factor disappears in the dynamic equation for tAV(t, 0) [Eq. (279)] and
in the subsequent equations of the relaxation theory. Also, if the latter is
restricted to terms linear in the density, the P operators under the integral
and in the Green's function can be left out [52]. However, in the first term
on the right-hand side of Eq. (279), H is now replaced by H + ((7(r, 0) ) a v ,
i.e., the lines will be displaced by an amount corresponding to the average
interaction (or rather to the difference of the average interactions for the
radiator in upper and lower states of the line), in addition to being broad-
ened and shifted by (mostly) second order interactions which are ac-
counted for by the integral term.

The separation of the two contributions to the shift is more subtle in cases
requiring higher than second-order perturbation theory for the integral
term. (Then the integral contributes the usual impact-produced shift minus
the first Born approximation for this shift, provided always that the impact
approximation is reasonably valid.) Higher-order terms are certainly re-
quired in the quasi-static limit. This causes no serious difficulty, however,
because then the line profile, so to speak, directly traces f/(r, t)) and there

148 I I . THEORY

is no question that a shift corresponding to monopole interactions should
be superimposed on profiles calculated, e.g., with the monopole-dipole
interactions only, which is standard procedure in quasi-static Stark profile
calculations. The tentative conclusion of this rather general discussion is
therefore that shifts from the average interaction and those calculated from
the usual approximations discussed previously are more or less additive,
except in two extreme cases. One of them corresponds, e.g., to the fully
quantum-mechanical calculations [108] for the Ca II resonance lines,
where the entire electrostatic interaction Hamiltonian was considered, the
other to (future) calculations for ionized helium analogous to the "exact"
classical path calculations for hydrogen [62, 63]. (This is not to say that
such calculations account properly for initial correlations.)

Keeping the above exceptions in mind, we now proceed with the discus-
sion of various estimates for the average interaction energy between all
perturbers and the radiator which, as mentioned above, has to arise from
monopole-monopole interactions (for isotropie perturber distributions).
The relevant part of the perturbation Hamiltonian for a perturber of charge
Zp is accordingly

Uo = Zpe»[(l/rp) - ( 1 / r . ) ] , rp < r&,

UQ = 0, rp> r&, (371)

where ra and rp are the radial coordinates of radiating and perturbing elec-
trons measured from the nucleus. (The Coulomb interaction with the total

ionic charge represented by a point charge at the nucleus is usually included

in the perturber Hamiltonian.) Classically speaking, the second term in the

rv < r& expression contributes to measurable shifts through a screening of
the nuclear (or "core") charge acting upon the radiating electron, whereas

the first term accounts for the increase in the effective point charge acting

upon the perturber as it penetrates the wave function of the radiating elec-

tron. It is therefore tempting to consider this term as part of the perturber

Hamiltonian, which would leave

C/p = - Z P ( e 2 / r a ) (372)

as the effective Hamiltonian for the average interaction (for rp < ra and
Up = 0 otherwise). For electrons as perturbers, the difference between
average interaction energies for upper and lower states should thus be
positive, for ions as perturbers, negative. Also, to the extent that (un-
perturbed) product wave functions for the total (radiator plus perturber)
system can be used, it is immediately clear that for neutral radiators (plane
wave perturber states), the effects of ions and electrons cancel for macro-
scopically neutral plasmas. Slightly different deviations from plane wave

I I . 5 . CORRELATION EFFECTS 149

states near the radiator will cause a small net effect. However, calculations
[144] (for hydrogen) yield level shifts much smaller than Stark widths,
which makes observations of these shifts (also for other neutral atom lines)
extremely difficult, if not impossible. From now on, we therefore consider
ion lines only, for which the zero-order perturber wave functions are the
(continuum) hydrogen wave functions for nuclear charge Z — 1 with
Z = 2 for singly charged ions, etc. In the classical path picture, electrons
are therefore on "attractive" hyperbolas, perturbing ions on "repulsive"
hyperbolas, and the electron effect is favored already in the unperturbed
product wave function approximation.

It would be less than candid not to mention that the separation of the
Hamiltonian adopted above is rather ad hoc, if not actually incorrect in
many cases. For example, if the impact approximation were not invalidated
by sharp resonances, then in quantum-mechanical and classical path calcu-
lations, certainly the entire interaction Hamiltonian of Eq. (371) must be
used, which would obviously even change the sign of our effect. However,
should a quasi-static approximation be more appropriate because of reso-
nances, then the situation probably would be more closely described by
such a separated Hamiltonian, since there would be enough time for the
perturber to actually take up the energy corresponding to the l/rp term
in Eq. (371). Even in this case, we must not overlook the fact that the
quasi-static nearest-neighbor approximation for, e.g., ionized helium lines is
expressible in terms of doubly excited states of neutral helium, and that the
corresponding satellite lines tend to be at longer wavelengths than the
parent line of ionized helium. This difficulty can perhaps be overcome by
invoking the Stark mixing of these doubly excited states by fields of other
plasma particles, which would effectively break down the usual selection
rules. If true, this would also introduce some additional density dependence
into the plasma polarization shift, which may, however, be very weak. After
these cautionary remarks, we now return to some simple estimates of this
elusive, but potentially very important effect. (Remember that wavelengths
of lines from multiply ionized one-electron systems could serve in the deter-
mination of fundamental constants.)

Almost all of the previous estimates [7, p. 95; 130; 145-147] of the plasma
polarization shift were essentially based on the unperturbed product wave
function approximation, so that on summing over all perturbers, a matrix
element of Up is

(i | C/p I i)av = - <i | (e/r.) Γ±η* p{r) dr | <>, (373)

with p(r) being the average (quantum-mechanical or classical) perturber

150 I I . THEORY

charge density. Except in the work of Tsuji and Narumi [145], instead of

really evaluating the expectation value according to Eq. (373), the per-

turbing charge was taken to be concentrated at the nucleus and a hydro-

genic effective charge approximation employed to estimate the level shifts,

i.e.,

(i | Up I i>av « - ( 2 Z # H M 2 ) ΔΖ, (374)

with

AZi = (4τ/β) [ V2 p(r) dr. (375)

Here rt is a suitable cutoff radius given approximately by

ri = (η*/Ζ)α0, (376)

which takes care of the rp < ra condition in Eq. (371). [In [145], the full
interaction given by Eq. (371) was employed, but evidently a sign error

caused the shift still to be to the blue and close to the observed [147] mag-

nitude.] Accounting for Coulomb interactions between perturbers and

radiators, classical equilibrium statistics yields for the charge density

p(r) = e Σ ZPNP e x p [ - ZP(Z - 1 ) ^ 7 ] , (377)

V

where the Np are the (usual) mean perturber densities, averaged over the
whole volume. In the earliest estimates [7, p. 95; 130], the Boltzmann

factors were assumed to be near unity, so that we may use the expansion

P(r) - - e Σ iZ*2(Z - l)e*/rkTlNp + · · ·. (378)

P

(The zero-order term vanishes because of macroscopic neutrality.) With

Eqs. (374)-(376), this yields for the effective interaction

(i I Up | t V - 8TT[(Z - l)/Z]n.*JZH Σ (ZP*En/kT)ao*Np , (379)

V

with EK = e2/2a0.
In another estimate [146] of the plasma polarization shift, an attempt

was made to allow for cases where the perturbing electrons must be de-

scribed quantum-mechanically by considering them to be in highly excited

but low angular momentum states of the preceding atom or ion in the

ionization sequence, e.g., in doubly excited states of neutral helium. (The

ions were neglected, because Coulomb repulsion should then be very

strong.) From approximate wave functions [34], the average charge den-
sity wTell within the Bohr orbit of such an additional nl electron can be

estimated to

Pm(r) « -(β/2ττ2)[(Ζ - l)/2n2rao)]3/2, (380)

I I . 5 . CORRELATION EFFECTS 151

while the relative probability of having an additional electron is given by
the Saha equation

Pni = 8τ*/2(ΕΗβΤ)™α03Ν exp[(Z - lyEn/rfikT], (381)

(Z — l)2Eji/n2 being an estimate for its binding energy. At this point, we
deviate some from the work of Griem [146] by considering s states (I = 0)
only, but keep the approximation of small binding energy. In this way, the
perturber charge density becomes

P(r) « Σ Pno Pno(r) « - e [ a o ( Z - l)EH/rkTJ'W, (382)

n-l

and with Eqs. (374)-(376), the average interaction energy becomes

(i I Up | z)av « ( 1 6 T T / 3 ) Z [ ( Z - l ) / Z ] 3 / 2 n ^ H ( ^ H A T ) 3 / 2 a o W , (383)

which is a factor of about 2.5 larger than the original estimate [146].
Beginning the sum, e.g., at n = 2 would instead lead to a reduction by a
factor of about 2, which should be compensated for at least partially by
including I = 1. In cases involving highly excited radiator states, the sum
should probably begin with still larger n values, but the accompanying
increase in relevant I values must not be overlooked, so that Eq. (383) with
the factor 16x/3 replaced by about 6 may be of more general validity for
estimates of the electron effect than the present derivation might suggest.

Such suitably reduced estimates are typically almost an order of magni-
tude below those from Eq. (379), but neither is valid when the exponential
factors deviate much from unity. Simply reinserting such factors into the
second model would not be sufficient, because a free electron recombining
into states corresponding to large exponentials would have to give sub-
stantial energy to the radiating electron, raising it to a higher level and
thus increasing the screening effect. This of course physically corresponds
to a resonance in the scattering, a complicated process not likely to be
described at all quantitatively by our simple model. It is much easier to
account for strong interactions (Coulomb versus thermal energies) by a
slight extension [147] of the semiclassical model. Neglecting also the ion
effect, the perturber charge density is taken as

p(r) = -eN exp[(Z - l)e2/rkT^ (384a)

or, when the following is smaller in magnitude,

p(r) = -eN explx/kTJ (384b)

This x might be close to the ionization energy of the preceding atom or
ion in the ionization sequence, although there is the question (see Burgess

152 I I . THEORY

and Peacock [223] below) how the perturber disposes of its large kinetic
energy (by exciting the radiator?), and the corresponding limit on p{r)
is of course a quantum-mechanical effect. Assuming r « \n?a§/Z for Eq.
(384a), the corresponding average shifts are

(i | U» I i>.v « (8TT/3) (tti4/Z2)EHa0*N exp[4Z(Z - l)EH/m2kT^ (385a)

(i | t/p | Oav - (8ΤΓ/3) (η*/ΖηΕκαο*Ν exp[x//cT], (385b)

and it should be kept in mind that the smaller of these options should be
more correct. In practice, this restricts Eq. (385b), at best, to resonance
lines, e.g., of ionized helium or of three-times ionized beryllium, for both of
which corresponding shifts have been reported [147, 148], although there
is also some conflicting evidence [149]. (See also Section III.3.)

The one-electron sequence is particularly interesting in the present con-
text, because its usual Stark broadening is almost entirely symmetrical.
However, since all the observations were of optically thick emission lines,
the profiles were dominated by the shape of the absorption coefficient. The
latter may differ [150a] from that of the emission coefficient, which is
normally considered. For example, there may be absorption mainly on the
red wings due to doubly excited states of neutral helium, but little emission
from these in the absence of excitation equilibrium. (A suggestion [150b]
that because of the different symmetry properties of upper and lower states
for two- (and more-) like-ion ''molecules/' there may be a linear Stark
effect from dipole-monopole interactions only in the emission coefficient
line shape, while ion effects on the absorption line shape depend mostly
on dipole-dipole interactions, is not tenable [150c], basically because the
stationary states of the "molecule" correspond to definite excitation of one
of the ions.)

Short of an attempt at a general theory for the average interaction, which
is certainly lacking, it remains to give a more quantum-mechanical esti-
mate for the case of strong Coulomb interactions. Continuum functions for
the one-electron problem yield for the perturbing (electron) charge density
at the nucleus [151a]

rm 2π(Ζ - peVfo
1 — exp[—2T(Z — l)e2/nvj

For typical electron velocities, the exponential can be neglected, and as-

suming constant charge density over the volume occupied by the radiating

„,,,ο. ψ:«-„* (g)\electron, we obtain w <387)

I I . 5 . CORRELATION EFFECTS 153

after performing the Maxwell average. This is smaller than the correspond-
ing semiclassical estimate [Eq. (385b)] by a factor exp(x/kT) (wkT/
4Εκ)ι/2/2π(Ζ — 1), i.e., by a factor of about 30 for the experimental [147-
149] conditions, although Eq. (387) should be close to an upper limit for
the average interaction within the product wave function approximation.
The latter, however, may be a very poor approximation, at least for reso-
nance lines, and was of course not at all adhered to in deriving Eq. (385).
We close by emphasizing that almost everything said in this section is
rather preliminary and by referring to a quantum-mechanical calculation
[151b] of the shift based on the relaxation theory and the full interaction,
i.e., using Eq. (371) rather than Eq. (372). (The shift obtained in this
calculation is, of course, to the red.)

II.5c. Plasma Satellites

After discussing the possible effect of (equilibrium) radiator-perturber
correlations on the positions of spectral lines, we now return to phenomena
connected with perturber-perturber correlations, but in contrast to Section
II.5a, mostly nonequilibrium ones, in particular those that are more
naturally described in terms of plasma wave fields than in terms of more or
less stochastic fields from individual particles. In other words, up to this
point, perturbing fields had been assumed to vary with time in a rather
random fashion, while this section begins with a consideration of the effects
on line shapes from harmonically varying fields, the opposite extreme of
time variation. After that, we take up the question of how to relate these
fields to other plasma properties in the framework of plasma kinetic theory,
as already suggested in Section II.5a.

The simplest radiating system in the present context would have three
levels (ignoring magnetic sublevels), say, the ground state "0", the upper
level for the radiative transition " 1 " , and one perturbing level " 2 " . Both
"0" and " 2 " must be connected with " 1 " by dipole transitions, but " 1 "
and " 2 " are assumed to be spaced so closely that " 2 " entirely dominates
the perturbation of " 1 " (and so closely that while transitions to " 0 " pro-
duce "optical" photons, those from transitions between " 1 " and " 2 " are,
e.g., in the far infrared).

The effects of a sinusoidally varying macroscopic electric field on the
optical spectrum of such a system have been discussed in great detail by
Autler and Townes [152] for arbitrary frequency and amplitude of the
perturbing field. These authors solved the ensuing time-dependent Schrö-
dinger equation numerically, using a method of continued fractions, and
gave analytic results for some limiting cases. Their work thus constitutes a
generalization of the time-independent but arbitrary field amplitude Stark

154 I I . THEORY

effect discussed at the end of Section II.2b. Autler and Townes and more
recently Cooper and Ringler [153a] also verified this theory of time-
dependent Stark effect experimentally.

The plasma problem differs in one essential respect (besides the presence
of more relevant atomic levels in many cases) from the problem considered
by Autler and Townes, who were interested mostly in rf molecular spectros-
copy. Rather than being subjected to a single sinusoidal field, the radiator
is under the influence of an entire mode spectrum of fields, say,

F ( 0 = Σ F«Q cos(Oi - a ) , (388)

α,Ω

where the a's are random phases. (The difficult question of how to super-
impose field effects from collective degrees of freedom of the plasma with
particle-produced effects will be touched upon later in this section.) The
Schrödinger equation for the evolution operator of the radiator is accord-
ingly

ihû(t, 0) = —expl-Ht) era · Σ Fai2 cos(Qt — a) expi - - f f i j w ( i , 0 ) ,

(389)

assuming the fields to be homogeneous over the dimensions of the radiator,
i.e., making the dipole approximation. This imposes even less of a limitation
than for the single particle (near-field) perturbations, because wavelengths
of plasma waves always exceed the Debye length which, in turn, is almost
always much larger than the range of radiator wave functions.

An approximate solution of Eq. (389) can be obtained by the usual
iteration procedure of time-dependent perturbation theory, beginning, e.g.,
with (i' | w(0, 0) \ i) = ôi>i on the right-hand side. In second order this
procedure, with cos x = \{eix + e~ix) and averaging over all modes, which
are assumed to be isotropie and to have random phases, leads to

(i | u«, 0) | i) = 1 + \{£j Σ I (i'\ r |*> I* Σ

exp[—i(ù>i>i — Ω)£] — 1 exp[—ί(ω»'< + Ω)ί] — 1

(ω,ν,. - Ω)^ + (ω^. + Ω)2

it it ì <39°)

+ r + —T-sr+···'
Cdi'i — Il ÙJi'i -f- il)

since terms such as ^ Ω βχρ[ί(Ω' — Ω)ί] may be replaced by 5ΩΏ for, say,

I I . 5 . CORRELATION EFFECTS 155

Ω/ > 1. (This excludes, e.g., portions of the line profile well beyond the
plasma resonances. These regions are especially important for hydrogen or
ionized helium lines, which in any case are not properly described by the
nondegenerate perturbation theory used here, but should rather be treated
as discussed by Blochinzew [133a].) Three types of terms can be dis-
tinguished in Eq. (390), namely, constant terms corresponding to the un-
perturbed allowed line, oscillatory terms involving sums and differences of
atomic and plasma frequencies, and secular terms. The latter give a
quadratic Stark shift Acoq of level i corresponding to the time-averaged
square of the field, summed over all frequencies Ω but with the usual energy
denominators of the time-independent theory replaced by the mean of
[fi(o)i>i db Ω)]-1. In terms of spectral densities UQ of the electric field
(J i/o dil = è Σ Pia/8v), this shift therefore is

(As discussed in Section II.2c, deviations from hydrogen matrix elements
may approximately be accounted for by using effective quantum numbers
and by inserting Bates and Damgaard [42, 43] correction factors. Remem-
ber also that U>i stands for the larger of U* and U .) In fourth order, a cor-
responding shift would appear also for levels i' ^ i, e.g., in the three level
system mentioned above, the separation of the two upper levels increases
by an amount as given by the second version of Eq. (391) but with the
factor li>i/{2li + 1) replaced by 4 ^ - / ( 4 ^ - — 1). We should also mention
that, strictly speaking, quadratic Stark effects do depend on magnetic
quantum numbers. In addition to the average shifts estimated here, there
will therefore be some splitting or, in practice, broadening of allowed and
forbidden lines due to these terms in anisotropie cases.

The oscillatory terms are of greatest interest here. If perturbations of the
lower state of the line are negligible, their contribution to the line shape is

• Ilia (Ω = ω,ν,· - Δα>) + Ua (Ω = Δω - «<»<)]

" 25UΓ+Γ71 \(m-eΤZ ΓΑ-ωΤ/ tΣ·, Wn,2 - lh)L· · ·1 (392)

156 I I . THEORY

with Δω = ω — ω»·/ . [This result can be obtained in a formal way from
Eqs. (6), (87), (88), and (390), using UQ = h Σ (Fla/tor) 6(Q - Ω').]
In other words, as originally predicted by Baranger and Mozer [131],
"plasma satellites" appear, displaced by plus or minus Ω from so-called
forbidden components which, in the limit of zero fields, would indeed be at
the frequencies ω»ν or Δω = ω»·',·. [Note that if satellite positions are meas-
ured relative to the actual position of the allowed line, the latter ought to
be corrected for the quadratic Stark shift [154] of the upper level of the
allowed line according to Eq. (391) or generalizations thereof, before calcu-
lating the zero field position of the often not observable forbidden com-
ponent.] For simplicity, Baranger and Mozer [131] had assumed UQ to
be a δ function, in which case Eq. (392) yields for the relative intensities
of the two satellite lines associated with level i'

S™ = f L(«) dco = - ^ - ( - ^ M f - ^ '■*- Uadü

~ 8 2U + 1 \meZj I2-. ^ò)
(ω<>< ± a)* {* )av '

where (F2)&v is the (time, etc.) average of the square of the oscillating
electric field. (Note that S^M + >S?M reduces for Ω = 0 to the usual ex-
pression for the intensity of a weak forbidden component.) A little con-
sideration shows that the (three) constant terms in Eq. (390) give a main
line near œif whose integrated intensity is below unity just by the amounts
of SlM and S?M, so that the latter are best interpreted relative to the total
intensity of allowed and satellite lines. [The profile of the allowed line, of
course, cannot be calculated from Eq. (390), which ignores single-particle
fields and higher-order terms.]

The (integrated) satellite intensities can be expected to obey Eq. (393)
only as long as the S^M remain reasonably small. Otherwise, Eqs. (390),
etc., would need to be extended to higher order. Since interference terms
between various modes and frequencies then appear, no general theory
seems practical here. In a first attempt [155] to estimate such nonlinear ef-
fects, saturation was simulated by introducing a phenomenological damping
term in the two-state equivalent of Eq. (389), and an approximate solution
was obtained that showed the "near" satellite (for ω»'» > 0 at Δω = ω»^
— Ω) to be much weaker than expected from Eq. (393) for the experimental
conditions. In a more systematic treatment [156], the iterative solution
(for two interacting states) of Eq. (389) in a many mode but single fre-

I I . 5 . CORRELATION EFFECTS 157

quency field was carried to fourth order, yielding at Δω « ω»·'< + Ω

St - S™ + s ( O · [^ - 1 - ( ^ ä ) ' ] + · ■ · (S«)

and a t Δω = co»'» — Ω

S_ = S™ - 2(S*M)*LΓΩ^ + 1 +\ωf»-'»^+i -ΩY/ J] + . . . . (395)

(In the paper of Kunze et al. [156], the third term in the square brackets
was expanded incorrectly.) Clearly, the near satellite is much more affected
by higher order terms and may be reduced to well below the Baranger and
Mozer predictions, even though the correction for the far (weaker) satel-
lite remains small.

Besides such higher order effects on these originally predicted satellite
lines, one naturally also expects satellites corresponding to harmonics of
the plasma frequencies, i.e., at ΔΩ « ±2Ω, etc., still ignoring any fre-
quency spread in the spectrum of plasma waves. A fourth order solution of
Eq. (389), again for two interacting states only, gives for their relative
intensities

&± = ! ^ Μ θ 5 Μ [ ( ω ^ ± Ω)/Ω]2 + - - ·, (396)

always assuming ω»'» to be positive. We also note that the (relative) shift
of the allowed line in the two-state case is

W«."< « -2(S*MS*M)1/2 (397)

from Eqs. (391) and (393) and assuming Ω2 <3C (ω»·'»·)2. As mentioned above,
such shifts or those from Eq. (391) and its generalizations ought to be
accounted for in determinations of Ω from the position of the satellites
relative to the actual allowed line center rather than to the unperturbed
line center, but have usually been ignored in the early experiments. A cer-
tain (a posteriori) justification for this can be given in terms of the near
Baranger and Mozer satellite at Δω « ω*'» — Ω and especially the stronger
of the fourth order satellites at Δω « 2Ω, which, if not resolved, may mask
the above shift.

It has been suggested [157] that some of the observed [156] plasma satel-
lites have been fourth rather than second-order effects. Because of the
strong dependence of the relative intensities on the ratio ω»'»/Ω, this is not
impossible theoretically, but a much more plausible explanation of the
single satellite feature reported by Kunze et al. [156] is in terms of unre-
solved second-order (Baranger and Mozer) satellites. To be observable,

158 II. THEORY

the strongest fourth-order satellite should have S2+ « 0 . 1 . Figure 14 shows
schematically how, under this condition and for a variety of aosumed
ω»',·/Ω values, the entire satellite and allowed line spectrum should look
like according to Eqs. (394)-(396), with quadratic Stark effect included
and assuming some common broadening of the various features. Clearly,
only for Ω/ω»·',· > 0.75 would a single satellite feature be consistent with
Burgess' suggestion. However, such high plasma wave frequency appears
very unlikely, not to mention the high field amplitudes required for S2+ ~
0.1 in this case and the much too large separation from the allowed line.
The case Ω/ω»·'» = 0.25 probably corresponds much more closely to the ex-
perimental conditions, in particular if one keeps in mind that the presence
of lower frequency oscillations would lead to an enhancement in the region
between the two second order satellites. For such low frequencies, the ex-
pansions for S+ or S- converge rather poorly even at moderate field
strengths, but fortunately the then more relevant sum of S+ and S- has a
well-behaved expansion.

In the case of several interacting states i', the situation is still more
complicated, not only in regard to shifts but especially also as far as higher
order satellites are concerned. For example, for the 2P-5D lines of neutral
helium, one expects, in addition to the Baranger and Mozer satellites at

. J1 I 1\ ! Ω-075ωϋ·\/

! Ω«0.50ωίΓ

.s- ;

LÌ \ ^ r \ s*

j Ω=α25ω8·
i i' " ω

FIG. 14. Schematic profiles of a spectral line with second ÇS+ and SJ) and fourth-order
(S2+ and S2-) plasma satellites for various values of the ratio of ''plasma" frequency Ω
and atomic level splitting ω,ν. The perturbing field amplitude is always assumed to give
S2+/A = 0.1 for the ratio of the "near" fourth order satellite and the allowed line (A).
Quadratic Stark shifts are included.

I I . 5 . CORRELATION EFFECTS 159

±Ω from 5F and the higher order satellites at ±2Ω from 5D, also satellites
at ±2Ω from 5G. (These would almost coincide with Baranger and Mozer
satellites from anharmonicities in the plasma field, which should then also
appear with 4D as upper state.) Other higher order satellites at ±3Ω would
again be associated with 5F, etc. More interesting problems are questions
of polarization [153] in nonisotropic situations and the various complica-
tions due to magnetic fields [158, 159] (see also Section IV.3). Since these
aspects require a very specialized (and numerical) treatment [154, 158],
we shall now return to the Baranger and Mozer satellites and discuss their
connection with more general theories of line broadening. However, it
should be mentioned that the numerical solutions do not allow for the many-
mode and -frequency nature of the plasma problem, which makes them
somewhat unrealistic when fourth or higher order terms are important.

From Eqs. (282) and (358), there follows for the normalized line shape
in the vicinity of a "forbidden level'' %' that is well separated from the
"allowed level" i by expansion in terms of £(ω)/Δω

L(co) « [-1/ττ/ΚΔω)2]Ιπι£(ω)

= [e2/37r(ÄAco)2]| ( i ' l r l i ) I2

• Re / Θχρ[ί(Δω - ω,ν<)ί] {F(0 · F(0) }Λν dt
•'ο

= (4ΤΓ/3) (e/Ä Δω)2 | {%' \τ\%) |2[£/Ω (Ω = αν, - Δω)

+ I/o (Ω = Δω - ax'»·)]. (398)

To arrive at the last version of L(œ), which naturally agrees with Eq.
(392), we again used the Wiener-Kintchine theorem, namely

Γ cos(dzöi) {F(0 - F(0) }av dt = (2ΤΓ)2[/Ω . (399)

Before going on with the general discussion, it is interesting to note that in
the case of a "white" spectrum ( Un = const), we simply recover a Lorentz
profile centered at the allowed line whose width is given by π(Δω)2 L(œ).
In plasmas with such spectra reaching well beyond the line width w, we can
therefore determine UQ from the measured width, say, of a hydrogen line
[160], provided of course that second-order perturbation theory is valid for
all essential K values. If the spectrum of plasma waves, on the other hand,
does not extend to the observed width, this clearly indicates that the
approximations made here are invalid. Fortunately, then the quasi-static
approximation can be employed, in which there is no trouble with per-
turbation theory. Such an approach was, e.g., used to explain the profiles

160 I I . THEORY

of ionized helium lines in an unstable plasma [132], and further measure-
ments of this effect will be discussed in Section III. 10.

The first version of Eq. (398) makes contact with plasma kinetic theory,
because the autocorrelation function of the fields may of course be replaced
by the more widely calculated (spatially Fourier analyzed) spectral
density of density fluctuations, using

exp/(iüt) {F(0 . F(0) }av dt = 4e2 //■^raax S(K, Ω) dK (400)

in the notation of Section II.5a and assuming isotropy. However, care must
be exercised in taking over S(K, Ω) evaluated [136] for analysis of light
scattering from equilibrium or nonequilibrium plasmas, where one is clearly
only interested in electron charge density fluctuations. To the extent that
second order perturbation theory remains valid, ion charge density fluc-
tuations must now be included as well, although for them the value of
i^max « Pmin m Eq. (400) is generally much smaller than would be appro-
priate for the electron contribution. (Remember that pmin corresponds to
the minimum impact parameter in the impact theory.) For ab initio
calculations, it is thus necessary to separate electron and ion contributions
to S(K, Ω) and to perform the K integrations separately. For values
K > iCmax, the quasi-static approximation might be applicable for the
ions. (Dufty [118a, b ] has given a more formal discussion of the problems
associated with this transition regime.)

We finally note that S(K, Ω) is related to the (longitudinal) dielectric
constant of an equilibrium plasma through [161] (fluctuation-dissipation
theorem)

S(K, Ω) = - (K*kT/2Te2Sl) Im <rl(K, Ω) (401)

[in the classical limit and allowing for the absence of a factor 2π in our
definition of S(K, Ω)], so that the asymptotic line shape becomes with
Eqs. (398) and (400) in case of equilibrium plasmas,

^--hfajw vi* kT
Δω — osi'i |

X I '"" K2 dK Im «-»(£, ± Δω T «;<,). (402)

Ό

This result [or rather, the corresponding result for <£(ω) ] was first obtained
by Zaidi [116], who used diagrammatic techniques, and also by Klein

I I . 5 . CORRELATION EFFECTS 161

[141] with a Green's function method. (The latter reference has a different
normalization factor.) For equilibrium plasmas and for | Δω — ω<<< I much
larger than the ion plasma frequency, the dielectric constant is [138, 140]

e(X, Ω) = 1 + ( f ) { [ l - x e x p ( - / ) / % x p ( | 2 ) ^ ]

+îK0l/2exp(:?2)} (403)

with x and ΚΌ denned by

x2 = mtf/K2kT, (404)

ΚΌ* = lireW/kT, (405)

and we thus recover the formulas of Section II.5a. For nonequilibrium but
stable plasmas with electron velocity distributions/(v), on the other hand,
e(K, ω) may be calculated (for | Δω — αν» | >>> cop») from [136]

where η is the positive infinitesimal, but Eq. (402) per se is invalid. As an
example for a calculation based on a suitable generalization of Eq. (402),
Chappell et al. [139] considered/(v) to be the sum of two Maxwellians and
indeed obtained considerable enhancement of the intensity near
| Δω — covi | « ωρ . These authors actually used an expression corre-
sponding to

- Ι ι η ε - Η Κ , Ω ) = [Im €(K, Ω ) ] / | €(Κ, Ω) |2 (407)

with Im e(K, Ω) obtained from Eq. (406) by contour integration, namely

Im e(K, Ω) = -τ(ωρ/Κ)2 ί dv K · [θ/(ν)/<9ν] δ(Ω - K · v)

= T(mU/kT)(œp/Ky f dvK- K / ( v ) δ(Ω - K · v). (408)

The second version by itself applies only to Maxwellian plasmas. However,
after substitution into Eq. (402), the temperature factors cancel and one
obtains (near or above the electron plasma frequency) the proper non-

162 I I . THEORY

equilibrium but stable plasma result for the line shape. This may be verified,
e.g., by calculating

Re / βχρ(ίΩί) {F(t) · F(0)}avdi
•'o

directly, using the test particle method [137].
For more practical aspects of the effects of suprathermal fluctuations on

line profiles for both degenerate (hydrogen, etc.) and nondegenerate (neu-
tral helium) situations, the reader is referred to Section IV.3, and for a
review of experiments in this area, to Section III. 10. However, one remark
is still in order for the present section, namely that the results for the
relative satellite intensities are valid only if the relative populations of
levels i' and i are according to statistical weight ratios. This requirement is
obvious from the original derivation [131] in terms of two- and one-
quantum processes beginning from these levels but hidden in the present
treatment behind the implicit assumptions for the density matrix p (see
Section II.3), which are not generally consistent with the type of per-
turbation considered here. To assess whether or not statistical populations
might prevail, one should compare collisional rates i—>i', e.g., according
to — 2 Re φ from Eq. (175), with radiative and competing collisional rates.

II.6. MAGNETIC FIELD EFFECTS

Many laboratory and astrophysical plasmas not prone to producing
satellite features contain magnetic fields which may influence spectral line
profiles in a number of ways [162]. Since magnetic fields can often be
assumed to be homogeneous and time-independent, there is first of all the
quasi-static Zeeman effect superimposed, say, on the quasi-static Stark
effect from ion-produced electric microfields on hydrogen (or hydrogenic
ion) lines. In the Paschen-Back limit, which is usually relevant here, the
Lorentz triplet corresponding to the normal Zeeman effect on the upper
state of these lines has intervals of

Δωζ = (rii - l)eB/2mc. (409)

With Eq. (16) for the Stark width, therefore follows for the magnetic field

Bc at which Zeeman effects and Stark broadening are about equally
important,

Bc » 3.3 · 10° ^ P - r eN*» « 1.3 · 10° ^ ^ F0, (410)
ΔΔ

I I . 6 . MAGNETIC FIELD EFFECTS 163

if we subtract upper and lower state Zeeman effects and use the Holtsmark

field strength F0 defined by Eq. (36). (For lines with strong unshifted
components, e.g., Ha , this is actually an overestimate.) Since we are
interested only in combined Zeeman and Stark effects if the latter are more

important than thermal Doppler broadening, comparison with Eq. (18)

shows that these critical magnetic fields must also fulfill the inequality

\2mrEn/ rii2n/2 a02

where En = h2/2mao2 is again the ionization energy of hydrogen.
For the Balmer lines of hydrogen, the required fields are < 10 kG, for the

Paschen lines < 4 kG, etc., at kT ~ 1 eV. (In the case of ionized helium
and kT ~ 4 eV, the fields must be larger by a factor of 4 for analogous
lines). Since Eq. (411) is primarily an estimate for the field strength at
which Zeeman effects begin to exceed thermal Doppler broadening, it can
be used in that sense for isolated lines as well. However, one should remember
that for them Stark widths are smaller by, say, an order of magnitude, so
that densities from Eq. (410) must be multiplied by a similar factor to
estimate the minimum electron density at which combined Stark and
Zeeman effects can be important. Also, while Eq. (410) indicates, for fixed
density, only a linear increase of the critical field with principal quantum
number, an almost cubic increase would be expected for isolated lines
(until Stark widths exceed inherent level splittings). As first observed
experimentally [163], the effect under discussion here is thus, for a given
plasma, most pronounced for lines with relatively low excitation energy,
other things being equal. Nevertheless, the required magnetic fields tend to
be somewhat higher than those encountered in most laboratory devices
(except in experiments specifically constructed for controlled thermo-
nuclear fusion research), not to mention those prevailing in typical as-
tronomical objects.

The above discussion of combined Stark and Zeeman effects on hydrogen
lines can be substantiated by comparison of detailed calculations, in which
a Zeeman term is added to the linear Stark term in Eq. (101) and where
eigenfunctions of the radiator Hamiltonian plus (ion-produced) electric
field and magnetic field perturbation Hamiltonians are used as a basis, with
calculations of Eq. (101) in its original form [53]. The detailed calculations
must of course be performed separately for, say, parallel and perpendicular
polarizations, and the basis functions also depend on the angle between the
magnetic field and the instantaneous ion-produced electric field. Even if the
simplest approximation is used for the electron broadening operator,

164 I I . THEORY

namely the leading term in Eq. (110), the calculations are very involved
and have been completed only for the hydrogen La , Ι^ , and H« lines [164].
The most striking features of these results are the modifications of the
central portions of the line profiles corresponding to the appearance of the
components of a normal Lorentz triplet appropriate for the polarization and
direction of observation; this is shown schematically in Fig. 15. Depending
on relative magnitudes of magnetic and Holtsmark field strengths, these
characteristic features are more or less pronounced and can thus be used
for either magnetic field or electron density determinations [165]. Near the
line center, they are already noticeable for fields about an order of mag-
nitude weaker than might be suggested by Eq. (410). This can be seen
from Fig. 16, where again for Ha the calculated ratio of central minimum to
one of the intensity peaks (parallel observation) is given as function of
electron density for a variety of magnetic field strengths. These calculations
[166c] are for an electron temperature of 2 · 104 K, but are very insensitive
to this temperature.

Calculations analogous to those for hydrogen can naturally also be made,
e.g., for neutral helium lines ("isolated lines"). In this case, combined
Zeeman and quadratic quasi-static Stark effects have to be evaluated first
[166a, b ] , and the φ matrix in the corresponding representation can then
be calculated [77, 167a] as described in Section II.3cß for "overlapping
lines." At this point, the reader may wonder whether the assumption of
straight classical paths remains justified for the perturbing electrons in this
and the above cases. A safe answer (for equilibrium plasmas) is that no
difficulties should arise from this source as long as the gyro frequency is

(b)

F I G . 15. Schematic combined Stark and Zeeman profiles of the H a line: (a) observed
perpendicular to the direction of the magnetic field ; (b) observed parallel to the field or
perpendicular through a polarization filter to suppress the central component.

I I . 6 . MAGNETIC FIELD EFFECTS 165

H 1—i—i i ' i i

Io / Iz
0.5 h

0.1

10

FIG. 16. Calculated ratio h/Iz of central minimum intensity to one of the intensity
peaks of the Stark broadened Ha line emitted by a magnetized plasma, as a function of
the electron density and for a variety of magnetic field strengths (after Nguyen-Hoe and
Drawin [166c]). The conditions of observation are as for Fig. 15b and the temperature,
whose value is not critical, is assumed as 2·104 Κ.

well below the plasma frequency, i.e., as long as magnetic fields obey [167b]

B« (4twNmc2)1/2. (412)

This criterion is usually well fulfilled (if not, see Maschke and Voslamber
[168] for examples of detailed calculations) and is of course based on the
fact that collisions with impact parameters beyond the Debye radius are
normally negligible, and that for smaller impact parameters Eq. (412)
ensures very small curvatures of the classical paths. (For nonequilibrium
plasmas, the situation may be different, because long-range, cooperative
fields can then be very important. See Sections II.5c, III.10, and IV.3.)

One can also argue [167b] that magnetic field effects on the ion-field
distribution function should not be too important, at least in cases where
the Holtsmark distribution or the various distributions corrected for
equilibrium correlations as discussed in Section IL2a would constitute a
good approximation otherwise, and as long as Eq. (412) is fulfilled. (How-
ever, dynamical corrections to the Holtsmark theory as discussed in Section
II.4c could be affected in various ways [169].) Besides all the above

166 I I . THEORY

magnetic field effects and analogous effects on plasma satellites (see Section
IV.3), another effect must be considered [163, 170], namely the Stark
effect from the Lorentz electric field F L = (v X B)/c. Competition of this
effect with ordinary quasi-static Stark broadening would require Fj, > F0,
i.e., 4:TN/S < (vB/ec)3/2, which is not often attained when Stark broadening
is at all important. Two possible exceptions to this have been discussed
[170], namely, nearly thermonuclear plasmas with strong magnetic fields
and the tenuous plasmas in H II regions (see Section IV.5). Still, in the
former case, suprathermal field fluctuations (see Sections III. 10 and IV.3)
may well obscure the Lorentz effect, while the estimates of Galushkin [170]
for the radio-frequency lines from H II regions, etc., suffer from the
neglect of a near cancellation between upper and lower state Stark shifts
for these lines. (The error is of the order of the principal quantum number,
not to mention the error from the breakdown of the quasi-static approxi-
mation.)

For Maxwellian distributions of radiator velocities, the Lorentz effect
per se of course gives rise to (polarization- and angle-dependent) Gaussian
profiles or, in general, profiles that cannot easily be distinguished from
Doppler profiles.

CHAPTER III

Experiments

Although there may be only a few situations in spectral line broadening
from interactions between charged particles and radiating atoms or ions in
a plasma for which we have, at least in principle, no theoretical approach,
the complexity is often so great as to make experimental verifications of the
various theoretical approximations indispensable. This necessity for critical
experiments is also apparent from the large number, approaching ten, of
characteristic frequencies entering the general problem, which were dis-
cussed in Chapter I. In other words, the relevant parameter space has too
many dimensions to make reliable predictions of all phenomena a likely
prospect, especially in the transition zones between the validity domains of
the simpler and therefore more transparent approximations. Moreover, a
good deal of experimental scrutiny of theoretical error estimates is called
for in almost all cases, because such estimates cannot account for all
sources of error in the absence of a truly complete theory, not to mention
the very real possibility of numerical errors in the often very involved cal-
culations. In practice, also the experimental approach may consist of
successive approximations because of the need to define the relevant plasma
parameters. For this and other reasons, there can be no clear demarcation
between the present chapter and the following last chapter of this book.

167

168 I I I . EXPERIMENTS

III.1. GENERAL CONSIDERATIONS

The ideal experiment in our special field, as in many others, would have a
homogeneous parallel slab geometry, and the plasma would of course be
stationary. Also, the thickness I of the slab would be such as to compromise
between sufficient intensity and small distortion of the "optically thin"
profile by radiative transport phenomena. For example, if only spontaneous
emission and absorption are to be considered, the observed line shape
(normalized to some central intensity and not to unit area) for this geom-
etry is

J(«) = 7o{l ~ exp[-27r2(e2/mc)/iV/L(co)Z]}, (413)

if we can further neglect continuous absorption and other lines. Here Nf
is the number density of radiators in the lower state of the line, / its absorp-
tion oscillator strength, and Σι(ω) the usual, i.e., optically thin, line shape.
Only for small values of the exponent are Ι(ω) and L(co) proportional to
each other. Otherwise, a measurement of I (co) determines L(co) much more
indirectly and usually with considerable loss of accuracy. In cases of
inhomogeneous layers and for situations where the implicit assumption of
having equal shapes for emission and absorption coefficients is not war-
ranted, the analysis would be even more complicated and the conclusions
regarding L(co) accordingly still less reliable. (For an introduction to the
rather complicated subject of radiative transfer, the reader is referred to
special treatises mainly on the analogous astrophysical problem [171-173].)

Returning to the desirability of homogeneity and stationarity, we must
note that these are more or less mutually exclusive requirements. This is
readily understood, because, e.g., stationary density and temperature dis-
tributions in space would of necessity be solutions of appropriate diffusion
equations and therefore not resemble step functions. The corollary to this is
the fact that spatial step functions of density, etc., are only approximated
in some transient experiments, such as those involving almost ideally plane
shock waves. The practical trade-off is thus between spatial "unfolding"
of the measured profiles and the need for time resolution. In cylindrical
geometries, the former operation is achieved by the well-known Abel
inversion procedure [174-178]. This procedure requires profile measure-
ments along lines of sight which are chords at right angles to the axis with
variable distance from it. To obtain good spatial definition, apertures have
to be very small, resulting in small signals. These small signals and the
two-dimensional scan (with respect to wavelength and distance from the
axis) may mean durations of profile data runs of several hours, during which
all relevant macroscopic parameters must be kept constant.

U L I . GENERAL CONSIDERATIONS 169

A usually necessary corollary to this requirement of long-term stability
is of course the requirement of reproducibility in the case of transient light
sources, which must be checked by suitable monitors, as should be the
stability of stationary sources. Another practical problem in the transient
case concerns the achievement of both time and spectral resolution, which
sometimes demands unusual compromises in optical instrument design or
use, a subject far beyond the scope of this book. (See the literature [179-
184] for some particularly interesting solutions of this problem.) There
are essentially five approaches here, which involve taking "snapshots" of
the interesting part of the spectrum (photographically or using gated image
intensifiers), streaking a stigmatic spectrum photographically or with an
image converter, making multichannel photoelectric records, scanning the
spectrum "from shot to shot" photoelectrically, or scanning rapidly during
one operation of the transient source with Fabry-Perot or other rapidly
swept spectrometers. Reproducibility is absolutely essential only for the
"shot-to-shot" method, although it is desirable also for the other methods,
as plasma conditions can then be measured with greater ease and precision.
Also, one must not forget that homogeneity, e.g., of shock-heated plasmas
in directions perpendicular to the shock propagation, cannot be taken for
granted either.

On balance, the above genera) considerations show that there is no
obvious preference for using stationary or transient sources, other things
being equal. In practice, the choice is therefore often dictated by other
considerations, such as the relative ease of exciting neutral atom spectra in
stationary sources versus the difficulties encountered in using them for
measurements of ion line profiles because of the temperature and electron
density limitations of, e.g., stabilized arcs.

Before discussing specific experiments, some additional desirable char-
acteristics ought to be at least mentioned. Namely, the plasmas should lend
themselves to the application of sufficiently accurate "diagnostic" methods
[185, 186], and it should be possible to eliminate or to correct for con-
tributions from other line broadening mechanisms. Aside from Doppler
broadening, the most likely competition here is from neutral gas broadening,
either by way of Van der Waals and corresponding short-range interactions
or by way of resonance, i.e., excitation energy transfer, interactions. (Very
approximate estimates for these other pressure broadening effects can be
found, e.g., in a previous work [7].) While neutral gas broadening is
evidently statistically independent of Stark broadening so that the usual
deconvolution procedure is valid, more care is required in the rare case of
simultaneous ion-impact and Doppler broadening of ion lines. However, in
contrast to simultaneous neutral gas and Doppler broadening [187, 188]

170 I I I . EXPERIMENTS

(see also the literature [319, 320] below), it may not be necessary to also
describe the translational motions of the radiators quantum-mechanically,
because the velocity changes and therefore the Doppler effects are essen-
tially determined by Coulomb scattering, which is independent of the
internal radiator state. Finally, in the presence of strong magnetic fields,
the various effects discussed in Section II.6 must be kept in mind.

III.2. HYDROGEN LINES

Because of their relatively large widths and strengths and because of
their great importance both for applications (see Chapter IV) and theory,
hydrogen lines have been the subject of so many experimental studies that
an exhaustive discussion of them seems impossible. Moreover, there have
been several relatively recent reviews [6; 7, Chapter 15-3; 189] on these
and other Stark-broadened lines, so that essentially only measurements
published since 1965 need be considered here and below. In spite of a
repetition (with a number of improvements) of one [190a] of the two
Lyman-α line (L«) experiments, measurements [191a] and theory [63e,
113c] still disagree for this line, and there are no new results on other Lyman
lines (except for an internal consistency check between L« and Lß discussed
by Moo-Young et al [192]) ; so we can begin with the Balmer series. This
is not to say that no conclusions should be drawn, e.g., from the measured
[190a, 191a] red asymmetry of La (see Section III.9), but to suggest that in
these difficult measurements of optically thick lines in the vacuum ultra-
violet, systematic errors may have been overlooked. For example, were the
arcs burning in a low-pitched spiral rather than a completely straight
column, the effective electron density would be somewhat lower than as-
sumed and moreover would depend on the wavelength separation from the
line center. Furthermore, in experiments using argon as the carrier gas,
broadening by argon atoms may not be negligible, and then there is the
possibility of non-LTE radiative transfer effects [191c] or of gradient
effects near the pure argon (electrode) regions. This explanation is con-
sistent with the improved agreement found [191b] at lower hydrogen
concentrations. More conclusive, however, is probably the agreement with
theory and the other early experiment [190b] obtained in a high-tempera-
ture arc [19Id] in pure hydrogen.

III.2a. The Balmer ß Line

The most frequently measured Balmer line is H^ , which occupies a very
convenient region of the spectrum, has a rather characteristic (see Fig. 17)

I I I . 2 . HYDROGEN LINES 171

T 111111r

o ,—Λ—' ώ ' ώ ' ab—

Δλ LÄj

FIG. 17. Comparison of measured (Wiese et al. [198], N = 8.3-1016 cm"3, T = 1.34-
IO4 K) and calculated profiles of the Ήβ line. The impact theory curve corresponds to
that of Kepple and Griem [59a] (or Appendix AI.a), the relaxation theory curve was
calculated [121f] according to Smith et al. [52], and the asymptotic values correspond to
Section IV.4a with the increased Debye shielding.

and broad profile, and is far less sensitive to radiative transfer effects
than H«.

The first measurement of Hp to be discussed here was performed by
McLean and Ramsden [193a], who used a so-called T tube (electrically
driven "shock" tube) as a light source. The measurements were made after
the reflection of the luminous front from a reflector plate by scanning the
line profile on a shot-to-shot basis. This procedure at the same time gave a
temperature of 1.5-1.8 eV from the line-to-continuum ratio method in this
near-LTE (local thermal equilibrium) plasma. With an absolute calibra-
tion, the continuum intensity measurements also yielded a value ( ^ 2 · 1017
cm-3) for the electron density in their hydrogen plasma. However, more
reliance was placed on an independent determination of the electron density
from the interferometrically measured optical refractivity at two wave-
lengths, in order to eliminate the neutral atom contribution. Using the
calculated Stark profiles of Griem et al. [53b], the authors inferred a third
value for the electron density from their profile measurements. All three
determinations agreed, on the average, to within 6%, corresponding to 4%
accuracy in the H,* half-intensity width. (This conclusion is not changed
when the calculations of Kepple and Griem [59] are used instead and has
been substantiated by the work of Konjevic et al. [193b].)

Even better agreement for H^ at similar temperatures ( ~ 2 eV) but
somewhat lower densities, namely (1.3-8.5) · 1016 cm-3, was reported by
Hill and Gerardo [194], who used a critically damped pulsed-discharge
tube, observed end-on. They scanned line profiles during single shots with a

172 III. EXPERIMENTS

rapid-scan spectrometer [183] and employed two multiple-pass laser
interferometers for an independent determination of the electron density
from the optical refractivity. Besides half-widths, these authors also com-
pared quarter (Δλι/4) and three-quarter (Δλ3/4) intensity (relative to the
average of the two intensity maxima) widths and entire profiles (down to
about 10% of the intensity of the maxima). No significant deviations
occurred except near the central minimum, which was measured to be
much shallower than was predicted theoretically. This might have been
due, at least in part, to contributions from boundary layers of much lower
electron density near the end faces of the discharge tube, but may equally
well have been due to deficiencies in the various calculations [53, 59, 121]
(see below).

While the two preceding experiments were performed in almost pure
hydrogen plasmas and on microsecond time scales, two arc experiments
[195, 196] were done in argon plasmas having hydrogen admixtures of a
few percent. In one of these experiments [195], hydrogen was kept away
from the inhomogeneous electrode regions by suitable gas flows, and the
pulsed (on millisecond time scales) arc column was observed end-on, using
sufficiently small apertures to ensure homogeneity. The other experiment
[196] was truly steady state, having a smaller bore, and the plasma column
was observed side-on, which required an Abel inversion [176]. Besides
employing similar gas mixtures, the experiments had another essential
feature in common: the current was chosen to yield maximum electron
density for a given total pressure of 1 atm. For LTE plasmas, this maximum
electron density (almost exactly 2 · 1017 cm-3) can be calculated with great
precision and is thus seen to be attained at a temperature close to 1.5 eV.
(Note that these plasma conditions are very close to those of McLean and
Ramsden [193a].)

Morris and Krey [195] compared with theory [59b] (use of results by
Griem et al. [53b] would not have changed the conclusions significantly)
by assuming that there was no line contribution beyond 200 A from the
line center in order to determine the continuum level, subtracted this level
from their "line" signal, and then (area) normalized the corrected profile.
This was converted to the a scale (a = Δλ/Fo), using the calculated elec-
tron density plus or minus 10%, and could now be compared directly with
the theoretical profile. Agreement within these limits was obtained except
near the line center and on the points near the continuum, whose spectral
variation had been determined by measurements in pure argon. While the
latter discrepancy may not be real, that near the line center ( ^ 1 5 % ) is
difficult to explain experimentally in this case, suggesting that theory
indeed gives too much of a central dip. Since this dip arises from the low

III.2. HYDROGEN LINES 173

probability of small ion field strengths in the Holtsmark and similar
approximations, a most natural explanation would be to ascribe this
filling-in to a near-Gaussian distribution of collective fields, whose rms
thermal value, according to Eqs. (46) and (56), would amount to about
10% of the mean particle-produced field. (The latter field was accounted
for in the calculations, while any collective fields were not.) Other tentative
explanations involve inelastic electron collisions (see also the following
section), higher-order effects in the electron broadening [62c], or dynamical
corrections (see Section II.4c).

Shumaker and Popenoe [196] made their comparison with theory by
determining the least squares fitted theoretical profile, using the latter to
compute another value for the electron density. This density was con-
sistently about 10% below the value obtained from the LTE plasma com-
position calculations. (By half-width comparisons only, Morris and Krey
[195] found a 5-8% deficiency in the "H/s" electron density.) The con-
clusion, therefore, was that calculated line profiles [59a, b ] are too broad by
about 7% for these conditions. However, part of this disagreement may be
due to small deviations from LTE or to other uncertainties in the com-
position calculations, a point that could probably only be settled by an
independent measurement of the electron density. (That some argon arcs
show unexplained discrepancies, also in regard to measured transition
probabilities, by as much as 30% has been discussed in great detail by
Richter and his co-workers [197a].)

Another, more recent, measurement [198] of Ήβ and other Balmer line
profiles in an arc whose central section contained pure hydrogen has yielded
accurate profiles over a range of densities, N « (1.5-10) · 1016 cm-3, and
especially for H$ also over a large intensity range, down to about 1% of
peak intensity. The measurements were made side-on, and the plasma
conditions were analyzed with a minimal use of LTE assumptions. An H^
profile from this experiment is shown in Fig. 17 and can be seen to agree or
disagree with theory about as suggested by the earlier results [193-196],
at least in the region between the, say, 10% of peak intensity points covered
by all of these experiments. (The situation for smaller relative intensities
will be discussed below.) The dependence of agreement or disagreement on
electron density can be inferred from the fractional intensity widths plotted
in Fig. 18, the experimental values [198] being about as close to the cal-
culations of Kepple [59b] (see Appendix IH.a) as to those [121f] based
on work of Smith and others [52]. However, the remaining deviations are
somewhat larger than the degrees of agreement and accuracies estimated by
some of the other experimentors. To see whether or not there are real
differences in this regard between the various methods, Wiese et al. [198]

I I I . EXPERIMENTS

10 20 40 60 80 100

Δλ[Α]
FIG. 18. Comparison of measured (as for Fig. 17 but at varying density, i.e., radius)
and calculated i, \, and \ (of maximum intensity) widths of the H^ line with calculated
fractional intensity widths. The impact theory values [59a] correspond to Appendix III.a,
while the relaxation theory results are again those of Vidal et al. [121f].

collected relevant results of previous workers, together with their own,
in a table which is reproduced here (Table IV). The only systematic trend
corresponds to slightly larger widths in the pulsed experiments (in spite of
more pronounced central structure). A similar (to Wiese et al. [198])
hydrogen arc experiment [199] allows an extension of such comparisons up
to N « 1.7 · 1017 cm-3 and essentially confirms the above conclusions for
Hfl within the 10% intensity points.

Averaging over the results of the six experiments discussed here and the
earlier measurements discussed by Wiese et al. [198] (see Table IV), we
can therefore say that Hp profile calculations [59a, b ] for electron densities
of 1 · 1016 to 2 · 1017 cm-3 and temperatures of 1-2 eV are probably correct
within about 5%, excepting the central dip and asymmetries, and that true
profiles, more likely than not, are narrower than calculated profiles. This
accuracy (near the peaks and not far beyond the half-intensity points) is
somewhat better than expected from theoretical error estimates [59a] (see
Fig. 2) and could only be exceeded by calculations which, among other
things, treat the transformation from frequency scales to wavelength scales
more accurately, allow for the interference between different lines as dis-
cussed in the paragraph following Eq. (355), and include quadratic Stark

TABLE IV ÌS3

COMPARISON OF ELECTRON DENSITIES FROM Ήβ HALF-WIDTH MEASUREMENTS WITH ELECTRON DENSITIES FROM INDEPENDENT w
DIAGNOSTIC METHODS'1 owöGM

E

H

Plasma source Diagnostic method T [104 K] N [1016 cm"3] iVXdiagnosticViVXH,,)
Impact theory Relaxation theory Reference

Ttube Continuum 1.4 8.0 1.01 0.94 [130]
Pulsed discharge Interferometry 2.1 1.5-7.5 1.01-1.03 0.92-0.95 [200]
Ttube Interferometry 0.9-1.5 20-36 0.96-1.02 0.87-0.93 [193b]
Ttube (1) Interferometry 1.8-2.1 [193a]
(2) Continuum 20-27 (1) 1.01 0.95 [193a]
Pulsed arc Fowler-Milne 1.7 (2) 0.98 0.92 [195]
Wall stabilized arc Fowler-Milne 1.7 20 1.05-1.08 0.98-1.01 [196]
Wall stabilized arc Line intensity 1.2 20 1.05 [201]
Wall stabilized arc Continuum 1.0-1.3 6.8 1.12 1.06 [202]
Vortex arc Line intensity 1.2 3-7 0.96 [197b]
Wall stabilized arc Continuum 0.9-1.4 1.17 1.07 [198]
— 1.06 0.92-1.01
1.18
1.6-9.3 1.04-1.12

° Using impact theory calculations [59b] or relaxation theory calculations [121f].

176 I I I . EXPERIMENTS

effects in the ion broadening calculations. Also, the accuracy with which
such comparisons can be made of course deteriorates toward the line wings.

According to an earlier review [189], measured intensities of H^ about
300 A from the line center in an arc plasma were a factor of about 1.5 lower
than extrapolated theoretical profiles [53]. A more recent T-tube experi-
ment [203a] indicated a similar behavior near this wavelength. In both
cases, agreement with asymptotic formulas [59, 120] (see Section IV.4a)
is much better. [Note that the " J " term in the wing formula of Kepple and
Griem [59] should actually read Jexp( —7i).] Good agreement was
found closer to the line center in both cases, and in view of the difficulties
connected with continuum corrections, etc., it was not clear whether or not
the above deviations at intensities over two orders of magnitude below
peak intensities were indeed real. This question has been answered affirma-
tively by the more accurate measurements reported by Wiese et al. [198]
and it is certain that beyond the, say, 10% relative intensity points,
calculations [121f] based on the relaxation theory [52] [or Eq. (417)] are
in better agreement with experiment. (Some of this expected improvement
is due to an increase [135a] in the Debye shielding correction.) However,
Wiese et al. [198] and the work of Pavlov and Prasad [203a] still suggest
about ± 2 0 % agreement with the impact theory [53, 59] over an intensity
range of two decades, to be compared with the =t 10% agreement over one
decade (down from the peaks toward the wings) indicated by the combined
evidence from the experiments discussed earlier in this section, so that the
only substantial remaining disagreement between experiments and both
theories is in the region of the central minimum. As an aside, we may note
that a similar disagreement used to exist for the Stark effect of Hp in a sta-
tic field [204].

This disagreement near the line center is accentuated at lower densities
[203b] (N « 1015 cm-3) and, interestingly enough in view of the expected
ion-dynamical effects (see Section II.4c), does depend on the mass (mean
relative velocity) of the perturbing ions. With pure hydrogen in the 2-pinch
plasma, no dip was seen, while in the case of helium (with hydrogen as
impurity), there was a slight indication of its existence. However, an
interpretation of the results solely in terms of dynamical effects fails [203c],
because these should be accompanied by a steepening of the shoulders of
the profiles, which was not observed. Other possible causes for the dis-
crepancy as discussed above must be considered as well, and perhaps also
the possibility of small scale density inhomogeneities from residual plasma
or hydrodynamic instabilities. Such effects may be responsible [64b] for
some of the deviations found in arc experiments, in particular because they
could invalidate the Abel inversion procedure, which is extremely critical

I I I . 2 . HYDROGEN LINES 177

near the line center. Recently entire profiles of H^ and higher series members
have been measured over a wide range of intensities in another 2-discharge
[205a], and arc measurements [205b] of the central dip of Hß in various
gas mixtures have been interpreted in terms of ion dynamical effects.
These effects would then have to be much larger and have to scale differ-
ently with radiator mass than discussed in Section II.4c.

Two additional experiments [201, 206] of probably lower accuracy in
regard to electron density determinations have been interpreted mainly in
terms of profile asymmetries, and will therefore be discussed in Section
III.9. Since such asymmetries are usually ignored in the theory, all pre-
ceding comparisons have implied averages of measured "blue" and "red"
intensities (see Fig. 17), a procedure which should not introduce any
significant additional errors.

III.2b. Other Balmer Lines at High Electron Densities

Most measurements of the Ha line since 1965 were made using stabilized
arcs [198,199, 207-209], either with small admixtures of hydrogen to argon
or with pure hydrogen and allowing for radiative transfer effects. In few of
these experiments was the electron density measured in such an inde-
pendent manner and to such a high accuracy as was obtained in some of the
up experiments for this crucial parameter for "absolute" comparison with
theory. Relative comparisons based on using H/3 as an electron density
standard are thus probably just as significant. In any event, while devia-
tions from the original calculations [53a] reached about 20%, measured
widths being larger, remaining deviations [59a] appeared to be less than
or about 10%, which is comparable to combined experimental and theo-
retical errors. It has been pointed out [58, 209] that even better consistency
between Ha and H/j half-widths is obtained by using a somewhat larger
"strong collision term" than adopted here (or by Kepple and Griem [59]),
i.e., by using the equations of Griem et al. [53b] (GKS II method). The
theoretical significance of this observation is somewhat obscure, there being
several other correction terms of similar magnitude (quadrupole inter-
actions and inelastic contributions) in the electron broadening calculations.
Also, the experimental situation is not entirely clear, since the most recent
experiments [198, 199] gave lower intensities than theory [59] on the
wings and more so than [209a], while [207] showed no such deficiency.
Especially [199] implies correspondingly much higher peak intensities for
the S (a) profile than the earlier experiments.

Agreement [199] with calculations [121f] based on the relaxation theory
[52] must therefore be viewed with considerable caution, also because the

178 I I I . EXPERIMENTS

differences between the two sets of calculations are mostly due to the
(incorrect [64b]) inclusion in the relaxation theory of inelastic contribu-
tions to the upper-lower state interference term even for the line cores.
Keeping this in mind and remembering the situation in regard to the central
dip of H/s, the experimental trend thus seems to be for less smoothing of the
ion-produced Ha profiles but for more smoothing of those of Ήβ than is pre-
dicted by the theory of electron broadening. Since this would be very diffi-
cult to understand theoretically, one is tempted to search for systematic
experimental errors like those discussed for Hß and, in the pure hydrogen
plasma experiments, also errors connected with the radiative transfer
problem. In this connection recent measurements on shock-heated helium
plasmas containing small admixtures of hydrogen are of considerable
interest [210a]. They show that relative widths of Ha and H^ are con-
sistent with the older calculations [59b] but not with those based on the
relaxation theory [121f]. The analysis of these measurements also suggests
that the central dip of Hp is filled in considerably by emission from cooler
layers near the tube walls.

Agreement between entire profiles [209a] (see Fig. 19) reaching over
three decades of intensity is, for the impact theory, poorer than that be-
tween half-widths. In the case of the Ha line with its strong unshifted Stark
component, small changes in the electron broadening may still be felt as a

1

impact cale.

r"*"""^• relaxation cale.

φ φ ({»asymptotic rei.

Λ» \1
V

31 1 1 i^_j

103 10z 101 10 e
a

FIG. 19. Comparison between measured (Birkeland et al. [209a], N « 1.0 · IO17 cm-3,
T « 1.4 · 104 K) and calculated (Kepple and Griem [59a] or Appendix I.a and Vidal
et al. [121f]) reduced profiles of the Ha line. The asymptotic values are as for Fig. 17.

I I I . 2 . HYDROGEN LINES 179

change in the wing shape, and one should not overlook the possibility of

corrections to the (quasi-static) ion broadening calculations either. Elec-

tron densities and temperatures in almost all (except Behringer [199])
experiments were < 1017 cm-3 and < 1.6 eV, respectively, leading to values
of the parameter (coF/cos)2 of < 1 from Eq. (327) multiplied by the electron-
to-ion mass ratio. Dynamical corrections to the ion broadening might

therefore have to be considered according to Section II.4c, but their inclu-

sion [210b] does not improve the agreement between theory and experiment

substantially. Such effects are even less important for Κβ , etc., because for
given plasma conditions, (coF/cos)2 is then smaller by factors of 10 or more,
while the influence of thermally excited plasma wave fields on Ha is prob-
ably obscured by the unshifted component.

The Ηγ line has been measured over a slightly wider range of electron
densities, both "absolutely" [194, 198-200], and relative [211] to Hfl , in
pulsed discharges, arcs, and various shock tubes. Agreement between cal-

culated [59a] and measured half-widths according to these measurements

is just as good as for Η^ , i.e., again better than might have been expected

from theoretical error estimates (see Fig. 2). As a matter of fact, ratios of

Ήβ and H7 half-widths may be predictable to within 5% or less, experi-
mental evidence for which is summarized in Table V (from Wiese et al.

[198]). Perhaps not unexpectedly for a line of such peculiar shape (see

TABLE V

MEASURED AND CALCULATED RATIOS OF THE HALF-WIDTHS FOR THE Ηγ AND Ήβ L I N E S

Half-width ratio

Method N « 1016 c m - 3 N « 1017 cm"3 Reference

1. Γ « 10 000 K — 1.21 [130]
T tube 1.21 [211]
Shock tube 1.21 1.14 [202]
Stabilized arc 1.23 1.34 [198]
Stabilized arc 1.29 1.19 [59a]
Impact theory 1.18 1.15 [121f]
Relaxation theory 1.13
T « 20 000 K 1.32 [200]
1.15 1.19 [211]
Pulsed discharge 1.19 1.23 [59a]
Ttube 1.15 [121f]
Impact theory 1.17 —
Relaxation theory

180 III. EXPERIMENTS

Fig. 20), the agreement with respect to detailed line shapes is not anywhere
near as good as that, a fact which is reflected in the somewhat poorer agree-
ment between measured and calculated fractional intensity widths (see
Fig. 21). Since all three lines (Ηβ , Ηγ , Ha) measured in two of the experi-
ments [194, 198] showed less structure near the line center than was
predicted theoretically [59a, b ] , it has been attempted [59c, 200] to
smooth the theoretical line shapes by inclusion of inelastic electron colli-
sions (see Section II.3a). As discussed by Hill et al. [200], this modification
indeed improves the line shapes, but necessitates an ad hoc lowering of the
electron density by about 10% below the independently measured value.
Simply adding inelastic contributions to the impact broadening operator is
therefore not sufficient, probably because the "strong collision term" is
already accounting for such collisions [64a]. However, we may take the
improvement in line shapes as an indication of errors stemming from the
use of the same "strong-collision constant," etc., for all φ-matrix elements
of a given line, although the possible influence of plasma wave-produced
fields near the line centers should not be ignored either. (Different strong-
collision constants are of course also expected theoretically [62-64].) There
also appears to be some experimental uncertainty, especially in regard to
the "shoulders," which are weakly present in the profiles of Hill and Gerardo
[194], Behringer [199], and Bengtson et al. [211] but are absent on those
from Wiese et al. [198].

The Ηδ line was measured only in two [194, 198] of the above experi-
ments for N = (1-3) · 1016 cm~3 and kT ~ 1-2 eV and found to be about
5% narrower than predicted [59a] (or 8% when inelastic collisions are

T 11111111r

I i i i i i i i i i i1

0 20 40 60 80 100
Λλ[8]

FIG. 20. Comparison between measured (Wiese et al. [198]) and calculated profiles of
the H7 line. Plasma conditions and theoretical procedures are as for Fig. 17.

III.2. HYDROGEN LINES

FIG. 21. Comparison (from Wiese et al. [198]) of measured and calculated fractional
intensity widths of the Ηγ line (as for Fig. 18).

added [200]). This line has a profile similar to that of H^ , but with a less
distinct central minimum. Still, the measured dip is again much shallower
[194] than in the calculations [59a], which do not explicitly allow for
inelastic collisions, or even nonexistent [198]. Explicit inclusion of inelastic
collisions [200] gives near perfect agreement in this region but results in
too broad wings at the measured electron density.

These high density results for the first four Balmer lines therefore exhibit
a decreasing trend for the ratio of measured to calculated line widths toward
higher members of the series. Probably the reasons for this ratio to be, say,
near 1.05 for H« and near 0.95 for H$ are entirely different, as should be
clear from the above discussion. The above trend is supported further by
observations [206] of increased deficiencies in this ratio for H$ at higher
densities than 3 · 1016 cm-3 and an incipient deficiency also for Ηγ . At
such densities, these two lines begin to overlap, i.e., should have been
treated as one entity in the impact theory, while, e.g., quadratic Stark effect
should have been included in the quasi-static calculations, not to mention
the "dissolution" effect [206], i.e., forced ionization in the ion-produced
field. Still, except for asymmetries (see Section III.9), these effects do not
seem to invalidate the theoretical error estimates given by Kepple and
Griem [59a] (Fig. 2) in any significant way, even for the highest densities
considered there. (Complete experimental proof of this statement for H« is
not yet available.) Within these errors, the calculated Stark profiles in
Appendix I.a may thus be considered as experimentally comfirmed, ex-

182 I I I . EXPERIMENTS

empting, to be sure, some details like central dips, "shoulders," and
asymmetry and the far wings. For the latter, the relaxation theory [52,
121f] calculations (see Section II.4b) or asymptotic formulas as discussed
in Section IV.4a are usually superior (see Figs. 17 and 20). For H« , further
work is necessary to clear up the discrepancies [198, 199, 209a] nearer to
the line center and on the line wings. Should those near the line center be
settled in favor of the earlier experiments, the relaxation theory (unified
theory) calculations [121f] for this line and, to a lesser extent, the following
series members would then indeed be inferior to the impact theory cal-
culations [59] in the line cores, as surmised theoretically [64b, c] and veri-
fied experimentally for shock-heated plasmas [210a].

III.2c. Balmer and Paschen Lines at Low Electron Densities and
Temperatures

Most experiments discussed in the preceding sections were performed at
electron densities above N = 5 · 1015 cm-3 and temperatures of kT > 1 eV.
In many astrophysical applications, especially the densities are very much
lower, making experimental tests of the Stark broadening theory at
N <3C 1016 cm-3 very desirable. À suitable laboratory plasma source for such
purposes is the blue plasmoid formed near the mirror regions of a "magnetic
bottle" filled with rf excited plasma (filling pressures of 0.1-0.3 Torr). The
electron density in such plasmoids can be determined, e.g., by microwave
techniques, and was N ~ 1.3 · 1013 cm-3 in most of the experiments in this
category, which, together with some others, have recently been reviewed
by Schlüter [212] and will therefore only be discussed in regard to their
theoretical interpretation.

Before coming to this point, we note that these experiments were per-
formed not only at very low densities but also at exceptionally low tem-
peratures (^0.16 eV). As a consequence of this, the parameter R appearing
in the theory of ion-field distribution functions is not at all unusually small,
but rather R « 0.16 according to Eq. (46). Ion-ion correlations and Debye
shielding are therefore not negligible, and rms fields from thermally excited
plasma waves are estimated from Eq. (56) to be 7% of the mean particle-
produced field. Since electrons of much higher than thermal energies are
known to be present near or in the plasmoid, some suprathermal excitation
of plasma waves must also be expected. Still, these fields should not be
important outside the half-intensity points, unless wave energies are
enhanced by an order of magnitude or more.

Measurements were made in these plasmas of Ha and higher Balmer lines
(up ton = 16) and also of corresponding Paschen lines [213] (see Schlüter

I I I . 2 . HYDROGEN LINES 183

[212] for other literature citations). Because of the low electron tempera-
ture and the relatively narrow lines, ratios of line to continuum intensities
Avere rather high, so that the line profiles could be scanned over as many as
three decades in intensity. This allowed coverage of the transition regime
between impact and quasi-static approximations for the electron broadening
of the higher members of the two series, which, as long as this transition
occurs well outside the half-widths, should be properly accounted for by
the relaxation theory [121c] (Section II.4b) and probably, in these outer
regions of the line profiles, just as well by a simple asymptotic formula
[120]. (The latter interpolates between the two extreme approximations
according to electron velocity, etc.; see Section IV.4a.)

The most striking facet of the measurements at first was their good
agreement with quasi-static profiles calculated for normal field strengths
corresponding to both ions and electrons, although better agreement was
obtained [214] by arbitrarily increasing the shielding parameter R by a
factor of 2. Such increase in the Debye shielding causes a narrowing of the
line cores, an effect which is more naturally explained in terms of the impact
approximation for electrons or, even better, in terms of the relaxation
theory. For many years, only extrapolations of the early impact theory
calculations [53a] for Ha to Hs had been available, which were later
subjected to a number of corrections (see Griem [120]). These extrapola-
tions had resulted in too high intensities on the line wings, and the failure
of the experiments to give more than twice the asymptotic Holtsmark
value had been considered as an indication of some basic defect in the
theory used so successfully for the lower Balmer lines at high densities (see
the two preceding sections). That this is not at all the case was shown by
Bengtson et al. [215] at least for the lines H6 (Ha) to Hi2. They both
verified the earlier measurements summarized by Schlüter and his co-
workers [212] and of Vidal [32], and extended the calculations of Kepple
and co-workers [59] to all of these lines, finding agreement within mutual
tolerances of about 20% for intensities of 10% or more of the peaks. The
former authors further noted good agreement with the asymptotic wing
broadening approximation everywhere outside the half-intensity points
and found that for n > 8 the electrons are just as effective as the ions,
although most of their contribution still lies in the impact domain. Essen-
tially the same conclusions, but again better agreement on the far wings
(including the Δλ_5/2 power law), can be reached by comparison with
relaxation theory calculations [121c], which automatically incorporate the
asymptotic behavior, so that earlier semiempirical descriptions [216] of the
profiles no longer seem necessary. (These semiempirical profiles of course
cannot be expected to be correct at significantly different temperatures

184 I I I . EXPERIMENTS

[120b], etc., and their near agreement with one of the La measurements,
namely that of Boldt and Cooper [190a], wras more likely than not for-
tuitous. See the discussion in the first paragraph of Section III.2.)

A tendency for somewhat larger deviations between experiment and
theory for, say, n = 15 as compared to n — 10 may be due to the proximity
of the impact to quasi-static transition to the half-width. This invalidates
the relaxation theory in its present "impact approximation" form (see
Section II.4b), and it may then be better to treat the electron broadening
in the quasi-static approximation corrected for dynamical effects according
to Section II.4c, i.e., following Kogan [15]. Comparison with recent
measurements [205a] at intermediate densities and for somewhat lower
n-values might be interesting in this context, as would be a discussion of
the trend for measured widths to be smaller than calculated.

III.3. IONIZED HELIUM LINES

Compared to the considerable amount of experimental data for hydrogen
lines, there is still very little laboratory material available for Stark-
broadened lines from the next member in the one-electron sequence.
Practically all experiments in this category carried out in the period
covered here employed pulsed light sources because of the high tempera-
tures required. However, in view of the relatively good success with such
sources in the case of hydrogen lines (see Tables IV and V), this should not
by itself be considered a drawback, although the effects of higher tempera-
tures on the walls of the various discharge vessels can be serious. For
example, as suggested by Berg [217], earlier measurements of the most
prominent visible He II, X4686 A {rì = 3, n = 4) line in a T-tube experi-
ment [130] may well have been affected by unresolved impurity lines,
mostly of Si III and O II.

Berg, using a hydrogen-helium mixture, also found that the impurity
line intensities were relatively small near the reflector in his improved
T tube, and then allowed for them by monitoring well-resolved lines of the
appropriate multiplets, whose relative intensities in regard to lines which
overlapped the He II, X4686 A line were determined in auxiliary data
runs with pure hydrogen fillings. The He II line profiles thus obtained
were consistent with theory [57a] to about ± 2 0 % over an intensity range
of about one decade and for N « 3 · 1017 cm-3 and kT « 4 eV, but no
evidence was found for the plasma polarization shift (Section II.5b) to
within ± 0 . 3 A at a full half-width of about 8 A. (Comparison with the more
recent profile calculations [61] presented in Appendix I.b would be just
as satisfactory.)

I I I . 3 . IONIZED HELIUM LINES 185

Measurements of the vacuum ultraviolet X1640Â (η' = 2, n = 3),
X1215A {η' = 2, n = 4), and λ 1085 A (η' = 2, n = 5) lines from a
capillary discharge (iV « 2 · 1018 cm"3, kT « 4 eV) were reported by
Hessberg and Bötticher [218]. Their measured profiles had to be corrected
for self-absorption and, in the absence at that time of detailed calculations
for these lines, could not be compared directly with theory. However, as
discussed by Eberhagen and Wunderlich [219], they do, within mutual
error brackets, support the latter authors' measurements in a pure helium
theta pinch of the X1215 A and X1085 A lines (after reduction to the
a = Αλ/Fo scale). The theta pinch experiment [219] (at N « 1017 cm-3
and kT « 20 eV) covered many more lines, namely, in addition to the
above two vacuum ultraviolet lines, also X1025 A (η' = 2, n = 6) and
X992 A (ri = 2, n = 7), and the lines X4686 A (ri = 3, n = 4), X3203 A
(n' = 3, n = 5), X2733 A (η' = 3, n = 6), and X2511 A (η' = 3, n = 7).
Comparison with theory [61] (except for differences in the temperature
and the charge of perturbing ions) was possible only for the lines from the
n — 4 and n — 5 upper states, the computational effort still being rather
prohibitive for larger principal quantum numbers. The measured profiles
show a decidedly steeper decay on the line wings than the calculated
profiles, falling even below the Holtsmark values. Only a fraction of this
deficiency could be due to dynamical corrections in the ion broadening
(Section II.4c), to larger upper-lower-state interference terms [64b] or to
the difference in perturber charge [28]; there remains a factor of about 2
disagreement, e.g., for points corresponding to less than 5% of the peak
intensity. Relaxation theory (Lewis [65]) corrections should not be all
that large either, but would decrease the discrepancy some, especially for
the lowest relative intensities.

Another theta pinch measurement [220] at slightly higher densities and
lower temperatures (N « 6 · 1017 cm"3, kT « 5 eV) of the X4686 A line
over a factor of about 50 in relative intensities, although also falling on the
line wings somewhat below original predictions [57] and recent impact
theory calculations [61], does yield a factor of about 2 above the Holtsmark
ion-broadening contribution, and more than that above the first theta
pinch profiles [219]. Probably Bogen's results [220] would entirely agree
with theory, were the impact theory on the wings replaced by a relaxation
theory calculation, or were the numerical results of Kepple [61] replaced
by an asymptotic formula [120] beyond, say, the 10% relative intensity
points (see also Section IV.4a).

One measurement [221] of the X3203 A line in a z-pinch discharge at
N « 2 · 1016 cm-3 and T < 4 eV further supports the rather satisfactory
agreement with theory [57, 61] between the 10% relative intensity points

186 I I I . EXPERIMENTS

found by Bogen [220] for the preceding member of this series. As a matter
of fact, significant deviations were seen only near the central minimum of
the X3203 A line, analogous to the situation for Ή.β , whose predicted shape
is rather similar to that of this ionized helium line. In this connection, we
might mention that the generally accepted fair agreement between the, say,
25% relative intensity points of the X4686 A line, whose shape is similar to
that of Ha , may be considered a verification of the electron impact broad-
ening calculations. One can make such statement because this region is
dominated by the unshifted component (which is not affected by per-
turbing ions as long as the quasi-static approximation is valid), but should
not extend it to the wings.

To decide between the conflicting evidence regarding the adequacy of
theory [57, 61] for ionized helium lines presented above, an experiment
[222] was performed in a T tube at N = (1-3) Q· 1017 cm"3 and T = 4-5 eV
on the He II, X304 A (ri = 1, n = 2), X1640 A (ri = 2, n = 3), X4686 A
(ri = 3, n = 4), and X10 123 A (ri = 4, n = 5) lines. These first four
n-a lines were chosen because they are especially sensitive to electron
broadening, which is afflicted by the largest theoretical error. The electron
density was inferred from the widths of neutral helium lines, which in turn
had been calibrated against Kß in separate experiments (see Section III.6).
The resonance line in the essentially pure helium plasma was peculiar in
that it was optically thick and therefore had to be analyzed according to
Eq. (413). The next line in this "series" was not significantly affected by
radiative transfer processes, as evidenced by comparison with profiles
measured at half the usual geometrical depth. However, as for the resonance
line, its core was obscured by instrumental broadening, so that only the
wings of the Stark profiles could be determined. For the resonance line,
these turned out to be about a factor of 2 below the calculated values [61].
This was first explained [150b, 222] by the absence of linear Stark effect
for two helium ions described by a symmetrized (with respect to excitation
energy, but not necessarily particle exchange [150c]) wave function, which
seemed appropriate in the nearest neighbor approximation for the profile of
the absorption coefficient, since the two ground state ions involved are
equally likely to be excited in the absorption process. [Remember that
according to Eq. (413), an optically thick emission profile is essentially
determined by the shape of the absorption coefficient.] However, the
stationary states of the two-ion (molecular) system actually do not corre-
spond to such sharing of the excitation energy, and we now therefore have
no explanation for the above discrepancy [150c].

For the next line, X1640 A, there was agreement within estimated errors,
although measured points on the far wings (corresponding to about 1% or

I I I . 3 . IONIZED HELIUM LINES 187

more relative intensity of the maximum intensity for the optically thin
Stark profile) tended to be about 50% above the calculated values, prob-
ably because of density gradients in the line of sight for the conditions of
these particular data runs. This suggestion is supported by a 20% difference
in the electron density inferred from widths of the lines He II, X4686 Â
and He I, X3889 A and 5876 A.

In case of the X4686 A line, agreement with theory [57, 61] was typically
within 20% between the 10% relative intensity points. For this line, some
of the experimental points were believed to be systematically low because
they were purposely picked between various impurity lines. Nevertheless,
these data and those for the XI640 A line do not at all support the low wing
intensities found in the work of Eberhagen and Wunderlich [219], whereas
they are entirely consistent with those of Bogen [220] and Jenkins and
Burgess [221]. (It seems very unlikely that these discrepancies have their
origin, to any great extent, in the different temperatures or ionic charges
in the various experiments, nor in the different intensity ranges covered.)
The best overall agreement with theory [61] was obtained for the MO 123 A
line, whose profile is shown in Fig. 22. This might be connected with the

01 1 1 1 l_i_J—! 1
0 0.1 0.2 Q3 0.4 0.5
a

FIG. 22. Comparison of a measured (Jones et al. [222]), N « 1.0 · IO17 cm"8, T «
4.0 · 10 K) reduced profile of the He II, X10 123 Â (η' = 4, n = 5) line with theory
(Kepple [61] or Appendix I.b). Asymptotic values (not shown) according to Section IV.4a
fall a factor of about 2 below the calculated wing intensity, because they contain the full
n-n' interference term while those of Kepple [61] and Appendix I.b only have the z$f
term.

188 I I I . EXPERIMENTS

fact that inelastic electron collisions were not considered explicitly in the
calculations for this He II line, in contrast to all other lines [61].

The first ionized helium line for which there was recent experimental
evidence [147] for the blue shift discussed in Section II.5b (Δλ « — 0.05 A
at N « 3 . 1017 cm-3 and kT « 4 eV) is the X304 A resonance line, ob-
served under optically thick conditions from a T tube [222]. The observed
shift, which is larger than the optically thin (Stark) half-width by more
than an order of magnitude, is in reasonable agreement with Eq. (385b),
but much more experimental and theoretical work is needed to verify this
effect and to provide a full understanding of it. One attempt [149a] in this
direction was inconclusive for this particular line, because of impurities
superimposed on the somewhat broader (than in Greig et al. [147])
optically thick resonance line, but the next line in the resonance series
showed a definite red shift. This raised the question whether the blue shift
of Greig et al. [147] was caused by radiative transfer effects, in particular,
by a different behavior of absorption and emission line shapes [150a].
(The former may involve contributions by transitions from singly to
doubly excited states of neutral helium, broadened by Stark effect.) The
relatively large blue shift of Greig et al. [147] is made even more suspect
by the fact that experimental upper limits [223] for the shifts of lines from
multiply ionized systems are well below theoretical extrapolations accord-
ing to Eq. (385b). [Note here that this formula was meant only for reso-
nance lines for which the condition for the validity of Eq. (384b) is not
seriously violated.] As a matter of fact, the most recent experiment [149b]
gives small red shifts or no shifts at all for the first two lines of the He II
resonance series and rather small blue shifts for the next two lines. The red
shifts can be attributed to asymmetries on the line wings, while the blue
shifts may scale in accordance to Eq. (385b) but are much smaller in mag-
nitude. They are, however, two or three times larger than one would
estimate using Eq. (384a) instead of Eq. (384b), and a factor about 10
larger than suggested by the estimates of Burgess and Peacock [223]. For
n = 5, agreement is best if comparison is made with Eq. (387).

All experiments on He II lines discussed up to this point were per-
formed at densities exceeding N = 2 · 1016 cm*3, usually by a considerable
factor, and involved transient sources. In addition, one measurement in a
magnetically confined steady state arc plasma of the X3203 A line at low
densities but high temperatures (N « 1014 cm"3, kT ~ 20 eV) has been
reported [224]. Because of Doppler broadening and fine structure, com-
parison in this case was possible only with asymptotic wing formulas, which
were verified out to about 1 A from the center.

On balance, the experimental evidence thus speaks for about 10%

I I I . 4 . HYDROGENIC LINES FROM HEAVY IONS 189

accuracy of impact theory [61] half-width calculations (Appendix Ill.b)
for ionized helium lines and for, say, 20% accuracy of the calculated line
shapes (Appendix I.b) within the 10% relative intensity points. Beyond,
asymptotic formulas (Section IV.4a) should have similar accuracy. How-
ever, in the case of absorption from the ground state, the calculations have
yet to be verified by experiments allowing for a more accurate assessment
of radiative transfer effects. Nothing definite can be said about shifts and
asymmetries, except that they are rather small, and more accurate profile
measurements are required, especially for n-a lines, to verify the transition
from having only strictly elastic to full n-nf interference terms in the elec-
tron broadening as discussed, e.g., at the end of Section II.3a.

III.4. HYDROGENIC LINES FROM HEAVY IONS

Besides the reinterpretation [148] of measurements of the Be IV
resonance line in terms of a possible plasma polarization shift, there seems
to be only one Stark broadening experiment with independent density
determinations concerning lines from multiply charged one-electron or
spectroscopically similar ions. This experiment [225], performed on a laser-
produced carbon and hydrogen plasma (N < 3 · 1018 cm"3, kT < 35 eV),
dealt with the 6-7 and 7-8 transitions in C VI (at 3434 and 5290 A,
respectively) and with the 5-6 and 6-7 transitions in C V (at 2982 and
4945 A). (Since the C V lines are much broader than the inherent separa-
tions of levels with different orbital angular momenta, they should also
behave like lines from one-electron systems, for all practical purposes.)
The lines observed are therefore so-called n-a lines, for which considerable
cancellation of upper and lower state broadening must be expected,
although not to the extreme extent as for the radio-frequency n-a lines of
hydrogen [111] (see Section IV.5). Their upper level populations should
approximately be in LTE with respect to the next higher ionization stage,
so that the absolute intensities can be used to infer electron densities from
Saha equations and charge neutrality relations.

Using essentially Eq. (16) for the quasi-static half-width, Irons [225]
concluded from the relative values of the measured widths that most of the
Stark broadening was caused by ions. Since the parameter (COF/<OS)2,
according to Eq. (327), suitably modified [126] for a perturber charge
Zp « 5 and the mass of carbon ions, is small ( ~ 0 . 1 ) , the quasi-static
approximation is indeed expected to be valid here, but it remained to show
that electron impact broadening should be small enough not to invalidate
the quasi-static scaling of the line widths. This was done using Eq. (110)

190 III. EXPERIMENTS

with the full interference term, and the situation was found to be the
inverse of that for the radio-frequency lines of hydrogen. (For the latter,
the ion broadening also must be treated in the impact approximation, but
then turns out to be small compared to the electron impact broadening
from inelastic collisions [111].)

After correcting for electron impact broadening, the absolute values of
the measured line width were smaller by a factor of about 4 than would be
predicted using Eq. (16). However, the Debye length in this dense plasma
follows from Eq. (44) to PO' ~ 4 · 10~6 cm, to be compared with a mean
separation between carbon ions of r5 « 2 · 10~6 cm from Eq. (53). Ion-ion
correlations and Debye shielding by electrons are thus both large and may
well explain, say, a factor of about 1.5 discrepancy in the absolute values
of the line widths. A corresponding reduction in the quasi-static (ion-field
produced) line width does not yet invalidate the above statement that
electron impact broadening is small enough for the scaling of the widths of
different lines to follow Eq. (16), while a factor of 4 reduction, as suggested
by a comparison of Eq. (16) with detailed calculations [61] for the 4-5
transition of ionized helium, could imply an important correction from
electron impact broadening in this scaling.

There thus remains a measure of unexplained behavior, although some
errors from density gradients cannot be ruled out either. Much more
experimental and theoretical work is needed before the accuracies of about
10-20% for hydrogen or ionized helium lines can be approached for multiply
charged ions exhibiting linear Stark effect or, for that matter, for any
multiply charged ion. Meanwhile, we are obliged to depend on theory (see
Section IV.6) to estimate widths and, more important, wing shapes of
such lines as required for various applications.

III.5. LINES WITH FORBIDDEN COMPONENTS

Strictly speaking, the C V lines mentioned in the preceding section do
include forbidden components, i.e., transitions with ΔΖ j* dbl. However,
since the widths of these lines are much larger than the relevant unper-
turbed level separations, the levels involved could for our purposes be
considered as completely degenerate, so that the lines are analogous to
lines from one-electron systems. We will not contemplate such extreme
cases of overlapping lines in this section, but rather only those situations
for which the allowed lines are reasonably isolated and for which the for-
bidden components appearing on the wings of such lines are relatively
weak. Quite naturally, most measurements (and calculations) in this class

I I I . 5 . LINES WITH FORBIDDEN COMPONENTS 191

were made on the simplest radiating system capable of emitting lines with
forbidden components in this sense, namely neutral helium.

The first extensive experimental study of helium lines with forbidden
components to be reviewed here was performed by Vidal [32], who meas-
ured diffuse (2P-nD) lines emitted from a radio frequency discharge
(N « 3 · 1013 cm-3, kT « 0.16 eV) similar to that used for the low-density
measurements of hydrogen Balmer and Paschen lines mentioned in Section
III.2c. The forbidden components observed belonged to the series 2 Ψ-η 3Ρ,
2 3P-n 3F and the corresponding singlet series. (Vidal also reported meas-
urements of forbidden components associated with 2S-nP lines, but gave no
detailed profiles of these much narrower lines.) As for the low-density
hydrogen lines, an attempt [25] was made to describe both ion and electron
broadening of the diffuse helium lines by the quasi-static theory. It was
recognized that this approximate theory could not possibly describe the
electron broadening in the vicinity of forbidden components or the peaks of
the main lines, and that the 2 lP-n XD line wings not perturbed by for-
bidden components were strongly blended with wings of the more intense
triplet lines. With respect to such calculation [25], this left the violet
wings of the diffuse triplet lines for comparison, which was favorable as far
as the relative (not normalized) line shapes are concerned. However, when
area-normalized, the measured intensities near the line centers were smaller
than calculated, and the measured intensities on the far wings were seen
to decrease more slowly with the principal quantum number of the upper
level than predicted by the quasi-static approximation.

Both of these deviations are clear indications that, at least for the lower
series members and near the line centers, the impact approximation would
have been more appropriate for the description of the electron broadening.
And indeed, a calculation [92] proceeding along the lines of Section II.3cß
reproduced the entire measured profiles of the diffuse triplet series with
n > 5 very satisfactorily, inclusive of forbidden components. To make this
comparison, a (slightly) corrected value [25] for the electron density was
used by Gieske and Griem [92] which, however, had not been determined
independently of line broadening theory in the experiment [32], but rather
from an improved version of the Inglis-Teller method [18]. Measured and
calculated profiles for the singlet lines, on the other hand, are inconsistent,
measured wing intensities being higher by a factor of about 2, probably
because of the difficulties encountered in separating contributions from the
far wings of triplet lines mentioned above. Whether or not the tendency,
with increasing principal quantum number, for measured triplet profiles to
become narrower than calculated profiles [92] (by <20%) is also due to
these experimental difficulties or is real remains to be seen. (As discussed

192 I I I . EXPERIMENTS

in Section III.2c, there may well be theoretical reasons for such an effect.)
One might also question the existence of thermal equilibrium in the plasma
in regard to the excitation of plasma waves, because the plasma is known
to contain a small population (~0.03%) of suprathermal electrons. How-
ever, the observed forbidden components do not seem to be any stronger
than calculated on the basis of particle-produced fields only.

Before passing on to measurements of helium lines with forbidden com-
ponents at higher densities and temperatures, we should note that plasma
conditions and profiles very similar to those of Vidal [32] can be seen in
cesium (for the fundamental series), for which Stone and Agnew [226]
and Majkowski and Donohue [227] (see also Sassi [94], Sassi and Coulaud
[95], and Wu and Shaw [228]) demonstrated rather far-reaching agree-
ment with electron impact broadening calculations. Besides two high
density experiments on helium lines reviewed before [189], there seem to be
only very recent experimental studies in the present category. (In this
section, we exclude experiments on plasmas with strong suprathermal field
fluctuations; these will be discussed in Section III. 10.) One of these recent
studies [229] was primarily concerned with widths and shifts of allowed
helium lines and their shapes (see the following section). However, at the
high densities (N « 1.4 · 1017 cm-3) in this T-tube experiment (kT « 2
eV), the X5016 A (2 l8-S Φ ) line had developed a forbidden component
near the X5042 A (2 χ8-3 *D) transition, whose profile was the mirror
image of that of the allowed line. The relative intensity of the forbidden
component was near 10%, in good agreement with a calculation [93] using
the procedures discussed in Section II.3cß.

The other high density helium experiments [221, 230-234] were specifi-
cally designed to check calculations of line profiles with forbidden com-
ponents. In the first [221] of this series, the He I, X4471 A (2 3P-4 3D, 3F)
line was measured (end-on) in a z pinch (N « 3 · 1016 cm"3, kT « 4 eV).
No'clearly separated forbidden component was observed, in contradiction
to theoretical expectations [90, 91]. Much of this disagreement could be
traced to density gradients in the source, which are particularly critical in
the vicinity of forbidden components, but some residual disagree-
ment is believed to be due to other causes (see also below). A dc arc
[jV = (0.4-2) · 1016 cm""3, kT « 1.5 eV] was used [230a] to measure the
profiles of the He I, X4922 A (2 Φ - 4 »D, XF) and He I, X4471 A
(2 3P-4 3D, 3F) transitions, side-on and Abel-inverted. Relative intensities
and shapes of allowed and forbidden components were found to agree fairly
well with calculations [90, 91], except that for X4922 A the components
were perhaps slightly closer together than was predicted theoretically,
while for both lines the intensity minima between the components were