On the quadrupole contributions to electron broadening of spectral lines

J. Quant. Spectrosc. Radiat. Transfer Vol. 31, No. 3, pp. 247-258, 1984 Printed in Great Britain.

0022-4073/84 $3.00+ .00 © 1984 Pergamon Press Ltd.

ON THE QUADRUPOLE CONTRIBUTIONS TO ELECTRON BROADENING OF SPECTRAL LINES

H. P~NNIG and ILUNGA S. K. MWANA UMBELA Max-Planck-Institut fiir Physik und Astrolahysik, Institut fiir Astrophysik, Karl-Schwarzschild-Str. l, D-8046 Garching bei Miinchen, Federal Republic of Germany

(Received23 May 1983) A~tract The quadrupole-quadrupole contribution to the electron-collision matrix is averaged over the directions of the impact parameters and velocities of the plasma electrons. The result is sufficiently general to cover such special cases as hydrogen, overlapping, isolated lines. Some errors in the published literature are corrected. INTRODUCTION

The present paper refers to that version of the theory of the broadening and shift of spectral lines in which the contributions of the plasma electrons and ions are assumed to be appropriately described by the impact theory and the quasistatic theory, respectively. A review of this version of the theory, from the pioneering work of Baranger 1-3 and Kolb and Griem4 to approximately 1973 may be found in Griem's monograph. 5 In the early papers, only the dipole-dipole term coming from the multipole expansion of the interaction energy between a plasma and one of the atomic electrons was taken into account in the second order approximation to the scattering matrix. In a paper concerning the hydrogen line Ly-o:, Griem6 first introduced the quadrupole-quadrupole term as a correction. Cooper and Oertel 7'8 and Deutsch and Klarsfeld9 followed him in the cases of isolated and of overlapping lines, respectively, for neutral more-electron atoms. Since the more recent papers differ in their respective ranges of application and since some of these are not free from mistakes concerning the angular average of the quadrupole-quadrupole contribution to the electron collision matrix, a rather confusing state of affairs has developed. The purpose of the present paper is to clarify the situation. It is organized as follows. In Section 1, the basic formulae are taken over from the literature and the major part of the notation as used in the paper is explained. The averaging of the quadrupole-quadrupole contribution over directions is the subject of Section 2. The results obtained in Section 2 are compared with those of Griem, 6 Cooper and Oertel, 7'8 and Deutsch and Klarsfeld9 in Section 3. 1. B A S I C F O R M U L A E

The following basic formulae apply to a neutral atom with an electronic structure consisting of a spherically symmetric atomic core and a single excited (optical) electron. The Hamiltonian for the optical electron in the free atom has the form h2

H0:-

3

~2

2m, j~l-ff-~E~+ eU(R),

(1.1)

where R = (X1, X2, )(3) = position vector of the optical electron, e = - lel = electric charge of the electron, U(R) = spherically symmetric potential fulfilling appropriate boundary conditions for R ~ 0 and R ~oo. In order to take into account the quasistatic electric field, which is produced by the plasma ions, we replace the Hamiltonian H0 by

H(F): =/4o + lelFR,

(1.2)

if this quasistatic field F may be assumed not to influence the atomic core and to be 247

248

H. PFENNIG et al.

homogeneous within the range of the atom. The eigenstates [a(F)), [a'(r)) . . . . and eigenvalues E,(F), E,,(F) . . . . of the Hamiltonian H(F) are understood to belong to the upper (initial, i) level of the spectral line under consideration. An analogous convention holds for the eigenstates ]fl(r)), [fl'(F)) . . . . and eigenvalues E~(F), E~,(F). . . . of the lower (final, f ) level. Since an average is to be performed over all directions of electron motion, the quasistatic ion field F may be taken along the z-axis without any loss of generality. Due to this axial symmetry about the z-axis, the energy-eigenstates [ct(F)) . . . . . [ f l ( F ) ) , . . . may (and will) be assumed to be eigenstates of L , the z-component of the angular momentum operator. No further assumptions on the eigenfunctions will be needed for the calculations in Section 2. It is only the special case r = 0 that one may assume the let), [fl) to be, in addition, eigenfunctions of the operator L 2 and of the parity-operator. For the interaction energy e V(r(t), R) between the plasma electron and the atom the following equation holds:

eV(r(t), R) =

e2

e2

-- r(t--) +

[r(t) -- R I

(1.3)

In Eq. (1.3), r(t) = [x~(t), x2(t),

x3(t)] =

position vector of the plasma electron.

The plasma electron may be assumed to pass the neutral atom with constant velocity, viz. r(t) = p + vt,

(pv) = 0,

(14a,b)

where p = (p~, P2, P3) ---- collision parameter, v = (vl, v2, v3) = velocity. Expanding the expression on the right hand side of Eq. (1.3) into multipoles yields

eV(r(t), R) = eVd(r(t), R) + eVq(r(t), R) + . . . .

0.5)

r(t)>R,

(1.6)

If

the dipole- and the quadrupole-terms are defined by

e Va(r(t ), R): = eZ[(r(t )R )/r3( t )]

(!.7a)

and

eVq(r(t),R):=e 2 1E3(r(t)R)2 ×2L rS(t)

(RR)~,

(1.7b)

r~3

respectively. The condition (1.6) may always be considered as being fulfilled in the case of weak collisions [see the remarks following Eq. (1.10)]. The electron collision matrix may be expressed in terms of the scattering matrix as (c([~,[~') = Ne

0°0

vf(v)dv

i Pma×27zp{(ot[S,(p, V ) - 0

l[~')}anglesdp,

(1.8)

where N~ = electron density in the plasma, f(v) = (Maxwellian) probability density of v, Pmax=appropriate upper cut-off for the collision parameter (el Griem, 5 p. 44), { }angles= average over'the directions of p and v (angular average) under the assumption, that they are isotropically distributed and fulfill the condition (1.4b). The perturbation of the states of the lower level will not be taken into account because it is usually small compared to that of the upper level. Calculating the scattering matrix

Quadrupole contributions to electron broadening of spectral lines

249

by means of time-dependent perturbation theory leads to the Neumann or, in more modern terms, Dyson series <=Is,o,,

= a. +

ef +~(~[~7(tl)[~')

dt,

--o0

(e)Z~;+°°;

''

+ N

--o0

(1.9)

(~ [ e(t,)[~")(~"l ~7(t=)l=') d/2 dtl +""

--oo

with (1.10) By definition, the calculation of the scattering matrix of weak collisions may be broken off after the second-order term in V(t), which is still explicitly given in Eq. (1.9). Although it is only in the case of hydrogen lines, that over a wide range of electron densities and temperatures these weak collisions yield the major contribution to the electron collision matrix, they alone will be considered in the following. For isolated neutral atom lines the contributions of weak and strong collisions tend to be of the same order of magnitude (cf. Tables IV,5). As already mentioned in the introduction, it is the purpose of the present paper to average the quadrupole--quadrupole-terms over angles. The result of the averaging of the other terms considered in the context of second-order perturbation theory (dipole, quadrupole, dipole-dipole and dipole-quadrupole) is well known and not controversial. All of the averages relative to these terms vanish, except for the dipole-dipole-term

{ (~leVXt,)l~")(~")le

~ , , )},.#, Va(t2)[~ 3

e 4 exp(i [eo,,,,(F)t. - co,,¢,(F)t2])

r3(fi)r3(t2 )

3

x L E {(PJ + vjt,)(pj, + ~j.tO}...,o,K~lxjl~, ") <~"lx~.l~'> j=lj'=l

=

e4 exp(i[t°~,*,"(F)ti - t°¢~"(F)I2]) (p2 3r3(tOrS(t2) + v2t, t2) ~. (~IX~I~">(~"Ix~I~'). j=l

(1.11) This result is obtained by using the equations {pjpj,}~.o,

= (p~/3),~,

(1.12a) (1.12b)

= 0.

(1.12c)

2. A V E R A G I N G OF T H E Q U A D R U P O L E - Q U A D R U P O L E - T E R M OVER T H E D I R E C T I O N S OF THE C O L L I S I O N P A R A M E T E R A N D T H E V E L O C I T Y

From Eq. (1.7b),

( O~[e Vq(ll)[o~ " ) ( • "[e 17q(t2)l~ ' ) =

e 4 exp(i[og~.(F)t I -- co¢~.(F)t2]) [9A (a, ~"; or", a') 4rS(tOr5(t2)

- 3B(~, ~"; a", ~') - 3C(~, or"; od', a')] + D(oq ~"; a", ~')] + terms which vanish after averaging over the angles.

(2.1)

250

H. PFENNIG

e t al.

In Eq. (2.1), the quantities A, B, C have the following forms: 5

A (~,

'. . . .

(2.2)

~'): = y. A (~(~, ~"; ~ , ~ 9, s=l 2

B(~, ~'; c~", a'): = ~ B(S)(~, ~ . ., .~. . , (X'),

(2.3)

s=l 2

(2.4)

C(~, ~"; ~", ~'): = y~ C('(~, ~"; ~", ~'). s 1

The quantities on the r.h.s, these equations are defined as

A °)(~,c~"; ~", a'): =

(2.5a)

A (2)(~, ~,,; ~,,, ~,): = t.2(~ I(vR)2[~"}(~"l(p R)zI:~'},

(2.5b)

d (3)(a, ~,,; ~,,, ~ ,): = at,h(ccl(pR)(vR)l~")~"l(pR)(vR)l='), A (4)((X, ~ tt; O~n , O~ ' ) : = t=2(~ IKpR)zI~")(~"IKvR)zI~' ~ ,

(2.5c) (2.5d)

A (5)(~, ~ ,,; ~,,, ~ ,): = tl 2 t 2 2 ( ( Z I K v R ) = I ~ g K ~ ' I ( v R ) ~ I ~ 9 ,

(2.5e)

B<°(c~, ~'; ~", ~'): = r2( t , ) ( ~ l(RR )l~ " ) ( a "[(p R )(p R )]a ' ),

(2.6a)

B(2)(o~, ~"; ~", ~

C(I)(~, (~";

'):

O~ n , 0 ~ ' ) :

=

122r2(/ l) ( O[I(RR )I~ 9 ( ~,.I(vR )(vR )I~ 9,

(2.6b)

=

r 2(t2)( ~ I(P R)(p

R)I 9

(2.7a)

C(2)(o{, O~'; 0~', O{'): =

D(a, ~"; a"; c~'): =

"I(RR)I 9,

112r2(12)(~ [(vR)(vR)l~ ")(~"I(~R)I~' 9, )r2( t 2)( ~

r2(t~

[(RR)[~ ") ( ~ "[(RR )Is ' ) .

(2.7b) (2.8)

The following remarks refer to the terms A ~'(a, ~"; c~", a'). (!) The term A o)(~, :¢,,; ~,,, :¢,) has arisen from 4 terms because

(~l(pR)(vR)l=" > : (~[(vR)(pR)l=" >

(2.9a)

and (2.9b)

=

(2) Except for the time factors, the terms A('(a, e"; a", e') and A(5)(~, a"; a", a') are of the same structure. They change into one another by substituting v for p. The same statement applies to the terms A (2)(e, ~ ,; c~", c~') and A (4)(c~,e "; a ", ~ '). Written in Cartesian components of p and v, Eqs. (2.5)-(2.8) read as follows: = E

a (s)

.

.

Ixkxk.l , ),

(2.1o)

f f 'kk'

it.

tt

t!

¢!

t

(2.11)

jkk'

C(S)(ot,

".

,, jj'k

jj

(2.12)

.

D(~, ~"; ~", ct') = r2(t,)r2(t2) Z <~IXjXjI~ . >(~ . . Ix, . . ik

with the factors a(j})kk,, b kk,~ (~) c!S? , jy given in Tables 1 and 2.

>,

(2.13)

Quadrupole contributions to electron broadening of spectral lines

251

Table 1. The coefficients ,7o) ~/,rkk'(s)

s

ajj,k k

Pj Pj' Pk Pk' t 2I vj vj, Pk Pk' 4 t i t 2 pj vj, Pk Vk' 2 t 2 Pj Pj' v k v k, t l2t 22 vj vj, vk Vk,

Table 2. The coefficients b~, and e~9.

b(S) kk'

c!~.! Jl

r2(tl ) Ok

r2(t2 ) Pj

Pk'

t22 r 2 ( t l ) Vk v k ,

t2 I

Pj,

r2(t2 ) vj vj,

In order to calculate average values over the angles, we use Eqs. (1.12) and the following well known relations:

p4 {pflpj2}a=sles = ~ (1 + 265,),

(2.14a)

13 4

2 --__ {vj2vj,}a.#= - 15 (1 + 26~,),

(2.14b)

2 {pj2vr}..g,~

fin"),

(2.14c)

3-0 'J ~J'"

(2.1@l)

P 2v2

= ~

(2

p2v2 {PJVJP/Vf}an~e' =

-

.

The averages of all other 4-fold products of components of p and v vanish. This fact has already been used in Eq. (2.1), when terms of this kind were omitted. The quantity A(~t, ~"; ct", ~') of Eq. (2.2) is the only one which requires some care in averaging. From Eqs. (2.10) and (2.14), 4

{A (')(~, ~"; ~ ", ~')}a.#~ = E a~(S)f~')A~( ~, ~"; ~', ~ ');

(2.15)

x=l

the coefficients a, °) have the values given in Table 3. The functions j~)(p, v, fi, t~) are defined in Table 4. By definition, the indices x = 1, 2, 3, 4 correspond to average values which do not vanish, as follows: =1

Jl j'l kl .... ;

= Z
(2.16a)

Jk' 0~">< o e "IX

>"

0~" ,

252

et al.

H. PFENNIG

x=2

j

j,

(2.16b)

k k' L_J

j#k u=3 ¢1.

~

j'

[

k

I [ _ _ 1

(2.16c)

k"

I¢

tt

tt

!

.

j#j' K

=4

i

J'

k

(2.16d)

k'

[__l

[

J

J # i"

= ~ ( ~ I X j X j , I ~ " > ( ~ " I X j X j , ] ,~') J #i' =

A3(~

~ ", ~ ' ) .

, ~";

T h e d i a g r a m s a t t a c h e d to Eqs. (2.16) p o i n t at the indices w h i c h are, in each case, equal to o n e another. Table 3. The coefficients a~{-o and ~,0. (s) aI

aJS)

1

(s) a3

I

T5

I

2

73-

7-5

(s) a6

]

l

73-

7~

1

+

-5

4

+

4

8

3o

7~

-

I T5-

2 IS

t 30

l - T6

I

I

]3-

2

- $6

4

I

Tg

T~

4

5

1

3o

73-

l

~(s)

5 2

+

5

Table 4. Thl functions f '(~)(p, t, tt, t2). s

f(S)(p,

v,

tl,

t 2)

4 F, (i, v t l ) 2 9 (:? v ) - C l t 2 (I:: v t 2 ) v

4

2 2 t]t 2

F r o m Eqs. (1.12), (2.3), (2.4), (2.11) a n d (2.12), it f o l l o w s that , 0~'; ~

",

angles --~

{c(

(X, 0~"

,

angles

r2(tOr2(t2) jk r2(tl)r2(tz) _

3

[AI(~' ~'; ~ ,, ' ~ ') + A2(~,

~

'. ,. .~. ,

~%

(2.17)

Quadrupole contributions to electron broadening of spectral lines

253

with AI, A2 defined in Eqs. (2.16a,b). The term D(~t, ~t"; ~", ct') of Eq. (2.13) does not depend on p and v. Therefore, it coincides with its average value {D(a, ~"; a", :¢')},,g1~ = rZ(tt)r2(t2)[Al( or, a"; a", a') + A2(~, a"; a", a')].

(2.18)

Combining the results (2.15)-(2.18) yields 1

4rS(fi)rS(t2) {[9A(c¢, ~"; ¢¢", ~')

--

¢!.

t¢

t¢

t

3B(~, ~ , c¢", ~') - 3C(~, 0~ '.,.( .2 . , ~') + D(~, ~ ; c¢ , ~ )]},,gl,s

5

_

~= 1

4rS(h)rS(t2)

E(ct, ~t"; ct", ~').

(2.19)

The coefficients 8 (') are given in the last column of Table 3 and the quantity E(~, ~"; ~", ~') is defined by (2.20)

E(ct, at ;~",~t'): = - - 2 ~ k The summation over s is easily performed and is found to be 5

[4rS(tOrS(t2)]-i ~ g(s)]~)= [lOrS(tOrS(t2)]-t[2p4 _ (pvt02 + 6(pv)2tdz _ (pvt2)2.+ 2v4t12t22] s=l

1 1 -

f0

(2.21)

L(x. x2),

where fq(Xl, x2): =

2(1 + xl2)(1 + x22) -- 3(x1 -- x2)2 (1 + x12)5/2(1 + x22)5/2

(2.22a)

and xl" = vq/p,

x2 : = vtz/p.

(2.22b,c)

In order to facilitate comparison with previous work (see Section 3), it is appropriate to rewrite the expression of the r.h.s, of Eq. (2.20) in two ways. (i) The first way amounts to a trivial rearranging of terms, i.e.,

J

J

3 2j~:

(2.23)

(ii) The second procedure involves introduction of the spherical components [Edmonds, 1° Eq. (5.1.3)] 1

__l R

R_I:= +]~/z'~(XI-iX2)=x/2

s i n 0 e -i*,

Ro: = X3 = R cos 0, R+ 1: = - ~

1

x/z

(2.24)

( Xl + iX2) = - ~

1

x/z

R sin 0 e +'~,

of the vector R and expressing the resulting products R_~R_I, R_IR0 . . . . (or appropriate QSRT Vol. 31, No. ~ E

H. PFENNIGet al.

254

linear combinations of these) in terms of the spherical harmonics defined in the manner of Racah, i.e.,

Cq(k'(O,4)): = 4 2 k ~

1 Ykq(O,49).

(2.25)

The functions Ykq(O,cb) on the r.h.s, of this equation are understood to be the usual spherical harmonics defined in the manner of Edmonds [Eq. (2.5.6)], l° i.e. they satisfy the relation

Y,_q(O, (9) = (-- 1)qY~q(O, ~ ).

(2.26)

In this way, we obtain (2.27a) q

= ~ (~lRZfq~2'l~")(~'lRZff'l~") *

(2.27b)

q

and, by taking into account Eqs. (1.9), (1.10), (2.1), (2.19), and (2.21),

(h y ( a o ~ 2 ~_~T +~ f_xfq(XI, x. x2)exp{i(e)~,,~2_ °.)~'ct"px2)}dx2dxl (o~[sqq'o~')=--l\~ep~/l\-p-/l f_,x, × 2 (°~[(R/ao)2Cq(2)l°~")(°~'l(R/ao)2Cq(2)[°~')* (2.28) +2

q=

2

for the quadrupole~luadrupole (qq) contribution to the elements of the electron scattering matrix. This result must then be integrated over collision parameters and averaged over the absolute values of the velocities according to Eq. (1.8). Unless the rotational symmetry about the z-axis, i.e. the direction of the quasistatic ion field, is destroyed, e.g., by magnetic field effects or anisotropies in the electron velocity distribution, the elements of the electron collision matrix in Eq. (1.8) should vanish unless

Am : = m~ - m,, = 0.

(2.29)

This selection rule holds for the quadrupole-quadrupole contribution (2.28), as is readily seen from Eqs. (2.27). That the selection rule (2.29) holds for the dipole-dipole contribution as well, is easy to reaffirm by the same technique. We merely rewrite the sum on the right-hand side of Eq. (1.11) as 3 j=l

+1

<~lRfq"'l~">(-1)q(~n[Rf(l-)qlO~/)(2.29a)

q = --I +1

(2.29b) q=--I

3. COMPARISON WITH PREVIOUS WORK By neglecting the 2nd and 3rd terms on the r.h.s, of Eq. (2.23), we obtain the Deutsch and Klarsfeld result 9 3

E(e, e"; e", e') = Z (elXjXJle")(~"l~XJ[C~') -

(3.1)

j=l

These author.s do not give any details on how they performed the average over the angles.

Quadrupole contributions to electronbroadeningof spectral lines

255

The expression on the r.h.s, of Eq. (3.1) may be non-zerio even for

[Am[ = 2 or 4,

(3.2)

i.e., it may violate the selection rule (2.29). The width w1" and the shift d of an isolated fine can be expressed (cf. Griem, 5 p. 51 et seq.) immediately, i.e. without inverting matrices, in terms of an average of diagonal elements of the electron collision matrix over magnetic quantum numbers as 1

w + id =

21,+ 1 ~

(nAm'lq"~'lnAm')"

(3.3)

me

Cooper and Oertel7,s calculated the quadrupole-quadrupole contribution (w + / d ) qq from this expression. In Eq. (3.3), the notation appropriate to the case of an isolated line has been anticipated. Thus, the eigenstates [a) can be chosen to be common eigenstates

lot) = In,lgn,)

(3.4)

of the operators H0, L 2 and L~, their energy-eigenvalues depending only on the principal and orbital, but not on the magnetic quantum numbers, i.e.,

E,=E,~.

(3.5)

These assumptions imply that both the eigenstates and eigenvalues are independent of the quasistatic ion field F. Under these conditions, Eqs. (2.28) and (3.3) yield

(w + i d :

= - 1-6" " , , ~ - - ~ : \ - ~ :

. . ,..

_~

1 1 x exp { i(o~,,t, - co,,.t,.) p-p-v(xl -- x2) } dx2 dxl 1 X 21,q-

× ~ Z 1 E(n,ld.n,, nell.me; n¢l,.m¢, n,lgn,),

(3.6)

m,. ao

where ... stands for integration over the collision parameter and the velocity. By taking into account the relations [see, e.g., Eq. (107.11), 12 and Eq. (5.4.6), t°, respectively] 1

E Il~= 21 + q,m"

1 I<111C(k)l Ir'>l ~

(3.7)

and

0 0]'

(3.8)

we may express the double sum on the r.h.s, of Eq. (3.6) in terms of reduced matrix elements 1

1

~ ~ E(n,l~1n,, n¢lcm¢; n¢l,.m¢, n,l~,n,)

21,+ 1 m, m,.uo = t = half of

2t,+

the ~o-callcdfull width betweenhalf-intensitypoints.

Ic(2)l It..>l 2

(3.9a)

H. PFENNIGet al.

256

and eventually in terms of 3j--symbols -- (2z=,, + l)KnJ=l(R/ao):ln3=,)l 2 ×

0

,0)

(3.9b)

•

From Eqs. (3.6) and (3.9a) follows exactly the same result for (w + / d ) qq as from Cooper and Oertel's 7'8 Eqs. (10) and (13). Eqs. (3.6) and (3.9b) yield exactlyt Eq. (151), 5. This means, that these results are proven to follow from the present Eq. (2.28) under the special conditions of an isolated line. An analogous statement applies to Eq. (17) in Griem's TM papers on the theory of wing broadening of the hydrogen line Lyman-a. In this equation, the matrix element (2.28) is calculated between spherical eigenstates [cf Eq. (3.4)] I~)=la')=121m),

m=___l

(3.10)

under the simplifying assumption % , , , = ~o,,~, = 0

(3.11)

and with the understanding that the sum E must be extended only over eigenstates with n = 2. The result is "" I

kpJ

+l"

+2

~

E

I( 2 l m[(R/ao)2Cq'2)12 1" m")l 2,

Y=Om"=-l"q=-2

(3.12a) where use has been made of

f

+cX~f X1 L ( x l ' X:) dx: dxl = 4/3.

(3.13)

The sums on the r.h.s, of Eq. (3.12a) are easy to calculate [cf Eqs. (3.7) and (3.8)]. We find ,

E

+,,,

E

+2

Z [(21m[(R/ao)2C, ml21"m")[ z=

l" = 0 m" = --l" q = --2

~ [(2 l](R/ao)212 1")12 3

IP

/" = 0

= 1(2 ll(R/a0):12 l)l: × 3 × = 30: x (3 × (2/15)).

0 (3.14)

By combining Eqs. (3.12a) and (3.14) we find exactly Griem's corrected ~ result in his earlier Eq. (17), 6 i.e., (2 1 mlS,qq[2 1 m ) = - 4 8

a0

\mepV l \ p I

m = ~ 1.

(3.12b)

Whereas in Eqs. (3.9) and (3.12b) some results of Cooper and Oertel and of Griem, respectively, have turned out to be special cases of the more general results in the present paper, Eq. (107) in Griem's monograph 5 is obviously incorrect.t The latter equation reads,

tOne should take notice of the factor (3/4) in Griem's definition of the quadrupole broadening and shift functions!

Quadrupole contributions to electron broadening of spectral lines

257

in the notation of the present paper, as follows~f 2 (

h

~2(a0~2~-,

(~ls:ql~> = - ~ \ ~ , p v / \-~ / ~ (~l(R/a°)~l°'")(~"l(R/a°)~l°'>'

(3.15)

This equation is meant to hold for the diagonal elements of the electron collision operator of hydrogen lines. It is based, as were Eqs. (3.12), on the assumption (3.11), but the {~), I~t") are now understood to be eigenstates, with one and the same principal quantum number and in parabolic coordinates, of the Hamiltonian H0 in Eq. (1.1). It is easy to find a counter-example which proves Eq. (3.15) to be inconsistent with Eq. (2.28). We need only choose the parabolic eigenstate [0 0 + 1) as the initial and final state. Because the transformation between the parabolic eigenstates and the spherical eigenstates is unitary, we find in particular

IO0 +1>=121spherical +1>. parabolic

(3.16)

However, we could obtain from Eq. (3.15)

2(

(°°+lls:ql°°+l}= _-

~h

)2(a°) -p 2I ( 0 0 + ll(R/ao)2lO0+l), 2 (3.17)

\mepv/ \ p /

in which a factor 3 x (2/15), i.e., (1/(21 + 1)) I12 for I = l " = 1, is missing in comparison with Eq. (3.12b). This discrepancy is caused solely by Griem's 5 method of averaging over the directions, as the comparison of his calculations with the corresponding ones in the present paper unambiguously shows. The case s = 1 (see Eq. (2.5a) provides a typical example: Griem starts from the--completely correct---equation 4

{(p R)2(p R) 2}angles= -~ (RR) (RR)

(3.18)

and goes on by tacitly using the relation p4

=

I(RR)I ">

(3.19a)

whereas Eqs. (2.15), (2.16) yield •nsl*s=

~=l a~tl)A~(~,~t"; ~", 0t').

(3.19b)

The expressions on the r.h.s, of Eqs. (3.19a,b) would agree, if the eigenstates I• }, I~t') . . . . constituted a complete set: Z Ict")(°t"l = 1

(3.20)

in the Hilbert space. However, they do not, because they belong only to the upper level of the spectral line under consideration. We might indeed derive Griem's Eq. (3.15) from the assumption (3.20) and from Eqs. (2.28), (3.11) and (3.13), because +2

E Cq~2)(- l)qf,..._q "(~) =

1.

q=-2

t = as far as the quadrupole-quadrupole contribution is concerned.

(3.21)

258

H. PI~NNIG et al.

Acknowledgements--For having twice been a guest and for having enjoyed very good working conditions on each occasion, one of us (llunga S. K. Mwana Umbela) is greatly indebted to the Max Planck Institute, Prof. R. Kippenhahn, and Dr. E. Trefftz. We thank Dr. J. Kirk for help with the preparation of this English version.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

REFERENCES M. Baranger, Phys. Rev. 111, 481 (1958). M. Baranger, Phys. Rev. 111, 494 (1958). M. Baranger, Phys. Rev. 112, 855 (1958). A. C. Kolb and H. R. Griem, Phys. Rev. 111, 514 (1958). H. R. Griem, Spectral Line Broadening by Plasmas. Academic Press, New York (1974). H. R. Griem, Phys. Rev. 140, A 1140 (1965). J. Cooper and G. K. Oertel, "Stark Broadening of Isolated Lines of Neutral Atoms", JILA Rep. No. 99; Boulder, Colorado; Joint Institute for Laboratory Astrophysics (1969). J. Cooper and G. K. Oertel, Phys. Rev. 180, 286 (1969). C. Deutsch and S. Klarsfeld, Phys. Rev. A 7, 2081 (1973). A. R. Edmonds, Drehimpulse in der Quantenmechanik, BI-Hoehsehuitasehenbuch 53/53 a. Bibliographisches Institut, Mannheim (1964). H. R. Griem, Phys. Rev. 144, 366 (Erratum) (1966). L.D. Landau and E. M. Lifschitz, Lehrbuch der theoretischen Physik, Band III: Quantenmechanik, 6. Auflage. Akademie-Verlag, Berlin (1979).