J . Phys. 8: At. Mol. Opt. Phys. 28 (1995) 565-517. Printed m the UK

Wing formulae for plasma-broadened spectral lines of hydrogenic ions

M J Seaton Depanment of Physics and Astronomy, University College London, Cower St. London WClE 6BT. UK

Received 24 August 1994

Abstract. Spectrum lines of hydrogenic ions in plasmas xe broadened by both ion and electron perturbers. In the present paper the ion contributions are calculaled using conventional quxi . static theory, and the eleclron contributions using n quantum-mechanical one-perturber theory. In many previous papers it has been assumed that quasi-static theory can be used to calculate the electron contributions in the line wings. The validity of that assumption is discussed Comparison with the results of the quantum.mechanical calculations shows that the quosi-static theory gives electron wing profiles in error by factors ranging from abour 0.5 to 2.0.

1. Introduction

Broadening of spectral lines in plasmas is caused by interactions with both ions and electrons. The ions have small thermal velocities and are usually assumed to be quasi-static and to produce electric microfields F which give Stark shifts in the radiating atoms. For the electrons, which have higher velocities, dynamical effects must be taken into account.

Hydrogenic ions present a special case due to the degeneracies of their ( n , I ) states. These degeneracies have two consequences: for the ion contributions they give Stark shifts which are linear in F for F sufficiently small, and for the electrons they give large contributions from many large impact parameters, which makes it practical to use comparatively simple theoretical treatments.

The use of quasi-static theory for the ions gives good results except for regions close to line centres, where ion dynamics may be important (Stamm and Voslamber 1979, Griem 1979). In most previous treatments the electron contributions to the profiles have been calculated using semi-classical theory but that can lead to some difficulties in line wings (where the outgoing electron has an energy different from that of the incoming electron). Some use has been made of quasi-static theory in calculating the electron wing-contributions. In a previous paper (Seaton 1990, to be referred to as ADOC XII-paper XIII in a series ‘atomic data for opacity calculations’) the electron contributions were calculated using a purely quantum-mechanical approach (use of Born and Coulomb-Born methods of atomic collision theory). The purpose of the present paper is to give a more detailed discussion of the profiles in the line wings.

2. Units and notations

Atomic units are defined by taking e = m = h = I. where e is the electrostatic charge on the electron. The atomic unit of energy is the Hartree (1 Hartree = 27.212 eV). In

09534075/95/040S65t13%19.50 0 1995 IOP Publishing Ltd 565

566 M J Scaton

order to avoid many factors of f we use Rydberg units of energy (2 Rydberg = I Hartree, I Rydberg = 13.606 eV). The energy of a hydrogenic ion of nuclear charge Z in Rydberg units is E(n) = - Z 1 / n 2 . With our adopted units the angular frequency w of a photon is numerically equal to the photon energy. We use thermal energies, ( k T ) , in Rydberg, (kT)Rydberg = T/157 887 with T in K. In formulae we use electron densities, Ne, in atomic units, N,(atomic units) =.:Ne

In plasmas more highly charged ions occur at higher temperatures and it is convenient to use a Z-scaled temperature variable ( k T ) / Z * . In most plasmas an ion with charge Z has maximum abundance for ( k T ) / Z z less than unity, typically ( k T ) / Z 2 Y 0.1.

Let WO be the energy of a photon at the line centre for the line n + nr9 WO = Z211/n2 - l/n‘*I. We use the variable

U = (w- -00) ( 1 )

a = u / ( k T ) (2)

where 4 is the Bohr radius in cm.

for the shift from the line centre. The dimensionless variable

is also convenient. We take the line profile to be $(U) normalized to +m

$(u)du = I . (3) L In the present paper we assume the line profiles to be symmetrical, $(-U) = @(+U), and consider values of U > 0.

3. Ion perturbers

3.1. The Stark effecf

We define linear Stark theory to be valid when the electric microfield is constant over the dimensions of the radiating atom, and sufficiently small for first-order perturbation theory to be used, so that the Stark shifts are linear in field strength. The shifts are obtained using Stark states (n, K. m ) . Taking F to be in atomic units the energy shift is

u ( n , K , m ) = a ( n , K , m ) F (4) where

(5 ) 3n Z

a(n, K, m ) = -(2K - n + Iml + 1 ) .

For a given n, the largest value of a is

a(n, max) = 3n(n - I ) /Z , (6)

Following Hummer and Mihalas (1988) we define a critical field F&) which gives the energy of the highest component of level n to be equal to that of the lowest component of level (n + I ) :

(7) Fc(n) = Z3(n + 4)/(3n4(n + I ) ’ ) .

At F = F,(n) we have a shift for the highest component of level n of U,@) = a(n. max)F&),

U&) = ~ ’ ( n + f)(n - + I)’). (8)

uC, that is to say when the Stark states for level n (calculated using first-order theory) start to overlap those for the next

The use of linear Stark theory begins to fail for U

Wing formulae for plasma-broadened spectral lines 567

higher level. The failure is complete for U >> U,. The difficulty cannot be overcome by using a more exact theory of Stark splitting in constant fields, since for U 2 n, the fields also vary significantly over the dimensions of the radiating atom. For a singly-charged perturber (a proton) the field at distance r is F = I/r*. We put Fc = l / r : , defining r,. The mean radius of an atom in level n is r, = (5n2 + 1)/(4Z) and the condition for the field Fc to be constant over the dimensions of the radiator is r, >> r,. We obtain

For 2 = 1 this gives rJrn = 2.504 for n = 2 and 4.694 for n = 10; smaller values are obtained for larger values of Z.

In the present paper we consider profiles @(U) only for 0 < U < uc. It is generally convenient to use a Z-scaling of the shift U, which is achieved using the variable u / u c and noting that uC o( 2’.

3.2. The Stark profiles

A line for transitions between two levels n and n’ has components j for transitions ( n , K , m) -+ (n’, K’, m’) .

Each component j has a fractional transition probability A,,

C A j = l

and a shift U = a j F where

a, = a(n , K , m ) - a(n‘, K‘, m ‘ ) .

The complete Stark profilet for U > 0 is

where &(U) is a Dirac delta-function, E;’) a sum over unshifted components (those with a, = 0) and cy a sum over components with Q, > 0.

3.3. The microfield

The microfield distribution function P(F) is normalized to som P ( F ) d F = 1. It depends on: the net charge on the radiating atom, z = (2 - I ) ; the charge on the perturber; and the plasma parameter

a = ro/rD (13) where r D is the Debye radius and (4ir/3)riNe = I . With ( k T ) in Rydberg and Ne in atomic units,

U = ( 2 8 8 ~ 1 N , ) ~ / ~ ( k T ) - ’ ~ ’ . (14)

PH(F) = ( ~ T I N , ) F - ~ ” { 1 + (512~/75)(N,F-~’*) + , , ,} (15)

For low density plasmas with small values of a one can use the Holtsmark distribution P H ( F ) (see Griem 1974) which, for singly-charged perturbers, has an asymptotic expansion

t We use the term ‘Stark profile’ in referring to the profile produced by the Stwk effect in the ion microfield. The term is sometimes used in referring lo the complete profile produced by charged perturbers.

568 M J Seaton

where we have taken the density of singly-charged perturbers to be equal to the electron density. For the Stark profile in the wing this gives

@s(u) - AN,u-’I2 (U + CO) (16) with

A = 21 A,a:’’. (17) j

It follows from (5) that

A a Z-’12. (18) Microfield distributions for higher densities have been discussed by many authors (for

some earlier papers see Baranger and Mozer 1959, Mozer and Baranger 1960. Pfenning and Trefftz 1966, Hooper 1968). It is usual to use the quantities FO = 27r(4Ne/15)2/3, E = F/Fo, W ( ~ , a , z ) = FoP(F) and z = (2 - I). For neutral radiators, i = 0, one obtains for all values of a the asymptotic form W(c,a ,O) - (27r)’/2(15/8)~-5/2 giving P ( F ) - (2nNe)F-’lz as for the Holtsmark distribution. For charged radiators the asymptotic form is modified. Using the tabulated functions of Mozer and Baranger, or of Hooper, one finds that for larger values of E it is a very good approximation to take W ( B , a, I ) = B(a)exp(-b(a)& x W ( c , a, 0) where B(0) = 1 and b(0) = 0, giving

P(F) - CF-5’2e~p(-~F’’Z). (19) This more rapid decrease in the limit of F large is a consequence of Coulomb repulsions between the charged perturbing ions and the charged radiator.

In practice we consider only fields F < Fc3 since the whole Stark theory begins to fail for F > Fc. For small values of the plasma parameter a one can use the Holtsmark distribution for all F < Fc for both neutral and charged radiators but for larger values of a one must use more refined calculations of the microfields and take account of the difference in asymptotic forms for the neutral and charged cases.

4. Electron perturbers

We give a summary of the theory described in more detail in ADOC XIII.

4.1. The one-perturber theory

In line wings we may assume the electron profile to be determined by single collisions between the perturber and one electron. This is known as the ‘one perturber theory’. The profile depends on matrix elements

(20) (‘VI In + TZI ‘V!) where the ‘v, and Vi are wavefunctions for the system of radiator plus colliding electron in the final state f and initial state i. Neglecting exchange we can use unsymmetrized functions

‘ V c = *,,(R)Qi,i(4 (21) i’

where the $(R) are functions for the radiator and the O(r) those for the colliding electron. Using (21) and (22) and putting (T I + 7 2 ) = (R+ r ) in (20) we obtain contributions from

Wing formulae for plasm-broadened spectral lines 569

the operator R (the atom radiates) and T (the colliding electron radiates). We consider here only the former contribution: the latter one gives probabilities for free-free processes.

We make the no qiienching approximation, of retaining in (21) only those states i' which belong to the initial energy level, and in (22) only states f' belonging to the final level. This means, in effect, that we allow for collisions (n , 1) + ( n , I ' ) but neglect collisions n --t n' with n' # n.

After some manipulations involving the equations satisfied by the functions 0 and use of Green's theorem we obtain expressions in terms of matrix elements

(h(R)ep(r)l IR - TI- ' lh(R)0dT)) . (23)

Since the interaction V = IR - T I - ' occurs as the operator in (23) we can calculate the matrix elements correct to first order in V on using zero-order functions 8 : plane waves for z = 0 (the Born approximation); and Coulomb waves for z > 0 (the Coulomb-Born approximation).

We use a further approximation, due to Bethe,

r > R IR-TI-' + R..F./r2 (24)

giving, in place of (23),

(@a IRI W . (08 l.F./r?l&) . (25)

Considering a transition between levels n and n' where n is the upper level, and taking 0, we obtain finally for the wing profile U

&(U) = ( Y " , " , ( U ) / 7 W 2 (26)

where G(n, n') is given in terms of purely algebraic coefficients (G = Z 2 G where G is defined in ADOC XIII); E is the initial energy of the colliding electron, E' = E + U its final energy; and the ( 6 , llr-2]6', I') are radial matrix elements for angular momenta I , 1'.

The expression for U"(€, E') involves lower and upper cut-offs, l&) and Ij(n). The upper cut-off depends on plasma conditions and is required only near the line centre (with U = 0 and lI = CO the sum in (29) is divergent). We define w,(O, T ) , the line- centre value calculated with inclusion of the upper cut-off, and wA(u, T ) calculated for U > 0 and 11(n) = CO and we take U I to be such that w,(O. T ) = wL(u1, T ) . We use w.(u, T ) = ~ ~ ( 0 , T ) for U < U I and w n ( u , T) = wL(u, T ) for U 2 U ] . That approximation introduces only a small error, in the immediate vicinity of U = u I (see figure 2 of ADOC xm).

The lower cut-off in (29) is determined by (i) unitarity conditions (validity of first-order theory), and (ii) validity of the Bethe approximation, r > R. The lower cut-off depends only on the properties of the upper level n , which is why we put w = W,,(U. T ) . The validity of the whole method depends on there being many values of I in the range 10 < 1 < I ] . The dependence of yA,nr on n'. the quantum number for the lower level, is given by the factor G(n , n') in (27).

570 M J Seaton

4.2. The conzplere electron profile

On choosing a suitable representation k for the transitions between quantum states, we may Put

Y = Z & Y ~ (30) A

and

$bk(u) = (Yk(u) / z )u - ’ (32) i n the line wing.

As one goes to the line centre one obtains finite values for n(0). Expressions such as (32) are clearly not valid in that limit, since they would give @1(0) to be infinite. In the line-centre region one must consider many collisions, giving the impact approximation (see Baranger 1962),

(33) &(U) = (” ( 1 2 + vk(o)2)-2 . We note that it is only the formula for +k in terms of yk which has been changed; the one-perturber theory is still used for the calculation of n(0).

The profiles must satisfy the normalization condition t m

@k(u)du = I . (34) 1, It is shown in ADOC XIJI that a convenient ‘unified’ expression for the entire profile is

(35)

This is consistent with the wing formula (32), the core formula (33) and the normalization (34). The complete profile is

&(U) = A&(u). (37) k

4.3. Numerical results for the electron contributions

The methods of ADOC have been used to write a FORTRAN code HYDPROF which is used in the opacity work. Numerical results given in the present paper are obtained using that code.

4.3.1. The electron integrals for neutrals. For neutrals, Z = I , i t is a very good approximation to take the integrals w.(u, T ) to depend only on a = u / ( k T ) . Figure 1 gives wn(a) against a for Z = I and upper levels n = 2 and n = 5. The differences between w2 and w5 are due to the different values for the lower cut-off parameter lo(n) (larger values of n require larger values of lo(n)).

The use of plane waves for 2 = I gives integrals U,(€, E + U) which go to zero in the limit of E + 0. It follows that for fixed shifts U the w, are small for small values of T. We see from figure I that the w,) are monotonic decreasing functions of a; hence for fixed U they are monotonic increasing functions of T .

Wing formulae for plasma-broadened spectral lines 57 I

Figure 1. The integrals wz and w5 for neutrals (2 = I ) against (I = u j ( k T ) . Limit of low densities (taking the upper cut-off to be Ir(n) = m in (29)). The results for

0.00 0.02 0.04 0.06 0.08 n = 2 and n = 5 differ due IO the differences in values of the lower cut-off. hdn) in (29).

0 'ir a=u/(kT)

Figure 2. The integrals tu.(", T) for n = 2 and Z = I (full curve), 2 (short.broken curve and 4 (long-broken curve) against ulu,, For temperatures ( k T ) / Z z = 0.1. Limit of low densities.

Figure 3. AS for figure 2 for n = 5.

4.3.2. The electron integrals for positive ions. Using Coulomb functions for Z > 1 (i.e. z > 0), the radial integrals ( E , llr-21s + U , 1') remain finite in the limit of E -+ 0, and the integrals w.(u, 7') remain finite in the limit of T + 0. The w,(u, T ) for positive ions cannot be expressed as functions of the single variable (Y = u / ( k T ) .

Figure 2 shows values of w,(u, T ) for n = 2, Z = 1 , 2 and 4, and ( k T ) / Z z = 0.1 plotted against u/u,(2) in the range 0 < u/u,(2) < 1, We see that use of Coulomb functions for Z = 2 and 4 gives values of wz which are larger than those for 2 = 1. For Z > 1 the results for wz are insensitive to the lower cut-off 10(2) and we obtain similar results for 2 = 2 and Z = 4: and those for Z = 00 are close to those for Z = 4.

Figure 3 shows similar results for n = 5 . For this larger value of n results are more sensitive to the lower cut-off, giving larger differences between values of w5 for Z = 2 and Z = 4.

It should be noted that the accuracy of the calculations improves with increasing Z since (i) use of perturbation theory is a bener approximation, the interaction potential V = IR - T I - ' becoming smaller compared with the Coulomb potential z/r, and (ii) the atomic radii become smaller and hence less error is introduced by assuming r > R.

4.3.3. Temperature dependence of the electron contribution. The quantity yn,p defined by (27) is proportional to Ne and has a dependence on T through the factor (kT) - ' /2 in (27)

512 EA J Seaton

n=2, n '= l . u=OSu. n-5, n'=l. u=O.5uc 3

h

2 2 - z=2 )

cl 0 ' - z=4 ,. cl

2- an v - z=4 M

0 3

Z=6

0 .a5 .I .15 .05 .1 .15

(kT)/ZZ (kT)/Z*

Figure 4. Temperature dependence of the quantities y,,,nc/Nc for n = 2, n' = I . Results for Z = 1.2.4 and 6 , and u/uc = 0.5. Limit of low densities.

Figure 5. As for figure 4, for n 5 , n' = I

and through the temperature dependence of the integrals w,(u, T ) . The dependence on T of (yn,n~/Ne) is shown on figures 4 and 5 for n = 2 and n = 5; and n' = I . We here consider just one value of photon energy, U = OSu,, and temperatures in the range ( k T ) / Z * = 0.05 to 0.15, which is of interest for many plasma experiments. For neutrals, we have already noted that the integrals U"(€, E + U ) tend to zero in the limit of E + 0 it follows that the yn,", tend to zero in the limit of T + 0. For the range of temperatures considered in figures 4 and 5, we see that the yn,", are slowly increasing functions of T for neutrals (2 = l), and slowly decreasing functions for positive ions.

5. Convolutions

If each line had just one component, convolutions would be very simple,

@(U) = 1 @ss(~ ' )@du - U') du' . (38)

A practical difficulty is that the representation of transitions j used in section 3.2 is not the same as the representation k used in section 4.2. The required formulae are given in ADOC XIII and in many previous papers.

It is shown in ADOC XIII that in the wings, sufficiently far from line centres, the complete line profile is just the sum of the ion and electron contributions.

@(Lo = @s@) +&(U). (39)

In the present paper we make little use of convolutions. When they are used (see section 7.2.2) we employ subsidiary approximations described in ADOC XIII, which avoid the need for matrix inversions.

6. Validity criteria for the use of quasi-static theory

The validity of quasi-static theory was discussed by Burkhardt (1940) and Spitzer (1940). We define a shift u p such that quasi-static theory should be a good approximation for

W i g formulae for plasm-broadened spectral lines 573

U >> up. The working formula for electron perturbers adopted by Griem (1974) is

(we take mu2 = 2 ( k T ) whereas Griem takes mu' = 3 ( k T ) ) . The formula adopted by Traving (1968) gives somewhat smaller values of U Q , by factors between 0.5 and 0.7. I n section 3.1 we introduced a critical shift U, such that linear Stark theory fails for U significantly larger than uc. Considering Lyman lines, Z = I and ( k T ) Y 0.1 and using (40) we obtain U Q u c . For most other lines one will obtain U Q > uc. It follows that for neutrals there will, at best, be only a small region in which the criteria for use of linear Stark theory and for electron quasi-static theory are both satisfied. If for positive ions we use a fixed value of the Z-scaled temperature. (kT)/Zz, we obtain U Q a Z3 compared with uc cx Zz. We then expect to obtain U Q > U, and no reg imin which one can use both linear Stark theory and quasi-static theory for the electrons.

7. Comparison of the ion and electron profiles

We compare the Stark profiles, +s, for the ions with the one-perturber profiles, &, for the electrons. We assume the perturbing ions to have unit charge and number density equal to the electron density. We consider profiles relative to the asymptotic Stark profiles with the Holtsmark distribution. @$(U) - A N , I ~ - ~ / ~ , and define

Xs = ~ ~ S U ~ / ~ / ( A N , ) Xe = & U ~ / ~ / ( A N ~ ) . (41)

7.1. The limit of low densities

For line wings in the limit of low densities we use (16) and (26) (4s = A N , u - ~ / ~ and = ( y / a ) ~ - ~ ) giving

xs = 1 x, = y ( u ) u 1 / 2 / ( ~ ~ ~ e ) . (42) Since y is proportional to Ne, X, is independent of Ne.

We note the following points.

(i) Dependence on temperature. The Stark profile has no dependence on temperature, since the ions are assumed to be static. The quantities yn,", for the electron profiles have a dependence on T discussed in section 4.3.3.

(ii) Dependence on atomic properties. The basic atomic properties required for both electron and ion perturbers depend on the matrix elements (n , 1 . mlRln, 1 & I . m'). These enter (26) for the electron profile. and are included in the calculation of the factor B(n, n') in (27). The diagonal elements of the z-components of (n. I , mlRln, 1 ;t 1, m') give the coefficients a(n. K , m ) used in the Stark theory to calculate A.

(iii) Dependence on principai quantum numbers. The dependence of 4s on quantum numbers n is contained in the quantities A (see (17)). The dependence for @e is contained in the factor G and in the n-dependence of the w, integrals (see sections 4.2 and 4.3).

(iv) Dependence on radiator charge, 2. The entire dependence on Z for 4s is given by A a Z-3/2, see (18). For & there is an explicit factor of Z-' in (27) and a further factor of Z-' if one uses a fixed value of the Z-scaled temperature, (kT ) /Z2 . If one also uses the z-scaled variable u / u c , the factor .Iiz in (42) gives a further proportionality to 2. Taking these three factors together gives a proportionality to Z-'. The dependence of the U), integrals on Z has been discussed in sections 4.3.2 and 4.3.3. In the limit of Z --f CO,

the w, are independent of Z and we then obtain X, cx Z-?

574 M J Seatort n=2. a=O

_ _ _ _ _ _ _ _ - - - - - - _ -_+ -_------ r,

2-4 .. ,,.

0 .5 .6 .7 .8 .9 1

" C 4 2=2

U 2 .5 .6 .7 .E .9 1 x'

0 S .6 .T .E .9 1

U/..

Figure 6. The qumtity X, (ratio of aymptotic electron profiles to asymptotic Holtsmmk profile) for I + 2 transitions and Z = 1.2 and 4 against u/u,. Three different temperatures are considered shon-broken curve, (kT) /Zz = 0.5; full curve. ( k T ) / Z 1 = 0.1; longbroken CYNC, ( k T ) / Z z = 0.15. Limit of low densities. The electron profile is taken to be @c(u) = ( v c ( u ) / n ) u - l .

0 .5 .6 .7 .8 .9 1

------------ . . . . . . . . . . . . . . . . . . . . .

.5 .6 .T .E .9 1

4% Figure 7. As for figure 6, for I -, S.

(v) Dependence on photon energy, U. For neutrals, the Stark wing profiles are proportional to u-5/2, The expression (26) for the electron profile contains a factor u - ~ multiplied by yn,",(u) which decreases with increasing U .

Values of X&) against u/uc are shown on figures 6 and 7 for 2 = I , 2 and 4; ( k T ) / Z z = 0.05, 0.10 and 0.15; and 0.5 < u / u , < 1.0: figure 6 for the transition I + 2 and figure 7 for I --c 5. If the quasi-static theory were valid for the electron we would have X,(u) = I in all cases.

Two trends can be noted. Firstly, X, increases with increasing T for the neutrals but decreases for the positive ions (except for the 1 -+ 5 line, 2 = 2, where there is little sensitivity). Secondly, values of X, for 2 =2 and 4 are larger than those for Z = I . That trend is halted when one goes to larger values of Z: one eventually obtains X, o( Z-'/'.

7.2. The limit of high densities

In the limit of high densities one must use microfield distributions which differ from the Holtsmark distribution and. for the electrons one must use the formulae given i n section

Wing formulae for plasma-broadened spectral lines 575

Z=1, a=0.8

_ _ _ _ - - - - - -I .A

t "=z

0 .5 1

u/u,

Figure 8. limit of high densities, plama parnmerer a = 0.8. The panmecer X for Stxk profiles (full curves) and electron profiles (broken curves). The Stmk profiles are calculaled using the microfieid distribution of Hooper (1969). The heavy vertical lines mark the point U = UI for the electron profiles (w,(u. T) = w,(O. T) for U < U ) ) . .Transitions 1 + n. ResulrS for z = I.

Z=2. a=0.8

n=4 t lt 1 0 0 - .5 1

0 I 0 .5 1

u/uc

Figure 9. As for figure 8 for Z = 2

4.2. We consider two cases of Z = 1 and Z = 2. For both we use a plasma parameter

of a = 0.8 and take ( k T ) / Z 2 = 0.1. These correspond to T = 15789 K and Ne = 1.955 x IO'* cm-3 for Z = 1, and T = 63 155 K and Ne = 1.251 x lozo c n r 3 for Z = 2. We do not allow for possible effects of electron degeneracy. We use the microfield distributions of Hooper (1968) for a neutral radiator and a singly-charged radiator.

7.2.1. Neutrals. Figure 8 shows values of X ( u ) for Z = 1: Xs for the ions (full curves) and X, for the electrons (broken curves). The vertical bars on the electron curves show values of u I defined in section 4.1: for U < U ] the electron profiles reduce to their core Lorentz form (33) and additional screening is included in the calculation of the lower cut-off L,(n).

7.2.2. Ions. Figure 9 show similar results for Z = 2. Repulsion of the perturbing ions by the charged radiator gives a reduction in the values of Xs. In the far wings for the I --f 2

516 M J Seaton

0 ' ........................... I , I , I , I , l

0 .2 .4 .6 .E 1 u/u.

100 -a

50 Figure 10. Complete profiles for the high-density c a e s of a = 0.8, Z = 2 and 1 -+ n transitions. Full curves,

long-broken curves, electron profiles. Note that the ion 0 .05 .1 .15 profile, as plotted for n = 2, is not normalized without

inclusion of the delta-function in (12).

convolved profiles, short-broken CUNCS, ion profiles;

u/u.

transition, X, is a good deal larger than XS. Figure 10 shows (for a = 0.8, Z = 2 and transitions 1 --f 3 and 1 -+ 3) the full

convolved profiles (full curves), the ion profiles (short-broken curves) and the electron profiles (long-broken curves). For the 1 -f 2 transition we consider the restricted range of U < 0 . 1 5 ~ ~ since the profiles are very small for U > uc.

8. Conclusions

We use U = (w - 00) for shifts from line centres and consider plasmas in which the perturbers are either electrons or protons. We define linear Stark theory to be that for which the elecoic microfield is constant over the dimensions of the radiator and sufficiently weak for Stark shifts to be linear in the field strengths. That theory is valid for U < U, with uc given by (8). Except for regions close to line centres, the protons can be considered to be quasi-static and to give profiles @s(u) calculated for U < uc using linear Stark theory.

In many previous papers it has been assumed that, i n line wings, quasi-static theory can also be used to calculate profiles &,(U) produced by the electrons. If that is correct one has &(U) = +&) in the wings. We have adopted a standard expression, equation (40), for a critical shift U Q such that quasi-static theory is valid for the electrons for U >> uQ. We find that U Q is either not much smaller than uc or is larger than uc. It follows that there are no regions in which linear quasi-static theory can be used with confidence for the electron perturbers.

We have calculated electron profiles using a quantum-mechanical oneperturber theory and have compared the profiles &(U) and @$(U) for both low densities and high densities. We find that, in the line wings, the ratios @&)/~s(u) vary from about 0.5 to 2.0 (but may be smaller for highly-charged radiators at low densities). Our general conclusion is that use of quasi-static theory for electron wing profiles will not give results of high accuracy but,

Wing formulae for plasma-broadened spectral lines 577

on the other hand, will not lead to gross errors. The errors which result from uncertainties in the cut-off parameters 10 and 11 in the Bethe

approximation are discussed in appendix D.3 of ADOC XnI. Further errors will result from the neglect of higher-order multipoles for the (nl + nl k 1) transitions and from the use of the no-quenching approximation. It would now be possible to make much more accurate calculations (at least for n and n' not too large) using matrix elements of the type (20) and accurate functions 'v, and Yi from close-coupling calculations. Such calculations are being attempted using the code STGFF from the opacity project package (see Benington et a1 1987).

Acknowledgments

I thank Drs Carlos Iglesias and Gillian Peach for helpful comments on the present work

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