J. Phys. B: At. Mol. Phys. 17 (1984) 1477-1489. Printed in Great Britain

Stark broadening of the H a line of hydrogen at low densities: quantal and semiclassical results

C StehlC and N Feautrier D6partement d'Astrophysique Fondamentale, Observatoire de Paris, 92190 Meudon, France

Received 16 June 1983, in final form 2 December 1983

Abstract. Stark profiles of the Ha lines of hydrogen are computed at low densities (from 10" to l O I 3 in the 'impact' theory. By a comparison with quantal results we show that a simple semiclassical perturbational approach with appropriate cutoffs is sufficient to give accurate profiles in the line centre. Neglecting the natural broadening and the fine-structure effects, we prove that the electronic broadening is negligible and that the profile has a Lorentzian shape, and we give an analytical expression of the half width.

1. Introduction

In the past years, many experimental works have been devoted to the Stark broadening of hydrogen lines but all were 'restricted to electronic and ionic densities larger than 1014 ~ m - ~ . Beyond this density, the Doppler width begins to mask the Stark broadening. Nowadays, the advent of high monochromatic tunable laser sources has led to the development of new non-linear spectroscopic techniques which allow high-resolution Doppler-free observations of usually unresolved profiles in plasmas (Weber and Hum- pert 198 1).

In such spectroscopic experiments, the hydrogen atoms are submitted to simul- taneous interactions with the radiating field and multiple perturbers. The difficulties encountered in the calculation of the resulting profile are of two kinds. The many-body problem of the fluctuating interactions between the radiator and all the surrounding perturbers has to be treated. The second difficulty arises from the simultaneous radiative and collisional processes. In the impact limit, these two problems are avoided (Baranger 1958, Cooper 1967). As proved elsewhere (StehlC et a1 1984) by a comparison with model microfield method results, the impact approximation is suitable at low perturber densities (Npd 1013 cm-3 at T = lo4 K for H a ) in the line centre, but fails in the wings and at higher densities.

Our purpose here is to prove that a semiclassical perturbational description (SCP) of the collisions is sufficient to give accurate profiles. This picture leads to divergencies at short impact parameters and as usual this difficulty is avoided by appropriate cutoff procedures. The strong collisions contribution can alter significantly the shape of the profile, therefore we test the validity of this approximate method by a comparison with the results of a non-perturbative quantum approach (exact resonance method QER) (Seaton 1961).

0022-3700/84/081477+ 13$02.25 @ 1984 The Institute of Physics 1477

1478 C Stehle' and N Feautrier

We have introduced some simplifications in order to obtain more tractable calcula- tions: neglecting the natural broadening, we have assumed a complete degeneracy of the states with the same n number. These assumptions usually made (Cooper et a1 1974, Vidal et a1 1973), but invalid at low densities, do not allow the calculated profile to be compared with the experimental ones. However, the conclusions obtained here concerning the description of the strong collisions in the SCP theory are useful since an extension of this method to the realistic situation where the preceeding effects are taken into account is relatively easy.

2. General formulation

Our first task is to relate the line profile expression to the collision S-matrix elements in the impact limit (Fano 1957,1963, Blum 1981, Omont 1965, Cooper 1967, Lewis 1980).

2.1. The line shape

Hereafter we use greek and roman letters to indicate the upper and lower spectroscopic states, with energies satisfying the relation E, - E a 5 hw,, 5 Aw.

Let us define the relaxation operator G of the line space spanned by the double-atom states la.)) (Baranger 1958) by:

G = 2 N ~ U ~ P

and

(+;,,bp = -Trppp(SabSzp - aabaup). (1) In this expression N p is the perturber density, pp the diagonal density matrix of the perturber bath in thermal equilibrium. From energy conservation considerations, the scattering operator elements sa, and Sap have to be calculated at the same relative kinetic energy.

It is easy to show that the w-normalised line shape is given by: -1

F ( w ) = 77 c Darr,aa(Paa -pa,)) Im ( @ ( w ) ) a a , b p D b p , a n ( P a a - P a a ) (2) ( aa a d p

where paa(p,,) are the population terms of the states a(a) ; Q, and D are operators in the line space defined by:

2.2. Expression of the relaxation operator G

Equation (2) gives the general expression of the profile in the impact limit. Collisional effects are given through the operator G which includes an average over the Maxwellian velocity distribution f ( v ) at temperature T and an integration over all impact para- meters b in a semiclassical treatment or a sum over all partial waves in a quantal description of the collision.

Stark broadening of the Ha line of hydrogen 1479

As discussed by Omont (1977), the correct procedure for the velocity average is to assume a dependence of F ( w ) and G in ( 2 ) and (3) on the atom velocity uA and to average the signal over uA. This anisotropy is of a large effect for perturber of infinitive mass (Seidel 1979), but may be neglected for lighter perturbers (protons for example). Therefore, we shall neglect this effect and use the mean relaxation operator defined by:

G = Np lom uf( v)q ( v ) dv (4)

In this expression, the relative velocity U = Q.- uA corresponds to the particle of reduced mass m = mAmp/ (mA+ mp) at the distance r = rp-rA from the mass centre. T = TI( TE) for ionic (electronic) perturbers. q(v) is the average of (1-SS-’) over all collisions at the relative velocity U. It includes an average over the three Euler angles which relate the collision frame to the laboratory frame. We can inversely fix the collision axes and rotate the mass centre coordinate frame. The rotational invari- ance of G, U and q make this angular average easy (Baranger 1958, Lam 1977, Shafer and Gordon 1973, Peach 1981) in terms of the reduced operators related to G, U and q (Messiah 1961). The profile expression becomes in this reduced line space (Lam 1977):

-1

~ ( w ) = ( r yam) ~m all, c ((ida Ile2Dllibip>>((ibip Il*(w)llida>> ( 5 )

with ((ida Ile2Dllibip)) = ( i Q l l ~ ~ l l t > ( i b l l e ~ l I i ~ )

((idp II *( 0 - II ida >) = ( 0 - waa aabSap - iNpa jds.jd,

j a ( j a ) and j b ( j p ) denote the angular momentum of the radiating atom in its spectropic states before and after the collision and Yo, the line strength of the transition (aa).

In the semiclassical approach, the relative rectilinear motion is defined by the velocity U and the impact parameter b. The reduced operator (I becomes

with:

In the quantum approach, the integration over the impact parameter is replaced by a summation over the orbital angular momentum Z of the relative motion. After the rotational average, (I is given by (Shafer and Gordon 1973)

with:

1480 C Stehle‘ and N Feautrier

where J,(J,) is the total angular momentum invariant in the collisions and l (1’) the angular momentum of the relative motion

J, = j , + 1 = j b + I’ and J, = j , + 1 = j p + 1‘ .

For large values of the angular momentum 1, J, = J, = 1 = 1’ and the comparison between the quantal and the semiclassical approaches is easily performed using the relation 1 = kb, where k is the wavenumber corresponding to v.

As usual, we assume that the interaction potential V(r) between the radiating atom and the charged perturber vanishes for r > rD ( rD, Debye radius) and we introduce an upper limit bD = rD in the integration on the impact parameter b in (6) and similarly an upper J, value corresponding to lD = kbD in (8). The slow ionic motion is screened by the reaction of electrons and ions in the plasma, in contrast to the fast electronic motion screened by electrons only. We have thus: rD = ( k T D / 4 ~ & e 2 ) ” 2 with TD = T E

for electrons and TD = TETI/(TE+ TI) for ions (Baranger and Mozer 1959, Mozer and Baranger 1960, Chen 1977).

3. Relaxation matrix of the hydrogen H a line

The SCP treatment leads to a very simple analytical formula for the halfwidth but this method, afflicted by singularities at short impact parameters (breakdown of the per- turbation theory), requires the introduction of lower cutoffs. The quantum collisional approach avoiding this problem (no divergency at low 1 values) is useful to test the semiclassical results and the cutoff procedure. Thus we compare the variations of P,a,bp(b, v ) and p?,,bp(L,, v ) as functions of b and L, in the semiclassical (7) and quantal (9) methods. Owing to the assumed degeneracy of the sublevels of the hydrogen atom, the coupling between the states with different principal quantum number is negligible and the dipolar part of the interaction is predominent. This usual dipolar approximation will be justified farther. The nlm representation is now used.

3.1. Perturbational calculation of the relaxation matrix in the semiclassical ( SCP) method (Sahal-Brbchot 1969, Peach 1981 )

S matrices are calculated using the second-order perturbation expansion: +m +m

S= l - ih - l v(t) dt-h-’ [-, dt, dbV(t1) V ( ~ Z ) (10)

where V denotes the electrostatic interaction in the dipolar approximation for large relative distances r. The angular average (7) is readily performed and the atomic part of Pa,,bp can be expressed as a function of the atomic line strengths for n = 2 and n = 3 (see Appendix). We obtain the following result in the reduced line space of states I S T ) ) , Ipcr)), and Ips)):

and

with

kP.7.P.7 = 114 kp6,p8 = 33 - k,,,,, = 99

kS,,Pw = 1246 k,,,,, = -12J7.5 kPW,,* = 0

Stark broadening of the Ha line of hydrogen 1481

where E = i m v 2 is the collision energy, a. the Bohr radius, IH the ionisation energy of hydrogen and me the electronic mass.

The b integral of Paa,bp(b, E ) in ( 6 ) leads to a logarithmic divergency for low impact parameters b corresponding to strong collisions. This divergency is avoided by a cutoff procedure. The choice of this cutoff is somewhat arbitrary and has been discussed in several papers (Griem et a1 1959, Kepple and Griem 1968, Bacon and Edwards 1968, Bacon eta1 1969, Bacon 1971). Following Omont and Meunier (1968) , we assume that the optical coherences are completely destroyed after a strong collision, and that the probabilities for excitation

gfm,,,,,,,(b, E ) = I(nlm(l-S(b, E)(nl’m’)(’ (12) are equal for each transition lm # l’m’.

Using the unitary properties of the S matrix and the second-order perturbation expansion, the diagonal matrix elements of P are simply connected to these excitation probabilities by:

Using the conservation relation for a given level n, we define two cutoff parameters rc(E) for n = 2 and p , (E) for n = 3:

p c ( E ) =J6 rc(E) = a. 108 - ( :=y2 and we use the following strong collision contributions:

and the equivalent form for n = 3 . It follows that Pa,,,, (b, E ) = 1 for b s rc(E) . This cutoff procedure will be discussed in the following section by a comparison with the corresponding quantum results.

3.2. Relaxation matrix in the exact resonance approach ( QER)

The exact resonance method, proposed by Seaton ( 1 961), gives explicit expressions of the S-matrix elements in the approximation of retaining only the long-range part of the dipole potential ( r -*V(r ) ) . It is unnecessary to explain here this method which was previously described with details by Seaton (1961) and has already been used in Stark broadening of hydrogen lines (Tran Minh et a1 1976, Feautrier et a1 1976, de Kertanguy et a1 1979). Let us notice only that the resulting S matrix is independent of the kinetic energy E for a given value of L, and, cansequently, PQ(La, v ) defined by ( 9 ) will be denoted farther by PQ(La) . To obtain the relaxation matrix, S matrices have to be computed for the two groups of states nlm associated to n = 2 and n = 3 . Contrary to the SCP approach, the imaginary part of PQ(La) is not equal to zero.

The QER approach excludes in the calculation of the S matrix the low L values corresponding to the possibility of attachment. The lowest values are for n = 3Linr = 6 for electrons and Linf = 181 for Ar+ perturbers. However, due to their small contribu- tion, the exact inclusion of the lower values of L is of little interest for the Stark broadening and would in addition lead to the introduction of short-range potentials.

1482 C Stehle' and N Feautrier

For large values of L, the trajectory notion has a physical meaning and, it is easy to show that the relaxation matrix obtained in the SCP method corresponds to the exact resonance results.

3.3. Results

We shall first note the very large Debye radius at low densities (2.92 X lo4 a. at lo4 K, l O I 3 ~ m - ~ , for ionic perturbers) in comparison with the average atomic radius in the states n = 2 (-6 ao) or n = 3 (-13.5 ao). This justifies the use of the dipolar approxi- mation.

In order to test the validity of the results of the SCP theory, we compare the contributions PQ(La) of each angular momentum La (or of the corresponding impact parameter b) obtained in the QER and SCP approaches. As an example, we plot the real and imaginary parts of P$,ps and PE,pB against La for electrons (figures 1 and 2) and Ar+ perturbers (figures 3 and 4). We found a fair agreement between the results at large values of L,(L, > 10 for electrons and La > 10 000 for Ar+) and we observe that the imaginary contributions are negligible in comparison with the real ones.

Let us now discuss the cutoff procedure proposed by equation ( 5 ) . In the electronic case, this relation gives a corresponding angular momentum L2,, = 5 for n = 2 and L3,, = 10 for n = 3 and this choice seems to be correct. However, for La < L,, our assumption of vanishing coherences in the strong collisions regime fails in agreement with the results of Bacon and Edwards (1968) and Bacon et a1 (1969). Nevertheless, this cannot affect the resulting profile because:

(i) the contribution of the small values of La (or b ) is not significant;

I I I I I

a% -3 : - & y , - , , , j 6 10 15 20 25

LO

Figure 1. Real and imaginary parts of P$,,, (L,) against La for electronic perturbers (full curves). - - -, the asymptotic behaviour 33(m/m,)2L,2; --- , P,,,,, ( b , v ) expressed in terms of La (real quantities).

Stark broadening of the H a line of hydrogen 1483

Figure 2. -, real and imaginary parts of Pz, ---, the real asymptotic behaviour -12 ( 7 . 5 ) ' l ~ m / m , ) 2 L ; Z .

against La for electronic perturbers.

I I I I I

1 I I I 1 So00 loo00 lSOO0 20000

Lu

Figure 3. Same as figure 1 for Ar+ perturbers (m = 1792 me),

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40-‘

C Stehle‘ and N Feautrier

14 CI L3,c 1,’ I

I 1 I

4 - -

I I I I I I I 10000 20000 30000

L O

Figure 4. Same as figure 2 for Ar+ perturbers (m = 1792 me).

(ii) owing to the rapid decrease of PQ for large L, values, the mean electronic contribution is far less than the ionic one, as can be observed by comparing the figures 1 and 2 with 3 and 4.

In the case of Arf perturbers, the cutoff angular momenta are L2,+ = 8000 for n = 2 and L3,c= 19 000 for n = 3. The diagonal terms (figure 3) oscillate about 1 for L, < L2,+ and the mean value of the coherence terms is zero in agreement with Omont’s assumption. Similar features are obtained for the other elements of PQ.

4. The line profile

The line profile is easily obtained after a summation over L, (or over the corresponding impact parameter b ) and an average over the velocities distribution. As expected, the electronic contribution to the broadening is negligible (< 10%) compared with the ionic one. The results obtained in the QER and SCP methods are in good agreement, which corresponds to reliable cutoff procedures (figure 5 ) . The principal features of these profiles are their Lorentzian shapes. This may be surprising since the line results from several interfering components. We analyse now this point and give an analytical formula for the line width.

The matrix corresponding to the operator e2D in the reduced line space can be expressed in terms of the line strengths YQa. The computation of the profile (equation 5 ) is easier if the eigenvalues (Yl, 0,O) and the eigenvectors 5 of the matrix are introduced:

Stark broadening of the Ha line of hydrogen 1485

I I I I I 0 1 2 3 4 x10

A A (A)

Figure 5. Stark Ha profiles for Ar+ perturbers at T, = TE = lo4 K and Np = 10" cm-? quantal and SCP calculations. The Lorentzian profile of halfwidth given by (20) is undistin- guishable from the quantal profile.

Using (16 ) , the profile is given by:

~ ( w ) = v-l ~m{[(w-o,,)l-i~~~.]-~}~~ (17)

d = g-lag. with

Each element eij of 6 can be computed quantally or semiclassically using the results of the previous sections. Let us introduce the operator PQ = &-'PQg of the line space, where Po is defined by expression (9); P z , P$ and P$ are the only relevant contributions to the profile calculation. These quantities calculated in the QER approach for electronic and Ar+ pcrturbers are compared in figures 6 and 7 with the same quantities evaluated in the SCP treatment and given by:

P12( b, E ) = P13( b, E ) = 0 (18 )

and

mIHaE P11(b ,E)=18- m,Eb2

for large values of b.

We observe: (i) the good agreement between the two approaches at large values of La; (ii) the very small contribution (equal to zero in the SCP approach) of p$ and P$

in comparison with PE, which explains the Lorentzian shape of the profile; (iii) the rapid oscillations of P g about one for strong collisions in the ionic

perturbers case. This authorises us to take as the cutoff value in the SCP approach, the r, value satisfying Pll(rc, E ) = l . In fact, this value corresponds exactly to the cutoff parameter r2,, for n = 2 (14) . Then, it is interesting to notice that cancellations between the profile contributions of the diff erent terms of the relaxation matrix enhance the validity of the SCP approach so that the results are almost insensitive to the cutoff choice.

1486 C Stehle‘ and N Feautrier

I I I I I I

LO

Figure 6. -, real parts of pz(L,), pg(L,) and p$(L,) for electronic perturbers. the semiclassical term pll( b, v ) = 18(m/m,)’L;’ expressed in terms of Lo.

I I I

1.5

t i

I I I I 5000 10000 15000

L O

Figure 7. Same as figure 6 for Ar+ perturbers ( m = 1792 me).

Taking PI, = 1 for b -= rc( E ) , the SCP method leads to the ‘following expression for the ionic contribution to halfwidth:

Stark broadening of the Ha line of hydrogen 1487

The term 1 corresponds here to the mean strong collision contributions ( r < r,) and 2 In( rD/rc) to the weak ones. Performing the integration over the velocities distribution, we obtain finally:

yi = 1.55 x 10-~ zvP (:)1'2k[27.54+ln(y:)] - ( H z c ~ - ~ K ) .

The results obtained with this analytic expression are in excellent agreement with the quantum ones (figure 5 ) and it is easy to show that electrons have a negligible contribution ye to the total halfwidth y = yi+ ye.

5. Conclusion

By comparing the quantum and semiclassical approaches, we have shown that the semiclassical model is well adapted to the line broadening calculation: this results from the rapid oscillations of the ionic contribution to P'(L,) with decreasing L, values and from the description of the strong collisions in the semiclassical method which agrees with the mean value of these oscillations.

This comparison provides also a good test of the cutoff procedure. It is interesting to note that the SCP method is very easy to generalise to more complicated situations (other Balmer lines, inclusion of the exact energy structure).

In the Ha case, we have proved that the profile has a Lorentzian shape, whose halfwidth is given by analytic formula. Obviously, since the fine-structure separation and the natural broadening have been neglected these results provide only a good estimation of the broadening. These assumptions will be removed in a forthcoming paper. Moreover the density range for which the impact approximation assumed here is valid, will be given by a comparison with the model microfield method (Brissaud et a1 1971, StehlC et a1 1984).

Acknowledgments

We would like to thank V Bommier and S Sahal for fruitful discussions.

Appendix. Expression of P(b, U) in the SCP approach

Introducing the second-order development of the T = 1 - S matrix (10) in the relation (7), we obtain the following.

(i) Diagonal terms

where 9(l,, 1,) and 9(1,, 1,) are the transition integrals for n = 2 and n = 3, connected

1488 C Stehle‘ and N Feautrier

to the radial wavefunction of hydrogen Rnl(r) by:

$(la, 1,) = JOm ~ z l , ( r ~ ~ , ( r > r 3 dr.

In (A.l ) the time integral reduces to 2b-’v-’ and we simplify this expression in terms of the line strengths Y(liZj):

1, 1 lp b2+ v2tlt2 ( 0 0 0 ) 1-r dtl 1-r dt’ ( b 2 + v2t:)3/2(b2+ v2tz)3/2‘

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Stark broadening of the Ha line of hydrogen 1489

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