Short-range potentials in quantal and semi-classical calculations for the Ly α line

J. Quant. Spectrosc. Radiat. Trans[er Vol. 23, pp. 377-386

0022-4073/8010401-0377/$02.00/0

~ Pergamon Press Ltd.. 1980. Printed in Great Britain

SHORT-RANGE POTENTIALS IN Q U A N T A L AND SEMI-CLASSICAL C A L C U L A T I O N S FOR THE Ly a L I N E N. TRAN MINHand N. FEAUTRIER Observatoirede Meudon-Daphe-5,place Jules Janssen, 92190 Meudon, France and A. R. EDMONDS The Blackett Laboratory,ImperialCollegeof Science and Technology,London SW7,England

(Received 1 August 1978) Abstract--The "exact resonance" approximationapplied to the calculation of the electronic profile of Lyman a gives an analytical expression, which converges for large angular momentum towards the classical limit of Lisitsa and Sholin. In order to study the validityof classical treatments, we comparethe quantum "exact resonance" results with the classical limit of this "exact resonance" approach and the classical method of Lisitsa and Sholin. We then analyse the contribution of quadrupole and polarization potentials in the quantumdescription and we comparethese with the results of Caby Eyraud et al. As was pointed out by Griem,quenching effectsmay be important. 1. INTRODUCTION In the study of hydrogen-line broadening, many recent theoretical and experimental studies are focused on the wing profiles, j-3 Simultaneous strong collisions are improbable; therefore, the one-perturber approximation is valid. In two previous by published papers Tran Ming et al. 4"5, we gave the quantum expression for the hydrogenic line profile and we pointed out the necessity of a correct dynamical treatment for the collision. The profile is given in terms of overlap integrals of the perturber wave functions. The exact-resonance approximation, described by Seaton 6 and valid for Lir I> 3 in the case of Lyman a, was applied by making exact allowance for the long-range dipole interaction. An analytical expression for the profile was obtained, which converges towards the classical limit of Lisitsa and Sholin I and of Voslamber7 and Pfennig) Finally, we have shown that quadrupole and polarization interactions can be taken into account. In the present paper, we are primarily concerned with the validity of semi-classical treatments and also with the contribution of strong interactions in the wings of the Lyman a line of hydrogen broadened by electrons. Quantum calculations restricted to the first three states of the atom were made 9 for close interactions (small values of the total angular momentum Lit). For more distant interactions, the exact-resonance approximation gives good results. We compare this reference calculation (Q.E.R.) with the results of the semi-classical limit of this exact-resonance approach (S.C.E.R.) and also of the classical method of Lisitsa and Sholin (LS). We give in Section 2 the analytical expression of the Lyman a profile. In Section 3, we analyse the results for the three approaches described above in order to determine the range of validity of semi-classical theories. Finally, we show the contribution of quadrupole and polarization potentials in the quantum description. Comparisons with the semi-classical results of Caby-Eyraud et al? are made. 2. THE LINE PROFILE OFLy t~ IN THE ONE ELECTRON APPROXIMATION In the one-perturber approximation valid in the wings of a line, the quantum line shape of Lyman a at the point Ato of the profile is I(Ato)

N,N, [(L~StlId'IIL'S')I ~ ~ ( h p ( ~ ) ' "(2Si + l)(2Li + 1) ~" Jo® x (2S? + 1)(2L,r

io

+ 1)1

377

dr a*(r

J ' (2~r)3 a~4K~Kf

lSIr)a(r.lr,lr)5

(1)

N. TRAN MINH et al.

378

Here N~ is the density of atoms in the initial state, Ne is the electron density; (t-#SlltddlL~S~) is the reduced matrix element of the atomic dipole, E~ is the kinetic energy of the perturber when the atom is in the state i', K~ and Kt are the corresponding wave numbers; p(~) is the density of initial states and f(e~) the Maxwell distribution. We define a channel F by the set of quantum numbers nLSI1/2LrSrMLrMs r, where nLS denotes the atomic state, l the angular momentum of the incident electron, L r and S r are, respectively, the total orbital angular momentum and the total spin of the system, MLr and Ms r are their projections. The sum E extends over l~LirS~rn~L~l~; Ato is connected to kinetic energies of the perturber in the initial and final states i and f by the relation

1

haw = ~ m( v~2 - vf).

(2)

The radial wave functions G(l~lF~lr) are defined in terms of the initial and final channels F} and r~ for the collision problem of the upper level i. Expanding the total wave function in terms of the set of atomic wave functions [Percival and Seaton~°], we find for each Lir and Sir the following well known set of equations for the functions G(r~lr'~[r):

[dd-~-~

+

K~2]G(F~,IFilr)= ~(Vq+ W~j)G(F~,IFi[r)

(3)

I

here Vq and Wq are direct and exchange potentials. It is well known that the interaction between the perturber and the atom is negligible for the lower state [ of Lyman a. Therefore, G(rrlrrir) is the radial part of a plane wave, viz.

(4) where Jl+l/2 is the cylindrical Bessel function. The problem is how to calculate the overlap integrals

I(i', i) = I ®drG*(r lr lr)O(r,,Ir,lr). J0

(s)

2.1. Quantum expression for overlap integrals Leaving the problem of the first three angular momenta Lir, for which special treatments are necessary [Feautrier et a/.9'11], we focus our attention on larger values of Lir. Exchange is not important and the dipole part of the potential is predominant. We have shown previously that [Tran Mirth et al. 5, Eq. (46)]

I(i', i) = 1(,2- K~

drG*(rrlr~lr) ~, U,jG(r,,Irjlr)

(6)

Because of the degeneracy of the levels 2s and 2p of the hydrogen atom, the couplings between the sublevels of a given level are predominant. For the Ly a line, it is possible to limit the expansion of the total wave-function to the three terms Is, 2s, and 2p of hydrogen; therefore, the interaction potentials Uq contains only dipole (A = 1) and quadrupole (A = 2) contributions. In order to take into account the contribution of the closed channels and for large values of r, it might be possible to replace the coupling terms Uq G(F~IFjIr ) in Eq. (3) by a single diagonal potential term: the polarization potential [Castillejo et al., 12 Feautrier et al.13]. The exact resonance approximation of Seaton6 gives analytic radial wave functions when we retain only asymptotic forms of the dipole potential. Equations (3) may be written as d2

(7)

Short-range potentialsin quantal and semi-classicalcalculationsfor the Ly a line

379

where Air 2 contains potentials V~j and centrifugal terms. To solve Eq. (7), we diagonalize A. We obtain eigenvalues am and the unitary matrix of eigenvectors X. Setting

(8)

am =l~m(t~m+l) with 1

~t-L, = 1;

r/

lX 2

].¢2=--~+ t[kl--~) -CJ

"1v2

1

; I$3=--~+

l-t-

2_

C

]'"

(9)

we have shown that G(F~IFilr) becomes [Tran Minh et al. 5 Eqs. (41) and (36)]

/ ,fir\ 1/2 G(rrlr,lr)=-2i(T ) ~XrmX.im e mi-t')'~12].,. +,,,21,tkr'i,

(10)

To calculate integrals in Eq. (6), it is possible to replace exact radial functions by their approximate expression (10). Thus, we introduce perturbation theory in order to calculate the quadrupole contribution and to take into account the effect of closed channels (polarization potential). With this approximation, we have shown [Tran Minh et al., 5 Eq. (49)] that

l(i', i) - Ki

Z Xi'mX]m e-i~"~/2 f f K7 eU"n ixrn 22•_

d rJq+,n( Ktr)J~, m+m( Kir ") u" ~,i

(11)

We set U~ ) -_A~,~ r~+l~ and M q = f o drJtt+ln(Ktr)r-*J~'+ln(Kir)

(12)

Evaluation of these integrals M q leads to hypergeometric functions [Cradshteyn and Ryzhik~4]. Expressions of I(i', i) for contributions of A = 1, 2, and 3 are given elsewhere [Eqs. (50), (51), (52), (55), (56) of Tran Mirth et al.5]. 2.2. Classical limit o[ the quantum-mechanical radial matrix elements For large values of the perturber angular momentum, one can use confluent hypergeometric functions [Erdelyi et al. ~5] to obtain the classical limit of M ~

Mq._.~Msemi_elassieal=h ~pmo)~(A-I)/2 ~,-~'-v]

1

W_(v/2)_(A/2)(2p_~)

(13)

F( "A+ 2 )- 1 ),

where p is the usual impact parameter defined from the angular momentum l of the perturber by the relation

l = Kp

(14)

and u = If - ~., for the blue wing and u = ~,, - lt for the red wing. W and 1" are Whittaker and Gamma functions [Abramowitz and Stegun16]. Using the JWKB approximation and assuming a straight trajectory for the perturber, the same expression for M ~"~-¢1~i¢~ is obtained ~/s a limiting case for the charge Z ~ 0 of the Alder et al. ~7theory of Coulomb excitation of ions. It is obvious that this classical limit assumes that ! is large and it is easy to obtain the semi-classical limit given by Eq. (13) directly from the limit of hypergeometric functions for large angular momenta. The summation over angular momenta in the expression of the profile is transformed into an integral on the impact parameter, according to

KTKtZr~L~r(2L;r + 1)---~

f

2¢rp do.

(15)

380

N. T ~ s Miss et al.

This semi-classical limit of the profile should be compared in the dipole approximation (A = 1) with the expression given by Lisitsa and Sholin, ~

I(O)= l h ~2

(16)

where

\ Vo/

with e2o0 ~=~h A o.~ot

B = ..-zTr-(Vois mean velocity) vo

and ¢r3 oo

dx

2

2

= fo x(i Wx,)[K,/.+x%_l(x/3) + Kv.+~-2)+1(x ~) + 2x

2

KI(xB)].

(17)

Here K~(a) is the Bateman function, which is related to the Whittaker function by K,(a) = Wcz,ln(2a)/F(1 + 3,/2)

(18)

x is connected to the usual impact parameter by x=

mvop 3h"

(19)

Assuming large values of angular momentum l (or p-> I/K), it is easy to show that our semi-classical limit of the exact resonance approximation is identical to that of Lisitsa and Sholin by setting v = - YIt is interesting to observe that integration over x (the impact parameter) is limited to p > Pmin (Pmi, corresponding to Lir = 3) in our semi-classical approach but can be performed from x = 0 in the theory of Lisitsa and Sholin using an exact solution of the dynamical problem in the semi-classical description. The difficulty in the exact resonance approximation comes from the strongly attractive dipole potential g ( g + 1)/r2 (with g complex) between the electron and the atom for Lir < 3. This potential gives oscillations of the inelastic cross section in the neighbourhood of the threshold and resonances in the elastic electron scattering by hydrogen. These purely quantum effects were taken into acccount in Feautrier and Tran Minh," but such phenomena cannot be found in a semi-classical theory. 3. RESULTS

In a previous paper [Feautrier et a/.9], we have shown that the correct average over the Maxwellian distribution of velocities must be evaluated. Therefore, we replaced v0 by the correct average in the expression of Sholin and Lisitsa for the profile. Calculations are carried out at the density and temperature of Boldt and Cooper's experiment ~8 (T = 12,200°K, Ne = 8.4 × 1016 cm-3). The hypergeometric functions were calculated by two different methods with a relative accuracy better than 10-4. Whittaker functions present computational difficulties. We use a method described by Edmonds and Kelly. 19 This method is based on use of a recurrence relation between contiguous Whittaker functions and a Chebyshev series expansion of the starting functions.

Short-range potentials in quantal and semi-classical calculations for the Ly a line

381

3.1. Comparison between quantum and semi-classical approach We write the profile as a sum of the contributions ILF of each angular momentum L~r. The usual correspondence Lir-,K~o allows us to define the contributions ILF in the classical description by an integral over the impact parameter from PLF-~ to p~T with PLF-~ = (/~.r_ 1)/(Ki) and PL7 = (I~'T)/(Ki)• In Figs. l(a) and l(b), we compare the contributions ILF obtained by the three methods (Q.E.R., S.C.E.R., L.S.) described above. We plot ILF relative to the asymptotic Holtsmark profile IH against L,.r for AA = + 4 ~ (Fig. l(a)) and AA = - 4 ~, (Fig. 1Co)). Triangles correspond to results of the quantum calculation restricted to the three first states of the atom with the exact direct and exchange interaction [Feautrier et al.S]. We found fair agreement between the results for ~ r > 3, especially between the quantum method and the semi-classical approach of Lisitsa and Sholin. Otherwise the L.S. results give a very good estimate of contributions by the small impact parameters (L; r < 3) towards the line centre and intermediate wings, although the usual conditions of validity of the semi-classical description of the collision with straight trajectories are not realized. Figures 2(a) and 2(b) give the same comparison for larger values of AA : AA = + 60/~ and - 60 .~, respectively. We observe that the agreement is not good. Semi-classical approaches overestimate the profile for L~r < 10, parti-

!

I

'~ O.C

/ 0

I 0

I

5

I0

5

(a)

I0

(b)

Fig. 1. Relative contribution of angular momentum / r to the electronic profile for Ak = 4 ~. (la) and Ak = - 4 ~ (Ib): small broken curves, this work using quantal description (Q.E.R.); full curves, this work using the semi-classical approach of Lisitsa and Sholin (L.S); large broken curves, this work tot the classical limit of the exact resonance method (S.CE.R.). Triangles correspond to exact quantum calculation Feautrier et aL5.

I

1

o4t

AX= ÷6o~

0.4

0,3

0.3

0.2

";" o.z

0.1

0.1

O' 0

I 5

(a)

I0

0

AX--60/~

1 0

5

I0

(b)

Fig. 2. Relative contribution of anfular momentum Lir to the electronic profiles for Ak = + 60/k (2a) and AA = - 6 0 A, (2b) with the same analysis as in Fig. I.

N. TRAN Mlr~.et al.

382

cularly for the blue wing. The semi-classical profile is symmetric and the asymmetry observed in the quantum profile, within the dipole approximation, comes from the appropriate dynamical treatment with correct relations of conservation of energy, which gives a lower limit E0= hAto of integration over the initial energy El for Ato > 0 [Feautrier et al:]. For L i t > 10 and AA = +_60 ,~, our calculations show that the semi-classical results are lower than the quantum results. Therefore, the difference between quantum and semi-classical approaches is not so large when we add contributions of all angular momenta from Li T = 3. Figures 3 and 4 show for Lraax

the red and blue wings, respectively, the variation of the sum

IL,r/Iu with distance AA

~ LIT=3

from the line centre. Lmax is the impact parameter corresponding to the Debye radius p~ For A,~ > 4 A, the summation has converged before this upper cut-off. We observe good agreement between the three results after summation. Therefore, the discrepancy between quantum and semi-classical profiles comes only from the contribution of the three first angular momenta (or corresponding impact parameters). We compare in Fig. 5 red wing electronic profiles calculated by the semi-classical theory of Lisitsa and Sholin with various lower impact parameter cut-offs po. The dotted line shows the exact quantum profile [Feautrier et al.9]. As we saw previously in our quantum results, we observe the large contribution of the low impact parameters. Of course, the semi-classical limit for large AA is the Holtsmark profile (Ie/IH ~ I). Nevertheless,

I

I 0

I

--

Red

wing

05 --

- -

1

I

IO

tOO

Ax,~ L~x

Fig. 3. Variation of the sum

E

LIT=3

/L.d/n

with ~ for the red wing using three approaches: - - - Q.E.R.; - -

'

L.S.; . . . . . . S.C.E.R.

[

I

IO

Blue wing

\ \.~x\ 05

--

\

1. -I

-I0

--

~

\1 -I00

L~x

Fig. 4. Variationof the sum ~ IL:/ln withAAfor the bluewingusingthreeapproaches:- - - Q.E.R.;- Li T = 3 L.S.; . . . . . S.C.E.R.

Short-rangepotentialsin quantaland semi-classicalcalculationsfor the Ly a line 1

383

1 Red wing

~,

Po=~

0.5

--

~

P

o

-

I

-

3 ]

I0

tOO

Ax. Fig. 5. Variation of the red wing electronicprofile with a lower cut off po in the (L.S.) approach. The broken curverepresentsthe exact quantumprofileof Eq. (5). the semi-classical description gives a correct estimation of these close interactions for AA < 60,~. But this is not the case for the blue wing (Fig. 6), for which the dipole semi-classical approximation overestimates short range interactions for IAAI> 10 A,. 3.2. Contribution of short range potentials Using perturbation theory, we introduce quadrupole and polarization potentials Eq. (12). This study is made only for L f > 3. It is seen (Figs. 7(a) and 7(b)) for AA--- _+4 A that the contribution of these potentials is not very large and becomes negligible for L f > 10. The results of the same study are given in Figs. 8(a) and 8(b) for AA = + 60,~ and -60,~. We observe that the quadrupole potential gives a small decrease of the intensity, but the polarization increases the contribution of small angular momenta (Lir -< 5) very strongly. These angular momenta make a large contribution to the total profile when AA is large. Therefore, the polarization potential effect is important in the wings. In Figs. 9 and 10, we compare the sum

I

1 Blue wing

I.O

--

0.5

--

I

I

-I0

-I00

~x,~

Fig. 61 As in Fig. S for the blue wing. QSRT Vol. 23. No. 4--C

384

N. TRANMINH et al. 1

T

-il

1

r

I

I

I

~0

A x =4~,

'\ 0

I0

20

30

40

0

50

l

l

ro

20

--T

--

30

l 40

50

(b)

(o)

Fig. 7. Contribution of short-range potentials to various angular m o m e n t a / r for AA = + 4 ~, (Fig. 7(a)) and Ak = - 4 ,~ (Fig. 7(b)): the small broken curves give the dipole result, the large broken curves correspond to dipole + quadrupole interaction, and the full curve to the complete interaction potential.

0.4

0,4

AX=60•

0.3

a x .-6o~,

0.3

"~o.2

,q o.z

0.1

-

OI

1

OO 0

5

I0

0.0

5

(o)

I0

(b)

Fig. 8. As in Fig. 7 for AA = +60 ~ (Fig. 8(a)) and AA = - 60 A (Fig. 8(b)).

I J.O

--

0.5

--

I Red

wing

~--4" \

I

I

IO

IOO

AX,~ Lma

Fig. 9. Variation of the sum ~=3 IL7/Iu with AA for the red wing using the dipole approach (. . . . ), dipole + quadrupole potentials (-.-.-), and dipole + quadrupole + polarization potentials ( - - ) .

Short-rangepotentialsin quantaland semi-classicalcalculationsfor the Ly a line I I.O

385

I

m

Blue wing

-....o 0.5 B

\ I

I

- I0

- IO0

Fig. 10. As in Fig. 9 for the blue wing. L~

for the red and blue wings, obtained with the dipole potential, dipole plus quadrupole Le.T=3

potentials and dipole plus quadrupole plus polarization potentials. The polarization contribution is very large and we believe that a perturbation treatment~ of this potential overestimates its contribution and is certainly not valid. In order to compare with the results of Caby-Eyraud et al. 2 we add a contribution of the three first angular momenta calculated with quadrupole and exchange potentials [Feautrier et al. 9] but without polarization. Results are shown in Fig. 11. Caby-Eyraud et al. 2 have investigated the contribution of quadrupole and polarization potential by a semi-classical perturbation theory. We observe quite good agreement with our results, especially for the large intensity of the red wing. But, owing to the breakdown of the perturbation treatment used in the two approaches, we believe that these results give only indications about the effects of short-range potentials and not quantitative information. i

I Red

I0

-

wing

-

//

~

P #

B

l

u

e

wlng

0 5 --

I [0

I00

~x.~ Fig. I1. The electronicquantumprofilefor the completeinteractionpotential(fullcurves) in comparison with the semi-classicalresultsof Caby-Eyraudet al.6 CONCLUSION By comparing quantum and semi-classical approaches, we have shown that the semiclassical results are good, owing to a compensation between the contributions of small and large

386

N. TRANMINHet al.

angular m o m e n t a . Therefore, semi-classical treatments are valid for larger AA than we expected a priori. But the more important aspect of our results is certainly the large contribution of polarization potentials for the lower angular m o m e n t a and for large values of AA. Our perturbation approach for this p r o b l e m is not valid and it is necessary to include pseudo states to give a better description. Acknowledgements--We would like to thank Dr. Van Regemorter for fruitful discussions.

REFERENCES 1. V. S. Lisitsa and G. V. Sholin, Soy. Phys. JETP 34, 484 (1972). 2. M. Caby-Eyraud, G. Coulaud, and Nguyen-Hoe, JQSRT 15, 593 (1975). 3. H. R. Griem, Adv. Arm. Molec. Phys. 11,331 (1975). 4. N. Tran Minh, N. Feautrier, and H. Van Regemorter, 3. Phys. B: Atom. Molec. Phys. 8, 1810 (1975). 5. N. Tran Minh, N. Feautrier, and H. Van Regemorter, JQSRT 16, 849 (1976). 6. M. J. Seaton, Proc. Phys. Soc. 77, 174 (1%1). 7. D. Voslamber, Z. Natur]orsch. 27a, 1783(1972). 8. H. Pfennig, Z. Natur[orsch. 26a, 1071 (1971). 9. N. Feautrier, N. Tran Minh, and H. Van Regemorter, J. Phys. B: Atom. Molec. Phys. 9, 1871 (1976). 10. I. C. Percival and M. J. Seaton, Proc. Camb. Phil. Soc. 53, 654 (1957). 11. N. Feautrier and N. Tran Minh, J. Phys. B: Atom. Molec. Phys. 10, 3427 (1977). 12. L. Castillejo, I. C. Percival, and M. J. Seaton, Proc. Roy. Soc. A 254, 259 (1960). 13. N. Feautrier, H. Van Regemorter, and Vo Ky Lan, J. Phys. B: Atom. Molec. Phys. 4, 670 (1971). 14. I. S. Gradshteyn and I. M. Ryzhik, Table of Series, Products and Integrals. Academic Press, New York (!%5). 15. A. Erdelyi, W. Magnus, F. Oberheninger, and F. G. Tricomi, Higher Transcendental Functions, p. 248. McGraw-Hill, New York (1953). 16. M. Abramovitch and I. A. Stegun, Handbook of Mathematical Functions. Dover, New York (1970). 17. K. Alder, A. Bohr, T. Huus, B. Mottelson, and A. Winter, Rev. Mod. Phys. 28, 432 (1956). 18. G. Boldt and W. S. Cooper, Z. Natur[orsch 19a, 968 (1964). 19. A. R. Edmonds and B. A. Kelly, to be published.