The photoelectron angular-distribution β parameter in the region of the 3s-1 4p resonance in Ar

Volume 89A, number 9

PHYSICS LETTERS

14 June 1982

THE PHOTOELECTRON ANGULAR-DISTRIBUTION g~PARAMETER IN THE REGION OF THE 3s’4p RESONANCE IN Ar M.Ya. AMUSIA and A.S. KHEIFETS A.F. Joffe Physico-Technical Institute, Leningrad, USSR Received 29 March 1982

14p resonance in argon is calculated. It is The asymmetry parameter of the 3p photoelectrons in the region ofthe demonstrated that going beyond the random phase approximation frame by3s inclusion of the “two-electron—two-hole” excitation gives results which are in good agreement with experiment.

As it was shown earlier [1], a correct calculation

of the photoionization cross section near the autoionization 3~_14p level in argon needs inclusion of the complicated “two-electron—two-hole” excitation. In this paper the essential influence of that excitation on the photoelectron angular-distribution parameter in the resonant region is demonstrated. The angular distribution of photoelectrons ejected from an atomic subshell by unpolarized light is given by the following formula: da/dE = [a(E)/4ir][1 +j3P 2(cos ~ (1) Here a(E) is the total photoionization cross section, z~is the angle between the direction of the incoming light and the electron velocity vector, E is the photoelectron energy. Asymmetry parameter j3 is expressed via photoionization amplitudes Tm and elastic scattering phases ~m in the following way [2]: 2. fX~~R~[Tj;~Tjç exp{i(ók ~m)}1/E Tm1 Here the indices m, k denote one of two dipole excited atomic configurations being either (nl)1E(l — 1) or (nl)~E(l+ 1). The notation (ni) specifies the

~

quantum numbers of the ionized subshell. The matrix amk is defined by the angular momentum of the

ionized subshell:

amk

=

1

1

~j~jj

~

—

1

3~’kTi) ~+ 2

The smaller value of the indices m, k corresponds to

the configuration (nl)~E(l 1). If the photon energy is in the autoionization resonance region the amplitudes Tm may be presented in the following form: T = D +D /(~., F ) (4 —

~

m

m

r rm

—

‘

—

r

rr

Here the first term describes the direct photoionization while the second represents the ionization via excitation of the intennediate resonance. In (4) r de14p, and m specifies one of notes the resonance 3s spectrum states 3p Es or the excited contmuous 3p1Ed, c~and wr being the photon and resonance energy respectively; Dm and Dr are the amplitudes of direct ionization and excitation of the resonance, respectively; ~rm and F 55 are the amplitudes of the in~ teraction between the resonance and the continuum with the resonance itself. Using Fano parameters for the photoionization

cross section near the resonance [2] (4) may be transformedinto: Tm =Dm +(ImD~/ImF~)[~q—i)/O + e)JFrm, (5) where q = —Re Drum Dr is the so called profile index, e = (~~)~ Re rrr)/(—lin err) is the photon energy —

—

counted off from the resonance and measured in its half width. Substituting (5) into (2) and assuming that the am-

plitudes D and F are almost independent of the photon (3)

o 031-9163/82/0000—0000/$02.75 © 1982 North-Holland

energy in the vicinity of the resonance, the following 437

Volume 89A, number 9 dependence ofj3 on ~3(e) = (Xe

+

Ye

PHYSICS LETTERS is obtained

+ Z)/(Ae +

Be

+

C),

(6)

where X

m, k

amk

Re{D,~Dkexp[i(~k

urn D lm F

=

~k

Re{ [F~D~(q

—

~m)]},

+

i)

m,k * +F~D~(q —i)] exp[1(~k r .

.

m )]},

—~

m,k

Im D ~

[F~D~(iq

—

1)

fact a mixture of the state 3s14p of “one-electron— 23d4p of “two-elecone-hole” type and tron—two-hole” type. the state 3p

—

/ Irn D \2 ~1 ~m ‘•) (q~+ 1)F~Frkjexp[i(6k rr 2, A = oO /4ir, B = ~ /4ir)2qp C = ~ /4ir)(l p + q p ). +

—

sm)]

,

-

—

Here ~0 = ~m IDm 12 is the nonresonant photoiomzation cross section, p2 = —(Im Dr)2 /~~o urn F~is a parameter which determines the connection between different continuous spectra. The parametrical equation (6) was presented in a similar form in ref. [3]. Let us calculate the amplitudes D and F which determine the coefficient in (6), starting from the Hartree—Fock approximation (HF). In this approximation the amplitudes D and F are calculated using the functions, asinthe Hartree—Fock matrix elementssingle-particle of the dipolewave electromagnetic teraction operator and Coulomb interelectron interaction operator, respectively. As it was shown in ref. [2] there are strong interelectron correlations in Ar atoms. Therefore not only the single-particle Hartree— Fock matrix elements contribute to the amplitudes but also the correlational ones. The main contribution comes from matrix elements representing the excitation and independent propagation of one or several virtual electron—hole pairs. This matrix elements are taken into account in the random phase approximation with exchange (RPAE). The RPAE amplitudes are determined by a system of integral equations 438

-

in

+F

5~D,~(-4q 1)1

rr

-

structure of the resonance. As it was shown in ref. [4] the hole state 3~—1is strongly mixed with other configurations, mainly with 3p23d. So the resonance is

amk Re [D~Dk

+~

which is usually solved numerically [2]. The RPAE describes the photoionization cross section of outer subshells of noble gas atoms satisfactory. However, according to ref. [1] the RPAE photoionization cross section of the 3p subshell in Ar just near the photoionization resonance 3s14p differs strongly from experiment. Only inclusion of more complex excitation of “two-electron—two-hole” type which are beyond the RPAE frame leads to agreement with experimental data. Noneffectiveness of RPAE in describing the 3s~4p autoiomzation state demonstrates the comphcated -

r

Z=

14 June 1982

The mixing of 3~—1and 3p23d states may be taken into account by introducing of the multiplier ~ into the wave function ~D Here Z 35 is the sodecalled renormalization multiplier which physically termines the part of time during which the state 3s’ is of pure one-hole character. In the remaining part of time it is “smeared out” over more complex atomic excitations, the main of which is 3p—23d. The renormalization of tJJ3~leads to decrease of amplitudes Dr and Frm by a factor ~ and Frr by a factor of Z 3~.But according the definition of Fano parameters and eq. (4), such a simple renormalization of the amplitudes does not alter the photoionization cross section and the angular distribution of photoelectrons. Only the resonance energy and width are changed. It is necessary not only 23d to modify the hole state which is 3~—1 achieved state by mixing it with 3p by introducing the renormalization factor Z 35. The 23d4p mixing with the resonance configuration 3p 3s~4pis also proved to be significant and must be taken into account by inclusion of the 3p23d4p virtual excitation in the amplitudes of processes in which the resonance takes part. These are Dr, “rm and Frr. We insert in it the simplest matrix elements corresponding to only a single virtual excitation of the 3p23d state. Diagrammatically and analytically these matrix elements are presented in ref. [1]. The matrix elements of this type are included in the amplitudes D and F along with the RPAE matrix elements. Thus both “one-electron—one-hole” and “two-electron— ~

Volume 89A, number 9

PHYSICS LETFERS

14 June 1982

two-hole” virtual excitations were taken into consideration in this paper.

Si ii

2

(a)

I

The results of our calculation in such approximations as HF, RPAE and RPAE together with “two-

They contain the experimental electron—two-hole” correctionscurve are presented taken directly in fig. 1.

o

I

4s7

2

/

+66

~C5

~?~4

4~ A —

(b)

,-~

from ref. [5] and also the convolution of the theoretical curve with a gaussian of FWHM. It is seen that the RPAE correlation alters the.HF results in such a way that the agreement with experiment becomes much worse, while the inclusion of the “two-electron— two-hole” excitation improves the situation leading to quite satisfactory agreement with experimental data. This proves our assumption that the 3s~4p resonance is in fact a mixture of almost equal contributions of “one-electron—one-hole” and “two-elec-

tron—two-hole” excitations. It is interesting to compare our results with the Rmatrix calculations [61.The R-matrix method uses

I;!

o 468

‘167

I

‘

‘1~

‘~-~

‘1~’1

4~

2

~,

A’

(c)

~

the R-matrix method [7] and RPAE confirms this

5~

Ig5~

‘1

q~

4~

—I

Fig. 1. The variation in pnear the argon 3s~4presonance, (a) The dash—dotted curve is the HF calculation in this paper, the dashed curve is the convolution of the HF curve with a gaussian of FWHM, the full curve is the experimental results of Codling et al. [5]. (b) The dash—dotted curve is the RPAE calculation in this paper, the dashed curve is the convolution of the RPAE curve with a gaussian of FWHM, the dotted curve is the SC result of Taylor [6], the full curve is the experimental results of Codling etaL [5]. (c) The dash—dotted curve is the RPAE together with the “two-electron—two-hole” excitation result of this paper, the dashed curve is its convolu-

tion with a gaussian of FWHM, the dotted curve is the CI result ofTaylor [6], the full curve is the experimental results of Codling et al. [5].

two different types of the ionized atom wave functions. The first one, so called single configuration wave function (SC) is combined from the resonance and simplest continuum states of “one-electron—onehole” type. The second type configuration interaction wave function (CI) contains more complex excitations including that of “two-electron—two-hole” type. It is naturally to suggest that the SC-calculation is similar to RPAE whereas the Cl-calculation is analogous to tions. Comparison of Fano parameters obtained by RPAE corrected by “two-electron—two-hole” correlasuggestion (see table 1). The same conclusion may be drawn from the angular distribution of photoelectrons (see fig. ib, c). Comparing the results of~3calculations with experiment we observe that in the region of minimum even the best calculated data differ from experiment. This discrepancy might be explained by some defects of the calculation. Indeed, we limit ourself with simplest matrix elements of the virtual “two-electron—twohole” excitation. In more accurate calculations all matrix elements of this type should be taken into account as it is done in RPAE with matrix elements of “one-electron—one-hole” virtual excitations. Comparison of the results of calculations and measurements demonstrates that at present energy resolution theoretical curves convoluted with a gaussian of FWHM are strongly distorted and the difference be439

Volume 89A, number 9

PHYSICS LETTERS

14 June 1982

Table 1 RPAE

7 (eV)

q

R-matrix

Experiment [8]

one-electron— one-hole

two-electron— two-hole

SC length

CI form

0.028 1.15 0.94

0.085 —0.27 0.89

0.015 1.58 0.91

0.068 —0.33 0.86

0.08 ±0.005 —0.22 ±0.05 0.86 ±0.04

tween them is smeared out. It makes much more difficult to interprete the experimental data. So it would

References

be desirable to perform measurements with better energy resolution.

[1] M.Ya. Amusia and A.S. Kheifets, Phys. Lett. 82A (1981) [2] 407. M.Ya. Amusia and N.A. Cherepkov, Case Stud. At. Phys.

The authors would like to thank Dr. N.M. Kabachnik for discussion and Dr. L.V. Chernysheva for placing computer programs at their disposal.

5 (1975) 47. [3] N.M. Kabachnik and I.P. Sazhina, J. Phys. B9 (1976) 1681. [4] B.E. Luyken, FJ. De Heer and R.Ch. Baas, Physica 61 (1972) 200. J.B. West, A.C. Parr, J.L. Dehmer and R.L. K. Codling, Stockbauer, J. Phys. B13 (1980) L689. [6] K.T. Taylor, J. Phys. BlO (1977) L699. [7] P.G. Burke and K.T. Taylor, J. Phys. B8 (1975) 2620. [8] R.P. Madden, D.L. Ederer and K. Codling, Phys. Rev. [5]

177 (1969) 136.

440