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Angular distribution of photoelectrons in argon and xenon
Volume 7, number 3
ANGULAR
1 fiovember 1970
CHEMICAL PHYSICS LETTERS
DISTRIBUTION
OF
Department
PHOTOELECTRONS
IN
ARGON
AND
XENON
S. T. MANSON of Physics, Georgia State University, Atkznta, Georgi’a 30303, USA
and D. J. KENNEDY School of Physics, Atlanta,
Georgia Imtitute Georgia 30332,
of Technology, USA
Received 15 September 1970 The angu!ar distribution of photoelectrons from the argon 3p and xenon 4d subshells have been caIculated in several approximations that predict quite different photoionization cross sections but angular distributions which are substantially in agreement.
The technique of photoe!ectron spectroscopy is a powerful tool for obtaining information on atoms and molecules. Owing to recent advances in design of experimental apparatus [l], a number of experiments which measure the angular distribution of photoelectrons near threshold have been performed or are in progress [2-IO]. It is tinerefore of interest to investigate this energy region theoretically. In the dipole approximation, which is excellent for low-energy photons, the differential photoionization cross section for unpolarized incoming light is [ll, 121 -d&l =q dC’2
[1-&3(E) P,(cos
e)],
(1)
where E is the outgoing photoelectron electron energy, UT(E) is the total photoionization cross section. and P(E) is the asymmetry parameter. The ang1e 0 is the angle that the photoelectron velocity makes with the incoming photon direction and P2(cos 0) = $(3cos20 - 1). Using one-electron wavefunctions and Russell-Saunders coupling, it can be shown [ll, 121 that
W-
I)$$) +U+l) (Z+2)4+4~) - W+
B(c) =
with the dipole
W2z_1WRz+1(~)~~~[6z+1(+ 6z_lWl
(2z+1)[ER~_1(~)+(z+1)~~+1(~)l matrix
,
element
of the initial and final states of the photoelecwhere Y-~P,~Z(Y) and 7-l PC 1*1(y) areradiaiwavefunctions tron. Here the BZ*l(c) are (he phase shifts of the continuum Zkl partial waves witlr >-espect to free waves.
Numerical calculations have been carried out for ionization from the 3p shell of Ar and the 4d shell of Xe. In each case a calculation was performed using a Hartree-Slater (HS)* wavefunction, as tabulated by Herman and Skillman [13], for the initial state and continuum states were solved for in the same potential as the initial state. Details of this method can be found elsewhere [24,15]. * We use this terminology as suggested by Y.Kim rather than ‘Hartree-Fock-Slater’ functions are better than Hartree but not quite as good as Hartree-Pock, Fock as implied by the Hsrtree-Fock-Slater terminalogy.
to emphasize that these waverather than an improvement on Hartree-
387
1 November 1970
CiiEMICAL PHYSICS LETTERS
VO~IIII~ 7. number 3
In addition, for Ar 3p - Ed, we computed the ed wavefunction in the field of Ari using a self-consistent Hartree-Fock (HF) equation taking full account of exchange. The dipole matrix element was then
calculated using the 3p function of neutral Ar. All of the bound state wavefunctions were obtained from the Clemcnti (IS] HF tab&&ion. The calculations were performed using both the length and velocity forms of the photoionization matrix element. Finally, for Xe 4d -) ef, we used the results of Starace [17] * who did an HF calculation for the Ef state in a HS basis by considering the intrachannel coupling [la], taking the matrix element with the HS dd, again in both length and velocity formulation. In all cases the I- E- 1 channel has been considered only in the HS approximation since it is expected that the cross sections and phase shifts thus obtained are quite close to experiment as well as to HF results [15]. The total photoionization cross sections are shown in figs. 1 and 2 for Ar and Xe, respectively. The results of the HS, HF-length and HF-velocity cases differ substantially for both atoms, the HF results being much closer to experiment than the HS [l?, x9]. _ _ The energy dependence of the asymmetry parameter P(E) is shown in fig. 3 for Ar and fig. 4 for Xe. There is remarkably close agreement between the HF length and HF velocity results over the entire energy range considered even though the cross sections are very different in magnitude. For photoelectron energies below about 1 rydberg, the HS results are also in substantial agreement with the HF calculations. At higher energies, the agreement is not so good in Xe, though the general shapes are the same, and is very poor in Ar, We also investigated the sensitiviw of the HF results by doing the HF calculations in Ar, not oniy as described above, but also by calculating the Ed in the field of neutral Ar and in the field of HS orbitals. In each of these cases the length and velocity results are virtually identical to those of the earlier Ar HF calculation. These results can be explained in terms of eq. (2) which for an initial p-state reduces to 2 z[R~- !4RsRd~~~(6d6s5,'] -* P(E) = n n (4)
If the matrix element for the d-channel mated by j?(s) = I- 2(%#d) so P(E)
COS(6d
is much larger than for the s-channel,
eq. (3) can be approxi-
- 6s))
(5)
should not change much from one type of calculation
to another.
This is true for Ar at Iow en-
* The authors would like to thank Dr. Starace for providing them with the details of his calculation prior to publication.
n:rr L _D_,_. Total photoionization cross section for the 3p subshall of argon.
Fig. 2. Total photoionization cross section for the 4d subshell of xenon.
VoIume 7. number 3
CHEMICAL PHYSICS LETTERS
Fig. 3. Asymmetry parameter B(E) as 2 function of photoelectron energy E for Ar 3p.
f November 1970
Fig. 4. Asymmetry parameter &a as a function of photoelectron energy E for Xe 46.
e&es. At higher energies however, the s-wave matrix elements are becoming larger compared to the d-wave and thus the e-dependence of fl will not be given correctly by eq. (5). In the HS calculation the different d-wave phase shift as well as the greatly differing values of Rd account for the greater variation of the HS B from the HF B, below about 1 rydberg.
At higher energy, the HS Rd decreases rapidly to a “Cooper minimum” so that approximation (5) breaks down entirely; when Rd = 0 then p = 0. The fl from the HF calculation will also hit zero at the energy where the HF Rd change s&n, about 3 rydbergs above threshold. The explanation of the Xe results is substantially the same, but since the electron is initially in a d-state, rather than a p-state, there is an extra contribution in the expression for B making things slightly more complicated. For energies well above the “Cooper minimum” it is expected that the results of all three calculations, HS, HF-length and HF-velocity, will be in good agreement. Further studies of the angular dis-_tributions
of atomic
photoelectrons
are in progress
and will be presented
elsewhere.
REFERENCES [l] K.Sieghahn, C. Nordling. A.Fahlman, R.Nordherg, K. Hamrin, J.Hedman, G.Johansson. T.Bergmark. S. Karlsson, I. Lindgren and B. Lindberg, Nova Acta Reg. Sot. Sci. Upsalfensis. Ser. IV. 20 (1967) 1. [2] J. Berkowitz and H. Ehrhardt, Phys. Letters 21 (1966) 531. [3] J. Berkowitz, H. Ehrhardt and T. Tekaat, 2. Physik 200 (1967) 69. [4] M. 0. Krause, Phys. Rev. 177 (1969) 151 and private communicaticn. [5] T. Carlson, privattl communication. [6] J. W.McGowan, D.A. Vroom and A. R. Comeoux. J. Chem. Phys. 51 (1969) 5626. (71 J. A. R.Samson. Proc. Roy. Sot., to he published and private communication. [E] H.Harrison, J. Chem. Phys. 52 (1970) 901. 19) R. Schoen. private communication. [lo] A.Kupperman, private communication. [ll] J. Coopez and R.N. Zare, J. Chem. Phys. 48 (1968) 942. eds. S.GeLt[12] J. Cooper and R.N. Zare. in: Lectures in theoretical physics, Vol. I&z, Atomic collision processes. man, K. Mahanthappa and W. Brittin (Gordon and Breach, New York, 1969) p. 31’7. 1131 F-Herman and S.Skillman, Atomic structure calculations (Prentice-Hall. Englewocd CUES. 2983). [14] J. W. Cooper, Phys. Rev. 128 GS62) 621. [15] 5. T. Manson and J. W. Cooper, Phys. Rev. 165 (1968) 128. 1161 E. Clementi, IBM J. Res. Develop. 9 (l965) 2. [17] A. Starace, Phys. Rev., to be published and private communication. 1181 U.Fano, Phys.Rev. 124 (1961) 1866. [lS] M. Amusia, N. Cherepkov, L. V. Chernysheva and S. Sheftel. Zh. Eksperim. 1 Teor. Fiz. 56 (lS69) 1897. (English transl. Soviet Phys. JETP 29 (1969) 1018.)
389
ANGULAR
1 fiovember 1970
CHEMICAL PHYSICS LETTERS
DISTRIBUTION
OF
Department
PHOTOELECTRONS
IN
ARGON
AND
XENON
S. T. MANSON of Physics, Georgia State University, Atkznta, Georgi’a 30303, USA
and D. J. KENNEDY School of Physics, Atlanta,
Georgia Imtitute Georgia 30332,
of Technology, USA
Received 15 September 1970 The angu!ar distribution of photoelectrons from the argon 3p and xenon 4d subshells have been caIculated in several approximations that predict quite different photoionization cross sections but angular distributions which are substantially in agreement.
The technique of photoe!ectron spectroscopy is a powerful tool for obtaining information on atoms and molecules. Owing to recent advances in design of experimental apparatus [l], a number of experiments which measure the angular distribution of photoelectrons near threshold have been performed or are in progress [2-IO]. It is tinerefore of interest to investigate this energy region theoretically. In the dipole approximation, which is excellent for low-energy photons, the differential photoionization cross section for unpolarized incoming light is [ll, 121 -d&l =q dC’2
[1-&3(E) P,(cos
e)],
(1)
where E is the outgoing photoelectron electron energy, UT(E) is the total photoionization cross section. and P(E) is the asymmetry parameter. The ang1e 0 is the angle that the photoelectron velocity makes with the incoming photon direction and P2(cos 0) = $(3cos20 - 1). Using one-electron wavefunctions and Russell-Saunders coupling, it can be shown [ll, 121 that
W-
I)$$) +U+l) (Z+2)4+4~) - W+
B(c) =
with the dipole
W2z_1WRz+1(~)~~~[6z+1(+ 6z_lWl
(2z+1)[ER~_1(~)+(z+1)~~+1(~)l matrix
,
element
of the initial and final states of the photoelecwhere Y-~P,~Z(Y) and 7-l PC 1*1(y) areradiaiwavefunctions tron. Here the BZ*l(c) are (he phase shifts of the continuum Zkl partial waves witlr >-espect to free waves.
Numerical calculations have been carried out for ionization from the 3p shell of Ar and the 4d shell of Xe. In each case a calculation was performed using a Hartree-Slater (HS)* wavefunction, as tabulated by Herman and Skillman [13], for the initial state and continuum states were solved for in the same potential as the initial state. Details of this method can be found elsewhere [24,15]. * We use this terminology as suggested by Y.Kim rather than ‘Hartree-Fock-Slater’ functions are better than Hartree but not quite as good as Hartree-Pock, Fock as implied by the Hsrtree-Fock-Slater terminalogy.
to emphasize that these waverather than an improvement on Hartree-
387
1 November 1970
CiiEMICAL PHYSICS LETTERS
VO~IIII~ 7. number 3
In addition, for Ar 3p - Ed, we computed the ed wavefunction in the field of Ari using a self-consistent Hartree-Fock (HF) equation taking full account of exchange. The dipole matrix element was then
calculated using the 3p function of neutral Ar. All of the bound state wavefunctions were obtained from the Clemcnti (IS] HF tab&&ion. The calculations were performed using both the length and velocity forms of the photoionization matrix element. Finally, for Xe 4d -) ef, we used the results of Starace [17] * who did an HF calculation for the Ef state in a HS basis by considering the intrachannel coupling [la], taking the matrix element with the HS dd, again in both length and velocity formulation. In all cases the I- E- 1 channel has been considered only in the HS approximation since it is expected that the cross sections and phase shifts thus obtained are quite close to experiment as well as to HF results [15]. The total photoionization cross sections are shown in figs. 1 and 2 for Ar and Xe, respectively. The results of the HS, HF-length and HF-velocity cases differ substantially for both atoms, the HF results being much closer to experiment than the HS [l?, x9]. _ _ The energy dependence of the asymmetry parameter P(E) is shown in fig. 3 for Ar and fig. 4 for Xe. There is remarkably close agreement between the HF length and HF velocity results over the entire energy range considered even though the cross sections are very different in magnitude. For photoelectron energies below about 1 rydberg, the HS results are also in substantial agreement with the HF calculations. At higher energies, the agreement is not so good in Xe, though the general shapes are the same, and is very poor in Ar, We also investigated the sensitiviw of the HF results by doing the HF calculations in Ar, not oniy as described above, but also by calculating the Ed in the field of neutral Ar and in the field of HS orbitals. In each of these cases the length and velocity results are virtually identical to those of the earlier Ar HF calculation. These results can be explained in terms of eq. (2) which for an initial p-state reduces to 2 z[R~- !4RsRd~~~(6d6s5,'] -* P(E) = n n (4)
If the matrix element for the d-channel mated by j?(s) = I- 2(%#d) so P(E)
COS(6d
is much larger than for the s-channel,
eq. (3) can be approxi-
- 6s))
(5)
should not change much from one type of calculation
to another.
This is true for Ar at Iow en-
* The authors would like to thank Dr. Starace for providing them with the details of his calculation prior to publication.
n:rr L _D_,_. Total photoionization cross section for the 3p subshall of argon.
Fig. 2. Total photoionization cross section for the 4d subshell of xenon.
VoIume 7. number 3
CHEMICAL PHYSICS LETTERS
Fig. 3. Asymmetry parameter B(E) as 2 function of photoelectron energy E for Ar 3p.
f November 1970
Fig. 4. Asymmetry parameter &a as a function of photoelectron energy E for Xe 46.
e&es. At higher energies however, the s-wave matrix elements are becoming larger compared to the d-wave and thus the e-dependence of fl will not be given correctly by eq. (5). In the HS calculation the different d-wave phase shift as well as the greatly differing values of Rd account for the greater variation of the HS B from the HF B, below about 1 rydberg.
At higher energy, the HS Rd decreases rapidly to a “Cooper minimum” so that approximation (5) breaks down entirely; when Rd = 0 then p = 0. The fl from the HF calculation will also hit zero at the energy where the HF Rd change s&n, about 3 rydbergs above threshold. The explanation of the Xe results is substantially the same, but since the electron is initially in a d-state, rather than a p-state, there is an extra contribution in the expression for B making things slightly more complicated. For energies well above the “Cooper minimum” it is expected that the results of all three calculations, HS, HF-length and HF-velocity, will be in good agreement. Further studies of the angular dis-_tributions
of atomic
photoelectrons
are in progress
and will be presented
elsewhere.
REFERENCES [l] K.Sieghahn, C. Nordling. A.Fahlman, R.Nordherg, K. Hamrin, J.Hedman, G.Johansson. T.Bergmark. S. Karlsson, I. Lindgren and B. Lindberg, Nova Acta Reg. Sot. Sci. Upsalfensis. Ser. IV. 20 (1967) 1. [2] J. Berkowitz and H. Ehrhardt, Phys. Letters 21 (1966) 531. [3] J. Berkowitz, H. Ehrhardt and T. Tekaat, 2. Physik 200 (1967) 69. [4] M. 0. Krause, Phys. Rev. 177 (1969) 151 and private communicaticn. [5] T. Carlson, privattl communication. [6] J. W.McGowan, D.A. Vroom and A. R. Comeoux. J. Chem. Phys. 51 (1969) 5626. (71 J. A. R.Samson. Proc. Roy. Sot., to he published and private communication. [E] H.Harrison, J. Chem. Phys. 52 (1970) 901. 19) R. Schoen. private communication. [lo] A.Kupperman, private communication. [ll] J. Coopez and R.N. Zare, J. Chem. Phys. 48 (1968) 942. eds. S.GeLt[12] J. Cooper and R.N. Zare. in: Lectures in theoretical physics, Vol. I&z, Atomic collision processes. man, K. Mahanthappa and W. Brittin (Gordon and Breach, New York, 1969) p. 31’7. 1131 F-Herman and S.Skillman, Atomic structure calculations (Prentice-Hall. Englewocd CUES. 2983). [14] J. W. Cooper, Phys. Rev. 128 GS62) 621. [15] 5. T. Manson and J. W. Cooper, Phys. Rev. 165 (1968) 128. 1161 E. Clementi, IBM J. Res. Develop. 9 (l965) 2. [17] A. Starace, Phys. Rev., to be published and private communication. 1181 U.Fano, Phys.Rev. 124 (1961) 1866. [lS] M. Amusia, N. Cherepkov, L. V. Chernysheva and S. Sheftel. Zh. Eksperim. 1 Teor. Fiz. 56 (lS69) 1897. (English transl. Soviet Phys. JETP 29 (1969) 1018.)
389