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Photoionization cross-section calculations for 4d10 subshell of In in the RPAE
Volume 60A, number 1
PHYSICS LETTERS
24 January 1977
PHOTOIONIZATION CROSS-SECTION CALCULATIONS FOR 4d’° SUBSHELL OF In
IN THE RPAE*
M.Ya. AMUSIA and N.A. CHEREPKOV A.F. loffe Physical-Technical Institute of the Academy of Sciences of the USSR, 194021 Leningrad, USSR and
V. RADOJEVK~ Laboratory for Theoretical Physics, Boris Kidri~Institute, P.O.B. 522, 11001 Beograd, Yugoslavia Received 8 November 1976 1°subshell of In are calculated in Photoionization cross-section and angular distribution of photoelectrons for 4d the Random Phase Approximation with Exchange (RPAE). The cross section is in satisfactory agreement with the recent experimental data by Stanford synchrotron.
The method of photoelectron spectroscopy, widely used in recent years, enables measurement of partial
photoionization cross sections of separate subshells. This allows one to compare the theoretical predictions with the experiment in more details. In the present artide our results of calculation of the 4d10 partial photoionization cross section for atomic 49Inmeasurements are reported and compared with the results of recent performed on Stanford synchrotron [1]. The calculations are performed in the framework of the Random Phase Approximation with Exchange (RPAE) described elsewhere [2, 3] The zeroth order wave functions are taken in the single-configuration Hartree-Fock (HF) approximation [4, 5] Since there is one Sp electron above the closed subshells, it can be coupled with the electron excited from the 4d subshells to either singlet or triplet state. To simplify the calculations, the average-of-configuration [6] HF functions (calculated in the “frozen core” field of -
-
ground state electrons [5] are taken for excited states,
which, for definite configuration, does not depend on the total orbital and spin angular momenta. The cross section has been calculated for photon (excitation) energies from ionization threshold to about 25 Ry. Both length and velocity forms of the cross sections were evaluated, which, in contrast to the HF approximation, practically coincide in the *
Work in part supported by the Republic Science Association of the Socialist Republic Serbia.
24
RPAE. Contribution of both 4d -+ ef and 4d -~ transitions (particle-hole channels) has been taken into account. The correlations within each of these channels and between them, as well as the correlation influence of the 5s2 and 5p on the 4d10 subshell have been taken into account. However, the influence of 2 and Sp) on the 4d10 subshell the outer to subshells (5sin contrast to the rather strong appeared be weak influence of 5p6 electrons in the Xe atom [7] The resuits for the photoionization cross section and photoelectron angular distribution of the outer 5p electron have been published earlier [8]. The 4d10 partial photoionization cross section of In atom, calculated in the RPAE, is presented in fig. 1 .
together with the experimental data [1]. Since the ex-
perimental cross section is originally presented in relative units, it is normalized in fig.1 to calculated value of the cross section at the maximum. It should be noted that only the cross section of atomic 51Sb is given in ref. [1] where it has also been stated that the results for In are essentially identical. Besides, the measurements reported there were not performed on gaseous but on the solid state target, whereas our calculations were performed for an isolated atom. The agreement with the experiment is in the whole satisfactory; positions of the main maximum, the Cooper minimum, and the second maximum practically coincide. However, experimental cross section values for the energies above the position of the Cooper minimum are significantly higher than the calculated ones.
Volume 60A, number 1
PHYSICS LETFERS
PHOTON ENERGY tRy)
Fig. 1. The 4d’°partial photoionization
cross
PHOTON ENERGY tRy)
section of In as
a function of photon energy. The solid line represents our calculated RPAE results, and the circles are experimental data [1] normalizedto calculated RPAE value at the maximum. RPAE 1°ionization values were threshold calculated (2.126 withRy), the theoretical which is byvalue aboutof the 4d Ry higher than the experimental value [9]. 0.6
It is possible that this discrepancy results from the correlation influence of the 4p6 subshell being neglected in our calculations; the photoionization threshold of this subshell is in the vicinity of the Cooper minimum of the cross section. The calculations have been performed with the theoretical values of the ionization thresholds; the threshold of 4d10 subshell of In (2.126 Ry) being higher than the experimental value [9] for about 0.6 Ry.
~
20
b
1.5
-
1.0
-
. ~In
.
(4d1°)
Fig. 3. The 4d’°partial photoionization cross section of
46Pd
[2], 491n (present work), and 54Xe [2] atoms calculated in the RPAE as a function of photon energy are presented to illustrate the dependence of the cross section on atomic number. The angular asymmetry parameter, f3, in the RPAE for 4d10 subshell as a function of photoelectron energy is given in fig. 2. There are two minima, one, as usually [2] , close to the position of the Cooper cross
section minimum, and the other near the ionization threshold of the subshell under consideration. No experimental data on angular distribution are available. It is of interest to see the dependence of the 4d’0 partial cross section as a function of the atomic number. The results of the RPAE calculations for the three following atoms are presented in fig. 3: for 10 subshell is outermost, for 46Pd [2], where the 4d 49In (the 0, and for results presented here) which has three electrons over 4d’ [2] Itwhere there eight 10 54Xe subshell. can be seenare that the electrons main over the 4dof the cross section becomes higher and the maximum shape of the curve more narrow with increasing subshell effective charge. Analogous change has been observed in the 3p6 partial cross section for the sequence of Ar, K, and Ca atoms [2] The authors would like to acknowledge useful discussions with Dr. Dj. ~ivanovid. One of the authors
~ 00
05
24 January 1977
-____________
0
5
__________
10 PHOTO
15 ELECTRON
20
25
ENERGY CRy)
(N.A. Ch.) would like to thank “Boris Kidri~”Institute for the hospitality extended to him during his visit, when the present work was completed.
Fig. 2. The asymmetry parameter ji of the photoelectron angular distribution in the RPAE as a function of photoelectron energy. 25
Volume 60A, number 1
PHYSICS LETTERS
References [1J1. Lindau, P. Pianetta and W.E. Spicer, Phys. Rev. ~ 57A (1976) 225. [2] M.Ya. Amusia and N.A. Cherepkov, CaseStudies ~ Atomic Physics 5 (1975) 47 [3] M.Ya. Amusia, N.A. Cherepkov, Dj. ~ivanoviéand V. Radojevié, Phys. Rev. A 13(1976)1466. [4] L.V. Chernysheva, NA. Cherepkov and V. Radojevid, Computer Phys. Commun. 11(1976) 57 [5] L.V. Chernysheva, N.A. Cherepkov and V. Radojevi~,
26
24 January 1977
Atomic frozen core Hartree-Fock program for discrete and continuous excited states (to be submitted for publication in Computer Phys. Commun.).
[6] J.C. Slater, Quantum theory of atomic structure, Vol.11 (McGraw-Hill, New York, 1960) p. 27. [7] M.Ya. Amusia and V.K. Ivanov, Phys. Lett. 59A (1976) 194. [8] N.A. Cherepkov, Zh. Eksp. Teor. Fiz. 65 (1973) 933. (in Russian - English translation: Soy. Phys. JETP 38 (1974) 463).
[9] W. Lotz, J. Opt. Soc. Am. 60(1970) 206.
PHYSICS LETTERS
24 January 1977
PHOTOIONIZATION CROSS-SECTION CALCULATIONS FOR 4d’° SUBSHELL OF In
IN THE RPAE*
M.Ya. AMUSIA and N.A. CHEREPKOV A.F. loffe Physical-Technical Institute of the Academy of Sciences of the USSR, 194021 Leningrad, USSR and
V. RADOJEVK~ Laboratory for Theoretical Physics, Boris Kidri~Institute, P.O.B. 522, 11001 Beograd, Yugoslavia Received 8 November 1976 1°subshell of In are calculated in Photoionization cross-section and angular distribution of photoelectrons for 4d the Random Phase Approximation with Exchange (RPAE). The cross section is in satisfactory agreement with the recent experimental data by Stanford synchrotron.
The method of photoelectron spectroscopy, widely used in recent years, enables measurement of partial
photoionization cross sections of separate subshells. This allows one to compare the theoretical predictions with the experiment in more details. In the present artide our results of calculation of the 4d10 partial photoionization cross section for atomic 49Inmeasurements are reported and compared with the results of recent performed on Stanford synchrotron [1]. The calculations are performed in the framework of the Random Phase Approximation with Exchange (RPAE) described elsewhere [2, 3] The zeroth order wave functions are taken in the single-configuration Hartree-Fock (HF) approximation [4, 5] Since there is one Sp electron above the closed subshells, it can be coupled with the electron excited from the 4d subshells to either singlet or triplet state. To simplify the calculations, the average-of-configuration [6] HF functions (calculated in the “frozen core” field of -
-
ground state electrons [5] are taken for excited states,
which, for definite configuration, does not depend on the total orbital and spin angular momenta. The cross section has been calculated for photon (excitation) energies from ionization threshold to about 25 Ry. Both length and velocity forms of the cross sections were evaluated, which, in contrast to the HF approximation, practically coincide in the *
Work in part supported by the Republic Science Association of the Socialist Republic Serbia.
24
RPAE. Contribution of both 4d -+ ef and 4d -~ transitions (particle-hole channels) has been taken into account. The correlations within each of these channels and between them, as well as the correlation influence of the 5s2 and 5p on the 4d10 subshell have been taken into account. However, the influence of 2 and Sp) on the 4d10 subshell the outer to subshells (5sin contrast to the rather strong appeared be weak influence of 5p6 electrons in the Xe atom [7] The resuits for the photoionization cross section and photoelectron angular distribution of the outer 5p electron have been published earlier [8]. The 4d10 partial photoionization cross section of In atom, calculated in the RPAE, is presented in fig. 1 .
together with the experimental data [1]. Since the ex-
perimental cross section is originally presented in relative units, it is normalized in fig.1 to calculated value of the cross section at the maximum. It should be noted that only the cross section of atomic 51Sb is given in ref. [1] where it has also been stated that the results for In are essentially identical. Besides, the measurements reported there were not performed on gaseous but on the solid state target, whereas our calculations were performed for an isolated atom. The agreement with the experiment is in the whole satisfactory; positions of the main maximum, the Cooper minimum, and the second maximum practically coincide. However, experimental cross section values for the energies above the position of the Cooper minimum are significantly higher than the calculated ones.
Volume 60A, number 1
PHYSICS LETFERS
PHOTON ENERGY tRy)
Fig. 1. The 4d’°partial photoionization
cross
PHOTON ENERGY tRy)
section of In as
a function of photon energy. The solid line represents our calculated RPAE results, and the circles are experimental data [1] normalizedto calculated RPAE value at the maximum. RPAE 1°ionization values were threshold calculated (2.126 withRy), the theoretical which is byvalue aboutof the 4d Ry higher than the experimental value [9]. 0.6
It is possible that this discrepancy results from the correlation influence of the 4p6 subshell being neglected in our calculations; the photoionization threshold of this subshell is in the vicinity of the Cooper minimum of the cross section. The calculations have been performed with the theoretical values of the ionization thresholds; the threshold of 4d10 subshell of In (2.126 Ry) being higher than the experimental value [9] for about 0.6 Ry.
~
20
b
1.5
-
1.0
-
. ~In
.
(4d1°)
Fig. 3. The 4d’°partial photoionization cross section of
46Pd
[2], 491n (present work), and 54Xe [2] atoms calculated in the RPAE as a function of photon energy are presented to illustrate the dependence of the cross section on atomic number. The angular asymmetry parameter, f3, in the RPAE for 4d10 subshell as a function of photoelectron energy is given in fig. 2. There are two minima, one, as usually [2] , close to the position of the Cooper cross
section minimum, and the other near the ionization threshold of the subshell under consideration. No experimental data on angular distribution are available. It is of interest to see the dependence of the 4d’0 partial cross section as a function of the atomic number. The results of the RPAE calculations for the three following atoms are presented in fig. 3: for 10 subshell is outermost, for 46Pd [2], where the 4d 49In (the 0, and for results presented here) which has three electrons over 4d’ [2] Itwhere there eight 10 54Xe subshell. can be seenare that the electrons main over the 4dof the cross section becomes higher and the maximum shape of the curve more narrow with increasing subshell effective charge. Analogous change has been observed in the 3p6 partial cross section for the sequence of Ar, K, and Ca atoms [2] The authors would like to acknowledge useful discussions with Dr. Dj. ~ivanovid. One of the authors
~ 00
05
24 January 1977
-____________
0
5
__________
10 PHOTO
15 ELECTRON
20
25
ENERGY CRy)
(N.A. Ch.) would like to thank “Boris Kidri~”Institute for the hospitality extended to him during his visit, when the present work was completed.
Fig. 2. The asymmetry parameter ji of the photoelectron angular distribution in the RPAE as a function of photoelectron energy. 25
Volume 60A, number 1
PHYSICS LETTERS
References [1J1. Lindau, P. Pianetta and W.E. Spicer, Phys. Rev. ~ 57A (1976) 225. [2] M.Ya. Amusia and N.A. Cherepkov, CaseStudies ~ Atomic Physics 5 (1975) 47 [3] M.Ya. Amusia, N.A. Cherepkov, Dj. ~ivanoviéand V. Radojevié, Phys. Rev. A 13(1976)1466. [4] L.V. Chernysheva, NA. Cherepkov and V. Radojevid, Computer Phys. Commun. 11(1976) 57 [5] L.V. Chernysheva, N.A. Cherepkov and V. Radojevi~,
26
24 January 1977
Atomic frozen core Hartree-Fock program for discrete and continuous excited states (to be submitted for publication in Computer Phys. Commun.).
[6] J.C. Slater, Quantum theory of atomic structure, Vol.11 (McGraw-Hill, New York, 1960) p. 27. [7] M.Ya. Amusia and V.K. Ivanov, Phys. Lett. 59A (1976) 194. [8] N.A. Cherepkov, Zh. Eksp. Teor. Fiz. 65 (1973) 933. (in Russian - English translation: Soy. Phys. JETP 38 (1974) 463).
[9] W. Lotz, J. Opt. Soc. Am. 60(1970) 206.