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Single ionization of Xe by fast electrons in the vicinity of the 4d10 subshell threshold
Volume SlA, number 2
PHYSICS LETTERS
10 February 1975
SINGLE IONIZATION OF Xe BY FAST ELECTRONS IN THE VICINITY OF THE 4d*’ SUBSHELL THRESHOLD M.Ya. AMUSIA, N.B. BEREZINA and L.V. CHERNYSHEVA AX Zoffe Physico-Technical Institute, Academy of Sciences of the USSR Received 13 January 1975 The generalized oscillator strength of outer shell excitations has a pronounced maximum in the vicinity of 4d” subshell ionization threshold due to virtual excitation of the latter. This phenomenon may be experimentally observed.
The aim of this paper is to determine the generalized oscillator strengths (GOS) of the transitions from outer shells of Xe atoms, especially in the vicinity of the inner shell ionization threshold. The calculations were performed in single particle and in random phase approximation with exchange (RPAE) models. In the latter case along with the correlations inside the outer shell intershell correlations were taken into account. The dependence of GOS’es on transferred momentum and energy is obtained. Near to Xe 4d1° shell ionization threshald an essential increase of GOS has been found if intershell correlations are taken into account. The similar effect has been observed earlier at the study of the photoionization of Xe and Kr atoms [ 11. However, the study of inelastic scattering (IES) offers us the possibility of obtaining the variety of additional data as compared to photoionization study. Using IES data one can investigate not only dipole but also the transitions of other multipolarity as well as the dependence of the dipole matrix elements on momentum transferred. Cross-section of fast electron one can express via GOS [2]. Therefore observed peculiarity in the behaviour of GOS of outer shells can be studied directly in experiment measuring of single charged Xe+ ions yield dependence on energy transferred. Comparison of experimental data with our calculations would permit to confirm the validity of the used model. GOS corresponding to transitions rro lo + EZwere calculated with the help of the formula
(FG1 af
n&-*el
where q is the momentum transferred to atom, and En0 and E, are the Hartree-Fock energies corresponding to discrete and continuous spectra. In the frame of BPAE approach the matrix element of the transition (eZl&r~Zu) is represented in the form of two terms, first of which (EZI&Z~Z~)describes the direct transition from outer n, I, Xe shell accou$ing of the correlations inside it. The second term describes the intershell electron interaction taking into account the virtual transitions from inner subshell 4d’u.
~ez~~Ozo~=~ezln;lnOzO~ t gF - c
(
e”=4d
)(
e”>F e’=4d
)
k’Z’ln;l E”Z”NE”Z”, dli‘l E’Z’,nozo> o-(E,.-~~,)t ig(l-2ne,,) *
The matrix of effective interaction t is determined from the solution of integral equatton which is diagrammatically presented in fig. 1. U and I!?’are the matrix of Coulomb electron interaction inside the shell and the matrix of intershell interaction respectively, exchange being taken into account in both matrices [31.
= ~philnozo~~2, q2 w=Ee-
Eno.
Fig. 1.
101
Volume 51 A, number 2
10 February 1975
PHYSICS LETTERS
c OS(a.u)
-_ I
5
6
--_._ 7
H-f - -/ _ 8
\ -_
g
LRv
Fig. 2. The GOS of outer shell excitations in the vicinity of 4dr0 subshell. The solid curve is obtained in RPAE approximation and the smooth curve - in single-particle approximation.
In our calculations we have considered the interaction only with nearest 4dl” subshell, which mostly affects the process above. GOS behaviour of monopole, dipole and quadrupole transitions from outer Xe shells has been studied. The data obtained are compared’with GOS calculated from single particle approach (fig. 2). The GOS peak found in the vicinity of 4d1° shell ionization threshold is obtained at various values of transferred momentum q. The main contribution in GOS at small q is given by dipole transitions. One can anticipate that with’an increase of the momentum transferred the relative contribution of intershell correlation will grow, since matrix elements of direct transitions due to oscillations of exp (iqr) in (ellexp (iqr) Inolu) will drop. However, this is not the case, because-below considered range of transferred energies for the strongest 5p+ed dipole transition Cooper minimum displays, hence the modulus of matrix element with this 4 increases. The role of correlations in dipole transitions goes down, but peak in GOS maintains and even grows, since with an increase of 4 the more important role is placed by monopole transitions. The effect of intershell interaction on 5p-f ep transition is especially highly pronounced. With further increase of transferred momentum the relative contribution in GOS of dipole and monopole transitions decreases when quadrupole increases. However, since quadrupole transitions probability from 4dlu subshell are small in the considered range of transferred energies they do not affect the subsequent transitions from outer Xe subshells. Therefore the peak caused by intershell interaction in GOS be102
.4
i
,6
8 L
Fig. 3. The GOS of outer shell excitations obtained according to eq. (1) for some different values of the momentum traw ferred.
gins to drop and eventually disappears at q > 2 at. un. Results of calculations for some Q values are given in fig. 3. The growth of GOS seen in fig. 3 at 2.35 at. un. is of the same origin as the Cooper maximum in photoionization cross-section. It is also present in single particle approximation and thus cannot be attributed to the intershell interaction. Results of calculations for some q values are given in fig. 3. Experimental study of this process can confirm the regularities in GOS behaviour similar to which were earlier observed at the study of Xe and Kr oscillator strengths [4]. The authors are indebted to V.K. Ivanov for assistance and fruitful discussions. We also express our gratitude to N.A. Cherepkov and S.I. Sheftel for valuable comments. Part of wavefunctions used in our computations has been supplied by S.I. Sheftel.
References [ 1] M.Ya. Amusia, V.K. Ivanov, N.A. Cherepkov and L.V. Chernysheva, Zh.Eksp. i Teor. Fia. 66 (1974) 1537. [2] N. Inokuti, Rev. Mod. Phys. 43 (1971) 297. [ 31 M.Ya. Amusia, N.A. Cherepkov and L.V. Chemysheva, Zh. Eksp. i Teor. Fiz. 60 (1971) 160. [4] Th.M. El-Sherbini and MI. Van der Wiel, Physica 62 (1972) 119.
PHYSICS LETTERS
10 February 1975
SINGLE IONIZATION OF Xe BY FAST ELECTRONS IN THE VICINITY OF THE 4d*’ SUBSHELL THRESHOLD M.Ya. AMUSIA, N.B. BEREZINA and L.V. CHERNYSHEVA AX Zoffe Physico-Technical Institute, Academy of Sciences of the USSR Received 13 January 1975 The generalized oscillator strength of outer shell excitations has a pronounced maximum in the vicinity of 4d” subshell ionization threshold due to virtual excitation of the latter. This phenomenon may be experimentally observed.
The aim of this paper is to determine the generalized oscillator strengths (GOS) of the transitions from outer shells of Xe atoms, especially in the vicinity of the inner shell ionization threshold. The calculations were performed in single particle and in random phase approximation with exchange (RPAE) models. In the latter case along with the correlations inside the outer shell intershell correlations were taken into account. The dependence of GOS’es on transferred momentum and energy is obtained. Near to Xe 4d1° shell ionization threshald an essential increase of GOS has been found if intershell correlations are taken into account. The similar effect has been observed earlier at the study of the photoionization of Xe and Kr atoms [ 11. However, the study of inelastic scattering (IES) offers us the possibility of obtaining the variety of additional data as compared to photoionization study. Using IES data one can investigate not only dipole but also the transitions of other multipolarity as well as the dependence of the dipole matrix elements on momentum transferred. Cross-section of fast electron one can express via GOS [2]. Therefore observed peculiarity in the behaviour of GOS of outer shells can be studied directly in experiment measuring of single charged Xe+ ions yield dependence on energy transferred. Comparison of experimental data with our calculations would permit to confirm the validity of the used model. GOS corresponding to transitions rro lo + EZwere calculated with the help of the formula
(FG1 af
n&-*el
where q is the momentum transferred to atom, and En0 and E, are the Hartree-Fock energies corresponding to discrete and continuous spectra. In the frame of BPAE approach the matrix element of the transition (eZl&r~Zu) is represented in the form of two terms, first of which (EZI&Z~Z~)describes the direct transition from outer n, I, Xe shell accou$ing of the correlations inside it. The second term describes the intershell electron interaction taking into account the virtual transitions from inner subshell 4d’u.
~ez~~Ozo~=~ezln;lnOzO~ t gF - c
(
e”=4d
)(
e”>F e’=4d
)
k’Z’ln;l E”Z”NE”Z”, dli‘l E’Z’,nozo> o-(E,.-~~,)t ig(l-2ne,,) *
The matrix of effective interaction t is determined from the solution of integral equatton which is diagrammatically presented in fig. 1. U and I!?’are the matrix of Coulomb electron interaction inside the shell and the matrix of intershell interaction respectively, exchange being taken into account in both matrices [31.
= ~philnozo~~2, q2 w=Ee-
Eno.
Fig. 1.
101
Volume 51 A, number 2
10 February 1975
PHYSICS LETTERS
c OS(a.u)
-_ I
5
6
--_._ 7
H-f - -/ _ 8
\ -_
g
LRv
Fig. 2. The GOS of outer shell excitations in the vicinity of 4dr0 subshell. The solid curve is obtained in RPAE approximation and the smooth curve - in single-particle approximation.
In our calculations we have considered the interaction only with nearest 4dl” subshell, which mostly affects the process above. GOS behaviour of monopole, dipole and quadrupole transitions from outer Xe shells has been studied. The data obtained are compared’with GOS calculated from single particle approach (fig. 2). The GOS peak found in the vicinity of 4d1° shell ionization threshold is obtained at various values of transferred momentum q. The main contribution in GOS at small q is given by dipole transitions. One can anticipate that with’an increase of the momentum transferred the relative contribution of intershell correlation will grow, since matrix elements of direct transitions due to oscillations of exp (iqr) in (ellexp (iqr) Inolu) will drop. However, this is not the case, because-below considered range of transferred energies for the strongest 5p+ed dipole transition Cooper minimum displays, hence the modulus of matrix element with this 4 increases. The role of correlations in dipole transitions goes down, but peak in GOS maintains and even grows, since with an increase of 4 the more important role is placed by monopole transitions. The effect of intershell interaction on 5p-f ep transition is especially highly pronounced. With further increase of transferred momentum the relative contribution in GOS of dipole and monopole transitions decreases when quadrupole increases. However, since quadrupole transitions probability from 4dlu subshell are small in the considered range of transferred energies they do not affect the subsequent transitions from outer Xe subshells. Therefore the peak caused by intershell interaction in GOS be102
.4
i
,6
8 L
Fig. 3. The GOS of outer shell excitations obtained according to eq. (1) for some different values of the momentum traw ferred.
gins to drop and eventually disappears at q > 2 at. un. Results of calculations for some Q values are given in fig. 3. The growth of GOS seen in fig. 3 at 2.35 at. un. is of the same origin as the Cooper maximum in photoionization cross-section. It is also present in single particle approximation and thus cannot be attributed to the intershell interaction. Results of calculations for some q values are given in fig. 3. Experimental study of this process can confirm the regularities in GOS behaviour similar to which were earlier observed at the study of Xe and Kr oscillator strengths [4]. The authors are indebted to V.K. Ivanov for assistance and fruitful discussions. We also express our gratitude to N.A. Cherepkov and S.I. Sheftel for valuable comments. Part of wavefunctions used in our computations has been supplied by S.I. Sheftel.
References [ 1] M.Ya. Amusia, V.K. Ivanov, N.A. Cherepkov and L.V. Chernysheva, Zh.Eksp. i Teor. Fia. 66 (1974) 1537. [2] N. Inokuti, Rev. Mod. Phys. 43 (1971) 297. [ 31 M.Ya. Amusia, N.A. Cherepkov and L.V. Chemysheva, Zh. Eksp. i Teor. Fiz. 60 (1971) 160. [4] Th.M. El-Sherbini and MI. Van der Wiel, Physica 62 (1972) 119.