- Email: [email protected]
A simple treatment of two-electron one-photon transitions
Volume 58A, number 5
PHYSICS LETTERS
20 September 1976
A SIMPLE TREATMENT OF TWO-ELECTRON ONE-PHOTON TRANSITIONS N. MOISEYEV and J. KATRIEL Department of Chemistry, Technion —Israel Institute of Technology, Haifa 32000, Israel Received 22 June 1976 Two electron radiative transition probabilities from the L and M shells into a doubly vacant K shell are computed non-relativistically, introducing angular correlation in the simplest conceivable way. The relative cross-sections for the different possible processes are in agreement with recently reported experimental values for Fe and Ni and predict interesting changes in the spectra of heavier systems.
In a recent experimental study the cross-sections for the emission of photons due to two-electron transitions from the L and M shells to the vacant K shell produced in collisions of Fe and Ni were measured [11. The cross-sections obtained incorporate both the probability of formation of the doubly vacant K shell and the two-electron—-one-photon spontaneous emission. In order to concentrate on the emission process one should consider the relative cross-sections for the different transitions of the same ion. The experimental results indicate that among the possible two electron transitions into the doubly vacant K shell the transition of two electrons from the L shell is the strongest. The transition of one L and one M electron into the K shell has been detected with a smaller cross-section and that of two electrons from the M to K shell has not been observed at all. These two-electron transitions are a direct manifestation of the correlations among the electrons and are therefore unaccountable within the independent particle In order to account for the two electron transitions in the simplest possible way we have computed the dipole transition matrix element subject to the following assumptions: (1) The wave-function is constructed out of screened hydrogenic orbitals. (2) Interelectronic correlation is introduced via the angular factor (1 XcosO12) which with is roughly equi2 configuration the 1s2 one. valent to mixing a p shown to incorporate a considerThis factor has been able, part of the angular correlation in the case of two electron systems [2,3]. 3) Interaction of the doubly ionized system with
the continuum is neglected. This simplification seems justified in view of the fact that the experimentally obtained line shapes are sharp and symmetrical [4]. The parameter X could in principle be determined by a variational computation for the final state with a doubly filled K shell. However, as the two-electron transition is entirely due to the correlation factor cos 012, the corresponding matrix element is proportional to X. The ratio between the cross-sections for the different two-electron transitions of a given atom is, therefore, independent of X, which was consequently not evaluated. The computations were carried out up toZ’-~55, where relativistic effects start to be too important to be ignored [5]. The results obtained are presented in fig. 1, in which the experimental values [1] of u(K0+ K~)/ I
—~
0I
°°‘
~‘(2k9)fr(2ka)
0001 25
35
40
45
50
55
z Fig. 1. Computed relative cross sections for two electron transitions into the K shell.
303
Volume 58A, number 5
PHYSICS nETTERS
a(2K~)are shown for Fe and Ni. The ratio a(2K~)/ cr(2K~)is predicted to be exceedingly low, in agreement with the fact that the 2K~transition was not detected in the experimental spectrum. The accuracy of the crude model presented matches that of the presently available experimental results so that a more sophisticated treatment would not be justified. The main feature to be pointed out is that the computation suggests that for heavier atoms than those experimentally studied the relative cross-sections of the K0+ K~and, in particular, the 2K~transitions should increase rather rapidly. Hence 2K~is predicted to be detectable in heavier atoms, provided that a
304
20 September 1976
reasonably efficient mechanism of production of a doubly vacant K shell is available. The availability of such results may be a very valuable source of direct experimental information concerning the role of interelectronic correlation effects in heavier atoms.
References [11 W. Wdltli et al., Phys. Rev. Lctt. 35 (1975) 656. [21 J. Killingbeck, Mol. Phys. 23 (1972) 907. [31N. Moiscyev and J. Katriel,Chem. Phys. 10(1975)67. [41 U. Fano, Phys. Rev. 124 (1961) 1866. [5] E.J. McGuirc, Phys. Rev. A3 (1971) 587.
PHYSICS LETTERS
20 September 1976
A SIMPLE TREATMENT OF TWO-ELECTRON ONE-PHOTON TRANSITIONS N. MOISEYEV and J. KATRIEL Department of Chemistry, Technion —Israel Institute of Technology, Haifa 32000, Israel Received 22 June 1976 Two electron radiative transition probabilities from the L and M shells into a doubly vacant K shell are computed non-relativistically, introducing angular correlation in the simplest conceivable way. The relative cross-sections for the different possible processes are in agreement with recently reported experimental values for Fe and Ni and predict interesting changes in the spectra of heavier systems.
In a recent experimental study the cross-sections for the emission of photons due to two-electron transitions from the L and M shells to the vacant K shell produced in collisions of Fe and Ni were measured [11. The cross-sections obtained incorporate both the probability of formation of the doubly vacant K shell and the two-electron—-one-photon spontaneous emission. In order to concentrate on the emission process one should consider the relative cross-sections for the different transitions of the same ion. The experimental results indicate that among the possible two electron transitions into the doubly vacant K shell the transition of two electrons from the L shell is the strongest. The transition of one L and one M electron into the K shell has been detected with a smaller cross-section and that of two electrons from the M to K shell has not been observed at all. These two-electron transitions are a direct manifestation of the correlations among the electrons and are therefore unaccountable within the independent particle In order to account for the two electron transitions in the simplest possible way we have computed the dipole transition matrix element subject to the following assumptions: (1) The wave-function is constructed out of screened hydrogenic orbitals. (2) Interelectronic correlation is introduced via the angular factor (1 XcosO12) which with is roughly equi2 configuration the 1s2 one. valent to mixing a p shown to incorporate a considerThis factor has been able, part of the angular correlation in the case of two electron systems [2,3]. 3) Interaction of the doubly ionized system with
the continuum is neglected. This simplification seems justified in view of the fact that the experimentally obtained line shapes are sharp and symmetrical [4]. The parameter X could in principle be determined by a variational computation for the final state with a doubly filled K shell. However, as the two-electron transition is entirely due to the correlation factor cos 012, the corresponding matrix element is proportional to X. The ratio between the cross-sections for the different two-electron transitions of a given atom is, therefore, independent of X, which was consequently not evaluated. The computations were carried out up toZ’-~55, where relativistic effects start to be too important to be ignored [5]. The results obtained are presented in fig. 1, in which the experimental values [1] of u(K0+ K~)/ I
—~
0I
°°‘
~‘(2k9)fr(2ka)
0001 25
35
40
45
50
55
z Fig. 1. Computed relative cross sections for two electron transitions into the K shell.
303
Volume 58A, number 5
PHYSICS nETTERS
a(2K~)are shown for Fe and Ni. The ratio a(2K~)/ cr(2K~)is predicted to be exceedingly low, in agreement with the fact that the 2K~transition was not detected in the experimental spectrum. The accuracy of the crude model presented matches that of the presently available experimental results so that a more sophisticated treatment would not be justified. The main feature to be pointed out is that the computation suggests that for heavier atoms than those experimentally studied the relative cross-sections of the K0+ K~and, in particular, the 2K~transitions should increase rather rapidly. Hence 2K~is predicted to be detectable in heavier atoms, provided that a
304
20 September 1976
reasonably efficient mechanism of production of a doubly vacant K shell is available. The availability of such results may be a very valuable source of direct experimental information concerning the role of interelectronic correlation effects in heavier atoms.
References [11 W. Wdltli et al., Phys. Rev. Lctt. 35 (1975) 656. [21 J. Killingbeck, Mol. Phys. 23 (1972) 907. [31N. Moiscyev and J. Katriel,Chem. Phys. 10(1975)67. [41 U. Fano, Phys. Rev. 124 (1961) 1866. [5] E.J. McGuirc, Phys. Rev. A3 (1971) 587.