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Effect of electronic correlations on KLL Auger lines in elements Mg-Sc
Volume 39A, number 5
PHYSICS LETTERS
EFFECT OF ELECTRONIC
CORRELATIONS
5 June 1972
ON KLL AUGER LINES IN
ELEMENTS Mg - Sc* D. CHATTARJI and B. TALUKDAR Department of Physics, Visva-Bharati University, Santiniketan, West Bengal, India
The effect of electronic correlations on KLL Auger line intensities is investigated for the elements 12<~Z~<21by usi~aga simple ansatz for the correlation of the bound state electrons. No correlation is assumed between the K-sheU electron and the continuum electron in the final state. Enhancement of the intensities of the iSo (KLzL2) and the ID2 (KL2L3) lines is obtained relative to the ISo (KL! L1 ) line.
In recent years, improved experimental data on the KLL Auger spectrum have been obtained for low- Z elements through use of electrostatic analysis by Mehlhorn [1] and others [2]. On the theoretical side, use of intermediate coupling theory [3] and, more recently, of configuration interaction [4] has led to a situation where the experimental and theoretical energies of the transition show reasonably good agreement, but discrepancies remain with regard to the relative intensities of the individual lines. These discrepancies happen to be larger for lower Z values. Significantly, results obtained by McGuire [5] using Hartree-FockSlater wave functions for total KLL Auger transition rates appear to be in good agreement with experiment, but even with such a sophisticated model the individual configuration intensities differ from experiment by as much as 50%. Looking at the range of theoretical resuits obtained so far [5,6], it is seen that the line intensities are extremely sensitive to the detailed nature of the wave functions used. Difficulties in measurements of individual line intensities for low Z values [7], makes comparison between theory and experiment even more difficult in this region. The above situation, particularly the fact that even Hartree-Fock-Slater calculations do not lead to good agreement, seems to warrant a more detailed investigation into the possible nature o f the wave function in a light atom. For example, one of the shortcomings of the Hartree Fock method is that it neglects details of * Work supported in part by the Department of Atomic Energy, Government of India.
the electronic repulsion. On the whole, it allows electrons to come close together more often than is actually the case. Since the Auger process arises basically from the interaction between the two participating electrons, the energy of this interaction should play an important role in the overall energetics of the system. Effectively, this interaction gives rise to a correction in the total Hamiltonian o f the system which is commonly called the correlation energy [8]. This correlation energy is quite appreciable for two electrons in the same spatial orbital, particularly so if they have opposite spins, as in the case of the K L 1 L 1 transition (which, incidentally, provides the basis for computing relative intensities). Coming down to wave functions, a logical step would be to construct a wave function in which the paired electrons in the bound state avoid each other more strongly. That the correlation term should be relatively more important in the lighter atoms is readily understood. The overall averaging effect of the other "spectator" electrons may not be quite so smooth in these atoms as to give rise to a central field and dependence of a physical process like the Auger effect on the interelectron distance r12 = Ir1 - r21 may be appreciable. On the basis of these considerations, we have tried here the following ansatz for the wave function of the participating bound-state electrons:
',Itcorr=q~A(nlnl';SLJM)(1 --~ exp(--/~r 12)) ,
(1)
where ~ and t~ are variational parameters, q'A is the antisymmetrized two-electron hydrogenic wave func399
Volume 39A, number 5
PHYSICS LETTERS
tion belonging to the SLJM representation (i.e. we have used in this paper an extreme LS coupling without configuration interaction). For the purpose of this paper, X is taken to be unity and IJ=Z/nao*, where a 0 is the first Bohr radius of hydrogen. Evidently, ~corr is small %r r12 small and becomes independent of r12 when the latter is large. Furthermore, ibr a given value o f r l 2 , "Pcorr is very nearly equal to q'A for large Z and very different from ~A for small Z. This is quite plausible, since at higher Z values the Coulomb field of the nucleus becomes strong so that the perturbation caused by the mutual repulsion of the electrons may be neglected. For lower Z values, this perturbation is more nearly comparable to the nuclear Coulomb field. For the final state of the system we do not assume any correlation between the two electrons, because they are now far apart and belong to states which are dynamically different in character, one being bound to the K-shell and the other raised to the continuum. For the continuum electron we use a screened Coulomb wave t~mction normalized to a current of one electron per unit time per unit energy range. Screening constants and energy values are chosen as in our earlier work [9]. Intensities o f the IS0 (KL2L2) and 11)2 (KL2L3) lines relative to that of the IS0 (KL 1 L I ) l i n e are shown for 12 ~
5 June 1972
20
I..~
- .....
I NTENSIT IES OF I,% ~KLiL2~ AND I[~ (KL2L~) LINES RELATIVE TO ~_ FUNCTION OF ATOMIC
PRESENTCALQULATIONS ASAADIS INTERMEDIATE COUPLING CALCULATIORS
0
I~ @LzL~
O
~D~ QKL2L
EXPERIM Ei~I'AL RESULTS
NUMBER ~E
ID2
................ ..........
IP
15
14
15
16 ATOMIC
0 .......
17
ISo
Ia
NUMBER
19
20
21
22
Z
Fig. 1. The intensities of ID2 (KL2L3) and ISo (KL2L2) lines relative to ISo (KL1L1) for 12
Volume 39A, number 5
PHYSICS LETTERS
with appropriate screening [12] may be one of the factors having to do with the discrepancies which persist. However, as conjectured by Callan [13] the simple e2/r12 interaction given originally by Wentzel for the Auger effect may not hold very nicely in the low Z region. In particular for small inter-electronic distance r12 the Hartree-Fock radial integrals overshoots its final steady value. This seems to indicate that the kind o f cut-off built into our model for small r12 is a venture in the right direction. That this "correlation" has been tagged onto an otherwise simple hydrogenic system serves to keep the field of vision clear for judging the relative importance o f the different elements of the calculation. For example, introduction of intermediate coupling and configuration interaction would be the next logical step enabling us to see how crucial this correlation is when compared to these other features. How one could go about introducing correlations into a full-grown Hartree-Fock system should be an interesting question. Intensities o f the other KLL lines will be reported shortly in a more exhaustive communication.
5 June 1972
References [1] B. Cleff and W. Mehlhom, Z. Physik 219 (1969) 311. [2] A. Fahlman, R. Nordberg, C. Nordling and K. Siegbahn, Z. Physik 192 (1966) 476. [3] W.N. Asaad and E.H.S. Burhop, Proc. Phys. Soc. (London) 71 (1958) 369. [4] W.N. Asaad, Nucl. Phys. 66 (1965) 494. [5] E.J. McGuire, Phys. Rev. 185 (1969) 1; Phys. Rev. A2 (1970) 273. [6] R.A. Rubenstein, Ph.D. Thesis (unpublished), University of lllinois. (1955). [7] I. Bergstrom and C. Nordling, in Alpha-, beta- and gamma ray spectroscopy, ed. K. Siegbahn (North-Holland, Amsterdam, 1965). [8] C.C.J. Roothaan and A.W. Weiss, Rev. Mod. Phys. 33 (1960) 194. [9] B. Talukdar and D. Chattarji, Phys. Rev. A1 (1970) 44; B. Talukdar, Ph.D. Thesis (unpublished), Visva-Bharati University ( 1970). [10] E.J. Callen, Phys. Rev. 124 (1961) 793. [11] W. Mehlhorn, in VIIth Intern. Conf. on the Physics of electronic and atomic collisions, Amsterdam, 1971. [12] V.O. Kostroun, M.H. Chen and B. Crasemann, Phys. Rev. A3 (1971) 533. [13] E.J. Callen, T.K. Krueger and W.L. McDavid, Bull. Am. Phys. Soc. 14 (1969) 830.
401
PHYSICS LETTERS
EFFECT OF ELECTRONIC
CORRELATIONS
5 June 1972
ON KLL AUGER LINES IN
ELEMENTS Mg - Sc* D. CHATTARJI and B. TALUKDAR Department of Physics, Visva-Bharati University, Santiniketan, West Bengal, India
The effect of electronic correlations on KLL Auger line intensities is investigated for the elements 12<~Z~<21by usi~aga simple ansatz for the correlation of the bound state electrons. No correlation is assumed between the K-sheU electron and the continuum electron in the final state. Enhancement of the intensities of the iSo (KLzL2) and the ID2 (KL2L3) lines is obtained relative to the ISo (KL! L1 ) line.
In recent years, improved experimental data on the KLL Auger spectrum have been obtained for low- Z elements through use of electrostatic analysis by Mehlhorn [1] and others [2]. On the theoretical side, use of intermediate coupling theory [3] and, more recently, of configuration interaction [4] has led to a situation where the experimental and theoretical energies of the transition show reasonably good agreement, but discrepancies remain with regard to the relative intensities of the individual lines. These discrepancies happen to be larger for lower Z values. Significantly, results obtained by McGuire [5] using Hartree-FockSlater wave functions for total KLL Auger transition rates appear to be in good agreement with experiment, but even with such a sophisticated model the individual configuration intensities differ from experiment by as much as 50%. Looking at the range of theoretical resuits obtained so far [5,6], it is seen that the line intensities are extremely sensitive to the detailed nature of the wave functions used. Difficulties in measurements of individual line intensities for low Z values [7], makes comparison between theory and experiment even more difficult in this region. The above situation, particularly the fact that even Hartree-Fock-Slater calculations do not lead to good agreement, seems to warrant a more detailed investigation into the possible nature o f the wave function in a light atom. For example, one of the shortcomings of the Hartree Fock method is that it neglects details of * Work supported in part by the Department of Atomic Energy, Government of India.
the electronic repulsion. On the whole, it allows electrons to come close together more often than is actually the case. Since the Auger process arises basically from the interaction between the two participating electrons, the energy of this interaction should play an important role in the overall energetics of the system. Effectively, this interaction gives rise to a correction in the total Hamiltonian o f the system which is commonly called the correlation energy [8]. This correlation energy is quite appreciable for two electrons in the same spatial orbital, particularly so if they have opposite spins, as in the case of the K L 1 L 1 transition (which, incidentally, provides the basis for computing relative intensities). Coming down to wave functions, a logical step would be to construct a wave function in which the paired electrons in the bound state avoid each other more strongly. That the correlation term should be relatively more important in the lighter atoms is readily understood. The overall averaging effect of the other "spectator" electrons may not be quite so smooth in these atoms as to give rise to a central field and dependence of a physical process like the Auger effect on the interelectron distance r12 = Ir1 - r21 may be appreciable. On the basis of these considerations, we have tried here the following ansatz for the wave function of the participating bound-state electrons:
',Itcorr=q~A(nlnl';SLJM)(1 --~ exp(--/~r 12)) ,
(1)
where ~ and t~ are variational parameters, q'A is the antisymmetrized two-electron hydrogenic wave func399
Volume 39A, number 5
PHYSICS LETTERS
tion belonging to the SLJM representation (i.e. we have used in this paper an extreme LS coupling without configuration interaction). For the purpose of this paper, X is taken to be unity and IJ=Z/nao*, where a 0 is the first Bohr radius of hydrogen. Evidently, ~corr is small %r r12 small and becomes independent of r12 when the latter is large. Furthermore, ibr a given value o f r l 2 , "Pcorr is very nearly equal to q'A for large Z and very different from ~A for small Z. This is quite plausible, since at higher Z values the Coulomb field of the nucleus becomes strong so that the perturbation caused by the mutual repulsion of the electrons may be neglected. For lower Z values, this perturbation is more nearly comparable to the nuclear Coulomb field. For the final state of the system we do not assume any correlation between the two electrons, because they are now far apart and belong to states which are dynamically different in character, one being bound to the K-shell and the other raised to the continuum. For the continuum electron we use a screened Coulomb wave t~mction normalized to a current of one electron per unit time per unit energy range. Screening constants and energy values are chosen as in our earlier work [9]. Intensities o f the IS0 (KL2L2) and 11)2 (KL2L3) lines relative to that of the IS0 (KL 1 L I ) l i n e are shown for 12 ~
5 June 1972
20
I..~
- .....
I NTENSIT IES OF I,% ~KLiL2~ AND I[~ (KL2L~) LINES RELATIVE TO ~_ FUNCTION OF ATOMIC
PRESENTCALQULATIONS ASAADIS INTERMEDIATE COUPLING CALCULATIORS
0
I~ @LzL~
O
~D~ QKL2L
EXPERIM Ei~I'AL RESULTS
NUMBER ~E
ID2
................ ..........
IP
15
14
15
16 ATOMIC
0 .......
17
ISo
Ia
NUMBER
19
20
21
22
Z
Fig. 1. The intensities of ID2 (KL2L3) and ISo (KL2L2) lines relative to ISo (KL1L1) for 12
Volume 39A, number 5
PHYSICS LETTERS
with appropriate screening [12] may be one of the factors having to do with the discrepancies which persist. However, as conjectured by Callan [13] the simple e2/r12 interaction given originally by Wentzel for the Auger effect may not hold very nicely in the low Z region. In particular for small inter-electronic distance r12 the Hartree-Fock radial integrals overshoots its final steady value. This seems to indicate that the kind o f cut-off built into our model for small r12 is a venture in the right direction. That this "correlation" has been tagged onto an otherwise simple hydrogenic system serves to keep the field of vision clear for judging the relative importance o f the different elements of the calculation. For example, introduction of intermediate coupling and configuration interaction would be the next logical step enabling us to see how crucial this correlation is when compared to these other features. How one could go about introducing correlations into a full-grown Hartree-Fock system should be an interesting question. Intensities o f the other KLL lines will be reported shortly in a more exhaustive communication.
5 June 1972
References [1] B. Cleff and W. Mehlhom, Z. Physik 219 (1969) 311. [2] A. Fahlman, R. Nordberg, C. Nordling and K. Siegbahn, Z. Physik 192 (1966) 476. [3] W.N. Asaad and E.H.S. Burhop, Proc. Phys. Soc. (London) 71 (1958) 369. [4] W.N. Asaad, Nucl. Phys. 66 (1965) 494. [5] E.J. McGuire, Phys. Rev. 185 (1969) 1; Phys. Rev. A2 (1970) 273. [6] R.A. Rubenstein, Ph.D. Thesis (unpublished), University of lllinois. (1955). [7] I. Bergstrom and C. Nordling, in Alpha-, beta- and gamma ray spectroscopy, ed. K. Siegbahn (North-Holland, Amsterdam, 1965). [8] C.C.J. Roothaan and A.W. Weiss, Rev. Mod. Phys. 33 (1960) 194. [9] B. Talukdar and D. Chattarji, Phys. Rev. A1 (1970) 44; B. Talukdar, Ph.D. Thesis (unpublished), Visva-Bharati University ( 1970). [10] E.J. Callen, Phys. Rev. 124 (1961) 793. [11] W. Mehlhorn, in VIIth Intern. Conf. on the Physics of electronic and atomic collisions, Amsterdam, 1971. [12] V.O. Kostroun, M.H. Chen and B. Crasemann, Phys. Rev. A3 (1971) 533. [13] E.J. Callen, T.K. Krueger and W.L. McDavid, Bull. Am. Phys. Soc. 14 (1969) 830.
401