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Effect of interchannel interaction on the neon KLL Auger transition rates
Volume 60A, number 5
PHYSICS LETTERS
21 March 1977
EFFECT OF INTERCHANNEL INTERACTION ON THE NEON KLL AUGER TRANSITION RATES 1
G. lIOWAT
Deportment of (~heonstrv.Edinburgh Unircrsitv, Edinburgh EH9 3.11. UK
T. ABERG2 Laboratory of Physics, helsinki Unircrsity of Technology, 02150 Espoo IS, Finland
0. GOSCINSKI3 Department of Quantum (‘heoustrr, Uppsala U,,irersity, 75120 Li~psalaI Sweden
S.C. SOONG4. C.P. I3HALLA4 and M. AFIMED4 Department of Physics, Kansas State University. Manhattan, Kansas 6650r5. LISA
Received 29 November 1976 Calculations of neon KLL Auger rates which include the mixing between the final 2S channels are reported. Flic results are compared with Kelly’s recent many—body—perturbation—theory calculations and with ~ibsolute rates, obtained from experiments of Gelius at al and Krause et al
In tile rare gas atoms, outershell autoioflizing states and their related photoabsorption spectra are strongly perturbed by interchannel interactions [II. Our letter shows that these interactions are significant for spectral distributions of K Auger electrons which are emitted with kinetic energies of several hundred electron volts. Hence we confirm the many-body-perturbation-theory (MBPT) calculation of the Ne KLL Auger rates by Kelly [21who took diagrams corresponding to the interaction between 2S final continuum states into account. A previous Hartree-Slater (US) calculation which did not include any channel mixing [3] gave nevertheless a good description of the experimental data. Our calculations utilize ooth HS and HartreeFock (HF) basis sets. We take the interaction between the final 2S continuum states into account by using the Fano-Prats variational configuration interaction approach [41Supported by Imperial Chemical Industries (UK). Supported by Finnish Academy of Sciences. ~ Supported by the Swedish Natural Sciences Research Coun2
cil. ~ Supported in part by the US Energy Research and Development Administration under Contract No. E(l1-1) -2753.
404
Since the Is-hole state is well separated trom any other hole state, it is sufficient to consider an isolated discrete state ~ degenerate with five continuous states i4i~..In LS coupling, these states are characterized by the following configurations and angular momenta: 2s 2 1Ses2S, 2s I 2p I 1’3Pe p2S, 2p2 1Ses2S, and 2p2 1Dcd2S. The corresponding antisymmetrized 2S wavefunctions are constructed from the appropriate one-electron spin orbitals by coupling the wavetinction of each term of the residual ion to the continuum orbital t~. The I lamiltonian suhmnatrix to be diagonalized is K~H EI~= ~ ~. (~ .,tH J;~ = M (1:’, 1:). iL
~.FIH
~-
/
‘~‘F”> = 1
-
(1)
iJ
•E”~~~~’~
+ ~‘F’ /
5.6 (E’ E”)(E’ 1:’) =
I
..
.1
where ~ is not necessarily orthogonal to Ji~ (~= I and the matrix V of the continuum interaction matrix elements VE’ E”(L) is not necessarily zero. In Fano’s .
1
work [5] it was assumed that = 0. and ~iE’,jE (E) = 0. The latter assumption is not valid for the basis sets employed in this and all works on Auger rates. Consequently. a prediagonalization of the Hamil-
Volume 60A, number
PHYSICS LETFERS
5
tonian matrix between continuum functions is necessary. The situation is analogous to the calculation of earths [6]. the 4d-subshell photoabsorption spectrum in rare If there is no intrachannel interaction, then it can be shown [7] that the Auger decay rate into channel i, F1~E~ (i = 1 5), is given by ...
5
F~(E)= 2~.(E, E) + ~p
_______
rd6
J~5j J
je,iE
~
E—e
I
(2)
21 March 1977
10,80
tion. In the HS calculations, initial ls-hole state spin orbitals provided there by theis X~method [8] interacwere emmethod as employed, no intrachannel ployed. The continuum orbitals were computed in the initial HS potential which approaches 2r~for large r. In the HF calculations initial state orbitals were also used. The Fock operator corresponding to the final state was constructed and the orthogonality of the continuum orbital to the bound state orbitals was imposed by the introduction of the appropriate Lagrangian multipliers. The calculation of the HS rates was performed by programs developed in Manhattan and the HF rates was based on programs developed in Edinburgh. The results of our calculations are compared with those of Kelly [2] in fig. 1. Kelly employs ls and ~ orbitals calculated with the final state configuration and 2s and 2p orbitals associated to the initial one. For each set of orbitals three rates are shown. The first does not include any correlation 2 1S 2 1Seffects, the second includes the 2s 0 —2p 0 mixing, and the third is based on eq. (2). Note that for Kelly’s basis set, the third rate is obtained by including only those diagrams which represent continuum interactions in the final state. 2 1SNote 2Salso andthat 2p2the 1Smixing 2S between the final 2s 0es 0e’s channels includes the mixing between the corresponding configurations of the residual ions. In fig. 1 the MBPT result of Kelly (labelled COOR I in ref. [2]) is also shown and compared with experimental results which have been obtained by scaling the width of the
1L1’S
7000 1040 960
775 095 085 7.05 075
~ ~
920
065 055
8 80 8.40 8 00
V
2.50
045
KL
E)12
to lowest order in V. In the sum which represents the interchannel interaction, P denotes the principal part of the integral. We shall use eq. (2) for both the HS and HF basis sets although the intrachannel interaction may not be negligible for the HS basis set. In the HF
IKL
rOTAL
3P 1L,3P
KL1L23
7.70 ~
230 2 70 1 90
100 0.80 0 70 090
7 70 730 770
.
V~060 050 040
. -
_________________
_________________
0.80
6580 00 5 60
10
~620 4Q 5.20 5 00 480
KL~L
~ KL2,3L2,1
1S 1,,
090 0 60 .1. 0.70 oso
040 ‘
1
030
0.20 HSHSHS’FtFHFMBPr I-lsHsHsHF,-IFHFP-lBPr
Fig. 1. Neon KLL Auger rates in units of i0~ au. The crosses correspond to the initial state HF basis, the circles to the liiitial state HS basis and the squares to Kelly’s mixed HF basis. For each basis are three rates which, from left to correspond to: there no correlation, the inclusion of 2s2 1S right, 2 5S 0 — 2p” 0 core mixing only, and the inclusion of interchannel mixing in the final state. The triangles represent Kelly’s CORR I MBPT calculation [2]. The diamonds correspond to experimental results with approximately a 9% error in the determination the absolute width of the neon ls photoelectron line 19,of 10].
Ne is photoelectronline measured by Gelius et al. [9] with the experimental relative rates of Krause et al. [10]. Fig. 1 reveals the manner in which interchannel interactions affect the Ne KLL rates. Although each set calculations similar the HS ratesofare uniformlyexhibits lower than the behaviour, HF rates. This suggests that the inclusion of the intrachannel interaction could raise the two first HS rates resulting in a larger deviation from the experimental data. This trend would be reversed by the interchannel interaction in the case of the 2s_2 1S and 2s~2p~1’3P rates. However, in 405
Volume 60A, number 5
PHYSICS LETTERS
the case of the 2p2 1D rate the final state interactions would add constructively but are compensated by 2p6 intrasheil correlations [2] corresponding to the interaction between initial ls2s22p4e 1Qe22 configurations. 2IS rate this compensating effect is smaller. For the Note also2p~ that the 2p6 intrashell correlation effect accounts for the main discrepancy between our HF total rate and the experimental rate. The sensitivity of the rates due to the choice of either initial or final HF spin orbitals indicates a small but significant relaxation effect. In conclusion, the interplay of the various correlation effects is strongly channel dependent.
21 March 1977
References II] U. lano and J.W. Looper, Rev. Mod. Phys. 40 (1965) 441: U. lano, J. Opt. Soc. Am. 65 (1975) 979. [2] H.P. 1(1975) [3] C.P. Kelly, Bhalla,Pimys. Phys. Rev. Lett.Al 44A (1973)556. 103. 141 U. Fano and F. Prats, Proc. NatI. Acad. Sci (India) A33 (1963) 553.
15]
U. Fano, Phys. Rev. 124 (1961) 1866.
16] A. Starace, Phys. Rev. B5 (1972) 1773. [71 G. Howat, T. Aberg 0. Goscinski, abstracts (Intern. Conf. on theand Physics of X-ray Extended spectra. NBS. Gaithersburg. 1976) p. 35, and tube published. 18] F. Herman, J.P. van Dyke and lB. Ortenburger, Phys. Rev. Lett. 22 (1969) 807; see also D.L. Walters and C.P. BhaIIa, PHys. Rev. A3 (1971) 1919.
19]
U. Gelius et al., (‘hem. Phys. Lett. 28 (1974) I.
110! MO. Krause et al., Phys. Lett. 31 A (1970) 81.
406
PHYSICS LETTERS
21 March 1977
EFFECT OF INTERCHANNEL INTERACTION ON THE NEON KLL AUGER TRANSITION RATES 1
G. lIOWAT
Deportment of (~heonstrv.Edinburgh Unircrsitv, Edinburgh EH9 3.11. UK
T. ABERG2 Laboratory of Physics, helsinki Unircrsity of Technology, 02150 Espoo IS, Finland
0. GOSCINSKI3 Department of Quantum (‘heoustrr, Uppsala U,,irersity, 75120 Li~psalaI Sweden
S.C. SOONG4. C.P. I3HALLA4 and M. AFIMED4 Department of Physics, Kansas State University. Manhattan, Kansas 6650r5. LISA
Received 29 November 1976 Calculations of neon KLL Auger rates which include the mixing between the final 2S channels are reported. Flic results are compared with Kelly’s recent many—body—perturbation—theory calculations and with ~ibsolute rates, obtained from experiments of Gelius at al and Krause et al
In tile rare gas atoms, outershell autoioflizing states and their related photoabsorption spectra are strongly perturbed by interchannel interactions [II. Our letter shows that these interactions are significant for spectral distributions of K Auger electrons which are emitted with kinetic energies of several hundred electron volts. Hence we confirm the many-body-perturbation-theory (MBPT) calculation of the Ne KLL Auger rates by Kelly [21who took diagrams corresponding to the interaction between 2S final continuum states into account. A previous Hartree-Slater (US) calculation which did not include any channel mixing [3] gave nevertheless a good description of the experimental data. Our calculations utilize ooth HS and HartreeFock (HF) basis sets. We take the interaction between the final 2S continuum states into account by using the Fano-Prats variational configuration interaction approach [41Supported by Imperial Chemical Industries (UK). Supported by Finnish Academy of Sciences. ~ Supported by the Swedish Natural Sciences Research Coun2
cil. ~ Supported in part by the US Energy Research and Development Administration under Contract No. E(l1-1) -2753.
404
Since the Is-hole state is well separated trom any other hole state, it is sufficient to consider an isolated discrete state ~ degenerate with five continuous states i4i~..In LS coupling, these states are characterized by the following configurations and angular momenta: 2s 2 1Ses2S, 2s I 2p I 1’3Pe p2S, 2p2 1Ses2S, and 2p2 1Dcd2S. The corresponding antisymmetrized 2S wavefunctions are constructed from the appropriate one-electron spin orbitals by coupling the wavetinction of each term of the residual ion to the continuum orbital t~. The I lamiltonian suhmnatrix to be diagonalized is K~H EI~= ~ ~. (~ .,tH J;~ = M (1:’, 1:). iL
~.FIH
~-
/
‘~‘F”> = 1
-
(1)
iJ
•E”~~~~’~
+ ~‘F’ /
5.6 (E’ E”)(E’ 1:’) =
I
..
.1
where ~ is not necessarily orthogonal to Ji~ (~= I and the matrix V of the continuum interaction matrix elements VE’ E”(L) is not necessarily zero. In Fano’s .
1
work [5] it was assumed that = 0. and ~iE’,jE (E) = 0. The latter assumption is not valid for the basis sets employed in this and all works on Auger rates. Consequently. a prediagonalization of the Hamil-
Volume 60A, number
PHYSICS LETFERS
5
tonian matrix between continuum functions is necessary. The situation is analogous to the calculation of earths [6]. the 4d-subshell photoabsorption spectrum in rare If there is no intrachannel interaction, then it can be shown [7] that the Auger decay rate into channel i, F1~E~ (i = 1 5), is given by ...
5
F~(E)= 2~.(E, E) + ~p
_______
rd6
J~5j J
je,iE
~
E—e
I
(2)
21 March 1977
10,80
tion. In the HS calculations, initial ls-hole state spin orbitals provided there by theis X~method [8] interacwere emmethod as employed, no intrachannel ployed. The continuum orbitals were computed in the initial HS potential which approaches 2r~for large r. In the HF calculations initial state orbitals were also used. The Fock operator corresponding to the final state was constructed and the orthogonality of the continuum orbital to the bound state orbitals was imposed by the introduction of the appropriate Lagrangian multipliers. The calculation of the HS rates was performed by programs developed in Manhattan and the HF rates was based on programs developed in Edinburgh. The results of our calculations are compared with those of Kelly [2] in fig. 1. Kelly employs ls and ~ orbitals calculated with the final state configuration and 2s and 2p orbitals associated to the initial one. For each set of orbitals three rates are shown. The first does not include any correlation 2 1S 2 1Seffects, the second includes the 2s 0 —2p 0 mixing, and the third is based on eq. (2). Note that for Kelly’s basis set, the third rate is obtained by including only those diagrams which represent continuum interactions in the final state. 2 1SNote 2Salso andthat 2p2the 1Smixing 2S between the final 2s 0es 0e’s channels includes the mixing between the corresponding configurations of the residual ions. In fig. 1 the MBPT result of Kelly (labelled COOR I in ref. [2]) is also shown and compared with experimental results which have been obtained by scaling the width of the
1L1’S
7000 1040 960
775 095 085 7.05 075
~ ~
920
065 055
8 80 8.40 8 00
V
2.50
045
KL
E)12
to lowest order in V. In the sum which represents the interchannel interaction, P denotes the principal part of the integral. We shall use eq. (2) for both the HS and HF basis sets although the intrachannel interaction may not be negligible for the HS basis set. In the HF
IKL
rOTAL
3P 1L,3P
KL1L23
7.70 ~
230 2 70 1 90
100 0.80 0 70 090
7 70 730 770
.
V~060 050 040
. -
_________________
_________________
0.80
6580 00 5 60
10
~620 4Q 5.20 5 00 480
KL~L
~ KL2,3L2,1
1S 1,,
090 0 60 .1. 0.70 oso
040 ‘
1
030
0.20 HSHSHS’FtFHFMBPr I-lsHsHsHF,-IFHFP-lBPr
Fig. 1. Neon KLL Auger rates in units of i0~ au. The crosses correspond to the initial state HF basis, the circles to the liiitial state HS basis and the squares to Kelly’s mixed HF basis. For each basis are three rates which, from left to correspond to: there no correlation, the inclusion of 2s2 1S right, 2 5S 0 — 2p” 0 core mixing only, and the inclusion of interchannel mixing in the final state. The triangles represent Kelly’s CORR I MBPT calculation [2]. The diamonds correspond to experimental results with approximately a 9% error in the determination the absolute width of the neon ls photoelectron line 19,of 10].
Ne is photoelectronline measured by Gelius et al. [9] with the experimental relative rates of Krause et al. [10]. Fig. 1 reveals the manner in which interchannel interactions affect the Ne KLL rates. Although each set calculations similar the HS ratesofare uniformlyexhibits lower than the behaviour, HF rates. This suggests that the inclusion of the intrachannel interaction could raise the two first HS rates resulting in a larger deviation from the experimental data. This trend would be reversed by the interchannel interaction in the case of the 2s_2 1S and 2s~2p~1’3P rates. However, in 405
Volume 60A, number 5
PHYSICS LETTERS
the case of the 2p2 1D rate the final state interactions would add constructively but are compensated by 2p6 intrasheil correlations [2] corresponding to the interaction between initial ls2s22p4e 1Qe22 configurations. 2IS rate this compensating effect is smaller. For the Note also2p~ that the 2p6 intrashell correlation effect accounts for the main discrepancy between our HF total rate and the experimental rate. The sensitivity of the rates due to the choice of either initial or final HF spin orbitals indicates a small but significant relaxation effect. In conclusion, the interplay of the various correlation effects is strongly channel dependent.
21 March 1977
References II] U. lano and J.W. Looper, Rev. Mod. Phys. 40 (1965) 441: U. lano, J. Opt. Soc. Am. 65 (1975) 979. [2] H.P. 1(1975) [3] C.P. Kelly, Bhalla,Pimys. Phys. Rev. Lett.Al 44A (1973)556. 103. 141 U. Fano and F. Prats, Proc. NatI. Acad. Sci (India) A33 (1963) 553.
15]
U. Fano, Phys. Rev. 124 (1961) 1866.
16] A. Starace, Phys. Rev. B5 (1972) 1773. [71 G. Howat, T. Aberg 0. Goscinski, abstracts (Intern. Conf. on theand Physics of X-ray Extended spectra. NBS. Gaithersburg. 1976) p. 35, and tube published. 18] F. Herman, J.P. van Dyke and lB. Ortenburger, Phys. Rev. Lett. 22 (1969) 807; see also D.L. Walters and C.P. BhaIIa, PHys. Rev. A3 (1971) 1919.
19]
U. Gelius et al., (‘hem. Phys. Lett. 28 (1974) I.
110! MO. Krause et al., Phys. Lett. 31 A (1970) 81.
406