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Calculated Auger transition probabilities — from free atom to solids
Journal of Electron Spectroscopy and Related Phenomena 72 (1995) 181- 185
Calculated Auger transition probabilities - from free atom to solids Yu.N.Kucherenko Institute of Metal Physics of the National Academy of Sciences of Ukraine, Vernadsky str. 36, 252142 Kiev, Ukraine
The behaviour of the calculated Auger transition probabilities crL has been studied in going from a free atom to an atom in solid. The Mg K L V transition matrix elements have been calculated using the wave functions of the core-ionized impurity embedded into the crystal. It has been shown that the energy dependence of the valence wave functions causes Auger transition probabilities increasing from the bottom of the valence band to the Fermi level. The dependence of the crL values on the energy of the valence electron state is very close to linear one.
1. I N T R O D U C T I O N It is well known that Auger electron spectroscopy can provide information on the local electronic structure of solids [1,2]. In particular this information is very useful for study of the impurity screening problem, electron correlation effects, alloying effects and others. However, in order to fully exploit the potential of Auger spectroscopy a thorough understanding of the factors contributing to the Auger spectral line shape is necessary. In order to interpret the energy distribution of the Auger electrons in the case of core-core-valence transitions (CCV) in solids the expression is widely used which describe the spectral profile as a superposition of the local partial densities of valence states (PDOS) N L weighted by the corresponding Auger transition probabilities crL, where L = (l, m). According to the final state rule [3] the PDOS N L is modified by the core-hole potential. The values of crL are usually calculated using atomic wave functions [4,51. In this approximation Auger transition matrix elements are independent of E v, the energy of the valence electron state involved in the Auger process. Taking into account a very weak dependence of crL values on the energy of the Auger electron E f one may consider them as constants. However, this can lead to a calculated Auger profile that differs significantly from experiment [6,7]. In order to explain the
discrepancies between the experimental and calculated Mg KLV spectra for different Mg containing alloys it has been supposed in [8] that Auger transition matrix elements change their values with changing the valence electron energy (Fig. 1): the transition probability to the final state fl must be different from that to the final state f2 because of different initial valence states. In [8] the dependence of the trL values on the energy has been approximated with a linear function and an excellent fitting to the experimental line shape has been achieved. In the present work we have performed ab initio calculations of the Auger transition probabilities for different energies of valence electron states. Effects of the embedding of the free atom into the cry,stal on the crL values have been also studied. 2. T H E O R E T I C A L M O D E L We describe the Auger line shape starting from Fermi's golden rule for the transitions from initial states I to the final state F : I(Ef)
:
( lwli)
6 ( E I. - E v + E c - E,).
X
(1)
Here W= W ( r l , r 2) is the operator of Coulomb interaction, and one has to sum over all final states and to average over all initial states.
0368-2048/95 $09.50 © 1995 Elsevier Science B.V. All rights reserved SSDi 0 3 6 8 - 2 0 4 8 ( 9 4 ) 0 2 3 1 2-3
182 Using the energy dependent (but not k-dependent) basis for the wave functions in the valence band (VB), we present the wave function corresponding to the energy band %(k) in the form: 'Pn(k,r)
FREE ATOM
m m--K.f2
Lf
y. C,L(k)qoL(r, Ev) ,
(2) L in the local region near the atom ejecting the Auger electron (within the atomic sphere). Consequently, the local PDOS can be introduced by the expression
NL(Ev)
=
=
1
.tv
SOLID
t
v
VB
Y.lc.z(k)l 2 k,n
(E v < cn(k) < E, + AE).
li i
(3)
In order to describe the Auger electron state the plane wave orthogonalized to all occupied (both valence and core) states has been used. After summation over the different spin states we obtain from (1)
Fig. 1. Sketch of the CCV Auger process
I ( E f ) = Z NL(Ev)o~Lc'V (E f ,Ev) (4) L under the condition E v = E / + E c - E l. Here the Auger transition probability depends on the valence state energy and may be written as
o~Lc'r(E/,Ev) = 4 ~ f
21, + 1 ~ ~ {IDc"~L'I2
+
coefficients. obtain
to(zv ( E / , E v ) =
= xf-E-f ~ { 9 1 ~ ( l s , i ; v , f ) + (2.)
+IEc,L.I= - Re D;,LvEe,LL,},
Especially for KLV-transitions we
(5)
where the direct and exchange integrals are defined
by DciLL, = I drtdr2 W~'(r~)z~'(r2' E/)W (r~,r2)V, (r,) tp£(r2, E,),
EciLL, = dr~dr2qZ:(r~)z'z.(r2, E/)W (r,, r2)~ (r2)~oL(r ,, e,). It should be noted that these expressions differ from those given in [8]. Here we have used the partial wave expansion of the normalized Auger electron wave function (ZL' (r, E l ) is the U th component of this function) and consequently we have given a new definition for D and E integrals. Separating the wave functions into radial and angular parts and evaluating the angular integrals we may obtain from (5) expressions for o-L in terms of the radial matrix elements and Clebsch-Gordan
91](ls, v;i,f)-
,
-91t,(ls, i;v,f)91t(ls,v;i,f)}c2(ltll';O,O).
(6)
l and l' are the quantum numbers for the angular momentum components of the valence state and Auger electron state, respectively. The values of radial matrix elements 9t can be obtained from the following formula:
91~(a,b;c,d) - - 8- x
22+1
1~ jdrlRa(rl)Rb(rx) x o
[r-~!dr2
• r#+2P,c(r2)Rd(r2) +
+r¢+2?dr2. r-~_l/~(r2)Rd (r2)].
(7)
rl 3. R E S U L T S A N D D I S C U S S I O N
For performing the Auger matrix elements calculations we have to choose the value of the cutoff radius Rcut to which the radial integrals in (7)
183 are evaluated. In the case of a free atom we may integrate to large values of radius till the convergence of the numerical integration procedure is achieved. In principle we have to use the same procedure for an atom in the solid. However, as it has been stated in 19], in this case it is satisfactory to integrate only up to the Wigner-Seitz (WS) radius. In order to study the changes of the a L values in going from a free atom to an atom in solid we have simulated the process of embedding an atom into the crystal. This process has been considered consisting of two stages: (i) an atom is compressed to the WS radius, and (ii) this compressed atom is embedded in one of the crystal lattice sites. The calculations have been performed for KLVtransitions in Mg using expressions (6) and (7). (i) We have used the renormalisation procedure introduced by Watson and co-workers [10]: The valence wave functions have been truncated at the cut-off radius and then renormalized within the Rcut sphere. It can be seen from Fig.2 that as the Rcut radius decreases the calculated a L values increase. This is caused by increasing valence electron density inside the Rcut radius. The oscillating a z dependence on the Rcut radius reflects the oscillations of the Auger electron wave function when it is truncated by the Rcut radius at its positive or negative value. This oscillating behaviour is predominantly displayed for a s values. (ii) Using the LMTO-Green function method [11] we have calculated self-consistently the wave functions of the core-ionized impurity in the crystal. This wave functions have been used then in (6), (7) to evaluate Auger transition matrix elements. The Rcut radius is equal in this case to the WS radius. The calculated results are presented in Fig.3. The energy dependence of the valence wave functions causes the o-L values to increase from the bottom of the VB to the Fermi level. It should be noted that the dependence of the Auger transition probabilities on the valence state energy is very close to linear. This fact may be a good justification for the linear function suggested in [8] as an approximation for the energy dependence of the crL values. The ab initio calculated results shown in Fig.3 agree well with those obtained semiempirically using experimental Auger spectra 18]. Some differences
~
30
tr,
25
•-,~ 2 0 ,~
,am 0 L
p
15
.e
5 ~-
O
I
3
L
I
4
~
I
5
~
I
t5
~
I
7
~
p
I
8
C u t - o f f radius, a.u. Fig.2. The crL values in the 'compressed' Mg atom as a function of the Rcut radius for KLIV (dashed lines) and KL23 V (solid lines) transitions. may be caused by the influence of the valence-hole life-time effects upon the shape of the experimental spectra. Consequently these effects are included in the semiempirical results but they have not been taken into account in the ab initio calculations. The life-time effects could only change to some extent (but not essentially) the slope of the lines. The as/a p ratio is an important parameter because it determines sufficiently the shape of the Mg KLV spectra. One can see in Fig.2 that the 'compressing' of atom changes the as/a p ratio because the ap value increases to greater degree than the a s one. Note, however, that the a s value in the 'compressed' atom approaches those a s values in the crystal which correspond to the bottom of the VB (the energy region where the valence sstates are localized). The ap value in the 'compressed' atom is very close to the ap values at the top of the VB. The values presented in Table 1 show that the as/a p ratio for the free (non'compressed') atom is very close to those for the solid. It should be noted that in solid the as/a p ratio changes its value in going from the bottom of
184
Table 1 Calculated
trs/crp ratio for Mg KL V transitions
¢Ys/crp
KL1V
KL23 V
free atom 'compressed' atom solid (VB bottom) solid (VB top)
2.19 1.54 2.35 2.29
0.68 0.39 0.61 0.61
the VB to the Fermi level very weak in spite of significant increasing both o-s and O-p values. These facts could explain why in many cases a satisfactory interpretation of the Auger spectra of solids has been achieved with making use of Auger transition probabilities calculated for free atom. Finally, we have studied the influence of the integration radius Rcut on the calculated crL values in solids. The valence wave functions was normalized within the WS sphere. Fig.4 shows the Auger transition probabilities as a function of the I
|
I
4. CONCLUSIONS We have performed ab initio calculations of the Auger transition probabilities for KLV process in the free Mg atom and in pure Mg metal. The behaviour of the calculated crL values in going from a free atom to an atom in solid has been studied. The Auger transition probabilities are strongly dependent on the energy of the valence electron
32
I
MgKL1V
16
Rcut radius. As it can be seen, the contributions to the integrals in (6), (7) may be considered as negligible only between r=3.0 a.u. and r=3.346 a.u., i.e. between the muffin-tin radius (a half of the distance to the nearest neighbouring atom, 3.033 a.u. for Mg) and the WS radius. This results are in contradiction with results presented in [9]. There it is concluded that the contributions to the radial integrals are negligible beyond 1 a.u. Our calculations show that the exchange integrals EaLL, have a good radial convergence due to the cut-off caused by localized core wave functions. The direct integrals DaLL,, however, change their values oscillating up to the muffin-tin radius.
I
I
I
MgKL23V
4 28
I
I
P
~ 24 12 "-~ 2 0
°,~,i
°.~-i
° v-,,l
,.Q
o
o o,,,~
16
8
P © ",I,,,,o
4
0
12 0 ". ~-.,I
/
8
4 [-
0
I
-8
-6
a
I
,
I
-4 -2 Energy, eV
,
I
O
0
i
-8
I
-6
~
I
,
I
-4 -2 Energy, eV
~
I
O
Fig.3. The o-L dependence on the energy of the valence electron state for Mg KLV transitions (the dashed lines show the o% dependence if only the Auger electron energy variation is taken into account).
185
i
i
The weak convergence of the radial integrals requires more detail investigations of the contributions to the transition matrix elements (including the region beyond the WS sphere) and consequently the further improvement of the theoretical description of the valence state (a Bitch function distorted in the potential of the core hole).
!
16 14 12 ~J °~ o~ ,.Q
10
REFERENCES /-"\
,.Q O I.
8
0
6
rc~ I.
4
S
\
f'\ \
/
[-
\
,
\_--
p
2 I
0 1.5
2.0
,
I
2.S
,
I
3.0
Ls
Cut-off radius, a.u. Fig.4. The crL dependence on the radius of integration for Mg KL1 V transitions from the bottom (dashed lines) and from the top (solid lines) of VB. state involved in the Auger process. For Mg KLV transitions they increase with increasing energy from the bottom of the VB to the Fermi level. The as~orp ratio however practically does not change and its value is very close to that in a free atom.
1. D.E. Ramaker, Critical Reviews in Solid State and Material Sciences / Ed. J.E. Greene, 17 (1991) 211. 2. P. Weightman, Physica Sctipta, T25 (1989) 165. 3. D.E. Ramaker, Phys. Rev. B, 25 (1982) 7341. 4. E.J. McGuire, Phys. Rev., 185 (1969) 1. 5. M.H Chen, F.P. Larkins and B. Graseman, Atomic Data and Nuclear Data Tables, 45 (1990) 1. 6. Yu.N. Kucherenko and A.Ya. Perlov, J. Electron Spectrosc. Relat. Phenom., 58 (1992) 199. 7. Yu.N. Kucherenko, A.Ya. Perlov and V.N. Antonov, J. Electron Spectrosc. Relat. Phenom., 60 (1992) 351. 8. Yu.N. Kucherenko, J. Electron Spectrosc. Relat. Phenom., 68 (1994) 339. 9. P.S. Fowles, J.E. Inglesfield and P. Weightman, J. Phys.: Condens. Matter, 3 (1991) 641. 10. R.E. Watson, J. Hudis and M.L. Perlman, Phys. Rev. B, 4 (1971) 4139. 11. O. Gunnarson, O. Jepsen and O.K. Andersen, Phys. Rev. B, 27 (1983) 7144.
Calculated Auger transition probabilities - from free atom to solids Yu.N.Kucherenko Institute of Metal Physics of the National Academy of Sciences of Ukraine, Vernadsky str. 36, 252142 Kiev, Ukraine
The behaviour of the calculated Auger transition probabilities crL has been studied in going from a free atom to an atom in solid. The Mg K L V transition matrix elements have been calculated using the wave functions of the core-ionized impurity embedded into the crystal. It has been shown that the energy dependence of the valence wave functions causes Auger transition probabilities increasing from the bottom of the valence band to the Fermi level. The dependence of the crL values on the energy of the valence electron state is very close to linear one.
1. I N T R O D U C T I O N It is well known that Auger electron spectroscopy can provide information on the local electronic structure of solids [1,2]. In particular this information is very useful for study of the impurity screening problem, electron correlation effects, alloying effects and others. However, in order to fully exploit the potential of Auger spectroscopy a thorough understanding of the factors contributing to the Auger spectral line shape is necessary. In order to interpret the energy distribution of the Auger electrons in the case of core-core-valence transitions (CCV) in solids the expression is widely used which describe the spectral profile as a superposition of the local partial densities of valence states (PDOS) N L weighted by the corresponding Auger transition probabilities crL, where L = (l, m). According to the final state rule [3] the PDOS N L is modified by the core-hole potential. The values of crL are usually calculated using atomic wave functions [4,51. In this approximation Auger transition matrix elements are independent of E v, the energy of the valence electron state involved in the Auger process. Taking into account a very weak dependence of crL values on the energy of the Auger electron E f one may consider them as constants. However, this can lead to a calculated Auger profile that differs significantly from experiment [6,7]. In order to explain the
discrepancies between the experimental and calculated Mg KLV spectra for different Mg containing alloys it has been supposed in [8] that Auger transition matrix elements change their values with changing the valence electron energy (Fig. 1): the transition probability to the final state fl must be different from that to the final state f2 because of different initial valence states. In [8] the dependence of the trL values on the energy has been approximated with a linear function and an excellent fitting to the experimental line shape has been achieved. In the present work we have performed ab initio calculations of the Auger transition probabilities for different energies of valence electron states. Effects of the embedding of the free atom into the cry,stal on the crL values have been also studied. 2. T H E O R E T I C A L M O D E L We describe the Auger line shape starting from Fermi's golden rule for the transitions from initial states I to the final state F : I(Ef)
:
( lwli)
6 ( E I. - E v + E c - E,).
X
(1)
Here W= W ( r l , r 2) is the operator of Coulomb interaction, and one has to sum over all final states and to average over all initial states.
0368-2048/95 $09.50 © 1995 Elsevier Science B.V. All rights reserved SSDi 0 3 6 8 - 2 0 4 8 ( 9 4 ) 0 2 3 1 2-3
182 Using the energy dependent (but not k-dependent) basis for the wave functions in the valence band (VB), we present the wave function corresponding to the energy band %(k) in the form: 'Pn(k,r)
FREE ATOM
m m--K.f2
Lf
y. C,L(k)qoL(r, Ev) ,
(2) L in the local region near the atom ejecting the Auger electron (within the atomic sphere). Consequently, the local PDOS can be introduced by the expression
NL(Ev)
=
=
1
.tv
SOLID
t
v
VB
Y.lc.z(k)l 2 k,n
(E v < cn(k) < E, + AE).
li i
(3)
In order to describe the Auger electron state the plane wave orthogonalized to all occupied (both valence and core) states has been used. After summation over the different spin states we obtain from (1)
Fig. 1. Sketch of the CCV Auger process
I ( E f ) = Z NL(Ev)o~Lc'V (E f ,Ev) (4) L under the condition E v = E / + E c - E l. Here the Auger transition probability depends on the valence state energy and may be written as
o~Lc'r(E/,Ev) = 4 ~ f
21, + 1 ~ ~ {IDc"~L'I2
+
coefficients. obtain
to(zv ( E / , E v ) =
= xf-E-f ~ { 9 1 ~ ( l s , i ; v , f ) + (2.)
+IEc,L.I= - Re D;,LvEe,LL,},
Especially for KLV-transitions we
(5)
where the direct and exchange integrals are defined
by DciLL, = I drtdr2 W~'(r~)z~'(r2' E/)W (r~,r2)V, (r,) tp£(r2, E,),
EciLL, = dr~dr2qZ:(r~)z'z.(r2, E/)W (r,, r2)~ (r2)~oL(r ,, e,). It should be noted that these expressions differ from those given in [8]. Here we have used the partial wave expansion of the normalized Auger electron wave function (ZL' (r, E l ) is the U th component of this function) and consequently we have given a new definition for D and E integrals. Separating the wave functions into radial and angular parts and evaluating the angular integrals we may obtain from (5) expressions for o-L in terms of the radial matrix elements and Clebsch-Gordan
91](ls, v;i,f)-
,
-91t,(ls, i;v,f)91t(ls,v;i,f)}c2(ltll';O,O).
(6)
l and l' are the quantum numbers for the angular momentum components of the valence state and Auger electron state, respectively. The values of radial matrix elements 9t can be obtained from the following formula:
91~(a,b;c,d) - - 8- x
22+1
1~ jdrlRa(rl)Rb(rx) x o
[r-~!dr2
• r#+2P,c(r2)Rd(r2) +
+r¢+2?dr2. r-~_l/~(r2)Rd (r2)].
(7)
rl 3. R E S U L T S A N D D I S C U S S I O N
For performing the Auger matrix elements calculations we have to choose the value of the cutoff radius Rcut to which the radial integrals in (7)
183 are evaluated. In the case of a free atom we may integrate to large values of radius till the convergence of the numerical integration procedure is achieved. In principle we have to use the same procedure for an atom in the solid. However, as it has been stated in 19], in this case it is satisfactory to integrate only up to the Wigner-Seitz (WS) radius. In order to study the changes of the a L values in going from a free atom to an atom in solid we have simulated the process of embedding an atom into the crystal. This process has been considered consisting of two stages: (i) an atom is compressed to the WS radius, and (ii) this compressed atom is embedded in one of the crystal lattice sites. The calculations have been performed for KLVtransitions in Mg using expressions (6) and (7). (i) We have used the renormalisation procedure introduced by Watson and co-workers [10]: The valence wave functions have been truncated at the cut-off radius and then renormalized within the Rcut sphere. It can be seen from Fig.2 that as the Rcut radius decreases the calculated a L values increase. This is caused by increasing valence electron density inside the Rcut radius. The oscillating a z dependence on the Rcut radius reflects the oscillations of the Auger electron wave function when it is truncated by the Rcut radius at its positive or negative value. This oscillating behaviour is predominantly displayed for a s values. (ii) Using the LMTO-Green function method [11] we have calculated self-consistently the wave functions of the core-ionized impurity in the crystal. This wave functions have been used then in (6), (7) to evaluate Auger transition matrix elements. The Rcut radius is equal in this case to the WS radius. The calculated results are presented in Fig.3. The energy dependence of the valence wave functions causes the o-L values to increase from the bottom of the VB to the Fermi level. It should be noted that the dependence of the Auger transition probabilities on the valence state energy is very close to linear. This fact may be a good justification for the linear function suggested in [8] as an approximation for the energy dependence of the crL values. The ab initio calculated results shown in Fig.3 agree well with those obtained semiempirically using experimental Auger spectra 18]. Some differences
~
30
tr,
25
•-,~ 2 0 ,~
,am 0 L
p
15
.e
5 ~-
O
I
3
L
I
4
~
I
5
~
I
t5
~
I
7
~
p
I
8
C u t - o f f radius, a.u. Fig.2. The crL values in the 'compressed' Mg atom as a function of the Rcut radius for KLIV (dashed lines) and KL23 V (solid lines) transitions. may be caused by the influence of the valence-hole life-time effects upon the shape of the experimental spectra. Consequently these effects are included in the semiempirical results but they have not been taken into account in the ab initio calculations. The life-time effects could only change to some extent (but not essentially) the slope of the lines. The as/a p ratio is an important parameter because it determines sufficiently the shape of the Mg KLV spectra. One can see in Fig.2 that the 'compressing' of atom changes the as/a p ratio because the ap value increases to greater degree than the a s one. Note, however, that the a s value in the 'compressed' atom approaches those a s values in the crystal which correspond to the bottom of the VB (the energy region where the valence sstates are localized). The ap value in the 'compressed' atom is very close to the ap values at the top of the VB. The values presented in Table 1 show that the as/a p ratio for the free (non'compressed') atom is very close to those for the solid. It should be noted that in solid the as/a p ratio changes its value in going from the bottom of
184
Table 1 Calculated
trs/crp ratio for Mg KL V transitions
¢Ys/crp
KL1V
KL23 V
free atom 'compressed' atom solid (VB bottom) solid (VB top)
2.19 1.54 2.35 2.29
0.68 0.39 0.61 0.61
the VB to the Fermi level very weak in spite of significant increasing both o-s and O-p values. These facts could explain why in many cases a satisfactory interpretation of the Auger spectra of solids has been achieved with making use of Auger transition probabilities calculated for free atom. Finally, we have studied the influence of the integration radius Rcut on the calculated crL values in solids. The valence wave functions was normalized within the WS sphere. Fig.4 shows the Auger transition probabilities as a function of the I
|
I
4. CONCLUSIONS We have performed ab initio calculations of the Auger transition probabilities for KLV process in the free Mg atom and in pure Mg metal. The behaviour of the calculated crL values in going from a free atom to an atom in solid has been studied. The Auger transition probabilities are strongly dependent on the energy of the valence electron
32
I
MgKL1V
16
Rcut radius. As it can be seen, the contributions to the integrals in (6), (7) may be considered as negligible only between r=3.0 a.u. and r=3.346 a.u., i.e. between the muffin-tin radius (a half of the distance to the nearest neighbouring atom, 3.033 a.u. for Mg) and the WS radius. This results are in contradiction with results presented in [9]. There it is concluded that the contributions to the radial integrals are negligible beyond 1 a.u. Our calculations show that the exchange integrals EaLL, have a good radial convergence due to the cut-off caused by localized core wave functions. The direct integrals DaLL,, however, change their values oscillating up to the muffin-tin radius.
I
I
I
MgKL23V
4 28
I
I
P
~ 24 12 "-~ 2 0
°,~,i
°.~-i
° v-,,l
,.Q
o
o o,,,~
16
8
P © ",I,,,,o
4
0
12 0 ". ~-.,I
/
8
4 [-
0
I
-8
-6
a
I
,
I
-4 -2 Energy, eV
,
I
O
0
i
-8
I
-6
~
I
,
I
-4 -2 Energy, eV
~
I
O
Fig.3. The o-L dependence on the energy of the valence electron state for Mg KLV transitions (the dashed lines show the o% dependence if only the Auger electron energy variation is taken into account).
185
i
i
The weak convergence of the radial integrals requires more detail investigations of the contributions to the transition matrix elements (including the region beyond the WS sphere) and consequently the further improvement of the theoretical description of the valence state (a Bitch function distorted in the potential of the core hole).
!
16 14 12 ~J °~ o~ ,.Q
10
REFERENCES /-"\
,.Q O I.
8
0
6
rc~ I.
4
S
\
f'\ \
/
[-
\
,
\_--
p
2 I
0 1.5
2.0
,
I
2.S
,
I
3.0
Ls
Cut-off radius, a.u. Fig.4. The crL dependence on the radius of integration for Mg KL1 V transitions from the bottom (dashed lines) and from the top (solid lines) of VB. state involved in the Auger process. For Mg KLV transitions they increase with increasing energy from the bottom of the VB to the Fermi level. The as~orp ratio however practically does not change and its value is very close to that in a free atom.
1. D.E. Ramaker, Critical Reviews in Solid State and Material Sciences / Ed. J.E. Greene, 17 (1991) 211. 2. P. Weightman, Physica Sctipta, T25 (1989) 165. 3. D.E. Ramaker, Phys. Rev. B, 25 (1982) 7341. 4. E.J. McGuire, Phys. Rev., 185 (1969) 1. 5. M.H Chen, F.P. Larkins and B. Graseman, Atomic Data and Nuclear Data Tables, 45 (1990) 1. 6. Yu.N. Kucherenko and A.Ya. Perlov, J. Electron Spectrosc. Relat. Phenom., 58 (1992) 199. 7. Yu.N. Kucherenko, A.Ya. Perlov and V.N. Antonov, J. Electron Spectrosc. Relat. Phenom., 60 (1992) 351. 8. Yu.N. Kucherenko, J. Electron Spectrosc. Relat. Phenom., 68 (1994) 339. 9. P.S. Fowles, J.E. Inglesfield and P. Weightman, J. Phys.: Condens. Matter, 3 (1991) 641. 10. R.E. Watson, J. Hudis and M.L. Perlman, Phys. Rev. B, 4 (1971) 4139. 11. O. Gunnarson, O. Jepsen and O.K. Andersen, Phys. Rev. B, 27 (1983) 7144.