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Statistical properties of Auger amplitudes and rates
Journal of Electron Spectroscopy and Related Phenomena 122 (2002) 181–193 www.elsevier.com / locate / elspec
Statistical properties of Auger amplitudes and rates A. Kyniene~ *, R. Karazija, V. Jonauskas ˇ State Institute of Theoretical Physics and Astronomy, A. Gostauto 12, 2600 Vilnius, Lithuania Received 6 August 2001; received in revised form 25 September 2001; accepted 28 September 2001
Abstract The statistical properties of Auger transitions are investigated for the first time. The fairly accurate approximate formula for the number of Auger amplitudes is derived. The symmetry property for this number and its approximation for semicomplementary arrays is determined. The results of calculations of the statistical characteristics (distribution function, initial and central moments, skewness, excess) for the distributions of Auger amplitudes and rates are presented in the case of transitions p 5 d N → p 6 d N 22 ´, sd N → s 2 d N 22 ´, d 9 p N → d 10 p N 22 ´ and their dependence on the number of electrons N in the sequences of atoms is investigated. It is shown that statistical properties of Auger spectra mainly depend on the orbital quantum numbers of shells involved in the transitions. For some characteristics the clearly expressed dependence on the even and odd numbers of electrons in outer open shell having integer or half-integer values of spins takes place. The rather large values of skewness and especially excess indicate a significant deviation of distribution of Auger amplitudes from the normal distribution. 2002 Elsevier Science B.V. All rights reserved. Keywords: Auger transitions; Statistical properties of spectra; Free atoms; Regularities of spectra
1. Introduction One of the perspective directions of Auger spectroscopy is the investigation of transitions in free atoms with open electronic shells [1]. The recently proposed Auger electron–photoelectron coincidence technique gives the possibility to achieve a resolution no longer limited by the lifetime of inner-shell hole and thus to resolve separate lines in Auger spectra [2]. During the last decade some studies have been devoted to the evaluation of the global and average characteristics of Auger spectra and their use for the theoretical generation of complex spectra [3–6]. *Corresponding author. ~ E-mail address: [email protected] (A. Kyniene).
However, as far as we know, the statistics of Auger amplitudes and lines was not considered till now. For this purpose, the results on the statistics of radiative transitions can be useful [7,8]. On the other hand, as we will see below, there exist essential differences between the statistical properties of radiative and Auger transitions. In this paper we consider the Auger transitions in single configuration and configuration interaction approximations. The Hartree–Fock method with relativistic corrections (Cowan code [9]) is used in the calculations. In Section 2 the main characteristics for the distribution of Auger amplitudes and rates are introduced. The approximate formulae for the number of amplitudes and lines are derived and their accuracy is investigated (Section 3). The results of
0368-2048 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S0368-2048( 01 )00356-5
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182
the systematic calculations of statistical properties of Auger spectra are presented and some of their regularities are discussed in Section 4.
2. The amplitudes of Auger transitions and characteristics of their distribution In the nonrelativistic approximation the amplitude of Auger transition is taken equal to the reduced matrix element of the Coulomb interaction operator H e: kKg JiH e iK9g 9J9´ljJl 5 (2J 1 1)1 / 2 kKg JuH e uK9g 9J9´ljJl
(1)
where K is the configuration of an atom, J is the quantum number of total angular momentum and g denotes all the additional quantum numbers, ´ is the energy of Auger electron. In the following, we assume that all transitions between two given configurations can be described by the same orbital of Auger electron with ´ as its average energy. The projection M of angular momentum J is not indicated in the matrix element (1) because it does not depend on this projection. Eq. (1) corresponds to the definition of amplitude for radiative transition. The distribution of Auger amplitudes can be characterized by its initial moments. The moment of the kth order is expressed as follows:
a ak (K 2 K9´)
O O (2J 1 1)
kKg JuH e uK9g 9J9´ljJl k g J g 9 J 9lj 5 ]]]]]]]]]]]] g(K, K9)
e
2
O O (2J 1 1) kKgJuH uK9g 9J9´ljJl k
e
g J g 9 J 9lj
2k
5 ]]]]]]]]]]]] 1 (K, K9)
(4)
Here 1 (K, K9) is the number of lines in the Auger spectrum. The sum in the numerator of Eq. (4) at k 5 1 as well as in the numerator of Eq. (2) at k 5 2 is equal to the total Auger transition rate A(K 2 K9´) between configurations K and K9. Only this sum of transition amplitudes has the explicit expression [5,10]. Thus, according to the definition of amplitude the following useful relations exist:
A(K 2 K9´) 5 ]]]] 1 (K, K9)
(2)
(2J 1 1)A(Kg J 2 K9g 9J9´)
O kKgJiH iK9g 9J9´ljl
a rk (K 2 K9´)
g(K, K9) a r1 (K 2 K9´) 5 ]]]a a2 (K 2 K9´) 1 (K, K9)
k/2
where g(K, K9) is the number of amplitudes. The intensities of Auger lines are determined by the transition rates as well as by the populations of initial levels. Often the assumption is made that all states are populated equally, then the populations of levels are proportional to their statistical weights (2J 1 1). Thus we will investigate the distribution of quantities
5
transition Kg J → K9g 9J9. In Eq. (3), the wave function of Auger electron is normalized to the unit density of the flux of electrons and the atomic units are used. By analogy with radiative transitions the quantity (3) can be named as the Auger line strength, but we will not introduce the new quantity and therefore the distribution of (2J 1 1)A(Kg J 2 K9g 9J9´) will be called the distribution of Auger rates. In the case of radiative transitions the sum over the channels is absent, then the simpler relations between the characteristics of the amplitudes and transition rates take place. The kth initial moment for the distribution of Auger rates is defined as follows:
(3)
lj
where A(Kg J 2 K9g 9J9´) is the rate of the Auger
(5)
Additionally to the initial moments ak the central moments mk can be introduced. They are defined with respect to the average energy and are expressed in terms of the initial moments [11]:
O (21) SkpDa k
p
mk 5
k2p
(a1 ) p
(6)
p 50
It is necessary to note that some arbitrary choice of signs for transition amplitudes is possible. In pure coupling the signs are defined by introducing some system of phases for irreducible tensors and wave functions. During the procedure of diagonalization of
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energy matrix, there also exists the possibility to choose the signs of some weights in the expansion of wave functions in intermediate coupling. For example, the maximal weights can be defined as positive numbers. Consequently, the signs of Auger as well as of radiative transition amplitudes in the intermediate coupling are not uniquely defined. Of course the values of the even initial moments do not depend on the signs of amplitudes, but they can influence the values of the odd moments and therefore all central moments. Thus, it is more grounded to investigate the distribution of the absolute values of amplitudes. The corresponding moments will be designated as uau a uau k and m k . The second central moment is the variance s 2 ; m2 . Its square root is proportional to the width of distribution. Instead of the third and fourth central moments the dimensionless coefficients skewness k1 and excess k2 are usually introduced
m3 m 3n m4 m 4n ]] ]] ]] k1 5 ]] 2 , k 5 2 2 (s 2 )3 / 2 (s n2 )3 / 2 (s 2 )2 (s n2 )2
(7)
F
a2 2 ]]2 2s a
G
(8)
we will take Œ2] ]] exp p rn (uau) 5 saŒ]
5
0
F
a2 ]] 2 2s 2a
G
at a $ 0,
(10)
and the coefficients in Eq. (7) have the following formulae ]] m n3 42p 2 ]] ]] ]], 5 2 3/2 p 2 2 p 2 2 (s n )
œ
2 m n4 3p 2 4p 2 12 ]] ]]]]] 5 (s 2n )2 (p 2 2)2
(11)
3. Number of Auger amplitudes and its symmetry property In the intermediate coupling the number of Auger amplitudes for a given channel ´l or ´lj can be obtained as the number of reduced matrix elements of the scalar operator acting between two configurations: g(K, K0) 5
O 1(KJ)1(K0J)
(12)
J
where s 2n and m n3 are the second and third central moments for the normal distribution. Then the proximity of the coefficients k1 , k2 to zero indicates the correspondence of distribution to the normal distribution. k1 is related to the asymmetry of distribution and k2 characterizes the density of values in it. In the case of radiative transitions, the assumption was made [7] that the distribution of amplitudes is normal with a zero average value. We will also compare the distribution of Auger amplitudes with normal distribution but for the absolute values of amplitudes, i.e. instead of the density of their probability 1 rn (a) 5 ]] ] exp Πsa 2p
p 22 s 2uau 5 ]]s a2 p
183
(9)
at a , 0
Then the variance of the distribution for the absolute values of amplitudes s 2uau is expressed by
where K0 denotes K9´l or K9´lj and K9 is the final configuration of the ion. 1 (KJ) is the number of levels with total momentum J in the configuration K. Its exact expression is not known, only its expansion in a series usually having a fairly fast convergence. Thus a realistic approximation of 1 (KJ) is obtained taking some first terms of this series [8]: g(K)[J] 1 (KJ) 5 ]]]]]] s 2z (K)(8ps 2z (K))1 / 2
S
2
[J] exp 2 ]]] 8s 2z (K)
F
DH
S GJ
D
z 1 m 4 (K) 1 1 ] ]] 23 24 s z4 (K)
5[J] 2 [J] 4 ]]] 15 2 ]]] 1 2s 2z (K) 16s 4z (K)
(13)
where s 2z (K) and m z4 (K) are the variance and the fourth central moment for the distribution of the projection M of total angular momentum J, g(K) is the statistical weight. All these quantities correspond to the considered configuration K and [J] means 2J 1 1. In the case of configuration with some open shells, s 2z (K) and m z4 (K) are expressed in terms of these quantities for separate shells or subshells:
A. Kyniene~ et al. / Journal of Electron Spectroscopy and Related Phenomena 122 (2002) 181 – 193
184
O s (l m (K) 5O m ( l s 2z (K) 5
2 z
Ni i
)
z 4
Ni i
)16
(14)
i
z 4
O s (l 2 z
Ni i
)s z2 ( l jNj )
(15)
i ,j
i
The same configuration can contain shells ( l 5 nl) and subshells ( l 5 nlj). The summation is performed over all open shells and / or subshells. The dependence of s 2z (K) and m z4 (K) on the number of electrons can be established using the group-diagrammatic method of summation of atomic quantities over all many-electron quantum numbers [3]: N(V 2 N) s 2z ( l N ) 5 ]]] B2 ( l) V (V 2 1)
(16)
N(V 2 N) m z4 ( l N ) 5 ]]]]]]] h[(V 2 N V (V 2 1)(V 2 2)(V 2 3) 2 1)(V 2 N 2 2) 2 (N 2 1)(V 2 N 2 1) 1 (N 2 1)(N 2 2)]B4 ( l) 1 3(N 2 1) 3 (V 2 N 2 1)B2 ( l)j
j
O m, k
O
l 11 / 2
Bk (l) 5
m 52j
Eqs. (21) and (22) were given in Ref. [8]. In the configuration K0, the Auger electron can be treated as the electron of discrete spectrum, thus 1 1 s 2z (´l) 5 ] B2 ( l), m z4 (´l) 5 ] B4 ( l) (23) V V The numerical application of the considered approximation for the configurations involved in various Auger transitions is demonstrated in Table 1. The correspondence to the exact numbers of levels is fairly satisfactory. The approximate expression for the number of Auger amplitudes is obtained replacing the summation in Eq. (12) by integration `
(17)
where V is the number of single-electron states equal to 4l 1 2 at l 5 nl and 2j 1 1 at l 5 nlj. B2 ( l) and B4 ( l) are the following sums: Bk (lj) 5
1 B2 (l) 5 ] (2l 1 1)(4l 2 1 4l 1 3) (21) 6 1 B4 (l) 5 ] (2l 1 1)(48l 4 1 96l 3 1 152l 2 1 104l 120 1 15) (22)
Bk (lj)
(18)
j 5l 21 / 2
They can be expressed by simple formulae: j( j 1 1)(2j 1 1) B2 (lj) 5 ]]]]] 3
E
g(K, K0) ¯ 1 (KJ)1 (K0J) dJ
and substituting Eq. (13) for 1 (KJ). After the standard integration and rearrangement of terms it gets: g(K)g(K0) g(K, K0) ¯ ]]]]]]]] 1 ] Œ 2 2p [s 2z (K) 1 s 2z (K0)] 3 / 2
H
5 z z 1 ]]]]]] 2 2 2 [ m 4 (K) 1 m 4 (K0) 8[s z (K) 1 s z (K0)]
(19)
2j 1 1 B4 (lj) 5 ]] [3(2j 1 1)4 2 10(2j 1 1)2 1 7] 240
(24)
0
2 3s 4z (K) 2 3s 4z (K0)]
(20)
J
(25)
Table 1 Numbers of levels with given J and their total number. Comparison of their exact values with that calculated by Eq. (13) J
0.5 5
p d
2
p 5d 4 sp 4 d3 d5
Statist. Exact. Statist. Exact. Statist. Exact. Statist. Exact. Statist. Exact.
7 7 21 21 3 3 3 2 5 4
1.5 11 11 36 35 4 3 5 5 8 7
2.5 12 11 41 39 1 2 5 5 9 10
3.5 9 9 36 36 0 0 4 3 8 7
4.5
5.5
6.5
7.5
5 5 25 26
2 2 14 15
0 0 6 6
0 0 2 2
2 3 5 5
1 1 3 3
0 0 1 1
0 0
8.5
0 0
oJ 1 (Kg J) 46 45 181 180 8 8 20 19 39 37
A. Kyniene~ et al. / Journal of Electron Spectroscopy and Related Phenomena 122 (2002) 181 – 193
Here the terms of the smallest order with the product of two coefficients m z4 (K) are omitted. The total number of the Auger amplitudes for the transitions between two configurations is obtained by the summation of the numbers of Auger amplitudes corresponding to the separate channels. The comparison of the calculated numbers of amplitudes with the exact value of this quantity for various transitions (Table 2) shows fairly close correspondence, especially for the more complex configurations with large number of levels. Then the uncertainty is of an order 4–5%. There exists the symmetry property for the number of Auger amplitudes: it does not change under the replacement N1 → V1 1 2 2 N1 for the transitions
l V1 1 21 l 2N2 → l 1V 1 l 2N2 22 ´l
in the basis of wave functions of intermediate coupling as well as for the number of amplitudes, determined only by the Coulomb interaction kKg LSiH e iK9g 9L9S9´lLSl
l V1 1 21 j 1 l N2 2g2 J2 J → l V1 1 l N2 2 22g 92 J 29 ´ljJ
(30)
and
l V1 1 21 j 1 l V2 2 122N2g 92 J 92 J → l V1 1 l V2 2 2N2g2 J2 ´ljJ (31)
as well as in the case of the same replacement Ni → Vi 1 2 2 Ni in one or both l 2N2 and l 3N3 shells ( l 5 nl) or subshells ( l 5 nlj) for the transitions
at fixed j 1 , J2 , J 29 , j and J values and all possible additional quantum numbers g2 and g 29 . In Eqs. (26) 2 2N 2 and (27), the complimentary shells l 2N2 and l V 2 as well as l N2 2 22 and l 2V 2 122N2 belong to different states. However, due to the triangular conditions of the total momenta with the same final momentum J, the recoupling does not change the number of matrix elements. A similar property for the number of amplitudes (Eq. (29)) follows from the equality of the numbers of amplitudes for the subsets with the
(27)
This symmetry property can be compared with the similar property for the radiative transitions [8]. The symmetry takes place for the number of amplitudes kKg JiH e iK9g 9J9´ljJl
(29)
Indicated symmetry property follows from the equality of the numbers of amplitudes for the subsets with given values of the total angular momentum J, for example, for
(26)
l V1 1 21 l 2N2 l 3N3 → l 1V 1 l 2N2 21 l 3N3 21 ´l
185
(28)
Table 2 Comparison of the numbers of Auger amplitudes in intermediate coupling, g(K, K9) calculated exactly and using the approximate formula (24) N p d →p d 5
N
6
N 22
´
p 51 / 2 d N → p 6 d N 22 ´ sd N → s 2 d N 22 ´ d 9 p N → d 10 p N 22 ´ d 9 s 2 p N → d 10 sp N 21 ´
Exact Approx. Deviation Exact Approx. Deviation Exact Approx. Deviation Exact Approx. Deviation Exact Approx. Deviation
(%)
(%)
(%)
(%)
(%)
2
3
4
5
6
7
8
9
10
45 52 13 29 33 12 16 17 6 28 32 13 22 24 8
860 928 7 557 598 7 273 284 4 195 216 10 259 273 5
5542 5838 5 3602 3783 5 1683 1729 3 349 383 9 770 804 4
15 889 16 587 4 10 344 10 774 4 4725 4837 2 195 216 10 770 804 4
22 376 23 300 4 14 575 15 145 4 6615 6762 2 28 32 13 259 273 5
15 889 16 587 4 10 344 10 774 4 4725 4837 2
5542 5838 5 3602 3783 5 1683 1729 3
860 928 7 557 598 7 273 284 4
45 52 13 29 33 12 16 17 6
22 24 8
186
A. Kyniene~ et al. / Journal of Electron Spectroscopy and Related Phenomena 122 (2002) 181 – 193
given values of orbital and spin momenta. This proof is easily extended to the general case. The number of Auger amplitudes corresponding to the transitions from the initial configuration with the number of electrons in the ith shell (subshell) Ni to the final configuration of the ion with the number of electrons in the same shell (subshell) N i9 is invariant with respect to the simultaneous substitutions Ni → 4l i 1 2 2 N 9i and N 9i → 4l i 1 2 2 Ni . The numbers in the other shells (subshells) can remain unchanged or be changed in the same way. No restrictions on the number of open shells (subshells) are imposed. This symmetry property is also fulfilled by the approximate expression, Eq. (25). It follows from the equality of moments for the partially and almost filled shells:
s 2z ( l N ) 5 s 2z ( l V 2N ),
m z4 ( l N ) 5 m z4 ( l V 2N )
(32)
as well as from the invariance of expressions
s 2z (K) 1 s 2z (K9) and m z4 (K) 1 m z4 (K9) 2 3s 4z (K) 2 3s 4z (K9)
(33)
with respect to the indicated replacement. In the expression for the number of Auger amplitudes (Eq. (25)), the second term within the brasses is usually essentially smaller than the first term equal to unity. Thus, this number is fairly closely approximated by the Gaussian function of N (Fig. 1). The summation of the numbers of Auger amplitudes over all possible channels gives the number of Auger lines 1 (K, K9). It is also possible to obtain the simple upper limit for this number. The operator of Coulomb interaction contains the summation over the ranks of spherical functions, therefore the selection rules for Auger transitions are not very constrained. The upper limit can be obtained making the assumption that the transitions from all initial levels to all final levels are allowed. Then
1 (K, K9) ¯ 1 (K)1 (K9)
Fig. 1. Number of Auger amplitudes in intermediate coupling for the transitions p 5 d N → p 6 d N 22 ´ (d) and sd N → s 2 d N 22 ´ (s) (exact values).
(34)
where 1 (K) is the total number of levels in configuration K. If the exact values of 1 (K) and 1 (K9) are taken, Eq. (34) becomes equality for d 9 p N 2 d 10 p N 22 ´ Auger transitions at a small number of
terms, but gives only approximate estimation for transitions between more complex configurations (Table 3). There exists also the approximate formula for 1 (K). This quantity has the following approximate expression [8]:
S
m z4 (K) 2 3 2g(K) 1 ]]] ]]] 1 (K) 5 ]]]] 1 2 1 8 (8ps 2z (K))1 / 2 24s 2z (K)
D
(35) at integer values of resultant J for configuration K and
S
m z4 (K) 2 3 2g(K) 1 1 (K) 5 ]]]] 1 2 ]]] 1 ]]] 2 1/2 2 8 (8ps z (K)) 6s z (K)
D
(36) at half integer values of J, where g, s 2z and m z4 have already been introduced in Eq. (13). However, Eq. (34) with the total numbers of levels taken from Eqs. (35) and (36), does not play already the role of the upper limit of 1 (K, K9). The symmetry considered above for the numbers of amplitudes is distorted for the numbers of lines due to the summation over the channels. Thus the ratio g(K, K9) /1 (K, K9) presented in the important relation (5) obtains the asymmetric dependence on N with two maxima (Fig. 2).
A. Kyniene~ et al. / Journal of Electron Spectroscopy and Related Phenomena 122 (2002) 181 – 193
187
Table 3 Comparison of the exact number of Auger lines 1 (K, K9) in intermediate coupling (a) with its values estimated from Eq. (34) (b) and calculated from Eqs. (35) and 36 (c) using values of total numbers of levels 1 (K), 1 (K9) N d p →d p 9
N
10
N 22
´
p 5 d N → p 6 d N 22 ´ d 9 s 2 p N → d 10 sp N 21
a b c a b c a b c
1
2
3
4
5
6
7
8
9
10
76 76 76 220 220 220 188 280 320
140 140 112 1602 1620 1424 140 168 280
60 60 60 3622 4066 4085 60 96 104
10 10 8 4470 6120 5696 6 8 8
3179 4400 4070
1125 1530 1408
204 228 228
18 18 16
12 12 12
28 28 28 45 45 44 82 112 112
4. Distribution of the Auger amplitudes and rates The systematic calculations of the distribution of Auger amplitudes and rates have been performed for one of the main types of Auger transitions
tional vacancy or without it) their mixing with configurations nd N21 (n 1 1)s and nd N 22 (n 1 1)s 2 takes place. In order to demonstrate the influence of such a mixing on the distributions of Auger amplitudes and rates, the results of calculations for the transitions between two complexes of configurations
1 12 n 1 l 14l 1 11 n 2 l 2N2 → n 1 l 4l n 2 l N2 2 22 ´l 1
Zr 2p 5 (3d 1 4s)4 → 2p 6 (3d 1 4s)2 ´l
(37)
in the series of atoms with an inner n 1 l 121 vacancy and a filling n 2 l N2 2 outer shell. In the case of configurations nd N (with an addi-
Fig. 2. Ratio of the number of Auger amplitudes g(K, K9) to the number of Auger lines 1 (K, K9) for the transitions p 5 d N → p 6 d N 22 ´ (d), sd N → s 2 d N 22 ´ (s), d 9 p N → d 10 p N 22 ´ (j) and d 9 s 2 p N → d 10 sp N 21 ´ (h).
(38)
in configurations interaction (CI) and noninteracting configuration approximations are presented in Figs. 3 and 4. In a double logarithmic scale, the distribution of the absolute values of amplitudes consists of two parts (Fig. 3a). The logarithm of density of small amplitudes (mainly caused by the mixing of different terms) varies linearly. From a certain value, the density begins to decrease more rapidly and the oscillations appear due to the smaller number of large amplitudes. Distribution approximately corresponds to the normal distribution, though the deviation is significant. In both approximations the density of states varies similarly, only some differences in the oscillations of the number of large amplitudes are noted. The distribution function is obtained by the integration of density from zero till the considered value of amplitude. This function is less sensitive to the approximation used and shows similar character (Fig. 3b). The distribution function of transition rates has the same average shape in both approximations, but some differences take place in the above-mentioned oscillation region (Fig. 4). In Table 4, the various characteristics of dis-
188
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Fig. 4. Distribution of the Auger rates for the transitions Zr 2p 5 (3d 1 4s)4 → 2p 6 (3d 1 4s)2 ´ : d CI approximation; s, single configuration approximation.
Fig. 3. Distribution of the amplitudes (a) and their distribution function (b) for the Auger transitions between two complexes of configurations Zr 2p 5 (3d 1 4s)4 → 2p 6 (3d 1 4s)2 ´ : ——— CI approximation; – – – single configuration approximation; ? ? ? normal distribution.
tribution in CI and noninteracting configuration approximations are compared; their differences are also quantitative but not qualitative. As the single configuration approximation usually determines the main regularities of atomic quantities and the average characteristics of distribution are not very sensitive to the approximation used, the following investigation of Auger transitions p 5 d N → p 6 d N22 ´, sd N → d N 22 ´
(39)
will be performed in single configuration approximation. In order to investigate the dependence of distribution on the number of electrons N, the initial configurations for all single ionized atoms were taken n 1 p 5 n 2 d N (n 2 1 1)s 2 (further the closed (n 2 1 1)s 2 shell is not indicated as the other closed shells). As was discussed in Section 2, the ambiguity in the determination of sign of Auger amplitudes can influence values of the odd initial moments. Indeed the nonmonotonic variation of such moments in the series of atoms with the same filling shell takes place. As an example in Fig. 5 the variation of the first initial moment is demonstrated for the Auger transitions 2p 5 4d N → 2p 6 4d N 22 ´l. On the other hand, the average of the absolute value of amplitude a uau 1 is a smooth function of N. In the following, we will use only absolute values of amplitudes and call them amplitudes. a uau 1 becomes minimal for the half-filled open shell. The reason is that the number of large amplitudes determined by Coulomb interaction increases towards N 5 5 slower than the number of smaller amplitudes determined by the mixing of terms by spin–orbit interaction in the intermediate coupling. The same regularity is also characteristic for the subsets of these transitions 2p 5 j4d 9 5s 2 → 2p 6 4d N ´l as well as for the transitions np 5 3d N → np 6 3d N 22 ´ (n 5 2, 3), and 3s3d N → 3s 2 3d N 22 ´. Two sides of the curve are not
A. Kyniene~ et al. / Journal of Electron Spectroscopy and Related Phenomena 122 (2002) 181 – 193
189
Table 4 Comparison of the various characteristics of distribution of Auger amplitudes and rates for Zr 2p 5 (3d 1 4s)4 → 2p 6 (3d 1 4s)2 ´ transitions in configuration interaction (CI) and separate configurations (SCA) approximations
CI SCA
a ua1 u
s 2ua u
k ua1 u
k ua2 u
a r1
s 2r
k r1
k r2
0.00318 0.00246
1.76310 25 2.02310 25
3.407 3.361
19.387 18.490
9.55310 25 9.55310 25
7.03310 28 9.45310 28
9.994 8.426
165.681 114.063
similar, the average amplitude is larger for the almost filled d 10 shell than for the partially filled d N shell. The investigation of other statistical moments reveals the regularity that the character of their dependence on N is mainly determined by the orbital but not principal quantum numbers of shells involved in the transitions. It is demonstrated by the results for the variance of amplitudes s 2uau (Fig. 6). In the case of various p 5 d N → p 6 d N 22 ´ transitions (including the transitions to the p 21 vacancy with the given j value of total angular momentum j) this quantity together with the average amplitude reaches a minimum at the half filled d N shell and increases to the ends of the interval of N. According to Eq. (5), the average transition rate a r1 is proportional to a 2uau with the coefficient equal to the number of Auger amplitudes g(K, K9) divided by the number of lines 1 (K, K9). However, the ratio
Fig. 5. First initial moment (average amplitude) of the distribution of amplitudes (d) and of their absolute values (s) for the Auger transitions 2p 5 4d N → 2p 6 4d N 22 ´ from the configurations with 2p 21 vacancy in single ions.
of these numbers has nonmonotonic character (Fig. 2). Thus the average value of rate a r1 for the 5 N 6 N22 transitions p d → p d ´ obtains an additional maximum at N 5 3 and further increases with N monotonically (Fig. 7). The dependence of s 2r on N has a similar character as for a 1r , as has been noticed for the amplitudes of considered transitions. The variation of these quantities with N for the transitions 3s3d N → 3s 2 3d N 22 ´ shows some specific changes: the additional maximum of a 1r is more expressed and the dependence of s 2r and a r2 on even and odd numbers of N manifests itself (Fig. 8). Probably, the last statistical property is related to the integer or half integer values of spin quantum numbers for d N shell with an even or odd number of electrons. We will meet this property considering the higher moments of distributions. The skewness and excess for the distribution of amplitudes (Figs. 9 and 10) obtain only positive values. The positiveness of k uau 1 is determined by the
Fig. 6. Variance of the amplitudes s 2ua u for the Auger transitions 2p 5 4d N → 2p 6 4d N 22 ´ (d) (3100), 2p 35 / 2 4d N → 2p 6 4d N 22 ´ (m) (3100) and 2p 5 3d N → 3p 6 3d N 22 ´ (s).
190
A. Kyniene~ et al. / Journal of Electron Spectroscopy and Related Phenomena 122 (2002) 181 – 193
Fig. 7. Average Auger transition rate a r1 (d) and variance s 2r (s) 5 N 6 N 22 for the transitions 2p 4d → 2p 4d ´.
distinct asymmetry of the distribution for the absolute values of amplitudes. The positiveness of k uau 2 shows that the distribution of values of Auger amplitudes in its main interval is more dense (the number of small amplitudes is larger) than in the case of normal distribution. k uau 1 varies in a comparatively narrow interval 2–4, but k 2uau has very large values exceeding 10 and even 15. Both these quantities are maximum for the transitions between configurations with d N shell having maximum of ua u states (near N 5 5). Since k uau 1 and k 2 are equal to
r 1 r 2
2 r
Fig. 8. Average Auger transition rate a (d), variance s (m) (3100) and second initial moment a (s) (3100) for the transitions 3s3d N → 3s 2 3d N 22 ´.
ua u
Fig. 9. Skewness k 1 of the amplitudes for the Auger transitions 5 N 6 N 22 5 N 6 N 22 2p 3d → 2p 3d ´ (s), 2p 4d → 2p 4d ´ (m) and 3s3d N → 3s 2 3d N 22 ´ (d).
zero for the normal distribution, their large values indicate a fairly significant deviation of the distribution of Auger amplitudes from normal especially for the configurations with a relatively high percentage of small amplitudes at N | 5. 2 As in the case of a 1uau and s uau the variation of k uau 1 uau and k 2 with N for all considered transitions has a similar character (Figs. 9 and 10). The specific feature of 3s3d N → 3s 2 3d N22 ´ transitions is a clearly expressed dependence on even and odd number of electrons. It is weakly marked out for transitions
Fig. 10. Excess k ua2 u of the amplitudes for the Auger transitions 2p 5 3d N → 2p 6 3d N 22 ´ (n), 3s3d N → 3s 2 3d N 22 ´ (h) as well as for transitions 2p 5 j4d N → 2p 6 4d N 22 ´ at j 5 1 / 2 (s) and j 5 3 / 2 (d) and for their common array 2p 5 4d N → 2p 6 4d N 22 ´ (m).
A. Kyniene~ et al. / Journal of Electron Spectroscopy and Related Phenomena 122 (2002) 181 – 193
p 5 d N → p 6 d N22 ´, however, this dependence also becomes significant, when the total moment quantum number j of 3p vacancy is fixed as for 3s vacancy (Fig. 10). The local minima for 3p 21 1 / 2 vacancy 21 correspond to the local maxima for 3p 3 / 2 vacancy, therefore the total dependence has a more continuous u character. The same is also valid for the k ua 1 coefficient. The skewness and excess of the distribution of transition rates fulfil the similar regularities: these quantities tend to obtain the maximum at the halffilled shell, k r2 by about one order exceeds k r1 . The dependence on the parity of N is characteristic to k r1 and k 2r for the transitions 3s3d N → 3s 2 3d N 22 ´ (Fig. 11), but only to k r2 in the case of transitions p 5 d N → p 6 d N22 ´. The dependence of k 1r on N becomes fairly irregular (Fig. 12). Due to the increased asymmetry and density of distribution, when the squares of small amplitudes are taken, the inequalities
k r1 4 k uau 1
k r2 4 k uau 2
(40)
are fulfilled. When the distribution of quantity x is normal, the distribution of x 2 is described by the Porter-Thomas law [7]. However, due to the sum contained in Eq. (3) even at the normal distribution of amplitudes, the distribution of rates corresponds to the indicated law only approximately. This correspondence becomes
191
Fig. 12. Excess of the transition rates k r2 for the Auger transitions 2p 5 j4d N → 2p 6 4d N 22 ´ at j 5 1 / 2 (d) and j 5 3 / 2 (s) as well as for their common array (m).
exact only when one channel of Auger transitions is possible or dominant. In the case of transitions d 9 p N → d 10 p N 22 ´ the configuration mixing has less influence compared with the transitions involving the outer d N shell. Due to a considerably smaller order of energy matrices for configurations with p N shell the relative number of small Auger amplitudes (caused by the mixing of terms) diminishes too and they do not have the essential influence on the average characteristics of distribution, their variation is mainly determined by the larger amplitudes. It causes some changes in the dependence of average characteristics on N. The calculations have been performed for the following Auger transitions in the sequences of atoms with one inner vacancy: 3d 9 4p N → 3d 10 4p N 22 ´ and 4d 9 5p N → 4d 10 5p N 22 ´ (41)
Fig. 11. Skewness k r1 (d) and excess k r2 (s) of the transition rates for the Auger transitions 3s3d N → 3s 2 3d N 22 ´.
The similarity of their characteristics presented in Fig. 13 confirms the conclusion that the distributions of Auger amplitudes and rates are mainly determined by the orbital not principal quantum numbers. The average Auger amplitude for transitions (41) as well as for transitions (39) tends to a minimum towards a half of p N shell, however, it obtains a maximum at the smallest not at the largest values of N (Fig. 13a). Such asymmetry between partially and almost filled shells is even more expressed for the
192
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numbers of electrons for these transitions does not manifest itself.
5. Conclusions
Fig. 13. Average amplitude a ua1 u (—) and excess k r2 (– –) (a) as well as variance s ua2 u (—) and skewness k 1r (– –) (b) of the for the Auger transitions 3d 9 4p N → 3d 10 4p N 22 ´ (d) and 9 N 10 N 22 4d 5p → 4d 5p ´ (s).
variance of amplitudes (Fig. 13b). Due to the nonmonotonic character of the ratio g(K, K9) /1 (K, K9) (Fig. 2) the dependence of a r1 on N has a broken shape. The variation of s r2 has a similar character as 2 s uau . The skewness and excess for the distributions of amplitudes and rates also fulfil the inequalities (40) as well as uau k uau 2 4k1
k r2 4 k r1
(42)
These characteristics are presented only for the distribution of rates (Fig. 13). Both the average rate and its variance do not show any symmetry with respect to a half of p N shell and have the largest values for the partially filled shell. The dependence of the statistical moments on the odd or even
The approximate formulae for the number of Auger amplitudes is derived with an uncertainty of about 5% for transitions between complex configurations with many levels. It is shown that the exact and approximate numbers of amplitudes fulfil the additional symmetry property. The dependence of this quantity on the number of electrons in the outer open shell is fairly closely approximated by the Gaussian function. On the other hand, the number of Auger lines contrary to the number of characteristic emission lines does not fulfil the symmetry property. The signs of Auger as well as of radiative transition amplitudes are not uniquely defined, therefore the odd initial moments can obtain nonmonotonic dependence on the number of electrons. For this reason, the distribution of the absolute values of amplitudes is considered in the present work. The systematic investigation of the average characteristics for distributions of Auger amplitudes and rates is performed for the transitions p 5 d N → p 6 d N22 ´, sd N → s 2 d N 22 ´ and 9 N 10 N 22 d p → d p ´ in the sequences of atoms with a filling outer open shell. The regularities of the dependence of average amplitude and transition rate as well as of variance, skewness and excess for their distributions are established. The findings are: (i) The distribution of amplitudes and rates is mainly determined by the orbital quantum numbers of the shells involved in the transitions and to a lesser extent by their principal quantum numbers. (ii) The distribution of Auger amplitudes, contrary to the amplitudes of radiative transitions, significantly deviates from the normal distribution that is indicated by large values of skewness and especially of excess. (iii) Some statistical properties of Auger transitions show a dependence on even or odd numbers of electrons in the open shell. This dependence is more expressed for ‘constrained’ transitions with one possible value for the total momentum of inner vacancy, for example, for 3s3d N → 3s 2 3d N 22 ´ or 2p 5 j4d N → 2p 6 4d N22 ´ transitions. Probably, such
A. Kyniene~ et al. / Journal of Electron Spectroscopy and Related Phenomena 122 (2002) 181 – 193
difference is related to the integer or half integer values of spin quantum numbers for the considered open shell.
References ¨ [1] H. Aksela, T. Makipaaso, S. Aksela, Phys. Rev. A31 (1985) 499; E. Kukk et al., J. Phys. B 27 (1994) 1965; G.B. Armen, H. Aksela, T. Aberg, S. Aksela, J. Phys. B 33 (2000) R49. [2] P. Lablanquie et al., Phys. Rev. Lett. 84 (2000) 47. [3] R. Karazija, Sums of Atomic Quantities and Mean Characteristics of Spectra, Mokslas, Vilnius, 1991, in Russian.
193
[4] A.G. Kochur, A.I. Dudenko, V.L. Sukhorukov, I.D. Petrov, J. Phys. B 27 (1994) 1709. ˇ R. Karazija, V. Jonauskas, Phys. Scripta 52 (1995) [5] S. Kucas, 639. ˇ [6] V. Jonauskas, R. Karazija, S. Kucas, J. Electron Spectrosc. Relat. Phenom. 107 (2000) 147. [7] C.E. Porter, R.G. Thomas, Phys. Rev. 104 (1956) 483. [8] J. Bauche, C. Bauche-Arnoult, J. Phys. B 20 (1987) 1659. [9] R.D. Cowan, The Theory of Atomic Structure and Spectra, University of California Press, Berkley, CA, 1981. [10] R. Karazija, A. Udris, J. Grudzinskas, Sov. Phys. Collect. 15 (4) (1975) 527. [11] M.G. Kendall, A. Stuart, The Advanced Theory of Statistics, Charles Griffin, London, 1963.
Statistical properties of Auger amplitudes and rates A. Kyniene~ *, R. Karazija, V. Jonauskas ˇ State Institute of Theoretical Physics and Astronomy, A. Gostauto 12, 2600 Vilnius, Lithuania Received 6 August 2001; received in revised form 25 September 2001; accepted 28 September 2001
Abstract The statistical properties of Auger transitions are investigated for the first time. The fairly accurate approximate formula for the number of Auger amplitudes is derived. The symmetry property for this number and its approximation for semicomplementary arrays is determined. The results of calculations of the statistical characteristics (distribution function, initial and central moments, skewness, excess) for the distributions of Auger amplitudes and rates are presented in the case of transitions p 5 d N → p 6 d N 22 ´, sd N → s 2 d N 22 ´, d 9 p N → d 10 p N 22 ´ and their dependence on the number of electrons N in the sequences of atoms is investigated. It is shown that statistical properties of Auger spectra mainly depend on the orbital quantum numbers of shells involved in the transitions. For some characteristics the clearly expressed dependence on the even and odd numbers of electrons in outer open shell having integer or half-integer values of spins takes place. The rather large values of skewness and especially excess indicate a significant deviation of distribution of Auger amplitudes from the normal distribution. 2002 Elsevier Science B.V. All rights reserved. Keywords: Auger transitions; Statistical properties of spectra; Free atoms; Regularities of spectra
1. Introduction One of the perspective directions of Auger spectroscopy is the investigation of transitions in free atoms with open electronic shells [1]. The recently proposed Auger electron–photoelectron coincidence technique gives the possibility to achieve a resolution no longer limited by the lifetime of inner-shell hole and thus to resolve separate lines in Auger spectra [2]. During the last decade some studies have been devoted to the evaluation of the global and average characteristics of Auger spectra and their use for the theoretical generation of complex spectra [3–6]. *Corresponding author. ~ E-mail address: [email protected] (A. Kyniene).
However, as far as we know, the statistics of Auger amplitudes and lines was not considered till now. For this purpose, the results on the statistics of radiative transitions can be useful [7,8]. On the other hand, as we will see below, there exist essential differences between the statistical properties of radiative and Auger transitions. In this paper we consider the Auger transitions in single configuration and configuration interaction approximations. The Hartree–Fock method with relativistic corrections (Cowan code [9]) is used in the calculations. In Section 2 the main characteristics for the distribution of Auger amplitudes and rates are introduced. The approximate formulae for the number of amplitudes and lines are derived and their accuracy is investigated (Section 3). The results of
0368-2048 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S0368-2048( 01 )00356-5
A. Kyniene~ et al. / Journal of Electron Spectroscopy and Related Phenomena 122 (2002) 181 – 193
182
the systematic calculations of statistical properties of Auger spectra are presented and some of their regularities are discussed in Section 4.
2. The amplitudes of Auger transitions and characteristics of their distribution In the nonrelativistic approximation the amplitude of Auger transition is taken equal to the reduced matrix element of the Coulomb interaction operator H e: kKg JiH e iK9g 9J9´ljJl 5 (2J 1 1)1 / 2 kKg JuH e uK9g 9J9´ljJl
(1)
where K is the configuration of an atom, J is the quantum number of total angular momentum and g denotes all the additional quantum numbers, ´ is the energy of Auger electron. In the following, we assume that all transitions between two given configurations can be described by the same orbital of Auger electron with ´ as its average energy. The projection M of angular momentum J is not indicated in the matrix element (1) because it does not depend on this projection. Eq. (1) corresponds to the definition of amplitude for radiative transition. The distribution of Auger amplitudes can be characterized by its initial moments. The moment of the kth order is expressed as follows:
a ak (K 2 K9´)
O O (2J 1 1)
kKg JuH e uK9g 9J9´ljJl k g J g 9 J 9lj 5 ]]]]]]]]]]]] g(K, K9)
e
2
O O (2J 1 1) kKgJuH uK9g 9J9´ljJl k
e
g J g 9 J 9lj
2k
5 ]]]]]]]]]]]] 1 (K, K9)
(4)
Here 1 (K, K9) is the number of lines in the Auger spectrum. The sum in the numerator of Eq. (4) at k 5 1 as well as in the numerator of Eq. (2) at k 5 2 is equal to the total Auger transition rate A(K 2 K9´) between configurations K and K9. Only this sum of transition amplitudes has the explicit expression [5,10]. Thus, according to the definition of amplitude the following useful relations exist:
A(K 2 K9´) 5 ]]]] 1 (K, K9)
(2)
(2J 1 1)A(Kg J 2 K9g 9J9´)
O kKgJiH iK9g 9J9´ljl
a rk (K 2 K9´)
g(K, K9) a r1 (K 2 K9´) 5 ]]]a a2 (K 2 K9´) 1 (K, K9)
k/2
where g(K, K9) is the number of amplitudes. The intensities of Auger lines are determined by the transition rates as well as by the populations of initial levels. Often the assumption is made that all states are populated equally, then the populations of levels are proportional to their statistical weights (2J 1 1). Thus we will investigate the distribution of quantities
5
transition Kg J → K9g 9J9. In Eq. (3), the wave function of Auger electron is normalized to the unit density of the flux of electrons and the atomic units are used. By analogy with radiative transitions the quantity (3) can be named as the Auger line strength, but we will not introduce the new quantity and therefore the distribution of (2J 1 1)A(Kg J 2 K9g 9J9´) will be called the distribution of Auger rates. In the case of radiative transitions the sum over the channels is absent, then the simpler relations between the characteristics of the amplitudes and transition rates take place. The kth initial moment for the distribution of Auger rates is defined as follows:
(3)
lj
where A(Kg J 2 K9g 9J9´) is the rate of the Auger
(5)
Additionally to the initial moments ak the central moments mk can be introduced. They are defined with respect to the average energy and are expressed in terms of the initial moments [11]:
O (21) SkpDa k
p
mk 5
k2p
(a1 ) p
(6)
p 50
It is necessary to note that some arbitrary choice of signs for transition amplitudes is possible. In pure coupling the signs are defined by introducing some system of phases for irreducible tensors and wave functions. During the procedure of diagonalization of
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energy matrix, there also exists the possibility to choose the signs of some weights in the expansion of wave functions in intermediate coupling. For example, the maximal weights can be defined as positive numbers. Consequently, the signs of Auger as well as of radiative transition amplitudes in the intermediate coupling are not uniquely defined. Of course the values of the even initial moments do not depend on the signs of amplitudes, but they can influence the values of the odd moments and therefore all central moments. Thus, it is more grounded to investigate the distribution of the absolute values of amplitudes. The corresponding moments will be designated as uau a uau k and m k . The second central moment is the variance s 2 ; m2 . Its square root is proportional to the width of distribution. Instead of the third and fourth central moments the dimensionless coefficients skewness k1 and excess k2 are usually introduced
m3 m 3n m4 m 4n ]] ]] ]] k1 5 ]] 2 , k 5 2 2 (s 2 )3 / 2 (s n2 )3 / 2 (s 2 )2 (s n2 )2
(7)
F
a2 2 ]]2 2s a
G
(8)
we will take Œ2] ]] exp p rn (uau) 5 saŒ]
5
0
F
a2 ]] 2 2s 2a
G
at a $ 0,
(10)
and the coefficients in Eq. (7) have the following formulae ]] m n3 42p 2 ]] ]] ]], 5 2 3/2 p 2 2 p 2 2 (s n )
œ
2 m n4 3p 2 4p 2 12 ]] ]]]]] 5 (s 2n )2 (p 2 2)2
(11)
3. Number of Auger amplitudes and its symmetry property In the intermediate coupling the number of Auger amplitudes for a given channel ´l or ´lj can be obtained as the number of reduced matrix elements of the scalar operator acting between two configurations: g(K, K0) 5
O 1(KJ)1(K0J)
(12)
J
where s 2n and m n3 are the second and third central moments for the normal distribution. Then the proximity of the coefficients k1 , k2 to zero indicates the correspondence of distribution to the normal distribution. k1 is related to the asymmetry of distribution and k2 characterizes the density of values in it. In the case of radiative transitions, the assumption was made [7] that the distribution of amplitudes is normal with a zero average value. We will also compare the distribution of Auger amplitudes with normal distribution but for the absolute values of amplitudes, i.e. instead of the density of their probability 1 rn (a) 5 ]] ] exp Πsa 2p
p 22 s 2uau 5 ]]s a2 p
183
(9)
at a , 0
Then the variance of the distribution for the absolute values of amplitudes s 2uau is expressed by
where K0 denotes K9´l or K9´lj and K9 is the final configuration of the ion. 1 (KJ) is the number of levels with total momentum J in the configuration K. Its exact expression is not known, only its expansion in a series usually having a fairly fast convergence. Thus a realistic approximation of 1 (KJ) is obtained taking some first terms of this series [8]: g(K)[J] 1 (KJ) 5 ]]]]]] s 2z (K)(8ps 2z (K))1 / 2
S
2
[J] exp 2 ]]] 8s 2z (K)
F
DH
S GJ
D
z 1 m 4 (K) 1 1 ] ]] 23 24 s z4 (K)
5[J] 2 [J] 4 ]]] 15 2 ]]] 1 2s 2z (K) 16s 4z (K)
(13)
where s 2z (K) and m z4 (K) are the variance and the fourth central moment for the distribution of the projection M of total angular momentum J, g(K) is the statistical weight. All these quantities correspond to the considered configuration K and [J] means 2J 1 1. In the case of configuration with some open shells, s 2z (K) and m z4 (K) are expressed in terms of these quantities for separate shells or subshells:
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184
O s (l m (K) 5O m ( l s 2z (K) 5
2 z
Ni i
)
z 4
Ni i
)16
(14)
i
z 4
O s (l 2 z
Ni i
)s z2 ( l jNj )
(15)
i ,j
i
The same configuration can contain shells ( l 5 nl) and subshells ( l 5 nlj). The summation is performed over all open shells and / or subshells. The dependence of s 2z (K) and m z4 (K) on the number of electrons can be established using the group-diagrammatic method of summation of atomic quantities over all many-electron quantum numbers [3]: N(V 2 N) s 2z ( l N ) 5 ]]] B2 ( l) V (V 2 1)
(16)
N(V 2 N) m z4 ( l N ) 5 ]]]]]]] h[(V 2 N V (V 2 1)(V 2 2)(V 2 3) 2 1)(V 2 N 2 2) 2 (N 2 1)(V 2 N 2 1) 1 (N 2 1)(N 2 2)]B4 ( l) 1 3(N 2 1) 3 (V 2 N 2 1)B2 ( l)j
j
O m, k
O
l 11 / 2
Bk (l) 5
m 52j
Eqs. (21) and (22) were given in Ref. [8]. In the configuration K0, the Auger electron can be treated as the electron of discrete spectrum, thus 1 1 s 2z (´l) 5 ] B2 ( l), m z4 (´l) 5 ] B4 ( l) (23) V V The numerical application of the considered approximation for the configurations involved in various Auger transitions is demonstrated in Table 1. The correspondence to the exact numbers of levels is fairly satisfactory. The approximate expression for the number of Auger amplitudes is obtained replacing the summation in Eq. (12) by integration `
(17)
where V is the number of single-electron states equal to 4l 1 2 at l 5 nl and 2j 1 1 at l 5 nlj. B2 ( l) and B4 ( l) are the following sums: Bk (lj) 5
1 B2 (l) 5 ] (2l 1 1)(4l 2 1 4l 1 3) (21) 6 1 B4 (l) 5 ] (2l 1 1)(48l 4 1 96l 3 1 152l 2 1 104l 120 1 15) (22)
Bk (lj)
(18)
j 5l 21 / 2
They can be expressed by simple formulae: j( j 1 1)(2j 1 1) B2 (lj) 5 ]]]]] 3
E
g(K, K0) ¯ 1 (KJ)1 (K0J) dJ
and substituting Eq. (13) for 1 (KJ). After the standard integration and rearrangement of terms it gets: g(K)g(K0) g(K, K0) ¯ ]]]]]]]] 1 ] Œ 2 2p [s 2z (K) 1 s 2z (K0)] 3 / 2
H
5 z z 1 ]]]]]] 2 2 2 [ m 4 (K) 1 m 4 (K0) 8[s z (K) 1 s z (K0)]
(19)
2j 1 1 B4 (lj) 5 ]] [3(2j 1 1)4 2 10(2j 1 1)2 1 7] 240
(24)
0
2 3s 4z (K) 2 3s 4z (K0)]
(20)
J
(25)
Table 1 Numbers of levels with given J and their total number. Comparison of their exact values with that calculated by Eq. (13) J
0.5 5
p d
2
p 5d 4 sp 4 d3 d5
Statist. Exact. Statist. Exact. Statist. Exact. Statist. Exact. Statist. Exact.
7 7 21 21 3 3 3 2 5 4
1.5 11 11 36 35 4 3 5 5 8 7
2.5 12 11 41 39 1 2 5 5 9 10
3.5 9 9 36 36 0 0 4 3 8 7
4.5
5.5
6.5
7.5
5 5 25 26
2 2 14 15
0 0 6 6
0 0 2 2
2 3 5 5
1 1 3 3
0 0 1 1
0 0
8.5
0 0
oJ 1 (Kg J) 46 45 181 180 8 8 20 19 39 37
A. Kyniene~ et al. / Journal of Electron Spectroscopy and Related Phenomena 122 (2002) 181 – 193
Here the terms of the smallest order with the product of two coefficients m z4 (K) are omitted. The total number of the Auger amplitudes for the transitions between two configurations is obtained by the summation of the numbers of Auger amplitudes corresponding to the separate channels. The comparison of the calculated numbers of amplitudes with the exact value of this quantity for various transitions (Table 2) shows fairly close correspondence, especially for the more complex configurations with large number of levels. Then the uncertainty is of an order 4–5%. There exists the symmetry property for the number of Auger amplitudes: it does not change under the replacement N1 → V1 1 2 2 N1 for the transitions
l V1 1 21 l 2N2 → l 1V 1 l 2N2 22 ´l
in the basis of wave functions of intermediate coupling as well as for the number of amplitudes, determined only by the Coulomb interaction kKg LSiH e iK9g 9L9S9´lLSl
l V1 1 21 j 1 l N2 2g2 J2 J → l V1 1 l N2 2 22g 92 J 29 ´ljJ
(30)
and
l V1 1 21 j 1 l V2 2 122N2g 92 J 92 J → l V1 1 l V2 2 2N2g2 J2 ´ljJ (31)
as well as in the case of the same replacement Ni → Vi 1 2 2 Ni in one or both l 2N2 and l 3N3 shells ( l 5 nl) or subshells ( l 5 nlj) for the transitions
at fixed j 1 , J2 , J 29 , j and J values and all possible additional quantum numbers g2 and g 29 . In Eqs. (26) 2 2N 2 and (27), the complimentary shells l 2N2 and l V 2 as well as l N2 2 22 and l 2V 2 122N2 belong to different states. However, due to the triangular conditions of the total momenta with the same final momentum J, the recoupling does not change the number of matrix elements. A similar property for the number of amplitudes (Eq. (29)) follows from the equality of the numbers of amplitudes for the subsets with the
(27)
This symmetry property can be compared with the similar property for the radiative transitions [8]. The symmetry takes place for the number of amplitudes kKg JiH e iK9g 9J9´ljJl
(29)
Indicated symmetry property follows from the equality of the numbers of amplitudes for the subsets with given values of the total angular momentum J, for example, for
(26)
l V1 1 21 l 2N2 l 3N3 → l 1V 1 l 2N2 21 l 3N3 21 ´l
185
(28)
Table 2 Comparison of the numbers of Auger amplitudes in intermediate coupling, g(K, K9) calculated exactly and using the approximate formula (24) N p d →p d 5
N
6
N 22
´
p 51 / 2 d N → p 6 d N 22 ´ sd N → s 2 d N 22 ´ d 9 p N → d 10 p N 22 ´ d 9 s 2 p N → d 10 sp N 21 ´
Exact Approx. Deviation Exact Approx. Deviation Exact Approx. Deviation Exact Approx. Deviation Exact Approx. Deviation
(%)
(%)
(%)
(%)
(%)
2
3
4
5
6
7
8
9
10
45 52 13 29 33 12 16 17 6 28 32 13 22 24 8
860 928 7 557 598 7 273 284 4 195 216 10 259 273 5
5542 5838 5 3602 3783 5 1683 1729 3 349 383 9 770 804 4
15 889 16 587 4 10 344 10 774 4 4725 4837 2 195 216 10 770 804 4
22 376 23 300 4 14 575 15 145 4 6615 6762 2 28 32 13 259 273 5
15 889 16 587 4 10 344 10 774 4 4725 4837 2
5542 5838 5 3602 3783 5 1683 1729 3
860 928 7 557 598 7 273 284 4
45 52 13 29 33 12 16 17 6
22 24 8
186
A. Kyniene~ et al. / Journal of Electron Spectroscopy and Related Phenomena 122 (2002) 181 – 193
given values of orbital and spin momenta. This proof is easily extended to the general case. The number of Auger amplitudes corresponding to the transitions from the initial configuration with the number of electrons in the ith shell (subshell) Ni to the final configuration of the ion with the number of electrons in the same shell (subshell) N i9 is invariant with respect to the simultaneous substitutions Ni → 4l i 1 2 2 N 9i and N 9i → 4l i 1 2 2 Ni . The numbers in the other shells (subshells) can remain unchanged or be changed in the same way. No restrictions on the number of open shells (subshells) are imposed. This symmetry property is also fulfilled by the approximate expression, Eq. (25). It follows from the equality of moments for the partially and almost filled shells:
s 2z ( l N ) 5 s 2z ( l V 2N ),
m z4 ( l N ) 5 m z4 ( l V 2N )
(32)
as well as from the invariance of expressions
s 2z (K) 1 s 2z (K9) and m z4 (K) 1 m z4 (K9) 2 3s 4z (K) 2 3s 4z (K9)
(33)
with respect to the indicated replacement. In the expression for the number of Auger amplitudes (Eq. (25)), the second term within the brasses is usually essentially smaller than the first term equal to unity. Thus, this number is fairly closely approximated by the Gaussian function of N (Fig. 1). The summation of the numbers of Auger amplitudes over all possible channels gives the number of Auger lines 1 (K, K9). It is also possible to obtain the simple upper limit for this number. The operator of Coulomb interaction contains the summation over the ranks of spherical functions, therefore the selection rules for Auger transitions are not very constrained. The upper limit can be obtained making the assumption that the transitions from all initial levels to all final levels are allowed. Then
1 (K, K9) ¯ 1 (K)1 (K9)
Fig. 1. Number of Auger amplitudes in intermediate coupling for the transitions p 5 d N → p 6 d N 22 ´ (d) and sd N → s 2 d N 22 ´ (s) (exact values).
(34)
where 1 (K) is the total number of levels in configuration K. If the exact values of 1 (K) and 1 (K9) are taken, Eq. (34) becomes equality for d 9 p N 2 d 10 p N 22 ´ Auger transitions at a small number of
terms, but gives only approximate estimation for transitions between more complex configurations (Table 3). There exists also the approximate formula for 1 (K). This quantity has the following approximate expression [8]:
S
m z4 (K) 2 3 2g(K) 1 ]]] ]]] 1 (K) 5 ]]]] 1 2 1 8 (8ps 2z (K))1 / 2 24s 2z (K)
D
(35) at integer values of resultant J for configuration K and
S
m z4 (K) 2 3 2g(K) 1 1 (K) 5 ]]]] 1 2 ]]] 1 ]]] 2 1/2 2 8 (8ps z (K)) 6s z (K)
D
(36) at half integer values of J, where g, s 2z and m z4 have already been introduced in Eq. (13). However, Eq. (34) with the total numbers of levels taken from Eqs. (35) and (36), does not play already the role of the upper limit of 1 (K, K9). The symmetry considered above for the numbers of amplitudes is distorted for the numbers of lines due to the summation over the channels. Thus the ratio g(K, K9) /1 (K, K9) presented in the important relation (5) obtains the asymmetric dependence on N with two maxima (Fig. 2).
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187
Table 3 Comparison of the exact number of Auger lines 1 (K, K9) in intermediate coupling (a) with its values estimated from Eq. (34) (b) and calculated from Eqs. (35) and 36 (c) using values of total numbers of levels 1 (K), 1 (K9) N d p →d p 9
N
10
N 22
´
p 5 d N → p 6 d N 22 ´ d 9 s 2 p N → d 10 sp N 21
a b c a b c a b c
1
2
3
4
5
6
7
8
9
10
76 76 76 220 220 220 188 280 320
140 140 112 1602 1620 1424 140 168 280
60 60 60 3622 4066 4085 60 96 104
10 10 8 4470 6120 5696 6 8 8
3179 4400 4070
1125 1530 1408
204 228 228
18 18 16
12 12 12
28 28 28 45 45 44 82 112 112
4. Distribution of the Auger amplitudes and rates The systematic calculations of the distribution of Auger amplitudes and rates have been performed for one of the main types of Auger transitions
tional vacancy or without it) their mixing with configurations nd N21 (n 1 1)s and nd N 22 (n 1 1)s 2 takes place. In order to demonstrate the influence of such a mixing on the distributions of Auger amplitudes and rates, the results of calculations for the transitions between two complexes of configurations
1 12 n 1 l 14l 1 11 n 2 l 2N2 → n 1 l 4l n 2 l N2 2 22 ´l 1
Zr 2p 5 (3d 1 4s)4 → 2p 6 (3d 1 4s)2 ´l
(37)
in the series of atoms with an inner n 1 l 121 vacancy and a filling n 2 l N2 2 outer shell. In the case of configurations nd N (with an addi-
Fig. 2. Ratio of the number of Auger amplitudes g(K, K9) to the number of Auger lines 1 (K, K9) for the transitions p 5 d N → p 6 d N 22 ´ (d), sd N → s 2 d N 22 ´ (s), d 9 p N → d 10 p N 22 ´ (j) and d 9 s 2 p N → d 10 sp N 21 ´ (h).
(38)
in configurations interaction (CI) and noninteracting configuration approximations are presented in Figs. 3 and 4. In a double logarithmic scale, the distribution of the absolute values of amplitudes consists of two parts (Fig. 3a). The logarithm of density of small amplitudes (mainly caused by the mixing of different terms) varies linearly. From a certain value, the density begins to decrease more rapidly and the oscillations appear due to the smaller number of large amplitudes. Distribution approximately corresponds to the normal distribution, though the deviation is significant. In both approximations the density of states varies similarly, only some differences in the oscillations of the number of large amplitudes are noted. The distribution function is obtained by the integration of density from zero till the considered value of amplitude. This function is less sensitive to the approximation used and shows similar character (Fig. 3b). The distribution function of transition rates has the same average shape in both approximations, but some differences take place in the above-mentioned oscillation region (Fig. 4). In Table 4, the various characteristics of dis-
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Fig. 4. Distribution of the Auger rates for the transitions Zr 2p 5 (3d 1 4s)4 → 2p 6 (3d 1 4s)2 ´ : d CI approximation; s, single configuration approximation.
Fig. 3. Distribution of the amplitudes (a) and their distribution function (b) for the Auger transitions between two complexes of configurations Zr 2p 5 (3d 1 4s)4 → 2p 6 (3d 1 4s)2 ´ : ——— CI approximation; – – – single configuration approximation; ? ? ? normal distribution.
tribution in CI and noninteracting configuration approximations are compared; their differences are also quantitative but not qualitative. As the single configuration approximation usually determines the main regularities of atomic quantities and the average characteristics of distribution are not very sensitive to the approximation used, the following investigation of Auger transitions p 5 d N → p 6 d N22 ´, sd N → d N 22 ´
(39)
will be performed in single configuration approximation. In order to investigate the dependence of distribution on the number of electrons N, the initial configurations for all single ionized atoms were taken n 1 p 5 n 2 d N (n 2 1 1)s 2 (further the closed (n 2 1 1)s 2 shell is not indicated as the other closed shells). As was discussed in Section 2, the ambiguity in the determination of sign of Auger amplitudes can influence values of the odd initial moments. Indeed the nonmonotonic variation of such moments in the series of atoms with the same filling shell takes place. As an example in Fig. 5 the variation of the first initial moment is demonstrated for the Auger transitions 2p 5 4d N → 2p 6 4d N 22 ´l. On the other hand, the average of the absolute value of amplitude a uau 1 is a smooth function of N. In the following, we will use only absolute values of amplitudes and call them amplitudes. a uau 1 becomes minimal for the half-filled open shell. The reason is that the number of large amplitudes determined by Coulomb interaction increases towards N 5 5 slower than the number of smaller amplitudes determined by the mixing of terms by spin–orbit interaction in the intermediate coupling. The same regularity is also characteristic for the subsets of these transitions 2p 5 j4d 9 5s 2 → 2p 6 4d N ´l as well as for the transitions np 5 3d N → np 6 3d N 22 ´ (n 5 2, 3), and 3s3d N → 3s 2 3d N 22 ´. Two sides of the curve are not
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189
Table 4 Comparison of the various characteristics of distribution of Auger amplitudes and rates for Zr 2p 5 (3d 1 4s)4 → 2p 6 (3d 1 4s)2 ´ transitions in configuration interaction (CI) and separate configurations (SCA) approximations
CI SCA
a ua1 u
s 2ua u
k ua1 u
k ua2 u
a r1
s 2r
k r1
k r2
0.00318 0.00246
1.76310 25 2.02310 25
3.407 3.361
19.387 18.490
9.55310 25 9.55310 25
7.03310 28 9.45310 28
9.994 8.426
165.681 114.063
similar, the average amplitude is larger for the almost filled d 10 shell than for the partially filled d N shell. The investigation of other statistical moments reveals the regularity that the character of their dependence on N is mainly determined by the orbital but not principal quantum numbers of shells involved in the transitions. It is demonstrated by the results for the variance of amplitudes s 2uau (Fig. 6). In the case of various p 5 d N → p 6 d N 22 ´ transitions (including the transitions to the p 21 vacancy with the given j value of total angular momentum j) this quantity together with the average amplitude reaches a minimum at the half filled d N shell and increases to the ends of the interval of N. According to Eq. (5), the average transition rate a r1 is proportional to a 2uau with the coefficient equal to the number of Auger amplitudes g(K, K9) divided by the number of lines 1 (K, K9). However, the ratio
Fig. 5. First initial moment (average amplitude) of the distribution of amplitudes (d) and of their absolute values (s) for the Auger transitions 2p 5 4d N → 2p 6 4d N 22 ´ from the configurations with 2p 21 vacancy in single ions.
of these numbers has nonmonotonic character (Fig. 2). Thus the average value of rate a r1 for the 5 N 6 N22 transitions p d → p d ´ obtains an additional maximum at N 5 3 and further increases with N monotonically (Fig. 7). The dependence of s 2r on N has a similar character as for a 1r , as has been noticed for the amplitudes of considered transitions. The variation of these quantities with N for the transitions 3s3d N → 3s 2 3d N 22 ´ shows some specific changes: the additional maximum of a 1r is more expressed and the dependence of s 2r and a r2 on even and odd numbers of N manifests itself (Fig. 8). Probably, the last statistical property is related to the integer or half integer values of spin quantum numbers for d N shell with an even or odd number of electrons. We will meet this property considering the higher moments of distributions. The skewness and excess for the distribution of amplitudes (Figs. 9 and 10) obtain only positive values. The positiveness of k uau 1 is determined by the
Fig. 6. Variance of the amplitudes s 2ua u for the Auger transitions 2p 5 4d N → 2p 6 4d N 22 ´ (d) (3100), 2p 35 / 2 4d N → 2p 6 4d N 22 ´ (m) (3100) and 2p 5 3d N → 3p 6 3d N 22 ´ (s).
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Fig. 7. Average Auger transition rate a r1 (d) and variance s 2r (s) 5 N 6 N 22 for the transitions 2p 4d → 2p 4d ´.
distinct asymmetry of the distribution for the absolute values of amplitudes. The positiveness of k uau 2 shows that the distribution of values of Auger amplitudes in its main interval is more dense (the number of small amplitudes is larger) than in the case of normal distribution. k uau 1 varies in a comparatively narrow interval 2–4, but k 2uau has very large values exceeding 10 and even 15. Both these quantities are maximum for the transitions between configurations with d N shell having maximum of ua u states (near N 5 5). Since k uau 1 and k 2 are equal to
r 1 r 2
2 r
Fig. 8. Average Auger transition rate a (d), variance s (m) (3100) and second initial moment a (s) (3100) for the transitions 3s3d N → 3s 2 3d N 22 ´.
ua u
Fig. 9. Skewness k 1 of the amplitudes for the Auger transitions 5 N 6 N 22 5 N 6 N 22 2p 3d → 2p 3d ´ (s), 2p 4d → 2p 4d ´ (m) and 3s3d N → 3s 2 3d N 22 ´ (d).
zero for the normal distribution, their large values indicate a fairly significant deviation of the distribution of Auger amplitudes from normal especially for the configurations with a relatively high percentage of small amplitudes at N | 5. 2 As in the case of a 1uau and s uau the variation of k uau 1 uau and k 2 with N for all considered transitions has a similar character (Figs. 9 and 10). The specific feature of 3s3d N → 3s 2 3d N22 ´ transitions is a clearly expressed dependence on even and odd number of electrons. It is weakly marked out for transitions
Fig. 10. Excess k ua2 u of the amplitudes for the Auger transitions 2p 5 3d N → 2p 6 3d N 22 ´ (n), 3s3d N → 3s 2 3d N 22 ´ (h) as well as for transitions 2p 5 j4d N → 2p 6 4d N 22 ´ at j 5 1 / 2 (s) and j 5 3 / 2 (d) and for their common array 2p 5 4d N → 2p 6 4d N 22 ´ (m).
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p 5 d N → p 6 d N22 ´, however, this dependence also becomes significant, when the total moment quantum number j of 3p vacancy is fixed as for 3s vacancy (Fig. 10). The local minima for 3p 21 1 / 2 vacancy 21 correspond to the local maxima for 3p 3 / 2 vacancy, therefore the total dependence has a more continuous u character. The same is also valid for the k ua 1 coefficient. The skewness and excess of the distribution of transition rates fulfil the similar regularities: these quantities tend to obtain the maximum at the halffilled shell, k r2 by about one order exceeds k r1 . The dependence on the parity of N is characteristic to k r1 and k 2r for the transitions 3s3d N → 3s 2 3d N 22 ´ (Fig. 11), but only to k r2 in the case of transitions p 5 d N → p 6 d N22 ´. The dependence of k 1r on N becomes fairly irregular (Fig. 12). Due to the increased asymmetry and density of distribution, when the squares of small amplitudes are taken, the inequalities
k r1 4 k uau 1
k r2 4 k uau 2
(40)
are fulfilled. When the distribution of quantity x is normal, the distribution of x 2 is described by the Porter-Thomas law [7]. However, due to the sum contained in Eq. (3) even at the normal distribution of amplitudes, the distribution of rates corresponds to the indicated law only approximately. This correspondence becomes
191
Fig. 12. Excess of the transition rates k r2 for the Auger transitions 2p 5 j4d N → 2p 6 4d N 22 ´ at j 5 1 / 2 (d) and j 5 3 / 2 (s) as well as for their common array (m).
exact only when one channel of Auger transitions is possible or dominant. In the case of transitions d 9 p N → d 10 p N 22 ´ the configuration mixing has less influence compared with the transitions involving the outer d N shell. Due to a considerably smaller order of energy matrices for configurations with p N shell the relative number of small Auger amplitudes (caused by the mixing of terms) diminishes too and they do not have the essential influence on the average characteristics of distribution, their variation is mainly determined by the larger amplitudes. It causes some changes in the dependence of average characteristics on N. The calculations have been performed for the following Auger transitions in the sequences of atoms with one inner vacancy: 3d 9 4p N → 3d 10 4p N 22 ´ and 4d 9 5p N → 4d 10 5p N 22 ´ (41)
Fig. 11. Skewness k r1 (d) and excess k r2 (s) of the transition rates for the Auger transitions 3s3d N → 3s 2 3d N 22 ´.
The similarity of their characteristics presented in Fig. 13 confirms the conclusion that the distributions of Auger amplitudes and rates are mainly determined by the orbital not principal quantum numbers. The average Auger amplitude for transitions (41) as well as for transitions (39) tends to a minimum towards a half of p N shell, however, it obtains a maximum at the smallest not at the largest values of N (Fig. 13a). Such asymmetry between partially and almost filled shells is even more expressed for the
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numbers of electrons for these transitions does not manifest itself.
5. Conclusions
Fig. 13. Average amplitude a ua1 u (—) and excess k r2 (– –) (a) as well as variance s ua2 u (—) and skewness k 1r (– –) (b) of the for the Auger transitions 3d 9 4p N → 3d 10 4p N 22 ´ (d) and 9 N 10 N 22 4d 5p → 4d 5p ´ (s).
variance of amplitudes (Fig. 13b). Due to the nonmonotonic character of the ratio g(K, K9) /1 (K, K9) (Fig. 2) the dependence of a r1 on N has a broken shape. The variation of s r2 has a similar character as 2 s uau . The skewness and excess for the distributions of amplitudes and rates also fulfil the inequalities (40) as well as uau k uau 2 4k1
k r2 4 k r1
(42)
These characteristics are presented only for the distribution of rates (Fig. 13). Both the average rate and its variance do not show any symmetry with respect to a half of p N shell and have the largest values for the partially filled shell. The dependence of the statistical moments on the odd or even
The approximate formulae for the number of Auger amplitudes is derived with an uncertainty of about 5% for transitions between complex configurations with many levels. It is shown that the exact and approximate numbers of amplitudes fulfil the additional symmetry property. The dependence of this quantity on the number of electrons in the outer open shell is fairly closely approximated by the Gaussian function. On the other hand, the number of Auger lines contrary to the number of characteristic emission lines does not fulfil the symmetry property. The signs of Auger as well as of radiative transition amplitudes are not uniquely defined, therefore the odd initial moments can obtain nonmonotonic dependence on the number of electrons. For this reason, the distribution of the absolute values of amplitudes is considered in the present work. The systematic investigation of the average characteristics for distributions of Auger amplitudes and rates is performed for the transitions p 5 d N → p 6 d N22 ´, sd N → s 2 d N 22 ´ and 9 N 10 N 22 d p → d p ´ in the sequences of atoms with a filling outer open shell. The regularities of the dependence of average amplitude and transition rate as well as of variance, skewness and excess for their distributions are established. The findings are: (i) The distribution of amplitudes and rates is mainly determined by the orbital quantum numbers of the shells involved in the transitions and to a lesser extent by their principal quantum numbers. (ii) The distribution of Auger amplitudes, contrary to the amplitudes of radiative transitions, significantly deviates from the normal distribution that is indicated by large values of skewness and especially of excess. (iii) Some statistical properties of Auger transitions show a dependence on even or odd numbers of electrons in the open shell. This dependence is more expressed for ‘constrained’ transitions with one possible value for the total momentum of inner vacancy, for example, for 3s3d N → 3s 2 3d N 22 ´ or 2p 5 j4d N → 2p 6 4d N22 ´ transitions. Probably, such
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difference is related to the integer or half integer values of spin quantum numbers for the considered open shell.
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[4] A.G. Kochur, A.I. Dudenko, V.L. Sukhorukov, I.D. Petrov, J. Phys. B 27 (1994) 1709. ˇ R. Karazija, V. Jonauskas, Phys. Scripta 52 (1995) [5] S. Kucas, 639. ˇ [6] V. Jonauskas, R. Karazija, S. Kucas, J. Electron Spectrosc. Relat. Phenom. 107 (2000) 147. [7] C.E. Porter, R.G. Thomas, Phys. Rev. 104 (1956) 483. [8] J. Bauche, C. Bauche-Arnoult, J. Phys. B 20 (1987) 1659. [9] R.D. Cowan, The Theory of Atomic Structure and Spectra, University of California Press, Berkley, CA, 1981. [10] R. Karazija, A. Udris, J. Grudzinskas, Sov. Phys. Collect. 15 (4) (1975) 527. [11] M.G. Kendall, A. Stuart, The Advanced Theory of Statistics, Charles Griffin, London, 1963.