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7. Матвеева Н.М, Козлов Э.В. Упорядоченные фазы в металлических системах. М.: Наука, 1989. - 248 с.
8. Flinn P.A.//Trans. Met. Soc.AIME.-1960,- V. 218 - P. 145-157.
9. Cupschalk S.G., Brown N. //Acta Met. - 1968. - V. 16. - № 5. -P. 657-666.
10. Marcinkowski M.J., Brown N. II J. Appl, Phys. - 1962. - V. 33. - № 2. - P. 537-552.
11. Голубенко Т.Я., Козлов Э.В. II Тезисы зонального семинара по прогрессивным методам упрочнения деталей машин и инструмента. - Томск. -1980. - С. 39.
12. Старостенков М.Д., Демьянов Б.Ф. II Металлофизика. -1985. - № 10. - С.128-130.
13. Козлов Э.В. II Кристаллическая структура и свойства металлических сплавов. - М.: Наука, 1978. - С. 110-118.
14. Старостенков М.Д. II Изв. вузов. Физика. -1992. - № 2. - С. 51-56.
15. Царегородцев А.И., Горлов Н.В., Демьянов Б.Ф., Старостенков М.Д. IIФММ. - 1984. - Т.58. - Вып. 2. - С. 336-343.
16. Штерн Д.М., Кормин Н.М., Козлов Э.В. II Изв. вузов. Физика. - 1980. - № 9. - С. 22-26.
17. Пушкареаа Г.В., Мартынов В.И., Емельянов В.Н., Голубенко Т.Я., Козлов Э.В. II Изв. вузов. Физика. - 1979. - № 3. - С. 59-61.
V.A. Kilin, D.A. Lazarev, Dm.A. Lazarev, V.M. Zelichenko, M. Ya. Amusia, G. Mentzel,
К. -H. ScharMer and H. Schmoranzer
THE SINGLE-PHOTON DOUBLE-IONIZATION OF NE VALENCE SHELL
Among the first theoretical attempts to interpret the single-photon double-ionization process in many-electron atoms. Carlson (1967) has proposed the shake off model. However, he found that the simple shake off model fails in the case of double ionization of the same shell. Fundamental results for the DPI cross sections of the Ne and Ar valence shells have been obtained by Carter and Kelly (1977) within the framework of the MBPT. Only qualitative agreement with the experiments has been achieved in this work as well as by Chang et al (1971. 1975). Namely, the cross sections for single-photon double ionization of the Ne L-shell into the 2s 2p5 "'P° and 'P° and the 2s° 2p6 'Sc states have been measured by Schartner et al (1993) for energies from threshold up to 150 eV, using the photon-induced fluorescence spectroscopy (P1FS) method (Schartner et al 1990 and further references therein). A comparison of the summed experimental 'P° and 'P° cross sections with the MBPT calculations by Carter and Kelly 1977 reveals a discrepancy qf 50%.
The general conclusion to be drawn from both the experimental and theoretical works is that the origin of the intensity for multiple ionization, especially in the valence region, is the electron correlations involved. Furthermore, the importance of the physical approximation used, in particular the choice of the potential for calculating photoelectron wavefunctions, is under discussion (Kelly 1977). To date, no state selective theoretical results have been presented which could distinguish the states of a doubly-photoionized valence shell and contain a detailed analysis of the emitted photoelectron spectra.
Some examples of different physical mechanisms of the single-photon multiple ionization are presented in Figure 1. In fact two essentially different mechanisms of multiple photoionization (PI) can be considered. The first one is the multi-step mechanism. It is valid if the exciting-photon energy is large enough to ionize an inner atomic shell. In thi§ case the Auger decay (Figure l.a) or a
cascade of Auger decays, the autoionization (1. b) and the double autoionization of a resonantly-excited state (1. c). the double Auger (l.d) and the satellite Auger (1. e) decay of an inner-shell vacancy may result in ejecting electron(s) in the second step of the DPI process, in addition to the primary photoelectron. Thus, the process of multiple ionization can be considered to proceed via two or more consecutive steps, of which the first is the single PI of the atom.
Another mechanism of the multiple PI occurs if the exciting-photon energy is relatively small, so that only valence shells can be ionized. Here the multi-step mechanism of ionization is forbidden because of the energy conservation law. Nevertheless the experimental data (Carlson 1967, Wight et al 1976. Samson and Haddad 1974, Schmidt et al 1976, El Sherbini and Van der Wiel 1972, Eckhardt and Schartner 1983, Schartner et al 1987, 1993) show that multiple photoinization can be even quite significant in this exciting-photon energy region.
nl £
< »- O—
>4 l\ С
i............... ...........nl................
-o— о—
M • /к M
Ni
Figure 1
For instance, the production ratio Xe+H/Xe" can reach up to 40% (El Sherbini and Van der Wiel 1972). Hence,
two or more electrons can be ejected simultaneously from outer shells because of direct multiple PI. Figure l.f describes'the direct double photoionization (DPI). Surely, this mechanism contributes to the multiple PI in the case of large exciting-photon energy as well.
In this work we present the results on the direct single-photon double-ionization cross sections, referred to as double photoionization (DPI) cross sections in the following, of valence shells and the corresponding photoelecton energy distributions. The direct DPI processes depend on the correlations between electrons in the ionized atom and as such their cross sections are much smaller than those for single photoionization, as a rule. This holds especially for the DPI experiments that are able to discriminate between different states of a double
ionized atom (Eckhardt and Schartner 1983, Schartner et al 1987, 1993, Lablanquie et al 1987, Price and Eland 1989).
The absolute partial DPI cross sections of the Ne L-shell into the 2s2 2p4 [3P, 'D, 'S], 2s1 2p5 [3P, 'p] and 2s° 2p6 ['S] states are presented here. The exciting-photon energy region from 60 eV (below the Nelll 2s2 2p4 fP] double ionization threshold 62.53 eV) up to 220 eV covers far and away the Nelll 2s° 2p6 [*S] double ionization threshold at 121.90 eV, the highest among the mentioned. The total cross section is obtained as a sum of the partial ones. The .theoretical results are compared with the experimental data on the total and the partial (state selective) DPI cross sections in the energy region available.
Theory and details of calculation
The DPI' cross section of an atom is an additive characteristic of both the properties of the final doubly charged ion and the two emitted electrons. According to the general formula the total DPI cross section for the
a" {(,)) --- 4n £
o >c ,,ec
Here co is the photon energy, c - 137.036 is the speed of light, ic denotes the quantum numbers of the residual ionic core states. /,'' is the corresponding ionization
closed shell atoms (see, e.g.. Amusia 1987. 1990) can be written as follows (atomic units are employed throughout the paper except where noted otherwise)
M0„ Asde, .
(1)
potential, and ee stands for the discrete quantum numbers of the two emitted photoelectrons of energies £{. ¿\ and satisfying the energy conservation law
CO
I
£ , + £
(2)
and the angular momentum selection rules. Continuum spectrum wavefunctions are normalized to energy scale and the transition amplitude M is implicated to be in the velocity form. To get the cross section in units of Mb one
has to multiply cr++ by aI =28.0 Mb.
Below we shall refer to a specific cross section for the
ionic state ic as to the partial cross section G** (co).
d(7 + < (co) _ 4n' d£, ox
After the irradiation of an atom by photons of energ\ co
has taken place the energy excess (co — I^) is
distributed continuously between two emitting electrons. Therefore, the cross section Eq.l is expressed by the
integral over the energy region from 0 to (co - /,'' ).
Correspondingly, the differential cross section
e2)
(3)
describes the energy distribution of the first emitted electron. In case an experiment does not discriminate the emitted electrons (e.g., their orbital momenta if they are different) one has to get a symmetric relation for the energy distribution function for the second electron by means of permutation of the emitted electron wavefunctions in the DPI transition amplitude Mtj K ce.
Then, the sum of the two functions could be related to the spectrum of emitted electrons.
The final formulae for the calculation of the DPI transition amplitudes, photoelectron energy distribution functions and cross sections are obtained within the framework of the perturbation theory (PT) and the LS-
coupling approximation. The non-relativistic Hartree-Fock approximation is employed to obtain the basis of single-electron wavefunctions that includes: the hole states, the discrete spectrum excited states with principle quantum numbers up'to n = 18 and the continuum spectrum excited states obtained at 203 points for the numerical integration over the intermediate continuum spectrum states.
Only the lowest non-vanishing order PT diagrams (see, e.g., Carter and Kelly 1977, Amusia 1987) are calculated directly. Also, certain classes of the higher-order PT diagrams are included semi-empiricaliy (Kelly 1968, Amusia 1987) by use of the experimental DPI thresholds in the energy denominators of the diagrams and
by means of calculating the emitted electron wavefunctions in a specific field of a residual ion.
The choice of the potential for calculating the photoelectron wavefunctions is expected to be very important if a limited (the lowest) order of PT is used to obtain the DPI cross section. The variations in the DPI cross sections expected for different choices of the potential were discussed in detail by Carter and Kelly (1977). In the calculations of the Ne valence-shells DPI cross section they have used the so- called V(N"" potential that behaves itself as l/'r, asymptotically. This has resulted in the overestimated screening effect in the potential of the escaping photoelectron and, hence, in the discrepancy with the experimental DPI cross section in the initial slope and in the position of the maximum and, presumably, in its lesser height (see Figure 2). For higher photon energies the agreement is generally better, excluding the later data of Holland et al (1979). More correctly then, the escaping electron wavefunctions could be calculated in a V^"-' potential, i.e.', in the field of the residual ion of plus two charge. On the other hand, the faster electron may be screened to some extent by the slower one, so that the real picture is more complicated and presumably lies somewhere between these two cases.
In a continuation of the discussion about the choice of the potential for two photoelectrons, it may be suggested to distinguish the "frozen atomic core" and the "frozen ionic core" V(N_|) potentials. The "frozen atomic core" approximation performs (presumes) the calculation of the photoelectron wavefunctions in the field of an atom having two vacancies, while the "frozen ionic core" approach employs the self-consistent field of the residual ion. The latter takes into account the relaxation effects in the residual ion that could be important at the near-threshold energy region.
For the. results presented below, the photoelectron wavefunctions are calculated within the "frozen atomic core" approximation. Various photoelectron pairs of orbital momenta (/, ,/2), / -- 0,1.2,..., (which are) LS-coupled in accordance with the angular momentum selection rules, are taken into account in the calculation of the DPI cross sections. Each curve for the 2s2 2p4 ('P, 'D. 'S), 2s1 2p5 ('P, JP) and 2s° 2p6 ('S) excitations is the direct sum of the lowest order cross sections represented by the transitions according to Table 1.
с
о
и ф
in 1/1 </> о
и
К Q
0.30 -
0.25 -
0,20 -
0,15 -
0,10 -
0,05 -
0,00
120 140 160 180 200 Exciting-photon energy / eV
Figure 2
Table 1
Neon dipole transitions 2s2 2p6 2s2"m 2p6"n ex/, e2h accounted for in the calculated DPI cross sections. I' ' is the double-ionization potential
Final ion state r (eV) Photoelectron pairs hh 2S+1l , Contribution, % for Aw= 5 50 100
sp 3P 6,7 2,9 2,6
2s2 2p4 3P 62,53 pd df 3P, 3P, 3D 3D 89,8 3,4 94,0 2,7 93,8 2,9
fg " gh 3P, 3P. 3D 3D 0,1 0,4 0,7
sp 'P 5,8 3,2 3,4
pd 'P, 'D, 'F 81,9 87,6 86,7
2s2 2p4 'D df 'P, 'D, 'F 4,8 3,5 3,8
65,73 fg gh 'P, 'P, 'D, 'F 'F 0,1 0,4 0,8
sf 'F 7,4 5.3 5,3
Pg dh 'F 'F
sp 'P 68,9 18,8 17,9
pd 'P 25,7 77,0 77,2
2s2 2p4 >S 69,44 df 'P 5,0 2,8 2,9
fg 'P 0,4 1,4 2,0
gh 'P ,!
ss .. JS 3D 0,3 0,5
PP 3P, 7 65,5 57,0
dd 3s, 3P, 3D 28,1 19,7 17,9
ff 3s, 3P, 3D 0,3 0.5
2s1 2p5 3P 87,93 gg hh 3S, 3S, 3P, 3P, 3D 'D
■ sd 3D 1,8 14,2 24,2
pf dg fh 3D 3D 3D
ss 's 1,0 1,6 1,6
PP 'S, 'P, 'D 43.7 43,0 46,7
dd 's, 'P, 'D 51,9 34.7 25,1
ff 's. 'P, 'D 0,1 0,5 0,7
2s12p5 'P 98,42 gg hh 'S, 's, 'P, 1P, 'D 'D 0,1
sd " 'D 3,3 2,6 25,8
Pf dg fh 'D 'D 'D
sp ■p 93,2 65,6 57,2
2s°2p6 'S pd 'p 6,6 33,0 39,5
121,90 df fg gh ■p ■p 'p 0,1 1,4 0,1 3,1 0,3
It was established by calculations that the emission of photoelectrons with / > 5 from the valence shells of rare gas atoms is practically negligible (in the DPI process). Nevertheless, the calculations show that the contribution of the higher-momentum photoelectrons to the corresponding partial cross section tends to increase with the increase of the exciting-photon energy. At least, the mentioned contributions have not reached their maximum
at co = 210 eV, see Figures 3a-3f. In the last column of the
Table 1 the relative contributions to the corresponding partial cross sections arising from different elecron pairs are performed for three values of the photon energy . co= I t Am, Acq = 5,50,100 eV.
Results and discussion
I. DPI cross sections for the 2,?" 2p" 'S state.
The calculated absolute DPI cross section for the 2s° 2pb 'S state and the components originating from the emission of the 5 various photoelectron pairs (1\. I2) are presented in Figure 3a. The main contributions to the cross section arise from the sp- and pd- electron pairs, of which the first essentially predominates in the energy region of 50-60 eV above the 2,v° 2p 'S threshold at 121.90 eV and comes to a maximum at about 160 eV. At higher photon energies the contribution of the pd- electron pair increases and reaches 2/3 of the sp-contribution, approximately, at 200 eV. The contributions of the rest of the accounted pairs df. fg and gh can be considered as negligible, although they show a tendency to grow with the increase of the photon energy.
The calculated partial cross section a^ + (2sa2ph \S) is in a good quantitative agreement with the experimental data points (Schartner et al 1993) in the energy rc,gion available for the comparison (from the threshold to 150 eV), see Figure 4. The calculated cross section has a flat plateau-like maximum of 0.0021 Mb around 185 eV that lies at somewhat higher energies than the maximum position around 160 eV. expected from the
experiment (Schartner et al 1993). The calculated a w is equal to 0.0019 Mb at co = 160 eV as well as at CO = 200 eV.
II. DPI cross sections for the 2.v' 2p l'~P states.
In the calculation of the DPI cross section a+< (co) for the 2.s' 2p configuration many more various photoelectron pairs (/1, /?) have been accounted for (see Table 1). Note that Carter and Kelly (1977) have included pp-, dd- and sd- electron pairs, only. As it is seen from Table 1, they contribute to the partial cross sections for the 2s' 2p5 ]P and 2i' 2p5 'P states most intensively.
Quantitative agreement was obtained for the sum of the 'P and P partial cross sections with the experimental data (Schartner et al 1993), see Figure 4. Both the qualitative and the quantitative agreement is seen in the exciting-photon energy region from the 2s' 2p 'P threshold at 87.93 eV up to 125 eV. At higher energies, the calculated curve lies under the experimental data points and reaches its maximum of 0.0165 Mb at 133 eV while the experimental maximum was expected at 140 eV (Schartner et al 1993). Note for the further discussion that the discrepancy begins just above the 2.v° 2pb 'S threshold.
The calculated partial cross sections <T++ (co) for the 2s' 2pJ 1P and 2s' 2p' 'P states as well as their components arising from the excitation of various photoelectron pairs (l\, l2) are presented in Figures 3b and
The contribution of the pp ''/'-excitation in the (X4' (co) for the 2.v' 2p~ 'P state essentially dominates for all exciting-photon energies considered, especially in the energy range near threshold. A relatively large contribution arises from the dd '/'-excitation within the energy interval of 50 eV above the threshold. The cross sections of the sd '£)- and pp 'D- excitations are rather small within 10-15 eV above the threshold. Then, the cross sections are increasing smoothly coming to values of around 0.001 Mb at 200 eV which seem to be the maximum points in the corresponding cross sections. The contribution of the pp 'S-. dd }S- and dd 'D- excitations, being rather small close to the threshold, increases with the increase of the photon energy, so that these excitations contribute noticeably for ¿¿>>120 eV.
Photon energy / eV Photon energy, eV
.2 0,015 o
w °.010 w
g 0,005 Ö
Photon energy / eV
a"(2s'2p5[1P])
-O
100 120 140 160 180 200 220 Photon energy / eV
o aj w
w 0,04 w
2 0,02 O
8Q 100 120 140 160 180 200 220 Photon energy / eV
c o
o a) w (0 (/) o
o
SS ,/ff_
100 120 140 160 180 200 220 Photon energy / eV
......-¿2."(2s12p5[sP];
Figure 3
Figure 4
Photon energy / eV
Figure 5
100 120 140 160 180 200 Exciting-photon energy / eV
s^2s'2pfPj.exp 2s,2ps['P]th.
>2sV['PJexp.
2s°2p6[,S] th.
°2p6['S] exp.
The higher-momentum photoelectron pairs (/|, 12) listed in Table 1 contribute almost negligibly for all excited photon energies considered. Nevertheless, their contribution increases slightly with the increase of the photon energy. The sum of such "negligible" contributions may become noticeable in the high-energy region, especially in the photoelectron angular distribution.
Singlet-coupled photoelectron pairs with the same orbital momenta l\ an U as above contribute to the partial cross section a" (co) for thé 2s1 2ps lP state. HowéVer, in this cross section, there is no predominant lf-excitation as the pp P- one in the <r "(co) cross section
for the 2s' 2!/?5 }P state. In the energy region close to the lP threshold, the dd lD- and pp lD- excitations prevail over the others. The latter also remains prevailing at higher photon energies while the sd 'D- excitation becomes the second in significance there. The curves corresponding to the dd '5- and pp 'S- excitations follow the behavior of the dd lD- and pp lD- curves but they lie 3 times lower, approximately. Also, the contribution of the higher-momentum photoelectron pairs in the partial cross section under consideration is almost negligible close to the threshold and increases with the increase of photon enerav.
si
'P =
P+ f
For comparison, the experimental (Schartner et al 1993) and the calculated partial DPI cross sections
cr++(<y) for the 2.v' 2p5 'P and 2s' 2pi }P states are presented in Figure 4. The qualitative and quantitative agreement of the experimental data and the calculated
cr++ (co) for the 2.v' 2p 'P state is seen in the region between the \P and 'P thresholds, i.e.. from 87.93 eV to 98.42 eV. Then, the very discrepancy between the experimental partial cross sections and the lowest-order PT ones appears and is clearly seen in Figure 4 (let us recall here that the sum of the above partial cross sections agrees well yvith the experimental data). Note, that the discrepancy in the cross sections begins at the 2tvl 2/?5 '/' threshold (98.42 eV) which shows the influence of the 2s1 2p~ (lP) £\l\ al2 doubl) excited states on the DPI cross section a f+ (2s' 2p3 V) and v.v. This inlluence has M, (a, ¿-,, £:) - M", (co, , t\) +
not been taken into account in our calculations within the lowest order of PT.
The examples of higher-order PT diagrams responsible for the influence mentioned above are drawn in Figure 6. Usual PT diagram notations are used: (— photon), (—<-— atomic election, or vacancy). (—■->— continuum spectrum electron) and wavy line stands for the Coulomb interaction V. In Figure 6 each section of a diagram shows the intermediate (real or virtual) many-electron state via which the DPI transition proceeds. The first diagram in
Figure 6 contributes the value to the lowest-order
PT amplitude and is accounted for in the cross sections presented above. The infinite series of the higher-order specific diagrams drawn in Figure 6 can be easily summed
resulting in the modified contribution M ■ in the total
DPI amplitude
+ M" (a,£ ,e2)
2 s 2pf,P)£l £z V 2s 2p\lP)£ i:2
1--/Л-2.У2ps('P)£ £2 Y.2s2p\]P)££2
(4)
Here, the term "specific diagrams" means that only the real intermediate continuum spectrum doubly-excited
states 2s 2p' (lP) £ £i are accounted for in Eq. 4, i.e., only the states satisfying the energy conservation law:
CO -=£• + £',. Such intermediate states are
expected to give the dominant correction to the corresponding amplitude (Amusia et al 1993).
If a real discrete excitation £ < 0 satisfies the energy conservation law the DPI process may be considered to proceed via two consecutive steps described above (see
Figure lb). Then, the energy denominator A = Q) - - £ - £2 tends to zero (or A is equal to zero) in a corresponding diagram. Therefore, the total width Ff of such intermediate state, the 2s2p (P)££2
state in the case under consideration, has to be taken into account. Following Amusia et al (1993), the modified
contribution M3 to the total DPI amplitude can be
written as follows:
The considered two-step transition mechanism may result in the appearance of a peak structure in the 'DPI
cross sections in the energy region near threshold. Similar equations can be obtained for the modified contributions M]p . as well.
Thus, the considered higher-order correlations describe the interaction of the 2s' 2/?' ('/') £\ ft and 2s1
2p CP)
¿V ¿V doubly-excited states which are expected to be important in the PT calculation of the DPI partial cross sections. It follows from Eqs.4 and 5 that in the energy region near and above the 2.v' 2p5 (]P) threshold the channel of additional population of the 2s' 2p CP) state is opened and leads to the increase of the DPI partial cross section
for the 2s' lp CP)
state. Correspondingly, the depopulation of the 2s' 2p5 ('?) state takes place which results in the decrease of the DPI partial cross section for the 2s' 2p~ ([P) state. We plan to include the above higher-order correlations in our calculations of the DPI partial cross sections in. the near future and expect that the non-lowest-order PT cross sections will follow the respective experimental ones more closely.
2s2piCP\el s2V 2s2p\'P)£ e: A + /I; 2
III. DPI cross sections for the 2s2 2p" 's . V , '¿> states.
The DPI channels leading to the final configuration 2s: 2p constitute the most significant contribution to the total DPI cross section. The calculated partial cross sections
cr++ (co) for the 2s2 2/?4 lS, }P, [D states as well as then-components arising from the excitation of various photoelectron pairs (/,, l2) are presented in Figures 3d, 3e and 3f. The main contributions to the corresponding cross sections arise from pd- electron pairs and, especially in the near-threshold photon energy region, from sp- ones. In our calculation the partial cross sections of this channels are found to be essentiallv lesser then in the experiment. However, the similar V(N"'' calculation appears to be quite close to the results performed by Chang and Poe (1975). Such a way the DPI cross section tends to decrease with increasing of the effective charge of the residual ion field used for calculation of the PT basis. Evidently, that it is necessary to account for higher-order PT diagrams to achieve an appropriate agreement with the experiment, for instance as it was discussed above.
Conclusions
The lowest-order PT results on the double photoionization cross sections of the Ne 2s- and 2p-valence shell are presented. For the calculation of the photoelectron wavefunctions the V(N"2' potential was used for the first time. The sum of the calculated DPI cross sections agrees well with the experimental data. That demonstrates that the lowest-order PT approach is sufficient to reproduce the main features of the DPI
process. On the other hand, the insufficiency of the lowest-order PT approach to reproduce properl) the partial cross sections was also shown. To remove the discrepancy between theoretical results and the experimental data, the ways of including the most important higher-order correlations in the calculation of the DPI cross sections are discussed as well.
Acknowledgments
We wish to thank Professor Dr. V. L. Sukhorukov and Dr. I. D. Petrov for helpful discussions. V.A. Kilin would like to express his gratitude to the Department of Physics, University of Kaiserslautern and to the Department of
Physics, University of Giessen for the possibility to work there during his stay and the Deutsche Forschungsgemeinschaft (DFG) for financial support.
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Д.Л. Лазарев, Дм.А. Лазарев, В.А. Килин, В.М. Зеличенко АВТОМАТИЗАЦИЯ РАБОТЫ С ОБЪЕКТАМИ КВАНТОВОЙ ТЕОРИИ УГЛОВОГО МОМЕНТА
Введение
В приложениях квантовой механики к исследованию атомов часто встречается задача вычисления многократных сумм произведений коэффициентов Клеб-ша-Гордана (ККГ) и Зрьсимволов Вигнера. В приближении сферической симметрии поля, создаваемого ядром и остальными электронами, квантовые числа орбитального момента и ею проекции на некоторое направление являются точными. Дальнейший учет апектостатического и спинорбитального взаимодействий оставляет точным лишь полный угловой момент атома. Собственные волновые функции оператора полного момента строятся как линейные комбинации произведений одноэлектроцных волновых функций [1]. Коэффициенты этих линейных комбинаций, как известно, представляют произведения коэффициентов
Клебша-Гордана или связанных с ними З^п-символов Вигнера. Поэтому выражения для матричных элементов различных операторов содержат многократные суммы ККГ и 3|1П-символов. Такие суммы в большинстве случаев можно привести к менее громоздкому виду путем выделения из них инвариантных по отношению к вращениям величин типа Зп|- символов Вигнера. Для упрощения и наглядного представления подобных преобразований разработан графический метод [2, 3, 4]. Однако и в этом виде задача остается весьма трудоемкой. Поэтому желательно каким-либо образом автоматизировать процесс преобразования. В настоящей статье описывается программа ОАМТ, предназначенная для решения поставленной задачи.
Теоретические сведения
В целях единообразия все ККГ удобно преобразо-вать в З'ци-символы. В основу формального процесса
« V0, ~а
К-1)
Ф+ч-х
Фх
а
-а
Р Ч
Ф X
преобразования сумм произведений З^и-символов могут быть положены три соотношения [3, гл. 12]:
У) Ч
= пАгАо;
Ф -х
а а'
(-1)" ri:
АА„<
(О (2)
