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Electron delocalization in trinuclear mixed-valence clusters
Volume 89, number 5
CHEMICAL PHYSICS LETTERS
2 July 1982
ELECTRON DELOCALIZATION IN TRINUCLEAR MIXED-VALENCE CLUSTERS
S.A. BORSHCH, I.N. KOTOV and I.B. BERSUKER Institute of Chemistry, Aeademy of Sciences of the Moldavian SSR, Kishinev 277028, USSR Received 4 May 1982
Possible trapped valence states in trimeric mixed-valence clusters are considered. The adiabatic potential in the space of a doubly-degenerate e vibration is calculated. Conditions under which minima with different electron distributions exist are determined. The coexistence of localized and delocalized distributions is possible.
1. Introduction
Mixed-valence compounds have been under intensive investigation during recent years. Such compounds contain two or more atoms of the same element in different valence states. In some cases the molecule may be considered as a cluster with equivalent centres, in which one or several "extra" electrons are introduced. The latter manifest themselves in experiments as either delocalized over all the centres (or over a part of them), or localized at single centres. If the wavefunction of the excess electron has a maximum at an individual centre and the lifetime of the localized state exceeds the characteristic time of measurement of the appropriate spectral method, then the non-equivalency of different centres is displayed in the spectrum. It has been shown by Hush [1] and Mayoh and Day [2] that for binuclear mixed-valence clusters the electron localization is determined by the relationship between three quantities: the matrix element of the interatomic electronic interaction, the constant of the electron-vibrational (vibronic) interaction (the vibronic constant) and the force constant of non-totally symmetric vibrations. Subsequently the vibronic theory for binuclear mixedvalence clusters, which revealed the origin of the optical charge-transfer band shapes, was worked out [3]. However, the origin of electron localization in systems containing more than two centres has not been considered. Meanwhile these systems have some peculiar features which cannot be reduced to binuclear ones. The electronic energy level spectrum for a trimer con0 009-2614/82/0000-0000/$ 02.75 © 1982 North-Holland
sists of a singlet and a doublet, thus resulting in a superposition of the Jahn-Teller and pseudo-Jahn-Teller effects [4,5]. This circumstance was first noted by Perrin and Gouterman [6]. The present paper is devoted to the investigation o f the origin and possible types of electron localization in mixed-valence trimeric cluster systems with one excess electron by means of adiabatic potential (AP) calculations (in the framework of the vibronic interaction theory) and evaluation of the electronic charge density distribution in the stable (AP minima) configurations.
2. Calculations of the adiabatic potential Consider a system with three equivalent molecular groups which transform one into another by symmetry transformations. The basic trinuclear carboxylate iron(II,III,III) mixed-valence clusters [7] may serve as such examples. Suppose that the "extra" electron when localized at the ith centre occupies a non-degenerate orbital ~0i. The wavefunction ~0i in general is a molecular orbital (MO) composed of the wavefunctions of the central atom and the neigbouring atoms. Tie electron in the non-degenerate state interacts only with the totally symmetric displacements of the first coordination sphere Qi. The model hamiltonian in the basis of the local wavefunctions can be written as
H =w
0 1
+A ~0
Q2 0 Q3
+ Hvib-
(1)
381
Volume 89, number 5
CHEMICAL PHYSICS LETTERS
The first term describes the interaction (between the localized electronic states) responsible for the electron transfer, the second is the interaction of localized states of each centre with the near-neighbour nuclear displacements Qi, and Hvib is the vibrational hamiltonian. Passing to the symmetrized wavefunctions and displacements of the cluster as a whole, which transform as the totally symmetric A and doubly degenerate E representations of the molecular symmetry group (we assume that the centres are widely spaced and the functions s0i may be taken mutually orthogonal),
2 July 1982 Ne
0 = 3-1/2A ~
~
n=l
cn
l)~qn7
-y
Ne
13 13 2 °°nqn7'
,l=1 7
(5)
where 1I= c
[°1 i] "0
[o1
0 -1/2 2 -1/2 0 e. 0 ;17"0=1 _2_1/2 0 0 -2-1/2j
(6)
~oA = 3-1/2(~01 + tp2 + s03); tPO = 6-1/2(2~°1
- -
~°2 -- ~3);
~oe = 2-1/2(~o2 -- ~o3).
(2)
Similar symmetrized linear combinations QA, Qo, Qe of the Qi displacements can be constructed. With the new basis and new vibrational coordinates, the hamiltonian (1) takes the form
H=w 0 -1 0
i QA X I Qo Qe
0 -
+ Hvib •
t
qn7 = C°nqn"/ one can rotate each group of normal coordinates transforming as the 3' row of the e representation in such a way as to orient one of the orthogonal transformations along the direction
+3-1/2A
Oo QA + Qo/21/2 -Qe/21/2
As seen from eq. (5), the problem is essentially a multimode one. However, in as far as only the potential energy is required, one can simplify the problem using the method of the "interaction mode". The details of this method first introduced by Toyozawa and Inoue [8] are given in ref. [9]. Using the scale transformation
Qe -Qe/21/2 QA - Qo / 21/2
'
Q 17
(3)
=
[G(On/C°n)qn~]FN/cn/con '
k
?Z
2
]-1/2
2
.
/ J 1.-n = 1
(7)
It can be shown that Q]7 is proportional to Q'r' Now eq. (5) can be rewritten in the form Ne
It can be seen that the hamiltonian (3) describes two electronic levels, A and E, interacting with the a and e displacements. In most cases the interaction with the syrmnetric mode may be excluded by means of a suitable definition of the origin. The displacements Qv (7 = e, 0) may be expanded over the e-type normal coordinates qn7 of the duster as a whole: Ne
Qv = n~l= Cnqnv'
(4)
where N e is the number of e vibrations. In the normal coordinates the potential energy matrix of the hamiltonian (3) may be written in the form
382
, I3 13 Q,2, nT
(S)
0 = U(Qo') + ~ n=2 "r where &r(QT) = 3-1/2A ~
I?~Q~' + ½a2 ~
3'
2 QT'
(9)
3'
The effective frequency of the "interaction mode" is given by the expression
=
In~=el 2
2.j7-1/2
Cn/COn[
•
(10)
Thus the problem is reduced to one interaction mode
Volume 89, number 5
CHEMICAL PHYSICS LETTERS
separated from the other vibrational modes (this operation, simplifying the potential energy, essentially complicates the expression for the kinetic energy, which is, however, not important here). The AP equals the sum of the electronic energy as a function of nuclear coordinates and the elastic energy of core interactions:
Ei(Q) ei(Q ) + ½22 ~] Q2.
\ \
(11)
=
2 July 1982
E{9
El 9)
"7
The electronic energies are the eigenvalues of the electronic hamiltonian Hel = w
-I 0 + 3-1/2A ~ 0 -1
~QT"
(12)
In polar coordinates Qo =/9 cos ~o, Qe = P sin ~0, the secular equation for this hamiltonian is e3 _(½A2p2 + 3w2) e - 2w 3 ,
A3p 3 cos 3tp
3/x/
(13)
p = (2/3) 1/2(1 w I/.4) [2,v/2 sinh(t) - 1]
Since the angular dependence of the AP is included in ei(P,¢), Ei(P,~) is a periodic function oftp with period 27r/3. As can be shown using eq. (13), Ei(p,~o) has extremal points at ~ = ~rn/3 (n = 0,1,...,5); I f A > 0 these points are maxima of the lower sheet of the AP for even values of n, and minima for odd ones, and vice versa if A<0. Consider the radial dependence of the AP in the extremal cross section Qe = 0 (or, equivalently, for ¢ = 0 and - ~ < p ~< +~o). The solutions ofeq. (13) for this case are
e16o) -Ap/v'g- w, :
e2,3(p) = ~ [W +Ap/v~
+ 3(A2p2/6 - 2Awp/3"vl'6+w2)1/2].
(14)
AS usual, we assume that the parameter of the intercentre electron transfer is negative [10] and the ground electronic state is singlet. It can be shown that the number of extremal points of the lowest sheet of the AP and their types are completely determined by the parameter
A=a21wl/A2. Performing the substitution
Fig. 1. The extremal profiles of the adiabatic potential for different A values: (a) 2x < 2/9; (b) 2/9 < & < 0.255; (c) 2x > 0.255.
(15)
(16)
we obtain the following equation for the extremal points of the AP: 4~A
sinh(2t) + (1 - 4A) cosh(t) - 3 sinh(t) = 0. (17)
In fig. 1 the numerically calculated extremal profiles of the AP for different A values are shown. The behaviour of the AP near the point 0 = 0 may be revealed by means of the expansion of the expression e2. 3 in powers of p up to the second order. For A < 2/9 the APhas the only minimum at p :~ 0 (fig. la; the extrenum at p > 0 is a saddle point). If the condition 2/9 < A < 0.255 is fulfilled, there are two minima at p = 0 and p 4~ 0. Finally, only one minimum at O = 0 is present, if A ~> 0.255 (fig. lc).
3. Results and discussion Let us consider the electronic density distribution at the minima of the AP. It is clear that the electronic ground-state wavefunction at 0 = 0 coincides with the first wavefunction of eq. (2). If/9 ~ 0, at ~0 = 0 or lr only the first and second functions of eq. (2) are intermixed, and the electronic wavefunction takes the form = b l ~ 1 +b2(~2 +~p3),
Ibll>~ Ib21.
(18) 383
Volume 89, number 5
CHEMICAL PHYSICS LETTERS
D~
Fig. 2. The degree of localization near the first centre of the cluster at the minimum of the lower sheet of its AP (~ = ~r) as a function of the vibronic constant (,4 and w are in units of r~a). Hence the minimum on the lowest sheet o f the AP (fig. la) corresponds with the electron localization mainly at centre 1. In the equivalent minima at ~0 = 7r/3 and 52r/3 the extra electron is mostly localized at the second and third centre, respectively. The probability amplitude b 1 as a function of the vibronic constant A is illustrated in fig. 2 for different values of w. A distinguishing feature o f the trimeric systems compared with the dimeric ones is the occurrence of an intermediate region where localized and delocalized states coexist. The special electron distribution is appropriate to the minimum of the AP, E 1 . Its electronic wavefunction in the extremal section is simply the third function of eq. (2). It means that the extra electron is delocalized over two of the three centres. If the energy of the minimum E 1 is not high enough and the corresponding energy levels may be populated at temperatures o f several hundred K (which requires ~hat Iwl ~ 100 cm-1), this electronic distribution m a y be observed, for instance, in the M6ssbauer spectra. It is probable that the population o f such minima is responsible for the observed temperature dependence o f the i r o n ( l l , I l I , I I I ) trifluoroacetate MiSssbauer spectra [11]. In these experiments, be-
384
2 July 1982
sides the usual spectrum averaged due to electron hopping in the ground state, two new quadrupole doublets occur with increasing temperature. The parameters for one o f these doublets are typical for the Fe(III) ion, the other obe being consistent with iron ions in the intermediate valence state. It is clear that the charge distribution in the low-lying excited state explains the origin of such a spectrum. To conclude, the electron localization problem in trimeric mixed-valence clusters has some features which distinguish them from the dimeric ones: (1) the alternatives "localization or delocalization" may be inappropriate due to the coexistence of two types of electron distribution, localized and delocalized; (2) in the excited ~tate the electron m a y be delocalized over only two of the three centres. These special features should manifest themselves in the intervalence-transfer spectra.
References [1] N.S. Hush, Chem. Phys. 10 (1975) 361. 12] B. Mayoh and F. Day, J. Am. Chem. Soc. 94 (1972) 2885. [3] P.N. Schatz, in: Mixed-valence compounds, ed. D.B. Brown (Reidel, Dordrecht, 1980) p. 115. [4] R. Englman, The Jahn-Teller effect in molecules and crystals (Wiley, New York, 1972). [5] I.B. Bersuker, The Jahn-TeUer effect and vibronic interactions in modern chemistry (Plenum, New York, 1982). [6] M.H. Perrin and M. Gouterman, J. Chem. Phys. 46 (1967) 1019. 17] R.A. Stukan, C.I. Turta, A.V. Ablov and S.A. Bobkova, Coord. Chem. 5 (1979) 95 in Russian. [8] Y. Toyozawa and M. Inoue, J. Phys. Soc. Japan 21 (1966) 1663. [9] I.B. Bersuker and V.Z. Polinger, Vibronic interactions in molecules and crystals (Nauka, Moscow, 1982) in Russian. [10] N.S. Hush, in: Mixed-valence compounds, ed. D.B. Brown (Reidel, Dordrecht, 1980) p. 151. [11] C.I. Turta, S.A. Borshch and S'A. Bobkova, in: Abstracts of the International Conference on Mgssbauer Spectroscopy (Portoroz, 1979)p. 127.
CHEMICAL PHYSICS LETTERS
2 July 1982
ELECTRON DELOCALIZATION IN TRINUCLEAR MIXED-VALENCE CLUSTERS
S.A. BORSHCH, I.N. KOTOV and I.B. BERSUKER Institute of Chemistry, Aeademy of Sciences of the Moldavian SSR, Kishinev 277028, USSR Received 4 May 1982
Possible trapped valence states in trimeric mixed-valence clusters are considered. The adiabatic potential in the space of a doubly-degenerate e vibration is calculated. Conditions under which minima with different electron distributions exist are determined. The coexistence of localized and delocalized distributions is possible.
1. Introduction
Mixed-valence compounds have been under intensive investigation during recent years. Such compounds contain two or more atoms of the same element in different valence states. In some cases the molecule may be considered as a cluster with equivalent centres, in which one or several "extra" electrons are introduced. The latter manifest themselves in experiments as either delocalized over all the centres (or over a part of them), or localized at single centres. If the wavefunction of the excess electron has a maximum at an individual centre and the lifetime of the localized state exceeds the characteristic time of measurement of the appropriate spectral method, then the non-equivalency of different centres is displayed in the spectrum. It has been shown by Hush [1] and Mayoh and Day [2] that for binuclear mixed-valence clusters the electron localization is determined by the relationship between three quantities: the matrix element of the interatomic electronic interaction, the constant of the electron-vibrational (vibronic) interaction (the vibronic constant) and the force constant of non-totally symmetric vibrations. Subsequently the vibronic theory for binuclear mixedvalence clusters, which revealed the origin of the optical charge-transfer band shapes, was worked out [3]. However, the origin of electron localization in systems containing more than two centres has not been considered. Meanwhile these systems have some peculiar features which cannot be reduced to binuclear ones. The electronic energy level spectrum for a trimer con0 009-2614/82/0000-0000/$ 02.75 © 1982 North-Holland
sists of a singlet and a doublet, thus resulting in a superposition of the Jahn-Teller and pseudo-Jahn-Teller effects [4,5]. This circumstance was first noted by Perrin and Gouterman [6]. The present paper is devoted to the investigation o f the origin and possible types of electron localization in mixed-valence trimeric cluster systems with one excess electron by means of adiabatic potential (AP) calculations (in the framework of the vibronic interaction theory) and evaluation of the electronic charge density distribution in the stable (AP minima) configurations.
2. Calculations of the adiabatic potential Consider a system with three equivalent molecular groups which transform one into another by symmetry transformations. The basic trinuclear carboxylate iron(II,III,III) mixed-valence clusters [7] may serve as such examples. Suppose that the "extra" electron when localized at the ith centre occupies a non-degenerate orbital ~0i. The wavefunction ~0i in general is a molecular orbital (MO) composed of the wavefunctions of the central atom and the neigbouring atoms. Tie electron in the non-degenerate state interacts only with the totally symmetric displacements of the first coordination sphere Qi. The model hamiltonian in the basis of the local wavefunctions can be written as
H =w
0 1
+A ~0
Q2 0 Q3
+ Hvib-
(1)
381
Volume 89, number 5
CHEMICAL PHYSICS LETTERS
The first term describes the interaction (between the localized electronic states) responsible for the electron transfer, the second is the interaction of localized states of each centre with the near-neighbour nuclear displacements Qi, and Hvib is the vibrational hamiltonian. Passing to the symmetrized wavefunctions and displacements of the cluster as a whole, which transform as the totally symmetric A and doubly degenerate E representations of the molecular symmetry group (we assume that the centres are widely spaced and the functions s0i may be taken mutually orthogonal),
2 July 1982 Ne
0 = 3-1/2A ~
~
n=l
cn
l)~qn7
-y
Ne
13 13 2 °°nqn7'
,l=1 7
(5)
where 1I= c
[°1 i] "0
[o1
0 -1/2 2 -1/2 0 e. 0 ;17"0=1 _2_1/2 0 0 -2-1/2j
(6)
~oA = 3-1/2(~01 + tp2 + s03); tPO = 6-1/2(2~°1
- -
~°2 -- ~3);
~oe = 2-1/2(~o2 -- ~o3).
(2)
Similar symmetrized linear combinations QA, Qo, Qe of the Qi displacements can be constructed. With the new basis and new vibrational coordinates, the hamiltonian (1) takes the form
H=w 0 -1 0
i QA X I Qo Qe
0 -
+ Hvib •
t
qn7 = C°nqn"/ one can rotate each group of normal coordinates transforming as the 3' row of the e representation in such a way as to orient one of the orthogonal transformations along the direction
+3-1/2A
Oo QA + Qo/21/2 -Qe/21/2
As seen from eq. (5), the problem is essentially a multimode one. However, in as far as only the potential energy is required, one can simplify the problem using the method of the "interaction mode". The details of this method first introduced by Toyozawa and Inoue [8] are given in ref. [9]. Using the scale transformation
Qe -Qe/21/2 QA - Qo / 21/2
'
Q 17
(3)
=
[G(On/C°n)qn~]FN/cn/con '
k
?Z
2
]-1/2
2
.
/ J 1.-n = 1
(7)
It can be shown that Q]7 is proportional to Q'r' Now eq. (5) can be rewritten in the form Ne
It can be seen that the hamiltonian (3) describes two electronic levels, A and E, interacting with the a and e displacements. In most cases the interaction with the syrmnetric mode may be excluded by means of a suitable definition of the origin. The displacements Qv (7 = e, 0) may be expanded over the e-type normal coordinates qn7 of the duster as a whole: Ne
Qv = n~l= Cnqnv'
(4)
where N e is the number of e vibrations. In the normal coordinates the potential energy matrix of the hamiltonian (3) may be written in the form
382
, I3 13 Q,2, nT
(S)
0 = U(Qo') + ~ n=2 "r where &r(QT) = 3-1/2A ~
I?~Q~' + ½a2 ~
3'
2 QT'
(9)
3'
The effective frequency of the "interaction mode" is given by the expression
=
In~=el 2
2.j7-1/2
Cn/COn[
•
(10)
Thus the problem is reduced to one interaction mode
Volume 89, number 5
CHEMICAL PHYSICS LETTERS
separated from the other vibrational modes (this operation, simplifying the potential energy, essentially complicates the expression for the kinetic energy, which is, however, not important here). The AP equals the sum of the electronic energy as a function of nuclear coordinates and the elastic energy of core interactions:
Ei(Q) ei(Q ) + ½22 ~] Q2.
\ \
(11)
=
2 July 1982
E{9
El 9)
"7
The electronic energies are the eigenvalues of the electronic hamiltonian Hel = w
-I 0 + 3-1/2A ~ 0 -1
~QT"
(12)
In polar coordinates Qo =/9 cos ~o, Qe = P sin ~0, the secular equation for this hamiltonian is e3 _(½A2p2 + 3w2) e - 2w 3 ,
A3p 3 cos 3tp
3/x/
(13)
p = (2/3) 1/2(1 w I/.4) [2,v/2 sinh(t) - 1]
Since the angular dependence of the AP is included in ei(P,¢), Ei(P,~) is a periodic function oftp with period 27r/3. As can be shown using eq. (13), Ei(p,~o) has extremal points at ~ = ~rn/3 (n = 0,1,...,5); I f A > 0 these points are maxima of the lower sheet of the AP for even values of n, and minima for odd ones, and vice versa if A<0. Consider the radial dependence of the AP in the extremal cross section Qe = 0 (or, equivalently, for ¢ = 0 and - ~ < p ~< +~o). The solutions ofeq. (13) for this case are
e16o) -Ap/v'g- w, :
e2,3(p) = ~ [W +Ap/v~
+ 3(A2p2/6 - 2Awp/3"vl'6+w2)1/2].
(14)
AS usual, we assume that the parameter of the intercentre electron transfer is negative [10] and the ground electronic state is singlet. It can be shown that the number of extremal points of the lowest sheet of the AP and their types are completely determined by the parameter
A=a21wl/A2. Performing the substitution
Fig. 1. The extremal profiles of the adiabatic potential for different A values: (a) 2x < 2/9; (b) 2/9 < & < 0.255; (c) 2x > 0.255.
(15)
(16)
we obtain the following equation for the extremal points of the AP: 4~A
sinh(2t) + (1 - 4A) cosh(t) - 3 sinh(t) = 0. (17)
In fig. 1 the numerically calculated extremal profiles of the AP for different A values are shown. The behaviour of the AP near the point 0 = 0 may be revealed by means of the expansion of the expression e2. 3 in powers of p up to the second order. For A < 2/9 the APhas the only minimum at p :~ 0 (fig. la; the extrenum at p > 0 is a saddle point). If the condition 2/9 < A < 0.255 is fulfilled, there are two minima at p = 0 and p 4~ 0. Finally, only one minimum at O = 0 is present, if A ~> 0.255 (fig. lc).
3. Results and discussion Let us consider the electronic density distribution at the minima of the AP. It is clear that the electronic ground-state wavefunction at 0 = 0 coincides with the first wavefunction of eq. (2). If/9 ~ 0, at ~0 = 0 or lr only the first and second functions of eq. (2) are intermixed, and the electronic wavefunction takes the form = b l ~ 1 +b2(~2 +~p3),
Ibll>~ Ib21.
(18) 383
Volume 89, number 5
CHEMICAL PHYSICS LETTERS
D~
Fig. 2. The degree of localization near the first centre of the cluster at the minimum of the lower sheet of its AP (~ = ~r) as a function of the vibronic constant (,4 and w are in units of r~a). Hence the minimum on the lowest sheet o f the AP (fig. la) corresponds with the electron localization mainly at centre 1. In the equivalent minima at ~0 = 7r/3 and 52r/3 the extra electron is mostly localized at the second and third centre, respectively. The probability amplitude b 1 as a function of the vibronic constant A is illustrated in fig. 2 for different values of w. A distinguishing feature o f the trimeric systems compared with the dimeric ones is the occurrence of an intermediate region where localized and delocalized states coexist. The special electron distribution is appropriate to the minimum of the AP, E 1 . Its electronic wavefunction in the extremal section is simply the third function of eq. (2). It means that the extra electron is delocalized over two of the three centres. If the energy of the minimum E 1 is not high enough and the corresponding energy levels may be populated at temperatures o f several hundred K (which requires ~hat Iwl ~ 100 cm-1), this electronic distribution m a y be observed, for instance, in the M6ssbauer spectra. It is probable that the population o f such minima is responsible for the observed temperature dependence o f the i r o n ( l l , I l I , I I I ) trifluoroacetate MiSssbauer spectra [11]. In these experiments, be-
384
2 July 1982
sides the usual spectrum averaged due to electron hopping in the ground state, two new quadrupole doublets occur with increasing temperature. The parameters for one o f these doublets are typical for the Fe(III) ion, the other obe being consistent with iron ions in the intermediate valence state. It is clear that the charge distribution in the low-lying excited state explains the origin of such a spectrum. To conclude, the electron localization problem in trimeric mixed-valence clusters has some features which distinguish them from the dimeric ones: (1) the alternatives "localization or delocalization" may be inappropriate due to the coexistence of two types of electron distribution, localized and delocalized; (2) in the excited ~tate the electron m a y be delocalized over only two of the three centres. These special features should manifest themselves in the intervalence-transfer spectra.
References [1] N.S. Hush, Chem. Phys. 10 (1975) 361. 12] B. Mayoh and F. Day, J. Am. Chem. Soc. 94 (1972) 2885. [3] P.N. Schatz, in: Mixed-valence compounds, ed. D.B. Brown (Reidel, Dordrecht, 1980) p. 115. [4] R. Englman, The Jahn-Teller effect in molecules and crystals (Wiley, New York, 1972). [5] I.B. Bersuker, The Jahn-TeUer effect and vibronic interactions in modern chemistry (Plenum, New York, 1982). [6] M.H. Perrin and M. Gouterman, J. Chem. Phys. 46 (1967) 1019. 17] R.A. Stukan, C.I. Turta, A.V. Ablov and S.A. Bobkova, Coord. Chem. 5 (1979) 95 in Russian. [8] Y. Toyozawa and M. Inoue, J. Phys. Soc. Japan 21 (1966) 1663. [9] I.B. Bersuker and V.Z. Polinger, Vibronic interactions in molecules and crystals (Nauka, Moscow, 1982) in Russian. [10] N.S. Hush, in: Mixed-valence compounds, ed. D.B. Brown (Reidel, Dordrecht, 1980) p. 151. [11] C.I. Turta, S.A. Borshch and S'A. Bobkova, in: Abstracts of the International Conference on Mgssbauer Spectroscopy (Portoroz, 1979)p. 127.