- Email: [email protected]
Investigations of EPR parameters for the trigonal Ti3+–Ti3+ pair in beryl crystal
Spectrochimica Acta Part A 67 (2007) 1281–1283
Investigations of EPR parameters for the trigonal Ti3+–Ti3+ pair in beryl crystal Fang Wang a , Zheng Wen-Chen a,b,∗ a
b
Department of Material Science, Sichuan University, Chengdu 610064, People’s Republic of China International Center for Materials Physics, Chinese Academy of Sciences, Shenyang 110016, People’s Republic of China Received 11 July 2006; accepted 16 September 2006
Abstract By using the complete diagonalization of energy matrix of 3d1 ions in trigonal symmetry, the EPR parameters (g factors g , g⊥ and zero-field splitting D) of the trigonal Ti3+ –Ti3+ pair in beryl crystal are calculated. In the calculations, the exchange interaction in the Ti3+ –Ti3+ pair is taken as the perturbation and the local trigonal distortion in the defect center is considered. The results (which are in agreement with the experimental values) are discussed. © 2006 Elsevier B.V. All rights reserved. Keywords: Electron parameters resonance; Crystal-field theory; Exchange interaction; Ti3+ –Ti3+ pair; Beryl
1. Introduction Beryl (Be3 Al2 Si6 O8 ) crystals containing transition- and alkali-metal impurities can present a lot of colour, depending upon the species and valence of impurity. So, they are used in jewelry and laser materials. In particular, the discovery of Ti3+ doped BeAl2 O4 tunable laser crystal [1] has simulated studies of Ti3+ -doped beryl crystals. Many spectroscopy investigations for the Ti3+ -doped beryl were made [2–4]. From the EPR measurements, Solntsev and Yurkin [2] found that in addition to the isolated Ti3+ centers at Al3+ and Si4+ sites, there is a weak trigonal (C3h ) Ti3+ center described by S = 1. This center is associated with the Ti3+ Al –Ti3+ Al pair located along the C3 -axis and ˚ [2]. The EPR parameters (g spaced by the distance r ≈ 4.597 A factors g , g⊥ and zero-field splitting D) of the Ti3+ –Ti3+ pair in beryl were given [2], until now no satisfactory theoretical explanations have been made for these EPR parameters. As is known, there is no zero-field splitting in usual 3d1 (or 3d9 ) systems. The zero-field splitting in this Ti3+ –Ti3+ pair is due to the exchange interaction of the pair. In this paper, we calculated the g factors by using the complete diagonalization of the energy
∗ Corresponding author at: Department of Material Science, Sichuan University, Chengdu 610064, People’s Republic of China. Tel.: +86 28 85412371; fax: +86 28 85416050. E-mail address: [email protected] (W.-C. Zheng).
1386-1425/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.saa.2006.09.038
matrix for 3d1 ion in trigonal symmetry and treat the exchange interaction as perturbation in the degeneracy state perturbation theory to calculate the zero-field splitting. In the calculations, the local trigonal distortion in the Ti3+ –Ti3+ pair is considered. The results are discussed. 2. Calculation For a Ti3+ –Ti3+ pair in crystals, the Hamiltonian can be expressed as 1 2 H = Hf1 + Hf2 + Hso (ζ) + Hso (ζ) + Vc1 + Vc2 + JS 1 · S 2 (1)
where the superscripts (or subscripts) 1 and 2 refer to two ions, respectively. Hf , Vc and Hso (ζ) denote, respectively, the freeion Hamiltonian, the crystal-field potential and the spin-orbit coupling term (where the ζ is the spin-orbit coupling parameter in crystal). S is the operator of spin momentum and J is the exchange parameter. For a 3dn ion in trigonal symmetry, the crystal-field potential is given as Vcλ = B20 O02 + B40 O04 + B43 O34
(2)
where Bkl are the crystal-field parameters. By using the weak field basis function φn = |j, M in trigonal symmetry [5], the 10 × 10 complete energy matrix for 3d1 ion in trigonal symmetry is derived. The eigenenergy and eigenvectors of the Hamiltonian can be obtained by diagonalizing the energy
1282
F. Wang, W.-C. Zheng / Spectrochimica Acta Part A 67 (2007) 1281–1283 Table 1 The EPR parameters (g factors and zero-field splitting D) of the trigonal Ti3+ –Ti3+ pair in beryl crystal
matrix. Thus, the doubly degenerated ground states are |G+ =
10
|G− =
an φ n
n=1
10
bn φn
(3)
n=1
where an and bn are the coefficients determined from the above diagonalization calculation. The anisotropic g factors can be written as g = 2G+ |kLz + ge S z |G+ g⊥ = 2G+ |kLx + ge S x |G− .
(4)
where ge (≈2.0023) is the free-electron value. Lγ (γ = x or y) and Sγ are the orbit and spin momentum operators, respectively. k is the orbital reduction factor. Because of the covalency reduction effect for 3dn ions in crystals, a covalency reduction parameter N is introduced [6–8]. Thus, we have k = N2 and ζ = N2 ζ 0 [6–8], where ζ 0 is corresponding parameter in free state. For a free Ti3+ ion, ζ 0 ≈ 154 cm−1 [9]. From the superposition model [10], the crystal-field parameters Bkl in the above matrix can be expressed as ¯ 2 (R) B20 ≈ 3A
2
(3 cos2 θi − 1)
i=1
¯ 4 (R) B40 ≈ 3A
2
(35 cos4 θi − 30 cos2 θi + 3)
(5)
i=1 2 √ ¯ 4 (R) sin3 θi cos θi B43 ≈ −6 35A
Calculation Experiment [2]
g
g⊥
D (10−4 cm−1 )
1.990 1.991(1)
1.853 1.842(1)
205.1 204.6(5)
the perturbation term) can be given as 1 (S1+ S2− + S1− S2+ ) + S1Z S2Z . HSS = JS1 · S2 = J 2
(7)
The wave functions of ground triplet for the Ti3+ –Ti3+ pair are ⎧ + MS = 1, ψ1 = |G+ ⎪ 1 G2 ⎪ ⎪ ⎨ + |G+ G− + |G− 1 G2 S = 1, (8) MS = 0, ψ0 = 1 2 √ ⎪ 2 ⎪ ⎪ ⎩ − MS = −1, ψ−1 = |G− 1 G2 Thus, we get the expression of zero-field splitting as D = ψ1 |HSS |ψ1 − ψ0 |HSS |ψ0
(9)
Applying the diagonalized wave functions |G± (i.e., those in Eq. (3)) to Eq. (9) and taking the exchange parameter J ≈ 14 cm−1 , the zero-field splitting D of the Ti3+ –Ti3+ pair in beryl is calculated. The result is also compared with the observed value in Table 1. 3. Discussion
i=1
where the subscripts 1 and 2 stand for the O2− ions in ¯ 4 (R) are the intrin¯ 2 (R) and A upper and lower triangles. A sic parameters with the metal–ligand distance R. In beryl ˚ [11]. For 3dn ions in crystals, crystal, R1 ≈ R2 ≈ R ≈ 1.904 A ¯ 4 (R) ≈ 3Dq/4 [10,12,13]. From the cubic field parameter A ˚ [14]) obtained from the optiDq ≈ 1910 cm−1 (with R ≈ 1.912 A cal spectra of Ti3+ in Al2 O3 [15]. We reasonably estimated ¯ 2 (R) ≈ (9–12)A ¯ 4 (R) ¯ 4 (R) ≈ 1453 cm−1 for Ti3+ in beryl. A A ¯ ¯ [13,16–18], we take A2 (R) ≈ 12A4 (R) here. θ i denotes the angle between the direction of Ri and C3 -axis. In the host beryl crystal, θ1h ≈ 55.30◦ and θ2h ≈ 59.68◦ [11]. Considering that the size and the nature of impurity are unlike those of the replaced host ion, the angle θ i in the impurity center may be different from the corresponding θih in the host crystal because of the impurity-induced local lattice relaxation. To decrease the adjustable parameter, for the Ti3+ –Ti3+ pair in beryl, we take only θ 2 as the adjustable parameter. Substituting all these parameters into the above matrix and formulas and fitting the calculated g factors g and g⊥ of the Ti3+ –Ti3+ in beryl to the observed values, we yield k ≈ 0.921,
θ2 ≈ 57.63◦
(6)
The calculated g factors are compared with the experimental values in Table 1. For the calculation of zero-field splitting D of Ti3+ –Ti3+ pair in beryl, the exchange interaction term HSS (which is used as
In the above calculations, we find that the exchange parameter J ≈ 14 cm−1 . The value of |J| is smaller than those in similar Cu2+ –Cu2+ pairs in crystals. For example, J ≈ −36, 36, 550 and 885 cm−1 were found for Cu2+ –Cu2+ pairs, respectively, in 4 AsCuCl3 , KCuCl3 , (pno)2 CuCl2 , and (pno)2 CuCl·H2 O crystals from the magnetic susceptibility analysis [19]. As in the case of Cu2+ –Cu2+ pairs in 4 AsCuCl3 crystal [20], the non-diagonal element ψ1 |HSS |ψ−1 is much smaller than those in the diagonal element ψ1 |HSS |ψ1 because of the small exchange parameter J. So, the exchange interaction in the Ti3+ –Ti3+ pair contributes little to g factors and can be omitted. Thus, the above calculations of g factors for the Ti3+ –Ti3+ pair from similar method of the single Ti3+ ion can be regarded as suitable. It should be pointed out that in beryl crystal the g factors (where g ≈ 1.9895 and g⊥ ≈ 1.9416 [2]) of single Ti3+ ion are slightly different from those of Ti3+ –Ti3+ pair (note: the experimental values are in Table 1). The reason may be due mainly to the slightly different local distortion of oxygen octahedra between the single Ti3+ center and the Ti3+ –Ti3+ pair in beryl. Considering that the anisotropic g factors for 3d1 ions in crystals are sensitive to the immediate environment, the above small difference of g factors between the single Ti3+ center and Ti3+ –Ti3+ pair can be understood. On the other hand, the zero-field splitting D is due to the exchange interaction (if the interaction is not considered, i.e., J = 0, from the above calculation, we have D = 0). Thus, by using the small exchange interaction as perturbation, the EPR parameters g , g⊥ and D are reasonably explained. The
F. Wang, W.-C. Zheng / Spectrochimica Acta Part A 67 (2007) 1281–1283
above method is expected to be effective to the calculations of EPR parameters in other trigonal d1 –d1 pairs in crystals. References [1] A.I. Alimpiev, G.V. Bukin, V.N. Matrosov, E.V. Pestroyakov, V.P. Solntsev, V.I. Trunov, E.G. Tsvetkov, V.P. Chebotave, Kvantovaya Elektron. 13 (1986) 885. [2] V.P. Solntsev, A.M. Yurkin, Cryst. Rep. 45 (2000) 128. [3] L.V. Bershov, Zh. Strukt. Khim. 10 (1969) 141. [4] D.R. Hutton, F.A. Darmann, G.J. Troup, Aust. J. Phys. 44 (1991) 429. [5] A.S. Chakravakty, Introduction to the Magnetic Properties of Solids, John Wiley and Sons, New York, 1980. [6] Z.Y. Yang, C. Rudowicz, J. Qin, Physica B 318 (2002) 188. [7] W.C. Zheng, Q. Zhou, Y. Mei, X.X. Wu, Opt. Mater. 27 (2004) 49.
1283
[8] W.C. Zheng, Q. Zhou, X.X. Wu, Y. Mei, Spctrochim. Acta A61 (2005) 1243. [9] A. Abragam, B. Bleaney, Electron Paramagnetic Resonance of Transition Metal Ions, Oxford University Press, London, 1970. [10] D.J. Newman, N. Betty, Rep. Prog. Phys. 52 (1989) 699. [11] B. Morison, Acta Crystallogr. B28 (1972) 1899. [12] W.L. Yu, M.G. Zhao, Phys. Rev. B37 (1988) 9254. [13] Q. Zhou, W.C. Zheng, X.X. Wu, Y. Mei, Spctrochim. Acta A62 (2005) 769. [14] D.S. McClure, J. Chem. Phys. 38 (1963) 2289. [15] S. Garcia-Revilla, F. Rodriguez, R. Valiente, M. Pollnau, J. Phys.: Condens. Matter 14 (2002) 447. [16] A. Edgar, J. Phys. C9 (1976) 4304. [17] C. Rudowicz, Y.Y. Zhou, J. Magn. Magn. Mater. 111 (1992) 153. [18] T.H. Yeom, S.H. Choh, M.L. Du, M.S. Jang, Phys. Rev. B53 (1996) 3415. [19] C. Chow, R.D. Willett, J. Chem. Phys. 59 (1973) 5903. [20] G.Y. Shen, C.Q. Xu, G.R. Bai, Phys. Status Solidi B158 (1990) k185.
Investigations of EPR parameters for the trigonal Ti3+–Ti3+ pair in beryl crystal Fang Wang a , Zheng Wen-Chen a,b,∗ a
b
Department of Material Science, Sichuan University, Chengdu 610064, People’s Republic of China International Center for Materials Physics, Chinese Academy of Sciences, Shenyang 110016, People’s Republic of China Received 11 July 2006; accepted 16 September 2006
Abstract By using the complete diagonalization of energy matrix of 3d1 ions in trigonal symmetry, the EPR parameters (g factors g , g⊥ and zero-field splitting D) of the trigonal Ti3+ –Ti3+ pair in beryl crystal are calculated. In the calculations, the exchange interaction in the Ti3+ –Ti3+ pair is taken as the perturbation and the local trigonal distortion in the defect center is considered. The results (which are in agreement with the experimental values) are discussed. © 2006 Elsevier B.V. All rights reserved. Keywords: Electron parameters resonance; Crystal-field theory; Exchange interaction; Ti3+ –Ti3+ pair; Beryl
1. Introduction Beryl (Be3 Al2 Si6 O8 ) crystals containing transition- and alkali-metal impurities can present a lot of colour, depending upon the species and valence of impurity. So, they are used in jewelry and laser materials. In particular, the discovery of Ti3+ doped BeAl2 O4 tunable laser crystal [1] has simulated studies of Ti3+ -doped beryl crystals. Many spectroscopy investigations for the Ti3+ -doped beryl were made [2–4]. From the EPR measurements, Solntsev and Yurkin [2] found that in addition to the isolated Ti3+ centers at Al3+ and Si4+ sites, there is a weak trigonal (C3h ) Ti3+ center described by S = 1. This center is associated with the Ti3+ Al –Ti3+ Al pair located along the C3 -axis and ˚ [2]. The EPR parameters (g spaced by the distance r ≈ 4.597 A factors g , g⊥ and zero-field splitting D) of the Ti3+ –Ti3+ pair in beryl were given [2], until now no satisfactory theoretical explanations have been made for these EPR parameters. As is known, there is no zero-field splitting in usual 3d1 (or 3d9 ) systems. The zero-field splitting in this Ti3+ –Ti3+ pair is due to the exchange interaction of the pair. In this paper, we calculated the g factors by using the complete diagonalization of the energy
∗ Corresponding author at: Department of Material Science, Sichuan University, Chengdu 610064, People’s Republic of China. Tel.: +86 28 85412371; fax: +86 28 85416050. E-mail address: [email protected] (W.-C. Zheng).
1386-1425/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.saa.2006.09.038
matrix for 3d1 ion in trigonal symmetry and treat the exchange interaction as perturbation in the degeneracy state perturbation theory to calculate the zero-field splitting. In the calculations, the local trigonal distortion in the Ti3+ –Ti3+ pair is considered. The results are discussed. 2. Calculation For a Ti3+ –Ti3+ pair in crystals, the Hamiltonian can be expressed as 1 2 H = Hf1 + Hf2 + Hso (ζ) + Hso (ζ) + Vc1 + Vc2 + JS 1 · S 2 (1)
where the superscripts (or subscripts) 1 and 2 refer to two ions, respectively. Hf , Vc and Hso (ζ) denote, respectively, the freeion Hamiltonian, the crystal-field potential and the spin-orbit coupling term (where the ζ is the spin-orbit coupling parameter in crystal). S is the operator of spin momentum and J is the exchange parameter. For a 3dn ion in trigonal symmetry, the crystal-field potential is given as Vcλ = B20 O02 + B40 O04 + B43 O34
(2)
where Bkl are the crystal-field parameters. By using the weak field basis function φn = |j, M in trigonal symmetry [5], the 10 × 10 complete energy matrix for 3d1 ion in trigonal symmetry is derived. The eigenenergy and eigenvectors of the Hamiltonian can be obtained by diagonalizing the energy
1282
F. Wang, W.-C. Zheng / Spectrochimica Acta Part A 67 (2007) 1281–1283 Table 1 The EPR parameters (g factors and zero-field splitting D) of the trigonal Ti3+ –Ti3+ pair in beryl crystal
matrix. Thus, the doubly degenerated ground states are |G+ =
10
|G− =
an φ n
n=1
10
bn φn
(3)
n=1
where an and bn are the coefficients determined from the above diagonalization calculation. The anisotropic g factors can be written as g = 2G+ |kLz + ge S z |G+ g⊥ = 2G+ |kLx + ge S x |G− .
(4)
where ge (≈2.0023) is the free-electron value. Lγ (γ = x or y) and Sγ are the orbit and spin momentum operators, respectively. k is the orbital reduction factor. Because of the covalency reduction effect for 3dn ions in crystals, a covalency reduction parameter N is introduced [6–8]. Thus, we have k = N2 and ζ = N2 ζ 0 [6–8], where ζ 0 is corresponding parameter in free state. For a free Ti3+ ion, ζ 0 ≈ 154 cm−1 [9]. From the superposition model [10], the crystal-field parameters Bkl in the above matrix can be expressed as ¯ 2 (R) B20 ≈ 3A
2
(3 cos2 θi − 1)
i=1
¯ 4 (R) B40 ≈ 3A
2
(35 cos4 θi − 30 cos2 θi + 3)
(5)
i=1 2 √ ¯ 4 (R) sin3 θi cos θi B43 ≈ −6 35A
Calculation Experiment [2]
g
g⊥
D (10−4 cm−1 )
1.990 1.991(1)
1.853 1.842(1)
205.1 204.6(5)
the perturbation term) can be given as 1 (S1+ S2− + S1− S2+ ) + S1Z S2Z . HSS = JS1 · S2 = J 2
(7)
The wave functions of ground triplet for the Ti3+ –Ti3+ pair are ⎧ + MS = 1, ψ1 = |G+ ⎪ 1 G2 ⎪ ⎪ ⎨ + |G+ G− + |G− 1 G2 S = 1, (8) MS = 0, ψ0 = 1 2 √ ⎪ 2 ⎪ ⎪ ⎩ − MS = −1, ψ−1 = |G− 1 G2 Thus, we get the expression of zero-field splitting as D = ψ1 |HSS |ψ1 − ψ0 |HSS |ψ0
(9)
Applying the diagonalized wave functions |G± (i.e., those in Eq. (3)) to Eq. (9) and taking the exchange parameter J ≈ 14 cm−1 , the zero-field splitting D of the Ti3+ –Ti3+ pair in beryl is calculated. The result is also compared with the observed value in Table 1. 3. Discussion
i=1
where the subscripts 1 and 2 stand for the O2− ions in ¯ 4 (R) are the intrin¯ 2 (R) and A upper and lower triangles. A sic parameters with the metal–ligand distance R. In beryl ˚ [11]. For 3dn ions in crystals, crystal, R1 ≈ R2 ≈ R ≈ 1.904 A ¯ 4 (R) ≈ 3Dq/4 [10,12,13]. From the cubic field parameter A ˚ [14]) obtained from the optiDq ≈ 1910 cm−1 (with R ≈ 1.912 A cal spectra of Ti3+ in Al2 O3 [15]. We reasonably estimated ¯ 2 (R) ≈ (9–12)A ¯ 4 (R) ¯ 4 (R) ≈ 1453 cm−1 for Ti3+ in beryl. A A ¯ ¯ [13,16–18], we take A2 (R) ≈ 12A4 (R) here. θ i denotes the angle between the direction of Ri and C3 -axis. In the host beryl crystal, θ1h ≈ 55.30◦ and θ2h ≈ 59.68◦ [11]. Considering that the size and the nature of impurity are unlike those of the replaced host ion, the angle θ i in the impurity center may be different from the corresponding θih in the host crystal because of the impurity-induced local lattice relaxation. To decrease the adjustable parameter, for the Ti3+ –Ti3+ pair in beryl, we take only θ 2 as the adjustable parameter. Substituting all these parameters into the above matrix and formulas and fitting the calculated g factors g and g⊥ of the Ti3+ –Ti3+ in beryl to the observed values, we yield k ≈ 0.921,
θ2 ≈ 57.63◦
(6)
The calculated g factors are compared with the experimental values in Table 1. For the calculation of zero-field splitting D of Ti3+ –Ti3+ pair in beryl, the exchange interaction term HSS (which is used as
In the above calculations, we find that the exchange parameter J ≈ 14 cm−1 . The value of |J| is smaller than those in similar Cu2+ –Cu2+ pairs in crystals. For example, J ≈ −36, 36, 550 and 885 cm−1 were found for Cu2+ –Cu2+ pairs, respectively, in 4 AsCuCl3 , KCuCl3 , (pno)2 CuCl2 , and (pno)2 CuCl·H2 O crystals from the magnetic susceptibility analysis [19]. As in the case of Cu2+ –Cu2+ pairs in 4 AsCuCl3 crystal [20], the non-diagonal element ψ1 |HSS |ψ−1 is much smaller than those in the diagonal element ψ1 |HSS |ψ1 because of the small exchange parameter J. So, the exchange interaction in the Ti3+ –Ti3+ pair contributes little to g factors and can be omitted. Thus, the above calculations of g factors for the Ti3+ –Ti3+ pair from similar method of the single Ti3+ ion can be regarded as suitable. It should be pointed out that in beryl crystal the g factors (where g ≈ 1.9895 and g⊥ ≈ 1.9416 [2]) of single Ti3+ ion are slightly different from those of Ti3+ –Ti3+ pair (note: the experimental values are in Table 1). The reason may be due mainly to the slightly different local distortion of oxygen octahedra between the single Ti3+ center and the Ti3+ –Ti3+ pair in beryl. Considering that the anisotropic g factors for 3d1 ions in crystals are sensitive to the immediate environment, the above small difference of g factors between the single Ti3+ center and Ti3+ –Ti3+ pair can be understood. On the other hand, the zero-field splitting D is due to the exchange interaction (if the interaction is not considered, i.e., J = 0, from the above calculation, we have D = 0). Thus, by using the small exchange interaction as perturbation, the EPR parameters g , g⊥ and D are reasonably explained. The
F. Wang, W.-C. Zheng / Spectrochimica Acta Part A 67 (2007) 1281–1283
above method is expected to be effective to the calculations of EPR parameters in other trigonal d1 –d1 pairs in crystals. References [1] A.I. Alimpiev, G.V. Bukin, V.N. Matrosov, E.V. Pestroyakov, V.P. Solntsev, V.I. Trunov, E.G. Tsvetkov, V.P. Chebotave, Kvantovaya Elektron. 13 (1986) 885. [2] V.P. Solntsev, A.M. Yurkin, Cryst. Rep. 45 (2000) 128. [3] L.V. Bershov, Zh. Strukt. Khim. 10 (1969) 141. [4] D.R. Hutton, F.A. Darmann, G.J. Troup, Aust. J. Phys. 44 (1991) 429. [5] A.S. Chakravakty, Introduction to the Magnetic Properties of Solids, John Wiley and Sons, New York, 1980. [6] Z.Y. Yang, C. Rudowicz, J. Qin, Physica B 318 (2002) 188. [7] W.C. Zheng, Q. Zhou, Y. Mei, X.X. Wu, Opt. Mater. 27 (2004) 49.
1283
[8] W.C. Zheng, Q. Zhou, X.X. Wu, Y. Mei, Spctrochim. Acta A61 (2005) 1243. [9] A. Abragam, B. Bleaney, Electron Paramagnetic Resonance of Transition Metal Ions, Oxford University Press, London, 1970. [10] D.J. Newman, N. Betty, Rep. Prog. Phys. 52 (1989) 699. [11] B. Morison, Acta Crystallogr. B28 (1972) 1899. [12] W.L. Yu, M.G. Zhao, Phys. Rev. B37 (1988) 9254. [13] Q. Zhou, W.C. Zheng, X.X. Wu, Y. Mei, Spctrochim. Acta A62 (2005) 769. [14] D.S. McClure, J. Chem. Phys. 38 (1963) 2289. [15] S. Garcia-Revilla, F. Rodriguez, R. Valiente, M. Pollnau, J. Phys.: Condens. Matter 14 (2002) 447. [16] A. Edgar, J. Phys. C9 (1976) 4304. [17] C. Rudowicz, Y.Y. Zhou, J. Magn. Magn. Mater. 111 (1992) 153. [18] T.H. Yeom, S.H. Choh, M.L. Du, M.S. Jang, Phys. Rev. B53 (1996) 3415. [19] C. Chow, R.D. Willett, J. Chem. Phys. 59 (1973) 5903. [20] G.Y. Shen, C.Q. Xu, G.R. Bai, Phys. Status Solidi B158 (1990) k185.