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Studies of the spin-Hamiltonian parameters and the defect structure for Cu2+ at the rhombic Be2+ site of beryl crystal
ARTICLE IN PRESS Physica B 404 (2009) 2025–2027
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Studies of the spin-Hamiltonian parameters and the defect structure for Cu2+ at the rhombic Be2+ site of beryl crystal Fang Wanga,, Zheng Wen-Chenb,c a
Department of Mathematics and Physics, Chongqing University of Science and Technology, Chongqing 401331, PR China Department of Material Science, Sichuan University, Chengdu 610064, PR China c International Centre for Materials Physics, Chinese Academy of Sciences, Shenyang 110016, PR China b
a r t i c l e in fo
abstract
Article history: Received 4 March 2009 Accepted 27 March 2009
The spin-Hamiltonian (SH) parameters (g factors gi and hyperfine structure constants Ai, where i ¼ x, y, z) for Cu2+ at the rhombic Be2+ site of beryl (Be3Al2Si6O18) crystals are calculated from both the perturbation theory method (PTM) and the complete diagonalization (of energy matrix) method (CDM). The two methods are based on the cluster approach and so both the contribution to the SH parameters from the spin–orbit coupling parameters of central dn ion and that of ligand ion are included. The calculated results from the two theoretical methods are in reasonable agreement with the experimental values, suggesting that both methods are effective for the studies of SH parameters of 3d9 ion in rhombic symmetry. The defect structural data (which are unlike the corresponding data in the host beryl crystal) of the rhombic Cu2+ center in beryl are also acquired from the calculations. The results are discussed. & 2009 Elsevier B.V. All rights reserved.
Pacs: 76.30.Fc 71.70.Ch 61.72.S Keywords: Electron paramagnetic resonance Crystal- and ligand-field theory Defect structure Cu2+ Beryl
1. Introduction Beryl (Be3Al2Si6O18) crystals doped with transition-metal (e.g., 3dn) ions have attracted much attention for the applications in gem and laser industries [1,2]. These transition-metal impurity ions in beryl are luminescence or laser active centers, so it is of interest to study the defect model (e.g., the substitutional site because there are several cationic sites in beryl) and defect structure of 3dn impurities in beryl crystal. EPR technique is a powerful tool to obtain the information on defect model and defect structure for a paramagnetic impurity in crystals, so many EPR spectral studies for 3dn-doped beryl crystals were made [3–8]. Among them, the EPR spectrum of Cu2+-doped beryl was measured and its spin-Hamiltonian (SH) parameters (g factors gi and hyperfine structure constants Ai, where i ¼ x, y, z) were given [7,8]. From the SH parameters, it is suggested that the Cu2+ ion occupies the rhombically distorted tetrahedral Be2+ site in beryl [7]. No theoretical calculations based on the defect model for these SH parameters have been carried out. In this paper, we calculate these SH parameters based on the above model and study the defect structural data for the rhombic Cu2+ center in
beryl from two theoretical methods, one of which is the perturbation theory method (PTM) and the other is the complete diagonalization (of energy matrix) method (CDM). The two methods are based on the cluster approach in which the contribution to SH parameters due to both the spin–orbit coupling parameters of center dn ion and that of ligand ion are included. The results are discussed.
2. Calculation using PTM For a 3dn MX4 cluster, the one-electron basis function can be expressed as the linear combination of the d orbital |dgS (where g ¼ e or t, the irreducible representation of Td group) of dn ion and the p orbitals |sgS, |pgS of ligand ion [9,10], i.e., jce i ¼ N e ðjde i þ ajpe iÞ jct i ¼ Nt ðjdt i þ bjst i þ gjpt iÞ
(1)
in which Ng is the normalization coefficient, a, b and g are the orbital mixing coefficients. These coefficients are related by the normalization correlation 2
Corresponding author. Tel.: +86 23 65025837.
E-mail address: [email protected] (W. Fang). 0921-4526/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2009.03.039
Nt ½1 þ b þ g2 þ 2bSdp ðsÞ þ 2lSdpt ðpÞ1=2 ¼ 1 Ne ½1 þ a2 þ 2aSdpe ðpÞ1=2 ¼ 1
(2)
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W. Fang, W.C. Zheng / Physica B 404 (2009) 2025–2027
where Sdp ðsÞ ¼ hdt jst i, Sdpt ðpÞ ¼ hdt jpt i and Sdpe ðpÞ ¼ hde jpe i are the group overlap integrals. From the g factors of Cu2+ in beryl [7], it can be expected that the ground state of this system is 2B1 (|dxyS). Thus, from the perturbation theory, the third-order perturbation formulas based on the cluster approach for the SH parameters of 3d9 ion in rhombic symmetry with the ground state |dxyS are derived as gx ¼ ge þ
0
impurity–ligand distance. yi and fi are the polar angle and azimuthal angle, respectively. For the host beryl crystal studied, ˚ yh1 45:45 and yh2 65:7 (fhi are not clear) Rh1 Rh2 1:653ð1Þ A, [21]. Considering that the nature and site of impurity ion are unlike those of the host ion it replaces, the defect structural data of the impurity center may be different from the corresponding data in the host crystal. For the impurity–ligand distance R, we can estimate it from the approximate formulas [22]
0
R Rh þ 12ðr i r h Þ
2Bk B2 k 2BB0 k 2B02 k B2 g e 2B02 g e þ þ E2 E3 E1 E1 E2 E1 E3 2E22 E23
2Bk B2 k 2BB0 k 2B02 k B2 g e 2B02 g e gy ¼ ge þ þ þ E1 E3 E2 E1 E2 E2 E3 2E21 E23 0 0
gz ¼ ge þ
2
0 0
0 0
2
2
8B k B k 2BB k 2BB k B g e B g e E3 E1 E3 E2 E3 E1 E2 2E21 2E22
(3)
3 ðg y g e Þ Ax ¼ Pðk þ 27Þ þ P 0 ½ðg x g e Þ 14 3 ðg x g e Þ Ay ¼ Pðk þ 27Þ þ P 0 ½ðg y g e Þ 14 3 3 Az ¼ Pðk 47Þ þ P0 ½ðg z g e Þ þ 14 ðg y g e Þ þ 14 ðg x g e Þ
(4)
where Ei (i ¼ 13) are the energy separators. geE2.0023 is the g values of free electron. k is the core dipolarization constant. The spin–orbit coupling parameters B, B0 , the orbital reduction factors k, k0 and the dipolar hyperfine constants P, P0 in the cluster approach should be pffiffiffi B ¼ N2t ½B0d þ gð 2b g=2ÞB0p pffiffiffi pffiffiffi B0 ¼ Nt N e ½B0d þ aðb= 6 þ g=2 3ÞB0p pffiffiffi k ¼ N2t ½1 g2 =2 þ 2bg þ 2bSdp ðsÞ þ 2gSdpt ðpÞ p pffiffiffi ffiffiffi 0 k ¼ Nt N e ½1 þ ag=2 3 þ ab= 6 þ aSdpe ðpÞ þ bSdp ðsÞ þ gSdpt ðpÞ P ¼ N2t P 0 ; 0 d
P0 ¼ Nt N e P0
(5)
where ri and rh are the ionic radii of impurity and the replaced host ion. From the ri(Cu2+) E0.72 A˚, rh( Be2+)E0.35 A˚ [23], we have R1ER2E1.838 A˚ for Cu2+ in beryl. As in the case of distance R, the angle yi in the impurity center may be unlike the corresponding angle yih in the host crystal. We assure yi Etyih, where t is a parameter to characterize the change of angle y and if t ¼ 1, the angle yi is unchanged. For the angle fi, we can reasonably assure f1Ep/4+e1 and f2E3p/4+e2, where ei represent the rhombic distortion in the impurity center. To decrease the number of adjustable parameter, we let e1 ¼ e2 ¼ e. For 3dn ion in tetrahedra, we have A¯ 4 ðR0 Þ 27=16Dq [17]. For 3dn clusters, we have DqpRn 0 , where nE571.5 [24,25]. Thus, from DqE570 cm1 [26] with R0E1.968 A˚ for (CuO4)6 cluster in ZnO: Cu2+ (note: Rh 1:978 A˚ in ZnO [27] and rh(Zn2+)E0.74 A˚ [23]) we obtain DqE750 cm1 for (CuO4)6 cluster in beryl with R0E1.838 A˚. From the distance R0, the group overlap integrals of (CuO4)6 cluster in beryl can be calculated from the SCF wave functions of Cu2+ given by Watson [28] and O2 given in Ref. [29], i.e., R3d ðrÞ ¼ 0:435f3 ð2:413Þ þ 0:531f3 ð4:706Þ
0 p
where B and B are the spin–orbit coupling parameters of free 3dn ion and free ligand ion, respectively. P0 is the corresponding constant of free dn ion. For the above system under study, we have B0d ðCu2þ Þ 829 cm1 [11], B0p ðO2 Þ 150 cm1 [12] and P0 (Cu2+) E388 10–4 cm1 [13]. The energy separators Ei should read E1 ¼ Eðjdxz iÞ Eðjdxy iÞ ¼ 3Ds þ 5Dt þ 3Dx 4DZ E2 ¼ Eðjdyz iÞ Eðjdxy iÞ ¼ 3Ds þ 5Dt 3Dx þ 4DZ E3 ¼ Eðjdx2 y2 iÞ Eðjdxy iÞ ¼ 10 Dq E4 ¼ Eðjdz2 iÞ Eðjdxy iÞ ¼ 10 Dq 4 Ds 5 Dt
(6)
in which the crystal parameters Ds, Dt, Dx and DZ in the superposition model [14] are given as " # 2 X 2 R0 t2 Ds ¼ A¯ 2 ðR0 Þ ð3 cos2 yi 1Þ Ri 7 i¼1 2 A¯ 4 ðR0 Þ 21 " # t4 2 X R0 ð35 cos4 yi 30 cos2 yi þ 3 7 sin4 yi cos 4fi Þ Ri i¼1
Dt ¼
" # 2 X 6 R0 t2 2 ¯ ðsin yi cos 2fi Þ A2 ðR0 Þ Dx ¼ Ri 21 i¼1 " # 2 X 10 R0 t4 2 2 ¯ ðsin yi ð7 cos yi 1Þ cos 2fi A4 ðR0 Þ DZ ¼ Ri 21 i¼1
(8)
þ 0:183f3 ð8:843Þ þ 0:003f3 ð15:523Þ R2p ðrÞ ¼ 0:6804f2 ð1:55Þ þ 0:4308f2 ð3:43Þ
(9)
Thus, we obtain Sdp(s)E0.09066; Sdpe(p)E0.07459 and Sdpt(p)E0.04307. The orbital mixing coefficients a, b and g can be taken as proportional to the negative of corresponding integral [9] i.e.,
a ¼ k0 Sdpe ðpÞ;
b k0 Sdp ðsÞ;
g k0 Sdpt ðpÞ
(10)
Thus, in the above formulas, we have four unknown parameters k0, t, e and k. They can be regarded as the adjustable parameters and obtained by matching the calculated SH parameters to the experimental values. From the calculations of the SH parameters, we obtain k0 6:9;
t 1:014;
0:5 ;
k 0:32
(11)
The comparisons between the calculated and the experimental SH parameters gi and Ai are shown in Table 1. Table 1 The spin-Hamiltonian parameters (g factors gi and hyperfine structure constants Ai, Ai are in units of 104 cm1) for Cu2+ in beryl crystal.
(7)
in which the power-low exponents t2E3 and t4E5 [15–17]. A¯ 2 ðR0 Þ and A¯ 4 ðR0 Þ are the intrinsic parameters with the reference ¯ Lots of studies found that for 3dn clusters, the distance R0 ( R). ratio A¯ 2 ðR0 Þ=A¯ 4 ðR0 Þ is in the range of 8–12 [15–20] and we take the average value, i.e., A¯ 2 ðR0 Þ=A¯ 4 ðR0 Þ 10 here. Ri (i ¼ 1, 2) is the
Calc.a Calc.b Expt. [7] [8] a b c
gx
gy
gz
Ax
Ay
Az
2.071 2.074 2.076 2.071
2.102 2.100 2.102 2.092
2.356 2.351 2.356 2.359
4.3 5.5 0 5.0c
15.9 15.2 23.5c 15.6c
–138.5 –139.8 106c 114.6c
Calculated using PTM. Calculated using CDM. The values are practically the absolute values.
ARTICLE IN PRESS W. Fang, W.C. Zheng / Physica B 404 (2009) 2025–2027
3. Calculation using CDM The Hamiltonian for 3d9 ion in rhombic crystal–field under an external magnetic field should read 0
H ¼ Hf þ HSO ðB; B0 Þ þ HCF ðDs; Dt; Dx; DZÞ þ HZe ðk; k Þ
(12)
where the four terms are, respectively, free-ion, spin–orbit interaction, crystal–field interaction and Zeeman terms. By means of the strong-field basis functions [30], the 10 10 complete energy matrix concerning the above Hamiltonian is established. Under an external magnetic field, the ground state 2B1(|dxyS) of 3d9 ion in rhombic field is further split into two sublevels E(71/2) with effective spin S ¼ 71/2. Thus, we have gi ¼
DEi
mB Hi
ðHi a0Þ
(13)
where DEi ¼ E(1/2)E(1/2) with the external magnetic field Hi along i direction. The hyperfine constant Ai can also be calculated from Eq. (4) with gi obtained from CDM (i.e. Eq. (13)). Thus, applying the same parameters as those in PTM to CDM, the SH parameters can be calculated by diagonalizing the complete energy matrix. The results are also shown in Table 1.
4. Discussion Many EPR studies suggested that for Cu2+ ions in crystals, the core dipolarization constant k is in the range of 0.2–0.6 [31–34]. Our result that kE0.32 for Cu2+ in beryl is within the range. So it seems to be rational. It is known that the signs of hyperfine structure constants Ai for dn and fn ions in crystals are unable to be determined solely from EPR experiment [13,35,36], so although the observed values of constants Ai for Cu2+ in beryl are written as positive, they are practically the absolute values. The above calculations suggest that for Cu2+ in beryl, Ax and Ay are positive and Az is negative (see Table 1). The defect structural data (e.g., Ri and yi) obtained from the above calculations for Cu2+ center in beryl are indeed different from the corresponding structural data in host crystals. It appears that by studying the EPR data (i.e., the SH parameters), some information on the defect structure of paramagnetic impurity center in crystals can be acquired.
2027
It will be seen from Table 1 by using the same parameters, the calculated SH parameters for Cu2+ in beryl from both PTM and CDM are in reasonable agreement with the experimental values. So, the two theoretical methods can be used to explain the SH parameters and to obtain the defect structure of 3dn impurity centers in crystals by analyzing their SH parameters. References [1] C.E. McManus, N.J. McMillan, R.S. Harmon, R.C. Whitmore, F.C. De Lucia, A.W. Miziolek, Appl. Opt. 47 (2008) G72. [2] L.A. Groat, G. Giuliani, D.D. Marshall, D. Turner, Ore Geol. Rev. 34 (2008) 87. [3] K. Krambrock, K.J. Guedes, M.V. B Pinheiro, Phys. Chem. Minerals 35 (2008) 409. [4] M. Dvir, W. Low, Phys. Rev. 119 (1960) 1587. [5] D.R. Hutton, F.A. Darmann, G.j. Troup, Austral. J. Phys. 44 (1991) 429. [6] V.P. Solntsev, A.M. Yurkin, Cryst. Rep. 45 (2000) 128. [7] J.M. Gaite, V.V. Izotov, S.I. Nikitin, S.Y. Prosvirhin, Appl. Mag. Reson. 20 (2001) 307. [8] V.P. Solncev, A.S. Lebeder, V.S. Pavluchenko, V.A. Kliakhin, Soviet Phys. Solid State 18 (1976) 1396. [9] J.T. Vallin, G.D. Watkins, Phys. Rev. 89 (1974) 2051. [10] W.C. Zheng, S.Y. Wu, W. Li, Physica B 253 (1998) 798. [11] J.S. Griffith, The Theory of Transition-Metal Ions, Cambridge University Press, London, 1964. [12] M.L. Du, C. Rudowicz, Phys. Rev. B 46 (1992) 8974. [13] B.R. McGarvey, J. Phys. Chem. 71 (1967) 51. [14] D.J. Newman, B. Ng, Rep. Progr. Phys. 52 (1989) 699. [15] C. Rudowicz, Z.Y. Yang, Y.Y. Yeung, J. Qin, J. Phys. Chem. Solids 64 (2003) 1419. [16] W.L. Yu, J. Phys. Condens. Matter 6 (1994) 5105. [17] Y. Mei, X.X. Wu, W.C. Zheng, Radiat. Eff. Def. Solids 163 (2008) 79. [18] T.H. Yeom, S.H. Choh, M.L. Du, M.S. Tang, Phys. Rev. B 53 (1996) 3415. [19] H.G. Liu, W.C. Zheng, L. He, Radiat. Eff. Def. Solids 163 (2008) 1. [20] C. Rudowicz, Y.Z. Zhou, J. Magn. Magn. Matter 11 (1992) 153. [21] B. Morosin, Acta Crystallogr. B 28 (1972) 1899. [22] W.C. Zheng, Physica B 215 (1995) 255. [23] R.C. Weast, CRC Handbook of Chemistry and Physics, CRC Press, Roca Raton, 1989, p.F.-187. [24] D. Hernandez, F. Rodriguez, M. Moreno, H.V. Gudel, Physica B 265 (1999) 196. [25] M. Moreno, M.T. Barriuxso, J.A. Arambura, J. Phys. Condens. Matter 4 (1992) 9481. [26] I. Boser, V. Scherz, M. Wohlecke, J. Lumin. 182 (1970) 39. [27] E.H. Kisi, M.M. Elcombe, Acta Cryst. C 45 (1989) 1867. [28] R.E. Watson, Phys. Rev. 119 (1960) 1934. [29] C.J. Ballhausen, H.B. Gray, Inorg. Chem. 1 (1962) 111. [30] A.S. Chakravary, Introduction to the Magnetic Properties of Solids, Wiley, New York, 1980. [31] A. Abragam, M.H.L. Pryce, Proc. Roy. Soc. (London) A 206 (1951) 173. [32] M.A. Hitchman, R.G. McDonald, D. Reinen, Inorg. Chem. 25 (1986) 519. [33] S.O. Graham, R.L. White, Phys. Rev. B 10 (1974) 4505. [34] X.X. Wu, W.C. Zheng, S. Tang, Z. Naturforsch. A 59 (2004) 47. [35] A. Abragam, B. Bleaney, Electron Paramagnetic Resonance of Transition Ions, Oxford University Press, London, 1970. [36] V. Havliek, P. Novak, B.V. Mill, Phys. Status Solidi B 64 (1974) k19.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Studies of the spin-Hamiltonian parameters and the defect structure for Cu2+ at the rhombic Be2+ site of beryl crystal Fang Wanga,, Zheng Wen-Chenb,c a
Department of Mathematics and Physics, Chongqing University of Science and Technology, Chongqing 401331, PR China Department of Material Science, Sichuan University, Chengdu 610064, PR China c International Centre for Materials Physics, Chinese Academy of Sciences, Shenyang 110016, PR China b
a r t i c l e in fo
abstract
Article history: Received 4 March 2009 Accepted 27 March 2009
The spin-Hamiltonian (SH) parameters (g factors gi and hyperfine structure constants Ai, where i ¼ x, y, z) for Cu2+ at the rhombic Be2+ site of beryl (Be3Al2Si6O18) crystals are calculated from both the perturbation theory method (PTM) and the complete diagonalization (of energy matrix) method (CDM). The two methods are based on the cluster approach and so both the contribution to the SH parameters from the spin–orbit coupling parameters of central dn ion and that of ligand ion are included. The calculated results from the two theoretical methods are in reasonable agreement with the experimental values, suggesting that both methods are effective for the studies of SH parameters of 3d9 ion in rhombic symmetry. The defect structural data (which are unlike the corresponding data in the host beryl crystal) of the rhombic Cu2+ center in beryl are also acquired from the calculations. The results are discussed. & 2009 Elsevier B.V. All rights reserved.
Pacs: 76.30.Fc 71.70.Ch 61.72.S Keywords: Electron paramagnetic resonance Crystal- and ligand-field theory Defect structure Cu2+ Beryl
1. Introduction Beryl (Be3Al2Si6O18) crystals doped with transition-metal (e.g., 3dn) ions have attracted much attention for the applications in gem and laser industries [1,2]. These transition-metal impurity ions in beryl are luminescence or laser active centers, so it is of interest to study the defect model (e.g., the substitutional site because there are several cationic sites in beryl) and defect structure of 3dn impurities in beryl crystal. EPR technique is a powerful tool to obtain the information on defect model and defect structure for a paramagnetic impurity in crystals, so many EPR spectral studies for 3dn-doped beryl crystals were made [3–8]. Among them, the EPR spectrum of Cu2+-doped beryl was measured and its spin-Hamiltonian (SH) parameters (g factors gi and hyperfine structure constants Ai, where i ¼ x, y, z) were given [7,8]. From the SH parameters, it is suggested that the Cu2+ ion occupies the rhombically distorted tetrahedral Be2+ site in beryl [7]. No theoretical calculations based on the defect model for these SH parameters have been carried out. In this paper, we calculate these SH parameters based on the above model and study the defect structural data for the rhombic Cu2+ center in
beryl from two theoretical methods, one of which is the perturbation theory method (PTM) and the other is the complete diagonalization (of energy matrix) method (CDM). The two methods are based on the cluster approach in which the contribution to SH parameters due to both the spin–orbit coupling parameters of center dn ion and that of ligand ion are included. The results are discussed.
2. Calculation using PTM For a 3dn MX4 cluster, the one-electron basis function can be expressed as the linear combination of the d orbital |dgS (where g ¼ e or t, the irreducible representation of Td group) of dn ion and the p orbitals |sgS, |pgS of ligand ion [9,10], i.e., jce i ¼ N e ðjde i þ ajpe iÞ jct i ¼ Nt ðjdt i þ bjst i þ gjpt iÞ
(1)
in which Ng is the normalization coefficient, a, b and g are the orbital mixing coefficients. These coefficients are related by the normalization correlation 2
Corresponding author. Tel.: +86 23 65025837.
E-mail address: [email protected] (W. Fang). 0921-4526/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2009.03.039
Nt ½1 þ b þ g2 þ 2bSdp ðsÞ þ 2lSdpt ðpÞ1=2 ¼ 1 Ne ½1 þ a2 þ 2aSdpe ðpÞ1=2 ¼ 1
(2)
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W. Fang, W.C. Zheng / Physica B 404 (2009) 2025–2027
where Sdp ðsÞ ¼ hdt jst i, Sdpt ðpÞ ¼ hdt jpt i and Sdpe ðpÞ ¼ hde jpe i are the group overlap integrals. From the g factors of Cu2+ in beryl [7], it can be expected that the ground state of this system is 2B1 (|dxyS). Thus, from the perturbation theory, the third-order perturbation formulas based on the cluster approach for the SH parameters of 3d9 ion in rhombic symmetry with the ground state |dxyS are derived as gx ¼ ge þ
0
impurity–ligand distance. yi and fi are the polar angle and azimuthal angle, respectively. For the host beryl crystal studied, ˚ yh1 45:45 and yh2 65:7 (fhi are not clear) Rh1 Rh2 1:653ð1Þ A, [21]. Considering that the nature and site of impurity ion are unlike those of the host ion it replaces, the defect structural data of the impurity center may be different from the corresponding data in the host crystal. For the impurity–ligand distance R, we can estimate it from the approximate formulas [22]
0
R Rh þ 12ðr i r h Þ
2Bk B2 k 2BB0 k 2B02 k B2 g e 2B02 g e þ þ E2 E3 E1 E1 E2 E1 E3 2E22 E23
2Bk B2 k 2BB0 k 2B02 k B2 g e 2B02 g e gy ¼ ge þ þ þ E1 E3 E2 E1 E2 E2 E3 2E21 E23 0 0
gz ¼ ge þ
2
0 0
0 0
2
2
8B k B k 2BB k 2BB k B g e B g e E3 E1 E3 E2 E3 E1 E2 2E21 2E22
(3)
3 ðg y g e Þ Ax ¼ Pðk þ 27Þ þ P 0 ½ðg x g e Þ 14 3 ðg x g e Þ Ay ¼ Pðk þ 27Þ þ P 0 ½ðg y g e Þ 14 3 3 Az ¼ Pðk 47Þ þ P0 ½ðg z g e Þ þ 14 ðg y g e Þ þ 14 ðg x g e Þ
(4)
where Ei (i ¼ 13) are the energy separators. geE2.0023 is the g values of free electron. k is the core dipolarization constant. The spin–orbit coupling parameters B, B0 , the orbital reduction factors k, k0 and the dipolar hyperfine constants P, P0 in the cluster approach should be pffiffiffi B ¼ N2t ½B0d þ gð 2b g=2ÞB0p pffiffiffi pffiffiffi B0 ¼ Nt N e ½B0d þ aðb= 6 þ g=2 3ÞB0p pffiffiffi k ¼ N2t ½1 g2 =2 þ 2bg þ 2bSdp ðsÞ þ 2gSdpt ðpÞ p pffiffiffi ffiffiffi 0 k ¼ Nt N e ½1 þ ag=2 3 þ ab= 6 þ aSdpe ðpÞ þ bSdp ðsÞ þ gSdpt ðpÞ P ¼ N2t P 0 ; 0 d
P0 ¼ Nt N e P0
(5)
where ri and rh are the ionic radii of impurity and the replaced host ion. From the ri(Cu2+) E0.72 A˚, rh( Be2+)E0.35 A˚ [23], we have R1ER2E1.838 A˚ for Cu2+ in beryl. As in the case of distance R, the angle yi in the impurity center may be unlike the corresponding angle yih in the host crystal. We assure yi Etyih, where t is a parameter to characterize the change of angle y and if t ¼ 1, the angle yi is unchanged. For the angle fi, we can reasonably assure f1Ep/4+e1 and f2E3p/4+e2, where ei represent the rhombic distortion in the impurity center. To decrease the number of adjustable parameter, we let e1 ¼ e2 ¼ e. For 3dn ion in tetrahedra, we have A¯ 4 ðR0 Þ 27=16Dq [17]. For 3dn clusters, we have DqpRn 0 , where nE571.5 [24,25]. Thus, from DqE570 cm1 [26] with R0E1.968 A˚ for (CuO4)6 cluster in ZnO: Cu2+ (note: Rh 1:978 A˚ in ZnO [27] and rh(Zn2+)E0.74 A˚ [23]) we obtain DqE750 cm1 for (CuO4)6 cluster in beryl with R0E1.838 A˚. From the distance R0, the group overlap integrals of (CuO4)6 cluster in beryl can be calculated from the SCF wave functions of Cu2+ given by Watson [28] and O2 given in Ref. [29], i.e., R3d ðrÞ ¼ 0:435f3 ð2:413Þ þ 0:531f3 ð4:706Þ
0 p
where B and B are the spin–orbit coupling parameters of free 3dn ion and free ligand ion, respectively. P0 is the corresponding constant of free dn ion. For the above system under study, we have B0d ðCu2þ Þ 829 cm1 [11], B0p ðO2 Þ 150 cm1 [12] and P0 (Cu2+) E388 10–4 cm1 [13]. The energy separators Ei should read E1 ¼ Eðjdxz iÞ Eðjdxy iÞ ¼ 3Ds þ 5Dt þ 3Dx 4DZ E2 ¼ Eðjdyz iÞ Eðjdxy iÞ ¼ 3Ds þ 5Dt 3Dx þ 4DZ E3 ¼ Eðjdx2 y2 iÞ Eðjdxy iÞ ¼ 10 Dq E4 ¼ Eðjdz2 iÞ Eðjdxy iÞ ¼ 10 Dq 4 Ds 5 Dt
(6)
in which the crystal parameters Ds, Dt, Dx and DZ in the superposition model [14] are given as " # 2 X 2 R0 t2 Ds ¼ A¯ 2 ðR0 Þ ð3 cos2 yi 1Þ Ri 7 i¼1 2 A¯ 4 ðR0 Þ 21 " # t4 2 X R0 ð35 cos4 yi 30 cos2 yi þ 3 7 sin4 yi cos 4fi Þ Ri i¼1
Dt ¼
" # 2 X 6 R0 t2 2 ¯ ðsin yi cos 2fi Þ A2 ðR0 Þ Dx ¼ Ri 21 i¼1 " # 2 X 10 R0 t4 2 2 ¯ ðsin yi ð7 cos yi 1Þ cos 2fi A4 ðR0 Þ DZ ¼ Ri 21 i¼1
(8)
þ 0:183f3 ð8:843Þ þ 0:003f3 ð15:523Þ R2p ðrÞ ¼ 0:6804f2 ð1:55Þ þ 0:4308f2 ð3:43Þ
(9)
Thus, we obtain Sdp(s)E0.09066; Sdpe(p)E0.07459 and Sdpt(p)E0.04307. The orbital mixing coefficients a, b and g can be taken as proportional to the negative of corresponding integral [9] i.e.,
a ¼ k0 Sdpe ðpÞ;
b k0 Sdp ðsÞ;
g k0 Sdpt ðpÞ
(10)
Thus, in the above formulas, we have four unknown parameters k0, t, e and k. They can be regarded as the adjustable parameters and obtained by matching the calculated SH parameters to the experimental values. From the calculations of the SH parameters, we obtain k0 6:9;
t 1:014;
0:5 ;
k 0:32
(11)
The comparisons between the calculated and the experimental SH parameters gi and Ai are shown in Table 1. Table 1 The spin-Hamiltonian parameters (g factors gi and hyperfine structure constants Ai, Ai are in units of 104 cm1) for Cu2+ in beryl crystal.
(7)
in which the power-low exponents t2E3 and t4E5 [15–17]. A¯ 2 ðR0 Þ and A¯ 4 ðR0 Þ are the intrinsic parameters with the reference ¯ Lots of studies found that for 3dn clusters, the distance R0 ( R). ratio A¯ 2 ðR0 Þ=A¯ 4 ðR0 Þ is in the range of 8–12 [15–20] and we take the average value, i.e., A¯ 2 ðR0 Þ=A¯ 4 ðR0 Þ 10 here. Ri (i ¼ 1, 2) is the
Calc.a Calc.b Expt. [7] [8] a b c
gx
gy
gz
Ax
Ay
Az
2.071 2.074 2.076 2.071
2.102 2.100 2.102 2.092
2.356 2.351 2.356 2.359
4.3 5.5 0 5.0c
15.9 15.2 23.5c 15.6c
–138.5 –139.8 106c 114.6c
Calculated using PTM. Calculated using CDM. The values are practically the absolute values.
ARTICLE IN PRESS W. Fang, W.C. Zheng / Physica B 404 (2009) 2025–2027
3. Calculation using CDM The Hamiltonian for 3d9 ion in rhombic crystal–field under an external magnetic field should read 0
H ¼ Hf þ HSO ðB; B0 Þ þ HCF ðDs; Dt; Dx; DZÞ þ HZe ðk; k Þ
(12)
where the four terms are, respectively, free-ion, spin–orbit interaction, crystal–field interaction and Zeeman terms. By means of the strong-field basis functions [30], the 10 10 complete energy matrix concerning the above Hamiltonian is established. Under an external magnetic field, the ground state 2B1(|dxyS) of 3d9 ion in rhombic field is further split into two sublevels E(71/2) with effective spin S ¼ 71/2. Thus, we have gi ¼
DEi
mB Hi
ðHi a0Þ
(13)
where DEi ¼ E(1/2)E(1/2) with the external magnetic field Hi along i direction. The hyperfine constant Ai can also be calculated from Eq. (4) with gi obtained from CDM (i.e. Eq. (13)). Thus, applying the same parameters as those in PTM to CDM, the SH parameters can be calculated by diagonalizing the complete energy matrix. The results are also shown in Table 1.
4. Discussion Many EPR studies suggested that for Cu2+ ions in crystals, the core dipolarization constant k is in the range of 0.2–0.6 [31–34]. Our result that kE0.32 for Cu2+ in beryl is within the range. So it seems to be rational. It is known that the signs of hyperfine structure constants Ai for dn and fn ions in crystals are unable to be determined solely from EPR experiment [13,35,36], so although the observed values of constants Ai for Cu2+ in beryl are written as positive, they are practically the absolute values. The above calculations suggest that for Cu2+ in beryl, Ax and Ay are positive and Az is negative (see Table 1). The defect structural data (e.g., Ri and yi) obtained from the above calculations for Cu2+ center in beryl are indeed different from the corresponding structural data in host crystals. It appears that by studying the EPR data (i.e., the SH parameters), some information on the defect structure of paramagnetic impurity center in crystals can be acquired.
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It will be seen from Table 1 by using the same parameters, the calculated SH parameters for Cu2+ in beryl from both PTM and CDM are in reasonable agreement with the experimental values. So, the two theoretical methods can be used to explain the SH parameters and to obtain the defect structure of 3dn impurity centers in crystals by analyzing their SH parameters. References [1] C.E. McManus, N.J. McMillan, R.S. Harmon, R.C. Whitmore, F.C. De Lucia, A.W. Miziolek, Appl. Opt. 47 (2008) G72. [2] L.A. Groat, G. Giuliani, D.D. Marshall, D. Turner, Ore Geol. Rev. 34 (2008) 87. [3] K. Krambrock, K.J. Guedes, M.V. B Pinheiro, Phys. Chem. Minerals 35 (2008) 409. [4] M. Dvir, W. Low, Phys. Rev. 119 (1960) 1587. [5] D.R. Hutton, F.A. Darmann, G.j. Troup, Austral. J. Phys. 44 (1991) 429. [6] V.P. Solntsev, A.M. Yurkin, Cryst. Rep. 45 (2000) 128. [7] J.M. Gaite, V.V. Izotov, S.I. Nikitin, S.Y. Prosvirhin, Appl. Mag. Reson. 20 (2001) 307. [8] V.P. Solncev, A.S. Lebeder, V.S. Pavluchenko, V.A. Kliakhin, Soviet Phys. Solid State 18 (1976) 1396. [9] J.T. Vallin, G.D. Watkins, Phys. Rev. 89 (1974) 2051. [10] W.C. Zheng, S.Y. Wu, W. Li, Physica B 253 (1998) 798. [11] J.S. Griffith, The Theory of Transition-Metal Ions, Cambridge University Press, London, 1964. [12] M.L. Du, C. Rudowicz, Phys. Rev. B 46 (1992) 8974. [13] B.R. McGarvey, J. Phys. Chem. 71 (1967) 51. [14] D.J. Newman, B. Ng, Rep. Progr. Phys. 52 (1989) 699. [15] C. Rudowicz, Z.Y. Yang, Y.Y. Yeung, J. Qin, J. Phys. Chem. Solids 64 (2003) 1419. [16] W.L. Yu, J. Phys. Condens. Matter 6 (1994) 5105. [17] Y. Mei, X.X. Wu, W.C. Zheng, Radiat. Eff. Def. Solids 163 (2008) 79. [18] T.H. Yeom, S.H. Choh, M.L. Du, M.S. Tang, Phys. Rev. B 53 (1996) 3415. [19] H.G. Liu, W.C. Zheng, L. He, Radiat. Eff. Def. Solids 163 (2008) 1. [20] C. Rudowicz, Y.Z. Zhou, J. Magn. Magn. Matter 11 (1992) 153. [21] B. Morosin, Acta Crystallogr. B 28 (1972) 1899. [22] W.C. Zheng, Physica B 215 (1995) 255. [23] R.C. Weast, CRC Handbook of Chemistry and Physics, CRC Press, Roca Raton, 1989, p.F.-187. [24] D. Hernandez, F. Rodriguez, M. Moreno, H.V. Gudel, Physica B 265 (1999) 196. [25] M. Moreno, M.T. Barriuxso, J.A. Arambura, J. Phys. Condens. Matter 4 (1992) 9481. [26] I. Boser, V. Scherz, M. Wohlecke, J. Lumin. 182 (1970) 39. [27] E.H. Kisi, M.M. Elcombe, Acta Cryst. C 45 (1989) 1867. [28] R.E. Watson, Phys. Rev. 119 (1960) 1934. [29] C.J. Ballhausen, H.B. Gray, Inorg. Chem. 1 (1962) 111. [30] A.S. Chakravary, Introduction to the Magnetic Properties of Solids, Wiley, New York, 1980. [31] A. Abragam, M.H.L. Pryce, Proc. Roy. Soc. (London) A 206 (1951) 173. [32] M.A. Hitchman, R.G. McDonald, D. Reinen, Inorg. Chem. 25 (1986) 519. [33] S.O. Graham, R.L. White, Phys. Rev. B 10 (1974) 4505. [34] X.X. Wu, W.C. Zheng, S. Tang, Z. Naturforsch. A 59 (2004) 47. [35] A. Abragam, B. Bleaney, Electron Paramagnetic Resonance of Transition Ions, Oxford University Press, London, 1970. [36] V. Havliek, P. Novak, B.V. Mill, Phys. Status Solidi B 64 (1974) k19.