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Investigations of EPR parameters and defect models for three Yb3+ impurity centers in ThO2 and CeO2 crystals
ARTICLE IN PRESS Physica B 405 (2010) 2326–2328
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Investigations of EPR parameters and defect models for three Yb3 + impurity centers in ThO2 and CeO2 crystals Hong-Gang Liu a,, Wen-Chen Zheng a,c, Wei-Qing Yang a,b a
Department of Material Science, Sichuan University, Chengdu 610064, PR China Department of Optics and Electronics, Chengdu University of Information Technology, Chengdu 610225, PR China c International Center for Materials Physics, Chinese Academy of Sciences, Shenyang 110016, PR China b
a r t i c l e in f o
a b s t r a c t
Article history: Received 14 December 2009 Accepted 16 February 2010
The electron paramagnetic resonance (EPR) parameters (g factors g//, g? and hyperfine structure constants 171A//, 171A?, 173A// and 173A?) for three Yb3 + centers, one cubic (center I) and two trigonal (centers II and III), in ThO2 and CeO2 crystals are studied from a complete diagonalization (of energy matrix) method. In the method, the Zeeman and hyperfine interaction terms are added to the classical Hamiltonian and the crystal-field parameters are calculated from the empirical superposition model. From the studies, the EPR parameters for these Yb3 + centers in ThO2 and CeO2 crystals are reasonably explained, the defect model for the trigonal Yb3 + center II suggested in the previous paper is confirmed and the defect structure of center II are also obtained. The theoretical hyperfine structure constants A of two Yb isotopes for centers II and III are suggested and remain to be checked by further experiments. The results are discussed. & 2010 Elsevier B.V. All rights reserved.
Keywords: Electron paramagnetic resonance Crystal-field theory Defect structure Yb3 + ThO2 and CeO2 crystals
1. Introduction Fluoride-type oxides ThO2 and CeO2 are interesting host materials for a number of measurements of rare earth ions. Many studies of optical and electron paramagnetic resonance (EPR) spectra for rare earth doped ThO2 and CeO2 crystals were made because of their possible applications in luminescence and fluorescence materials [1–4]. The EPR spectra of Yb3 + in ThO2 and CeO2 crystals were measured [5–7]. It is found from the measurements that there are three Yb3 + centers, one cubic (center I) and two trigonal (centers II and III) centers, in both the crystals [5–7]. Yb3 + ions in ThO2 and CeO2 crystals occupy the cubic tetravalent cation sites. Thus, the cubic or trigonal Yb3 + centers are dependent upon the proximity of the necessary compensating charge. In center I, the charge compensator is far from the Yb3 + ion and in center II, a positive charge located at the center of the adjacent oxygen cube along [1 1 1] axis was suggested in Refs. [5,6]. From the electron–nuclear double resonance (ENDOR) experiment, Baker et al. [7] attributed center III to a substitutional F ion in the (YbO8)13 cube to form the (YbO7F)12 clusters in ThO2 and CeO2 crystals. Although the EPR (g factors) for the three Yb3 + centers in the two crystals were reported decades ago [5–7], no satisfactory theoretical calculations based on the above defect models for these EPR g factors
have been made. In this paper, we calculate these g factors and also the hyperfine structure constants A of 171Yb3 + and 173Yb3 + isotopes in ThO2 and CeO2 crystals from a complete diagonalization (of energy matrix) method. In this method, the Zeeman and hyperfine interaction terms are added to the classical Hamiltonian and the crystal-field parameters used in calculations are obtained from the empirical superposition model. The results (including the checking of the defect model for center II) are discussed.
2. Calculation A Yb3 + ion has the electron configuration 4f13 with a ground state 2F7/2 and an excited state 2F5/2. According to the group theory, under a cubic crystal field (e.g., in center I), the 2F7/2 state is split into three components (G8 + G6 + G7) and the 2F5/2 state is split into two sublevels (G8 + G7) [8]. When the cubic site symmetry descends to a lower one, e.g., in trigonal center II or III, the G8 multiplet is reduced to two doublets. Thus, the 2F7/2 and 2 F5/2 states under trigonal crystal field are split into four and three Kramers doublets, respectively [9]. The complete Hamiltonian for Yb3 + ion in a crystal field and under an external magnetic field can be written as H ¼ Hf þ HSO þ HCF þ HZe þHhf ;
Corresponding author. Tel.: + 86 28 85412371; fax: + 86 28 85416050.
E-mail address: [email protected] (H.-G. Liu). 0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.02.037
ð1Þ
where Hf is free ion term, HSO[ = z(L S)] is the spin–orbit interaction term including the spin–orbit coupling parameter z
ARTICLE IN PRESS H.-G. Liu et al. / Physica B 405 (2010) 2326–2328
2327
and HCF is the crystal-field interaction term. The Zeeman and hyperfine interaction terms HZe and Hhf can be written as [9,10]
the experimental values, we can obtain that for center I of ThO2:Yb3 +
HZe ¼ gJ mB JHm ;
A 4 ðR0 Þ 71 cm1 ;
Hhf ¼ PðNIÞ ¼ PNJ JI
ð2Þ
where Hm is the external magnetic field, P is the dipolar hyperfine structure constant, NJ is the diagonal matrix element for 2S + 1LJ state, N is the equivalent operator of magnetic hyperfine structure and the other symbols are standard [9,10]. For Yb3 + ion, P(171Yb)E392 10 4 cm 1 and P(173Yb)E 108 10 4 cm 1. They are obtained from the formula P ¼ gs gN bbN /r 3 S[11] with /r 3SE12.50 a.u. and gN¼m/I, where m E0.49188, I= 1/2 for 171 Yb and m E 0.67755 I=5/2 for 173Yb [9]. It is noted that for the matrix elements between different J-manifold, the gJ and NJ in Eq. (2) should be replaced by gJ0 and NJ0 , respectively [9,10]. By means of the irreducible tensor operator and/or equivalent operator methods, we obtain the 14 14 energy matrix corresponding to the Hamiltonian shown in Eq. (1). It should be pointed out that the dimension D for hyperfine interaction matrix of Yb3 + ion is exactly dependent on nuclear spin I of different isotopes (e.g., for I= 1/2 (171Yb), D=14 2=28 or for I =5/2(173Yb), D= 14 6= 84) [12]. Thus, the 14 14 hyperfine interaction matrices are obtained by equivalent operator methods. Diagonalizing the energy matrix, the EPR parameters can be calculated from the following formulas
DEZe ðiÞ
gi ¼
mB Hm ðiÞ
;
Ai ¼ DEhf ðiÞ;
ð3Þ
where DEZe(i) represents the Zeeman splitting under the external magnetic field Hm along i axis and DEhf(i) stands for the hyperfine splitting in the case of the equivalent operator N of magnetic hyperfine structure along i axis. If we choose the threefold rotation axis ([1 1 1] direction) as the z-axis, the crystal-field interaction HCF for 4fn ions in cubic and trigonal site symmetries can be expressed as [13] " HCF ðcubicÞ ¼ B40 " þB60
# rffiffiffiffiffiffi 10 4 C04 ðC3 C34 Þ 7
# pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 210 6 231 6 6 6 ðC3 C3 Þ þ ðC6 þC6 Þ 24 24
C06
ð4Þ
and for center I of CeO2:Yb A 4 ðR0 Þ 87 cm1 ;
6 6 þB60 C06 þB63 ðC3 C36 Þ þB66 ðC66 þ C6 Þ
448 A 4 ðR0 Þ; 27
B60 ¼
4096 A 6 ðR0 Þ 81
ð7Þ
A 6 ðR0 Þ 28 cm1 ;
k 0:972
ð8Þ
in which the power-law exponents t2 E5, t4 E6 and t6 E10 [18,19]. A 4 ðR0 Þ, A 6 ðR0 Þ and k are the same as those in center I (i.e., those in Eqs. (7)–(8)) because the reference distance R0 is not changed. A 2 ðR0 Þ and DZ, which are treated as adjustable parameters can be obtained by fitting the calculated g factors to the experimental values. Using the above diagonalization (of energy matrix) method, we find that for center II of ThO2:Yb3 + ˚ DZ 0:205 A
ð10Þ
and for center II of CeO2:Yb3 + ð5Þ
in which Bkq are the crystal-field parameters and Cqk are the Racah spheric tensor operators. The superposition model [14,15], which is related to the local structure data is often used to calculate the crystal-field parameters Bkq . For center I of Yb3 + in ThO2 and CeO2 crystals, the crystal-field parameters can be written as B40 ¼
k 0:976
3+
The comparisons of g and A factors between calculation and experiment are shown in Tables 1 and 2. For the trigonal center II in ThO2/CeO2:Yb3 + , the trigonal distortion is caused by the interstitial charge q located at the center of the adjacent oxygen cube along [1 1 1] axis. If the charge q is positive, the O2 ion intervening in the Yb3 + and charge q should be displaced towards q by DZ (the positive direction of displacement is defined from O2 to q) due to the electrostatic interaction between the intervening O2 and charge q, whereas if the charge q is negative, the displacement direction of the intervening O2 should be opposite. Thus, according to superposition model, the crystal-field parameters for the trigonal Yb3 + centers in ThO2 and CeO2 crystals can be given as t2 R0 ; B20 ¼ 2A 2 ðR0 Þ þ 2A 2 ðR0 Þ R0 þ DZ t4 232 R0 B40 ¼ A 4 ðR0 Þ þ 8A 4 ðR0 Þ ; R0 þ DZ 27 64 pffiffiffiffiffiffi B43 ¼ 70A 4 ðR0 Þ; 27 t6 2800 R0 A 6 ðR0 Þ þ 16A 6 ðR0 Þ ; B60 ¼ R0 þ DZ 81 p ffiffiffiffiffiffiffiffi ffi 512 B63 ¼ 210A 6 ðR0 Þ; 243 p ffiffiffiffiffiffiffiffiffi 512 B66 ¼ ð9Þ 231A 6 ðR0 Þ 243
A 2 ðR0 Þ 751 cm1 ; 4 HCF ðtrigonalÞ ¼ B20 C02 þB40 C04 þ B43 ðC3 C34 Þ
A 6 ðR0 Þ 20 cm1 ;
ð6Þ
in which A k ðR0 Þ (k =4,6) are the intrinsic parameters with the reference distance R0. Here we take R0 E2.425 A˚ [7] for ThO2:Yb3 + and R0 E2.343 A˚ [7] for CeO2:Yb3 + . R0 is the metal–ligand distance in the host crystal. In the calculations, the spin–orbit coupling parameters z of Yb3 + ions can be taken as 2950 cm 1 [9]. But considering the covalency reduction effect for Yb3 + ions in crystals, an orbit reduction factor k [16,17] has to be introduced and the spin–orbit coupling parameter z should be multiplied by it (note: k is an adjustable parameter here, if the covalency effect is neglected, k=1). Then, by fitting the calculated g and A factors to
A 2 ðR0 Þ 892 cm1 ;
˚ DZ 0:214 A
ð11Þ
The comparisons of g factors between calculation and experiment are shown in Tables 1 and 2. The calculated A factors of two Yb isotopes are also given in Tables 1 and 2. For trigonal center III in ThO2:Yb3 + and CeO2:Yb3 + , an O2 ion at the corner of the oxygen cube is replaced by an F ion. For simplicity, we suppose that there is no local structural distortion after the substitution. Thus, by the superposition model, the crystal-field parameters for this trigonal Yb3 + center in ThO2 and Table 1 The EPR parameters (g factors g//, g? and hyperfine structure constants 171A and 173 A, A are in units of 10 4 cm 1) for three Yb3 + centers in ThO2 crystal.
Center I Center II Center III
Calc. Expt. [5] Calc. Expt. [5] Calc. Expt. [7]
171
g//
g?
3.423 3.423(1) 4.773 4.772(2) 4.493 4.495(2)
877 877(1) 2.724 1245 2.724(1) – 2.873 1169 2.872(1) –
A//
171
A?
691 – 731 –
173
A//
173
A?
242 241.8(3) 343 190 – – 322 201 – –
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H.-G. Liu et al. / Physica B 405 (2010) 2326–2328
14.90 cm 1, respectively. The apparent reason is that the Yb3 + –F distance (E2.3478 A˚ [25]) in BaF2 crystal is close to ˚ in CeO2 crystal. Thus, our the reference distant R0 ( E2.343 A) results for the trigonal center III in Eqs. (13) and (14) can be regarded as rational.
Table 2 The EPR parameters (g factors g//, g? and hyperfine structure constants 171A and 173 A, A are in units of 10 4cm 1) for three Yb3 + centers in CeO2 crystal. g// Center I Center II Center III
Calc. Expt. [6] Calc. Expt. [6] Calc. Expt. [7]
g?
3.425 3.424(1) 4.735 4.733(4) 4.148 4.152(2)
2.749 2.744(2) 3.057 3.060(2)
171
A//
171
A?
870 877.1(1) 1230 690 – – 1069 772 – –
173
A//
173
A?
240 242.0(3) 339 190 – – 295 213 – –
CeO2 crystals can be given as 232 B20 ¼ 2A 2 ðOR0 Þ þ 2A 2 ðFR0 Þ; B40 ¼ A 4 ðOR0 Þ þ 8A 4 ðFR0 Þ; 27 pffiffiffiffiffiffi 64 70 2800 A 4 ðOR0 Þ; B60 ¼ A 6 ðOR Þ þ 16A 6 ðFR0 Þ; B43 ¼ 27 81 pffiffiffiffiffiffiffiffiffi 0 pffiffiffiffiffiffiffiffiffi 512 210 512 231 A 6 ðOR0 Þ; B66 ¼ A 6 ðOR0 Þ; B63 ¼ 243 243
ð12Þ
where A k ðOR0 Þ and A k ðFR0 Þ are the intrinsic parameters for O2 and F ligands. R0 is the reference distance and is still taken as R0 E2.425 A˚ for the ThO2 crystal and R0 E2.343 A˚ for the CeO2 crystal. Thus, A 4 ðOR0 Þ and A 6 ðOR0 Þ are the same to the corresponding values in Eqs. (7)–(8) and A 2 ðOR0 Þ should also be identical with those in Eqs. (10)–(11). Thus, the adjustable parameters are A 2 ðFR0 Þ, A 4 ðFR0 Þ and A 6 ðFR0 Þ. They can be estimated by fitting the g factors to the experimental values. By means of the diagonalization (of energy matrix) method, we find that for center III of ThO2:Yb3 + A 2 ðFR0 Þ 552 cm1 ;
A 4 ðFR0 Þ 48 cm1 ;
ð13Þ and for center III of CeO2:Yb
A 4 ðFR0 Þ 59 cm1 ;
References
A 6 ðFR0 Þ 11 cm1
3+
A 2 ðFR0 Þ 717 cm1 ;
The smaller orbit reduction factor k for CeO2:Yb3 + than that for ThO2:Yb3 + shows that the covalence in CeO2 is somewhat greater than in ThO2 because of the smaller metal–ligand distance R0, which conforms to the results in Ref. [7]. The calculations show that for the trigonal center II in CeO2 and ThO2 crystals, both displacements DZ 40, suggesting that the interstitial charge q in center II is positive. Hence, the defect models for the center II in ThO2:Yb3 + and CeO2:Yb3 + suggested in Refs. [5,6] are confirmed and the defect structures (i.e., the displacement of the intervening O2 ion) of the trigonal center II for the two host crystals are obtained. In Tables 1 and 2, the calculated g factors, g// and g? for three Yb3 + impurity centers and hyperfine structure constants 171A and 173 A for center I are in good agreement with the experimental values (note that the difference in sign between 171A and 173A for center I is due to the different signs of constant P for two Yb isotopes [26]). Thus, the EPR parameters of the three Yb3 + centers in ThO2 and CeO2 crystals are explained reasonably. It is also worthwhile to point out that the calculated hyperfine structure constants 171A and 173A for centers II and III remain to be checked by further experiments, since there were no experimental values reported.
A 6 ðFR0 Þ 15 cm1 : ð14Þ
The comparisons of g factors between calculation and experiment can be found in Tables 1 and 2 and the calculated A factors of two Yb isotopes are also shown in Tables 1 and 2.
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
3. Discussion In Section 2, we obtain the intrinsic parameters A k ðR0 Þ (k= 2, 4, 6) for O2 ligand and F ligand in ThO2:Yb3 + and CeO2:Yb3 + crystals. For these parameters, we can find that:
[11] [12] [13]
[14] [15]
3+
(i) The parameter A 2 ðR0 Þ 4 A 4 ðR0 Þ 4A 6 ðR0 Þ for both ThO2:Yb and CeO2:Yb3 + crystals is consistent with an empirical rule obtained from other superposition model analyses for the optical and EPR spectra of 4fn ions in crystals [20–23]. Hence, these parameters A k ðR0 Þobtained in the present paper are reasonable. (ii) Eqs. (7) and (8), Eqs. (10)–(14) indicate that each A k ðR0 Þ of CeO2:Yb3 + is greater than that of ThO2:Yb3 + . The reason is that crystal-field strength and hence the parameters A k ðR0 Þ increase with a decrease in reference distant R0 [24]. (iii) The values of A 4 ðFR0 Þ ( E59 cm 1) and A 6 ðFR0 Þ ( E15 cm 1) for CeO2:Yb3 + in Eq. (14) are very close to those for BaF2:Yb3 + in Ref. [25] where A 4 ðFR0 Þ and A 6 ðFR0 Þ are 58.68 and
[16] [17] [18] [19] [20] [21] [22] [23] [24]
[25] [26]
W. Kolbe, N. Edelstein, C.B. Finch, M.M. Abraham, Chem. Phys. 60 (1974) 607. S. Hubert, P. Thouvenot, J. Alloys Compd. 180 (1992) 193. S. Hubert, P. Thouvenot, N. Edelstein, Phys. Rev. B 48 (1993) 5751. M. Yin, J.C. Krupa, E. Antic-Fidancev, A. Lorriaux-Rubbens, Phys. Rev. B 61 (2000) 8073. M. Abraham, R.A. Weeks, G.W. Clark, C.B. Finch, Phys. Rev. 137 (1965) A138. M. Abraham, R.A. Weeks, G.W. Clark, C.B. Finch, Phys. Rev. 148 (1966) 350. J.M. Baker, T. Rs. Reddy, R.L. Wood, J. Phys. C. Solid State Phys. 12 (1979) 3317. V. Petit, P. Camy, J.-L. Doualan, X. Portier, R. Moncorge´, Phys. Rev. B 78 (2008) 085131. A. Abragam, B. Bleaney, Electron Paramagnetic Resonance of Transition Ions, Oxford University Press, London, 1970. L.A. Sorin, M.V. Vlasova, Electron Spin Resonance of Paramagnetic Crystals, Plenum Press, New York, 1973 (translated from Russian by P. Gluck). B.K. McGarvry, J. Phys. Chem. 71 (1967) 51. H.G. Liu, W.C. Zheng, W.L. Feng, J. Lumin. 130 (2010) 103. ¨ C. Gorller-Walrand, K. Binnemans, in: K.A. Gschneidner Jr, L. Eyring (Eds.), Handbook of the Physics and Chemistry of Rare Earths, vol. 23, Elsevier, Amsterdam, 1996. D.J. Newman, B. Ng, Rep. Prog. Phys. 52 (1989) 699. D.J. Newman, B. Ng, Crystal Field Handbook, Cambridge University Press, Cambridge, 2000. J.D. Axe, G. Burns, Phys. Rev. 152 (1966) 331. K.I. Gerasimov, A.M. Leushin, M.L. Falin, Phys. Solid State 43 (2001) 1675. N. Magnani, G. Amoretti, A. Baraldi, R. Capelletti, Radiat. Eff. Defects Solids 157 (2002) 921. N. Magnani, G. Amoretti, A. Baraldi, R. Capelletti, Eur. Phys. J. B 29 (2002) 79. D.J. Newman, G.E. Stedman, J. Chem. Phys. 51 (1969) 3013. D.J. Newman, Aust. J. Phys. 30 (1977) 315. C. Linares, A. Louat, J. Phys. Paris 36 (1975) 717. N. Magnani, A. Baraldi, E. Buffagni, R. Capelletti, M. Mazzera, S. Brovelli, A. Lauria, Phys. Status Solidi (c) 4 (2007) 1209. ¨ T. Troster, in: K.A. Gschneidner, J.-C.G. Bunzli, V.K. Pecharsky (Eds.), Handbook of the Physics and Chemistry of Rare Earths, vol. 33, Elsevier, Amsterdam, 2003. M.L. Falin, K.I. Gerasimov, V.A. Latypov, A.M. Leushin, J. Phys.: Condens. Matter 15 (2003) 2833. H.G. Liu, W.C. Zheng, W.L. Feng, Philos. Mag. 88 (2008) 3075.
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Physica B journal homepage: www.elsevier.com/locate/physb
Investigations of EPR parameters and defect models for three Yb3 + impurity centers in ThO2 and CeO2 crystals Hong-Gang Liu a,, Wen-Chen Zheng a,c, Wei-Qing Yang a,b a
Department of Material Science, Sichuan University, Chengdu 610064, PR China Department of Optics and Electronics, Chengdu University of Information Technology, Chengdu 610225, PR China c International Center for Materials Physics, Chinese Academy of Sciences, Shenyang 110016, PR China b
a r t i c l e in f o
a b s t r a c t
Article history: Received 14 December 2009 Accepted 16 February 2010
The electron paramagnetic resonance (EPR) parameters (g factors g//, g? and hyperfine structure constants 171A//, 171A?, 173A// and 173A?) for three Yb3 + centers, one cubic (center I) and two trigonal (centers II and III), in ThO2 and CeO2 crystals are studied from a complete diagonalization (of energy matrix) method. In the method, the Zeeman and hyperfine interaction terms are added to the classical Hamiltonian and the crystal-field parameters are calculated from the empirical superposition model. From the studies, the EPR parameters for these Yb3 + centers in ThO2 and CeO2 crystals are reasonably explained, the defect model for the trigonal Yb3 + center II suggested in the previous paper is confirmed and the defect structure of center II are also obtained. The theoretical hyperfine structure constants A of two Yb isotopes for centers II and III are suggested and remain to be checked by further experiments. The results are discussed. & 2010 Elsevier B.V. All rights reserved.
Keywords: Electron paramagnetic resonance Crystal-field theory Defect structure Yb3 + ThO2 and CeO2 crystals
1. Introduction Fluoride-type oxides ThO2 and CeO2 are interesting host materials for a number of measurements of rare earth ions. Many studies of optical and electron paramagnetic resonance (EPR) spectra for rare earth doped ThO2 and CeO2 crystals were made because of their possible applications in luminescence and fluorescence materials [1–4]. The EPR spectra of Yb3 + in ThO2 and CeO2 crystals were measured [5–7]. It is found from the measurements that there are three Yb3 + centers, one cubic (center I) and two trigonal (centers II and III) centers, in both the crystals [5–7]. Yb3 + ions in ThO2 and CeO2 crystals occupy the cubic tetravalent cation sites. Thus, the cubic or trigonal Yb3 + centers are dependent upon the proximity of the necessary compensating charge. In center I, the charge compensator is far from the Yb3 + ion and in center II, a positive charge located at the center of the adjacent oxygen cube along [1 1 1] axis was suggested in Refs. [5,6]. From the electron–nuclear double resonance (ENDOR) experiment, Baker et al. [7] attributed center III to a substitutional F ion in the (YbO8)13 cube to form the (YbO7F)12 clusters in ThO2 and CeO2 crystals. Although the EPR (g factors) for the three Yb3 + centers in the two crystals were reported decades ago [5–7], no satisfactory theoretical calculations based on the above defect models for these EPR g factors
have been made. In this paper, we calculate these g factors and also the hyperfine structure constants A of 171Yb3 + and 173Yb3 + isotopes in ThO2 and CeO2 crystals from a complete diagonalization (of energy matrix) method. In this method, the Zeeman and hyperfine interaction terms are added to the classical Hamiltonian and the crystal-field parameters used in calculations are obtained from the empirical superposition model. The results (including the checking of the defect model for center II) are discussed.
2. Calculation A Yb3 + ion has the electron configuration 4f13 with a ground state 2F7/2 and an excited state 2F5/2. According to the group theory, under a cubic crystal field (e.g., in center I), the 2F7/2 state is split into three components (G8 + G6 + G7) and the 2F5/2 state is split into two sublevels (G8 + G7) [8]. When the cubic site symmetry descends to a lower one, e.g., in trigonal center II or III, the G8 multiplet is reduced to two doublets. Thus, the 2F7/2 and 2 F5/2 states under trigonal crystal field are split into four and three Kramers doublets, respectively [9]. The complete Hamiltonian for Yb3 + ion in a crystal field and under an external magnetic field can be written as H ¼ Hf þ HSO þ HCF þ HZe þHhf ;
Corresponding author. Tel.: + 86 28 85412371; fax: + 86 28 85416050.
E-mail address: [email protected] (H.-G. Liu). 0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.02.037
ð1Þ
where Hf is free ion term, HSO[ = z(L S)] is the spin–orbit interaction term including the spin–orbit coupling parameter z
ARTICLE IN PRESS H.-G. Liu et al. / Physica B 405 (2010) 2326–2328
2327
and HCF is the crystal-field interaction term. The Zeeman and hyperfine interaction terms HZe and Hhf can be written as [9,10]
the experimental values, we can obtain that for center I of ThO2:Yb3 +
HZe ¼ gJ mB JHm ;
A 4 ðR0 Þ 71 cm1 ;
Hhf ¼ PðNIÞ ¼ PNJ JI
ð2Þ
where Hm is the external magnetic field, P is the dipolar hyperfine structure constant, NJ is the diagonal matrix element for 2S + 1LJ state, N is the equivalent operator of magnetic hyperfine structure and the other symbols are standard [9,10]. For Yb3 + ion, P(171Yb)E392 10 4 cm 1 and P(173Yb)E 108 10 4 cm 1. They are obtained from the formula P ¼ gs gN bbN /r 3 S[11] with /r 3SE12.50 a.u. and gN¼m/I, where m E0.49188, I= 1/2 for 171 Yb and m E 0.67755 I=5/2 for 173Yb [9]. It is noted that for the matrix elements between different J-manifold, the gJ and NJ in Eq. (2) should be replaced by gJ0 and NJ0 , respectively [9,10]. By means of the irreducible tensor operator and/or equivalent operator methods, we obtain the 14 14 energy matrix corresponding to the Hamiltonian shown in Eq. (1). It should be pointed out that the dimension D for hyperfine interaction matrix of Yb3 + ion is exactly dependent on nuclear spin I of different isotopes (e.g., for I= 1/2 (171Yb), D=14 2=28 or for I =5/2(173Yb), D= 14 6= 84) [12]. Thus, the 14 14 hyperfine interaction matrices are obtained by equivalent operator methods. Diagonalizing the energy matrix, the EPR parameters can be calculated from the following formulas
DEZe ðiÞ
gi ¼
mB Hm ðiÞ
;
Ai ¼ DEhf ðiÞ;
ð3Þ
where DEZe(i) represents the Zeeman splitting under the external magnetic field Hm along i axis and DEhf(i) stands for the hyperfine splitting in the case of the equivalent operator N of magnetic hyperfine structure along i axis. If we choose the threefold rotation axis ([1 1 1] direction) as the z-axis, the crystal-field interaction HCF for 4fn ions in cubic and trigonal site symmetries can be expressed as [13] " HCF ðcubicÞ ¼ B40 " þB60
# rffiffiffiffiffiffi 10 4 C04 ðC3 C34 Þ 7
# pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 210 6 231 6 6 6 ðC3 C3 Þ þ ðC6 þC6 Þ 24 24
C06
ð4Þ
and for center I of CeO2:Yb A 4 ðR0 Þ 87 cm1 ;
6 6 þB60 C06 þB63 ðC3 C36 Þ þB66 ðC66 þ C6 Þ
448 A 4 ðR0 Þ; 27
B60 ¼
4096 A 6 ðR0 Þ 81
ð7Þ
A 6 ðR0 Þ 28 cm1 ;
k 0:972
ð8Þ
in which the power-law exponents t2 E5, t4 E6 and t6 E10 [18,19]. A 4 ðR0 Þ, A 6 ðR0 Þ and k are the same as those in center I (i.e., those in Eqs. (7)–(8)) because the reference distance R0 is not changed. A 2 ðR0 Þ and DZ, which are treated as adjustable parameters can be obtained by fitting the calculated g factors to the experimental values. Using the above diagonalization (of energy matrix) method, we find that for center II of ThO2:Yb3 + ˚ DZ 0:205 A
ð10Þ
and for center II of CeO2:Yb3 + ð5Þ
in which Bkq are the crystal-field parameters and Cqk are the Racah spheric tensor operators. The superposition model [14,15], which is related to the local structure data is often used to calculate the crystal-field parameters Bkq . For center I of Yb3 + in ThO2 and CeO2 crystals, the crystal-field parameters can be written as B40 ¼
k 0:976
3+
The comparisons of g and A factors between calculation and experiment are shown in Tables 1 and 2. For the trigonal center II in ThO2/CeO2:Yb3 + , the trigonal distortion is caused by the interstitial charge q located at the center of the adjacent oxygen cube along [1 1 1] axis. If the charge q is positive, the O2 ion intervening in the Yb3 + and charge q should be displaced towards q by DZ (the positive direction of displacement is defined from O2 to q) due to the electrostatic interaction between the intervening O2 and charge q, whereas if the charge q is negative, the displacement direction of the intervening O2 should be opposite. Thus, according to superposition model, the crystal-field parameters for the trigonal Yb3 + centers in ThO2 and CeO2 crystals can be given as t2 R0 ; B20 ¼ 2A 2 ðR0 Þ þ 2A 2 ðR0 Þ R0 þ DZ t4 232 R0 B40 ¼ A 4 ðR0 Þ þ 8A 4 ðR0 Þ ; R0 þ DZ 27 64 pffiffiffiffiffiffi B43 ¼ 70A 4 ðR0 Þ; 27 t6 2800 R0 A 6 ðR0 Þ þ 16A 6 ðR0 Þ ; B60 ¼ R0 þ DZ 81 p ffiffiffiffiffiffiffiffi ffi 512 B63 ¼ 210A 6 ðR0 Þ; 243 p ffiffiffiffiffiffiffiffiffi 512 B66 ¼ ð9Þ 231A 6 ðR0 Þ 243
A 2 ðR0 Þ 751 cm1 ; 4 HCF ðtrigonalÞ ¼ B20 C02 þB40 C04 þ B43 ðC3 C34 Þ
A 6 ðR0 Þ 20 cm1 ;
ð6Þ
in which A k ðR0 Þ (k =4,6) are the intrinsic parameters with the reference distance R0. Here we take R0 E2.425 A˚ [7] for ThO2:Yb3 + and R0 E2.343 A˚ [7] for CeO2:Yb3 + . R0 is the metal–ligand distance in the host crystal. In the calculations, the spin–orbit coupling parameters z of Yb3 + ions can be taken as 2950 cm 1 [9]. But considering the covalency reduction effect for Yb3 + ions in crystals, an orbit reduction factor k [16,17] has to be introduced and the spin–orbit coupling parameter z should be multiplied by it (note: k is an adjustable parameter here, if the covalency effect is neglected, k=1). Then, by fitting the calculated g and A factors to
A 2 ðR0 Þ 892 cm1 ;
˚ DZ 0:214 A
ð11Þ
The comparisons of g factors between calculation and experiment are shown in Tables 1 and 2. The calculated A factors of two Yb isotopes are also given in Tables 1 and 2. For trigonal center III in ThO2:Yb3 + and CeO2:Yb3 + , an O2 ion at the corner of the oxygen cube is replaced by an F ion. For simplicity, we suppose that there is no local structural distortion after the substitution. Thus, by the superposition model, the crystal-field parameters for this trigonal Yb3 + center in ThO2 and Table 1 The EPR parameters (g factors g//, g? and hyperfine structure constants 171A and 173 A, A are in units of 10 4 cm 1) for three Yb3 + centers in ThO2 crystal.
Center I Center II Center III
Calc. Expt. [5] Calc. Expt. [5] Calc. Expt. [7]
171
g//
g?
3.423 3.423(1) 4.773 4.772(2) 4.493 4.495(2)
877 877(1) 2.724 1245 2.724(1) – 2.873 1169 2.872(1) –
A//
171
A?
691 – 731 –
173
A//
173
A?
242 241.8(3) 343 190 – – 322 201 – –
ARTICLE IN PRESS 2328
H.-G. Liu et al. / Physica B 405 (2010) 2326–2328
14.90 cm 1, respectively. The apparent reason is that the Yb3 + –F distance (E2.3478 A˚ [25]) in BaF2 crystal is close to ˚ in CeO2 crystal. Thus, our the reference distant R0 ( E2.343 A) results for the trigonal center III in Eqs. (13) and (14) can be regarded as rational.
Table 2 The EPR parameters (g factors g//, g? and hyperfine structure constants 171A and 173 A, A are in units of 10 4cm 1) for three Yb3 + centers in CeO2 crystal. g// Center I Center II Center III
Calc. Expt. [6] Calc. Expt. [6] Calc. Expt. [7]
g?
3.425 3.424(1) 4.735 4.733(4) 4.148 4.152(2)
2.749 2.744(2) 3.057 3.060(2)
171
A//
171
A?
870 877.1(1) 1230 690 – – 1069 772 – –
173
A//
173
A?
240 242.0(3) 339 190 – – 295 213 – –
CeO2 crystals can be given as 232 B20 ¼ 2A 2 ðOR0 Þ þ 2A 2 ðFR0 Þ; B40 ¼ A 4 ðOR0 Þ þ 8A 4 ðFR0 Þ; 27 pffiffiffiffiffiffi 64 70 2800 A 4 ðOR0 Þ; B60 ¼ A 6 ðOR Þ þ 16A 6 ðFR0 Þ; B43 ¼ 27 81 pffiffiffiffiffiffiffiffiffi 0 pffiffiffiffiffiffiffiffiffi 512 210 512 231 A 6 ðOR0 Þ; B66 ¼ A 6 ðOR0 Þ; B63 ¼ 243 243
ð12Þ
where A k ðOR0 Þ and A k ðFR0 Þ are the intrinsic parameters for O2 and F ligands. R0 is the reference distance and is still taken as R0 E2.425 A˚ for the ThO2 crystal and R0 E2.343 A˚ for the CeO2 crystal. Thus, A 4 ðOR0 Þ and A 6 ðOR0 Þ are the same to the corresponding values in Eqs. (7)–(8) and A 2 ðOR0 Þ should also be identical with those in Eqs. (10)–(11). Thus, the adjustable parameters are A 2 ðFR0 Þ, A 4 ðFR0 Þ and A 6 ðFR0 Þ. They can be estimated by fitting the g factors to the experimental values. By means of the diagonalization (of energy matrix) method, we find that for center III of ThO2:Yb3 + A 2 ðFR0 Þ 552 cm1 ;
A 4 ðFR0 Þ 48 cm1 ;
ð13Þ and for center III of CeO2:Yb
A 4 ðFR0 Þ 59 cm1 ;
References
A 6 ðFR0 Þ 11 cm1
3+
A 2 ðFR0 Þ 717 cm1 ;
The smaller orbit reduction factor k for CeO2:Yb3 + than that for ThO2:Yb3 + shows that the covalence in CeO2 is somewhat greater than in ThO2 because of the smaller metal–ligand distance R0, which conforms to the results in Ref. [7]. The calculations show that for the trigonal center II in CeO2 and ThO2 crystals, both displacements DZ 40, suggesting that the interstitial charge q in center II is positive. Hence, the defect models for the center II in ThO2:Yb3 + and CeO2:Yb3 + suggested in Refs. [5,6] are confirmed and the defect structures (i.e., the displacement of the intervening O2 ion) of the trigonal center II for the two host crystals are obtained. In Tables 1 and 2, the calculated g factors, g// and g? for three Yb3 + impurity centers and hyperfine structure constants 171A and 173 A for center I are in good agreement with the experimental values (note that the difference in sign between 171A and 173A for center I is due to the different signs of constant P for two Yb isotopes [26]). Thus, the EPR parameters of the three Yb3 + centers in ThO2 and CeO2 crystals are explained reasonably. It is also worthwhile to point out that the calculated hyperfine structure constants 171A and 173A for centers II and III remain to be checked by further experiments, since there were no experimental values reported.
A 6 ðFR0 Þ 15 cm1 : ð14Þ
The comparisons of g factors between calculation and experiment can be found in Tables 1 and 2 and the calculated A factors of two Yb isotopes are also shown in Tables 1 and 2.
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
3. Discussion In Section 2, we obtain the intrinsic parameters A k ðR0 Þ (k= 2, 4, 6) for O2 ligand and F ligand in ThO2:Yb3 + and CeO2:Yb3 + crystals. For these parameters, we can find that:
[11] [12] [13]
[14] [15]
3+
(i) The parameter A 2 ðR0 Þ 4 A 4 ðR0 Þ 4A 6 ðR0 Þ for both ThO2:Yb and CeO2:Yb3 + crystals is consistent with an empirical rule obtained from other superposition model analyses for the optical and EPR spectra of 4fn ions in crystals [20–23]. Hence, these parameters A k ðR0 Þobtained in the present paper are reasonable. (ii) Eqs. (7) and (8), Eqs. (10)–(14) indicate that each A k ðR0 Þ of CeO2:Yb3 + is greater than that of ThO2:Yb3 + . The reason is that crystal-field strength and hence the parameters A k ðR0 Þ increase with a decrease in reference distant R0 [24]. (iii) The values of A 4 ðFR0 Þ ( E59 cm 1) and A 6 ðFR0 Þ ( E15 cm 1) for CeO2:Yb3 + in Eq. (14) are very close to those for BaF2:Yb3 + in Ref. [25] where A 4 ðFR0 Þ and A 6 ðFR0 Þ are 58.68 and
[16] [17] [18] [19] [20] [21] [22] [23] [24]
[25] [26]
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