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Theoretical studies of spin-Hamiltonian parameters of Mo5+ ion doped in K2SnCl6 crystal
Physica B 429 (2013) 24–27
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Theoretical studies of spin-Hamiltonian parameters of Mo5+ ion doped in K2SnCl6 crystal Wang Fang n, Da-Xiao Yang, Heng-Jie Chen, Hai-Yan Tang Department of Mathematics and Physics, Chongqing University of Science and Technology, Chongqing 401331, People's Republic of China
art ic l e i nf o
a b s t r a c t
Article history: Received 8 July 2013 Received in revised form 25 July 2013 Accepted 29 July 2013 Available online 3 August 2013
The spin-Hamiltonian (SH) parameters (g factors g//, g ? and hyperfine structure constants A//, A ? ) of K2SnCl6: Mo5+ (4d1) crystal are theoretically studied by the use of two microscopic spin-Hamiltonian (SH) methods, the high-order perturbation theory method (PTM) and the complete diagonalization (of energy matrix) method (CDM) within the molecular orbital (MO) scheme. The contributions arising both from the crystal field and charge transfer excitations are taken into account. The investigations show that the charge transfer mechanism plays a decisive role in the understanding of the spin-Hamiltonian (SH) parameters for 4d1 ions in crystals with the strong coordinate covalence, especially for g// 4g ? which cannot be explained in the frame work of traditional crystal field approximation (CFA). The local defect structure around Mo5+ impurity ion center is determined to be D4 h point group symmetry. & 2013 Elsevier B.V. All rights reserved.
Keywords: Charge transfer mechanism Defect structure [MoCl6] 1 cluster Ligand field theory
1. Introduction A2XY6 (A¼K, Cs, Rb, Tl; X¼ tetravalent cation, Y¼ F , Cl , Br , I ) crystals doped with transition metal or rare earth ions have attracted a great deal of attention [1–8] because of the applications in luminescence and, in particular, in upconversion luminescent materials [6–8], and of the merits of being doped easily with different impurities (especially at the 6-fold coordinated tetravalent cation site) [1]. When the pentavalent Mo5+ ion is doped into K2SnCl6 crystal (Fig. 1), it will substitute for the tetravalent Sn4+ ion and a vacancy with negative charge will be introduced along one of the C4 axis at the interstitial site because of the charge compensation effect [9].The experimental EPR results (g factors g//, g ? and hyperfine structure constants A//, A ? ) show that [MoCl6] 1 clusters are in the compressed tetragonal symmetry and the ground state of Mo5+ ion is B2g (dxy) [9,10]. There are two possible defect models for the compressed tetragonal symmetry proposed in the investigation, one is C4v symmetry caused by the repulsion of one Cl with the negative compensating hole along the C4 axis, another is D4h symmetry due to the static Jahn–Teller effect. To find which one is realistic, the calculations have been carried out by the use of those two defect models proposed, and the calculated results show that when D4h defect model is used, the results from both PTM and CDM methods are in good agreement with the experimental dada, while for C4v defect model, no plausible results can be obtained in any case. So, we carry out the calculations by the use D4h defect model in the following
n
Corresponding author. Tel.: +86 23 6502 3914. E-mail address: [email protected] (W. Fang).
0921-4526/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2013.07.030
section. In the calculation, the MO theory in the ligand-field (LF) scheme is used and the contributions from the CT excitation states are taken into account. The investigation show that this CT mechanism plays a decisive role in accounting for the SH parameters of 4dn transition metal (TM) ions with Cl ligands. 2. Calculations The ground state of Mo5+ ion in K2SnCl6 crystal is B2g (dxy) and the expressions of SH parameters g factors and hyperfine structure constants A factors for the single orbital term can be derived by the PTM which have been discussed in detail elsewhere [11–13]. The one-electron basis functions for antibonding orbitals belonging to t2g and eg state for 4dn ions in octahedral field can be expressed as the liner combination of atomic orbitals (LCAO) [14], ψ at ¼ ðN at Þ1=2 ðφt λπ χ π Þ;
ψ ae ¼ ðN ae Þ1=2 ðφe λs χ s λs χ s Þ;
ð1Þ
respectively, and the bonding orbital can be written as [14] ψ bt ¼ ðNbt Þ1=2 ðχ π þ γ π φt Þ; ψ be ¼ ðN be Þ1=2 ðχ s þ γ s φe Þ;
ð2Þ
where φt and φe denote the pure d orbital of the metal ion, χ i (i¼π, s, s) are group orbitals of ligands, ðNbt Þ1=2 , ðN be Þ1=2 and ðN at Þ1=2 , ðNae Þ1=2 are the normalization constants, and λi (i¼π, s, s) are covalency parameters. The third-order perturbation formulas of g factors g//, g ? and second-order hyperfine structure constants A factors A//, A ? based on CF mechanisms for 4d1 ion in tetragonal symmetry
W. Fang et al. / Physica B 429 (2013) 24–27
25
reduction factors k, k′ related to CF mechanisms in Eq. (3) are [14] ζ CF ¼ ðNat Þ1 ½ζ d þ ðλπ Þ2 ζ p =2; 0 ¼ ðN t a N ae Þ1=2 ½ζ d λπ λs ζ p =2; ζ CF
kCF ¼ ðN at Þ1 ½12λt Sπ þ ðλπ Þ2 =2
ð7Þ
0 ¼ ðNat N ae Þ1=2 ½1λs Ss λs Ss λπ Sπ λπ ðλs þ Aλs Þ=2; kCF
where ζd and ζp are the SOC parameters of free 3dn ion and that of ligand ions in free state, respectively. For Mo5+, ζd E1013 cm 1 [20] and for Cl , ζp E587 cm 1 [21], Ss ¼ 〈φe jχ s 〉, Ss ¼ 〈φe jχ s 〉, Sπ ¼ 〈φt jχ π 〉 are the group overlap integrals, which can be calculated using the Clementi's SCF functions [22,23] and impurity-ligand distance R in crystal. Considering that the ionic radius ri (E0.061 nm [24]) of impurity Mo5+ is unlike the radius rh (E0.069 nm [24]) for the replaced host ion Sn4+, the impurity-ligand distance R for Mo5+ ion in K2SnCl6 should be different from the corresponding Rh (E0.2448 nm [25]) in the host K2SnCl6 crystal, we estimate approximately the distance R from the empirical formula R0 Rh þ 1=2ðr i r h Þ [26,27] and obtain R0 ¼ 0.2408 nm. Thus, the group overlap are calculated to be
Fig. 1. One unit cell of K2SnCl6 crystal.
Ss 0:1028; Ss 0:1848; and Sp 0:0746:
are derived as (gs ¼2.0023) g z ¼ g s þ Δg CF z ; 0 0 0 0 8kCF ζ CF 4kCF ζ CF ζ CF kCF ζ 2 ζ 2 g 2 CF CF2 s ; Δg CF z ¼ E1 E1 E2 E2 E2
g x ¼ g s þ Δg CF x ; Δg CF x ¼
0 0 2kCF ζ CF kCF ζ 2CF 2kCF ζ CF ζ CF 2ζ 0 2 g ζ 2 g þ þ CF2 s CF 2 s ; 2 E2 E1 E 2 E2 E1 2E2
CF Az ¼ ACF z ð1Þ þ Az ð2Þ; 4 =Nat ; ACF ð1Þ ¼ P κ z 7 0
ð3Þ
1
Ax ¼
ACF x ð1Þ þ
Δg CT x
e 2
ACF x ð2Þ;
In Eq. (6) R ? represents equatorial metal-ligand distance and equal to R0, R J represents axial metal-ligand distance and taken to be (R0 ΔR) in which ΔR is the inward displacement of Cl along the C4 axis. In Eq. (4), P is dipolar hyperfine structure constants and has the value of 66.7 10 4 cm 1 for Mo5+ [28] and κ is the Femi-contact term which is taken to be 1.05 [9], A ¼ R0 〈φx j∂=∂xjχ s i, for Cl ion 〈φx j∂=∂xjχ s i ¼ 0:409 [29]. The contributions to the SH parameters from the CT levels come from the excitation procedure of promoting an electron from the occupied π bonding orbitals egn (dxz and dyz) or b1gn (dx2 y2) to dxy ground state, and the high-order perturbation formulas of the SH parameters due to this CT mechanism have the follow form 0 0 8kCT ζ CT ; E1C 2kCT ζ CT ¼ ; E2C
Δg CT z ¼
0 6ζ CF B 8ζ CF C ffi A; þ qffiffiffiffiffiffiffiffiffiffiffi ACF z ð2Þ ¼ P @ a Nt E1 7 N a N a E t
0
ð4Þ ACT z
B 11ζ CF C ACF A; x ð2Þ ¼ P @ qffiffiffiffiffiffiffiffiffiffiffiffi 7 Nat Nae E2
B 11γ t ζ CT C ACT A; x ¼ P @ qffiffiffiffiffiffiffiffiffiffiffiffi 7 N at N bt E2c
in which
where ð5Þ
Dq is cubic crystal field parameter and estimated to be 2350 cm 1 [15], Ds and Dt represent the tetragonal crystal field parameter and have the following form using the superposition model " 3 # 4 R0 3 R0 Ds ¼ A2 ðR0 Þ ; 7 R? R== " 5 # 16 R0 5 R0 Dt ¼ A4 ðR0 Þ ; ð6Þ 21 R? R== and A2 ðR0 Þ, A4 ðR0 Þ are intrinsic parameters with reference distance R0, A4 ðR0 Þ E3/4Dq for dn ions in cubic field [16,17]. The ratio A2 ðR0 Þ=A4 ðR0 Þ for mdn ions in clusters decreases with the increasing m value, for 3dn, A2 ðR0 Þ=A4 ðR0 ÞE9–12 are found for many ions in crystals [18,19] andA2 ðR0 Þ=A4 ðR0 Þ ¼ 8:3 is taken here in the calculations. The spin-orbit coupling (SOC) parameters ζ, ζ′ and the orbital
ð9Þ 1
0 6γ t ζ CT C B 8γ e ζ CT ffi ffi A; ¼ P @qffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffi a b Nt Ne E1C 7 N at N bt E2c 0 1
2 =Nat ; ACF x ð1Þ ¼ P κ þ 7 0 1
E1 ¼ 10Dq; E2 ¼ 3Ds þ 5Dt:
ð8Þ
ð10Þ
ζ CT ¼ ðN bt N at Þ1=2 ½γ π ζ d λπ ζ p =2; 0 ¼ ðN bt N ae Þ1=2 ½γ s ζ d þ λπ ζ p =2; ζ CT
ð11Þ
kCT ¼ ðN bt N at Þ1=2 ½γ π þ ð1λπ γ π ÞSπ λπ =2; 0 ¼ ðN bt N ae Þ1=2 ½γ s γ s λπ St Ss þ λt =2; kCT
and E1c, E2c are CT energies of egn (dxz and dyz) and b1gn (dx2 y2), respectively. We estimate E1c ¼41150 and E2c ¼28250 cm 1 [15]. The g factors g//, g ? and hyperfine structure constants A//, A ? from both the CF and CT mechanisms are expressed as CT g == ¼ g z þ g CT z ; g ? ¼ gx þ gx ; CT A== ¼ Az þ ACT z ð2Þ; A ? ¼ Ax þ Ax ð2Þ:
ð12Þ
All the normalization relationship and the relations between the LCAO coefficients and the group overlap used in the calculations can be found in reference [14].
26
W. Fang et al. / Physica B 429 (2013) 24–27
Table1 The spin-Hamiltonian parameters (g factors g//, g ? and hyperfine structure constants A//, A ? are in units of 10–4 cm–1) for Mo5+ in K2SnCl6 crystal.
Calc.a Calc.b Expt.[1]
Δg CF z
ΔgCT z
g==
ΔgCF x
Δg CT x
g?
0.1037
0.0753
1.9739 1.9743 1.97397 0.0006
0.0701
0.0064
1.9386 1.9388 1.9386 7 0.0006
ACF z (1)
ACF z (2)
ACT z (2)
A==
ACF X (1)
ACF X (2)
ACT X (2)
A?
59.7734
7.8033
2.8322
70.4089 69.9218 70.67 0.3
28.1751
2.9503
0.6672
31.7927 32.7721 31.7 7 0.4
a b
Calculated using PTM. Calculated using CDM.
The least-square fit procedure [13] is adopted to determine the MO coefficients, the displacement ΔR and the Femi-contact term κ by the use of the expressions in Eqs. (3),(4),(9), (10), and (12). A parameter to judge the quantity of the fitting methods is introduced [13] ω ¼ ∑ ðpiF piI Þ2 = ðpiI Þ2
ð13Þ
i
4. Conclusion
where piF stand for fitted values in Eq. (12) (g factors g//, g ? and hyperfine structure constants A//, A ? ) and piI for input experimental ones given in Refs [9,10]. The best-matching procedure mentioned above yields the value of g factors g//, g ? and hyperfine structure constants A//, A ? which are all listed in Table 1 with the parameters λe E0.7129, λs E0.6017, λt E0.9773, ΔR¼0.08 nm. The calculated value of parameter ωE1.58 10 5 and E2 E15502 cm 1 which is close to the optical data found in [MoCl6] clusters [15,30,31]. Taking those parameters λe, λs, λt and ΔR into complete energy matrix and using the CDM discussed previously [32], the theoretical values of g factors g//, g ? and hyperfine structure constants A//, A ? can be obtained and are also listed in Table 1.
3. Discussion It can be found from Table 1 that the calculated values from both PTM and CDM methods are not only close to each other, but also in good agreement with the experimental ones. It is worth noting that the investigations indicate the great importance of charge transfer mechanism in the understanding of the spinHamiltonian (SH) parameters for K2SnCl6: Mo5+, especially for g// 4g ? . The second-order g factors g//, g ? are usually expressed as following within CFA 8ζ 2ζ g == ¼ g s d ; g ? ¼ g s d : E1 E2
9.1, 36.3 and 22.6 for Δgz, Δgx, Az(2) and Ax(2), respectively, the magnitude for Δgz, is remarkable. The results come from the fact that the contributions from CT mechanism are proportional to SOC parameter (see Eqs. (9)–(11) )which depend mainly on the values of ligand coefficient ζp, and ζp is relatively large for Cl .
ð14Þ
With this equation and taking the fact that the experimental energy levels E1 is much less than 4E2 into account [15,30,31], the values g// should be expected to be less than g ? , which is inconformity with the experimental fact g// 4g ? for K2SnCl6: CF Mo5+. Similarly, if we take just the contribution of Δg CF z and Δg x from the CF mechanisms into account within the MO scheme, the calculated g// is also less than g ? , which as can be seen from the data given the Table 1. So the positive contributions of Δg CT z and Δg CT x from the CT mechanisms become the decisive element in the account for g// 4g ? for K2SnCl6: Mo5+ and Δg CT should be z distinctly larger than Δg CT x , as shown in Table 1. The magnitude of the CT contributions is characterized by the ratio jQ CT = Q CF j (Q¼ Δgz, Δgx and Az(2), Ax(2) and have the following values 72.6,
The investigations indicate that the D4h symmetry model due to the static Jahn–Teller effect is reasonable and the CT excitation states play a decisive role in the explanations of SH parameters for 4dn TM central ion with Cl ligands. In addition, this mechanism may also be applied to the explanation of the reverse of g factors g// and g ? for liang F ( g// og ? ) and Cl or Br ( g// 4g ? )found in another complex system containing Mo5+ ion [33].
Acknowledgments This project was supported by the Science and Technology Research Program of Education Commission of Chongqing Municipality (Grant no. KJ121419) and the Research Foundation of Chongqing University of Science and Technology (Grant no. CK2010B05). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
M.G. Brik, I.V. Kityk, J. Phys. Chem. Solids 72 (2011) 1256. M.G. Brik, J. Phys. Chem. Solids 72 (2007) 1341. R. Wernicke, H. Schmidtke, J. Chem. Phys. 72 (1980) 1938. I.N. Douglas, J. Chem. Phys. 51 (1969) 3066. I.N. Douglas, J.V. Nicholas, B.G. Wybourne, J. Chem. Phys. 48 (1968) 1414. M. Wermuth, C. Reber, H. Gudel, Inorg. Chem. 40 (2001) 3693. D.R. Gamelin, M. Wermuth, H. Gudel, J. Lumin. 83–83 (1999) 405. J.L. Pascual, Z. Barandiaran, L. Seijo, J. Chem. Phys. 124 (2006) 124315. W. Low, Chem. Lett. 24A (1967) 46. S. Maniv, W. Low, A. Gabay, J. Phys. Chem. Solids 35 (1974) 373. A. Abragam, M.H.L. Pryce, Proc. R. Soc. London A 205 (1951) 135. J.S. Griffith, The Theory of Transition-Metal Ions, Cambridge University Press, London, 1964. J.A. Aramburu, M. Moreno, J. Chem. Phys. 83 (1985) 6071. S. Sugano, Y. Tanabe, H. Kamimura, Multiplets of Transition Metal Ions in Crystals, Academic Press, New York, 1970. Sally M. Horner, S.Y. Tyree Jr., Inorg. Nucl. Chem. Lett. 1 (1965) 43. W.L. Yu, X.M. Zhang, L.X. Yang, B.Q. Zen, Phys. Rev. B 50 (1994) 6756. S. Pandey, R. Kripal, Acta Phys. Pol. A 123 (2013) 101. W.C. Zheng, W. Fang, Y. Mei, J. Appl. Phys. 101 (2007) 202. W.C. Zheng, W. Fang, G.Q. Qu, Solid State Commun. 144 (2007) 202. K. Chandrasekharan, V.S. Murty, F. Joseph Kumar, P. Ramaswamy, P. Sambasiva Rao, J. Mater. Sci. 35 (2000) 1561. G.L. Mcpherson, R.C. Koch, G.D. Stacky, J. Chem. Phys. 60 (1974) 1424. E. Clementi, D.L. Raimondi, J. Chem. Phys. 38 (1963) 2686. E. Clementi, D.L. Raimondi, W.P. Reinhardt, J. Chem. Phys. 47 (1967) 1300. R.D. Shannon, Acta Cryst. A 32 (1976) 751.
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Ralph W.G. Wyckoff, Proc. Natl. Acad. Sci. 9 (1923) 33. W.C. Zheng, X.X. Wu, Q. Zhou, Y. Mei, Chem. Phys. Lett. 406 (2005) 192. W. Fang, X.X. Wu, W.C. Zheng, J. Magn. Magn. Mater. 320 (2008) 2784. B.R. McGarevy, J. Phys. Chem. 71 (1967) 51. A.A. Misetich, R.E. Watson, Phys. Rev. 143 (1966) 335.
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Jay R. Winkler, Harry B. Gray, Struct. Bonding 142 (2012) 17. Howard H. Patterson, John L. Nims, Inorg. Chem. 11 (1972) 520. W. Fang, W.C. Zheng, Physica B 404 (2009) 2025. K.K. Sunil, J.F. Harrison, Max T. Roger, J. Chem. Phys. 76 (1982) 3087.
27
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Theoretical studies of spin-Hamiltonian parameters of Mo5+ ion doped in K2SnCl6 crystal Wang Fang n, Da-Xiao Yang, Heng-Jie Chen, Hai-Yan Tang Department of Mathematics and Physics, Chongqing University of Science and Technology, Chongqing 401331, People's Republic of China
art ic l e i nf o
a b s t r a c t
Article history: Received 8 July 2013 Received in revised form 25 July 2013 Accepted 29 July 2013 Available online 3 August 2013
The spin-Hamiltonian (SH) parameters (g factors g//, g ? and hyperfine structure constants A//, A ? ) of K2SnCl6: Mo5+ (4d1) crystal are theoretically studied by the use of two microscopic spin-Hamiltonian (SH) methods, the high-order perturbation theory method (PTM) and the complete diagonalization (of energy matrix) method (CDM) within the molecular orbital (MO) scheme. The contributions arising both from the crystal field and charge transfer excitations are taken into account. The investigations show that the charge transfer mechanism plays a decisive role in the understanding of the spin-Hamiltonian (SH) parameters for 4d1 ions in crystals with the strong coordinate covalence, especially for g// 4g ? which cannot be explained in the frame work of traditional crystal field approximation (CFA). The local defect structure around Mo5+ impurity ion center is determined to be D4 h point group symmetry. & 2013 Elsevier B.V. All rights reserved.
Keywords: Charge transfer mechanism Defect structure [MoCl6] 1 cluster Ligand field theory
1. Introduction A2XY6 (A¼K, Cs, Rb, Tl; X¼ tetravalent cation, Y¼ F , Cl , Br , I ) crystals doped with transition metal or rare earth ions have attracted a great deal of attention [1–8] because of the applications in luminescence and, in particular, in upconversion luminescent materials [6–8], and of the merits of being doped easily with different impurities (especially at the 6-fold coordinated tetravalent cation site) [1]. When the pentavalent Mo5+ ion is doped into K2SnCl6 crystal (Fig. 1), it will substitute for the tetravalent Sn4+ ion and a vacancy with negative charge will be introduced along one of the C4 axis at the interstitial site because of the charge compensation effect [9].The experimental EPR results (g factors g//, g ? and hyperfine structure constants A//, A ? ) show that [MoCl6] 1 clusters are in the compressed tetragonal symmetry and the ground state of Mo5+ ion is B2g (dxy) [9,10]. There are two possible defect models for the compressed tetragonal symmetry proposed in the investigation, one is C4v symmetry caused by the repulsion of one Cl with the negative compensating hole along the C4 axis, another is D4h symmetry due to the static Jahn–Teller effect. To find which one is realistic, the calculations have been carried out by the use of those two defect models proposed, and the calculated results show that when D4h defect model is used, the results from both PTM and CDM methods are in good agreement with the experimental dada, while for C4v defect model, no plausible results can be obtained in any case. So, we carry out the calculations by the use D4h defect model in the following
n
Corresponding author. Tel.: +86 23 6502 3914. E-mail address: [email protected] (W. Fang).
0921-4526/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2013.07.030
section. In the calculation, the MO theory in the ligand-field (LF) scheme is used and the contributions from the CT excitation states are taken into account. The investigation show that this CT mechanism plays a decisive role in accounting for the SH parameters of 4dn transition metal (TM) ions with Cl ligands. 2. Calculations The ground state of Mo5+ ion in K2SnCl6 crystal is B2g (dxy) and the expressions of SH parameters g factors and hyperfine structure constants A factors for the single orbital term can be derived by the PTM which have been discussed in detail elsewhere [11–13]. The one-electron basis functions for antibonding orbitals belonging to t2g and eg state for 4dn ions in octahedral field can be expressed as the liner combination of atomic orbitals (LCAO) [14], ψ at ¼ ðN at Þ1=2 ðφt λπ χ π Þ;
ψ ae ¼ ðN ae Þ1=2 ðφe λs χ s λs χ s Þ;
ð1Þ
respectively, and the bonding orbital can be written as [14] ψ bt ¼ ðNbt Þ1=2 ðχ π þ γ π φt Þ; ψ be ¼ ðN be Þ1=2 ðχ s þ γ s φe Þ;
ð2Þ
where φt and φe denote the pure d orbital of the metal ion, χ i (i¼π, s, s) are group orbitals of ligands, ðNbt Þ1=2 , ðN be Þ1=2 and ðN at Þ1=2 , ðNae Þ1=2 are the normalization constants, and λi (i¼π, s, s) are covalency parameters. The third-order perturbation formulas of g factors g//, g ? and second-order hyperfine structure constants A factors A//, A ? based on CF mechanisms for 4d1 ion in tetragonal symmetry
W. Fang et al. / Physica B 429 (2013) 24–27
25
reduction factors k, k′ related to CF mechanisms in Eq. (3) are [14] ζ CF ¼ ðNat Þ1 ½ζ d þ ðλπ Þ2 ζ p =2; 0 ¼ ðN t a N ae Þ1=2 ½ζ d λπ λs ζ p =2; ζ CF
kCF ¼ ðN at Þ1 ½12λt Sπ þ ðλπ Þ2 =2
ð7Þ
0 ¼ ðNat N ae Þ1=2 ½1λs Ss λs Ss λπ Sπ λπ ðλs þ Aλs Þ=2; kCF
where ζd and ζp are the SOC parameters of free 3dn ion and that of ligand ions in free state, respectively. For Mo5+, ζd E1013 cm 1 [20] and for Cl , ζp E587 cm 1 [21], Ss ¼ 〈φe jχ s 〉, Ss ¼ 〈φe jχ s 〉, Sπ ¼ 〈φt jχ π 〉 are the group overlap integrals, which can be calculated using the Clementi's SCF functions [22,23] and impurity-ligand distance R in crystal. Considering that the ionic radius ri (E0.061 nm [24]) of impurity Mo5+ is unlike the radius rh (E0.069 nm [24]) for the replaced host ion Sn4+, the impurity-ligand distance R for Mo5+ ion in K2SnCl6 should be different from the corresponding Rh (E0.2448 nm [25]) in the host K2SnCl6 crystal, we estimate approximately the distance R from the empirical formula R0 Rh þ 1=2ðr i r h Þ [26,27] and obtain R0 ¼ 0.2408 nm. Thus, the group overlap are calculated to be
Fig. 1. One unit cell of K2SnCl6 crystal.
Ss 0:1028; Ss 0:1848; and Sp 0:0746:
are derived as (gs ¼2.0023) g z ¼ g s þ Δg CF z ; 0 0 0 0 8kCF ζ CF 4kCF ζ CF ζ CF kCF ζ 2 ζ 2 g 2 CF CF2 s ; Δg CF z ¼ E1 E1 E2 E2 E2
g x ¼ g s þ Δg CF x ; Δg CF x ¼
0 0 2kCF ζ CF kCF ζ 2CF 2kCF ζ CF ζ CF 2ζ 0 2 g ζ 2 g þ þ CF2 s CF 2 s ; 2 E2 E1 E 2 E2 E1 2E2
CF Az ¼ ACF z ð1Þ þ Az ð2Þ; 4 =Nat ; ACF ð1Þ ¼ P κ z 7 0
ð3Þ
1
Ax ¼
ACF x ð1Þ þ
Δg CT x
e 2
ACF x ð2Þ;
In Eq. (6) R ? represents equatorial metal-ligand distance and equal to R0, R J represents axial metal-ligand distance and taken to be (R0 ΔR) in which ΔR is the inward displacement of Cl along the C4 axis. In Eq. (4), P is dipolar hyperfine structure constants and has the value of 66.7 10 4 cm 1 for Mo5+ [28] and κ is the Femi-contact term which is taken to be 1.05 [9], A ¼ R0 〈φx j∂=∂xjχ s i, for Cl ion 〈φx j∂=∂xjχ s i ¼ 0:409 [29]. The contributions to the SH parameters from the CT levels come from the excitation procedure of promoting an electron from the occupied π bonding orbitals egn (dxz and dyz) or b1gn (dx2 y2) to dxy ground state, and the high-order perturbation formulas of the SH parameters due to this CT mechanism have the follow form 0 0 8kCT ζ CT ; E1C 2kCT ζ CT ¼ ; E2C
Δg CT z ¼
0 6ζ CF B 8ζ CF C ffi A; þ qffiffiffiffiffiffiffiffiffiffiffi ACF z ð2Þ ¼ P @ a Nt E1 7 N a N a E t
0
ð4Þ ACT z
B 11ζ CF C ACF A; x ð2Þ ¼ P @ qffiffiffiffiffiffiffiffiffiffiffiffi 7 Nat Nae E2
B 11γ t ζ CT C ACT A; x ¼ P @ qffiffiffiffiffiffiffiffiffiffiffiffi 7 N at N bt E2c
in which
where ð5Þ
Dq is cubic crystal field parameter and estimated to be 2350 cm 1 [15], Ds and Dt represent the tetragonal crystal field parameter and have the following form using the superposition model " 3 # 4 R0 3 R0 Ds ¼ A2 ðR0 Þ ; 7 R? R== " 5 # 16 R0 5 R0 Dt ¼ A4 ðR0 Þ ; ð6Þ 21 R? R== and A2 ðR0 Þ, A4 ðR0 Þ are intrinsic parameters with reference distance R0, A4 ðR0 Þ E3/4Dq for dn ions in cubic field [16,17]. The ratio A2 ðR0 Þ=A4 ðR0 Þ for mdn ions in clusters decreases with the increasing m value, for 3dn, A2 ðR0 Þ=A4 ðR0 ÞE9–12 are found for many ions in crystals [18,19] andA2 ðR0 Þ=A4 ðR0 Þ ¼ 8:3 is taken here in the calculations. The spin-orbit coupling (SOC) parameters ζ, ζ′ and the orbital
ð9Þ 1
0 6γ t ζ CT C B 8γ e ζ CT ffi ffi A; ¼ P @qffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffi a b Nt Ne E1C 7 N at N bt E2c 0 1
2 =Nat ; ACF x ð1Þ ¼ P κ þ 7 0 1
E1 ¼ 10Dq; E2 ¼ 3Ds þ 5Dt:
ð8Þ
ð10Þ
ζ CT ¼ ðN bt N at Þ1=2 ½γ π ζ d λπ ζ p =2; 0 ¼ ðN bt N ae Þ1=2 ½γ s ζ d þ λπ ζ p =2; ζ CT
ð11Þ
kCT ¼ ðN bt N at Þ1=2 ½γ π þ ð1λπ γ π ÞSπ λπ =2; 0 ¼ ðN bt N ae Þ1=2 ½γ s γ s λπ St Ss þ λt =2; kCT
and E1c, E2c are CT energies of egn (dxz and dyz) and b1gn (dx2 y2), respectively. We estimate E1c ¼41150 and E2c ¼28250 cm 1 [15]. The g factors g//, g ? and hyperfine structure constants A//, A ? from both the CF and CT mechanisms are expressed as CT g == ¼ g z þ g CT z ; g ? ¼ gx þ gx ; CT A== ¼ Az þ ACT z ð2Þ; A ? ¼ Ax þ Ax ð2Þ:
ð12Þ
All the normalization relationship and the relations between the LCAO coefficients and the group overlap used in the calculations can be found in reference [14].
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W. Fang et al. / Physica B 429 (2013) 24–27
Table1 The spin-Hamiltonian parameters (g factors g//, g ? and hyperfine structure constants A//, A ? are in units of 10–4 cm–1) for Mo5+ in K2SnCl6 crystal.
Calc.a Calc.b Expt.[1]
Δg CF z
ΔgCT z
g==
ΔgCF x
Δg CT x
g?
0.1037
0.0753
1.9739 1.9743 1.97397 0.0006
0.0701
0.0064
1.9386 1.9388 1.9386 7 0.0006
ACF z (1)
ACF z (2)
ACT z (2)
A==
ACF X (1)
ACF X (2)
ACT X (2)
A?
59.7734
7.8033
2.8322
70.4089 69.9218 70.67 0.3
28.1751
2.9503
0.6672
31.7927 32.7721 31.7 7 0.4
a b
Calculated using PTM. Calculated using CDM.
The least-square fit procedure [13] is adopted to determine the MO coefficients, the displacement ΔR and the Femi-contact term κ by the use of the expressions in Eqs. (3),(4),(9), (10), and (12). A parameter to judge the quantity of the fitting methods is introduced [13] ω ¼ ∑ ðpiF piI Þ2 = ðpiI Þ2
ð13Þ
i
4. Conclusion
where piF stand for fitted values in Eq. (12) (g factors g//, g ? and hyperfine structure constants A//, A ? ) and piI for input experimental ones given in Refs [9,10]. The best-matching procedure mentioned above yields the value of g factors g//, g ? and hyperfine structure constants A//, A ? which are all listed in Table 1 with the parameters λe E0.7129, λs E0.6017, λt E0.9773, ΔR¼0.08 nm. The calculated value of parameter ωE1.58 10 5 and E2 E15502 cm 1 which is close to the optical data found in [MoCl6] clusters [15,30,31]. Taking those parameters λe, λs, λt and ΔR into complete energy matrix and using the CDM discussed previously [32], the theoretical values of g factors g//, g ? and hyperfine structure constants A//, A ? can be obtained and are also listed in Table 1.
3. Discussion It can be found from Table 1 that the calculated values from both PTM and CDM methods are not only close to each other, but also in good agreement with the experimental ones. It is worth noting that the investigations indicate the great importance of charge transfer mechanism in the understanding of the spinHamiltonian (SH) parameters for K2SnCl6: Mo5+, especially for g// 4g ? . The second-order g factors g//, g ? are usually expressed as following within CFA 8ζ 2ζ g == ¼ g s d ; g ? ¼ g s d : E1 E2
9.1, 36.3 and 22.6 for Δgz, Δgx, Az(2) and Ax(2), respectively, the magnitude for Δgz, is remarkable. The results come from the fact that the contributions from CT mechanism are proportional to SOC parameter (see Eqs. (9)–(11) )which depend mainly on the values of ligand coefficient ζp, and ζp is relatively large for Cl .
ð14Þ
With this equation and taking the fact that the experimental energy levels E1 is much less than 4E2 into account [15,30,31], the values g// should be expected to be less than g ? , which is inconformity with the experimental fact g// 4g ? for K2SnCl6: CF Mo5+. Similarly, if we take just the contribution of Δg CF z and Δg x from the CF mechanisms into account within the MO scheme, the calculated g// is also less than g ? , which as can be seen from the data given the Table 1. So the positive contributions of Δg CT z and Δg CT x from the CT mechanisms become the decisive element in the account for g// 4g ? for K2SnCl6: Mo5+ and Δg CT should be z distinctly larger than Δg CT x , as shown in Table 1. The magnitude of the CT contributions is characterized by the ratio jQ CT = Q CF j (Q¼ Δgz, Δgx and Az(2), Ax(2) and have the following values 72.6,
The investigations indicate that the D4h symmetry model due to the static Jahn–Teller effect is reasonable and the CT excitation states play a decisive role in the explanations of SH parameters for 4dn TM central ion with Cl ligands. In addition, this mechanism may also be applied to the explanation of the reverse of g factors g// and g ? for liang F ( g// og ? ) and Cl or Br ( g// 4g ? )found in another complex system containing Mo5+ ion [33].
Acknowledgments This project was supported by the Science and Technology Research Program of Education Commission of Chongqing Municipality (Grant no. KJ121419) and the Research Foundation of Chongqing University of Science and Technology (Grant no. CK2010B05). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
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