Studies of the local distortions for nd1 (Mo5+ and W5+) impurities in KTP crystals

Polyhedron 105 (2016) 71–76

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Studies of the local distortions for nd1 (Mo5+ and W5+) impurities in KTP crystals Chang-Chun Ding ⇑, Shao-Yi Wu ⇑, Li-Juan Zhang, Xian-Fen Hu, Guo-Liang Li Department of Applied Physics, School of Physical Electronics, University of Electronic Science and Technology of China, Chengdu 610054, PR China

a r t i c l e

i n f o

Article history: Received 22 September 2015 Accepted 10 December 2015 Available online 18 December 2015 Keywords: Defect structures Electron paramagnetic resonance (EPR) Mo5+ W5+ KTiOPO4 (KTP)

a b s t r a c t 4d1 (Mo5+) and 5d1 (W5+) impurities doped into nonlinear optical crystals KTiOPO4 (KTP) can occupy the inequivalent Ti1 and Ti2 sites and exhibit orthorhombically compressed octahedral nd1 (n = 4, 5) centres arising from the Jahn–Teller effect. Closely relevant to the electronic levels as well as optical and magnetic properties of these materials, the above local distortions can be suitably reflected by the anisotropic electron paramagnetic resonance (EPR) signals with the paramagnetic nd1 probes. This work stresses on the theoretical studies of the EPR parameters (g factors gx, gy, gz and hyperfine structure constants Ax, Ay, Az) and local distortions for the above nd1 impurity centres in a consistent way. The local distortions are characterised by the relative axial compression ratios q (
0.040, 0.032 and 0.031) and the relative planar bond length variation ratios s (
0.030, 0.015 and 0.012) for Mo5+ centre I and W5+ centres II and III, respectively, which are larger than those of host Ti1 and Ti2 sites. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Potassium titanyl phosphate (KTiOPO4 or KTP) crystals are wellknown optical materials with large nonlinear optical coefficients and a high optical damage threshold, affording the promising applications in optical parametric oscillation (OPO), optical parametric amplifier (OPA) and second harmonic generation (SHG) [1–5]. With the advantages of simple energy levels and significant spectroscopic (e.g., optical and magnetic resonance) signals, nd1 transition-metal (e.g., Mo5+(4d1) and W5+(5d1)) impurities can effectively influence the optical and magnetic properties of these materials [6,7]. These properties may sensitively depend upon local structures and electronic states of the impurities in the hosts and can be efficiently studied by means of electron paramagnetic resonance (EPR) technique [8–11]. To reveal structures and impurity behaviours of molybdenum and tungsten doped KTP crystals, the EPR experiments were employed, and the EPR parameters, i.e., g factors (gx, gy and gz) and hyperfine structure constants (Ax, Ay and Az), were measured for three orthorhombic centres, i.e., Mo5+ centre I and W5+ centres II and III [10,11]. The detailed EPR spectral diagrams of these centres are shown in Fig. 1. Associated with the ground state 2B1g (f) for a nd1 ion under orthorhombically compressed octahedra [11], g and A factors demonstrate the consistent anisotropies (gz < gy < gx < 2 and |Ax| < |Ay| < |Az|) for all ⇑ Corresponding authors. E-mail addresses: [email protected] (C.-C. Ding), [email protected] (S.-Y. Wu). http://dx.doi.org/10.1016/j.poly.2015.12.013 0277-5387/Ó 2015 Elsevier Ltd. All rights reserved.

these centres [10,11]. Although centres I, II and III were assigned to Mo5+ on Ti2 site and W5+ on Ti1 and Ti2 sites in KTP, respectively [10,11], no further theoretical analysis on the above EPR parameters has been carried out. Nor has the information related to the local structures of these centres been obtained. In essence, theoretical analysis of the EPR spectra would reveal important impurity behaviours in these materials. Moreover, information relevant to the local structures of the impurities in KTP can be helpful to understand properties of these materials containing nd1 dopants. Therefore, it is worthwhile to perform further investigations on the EPR parameters and the local structures of the above impurities centres. This paper is organised as follows. In Section 2, the EPR parameters of orthorhombically compressed octahedral nd1 clusters are described, and the related formulae and quantities are illustrated. In Section 3, the calculations of the EPR parameters are carried out for the nd1 centres. Section 4 deals with the discussion of the properties of the EPR parameters and local structures as well as the validity of the theoretical results. In the last section, the conclusion is proposed.

2. Theoretical formulae of the EPR parameters for orthorhombically compressed octahedral nd1 clusters 2.1. Local structures of the nd1 impurity centres KTP single crystal has orthorhombic structure with space group Pna21. In general, the KTP lattice has two inequivalent potassium

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Fig. 1. (a) EPR spectra of Mo5+ in KTP at 77 K, cited from Ref. [10]. (b) EPR spectra of two W5+ centres in KTP recorded at 90 K in three crystal planes ab, bc and ca, cited from Ref. [11].

sites, two inequivalent phosphorus, ten inequivalent oxygen sites and two inequivalent titanium sites (TiO6 octahedra) [12–14]. The two host TiO6 octahedra with orthorhombic compression distortions are combined together in a chain (–OT2–Ti1–OTi2–Ti2– OT2–) (see Fig. 2(a)) [12–14]. As mentioned in Ref. [11], the observed hyperfine structure constants characterised by |Az| > | Ax|, |Ay| for both Mo5+ and W5+ centres are ascribed to the ground state 2B1g (|dxyi) of the 4d1 and 5d1 impurities on substitutional octahedral Ti sites in KTP, corresponding to the principle Z axis roughly parallel to the direction of the compression along the shortest Ti–O bonds. Meanwhile, the measured g anisotropies gx, gy > gz of both centres in KTP are also supported by the similar orthorhombically compressed octahedral d1 ions (e.g., V4+ and Mo5+ in TiO2 [15,16] as well as WO3+ in K2[WO(NCS)5] [17], K2SnF6H2O [18] and (NH4)2GeF6 [19]) associated with the analogous g anisotropies and the same ground state 2B1g (|dxyi).

On the contrary, orthorhombically elongated octahedral d1 centres (e.g., Ti3+ and V4+ in CaYAlO4) would show different ground state 2 B3g (|dyz>) and opposite g anisotropies (gx, gy < gz) [20,21]. Thus, the studied systems are more likely attributed to the orthorhombically compressed octahedral Mo5+ and W5+ centres on Ti sites in KTP. Because of the size and charge mismatch, the local structures around the impurities Mo5+ and W5+ can be different from those of the host Ti4+ sites. Importantly, nd1 ions are Jahn–Teller ions with the ground orbital triplet 2T2g in an ideal octahedron, which may undergo the Jahn–Teller effect via the vibration interactions [15,22]. For example, two bonds of the oxygen octahedra coordinated to the nd1 impurities can suffer the different relative compression along Z axis and the other four perpendicular ones would experience the relative planar bond length variations along X and Y axes with respect to the host Ti sites, with the lowest 2 B1g(f) (|dxyi) level. The above local distortion of the Jahn–Teller

O2– in an ideal octahedron

Ti2

O2– under Jahn-Teller

Z X

distortion by ρ and τ

2Rρ Ti1 Y

Y

R(ρ–τ)

R(ρ + τ) Rz

Z

Y

Ry X Mo(W)

Z

X

Rx

Ti2

Ti1

(a)

(b)

Fig. 2. (a) The inequivalent Ti1 and Ti2 sites (TiO6 octahedra) in KTP, which are combined together in a chain (–OT2–Ti1–OTi2–Ti2–OT2–) [12–14]. (b) The local structures of the orthorhombic nd1 centres I, II and III in KTP, with the local impurity – ligand distances Ri (i = x, y and z) in terms of the reference distance R for an ideal octahedron and the local distortion parameters q and s due to the Jahn–Teller effect.

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nature can be conveniently expressed in terms of the relative axial compression ratio q and the relative planar bond length variation ratio s (see Fig. 2(b)). 2.2. Perturbation formulae of the EPR parameters As for nd1 ions in orthorhombically compressed octahedra, the higher orbital doublet 2E2g in original cubic case would split into two orbital singlets 2A1g (h) (|dz23r2i) and 2A01g (e) (|dx2y2i). Meanwhile, the ground orbital triplet 2T2g in cubic case can be separated into three orbital singles 2B1g(f) (|dxyi), 2B2g(g) (|dxzi) and 2B3g(n) (|dyzi), with 2B1g(f) (|dxyi) lying lowest (see Fig. 3) [23–26]. 2.2.1. Energy splittings for an orthorhombically compressed octahedral nd1 cluster The energy levels for an orthorhombically compressed octahedral nd1 cluster can be written as

E1 ¼ Eðjdxz iÞ  Eðjdxy iÞ ¼ 5Dt  3Ds þ 3Dn  4Dg ; E2 ¼ Eðjdyz iÞ  Eðjdxy iÞ ¼ 5Dt  3Ds  3Dn þ 4Dg ; E3 ¼ Eðjdx2 y2 iÞ  Eðjdxy iÞ ¼ 10Dq ;

ð1Þ

E4 ¼ Eðjdz2 3r2 iÞ  Eðjdxy iÞ ¼ 10Dq  Ds  10Dt þ 3Dn  4Dg : Here Dq is the cubic field parameter, which can be obtained from the optical spectral measurements of the studied or similar systems. Ds, Dt, Dn and Dg are the orthorhombic field parameters. 2.2.2. Formulae of the orthorhombic field parameters based on the superposition model The orthorhombic field parameters are usually determined from the local geometry of the systems and the superposition model [27,28]

h

i

Ds
ð4=7ÞA2 ðRÞ ðR=Rx Þt2 þ ðR=Ry Þt2  2ðR=Rz Þt2 ; h i Dt
ð2=21ÞA4 ðRÞ ðR=Rx Þt4 þ ðR=Ry Þt4  2ðR=Rz Þt4 ; h i Dn
ð2=7ÞA2 ðRÞ ðR=Rx Þt2  ðR=Ry Þt2 ; h i Dg
ð10=21ÞA4 ðRÞ ðR=Rx Þt4  ðR=Ry Þt4 :

ð2Þ

Here Rx
R (1 + q + s), Ry
R (1 + q  s) and Rz
R (1  2q) are the effective impurity–ligand distances along X, Y and Z axes, respectively. The power-law exponents t2 and t4 are normally taken as 3 and 5, respectively [27]. For transition-metal ions in octahedral crystal-field environments, the intrinsic parameters A2 (R) andA4 (R) satisfy the relationships A4 (R)
(3/4) Dq and A2 (R)
10.8 A4 (R) [28–30].

2.2.3. Perturbation formulae of the EPR parameters Due to the high valence states of the central ions Mo5+ and W5+, the [MoO6]7 and [WO6]7 clusters may contain strong covalency or admixture between impurity and ligand orbitals, which were usually neglected in the previous studies [23,24]. The perturbation formulae of the EPR parameters for an orthorhombically compressed octahedral nd1 cluster including the ligand contributions based on the cluster approach can be expressed as [15] 0

0

g x ¼ g s  2k f0 =E1 þ kf2 ½1=ð2E21 Þ  1=ð2E22 Þ þ 2k ff0 ½1=ðE2 E3 Þ  1=E23 ; 0

0

g y ¼ g s  2k f0 =E2  kf2 ½1=ð2E21 Þ  1=ð2E22 Þ þ 2k ff0 ½1=ðE1 E3 Þ  1=E23 ; 0 0

0 02

g z ¼ g s  8k f =E3  kf2 ð1=E21 þ 1=E22 Þ=2 þ 2k f =E23 : AX ¼ P½j þ j0 þ 2H=7 þ 11ðg x  g s Þ=14; Ay ¼ P½j þ j0 þ 2H=7 þ 11ðg y  g s Þ=14; Az ¼ P½j  4H=7 þ g z  g s þ 3ðg x  g s Þ=14 þ 3ðg y  g s Þ=14: ð3Þ Here gs (
2.0023) is the free electron spin-only value. f (or f0 ) and k (or k0 ) denote the spin–orbit coupling coefficient(s) and the orbital reduction factor(s), respectively, characteristic of the diagonal (or off-diagonal) matrix elements for the spin–orbit coupling and orbital angular momentum operators in a cubic octahedron [15,22]. P is the dipolar hyperfine structure parameter of the nd1 ion. j and j0 are the isotropic and anisotropic core polarisation constants. H is the reduction factor characteristic of the anisotropic nd–ns ((n + 1)s) orbital admixtures of the nd1 ion in crystals due to the axial (compression) distortions of the systems. So, the local distortions (q and s) are quantitatively involved in the orthorhombic field parameters and hence in the EPR parameters (from the energy denominators E1–E3) of the impurity centres. 2.2.4. Cluster approach formulae According to the cluster approach [15,22], the spin–orbit coupling coefficients f and f0 and the orbital reduction factors k and k0 are normally determined from the normalisation factors Nc and the orbital admixture coefficients kc (here the subscripts c = t and e stand for the irreducible representations T2g and Eg of Oh group):

f ¼ Nt ðf0d þ k2t f0p =2Þ; f0 ¼ ðNt Ne Þ1=2 ðf0d  kt ke f0p =2Þ; k ¼ Nt ð1 þ k2t =2Þ;

0

k ¼ ðNt Ne Þ1=2 ½1  kt ðke þ ks AÞ=2:

ð4Þ

Here f0d and f0p are the spin–orbit coupling coefficients of the nd1 and ligand ions in free states, respectively. A denotes the integral @ Rhnsj @y jnpy i, with the reference distance R. Based on the cluster approach, the relevant molecular orbital coefficients (Nc and kc) can be theoretically determined from the normalisation conditions and the approximation relationships in terms of the covalency factor N and the related group overlap integrals (St, Se and Ss) [15,22]

Nt ð1  2kt St þ k2t Þ ¼ 1; Ne ð1  2ke Se  2ks Ss þ k2e þ k2s Þ ¼ 1; N2 ¼ N t ð1 þ k2t S2t  2kt St Þ;

ð5Þ

N2 ¼ N e ð1 þ k2e S2e þ k2s S2s  2ke Se  2ks Ss Þ: 3. Calculation procedure and results

Fig. 3. The scheme of energy level splittings for an orthorhombically compressed octahedral nd1 cluster. Ei (i = 1–4) are the corresponding energy differences.

Since the optical spectral data were not reported for KTP:Mo5+ and KTP:W5+, the spectral parameters can be estimated from those of the similar octahedral nd1 systems in TiO2 [15] and other compounds [31]. Considering the relationship Dq / Rn (n
5 ± 1.5 [32,33] and the fact that covalency of isoelectronic systems in the same column of the period table increases with increasing principal quantum number, one can take the values Dq (
2400,

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2500 and 2570 cm1) and N (
0.78, 0.76 and 0.75) for Mo5+ centre I and W5+ centres II and III in KTP, respectively. By using the Slatertype self-consistent field (SCF) functions [34,35], the group overlap integrals St, Se, Ss and A can be obtained for Mo5+ and W5+ in KTP with the reference distances R (
1.9723 and 1.968 Å for Ti1 and Ti2 sites, respectively [28]) and listed in Table 1. Making use of Eq. (5), the molecular orbital coefficients are calculated and shown in Table 1. In the light of the free-ion values f0d
1030 and 3500 cm1 for Mo5+ and W5+ [23] and f0p
603 cm1 for O2– [36], the spin–orbit coupling coefficients f and f0 , and the orbital reduction factors k and k0 are determined from Eq. (4) and listed in Table 1. In the calculations of hyperfine structure constants, the dipolar hyperfine structure parameters are P
66.7  104 [23] and 76  104 cm1 [37] for Mo5+ and W5+, respectively. The core polarisation constants are taken as j
0.99, 0.87 and 0.98, respective, for centres I, II and III in view of the different magnitudes of A factors. From the central ion nd–ns ((n + 1)s) orbital admixtures due to the perpendicular orthorhombic distortions (relevant to the perpendicular anisotropy dA = |Ax  Ay|), the anisotropic core polarisation constants j0 can be estimated as 0.06, 0.12 and 0.12 for centres I, II and III, respectively, from their experimental dA (
8  104, 15  104 and 15  104 cm1 [10,11]). The reduction factors H due to the local compression distortions are approximately taken as 0.15, 0.35 and 0.5 for centres I, II and III in the light of the observed axial anisotropies DA (=|Az|  |Ax + Ay|/2
16  104, 51  104 and 62  104 cm1, respectively. Thus, merely the local distortion parameters (i.e., the relative axial compression ratio q and the relative planar bond length variation ratio s) are unknown in the formulae of the EPR parameters. Inputting the related values into Eq. (3) and matching the theoretical EPR parameters to the experimental data [10,11], the optimal local distortion parameters are obtained:

q
0:040; 0:032 and 0:031 and s
0:030; 0:015 and 0:012

ð6Þ

for centres I, II and III, respectively. The corresponding EPR parameters (Cal.c) are shown in Table 2. To further understand the influences of the Jahn–Teller effect and size and charge mismatch on the local structures of the impurity centres, the results (Cal.a) based on the host structural data [13] of Ti1 and Ti2 sites in KTP are collected in Table 2. In order to clarify importance of the covalency and ligand contributions, the EPR parameters (Cal.b) based on omission of the ligand orbital and spin–orbit coupling contributions (i.e., taking f = f’ = f0d and k = k0 = N in Eq. (2)) are also obtained and shown in Table 2. 4. Discussion From Table 2, the EPR parameters (Cal.c) based on the optimal local structural parameters (q and s) in Eq. (5) and inclusion of the ligand contributions show the best agreement with the

Table 2 The g factors and hyperfine structure constants (in 104 cm1) for the Mo5+ and W5+ centres in KTP. Centres

gx

gy

gz

I

Cal.a Cal.b Cal.c Expt. [10]

1.8915 1.8792 1.9002 1.905

1.8728 1.8174 1.8526 1.857

1.8313 1.7985 1.8372 1.837

Ax 65 66 64 64

Ay 72 77 75 76

Az 87 90 86 86

II

Cal.a Cal.b Cal.c Expt. [11]

1.6808 1.5191 1.6084 1.6037

1.6414 1.4075 1.5471 1.5406

1.4906 1.3429 1.4688 1.4634

70 85 77 77

86 96 91 92

131 149 136 135

III

Cal.a Cal.b Cal.c Expt. [11]

1.7036 1.5110 1.6096 1.6046

1.7280 1.4776 1.5612 1.5636

1.5313 1.3798 1.5071 1.5046

72 89 80 82

89 102 96 97

141 161 147 151

a Calculations based on the host structural data of Ti1 and Ti2 sites (i.e., in the absence of the Jahn–Teller effect) and inclusion of the ligand contributions. b Calculations based on the local structural data without the ligand contributions (i.e., f = f’ = Nf0d and k = k0 = N). c Calculations based on the local structural data from the optimal local distortion parameters q and s in Eq. (6) due to the Jahn–Teller effect.

measured data [10,11], with the discrepancies no more than 0.0065 and 5  104 cm1 for g factors and hyperfine structure constants, respectively. Thus, the experimental EPR spectra of the various nd1 impurity centres in KTP are reasonably interpreted in uniform way. Furthermore, the information of the impurity local structures is determined by analysing the EPR parameters in consideration of the Jahn–Teller effect and size and charge mismatch. (1) In view of the ionic radii of Mo5+ (
0.61 Å [38]) and W5+ (
0.62 Å [38]) comparable with that (
0.605 Å [38]) of host Ti4+, the impurities tend to occupy octahedral compressed Ti4+ sites in KTP and exhibit 2B1g(f) ground state, without inducing much local lattice instability. By contrast, d1 ions in tetragonally or orthorhombically distorted tetrahedra may exhibit 2A01 (e) (or 2A01 (h)) ground state arising from cubic orbital doublet 2E [23] and the cubic field parameters much smaller than those in octahedra, e.g., Dq
1250 and 2200 cm1 for tetrahedral and octahedral V4+(3d1) in Y3Al5O12 [39]. Meanwhile, tetrahedral d1 centres such as Ti3+-VO (3d1) centre in BaTiO3 can show different g anisotropy (g// > g\) [40,41]. In the light of the large Dq and g anisotropies (gx, gy > gz) [10,11], the present Mo5+ and W5+ in KTP are not likely attributed to tetrahedral d1 centres but octahedral ones. On the other hand, the EPR studies on d1 ions in various crystals [10,11,15–19,42,43] reveal that the anisotropies gx, gy > gz and |Az| > |Ax|, |Ay| should be assigned to the orthorhombically compressed octahedral d1 centres with the ground state 2B1g(|dxyi). Inversely, orthorhombically elongated octahedral d1 centres (e.g., Ti3+ and V4+ in CaYAlO4) can exhibit opposite g anisotropies (gx, gy < gz)

Table 1 The group overlap integrals, the normalisation factors, the orbital admixture coefficients, the spin–orbit coupling coefficients (in cm1) and the orbital reduction factors for Mo5+ centre I and W5+ centres II and III in KTP. Centres

St

Se

I II III

0.0411 0.0311 0.0315

0.1101 0.0875 0.0884

I II III

ke 0.4552 0.4690 0.4803

ks 0.3663 0.3768 0.3863

Ss

A 0.0886 0.0703 0.0711

f 839 2727 2695

1.2841 1.2841 1.2813 f0 832 2763 2731

Nt

Ne

kt

0.7979 0.7738 0.7642

0.8502 0.8150 0.8064

0.5460 0.5727 0.5879

k 0.9169 0.9007 0.8962

k0 0.6155 0.5774 0.5599

C.-C. Ding et al. / Polyhedron 105 (2016) 71–76

and different ground state 2B2g(|dyz>) [20,21]. Therefore, the present Mo5+ and W5+ in KTP having the anisotropies gx, gy > gz and |Az| > |Ax|, |Ay| [10,11] may be suitably ascribed to the substitutional orthorhombically compressed octahedral nd1 centres on Ti sites with ground state 2B1g(|dxyi). (2) The EPR parameters (Cal.a) based on the host structural data of Ti sites in KTP are in poor agreement with the experimental data, characterised by the larger average g values [=(gx + gy + gz)/3] for all centres and even opposite signs of the perpendicular anisotropies dg for W5+ centre III on Ti2 site. So, the structural data of the host Ti sites are unsuitable for the studies on the EPR spectra of Mo5+ and W5+ in KTP. In contrast, the local distortion parameters q and s should be involved in the EPR analysis for the [MoO6]7 and [WO6]7 clusters. From Eq. (2), the quantities q (related to Ds and Dt) and s (related to Dn and Dg) reflect the axial and perpendicular orthorhombic distortions and correlate to the axial and perpendicular anisotropies Dg and dg, respectively. The microscopic mechanism of the above local distortions is of the dominant Jahn–Teller character, while the size and charge mismatch between impurities and host cations would also induce some lattice modifications. Judging from the local distortion parameters q and s larger than those (qh
0.010 and 0.014 and sh
0.010 and 0.009) of host Ti1 and Ti2 sites, the Jahn–Teller effect and the size and charge mismatch may tend to enhance the local axial compression and perpendicular orthorhombic distortions for Mo5+ and W5+ in KTP, yielding the energetically favored defects with more intense orthorhombic distortions. Thus, the above analysis of the local structural distortions for the nd1 centres in KTP arising from the Jahn–Teller effect can be regarded as reasonable. (3) The results (Cal.b) without the ligand contributions demonstrate the much smaller g averages than the experimental data, and the discrepancies could hardly be cancelled even by adjusting q and s. Although the spin–orbit coupling coefficient (
603 cm1 [36]) of ligand oxygen is much smaller than that (
3500 or 1030 cm1 [23], respectively) of central ion W5+ or Mo5+, strong covalency (N
0.75–0.78  1) of the impurity centres can lead to significant impurity-ligand orbital admixtures (characterised by the obvious orbital admixture coefficients 0.4–0.6). Especially, the remarkable relative anisotropies (k  k0 )/k0 (
49–60%) for the orbital reduction factors also reveal the important anisotropic ligand contributions. Therefore, the ligand contributions should be involved in the analysis of EPR spectra for Mo5+ and W5+ (or other nd1) ions with high valence states in KTP (or similar orthorhombic oxygen octahedra). (4) The signs of the hyperfine structure constants were not experimentally determined for the nd1 centres in Refs. [10,11]. In the light of the present computations, the hyperfine structure constants are found to be positive and negative for Mo5+ and W5+ in KTP, respectively. Physically, the isotropic and anisotropic contributions to the A factors originate from the isotropic and anisotropic central ion nd–ns (or (n + 1)s) orbital admixtures (and also g anisotropies) in the crystals, characterised by the core polarisation constants j and j0 (and also g-shifts), respectively. The sequence (I < II < III) of the reduction factor H for the various centres is in accordance with that of the axial anisotropy |DA| (=|Az  (Ax + Ay)/2|) for the hyperfine structure constants. This can be ascribed to the declining order (I > II > III) for the local axial compression ratio q. On the other hand, the order (I < II
III) of j0 for these centres is in parallel with that of the perpendicular anisotropy |dA| (=|Ax  Ay|) and can be regarded as reasonable.

75

5. Conclusion Based on the perturbation calculations, the local distortions are described by the relative axial compression ratios q (
0.040, 0.032 and 0.031) and the relative planar bond length variation ratios s (
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