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Investigations on the g factors and superhyperfine parameters for various tetragonal Ni+ centers in RbCaF3
Journal of Molecular Structure: THEOCHEM 942 (2010) 104–109
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Investigations on the g factors and superhyperfine parameters for various tetragonal Ni+ centers in RbCaF3 Hua-Ming Zhang a, Shao-Yi Wu a,b,*, Li-Li Li a, Pei Xu a a b
Department of Applied Physics, University of Electronic Science and Technology of China, Chengdu 610054, PR China International Center for Materials Physics, Chinese Academy of Sciences, Shenyang 110016, PR China
a r t i c l e
i n f o
Article history: Received 5 November 2009 Received in revised form 26 November 2009 Accepted 4 December 2009 Available online 8 January 2010 Keywords: Electron paramagnetic resonance Crystal-fields and spin Hamiltonians Ni+ RbCaF3
a b s t r a c t The anisotropic g factors gk and g\ and the planar F superhyperfine parameters Ak and A\ in the tetragonal impurity Ni+ I, II and III centers (as well as the axial F superhyperfine parameters A0k and A0? in I center) on the substitutional Ca2+ site in RbCaF3 are theoretically investigated in a uniform way using the perturbation formulas of these parameters for a 3d9 ion in tetragonally elongated octahedra. In the calculations, the ligand unpaired spin densities are determined quantitatively from the related molecular orbital coefficients based on the cluster approach. These defects are attributed to the substitutional Ni+ associated with none, one and two axial nearest neighbour F vacancies for I, II and III centers, respectively. Ni+ I center is found to suffer the relative elongation (5%) along the [0 0 1] (or C4) axis due to the Jahn–Teller effect, while the tetragonal elongation distortions are mainly ascribed to the axial vacancies in II and III centers. The local structures of the various impurity centers are discussed. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Belonging to perovskite-type structure, RbCaF3 has attracted attentions of researchers due to the unique luminescence [1–3] and spectroscopic properties [4,5] as well as structure phase transition behaviours [6–9] when doped with some transition-metal ions. Usually, the optical properties depend strongly on the electronic states and local structures of the impurity ions in the host, and the phase transition behaviours can also be probed by these transition-metal ions. As is well known, electron paramagnetic resonance (EPR) is a powerful technique to study defect structures and electronic properties of paramagnetic ions in crystals, and extensive EPR investigations have been carried out for such systems as Ni+, Ni2+, Mn2+, Cr3+, Fe3+ and Gd3+ in RbCaF3 [8,10–12]. As an unusual system which appears only during irradiation process, Ni+ (3d9) is regarded as an important topic in spectroscopic studies because of the simple energy level structure with one ground state and one excited state under ideal octahedral environments. For example, EPR experiments were performed for RbCaF3:Ni+, and the anisotropic g factors gk and g\ and the planar F superhyperfine parameters Ak and A\ were measured for various tetragonal I, II and III centers (as well as the axial F superhyperfine parameters A0k and A0? for I center) [10]. These centers are * Corresponding author. Address: Department of Applied Physics, University of Electronic Science and Technology of China, Chengdu 610054, PR China. Tel.: +86 2883202586; fax: +86 2883202009. E-mail address: [email protected] (S.-Y. Wu). 0166-1280/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2009.12.005
attributed to the substitutional Ni+ on the host Ca2+ site in RbCaF3, associated with none, one and two nearest neighbour fluorine vacancies (VF) along [0 0 1] (or the C4) axis for I, II and III centers, respectively [10]. Until now, however, no satisfactory and uniform interpretation to the above EPR results has been made for these centers, and the information about local structures for Ni+ in RbCaF3 has not been obtained yet. On the other hand, the previous analysis of the g factors for similar systems (e.g., I center in CsCaF3:Ni+) was usually carried out from the g formulas for a 3d9 ion in tetragonally elongated octahedra using various adjustable parameters (i.e., five independent molecular orbital coefficients a0, a1, a2, b0 and l) [13]. Nevertheless, these treatments failed to connect the EPR spectra with the local structures (e.g., tetragonal distortions) of the systems, and various adjustable parameters were introduced to describe the structure deformations. It is noted that the local tetragonal elongation distortions due to the Jahn–Teller effect were qualitatively mentioned for the Ni+ I centers in RbCaF3 and CsCaF3 [10,13]. However, the above Jahn–Teller distortions for the [NiF6]5 clusters were not quantitatively treated by correlating the g factors to the local structures in the previous works [10,13]. With respect to determining superhyperfine parameters, the previous studies for 3dn (e.g., Ni+, Mn2+) ions in ABF3 compounds usually evaluated the unpaired spin densities fs and fr of the fluorine 2s and 2pr orbitals by fitting the two experimental superhyperfine parameters [14]. But the theoretical relationships were not established for the unpaired spin densities and the covalency (impurity-ligand orbital admixtures) of the systems. In addition, the previous
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calculations of the superhyperfine parameters for the similar Ni+ I center in CsCaF3 were based on the various adjustable molecular orbital coefficients [13]. In general, information about the local structures and the microscopic mechanisms of the EPR spectra for transition-metal ions in RbCaF3 would be useful to investigate the properties of this material (and other ABF3 type compounds) with dopants. In addition, the superhyperfine parameters arise mainly from the interaction between the unpaired electrons of a metal and the nuclear spin of a ligand and play an important role in understanding spin states and electronic structures for paramagnetic ions in crystals. Thus, further theoretical investigations on the g factors and the superhyperfine parameters as well as the local structures for the various impurity centers in RbCaF3:Ni+ are of scientific and practical significance. In this work, uniform calculations of these parameters are presented using their formulas for a 3d9 ion in a tetragonally elongated octahedron. The molecular orbital coefficients arising from covalent interactions between the Ni+ 3d orbitals and the F 2s (and 2p) orbitals are theoretically determined from the cluster approach.
both systems, the tetragonal distortion may be mainly due to the VF, corresponding to an elongated octahedron with one or two apical ligands moving to infinity. For a Ni+ (3d9) ion in tetragonally elongated octahedra, the ground 2Eg irreducible representation may be separated into two orbital singlets 2B1g(|x2 y2i) and 2A1g(|z2i), with the former lying lowest. Meanwhile, the upper 2T2g representation would split into an orbital singlet 2B2g(|xyi) and a doublet 2Eg(|xzi, |yzi) [15]. As mentioned before, the previous EPR analysis for a tetragonally elongated 3d9 cluster was usually based on the simple second-order perturbation formulas of the g factors. Later on, the high (fourth)-order perturbation formulas of the g factors for a 3d9 ion in tetragonally elongated octahedra were derived from the conventional crystal-field model by considering only the central ion orbital and spin–orbit coupling contributions [16]. For the studied Ni+ centers in RbCaF3:Ni+, the systems may exhibit some covalency and impurity-ligand orbital admixtures. In order to study the EPR spectra for these centers to a better extent, the ligand orbital and spin–orbit coupling contributions which were normally neglected in the previous works [13,16] are included here from the cluster approach. Utilizing the perturbation procedure similar to that in Ref. [16] and the cluster approach [17], the improved g formulas are established as follows:
2. Theory and formulas In the cubic phase of RbCaF3, the Ca2+ ion has six nearest neighbour fluorine ions forming the [CaF6]4 octahedron. When a Ni+ ion is doped into the lattice of RbCaF3, it can substitute the host Ca2+ due to their similar ionic radii. Since Ni+ has less positive charge as compared with the replaced Ca2+, some means of charge compensation may occur. As the charge compensation happens far away from the impurity Ni+, its influence on the impurity structure can be neglected and then the uncompensated I center (i.e., the [NiF6]5 cluster) is formed. For the Jahn–Teller ion Ni+, the [NiF6]5 cluster may experience the Jahn–Teller elongation via stretching the two Ni+–F bonds along [0 0 1] (or the C4) axis, which reduces the local symmetry from the original cubic (Oh) to tetragonal (D4h). Thus, the local structure of I center can be described as a relative elongation ratio s (see Fig. 1). The other two centers originate from one and two nearest neighbour fluorine vacancies (VF) along the C4 axis due to the charge compensation for II (i.e., [NiF5]4 cluster) and III (i.e., [NiF4]3 cluster) centers, respectively (see Fig. 1). In
C4 axis
0
0
g k ¼ g s þ 8k f0 =E1 þ kf02 =E22 þ 4k ff0 =ðE1 E2 Þ þ g s f02 ½1=E21 1=ð2E22 Þ kff02 ð4=E1 1=E2 Þ=E22 0
2k ff02 ½2=ðE1 E2 Þ 1=E22 =E1 g s ff02 ½1=ðE1 E22 Þ 1=ð2E32 Þ; 0
0
g ? ¼ g s þ 2k f0 =E2 4kf02 =ðE1 E2 Þ þ k ff0 ð2=E1 1=E2 Þ=E2 þ 2g s f02 =E21 0
þ ff0 ðkf0 k fÞ=ðE1 E22 Þ 0
ff0 ð1=E2 2=E1 Þð2kf0 =E1 þ k f=E2 Þ=ð2E2 Þ g s ff02 ½1=E21 1=ðE1 E2 Þ þ 1=E22 =ð2E2 Þ:
ð1Þ
Here gs(2.0023) is the spin-only value. E1 and E2 are the energy separations between the excited 2B2g and 2Eg and the ground 2B1g states [16], which can be expressed in terms of the cubic field parameter Dq and the tetragonal field parameters Ds and Dt: E1 10Dq and E2 10Dq + 3Ds 5Dt.
C4 axis
C4 axis
II center
III center
2τ R
τR
I center
F
−
Ni+ −
F vacancy (V F)
Fig. 1. The local structures for Ni+ I, II and III centers in RbCaF3. In I center, the ligand octahedron suffers the relative elongation s (5%) along the C4 axis due to the Jahn– Teller effect. On the other hand, one and two axial nearest neighbour F vacancies (VF) occur in II and III centers, respectively.
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Based on the cluster approach [17], the spin–orbit coupling coefficients f (and f0 ) and the orbital reduction factors k (and k0 ) in Eq. (1) are determined as follows:
f0 ¼ ðN t Ne Þ1=2 ðf0d kt ke f0p =2Þ;
f ¼ N t ðf0d þ k2t f0p =2Þ; k ¼ N t ð1 þ f0d
k2t =2Þ;
0
k ¼ ðN t Ne Þ1=2 ½1 kt ðke þ ks AÞ=2;
ð2Þ
f0p
where and are the spin–orbit coupling coefficients of the free 3d9 and ligand ions, respectively. A denotes the integral @ jnpy i, with the impurity-ligand distance R. Nc (here c = t Rhnsj @y and e stands for the irreducible representations t2g and eg of the group Oh) and kc (or ks) are, respectively, the normalization factors and the orbital admixture coefficients. They are usually obtained from the normalization conditions [17]
Nt ð1 2kt Sdpt þ
k2t Þ
¼ 1;
Ne ð1 2ke Sdpe 2ks Sds þ k2e þ k2s Þ ¼ 1;
ð3Þ
and the approximate relationships [17]
h i N2 ¼ N2t 1 þ k2t S2dpt 2kt Sdpt ; h i N2 ¼ N2e 1 þ k2e S2dpe þ k2s S2ds 2ke Sdpe 2ks Sds :
ð4Þ
Here N is the average covalency factor, characteristic of the covalency or impurity-ligand orbital admixture. Sdpc (and Sds) are the group overlap integrals. In general, orbital admixture and overlap between the central ion and ligands have consistent dependence on bond length, and one can approximately adopt the proportional relationship qke/Sdpe ks/Sds between the orbital admixture coefficients and the related group overlap integrals within the same irreducible representation eg, with the proportionality factor q taken as an adjustable parameter. This point is supported by some molecular orbital calculations [18] and can be regarded as reasonable. In the previous investigations of the superhyperfine parameters [14], the unpaired spin densities were normally obtained by fitting two experimental results. However, the influences of the covalency, the ligand 2pr orbitals and the dipole–dipole interactions were inadequately considered. In order to make further studies on the superhyperfine interactions, the improved formulas of the superhyperfine parameters can be similarly established for a 3d9 cluster by including the above contributions using the cluster approach. Thus, we have
Ak ¼ As þ 2Ar þ 2Ad ; A ? ¼ A s Ar Ad :
ð5Þ
Here As denotes the isotropic contributions arising from the ligand 2s orbitals. Ar and Ad indicate the anisotropic contributions due to the admixtures between the impurity 3d and ligand 2p orbitals and the dipole–dipole interactions between the metal electron and ligand nucleus, respectively. The isotropic and anisotropic parts of the superhyperfine parameters may be expanded as follows [14]:
As ¼ fs A0s ;
Ar ¼ fr A0p ;
Ad ¼ gg n bbn =R3 :
ð6Þ
In the above expressions, fs and fr are the unpaired spin densities of the ligands. A0s ¼ ð8p=3Þg s bg n bn j wð0Þj2 15; 192 cm1 and A0p ¼ g s bg n bn hr 3 i2p 463 cm1 [19] are the related nuclear parameters for the ligand F. Here gn is the nuclear g value. b and bn are the electron Bohr magneton and nuclear magneton. w(0) is the wave function of the fluorine 2s orbital at the nucleus. hr3i2p is the expectation value of the inverse cube of the radial wave function of the fluorine 2p orbital. In the calculations of the dipole–dipole interaction term Ad [20], the g factor is conveniently taken as the isotropic or average value [=(gk + 2g\)/3] of those in Eq. (1). In the present calculations, the unpaired spin densities are theoretically determined from the relevant molecular orbital coefficients based on the cluster
approach: fs ¼ N e k2s =3 and fr ¼ N e k2e =3 for the planar fluorine ions. As for the axial fluorine ligands in I center, the unpaired spin densities may be rewritten as fs0 =2 and fr0 =2 [21], corresponding to the axial bond length Rk and the proportionality factor q0 . 3. Application Now the above formulas are applied to the studies of the tetragonal centers in RbCaF3:Ni+. The different local structures for these centers can be characterized by the dissimilar tetragonal field parameters Ds and Dt, which depend upon the relative elongation ratio s in I center and the VF in II and III centers (see Fig. 1). 3.1. Ni+-I center For this uncompensated center with the Jahn–Teller elongation, the parallel and perpendicular impurity-ligand bond lengths may be written in terms of the reference distance R and the relative elongation ratio s as: Rk R(1 + 2s) and R\ R(1 s). Thus, the tetragonal field parameters can be expressed from the superposition model [22] as follows:
Ds ¼ ð4=7ÞA2 ðRÞ½ðR=R? Þt2 ðR=Rk Þt2 ; Dt ¼ ð16=21ÞA4 ðRÞ½ðR=R? Þt4 ðR=Rk Þt4 :
ð7Þ n
Here A2 ðRÞ and A4 ðRÞ are the intrinsic parameters. For 3d ions in octahedral crystal-fields, the relations A4 ðRÞ ð3=4Þ Dq and A2 ðRÞ 10:8A4 ðRÞ have been proved valid in many crystals [23– 25]. t2(3) and t4(5) are the power-law exponents [22]. Therefore, the g factors, especially the anisotropy Dg(=gk g\) can be correlated to the tetragonal field parameters and hence to the local structure of this center. Since the ionic radius ri (0.90 Å, extrapolated from those of Ni2+ and Ni3+ ions [26]) of the impurity Ni+ is smaller than the radius rh (1.14 Å [26]) of the host Ca2+, the reference distance R may be unlike the corresponding cation–anion distance RH (2.228 Å [27]) in the pure crystal. Fortunately, studies based on the extended X-ray absorption fine structure (EXAFS) measurements have verified that the empirical formula R RH + (ri rh)/2 is approximately valid for impurity ions in crystals [28]. Thus, we have R 2.107 Å for the studied Ni+ centers in RbCaF3. From the distance R and the Slater-type SCF functions [29,30], the group overlap integrals Sdpt 0.0045, Sdpe 0.0180, Sds 0.0144 and the integral A 1.5636 are calculated. Then, the molecular orbital coefficients Nc and kc can be obtained from Eqs. (3) and (4), provided the proportionality factor q is known. Using the free-ion values f0d 605 cm1 [31] for Ni+ and f0p 220 cm1 [32] for F, the spin–orbit coupling coefficients f and f0 and the orbital reduction factors k and k0 may be determined from Eq. (2). For RbCaF3:Ni+, to our knowledge, the spectral parameters Dq and N were not reported. However, they can be acquired from those (Dq 490 cm1 and N 0.94 [13]) for the similar [NiF6]5 cluster in CsCaF3:Ni+ (with R 2.142 Å [27]). According to the relationship Dq / R5 [33,34] and the fact that the covalency factor declines slightly with decreasing distance R [35], Dq 540 cm1 and N 0.93 is approximately obtained for RbCaF3:Ni+ here. Thus, there are only two unknown parameters, i.e., the proportionality factor q for the planar fluorine ligands (or q0 for the axial ones) and the relative elongation ratio s, in the formulas of the g factors and the superhyperfine parameters. Substituting these values into Eqs. (1) and (5) and fitting the calculated results to the experimental data, one can obtain
s 5%; q 0:528 ðor q0 0:380Þ:
ð8Þ
The related molecular orbital coefficients, the spin–orbit coupling coefficients and the orbital reduction factors as well as the unpaired
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spin densities are listed in Table 1. The corresponding g factors and the superhyperfine parameters (Cal.c) are given in Table 2. To clarify the importance of the ligand orbital and spin–orbit coupling contributions to the g factors, the theoretical g factors (Cal.b) based on the conventional formulas [16] in the absence of the ligand contributions (i.e., k = k0 = N and f ¼ f0 ¼ Nf0d Þ are also collected in Table 2. For comparisons, the theoretical results (Cal.a) for the similar Ni+ I center in CsCaF3 based on the various adjustable molecular orbital coefficients in the previous work [13] are also shown in Table 2 (note that theoretical calculations for the g factors and the superhyperfine parameters were not performed for the Ni+ centers in RbCaF3 in Ref. [10]). 3.2. Ni+-II center In this center, one nearest neighbour VF occurs along the C4 axis as charge compensation. In view of the positive effective charge of the VF, the central Ni+ may be displaced away from the VF by an amount along the C4 axis due to the electrostatic repulsion. However, in consideration of the monovalent states of Ni+ and F as well as the relatively larger distance (2.2 Å), the electrostatic interaction acting upon the impurity can be insignificant and the impurity displacement may be negligible for the sake of reduction in the number of adjustable parameters. Thus, the low symmetrical distortion of this center arises mainly from the nearest neighbour VF, yielding an elongated octahedron with one apical ligand missing. Similarly, the tetragonal field parameters can be obtained from the superposition model [22]:
Ds ¼ ð2=7ÞA2 ðR0 Þ;
Dt ¼ ð8=21ÞA4 ðR0 Þ:
ð9Þ
Since the presence of the VF may slightly modify the electronic states (and the planar superhyperfine parameters) of the [NiF5]4 cluster as compared with the [NiF6]5 cluster in I center, the covalency factor is adopted as N 0.89 here. Substituting these values into Eqs. (1) and (5) and matching the theoretical results to the experimental data, one can obtain the proportionality factor
q 0:440
ð10Þ
for II center. The molecular orbital coefficients, the spin–orbit coupling coefficients, the orbital reduction factors and the unpaired spin densities are shown in Table 1. The corresponding results (Cal.c) and those (Cal.b) based on the conventional formulas are also collected in Table 2. 3.3. Ni+-III center In this center, two nearest neighbour VF occur in the C4 axis, yielding the [NiF4]3 cluster (corresponding to an elongated octahedron with two apical ligands missing). From the superposition model [22], the tetragonal field parameters may be similarly expressed as follows:
Ds ¼ ð4=7ÞA2 ðR0 Þ;
Dt ¼ ð16=21ÞA4 ðR0 Þ:
ð11Þ
Here the covalency factor is taken as N 0.88 for III center in view of the different local environment. Substituting the related values
Table 2 The g factors and the superhyperfine parameters (in 104 cm1) for various Ni+ centers in RbCaF3:Ni+. Centers
gk
g\
Ak
A\
A0k
A0?
I Cal.a Cal.b Cal.c Expt. [10]
2.822 2.818 2.775 2.778
2.155 2.138 2.131 2.133
52.09 – 64.57 67.71
18.93 – 30.65 29.35
6.85 – 8.39 8.01
10.10 – 12.94 12.34
II Cal.b Cal.c Expt. [10]
2.748 2.687 2.688
2.127 2.117 2.115
– 73.94 73.71
– 30.68 30.02
– – –
– – –
III Cal.b Cal.c Expt. [10]
2.720 2.653 2.651
2.098 2.090 –
– 79.89 –
– 32.38 32.69
– – –
– – –
a Calculations based on various adjustable molecular orbital coefficients for the similar Ni+ I center in CsCaF3 of the previous work [13]. b Calculations based on the conventional formulas of the g factors by neglecting the ligand contributions (i.e., k = k0 = N and f ¼ f0 ¼ Nf0d ) in this work. c Calculations based on the improved formulas by considering the ligand contributions from the cluster approach in this work.
into Eqs. (1) and (5) and fitting the theoretical results to the observed values, the proportionality factor is obtained:
q 0:430:
ð12Þ
The related parameters such as the molecular orbital coefficients are also given in Table 1. The corresponding calculation results (Cal.c and Cal.b) are shown in Table 2. 4. Discussion From Table 2, one can find that the calculated g factors and the superhyperfine parameters (Cal.c) for all the tetragonal centers in RbCaF3:Ni+ are in good agreement with the experimental data. Thus, the observed EPR spectra for these Ni+ centers are satisfactorily explained in a uniform way, and the information about the defect structures are also obtained in this work. (1) As compared with the previous qualitative description of the Jahn–Teller elongations for the I centers in RbCaF3:Ni+ [10] and CsCaF3:Ni+ [13], the present studies quantitatively yield the relative elongation ratio s 5% for Ni+ I center in RbCaF3 by theoretically connecting the EPR spectra with the local structure of the system. The large anisotropy Dg (0.645) for I center may be ascribed to the large s due to the Jahn–Teller effect via stretching and compressing the parallel and perpendicular impurity-ligand bonds by about 0.02 Å and 0.01 Å, respectively. Interestingly, similar relative elongation ratio of about 3% is also reported for the tetragonally elongated [NiF6]5 cluster in CsCaF3:Ni+ based on the local density function method multiple scattering Xa [36]. In addition, the [RhCl6]4 and [RhH6]4 clusters (with the same spin S = 1/ 2) in NaCl:Rh2+ and LiH:Rh2+ were also found to exhibit relative elongation of about 3.4% [37] and 3% [38] along the C4 axis based on the density function theory calculations and the superhyperfine analyses, respectively. Here, the relatively larger s for I center in
Table 1 The molecular orbital coefficients Nc and kc (and ks), the spin–orbit coupling coefficients (in cm1), the orbital reduction factors and the ligand unpaired spin densities for various Ni+ centers in RbCaF3:Ni+.
I II III
Nt
Ne
kt
ke
ks
f
f0
k
k0
fs (%)
fr (%)
0.931 0.891 0.878
0.936 0.898 0.884
0.276 0.354 0.378
0.223 0.283 0.302
0.094 0.099 0.102
571 551 545
559 531 522
0.967 0.947 0.940
0.886 0.825 0.804
0.28 0.30 0.31
1.55 2.39 2.68
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RbCaF3:Ni+ than that in LiH:Rh2+ is in agreement with the weaker chemical bonding and lower elastic constant of the impurity-ligand bond due to the longer distance R and lower valence state of the central ion in the former. So, the relative elongation ratio s acquired in this work for I center may be regarded as valid in physics. On the other hand, the tetragonal elongations in II and III centers arise mainly from the presence of the nearest neighbour VF, corresponding to the relatively smaller anisotropies Dg (0.56– 0.57). (2) The studied Ni+ centers in RbCaF3 exhibit some covalency and impurity-ligand orbital admixtures, characterized by the covalency factors N (0.9 < 1) and the moderate orbital admixture coefficients (kt ke 0.3 and ks 0.1) obtained from the cluster approach. In view of the electrostatic attraction of the apical VF and the central Ni+ in II and III centers, the planar fluorine ions are expected to move slightly towards the center of the octahedron and lead to smaller impurity-ligand distances. This results in more significant electronic cloud overlap between Ni+ and the planar F and hence lower N for both centers [35]. In addition, shrinkage of the planar bond lengths may also induce an increase of Dq, which can largely compensate the decrease of Dq arising from the presence of the VF. Thus, the above slight modifications of the planar ligands in II and III centers may not produce obvious deviations of Dq from that for I center and the values of Ds and Dt from Eqs. (9) and (11) or significant errors for the final results. When the ligand orbital and spin–orbit coupling contributions are neglected, the theoretical g factors (Cal.b) are not as good as those (Cal.c) including these contributions, particularly gk are larger than the experimental data. Moreover, the above discrepancies cannot be removed by modifying the relative elongation ratio s for I center and the proportionality factors q for all the centers, because the variations of s or q can merely influence the third-order perturbation term (related to E2) in the formula of gk and hardly depress the increase of gk under omission of the ligand contributions. Therefore, the anisotropic contributions to the g factors from the different components of the orbital reduction factors (with the relative deviation k/k0 1 10–20%) and the spin–orbit coupling coefficients (with the relative deviation f/f0 1 2–5%) should be taken into account using the cluster approach. On the other hand, the advantages of adopting only two adjustable parameters (i.e., the relative elongation ratio s and the proportionality factor q) over the previous work [13] for I center may be discussed here. First, reduction in the number of adjustable parameters in this work is achieved by establishing the relationships among the related molecular orbital coefficients (e.g., Nc and kc) based on the cluster approach (see Eqs. ((2)–(4)). Second, the tetragonal field parameters are quantitatively determined from the local structures of the systems using the superposition model, and their contributions to the g factors are explicitly indicated in the energy denominator E2. However, various molecular orbital coefficients (e.g., a0, a1, a2, b0 and l) were taken as adjustable parameters, and the tetragonal distortion (local structure) was not quantitatively correlated to the EPR spectra in the previous treatments for the Ni+ I center in CsCaF3 [13]. (3) From Table 1, the unpaired spin densities fs and fr increase quadratically with the increase of the corresponding orbital admixture coefficients in the order of I < II < III, and the resultant superhyperfine parameters also show similar increasing trend on the whole. This is consistent with the decrease of the covalency factor N (or increase of the covalency effect) and the decline of the proportionality factor q from I to III centers. Meanwhile, the even smaller q0 (0.380) for the axial F superhyperfine parameters in I center also obey the same rule in view of the longer axial impurity-ligand distance Rk due to the Jahn–Teller elongation. The fitted proportionality factors q are lower than unity, suggesting that the ratio ks/Sds is about 50% smaller than ke/Sdpe. This can be attributed to the much less admixture of the 3d orbital of Ni+ with the 2s orbi-
tal of F than that with the 2p orbital, since the 2s orbital is usually more compact than the 2p one for the same ligand. The decrease of q may be interpreted as the modifications of the electronic clouds around the Ni+ due to the presence of the VF in II and III centers. The slightly shorter planar Ni+–F distances in both centers can induce stronger admixture between the Ni+ 3d (|x2 y2i) and the F 2p orbitals and thus result in relatively weaker Ni+ 3d–F 2s admixture. The small q (0.430) obtained for the [NiF4]3 cluster in III center compares favorably with low proportionality factors q (0.430 and 0.446) that were reported for the similar isoelectronic square-planar [CuX4]2 clusters in K2PdX4:Cu2+ (X = Cl, Br) based on the EPR analysis [39]. Thus, the proportionality factors and the resultant superhyperfine parameters in this work can be regarded as reasonable. As mentioned before for the g factors, only the proportionality factor q is adopted as the adjustable parameter in the superhyperfine parameters based on the present cluster approach treatments, which seem superior to the previous studies for the similar Ni+ I center in CsCaF3 [13] based on various fitted molecular orbital coefficients. (4) There are some errors in the above calculations. First, approximation of the theoretical model and the perturbation formulas adopted in this work can lead to some errors. Second, the spectral parameters Dq and N estimated from those of the similar CsCaF3:Ni+ [13] may also introduce some errors into the calculation results. When Dq changes by 10%, the fitted s and the g factors would vary by about 0.5%, suggesting that Dq is somewhat related to the tetragonal distortion and hence to the anisotropy Dg of the systems. As N changes by 10%, the errors for the resultant s and the g factors are estimated to be not more than 0.4%, because the covalency influences mainly the average of the g factors. Third, the errors may arise from the approximation of the relationship A2 ðRÞ 10:8A4 ðRÞ [23–25] for the tetragonal field parameters. The deviation of the relative elongation ratio s in I center is estimated to be about 3% as the ratio A2 ðRÞ=A4 ðRÞ varies within the widely accepted range of 9–12. Fourth, the reference distance R obtained from the empirical formula [28] would bring forward some errors to the group overlap integrals and hence to the resultant g factors and the superhyperfine parameters. When the host distance RH is adopted in the calculations, the errors of about 4% may be induced for the group overlap integrals, which lead to the deviations of no more than 0.5% for the final results. Fifth, in the calculations for II center, the impurity off-center displacement due to the nearest neighbour VF is not taken into account. When considering the above impurity displacement in Eq. (9), the optimal value of this displacement is found to be about 0.008 Å. This can be illustrated by monovalence of Ni+ and F and the relatively larger distance, only yielding weak electrostatic repulsion upon the central Ni+. In addition, the electrostatic attraction of the planar F ligands can make the Ni+ move towards the center of the octahedron, which largely cancels the above off-center displacement. As a result, the impurity displacement may be regarded as negligible here. References [1] A.N. Belsky, P. Chevallier, E.N. Mel’chakov, C. Pédrini, P.A. Rodnyi, A.N. Vasil’ev, Chem. Phys. Lett. 278 (1997) 369. [2] M.C. Marco de Lucas, F. Rodriguez, M. Moreno, A. Tressaud, J. Phys. Condens. Matter 6 (1994) 6353. [3] M.C. Marco de Lucas, F. Rodriguez, M. Moreno, J. Phys. Condens. Matter 5 (1993) 1437. [4] B. Vdlacampa, R. Alcalá, P.J. Alonso, J.M. Spaeth, J. Phys. Condens. Matter 5 (1993) 747. [5] M.C. Marco de Lucas, F. Rodriguez, M. Moreno, J. Phys. Condens. Matter 7 (1995) 7535. [6] A. Trokiner, H. Zanni-Theveneau, J. Phys. C 21 (1988) 4913. [7] L.L. Boyer, J.R. Hardy, Phys. Rev. B 24 (1981) 2577. [8] J.Y. Buzare, P. Foucher, J. Phys. Condens. Matter 3 (1991) 2535. [9] F.A. Modine, E. Sonder, W.P. Unruh, Phys. Rev. B. 10 (1974) 1623.
H.-M. Zhang et al. / Journal of Molecular Structure: THEOCHEM 942 (2010) 104–109 [10] R. Alcalá, E. Zorita, P.J. Alonso, Phys. Rev. B 38 (1988) 11,156. [11] F. Lahoz, P.J. Alonso, R. Alcalá, T. Pawlik, J.M. Spaeth, J. Phys. Condens. Matter 7 (1995) 8637. [12] C. Rudowicz, Phys. Rev. B 37 (1988) 27. [13] B. Villacampa, R. Alcalá, P.J. Alonso, Phys. Rev. B 49 (1994) 1039. [14] J. Owen, J.H.M. Thornley, Rep. Prog. Phys. 29 (1966) 675. [15] A. Abragam, B. Blenaney, Electron Paramagnetic Resonance of Transition Ions, Oxford University Press, London, 1970. [16] W.C. Zheng, S.Y. Wu, Z. Naturforsch. A 55 (2000) 915. [17] S.Y. Wu, X.Y. Gao, H.N. Dong, J. Magn. Magn. Mater. 301 (2006) 67. [18] J.T. Vallin, G.D. Watkins, Phys. Rev. B 9 (1974) 2051. [19] M.T. Barriuso, J.A. Aramburu, M. Moreno, J. Phys. Condens. Matter 2 (1990) 771. [20] R. Alcalá, B. Villacampa, Solid State Commun. 90 (1994) 13. [21] Z. Luz, A. Raizman, J.T. Suss, Solid State Commun. 21 (1977) 849. [22] D.J. Newman, B. Ng, Rep. Prog. Phys. 52 (1989) 699. [23] C. Rudowicz, J. Phys. C 20 (1987) 6033. [24] C. Rudowicz, Z.Y. Yang, Y.Y. Yeung, J. Qin, J. Phys. Chem. Solids 64 (2004) 1419. [25] Z.Y. Yang, J. Phys. Condens. Matter 12 (2000) 4091.
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Journal of Molecular Structure: THEOCHEM journal homepage: www.elsevier.com/locate/theochem
Investigations on the g factors and superhyperfine parameters for various tetragonal Ni+ centers in RbCaF3 Hua-Ming Zhang a, Shao-Yi Wu a,b,*, Li-Li Li a, Pei Xu a a b
Department of Applied Physics, University of Electronic Science and Technology of China, Chengdu 610054, PR China International Center for Materials Physics, Chinese Academy of Sciences, Shenyang 110016, PR China
a r t i c l e
i n f o
Article history: Received 5 November 2009 Received in revised form 26 November 2009 Accepted 4 December 2009 Available online 8 January 2010 Keywords: Electron paramagnetic resonance Crystal-fields and spin Hamiltonians Ni+ RbCaF3
a b s t r a c t The anisotropic g factors gk and g\ and the planar F superhyperfine parameters Ak and A\ in the tetragonal impurity Ni+ I, II and III centers (as well as the axial F superhyperfine parameters A0k and A0? in I center) on the substitutional Ca2+ site in RbCaF3 are theoretically investigated in a uniform way using the perturbation formulas of these parameters for a 3d9 ion in tetragonally elongated octahedra. In the calculations, the ligand unpaired spin densities are determined quantitatively from the related molecular orbital coefficients based on the cluster approach. These defects are attributed to the substitutional Ni+ associated with none, one and two axial nearest neighbour F vacancies for I, II and III centers, respectively. Ni+ I center is found to suffer the relative elongation (5%) along the [0 0 1] (or C4) axis due to the Jahn–Teller effect, while the tetragonal elongation distortions are mainly ascribed to the axial vacancies in II and III centers. The local structures of the various impurity centers are discussed. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Belonging to perovskite-type structure, RbCaF3 has attracted attentions of researchers due to the unique luminescence [1–3] and spectroscopic properties [4,5] as well as structure phase transition behaviours [6–9] when doped with some transition-metal ions. Usually, the optical properties depend strongly on the electronic states and local structures of the impurity ions in the host, and the phase transition behaviours can also be probed by these transition-metal ions. As is well known, electron paramagnetic resonance (EPR) is a powerful technique to study defect structures and electronic properties of paramagnetic ions in crystals, and extensive EPR investigations have been carried out for such systems as Ni+, Ni2+, Mn2+, Cr3+, Fe3+ and Gd3+ in RbCaF3 [8,10–12]. As an unusual system which appears only during irradiation process, Ni+ (3d9) is regarded as an important topic in spectroscopic studies because of the simple energy level structure with one ground state and one excited state under ideal octahedral environments. For example, EPR experiments were performed for RbCaF3:Ni+, and the anisotropic g factors gk and g\ and the planar F superhyperfine parameters Ak and A\ were measured for various tetragonal I, II and III centers (as well as the axial F superhyperfine parameters A0k and A0? for I center) [10]. These centers are * Corresponding author. Address: Department of Applied Physics, University of Electronic Science and Technology of China, Chengdu 610054, PR China. Tel.: +86 2883202586; fax: +86 2883202009. E-mail address: [email protected] (S.-Y. Wu). 0166-1280/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2009.12.005
attributed to the substitutional Ni+ on the host Ca2+ site in RbCaF3, associated with none, one and two nearest neighbour fluorine vacancies (VF) along [0 0 1] (or the C4) axis for I, II and III centers, respectively [10]. Until now, however, no satisfactory and uniform interpretation to the above EPR results has been made for these centers, and the information about local structures for Ni+ in RbCaF3 has not been obtained yet. On the other hand, the previous analysis of the g factors for similar systems (e.g., I center in CsCaF3:Ni+) was usually carried out from the g formulas for a 3d9 ion in tetragonally elongated octahedra using various adjustable parameters (i.e., five independent molecular orbital coefficients a0, a1, a2, b0 and l) [13]. Nevertheless, these treatments failed to connect the EPR spectra with the local structures (e.g., tetragonal distortions) of the systems, and various adjustable parameters were introduced to describe the structure deformations. It is noted that the local tetragonal elongation distortions due to the Jahn–Teller effect were qualitatively mentioned for the Ni+ I centers in RbCaF3 and CsCaF3 [10,13]. However, the above Jahn–Teller distortions for the [NiF6]5 clusters were not quantitatively treated by correlating the g factors to the local structures in the previous works [10,13]. With respect to determining superhyperfine parameters, the previous studies for 3dn (e.g., Ni+, Mn2+) ions in ABF3 compounds usually evaluated the unpaired spin densities fs and fr of the fluorine 2s and 2pr orbitals by fitting the two experimental superhyperfine parameters [14]. But the theoretical relationships were not established for the unpaired spin densities and the covalency (impurity-ligand orbital admixtures) of the systems. In addition, the previous
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calculations of the superhyperfine parameters for the similar Ni+ I center in CsCaF3 were based on the various adjustable molecular orbital coefficients [13]. In general, information about the local structures and the microscopic mechanisms of the EPR spectra for transition-metal ions in RbCaF3 would be useful to investigate the properties of this material (and other ABF3 type compounds) with dopants. In addition, the superhyperfine parameters arise mainly from the interaction between the unpaired electrons of a metal and the nuclear spin of a ligand and play an important role in understanding spin states and electronic structures for paramagnetic ions in crystals. Thus, further theoretical investigations on the g factors and the superhyperfine parameters as well as the local structures for the various impurity centers in RbCaF3:Ni+ are of scientific and practical significance. In this work, uniform calculations of these parameters are presented using their formulas for a 3d9 ion in a tetragonally elongated octahedron. The molecular orbital coefficients arising from covalent interactions between the Ni+ 3d orbitals and the F 2s (and 2p) orbitals are theoretically determined from the cluster approach.
both systems, the tetragonal distortion may be mainly due to the VF, corresponding to an elongated octahedron with one or two apical ligands moving to infinity. For a Ni+ (3d9) ion in tetragonally elongated octahedra, the ground 2Eg irreducible representation may be separated into two orbital singlets 2B1g(|x2 y2i) and 2A1g(|z2i), with the former lying lowest. Meanwhile, the upper 2T2g representation would split into an orbital singlet 2B2g(|xyi) and a doublet 2Eg(|xzi, |yzi) [15]. As mentioned before, the previous EPR analysis for a tetragonally elongated 3d9 cluster was usually based on the simple second-order perturbation formulas of the g factors. Later on, the high (fourth)-order perturbation formulas of the g factors for a 3d9 ion in tetragonally elongated octahedra were derived from the conventional crystal-field model by considering only the central ion orbital and spin–orbit coupling contributions [16]. For the studied Ni+ centers in RbCaF3:Ni+, the systems may exhibit some covalency and impurity-ligand orbital admixtures. In order to study the EPR spectra for these centers to a better extent, the ligand orbital and spin–orbit coupling contributions which were normally neglected in the previous works [13,16] are included here from the cluster approach. Utilizing the perturbation procedure similar to that in Ref. [16] and the cluster approach [17], the improved g formulas are established as follows:
2. Theory and formulas In the cubic phase of RbCaF3, the Ca2+ ion has six nearest neighbour fluorine ions forming the [CaF6]4 octahedron. When a Ni+ ion is doped into the lattice of RbCaF3, it can substitute the host Ca2+ due to their similar ionic radii. Since Ni+ has less positive charge as compared with the replaced Ca2+, some means of charge compensation may occur. As the charge compensation happens far away from the impurity Ni+, its influence on the impurity structure can be neglected and then the uncompensated I center (i.e., the [NiF6]5 cluster) is formed. For the Jahn–Teller ion Ni+, the [NiF6]5 cluster may experience the Jahn–Teller elongation via stretching the two Ni+–F bonds along [0 0 1] (or the C4) axis, which reduces the local symmetry from the original cubic (Oh) to tetragonal (D4h). Thus, the local structure of I center can be described as a relative elongation ratio s (see Fig. 1). The other two centers originate from one and two nearest neighbour fluorine vacancies (VF) along the C4 axis due to the charge compensation for II (i.e., [NiF5]4 cluster) and III (i.e., [NiF4]3 cluster) centers, respectively (see Fig. 1). In
C4 axis
0
0
g k ¼ g s þ 8k f0 =E1 þ kf02 =E22 þ 4k ff0 =ðE1 E2 Þ þ g s f02 ½1=E21 1=ð2E22 Þ kff02 ð4=E1 1=E2 Þ=E22 0
2k ff02 ½2=ðE1 E2 Þ 1=E22 =E1 g s ff02 ½1=ðE1 E22 Þ 1=ð2E32 Þ; 0
0
g ? ¼ g s þ 2k f0 =E2 4kf02 =ðE1 E2 Þ þ k ff0 ð2=E1 1=E2 Þ=E2 þ 2g s f02 =E21 0
þ ff0 ðkf0 k fÞ=ðE1 E22 Þ 0
ff0 ð1=E2 2=E1 Þð2kf0 =E1 þ k f=E2 Þ=ð2E2 Þ g s ff02 ½1=E21 1=ðE1 E2 Þ þ 1=E22 =ð2E2 Þ:
ð1Þ
Here gs(2.0023) is the spin-only value. E1 and E2 are the energy separations between the excited 2B2g and 2Eg and the ground 2B1g states [16], which can be expressed in terms of the cubic field parameter Dq and the tetragonal field parameters Ds and Dt: E1 10Dq and E2 10Dq + 3Ds 5Dt.
C4 axis
C4 axis
II center
III center
2τ R
τR
I center
F
−
Ni+ −
F vacancy (V F)
Fig. 1. The local structures for Ni+ I, II and III centers in RbCaF3. In I center, the ligand octahedron suffers the relative elongation s (5%) along the C4 axis due to the Jahn– Teller effect. On the other hand, one and two axial nearest neighbour F vacancies (VF) occur in II and III centers, respectively.
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Based on the cluster approach [17], the spin–orbit coupling coefficients f (and f0 ) and the orbital reduction factors k (and k0 ) in Eq. (1) are determined as follows:
f0 ¼ ðN t Ne Þ1=2 ðf0d kt ke f0p =2Þ;
f ¼ N t ðf0d þ k2t f0p =2Þ; k ¼ N t ð1 þ f0d
k2t =2Þ;
0
k ¼ ðN t Ne Þ1=2 ½1 kt ðke þ ks AÞ=2;
ð2Þ
f0p
where and are the spin–orbit coupling coefficients of the free 3d9 and ligand ions, respectively. A denotes the integral @ jnpy i, with the impurity-ligand distance R. Nc (here c = t Rhnsj @y and e stands for the irreducible representations t2g and eg of the group Oh) and kc (or ks) are, respectively, the normalization factors and the orbital admixture coefficients. They are usually obtained from the normalization conditions [17]
Nt ð1 2kt Sdpt þ
k2t Þ
¼ 1;
Ne ð1 2ke Sdpe 2ks Sds þ k2e þ k2s Þ ¼ 1;
ð3Þ
and the approximate relationships [17]
h i N2 ¼ N2t 1 þ k2t S2dpt 2kt Sdpt ; h i N2 ¼ N2e 1 þ k2e S2dpe þ k2s S2ds 2ke Sdpe 2ks Sds :
ð4Þ
Here N is the average covalency factor, characteristic of the covalency or impurity-ligand orbital admixture. Sdpc (and Sds) are the group overlap integrals. In general, orbital admixture and overlap between the central ion and ligands have consistent dependence on bond length, and one can approximately adopt the proportional relationship qke/Sdpe ks/Sds between the orbital admixture coefficients and the related group overlap integrals within the same irreducible representation eg, with the proportionality factor q taken as an adjustable parameter. This point is supported by some molecular orbital calculations [18] and can be regarded as reasonable. In the previous investigations of the superhyperfine parameters [14], the unpaired spin densities were normally obtained by fitting two experimental results. However, the influences of the covalency, the ligand 2pr orbitals and the dipole–dipole interactions were inadequately considered. In order to make further studies on the superhyperfine interactions, the improved formulas of the superhyperfine parameters can be similarly established for a 3d9 cluster by including the above contributions using the cluster approach. Thus, we have
Ak ¼ As þ 2Ar þ 2Ad ; A ? ¼ A s Ar Ad :
ð5Þ
Here As denotes the isotropic contributions arising from the ligand 2s orbitals. Ar and Ad indicate the anisotropic contributions due to the admixtures between the impurity 3d and ligand 2p orbitals and the dipole–dipole interactions between the metal electron and ligand nucleus, respectively. The isotropic and anisotropic parts of the superhyperfine parameters may be expanded as follows [14]:
As ¼ fs A0s ;
Ar ¼ fr A0p ;
Ad ¼ gg n bbn =R3 :
ð6Þ
In the above expressions, fs and fr are the unpaired spin densities of the ligands. A0s ¼ ð8p=3Þg s bg n bn j wð0Þj2 15; 192 cm1 and A0p ¼ g s bg n bn hr 3 i2p 463 cm1 [19] are the related nuclear parameters for the ligand F. Here gn is the nuclear g value. b and bn are the electron Bohr magneton and nuclear magneton. w(0) is the wave function of the fluorine 2s orbital at the nucleus. hr3i2p is the expectation value of the inverse cube of the radial wave function of the fluorine 2p orbital. In the calculations of the dipole–dipole interaction term Ad [20], the g factor is conveniently taken as the isotropic or average value [=(gk + 2g\)/3] of those in Eq. (1). In the present calculations, the unpaired spin densities are theoretically determined from the relevant molecular orbital coefficients based on the cluster
approach: fs ¼ N e k2s =3 and fr ¼ N e k2e =3 for the planar fluorine ions. As for the axial fluorine ligands in I center, the unpaired spin densities may be rewritten as fs0 =2 and fr0 =2 [21], corresponding to the axial bond length Rk and the proportionality factor q0 . 3. Application Now the above formulas are applied to the studies of the tetragonal centers in RbCaF3:Ni+. The different local structures for these centers can be characterized by the dissimilar tetragonal field parameters Ds and Dt, which depend upon the relative elongation ratio s in I center and the VF in II and III centers (see Fig. 1). 3.1. Ni+-I center For this uncompensated center with the Jahn–Teller elongation, the parallel and perpendicular impurity-ligand bond lengths may be written in terms of the reference distance R and the relative elongation ratio s as: Rk R(1 + 2s) and R\ R(1 s). Thus, the tetragonal field parameters can be expressed from the superposition model [22] as follows:
Ds ¼ ð4=7ÞA2 ðRÞ½ðR=R? Þt2 ðR=Rk Þt2 ; Dt ¼ ð16=21ÞA4 ðRÞ½ðR=R? Þt4 ðR=Rk Þt4 :
ð7Þ n
Here A2 ðRÞ and A4 ðRÞ are the intrinsic parameters. For 3d ions in octahedral crystal-fields, the relations A4 ðRÞ ð3=4Þ Dq and A2 ðRÞ 10:8A4 ðRÞ have been proved valid in many crystals [23– 25]. t2(3) and t4(5) are the power-law exponents [22]. Therefore, the g factors, especially the anisotropy Dg(=gk g\) can be correlated to the tetragonal field parameters and hence to the local structure of this center. Since the ionic radius ri (0.90 Å, extrapolated from those of Ni2+ and Ni3+ ions [26]) of the impurity Ni+ is smaller than the radius rh (1.14 Å [26]) of the host Ca2+, the reference distance R may be unlike the corresponding cation–anion distance RH (2.228 Å [27]) in the pure crystal. Fortunately, studies based on the extended X-ray absorption fine structure (EXAFS) measurements have verified that the empirical formula R RH + (ri rh)/2 is approximately valid for impurity ions in crystals [28]. Thus, we have R 2.107 Å for the studied Ni+ centers in RbCaF3. From the distance R and the Slater-type SCF functions [29,30], the group overlap integrals Sdpt 0.0045, Sdpe 0.0180, Sds 0.0144 and the integral A 1.5636 are calculated. Then, the molecular orbital coefficients Nc and kc can be obtained from Eqs. (3) and (4), provided the proportionality factor q is known. Using the free-ion values f0d 605 cm1 [31] for Ni+ and f0p 220 cm1 [32] for F, the spin–orbit coupling coefficients f and f0 and the orbital reduction factors k and k0 may be determined from Eq. (2). For RbCaF3:Ni+, to our knowledge, the spectral parameters Dq and N were not reported. However, they can be acquired from those (Dq 490 cm1 and N 0.94 [13]) for the similar [NiF6]5 cluster in CsCaF3:Ni+ (with R 2.142 Å [27]). According to the relationship Dq / R5 [33,34] and the fact that the covalency factor declines slightly with decreasing distance R [35], Dq 540 cm1 and N 0.93 is approximately obtained for RbCaF3:Ni+ here. Thus, there are only two unknown parameters, i.e., the proportionality factor q for the planar fluorine ligands (or q0 for the axial ones) and the relative elongation ratio s, in the formulas of the g factors and the superhyperfine parameters. Substituting these values into Eqs. (1) and (5) and fitting the calculated results to the experimental data, one can obtain
s 5%; q 0:528 ðor q0 0:380Þ:
ð8Þ
The related molecular orbital coefficients, the spin–orbit coupling coefficients and the orbital reduction factors as well as the unpaired
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spin densities are listed in Table 1. The corresponding g factors and the superhyperfine parameters (Cal.c) are given in Table 2. To clarify the importance of the ligand orbital and spin–orbit coupling contributions to the g factors, the theoretical g factors (Cal.b) based on the conventional formulas [16] in the absence of the ligand contributions (i.e., k = k0 = N and f ¼ f0 ¼ Nf0d Þ are also collected in Table 2. For comparisons, the theoretical results (Cal.a) for the similar Ni+ I center in CsCaF3 based on the various adjustable molecular orbital coefficients in the previous work [13] are also shown in Table 2 (note that theoretical calculations for the g factors and the superhyperfine parameters were not performed for the Ni+ centers in RbCaF3 in Ref. [10]). 3.2. Ni+-II center In this center, one nearest neighbour VF occurs along the C4 axis as charge compensation. In view of the positive effective charge of the VF, the central Ni+ may be displaced away from the VF by an amount along the C4 axis due to the electrostatic repulsion. However, in consideration of the monovalent states of Ni+ and F as well as the relatively larger distance (2.2 Å), the electrostatic interaction acting upon the impurity can be insignificant and the impurity displacement may be negligible for the sake of reduction in the number of adjustable parameters. Thus, the low symmetrical distortion of this center arises mainly from the nearest neighbour VF, yielding an elongated octahedron with one apical ligand missing. Similarly, the tetragonal field parameters can be obtained from the superposition model [22]:
Ds ¼ ð2=7ÞA2 ðR0 Þ;
Dt ¼ ð8=21ÞA4 ðR0 Þ:
ð9Þ
Since the presence of the VF may slightly modify the electronic states (and the planar superhyperfine parameters) of the [NiF5]4 cluster as compared with the [NiF6]5 cluster in I center, the covalency factor is adopted as N 0.89 here. Substituting these values into Eqs. (1) and (5) and matching the theoretical results to the experimental data, one can obtain the proportionality factor
q 0:440
ð10Þ
for II center. The molecular orbital coefficients, the spin–orbit coupling coefficients, the orbital reduction factors and the unpaired spin densities are shown in Table 1. The corresponding results (Cal.c) and those (Cal.b) based on the conventional formulas are also collected in Table 2. 3.3. Ni+-III center In this center, two nearest neighbour VF occur in the C4 axis, yielding the [NiF4]3 cluster (corresponding to an elongated octahedron with two apical ligands missing). From the superposition model [22], the tetragonal field parameters may be similarly expressed as follows:
Ds ¼ ð4=7ÞA2 ðR0 Þ;
Dt ¼ ð16=21ÞA4 ðR0 Þ:
ð11Þ
Here the covalency factor is taken as N 0.88 for III center in view of the different local environment. Substituting the related values
Table 2 The g factors and the superhyperfine parameters (in 104 cm1) for various Ni+ centers in RbCaF3:Ni+. Centers
gk
g\
Ak
A\
A0k
A0?
I Cal.a Cal.b Cal.c Expt. [10]
2.822 2.818 2.775 2.778
2.155 2.138 2.131 2.133
52.09 – 64.57 67.71
18.93 – 30.65 29.35
6.85 – 8.39 8.01
10.10 – 12.94 12.34
II Cal.b Cal.c Expt. [10]
2.748 2.687 2.688
2.127 2.117 2.115
– 73.94 73.71
– 30.68 30.02
– – –
– – –
III Cal.b Cal.c Expt. [10]
2.720 2.653 2.651
2.098 2.090 –
– 79.89 –
– 32.38 32.69
– – –
– – –
a Calculations based on various adjustable molecular orbital coefficients for the similar Ni+ I center in CsCaF3 of the previous work [13]. b Calculations based on the conventional formulas of the g factors by neglecting the ligand contributions (i.e., k = k0 = N and f ¼ f0 ¼ Nf0d ) in this work. c Calculations based on the improved formulas by considering the ligand contributions from the cluster approach in this work.
into Eqs. (1) and (5) and fitting the theoretical results to the observed values, the proportionality factor is obtained:
q 0:430:
ð12Þ
The related parameters such as the molecular orbital coefficients are also given in Table 1. The corresponding calculation results (Cal.c and Cal.b) are shown in Table 2. 4. Discussion From Table 2, one can find that the calculated g factors and the superhyperfine parameters (Cal.c) for all the tetragonal centers in RbCaF3:Ni+ are in good agreement with the experimental data. Thus, the observed EPR spectra for these Ni+ centers are satisfactorily explained in a uniform way, and the information about the defect structures are also obtained in this work. (1) As compared with the previous qualitative description of the Jahn–Teller elongations for the I centers in RbCaF3:Ni+ [10] and CsCaF3:Ni+ [13], the present studies quantitatively yield the relative elongation ratio s 5% for Ni+ I center in RbCaF3 by theoretically connecting the EPR spectra with the local structure of the system. The large anisotropy Dg (0.645) for I center may be ascribed to the large s due to the Jahn–Teller effect via stretching and compressing the parallel and perpendicular impurity-ligand bonds by about 0.02 Å and 0.01 Å, respectively. Interestingly, similar relative elongation ratio of about 3% is also reported for the tetragonally elongated [NiF6]5 cluster in CsCaF3:Ni+ based on the local density function method multiple scattering Xa [36]. In addition, the [RhCl6]4 and [RhH6]4 clusters (with the same spin S = 1/ 2) in NaCl:Rh2+ and LiH:Rh2+ were also found to exhibit relative elongation of about 3.4% [37] and 3% [38] along the C4 axis based on the density function theory calculations and the superhyperfine analyses, respectively. Here, the relatively larger s for I center in
Table 1 The molecular orbital coefficients Nc and kc (and ks), the spin–orbit coupling coefficients (in cm1), the orbital reduction factors and the ligand unpaired spin densities for various Ni+ centers in RbCaF3:Ni+.
I II III
Nt
Ne
kt
ke
ks
f
f0
k
k0
fs (%)
fr (%)
0.931 0.891 0.878
0.936 0.898 0.884
0.276 0.354 0.378
0.223 0.283 0.302
0.094 0.099 0.102
571 551 545
559 531 522
0.967 0.947 0.940
0.886 0.825 0.804
0.28 0.30 0.31
1.55 2.39 2.68
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RbCaF3:Ni+ than that in LiH:Rh2+ is in agreement with the weaker chemical bonding and lower elastic constant of the impurity-ligand bond due to the longer distance R and lower valence state of the central ion in the former. So, the relative elongation ratio s acquired in this work for I center may be regarded as valid in physics. On the other hand, the tetragonal elongations in II and III centers arise mainly from the presence of the nearest neighbour VF, corresponding to the relatively smaller anisotropies Dg (0.56– 0.57). (2) The studied Ni+ centers in RbCaF3 exhibit some covalency and impurity-ligand orbital admixtures, characterized by the covalency factors N (0.9 < 1) and the moderate orbital admixture coefficients (kt ke 0.3 and ks 0.1) obtained from the cluster approach. In view of the electrostatic attraction of the apical VF and the central Ni+ in II and III centers, the planar fluorine ions are expected to move slightly towards the center of the octahedron and lead to smaller impurity-ligand distances. This results in more significant electronic cloud overlap between Ni+ and the planar F and hence lower N for both centers [35]. In addition, shrinkage of the planar bond lengths may also induce an increase of Dq, which can largely compensate the decrease of Dq arising from the presence of the VF. Thus, the above slight modifications of the planar ligands in II and III centers may not produce obvious deviations of Dq from that for I center and the values of Ds and Dt from Eqs. (9) and (11) or significant errors for the final results. When the ligand orbital and spin–orbit coupling contributions are neglected, the theoretical g factors (Cal.b) are not as good as those (Cal.c) including these contributions, particularly gk are larger than the experimental data. Moreover, the above discrepancies cannot be removed by modifying the relative elongation ratio s for I center and the proportionality factors q for all the centers, because the variations of s or q can merely influence the third-order perturbation term (related to E2) in the formula of gk and hardly depress the increase of gk under omission of the ligand contributions. Therefore, the anisotropic contributions to the g factors from the different components of the orbital reduction factors (with the relative deviation k/k0 1 10–20%) and the spin–orbit coupling coefficients (with the relative deviation f/f0 1 2–5%) should be taken into account using the cluster approach. On the other hand, the advantages of adopting only two adjustable parameters (i.e., the relative elongation ratio s and the proportionality factor q) over the previous work [13] for I center may be discussed here. First, reduction in the number of adjustable parameters in this work is achieved by establishing the relationships among the related molecular orbital coefficients (e.g., Nc and kc) based on the cluster approach (see Eqs. ((2)–(4)). Second, the tetragonal field parameters are quantitatively determined from the local structures of the systems using the superposition model, and their contributions to the g factors are explicitly indicated in the energy denominator E2. However, various molecular orbital coefficients (e.g., a0, a1, a2, b0 and l) were taken as adjustable parameters, and the tetragonal distortion (local structure) was not quantitatively correlated to the EPR spectra in the previous treatments for the Ni+ I center in CsCaF3 [13]. (3) From Table 1, the unpaired spin densities fs and fr increase quadratically with the increase of the corresponding orbital admixture coefficients in the order of I < II < III, and the resultant superhyperfine parameters also show similar increasing trend on the whole. This is consistent with the decrease of the covalency factor N (or increase of the covalency effect) and the decline of the proportionality factor q from I to III centers. Meanwhile, the even smaller q0 (0.380) for the axial F superhyperfine parameters in I center also obey the same rule in view of the longer axial impurity-ligand distance Rk due to the Jahn–Teller elongation. The fitted proportionality factors q are lower than unity, suggesting that the ratio ks/Sds is about 50% smaller than ke/Sdpe. This can be attributed to the much less admixture of the 3d orbital of Ni+ with the 2s orbi-
tal of F than that with the 2p orbital, since the 2s orbital is usually more compact than the 2p one for the same ligand. The decrease of q may be interpreted as the modifications of the electronic clouds around the Ni+ due to the presence of the VF in II and III centers. The slightly shorter planar Ni+–F distances in both centers can induce stronger admixture between the Ni+ 3d (|x2 y2i) and the F 2p orbitals and thus result in relatively weaker Ni+ 3d–F 2s admixture. The small q (0.430) obtained for the [NiF4]3 cluster in III center compares favorably with low proportionality factors q (0.430 and 0.446) that were reported for the similar isoelectronic square-planar [CuX4]2 clusters in K2PdX4:Cu2+ (X = Cl, Br) based on the EPR analysis [39]. Thus, the proportionality factors and the resultant superhyperfine parameters in this work can be regarded as reasonable. As mentioned before for the g factors, only the proportionality factor q is adopted as the adjustable parameter in the superhyperfine parameters based on the present cluster approach treatments, which seem superior to the previous studies for the similar Ni+ I center in CsCaF3 [13] based on various fitted molecular orbital coefficients. (4) There are some errors in the above calculations. First, approximation of the theoretical model and the perturbation formulas adopted in this work can lead to some errors. Second, the spectral parameters Dq and N estimated from those of the similar CsCaF3:Ni+ [13] may also introduce some errors into the calculation results. When Dq changes by 10%, the fitted s and the g factors would vary by about 0.5%, suggesting that Dq is somewhat related to the tetragonal distortion and hence to the anisotropy Dg of the systems. As N changes by 10%, the errors for the resultant s and the g factors are estimated to be not more than 0.4%, because the covalency influences mainly the average of the g factors. Third, the errors may arise from the approximation of the relationship A2 ðRÞ 10:8A4 ðRÞ [23–25] for the tetragonal field parameters. The deviation of the relative elongation ratio s in I center is estimated to be about 3% as the ratio A2 ðRÞ=A4 ðRÞ varies within the widely accepted range of 9–12. Fourth, the reference distance R obtained from the empirical formula [28] would bring forward some errors to the group overlap integrals and hence to the resultant g factors and the superhyperfine parameters. When the host distance RH is adopted in the calculations, the errors of about 4% may be induced for the group overlap integrals, which lead to the deviations of no more than 0.5% for the final results. Fifth, in the calculations for II center, the impurity off-center displacement due to the nearest neighbour VF is not taken into account. When considering the above impurity displacement in Eq. (9), the optimal value of this displacement is found to be about 0.008 Å. This can be illustrated by monovalence of Ni+ and F and the relatively larger distance, only yielding weak electrostatic repulsion upon the central Ni+. In addition, the electrostatic attraction of the planar F ligands can make the Ni+ move towards the center of the octahedron, which largely cancels the above off-center displacement. As a result, the impurity displacement may be regarded as negligible here. References [1] A.N. Belsky, P. Chevallier, E.N. Mel’chakov, C. Pédrini, P.A. Rodnyi, A.N. Vasil’ev, Chem. Phys. Lett. 278 (1997) 369. [2] M.C. Marco de Lucas, F. Rodriguez, M. Moreno, A. Tressaud, J. Phys. Condens. Matter 6 (1994) 6353. [3] M.C. Marco de Lucas, F. Rodriguez, M. Moreno, J. Phys. Condens. Matter 5 (1993) 1437. [4] B. Vdlacampa, R. Alcalá, P.J. Alonso, J.M. Spaeth, J. Phys. Condens. Matter 5 (1993) 747. [5] M.C. Marco de Lucas, F. Rodriguez, M. Moreno, J. Phys. Condens. Matter 7 (1995) 7535. [6] A. Trokiner, H. Zanni-Theveneau, J. Phys. C 21 (1988) 4913. [7] L.L. Boyer, J.R. Hardy, Phys. Rev. B 24 (1981) 2577. [8] J.Y. Buzare, P. Foucher, J. Phys. Condens. Matter 3 (1991) 2535. [9] F.A. Modine, E. Sonder, W.P. Unruh, Phys. Rev. B. 10 (1974) 1623.
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