## Brazilian Journal of Physics

*Print version* ISSN 0103-9733

### Braz. J. Phys. vol.40 no.3 São Paulo Sept. 2010

#### http://dx.doi.org/10.1590/S0103-97332010000300020

**Theoretical studies of the EPR parameters for Ni^{2+} and Co^{+} in MgO **

**Zhi-Hong Zhang ^{I}; Shao-Yi Wu^{I, II, *}; Pei Xu^{I}; Li-Li Li^{I}**

^{I}Department of Applied Physics, University of Electronic Science and Technology of China, Chengdu 610054, P.R. China

^{II}International Centre for Materials Physics, Chinese Academy of Sciences, Shenyang 110016, P.R. China

**ABSTRACT**

The electron paramagnetic resonance (EPR) parameters (*g* factors and the hyperfine structure constants) for *Ni*^{2+} and *Co*^{+} in *MgO* are theoretically studied from the perturbation formulas of these parameters for a 3d^{8} ion in octahedral crystal-fields. In the computations, the ligand orbital and spin-orbit coupling contributions are taken into account using the cluster approach. The calculated EPR parameters are in good agreement with the experimental data. The larger g factor and the smaller magnitude of the hyperfine structure constant for Ni^{2+} as compared with those for Co^{+} can be attributed to the higher spin-orbit coupling coefficient and the lower dipolar hyperfine structure parameter of the former, respectively.

**Keywords:** Impurity and defects; Electron paramagnetic resonance; *Ni*^{2+}; *Co*^{+}; MgO.

**1. INTRODUCTION**

Magnesium oxide (MgO) is usually regarded as a model system to investigate magnetic [1,2], adsorption [3,4], structure [5,6] and optical properties [7] of doped transition-metal impurities. Particularly, MgO containing *Ni*^{2+} and *Co*^{+} can exhibit unique catalytic [8-10] and tunable laser properties [11,12]. Normally, these properties are closely correlated with the electronic states of the transition-metal ions in the hosts, which can be investigated by means of electron paramagnetic resonance (EPR) technique. For example, EPR experiments were carried out for *Ni*^{2+} and *Co*^{2+} doped MgO, and the EPR parameters (i.e., the isotropic g factors and the hyperfine structure constants) were also measured for the cubic *Ni*^{2+} and *Co*^{+} centers [13,14].

Up to now, however, the above EPR experimental results have not been satisfactorily explained. On the other hand, the EPR spectra and magnetic properties have been extensively investigated for *Ni*^{2+} in various chlorides by considering only the central ion orbital and spin-orbit coupling contributions [15]. Nevertheless, the contributions to the EPR parameters from the ligand orbital and spin-orbit coupling interactions were not taken into account in the previous studies. In fact, for the 3*d*^{8} ions in oxides, the systems may still show some covalency and impurity-ligand orbital admixtures. In addition, the EPR spectra for MgO:*Co*^{+} have not been interpreted until now. Considering that (i) investigations on the EPR parameters for *Ni*^{2+} and *Co*^{+} in MgO can reveal useful information about electronic structures which would be helpful to understand the properties of these systems and that (ii) 3*d*^{8} ions can be treated as model systems containing only two unpaired holes, further quantitative studies on the EPR spectra for the *Ni*^{2+} and *Co*^{+} centers are of scientific and practical significance. In this work, the improved perturbation formulas of the EPR parameters based on the cluster approach are applied to the theoretical analysis of the *Ni*^{2+} and *Co*^{+} centers in MgO. In the calculations, the ligand orbital and spin-orbit coupling contributions are considered in a uniform way. The results are discussed.

**2. CALCULATIONS **

Judging from the observed isotropic g factors and the hyperfine structure constants, the experimental EPR signals [13,14] can be assigned to the substitutional cubic *Ni*^{2+} and *Co*^{+} centers in MgO. When a 3*d*^{8} ion locates on an octahedral (*O _{h}*) site, the free-ion configuration

^{3}

*F*would be separated into two orbital doublets

^{3}

*T*

_{1g}and

^{3}

*T*

_{2g}and one singlet

^{3}

*A*

_{2g}, with the latter lying lowest corresponding to the isotropic

*g*and

*A*signals [16]. As for the

*g*factor of a 3

*d*

^{8}ion in octahedra, the perturbation formula was established using the conventional crystal-field model [17,18], by including only the contributions from the central ion orbital and spin-orbit coupling interactions. In order to study the EPR spectra of the 3

*d*

^{8}centers more exactly, the ligand orbital and spin-orbit contributions may be taken into account. Thus, the improved g formula based on the cluster approach is applied here. Meanwhile, the perturbation formula of the hyperfine structure constant for a 3

*d*

^{8}ion in regular octahedra can be similarly derived. Thus, we have [19] :

Here *g _{s}* ( ≈ 2.0023) is the spin-only value. ζ and ζ´ are the spin-orbit coupling coefficients,

*k*and

*k*´ are the orbital reduction factors, and

*P*and

*P*´ are the dipolar hyperfine structure parameters for a 3

*d*

^{8}ion in crystals. κ is the core polarization constant. The energy denominators

*E*(

_{i}*i*= 1 ~ 3) denote the energy separations between the excited

^{3}

*T*

_{2},

^{1}

*T*

_{2a}and

^{1}

*T*

_{2b}and the ground

^{3}

*A*

_{2}states [17-19]. They can be described in terms of the cubic field parameter

*Dq*and the Racah parameters B and C for the 3

*d*

^{8}ion in crystals:

*E*

_{1}≈ 10

*Dq, E*

_{2}≈ 10

*Dq*+ 12

*B*and

*E*

_{3}≈ 8

*B*+ 2

*C*+ 10

*Dq*[17-19]. From the cluster approach containing the ligand

*p*- and

*s*-orbital contributions [20], the spin-orbit coupling coefficients ζ and ζ´, the orbital reduction factors

*k*and

*k*' and the dipolar hyperfine structure parameters

*P*and

*P*' can be expressed as

Here and are the spin-orbit coupling coefficients of the free 3*d*^{8} and the ligand ions, respectively. *A* denotes the integral , where *R* is the impurity-ligand distance of the studied systems. *N*_{γ} and λ_{γ} (here the subscripts γ = *t* and e denote the irreducible representations *T*_{2g} and *E _{g}* of

*O*group, respectively) are the normalization factors and the orbital admixture coefficients. They are determined from the approximation relationships [20]

_{h}and the normalization conditions [20]

Here *N* is the average covalency factor, characteristic of the covalency of the studied systems. *S _{dp}*

_{γ}(and

*S*) are the group overlap integrals. In general, the orbital admixture coefficients increase with increasing the group overlap integrals, and one can approximately adopt the proportional relationship λ

_{ds}_{s}/λ

*≈*

_{e}*S*between the orbital admixture coefficients and the related group overlap integrals within the same irreducible representation Eg. Obviously, omission of the ligand contributions (i.e., ζ´ = ζ =

_{ds}/S_{dpe}*N*,

*k´ = k = N, P´ = P = P*

_{0}

*N*), the above g formula returns to that of the previous work based on the conventional crystal-field model [17,18].

Usually, the impurity-ligand distance *R* is different from the host cation-anion distance *R _{H}* in a pure crystal due to the difference between the ionic radius

*r*of impurity and the radius

_{i}*r*of host ions. Fortunately, studies based on experimental superhyperfine parameter and extended X-ray absorption fine structure (EXAFS) measurements have verified that the empirical formula

_{h}*R*≈

*R*+ (

_{H }*r*)/2 is approximately valid for impurity ions in crystals [21]. From

_{i }– r_{h}*R*( ≈ 2.105 Å [22] ) for MgO,

_{H}*r*( ≈ 0.69 and 0.82 Å [22] ) for

_{i}*Ni*

^{2+}and

*Co*

^{+}as well as

*r*( ≈ 0.66 Å [22] ) for

_{h}*Mg*

^{2+}, the distances

*R*are obtained and listed in Table 1. From the distances

*R*and the Slater-type self-consistent field (SCF) wave functions [23,24], the group overlap integrals are calculated and shown in Table 1. According to the optical spectra for

*Ni*

^{2+}in MgO [25,26], the cubic field parameter Dq ≈ 860

*cm*

^{-1}and the covalency factor

*N*≈ 0.90 can be obtained. Since the isoelectronic monovalent

*Co*

^{+}suffers weaker crystal-fields (i.e., lower

*Dq*) and covalency effect (i.e., higher

*N*) [27] when coordinated to the same oxygen ligands, the spectral parameters

*Dq*≈ 780

*cm*

^{-1}and

*N*≈ 0.91 may be estimated for

*Co*

^{+}in MgO. Then the Racah parameters are determined from the relationships

*B*≈

*N*

^{2}

*B*

_{0}and

*C*≈

*N*

^{2}

*C*

_{0}[28], with the corresponding free-ion values

*B*

_{0}≈ 1208 and 878

*cm*

^{-1}and

*C*

_{0}≈ 4459 and 3828

*cm*

^{-1}[27] for

*Ni*

^{2+}and

*Co*

^{+}, respectively. Using Eqs. (3) and (4), the molecular orbital coefficients

*N*

_{γ}and λ

_{γ}(and λ

*) can be calculated. From the free-ion values ≈ 649 and 456*

_{s}*cm*

^{-1}[27] for

*Ni*

^{2+}and

*Co*

^{+}and ≈ 151

*cm*

^{-1}[29] for

*O*

^{2-}, the parameters in Eq. (2) are obtained and shown in Table 1. In the formula of the hyperfine structure constant, the dipolar hyperfine structure parameters are

*P*

_{0}≈ 112 × 10

^{-4}cm

^{-1}and 228 × 10

^{-4}

*cm*

^{-1}[30] for

*Ni*

^{2+}and

*Co*

^{+}, respectively. The core polarization constant can be determined from the empirical relationship κ ≈ -2χ/(3 <

*r*

^{-3}> ), where χ is characteristic of the density of unpaired spins at the nucleus of the central ion and <

*r*

^{-3}> the expectation value of the inverse cube of the radial wave function of the 3

*d*

^{8}orbital [30]. Applying <

*r*

^{-3}> ≈ 7.094 and 5.388 a.u. [16] and χ ≈ -3.15 and -3.31 a.u. [30] for

*Ni*

^{2+}and

*Co*

^{+}in oxides, one can obtain κ ≈ 0.3 and 0.41 for MgO:

*Ni*

^{2+}and MgO:

*Co*

^{+}, respectively. Substituting the above values into the formulas of the EPR parameters, the corresponding theoretical results (Cal. b) are calculated and shown in Table 2. To clarify the importance and the tendency of the covalency and the ligand contributions for

*Ni*

^{2+}and

*Co*

^{+}in MgO, the results (Cal. a) based on omission of the ligand contributions are also collected in Table 2 for comparison.

**3. DISCUSSION**

Table 2 indicates that the theoretical EPR parameters (Cal. ^{b}) for *Ni*^{2+} and *Co*^{+} in MgO based on the improved formulas containing the ligand contributions are in better agreement with the observed values than those (Cal. *ª*) in the absence of the above contributions.

1) The experimental isotropic g and A factors are attributed to the cubic *Ni*^{2+} and *Co*^{+} centers on the substitutional *Mg*^{2+} site in MgO. It is noted that there are some low symmetrical 3*d*^{8} centers in other oxides, e.g., the trigonal *Ni*^{2+} and *Cu*^{3+} centers in α – *Al*_{2}*O*_{3} [31]. The anisotropic *g* factors *g*_{//} and *g*_{⊥} [31] for the trigonal centers may be similarly analyzed from the perturbation formulas [32] for a trigonally distorted octahedral 3*d*^{8} cluster, and the contributions from the trigonal distortion can be quantitatively considered in the calculations of the trigonal crystal-field parameters from the superposition model [33] and the local geometrical relationship of the impurity centers. Interestingly, the larger g factors [31] for *Cu*^{3+} than those for *Ni*^{2+} are attributable to the higher spin-orbit coupling coefficient ( ≈ 876 *cm*^{-1} [16] ) of the former than that ( ≈ 649 *cm*^{-1} [27] ) of the latter. In addition, *Co*^{+} replacing the host *Mg*^{2+} in MgO may induce some means of charge compensation due to the fewer charge of the impurity. On the other hand, larger size of *Co*^{+} than *Mg*^{2+} can lead to enhancement of the local metal-ligand interactions around the impurity and thus make *Co*^{+} stable on *Mg*^{2+} site. Further, local charge compensation (e.g., oxygen vacancy nearby) would break the original cubic symmetry of the ideal *Mg*^{2+} site and yield anisotropic EPR parameters. In view of the observed isotropic EPR signals [13,14], the charge compensation may occur in the outer ligand spheres far away from the impurity *Co*^{+}, and the possible disturbance of the local structure of this center can be regarded as very small and negligible for simplicity. Of course, further experimental investigations of possible charge compensation for *Co*^{+} in MgO seem necessary and meaningful.

2) From Eqs. (1) and (2), the *g* factor largely depends upon the spin-orbit coupling coefficient ζ' (related to the central ion spin-orbit coupling coefficient). Thus, the larger g factor for MgO:*Ni*^{2+} than that for MgO:*Co*^{+} can be illustrated by the higher spin-orbit coupling coefficient ( ≈ 649 *cm*^{-1}) of the former than that ( ≈ 456 *cm*^{-1}) of the latter. On the other hand, the hyperfine structure constant is sensitively related to the dipolar hyperfine structure parameter *P*_{0} and the dominant contribution proportional to the core polarization constant κ. So, the larger magnitude of the hyperfine structure constant for MgO:*Co*^{+} than that for MgO:*Ni*^{2+} can be understood in view of the higher values of *P*_{0} and κ in the former.

3) The average covalency factors *N* ( ≈ 0.90 and 0.91 for *Ni*^{2+} and *Co*^{+}) in this work still show some influences of the covalency on the EPR parameters, although the spin-orbit coupling coefficient ( ≈ 151 *cm*^{-1} [28] ) of the oxygen ligand is much smaller than that ( ≈ 649 or 456 *cm*^{-1} [26] ) of the impurity *Ni*^{2+} or *Co*^{+}. Thus, omission of the ligand contributions yields larger *g* factors and slightly lower hyperfine structure constants in magnitude (Cal. *ª*). It seems that the improved formulas of the EPR parameters adopted in this work are superior to the previous ones [17,18] based on the conventional crystal-field model. Further, the covalency and the strength of the crystal-fields exhibit the decreasing tendency from *Ni*^{2+} to *Co*^{+} in the same MgO host, i.e., *N* (*Ni*^{2+}) __<__ *N* (*Co*^{+}) and *Dq* (*Ni*^{2+}) < *Dq* (*Co*^{+}). This point is consistent with the lower valence state and hence weaker covalency and impurity-ligand interactions of the latter.

**4. CONCLUSION**

The EPR parameters for *Ni*^{2+} and *Co*^{+} in MgO are satisfactorily explained from the perturbation formulas based on the cluster approach. Inclusion of the ligand contributions yields better theoretical results as compared with those in the absence of these contributions. The larger g factor and the smaller magnitude of the hyperfine structure constant for *Ni*^{2+} than those for *Co*^{+} can be attributed to the higher spin-orbit coupling coefficient and the lower dipolar hyperfine structure parameter of the former.

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(Received on 21 June, 2010)

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