Investigations of the g factors and local structure for orthorhombic Cu^{2+}(1) site in fresh PrBa_{2}Cu_{3}O_{6+x} powders

The electron paramagnetic resonance (EPR) g factors g_x, g_y and g_z of the orthorhombic Cu^{2+}(1) site in fresh PrBa_{2}Cu_{3}O_{6+x} powders are theoretically investigated using the perturbation formulas of the g factors for a 3d^9 ion under orthorhombically elongated octahedra. The local orthorhombic distortion around the Cu^{2+}(1) site due to the Jahn-Teller effect is described by the orthorhombic field parameters from the superposition model. The [CuO6]^{10-} complex is found to experience an axial elongation of about 0.04 {\AA} along c axis and the relative bond length variation of about 0.09 {\AA} along a and b axes of the Jahn-Teller nature. The theoretical results of the g factors based on the above local structure are in reasonable agreement with the experimental data.


Introduction
PrBa 2 Cu 3 O 6+x (Pr123) compounds are useful materials with anomalous resistive and magnetic [1], vortex [2,3], friction [4], structural [5] and superconductive [6] properties and have attracted extensive interest of researchers. These properties are largely related to the local structure and electronic behaviours near the Cu 2+ site, which can be investigated by means of the electron paramagnetic resonance (EPR) technique. For example, EPR experiments were carried out for fresh PrBa 2 Cu 3 O 6+x powders, and the anisotropic g factors g x , g y and g z were also measured for the Cu 2+ (1) site [7]. Until now, however, the above experimental results have not been quantitatively interpreted, and the local structure around the Cu 2+ (1) site is not determined, either. Since the electronic properties and the local structure of the paramagnetic Cu 2+ as well as the microscopic mechanisms of its EPR spectra would be helpful in understanding the properties of the Pr123 systems, further theoretical investigations on the g factors and the local structure of the Cu 2+ (1) site for the fresh PrBa 2 Cu 3 O 6+x powders are of fundamental and

Calculation
In the orthorhombic phase of PrBa 2 Cu 3 O 6+x , the Cu 2+ (1) site belonging to orthorhombic point symmetry (D 2 ) has six nearest neighbour oxygen ligands, which construct a distorted octahedron with the approximately mutually vertical Cu-O bonds of the average (or reference) distance R ( ≈ 1.917 Å [8]). As a Jahn-Teller ion, Cu 2+ can suffer the Jahn-Teller effect via relaxation and compression of the Cu-O bonds parallel with and perpendicular to the c axis in terms of the relative axial elongation ∆Z . Meanwhile, the planar Cu-O bonds may suffer another relative bond length variation ∆X along a and b axes. Thus, the local structure of the orthorhombic Cu 2+ (1) site in the fresh PrBa 2 Cu 3 O 6+x powders can be characterized by axial elongation ∆Z and the perpendicular bond length variation ∆X (see figure 1).
For a Cu 2+ (3d 9 ) ion in an orthorhombically elongated octahedron, the original cubic ground orbital doublet 2 E g may split into two orbital singlets 2 A 1g and 2 A ′ 1g , with the latter lying lowest. Nevertheless, the upper cubic orbital triplet 2 T 2g can be separated into three orbital singlets 2 B 1g , 2 B 2g and 2 B 3g [9]. The high order perturbation formulas of the g factors for an orthorhombically elongated octahedron can be expressed as follows [10]: Here g s ( ≈ 2.0023) is the spin-only value. k is the orbital reduction factor, and ζ is the spin-orbit coupling coefficient of the 3d 9 ion in crystals. The denominators E i (i = 1−3) are the separations between the excited 2 B 1g , 2 B 2g and 2 B 3g and the ground 2 A ′ 1g states, which can be obtained in terms of the cubic field parameter D q and the orthorhombic field parameters D s , D t , D ξ and D η as [10]: As mentioned before, the orthorhombic distortion of the Cu 2+ (1) site may be described as the reference distance R as well as the relative axial elongation ∆Z and the planar bond length variation ∆X .

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Investigations of the g factors for fresh PrBa 2 Cu 3 O 6+x powders  figure 1). This means that the Cu-O bonds suffer a relative elongation and compression of 2∆Z and ∆Z along c and a (or b) axes, respectively. Meanwhile, the planar Cu-O bonds undergo an additional relative bond length variation of ∆X along a and b axes.
Usually, the crystal-field parameters can be determined from the superposition model which has been extensively adopted for transition-metal ions in crystals [11]. Moreover, this model can also be used for powder or polycrystal systems (e.g., the superposition model analysis of the EPR spectra for Fe 3+ modified polycrystalline PbTiO 3 [12] and PbZrO 3 [13] and Eu 2+ in polycrystalline A zeolite [14]) and may be suitably applied to the fresh PrBa 2 Cu 3 O 6+x powders studied here. From the local geometry and the superposition model [11], the orthorhombic field parameters can be determined as follows: It is noted that the angular dependence is reduced due to the Cu-O bond angles (0, π/2 and π related to Z axis and 0, π/2, π and 3π/2 related to X axis) and thus only the bond lengths R i are reserved in the above formulas. Here t 2 ≈ 3 and t 4 ≈ 5 are the power-law exponents [11] and they are the intrinsic parameters. For 3d n ions in octahedra, the relationships A 4 ≈ 3D q /4 and A 2 ≈ 10.8A 4 [11,15,16] were proved to be valid for many crystals and are suitably adopted here. Thus, the g factors (especially the axial anisotropy ∆g = g z −(g x + g y )/2 and the perpendicular anisotropy δg = g x − g y ) are correlated with the orthorhombic field parameters and hence with the local structure of the system studied.
From the optical spectra for Cu 2+ in some oxides [17], the spectral parameters D q ≈ 1400 cm −1 and k ≈ 0.77 are obtained for the Cu 2+ center in the fresh PrBa 2 Cu 3 O 6+x powders here. The spin-orbit coupling coefficient is usually expressed as ζ ≈ kζ 0 , where ζ 0 (≈ 829 cm −1 [18]) is the corresponding freeion value. Substituting these values into equation (1) and fitting the calculated g factors (especially the anisotropies) to the experimental data, one can determine the local axial elongation and the planar bond length variation: ∆Z ≈ 0.04 Å and ∆X ≈ 0.09 Å. (4) The corresponding theoretical g factors are shown in table 1.  Table 1 reveals that the theoretical g factors and the anisotropies for the fresh PrBa 2 Cu 3 O 6+x powders based on the local structural parameters ∆Z and ∆X are in good agreement with the experimental data.

Discussion
Therefore, the EPR spectra are satisfactorily interpreted for the fresh PrBa 2 Cu 3 O 6+x powders in this work.
The EPR g factors of the fresh PrBa 2 Cu 3 O 6+x powders can be characterized by the axial and perpendicular anisotropies ∆g ( ≈ 0.15) and δg ( ≈ 0.044), which are ascribed to the local axial elongation ∆Z (≈ 0.04 Å) and the planar bond length variation ∆X ( ≈ 0.09 Å), respectively. Thus, the Cu 2+ (1) site exhibits a moderate orthorhombic distortion of the Jahn-Teller nature. Similar axial elongations and planar bond length variations due to the Jahn-Teller effect were also found for some Jahn-Teller ions (e.g., Cr 5+ and Ti 3+ with the same spin S = 1/2) in oxygen octahedra [19,20]. It seems that Cu 2+ prefers to exhibit elongation distortions (i.e., orthorhombically elongated octahedra) under oxygen environments.
There are some errors in the above calculations. First, the approximations of the theoretical model and formulas may bring about some errors in this work. Second, the errors also arise from the approximation of the relationship A 2 (R) ≈ 10.8A 4 (R) [11,15,16], which would somewhat affect the orthorhombic field parameters and the final results. The errors for the local structural parameters ∆Z and ∆X are estimated to be no more than 1% as the ratio A 2 (R)/A 4 (R) changes by 10%. Third, the present calculations are based on the conventional crystal-field model containing only the central ion orbital and spin-orbit coupling contributions, while the ligand orbital and spin-orbit coupling contributions are not taken into account. Fortunately, although the studied system has some covalency (characterized by the covalency factor N ≈ 0.77 < 1), the spin-orbit coupling coefficient ( ≈ 151 cm −1 [21]) of the ligand O 2− is much smaller than that ( ≈ 829 cm −1 [18]) of Cu 2+ . According to various EPR studies for Cu 2+ under oxygen octahedra [22][23][24], the ligand contributions to the g factors may be very small and negligible. So, the present theoretical calculations can be regarded as reasonable. Moreover, the investigations in this work would be helpful in carrying out structural and magnetic studies on PrBa 2 Cu 3 O 6+x superconductors as well as applicable to other similar R123 systems.