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Condens. Matter 2019, 4, 86 10 of 15 Table 2 also contains results of hyperfine parameter calculations. One can see that Bhf is significantly overestimated (see Table 1 for comparison with experiment), which is likely the effect of neglecting other contributions to the hyperfine field, which will be yet discussed below (see also Reference [17]). The Vzz value is underestimated by about 15%, which is acceptable and a common feature when experimental and computational EFG results are related. The values of asymmetry parameter (η) are also underestimated compared to experiment. AF1 and AF2 show very similar hyperfine parameters, whereas AF3 differs slightly from them. Nevertheless, this indicates a weak sensitivity of the EFG and Fermi contact contribution to Bhf to various antiferromagnetic arrangements (Reference [18] reports similar observation). With regard to the evaluations of more complex experimental situations, we note that different Fe ions have generally different orientations of the EFG principal axes with respect to the lattice translation vectors. This fact is documented in Figure 8 where EFG axes are schematically shown for each of Fe1, Fe2, Fe3, and Fe4 ions together with the directions of translation vectors a, b, and c. Thus, on one hand, Fe1 and Fe2 have the same EFG axis orientations, and, on the other hand, Fe3 and Fe4 have also the same orientations, but distinct from that of Fe1 and Fe2. The Vzz axis is always along b, and Vyy and Vxx axes are always in the a-c plane, though they are tilted from the a and c directions. For Bext = 0, all Fe ions are equivalent with respect to hyperfine interactions. When the external magnetic field is applied along b, EFG principal axis orientations are unimportant since θ = 0◦ [15]. In this case, Fe1 and Fe3 are pairwise equivalent as well as are Fe2 and Fe4, because of the “magnetic” equivalence within the AF2 order, as discussed above. Considering the most general situation, when Bext is deviated from the easy magnetization axis b, results in the outcome that all Fe sites are nonequivalent regarding the hyperfine interactions. Figure 8. Electric field gradient (EFG) tensor principal axes for Fe1, Fe2, Fe3, and Fe4 cations (shown with their corresponding coordination octahedra; cf. Figure 1). Coloring of EFG Vzz, Vyy, and Vxx axes/vectors is illustrated for the Fe1 case and kept identical for other cations. The length of vectors corresponds to the size of Vzz, Vyy, and Vxx EFG tensor components. When the SOC is enabled in the DFT calculations, it is possible to find the easy and hard magnetization directions. Table 3 lists results of three calculations with the magnetization kept collinear with a, b, and c directions while retaining the AF2 order. The total energies per unit cell are given relative to the lowest energy in the second column. One can see that the easy magnetization direction is b ([010]), whereas c ([001]) appears to be the hard direction. The a ([100]) magnetization direction is somewhat softer than the previous one. This is in good agreement with experimental observations [13]. The spin magnetic moment of Fe ions calculated in the muffin-tin sphere is 3.49 μB, which is reasonable considering that no correction to account for correlated 3d-electrons of Fe was applied (cf., e.g., Reference [39]). The orbital contributions (ml) to the magnetic moment are also given in Table 3 for all three investigated cases (magnitude is shown only). The largest contribution 0.11 μB occurs for the easy magnetization direction. When the energy difference (per Fe ion) between the hard and soft direction is compared to the energy of the Fe2+ ion with a magnetic moment of 4 μB in the magnetic field Bext, one gets that these energies match at the field Bext ≃ 40 T. Then, a significant deviation from the easy magnetization direction may happen only whenPDF Image | Mossbauer Spectroscopy of Triphylite (LiFePO4) at Low Temperatures
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