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Condens. Matter 2019, 4, 86 12 of 15 with K(i) being the anisotropy constants of i-th atom/ion and Ylm are spherical harmonics expressed in lm terms of spherical coordinates (θ polar and φ azimutal angle [41]). The summation over atoms goes through all atoms contributing to the anisotropy. Since atomic sites have usually some non trivial symmetry, the effective number of anisotropy constants is reduced. The summation over l is typically limited by 2, 4, or 6. Using the work [42] and the so-called real spherical harmonics, we can deduce the formula describing magnetocrystalline anisotropy (up to l = 2) per one Fe atom reflecting its C1h (mirror) point symmetry in LiFePO4: E(1) (θ,φ) = K00 +K11c sinθcosφ+K11s sinθsinφ (3) MA + K20cos2θ+K22csin2θcos2φ+K22ssin2θsin2φ, where Klm[c|s] are anisotropy constants and θ, φ are defined with respect to a suitable coordinate system. Namely, θ is the deviation from the axis b, whereas φ does not need to be fixed at this moment. Equation (3) is valid for one Fe ion. We have, however, same equations for other ions where anisotropy constants are the same, but θ and φ angles are related using symmetry operations (rotations) which mutually transform Fe ion surroundings. In this way, there are just 6 anisotropy constants. They could, in principle, be obtained using ab initio calculations by taking into account the spin-orbit coupling. By considering the external magnetic field and single ion magnetic anisotropy, the direction of MMs at individual Fe sites can be determined. This may serve as a starting point to fit more precisely Mössbauer spectra considering also the anisotropic magnetic hyperfine interaction and EFG tensor principal axis geometrical relationships discussed above. 4. Conclusions and Outlook The LiFePO4 powder sample was first examined experimentally. Its crystal structure at room temperature and magnetic order below the Néel temperature were determined, demonstrating a fair agreement with previous investigations. The Mössbauer study of this LiFePO4 sample was then performed, showing the good potential of this technique to investigate complex magnetic structures. Measurements done above and below the Néel temperature agree with earlier Mössbauer studies. A doublet and an octet are detected, respectively, in the corresponding spectra. Applying an external magnetic field to the sample at liquid helium temperature results in the deviations of Fe magnetic moments from the crystallographic axis b, being the easy magnetization direction. A small average size of such deviations points to a strong magnetic anisotropy. Moreover, a rather flat distribution of the magnetic effective fields ranging from about 6 to 19 T shows that the external field is added to or subtracted from the hyperfine field, indicating again a large anisotropy. Whereas the distribution should be narrow and peaked at about 6.5 T when the anisotropy is weak—magnetic moments follow the external field, thus destroying the antiferromagnetic order, which was not observed. By studying theoretically/computationally various features of the magnetic order and hyperfine interactions in LiFePO4, we may state a good predictability of experimental results and suggest an advanced method of Mössbauer spectra evaluations, preferably from measurements carried out for single crystal specimens. It is observed that in addition to the Fermi contact term, the orbital magnetic moments of Fe ions contribute significantly to the hyperfine field. The spin-dipolar contribution is important too. The single Fe ion anisotropy is predicted to be high in agreement with experiment. An expression describing the single ion magnetic anisotropy is suggested and its parameters could be obtained in the future by means of SOC DFT calculations with varying directions of the magnetization. This would allow us to determine the direction of magnetic moments at individual Fe sites upon application of the magnetic field, which can be used to fit the Mössbauer spectra more adequately.PDF Image | Mossbauer Spectroscopy of Triphylite (LiFePO4) at Low Temperatures
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