#### 3.1. Structural and Magnetic Measurements

Figure 2 shows the XRD patterns of the LiFePO

${}_{4}$ sample. According to the powder XRD Rietveld analysis, the sample was assigned to be a single phase of an orthorhombic symmetry with the space group

$Pnma$ (No. 62). The observed lattice parameters

$a=10.3234\left(3\right)$ Å,

$b=6.0045\left(2\right)$ Å and

$c=4.6915\left(1\right)$ Å is are in a good agreement with the previous data [

1,

3,

34,

35] summarized also in Reference [

2]. In this way, the expected structure of the investigated sample was confirmed. Moreover, we can infer that the Li stoichiometry of the sample is likely good since reduced Li content results in a decrease of the parameters

a and

b and an increase of the parameter

c.

Figure 3 displays the temperature dependences of the zero field cooled (ZFC) and the field cooled (FC) magnetizations,

${\sigma}_{\left(Z\right)FC}$, under an applied magnetic field of 1 T. The results show a typical antiferromagnetic transition at

${T}_{\mathrm{N}}=52$ K and a decrease in the magnetization with the decreasing temperature below

${T}_{\mathrm{N}}$. For temperatures

$T>{T}_{\mathrm{N}}$, the molar susceptibility,

${\chi}_{m}$, see

Figure 3, was fitted to the Curie-Weiss law

${\chi}_{m}=C/(T+\Theta )$;

C and

$\Theta $ are constants related to the measured system (LiFePO

${}_{4}$) and observed magnetic phase transition. The Curie constant

$C={\mu}_{0}{N}_{\mathrm{A}}{\mu}_{\mathrm{eff}}^{2}/3{k}_{\mathrm{B}}$, with

${\mu}_{0}$ being the vacuum permeability,

${N}_{\mathrm{A}}$ the Avogadro constant,

${k}_{\mathrm{B}}$ the Boltzmann constant, and

${\mu}_{\mathrm{eff}}$ the effective magnetic moment. The fitted values are

$\Theta =-91\left(2\right)$ K and

$C=4.6\left(1\right)\times {10}^{-5}$ m

${}^{3}$K/mol. From the Curie constant

C, the value

${\mu}_{\mathrm{eff}}=5.5\left(1\right)$ Bohr magnetons (

${\mu}_{\mathrm{B}}$) is obtained, in good agreement with

$\Theta =-92\left(1\right)$ K and

${\mu}_{\mathrm{eff}}=5.58\left(1\right)$ ${\mu}_{\mathrm{B}}$ reported in Reference [

36] (see also Reference [

17]). The effective magnetic moment is slightly higher than

${\mu}_{\mathrm{eff}}=4.90$ ${\mu}_{\mathrm{B}}$ for Fe

${}^{2+}$ in the high-spin state (

$S=2$) with the orbital angular momentum quenched (

$L\approx 0$) by the crystal field. The theoretical value of the high spin state of the free Fe

${}^{2+}$ ion (

$S=2$,

$L=2$) is

${\mu}_{\mathrm{eff}}=6.71$ ${\mu}_{\mathrm{B}}$. A higher value of

${\mu}_{\mathrm{eff}}$ observed here thus indicates that the orbital angular momentum was not fully quenched by the crystal field. This observation is consistent with a non-zero orbital Fe

${}^{2+}$ magnetic moment deduced from ab initio calculations discussed below. We mention an increase of the magnetization below temperature

$\sim 17$ K (

Figure 3). Such an increase of the magnetization in LiFePO

${}_{4}$ has also been reported in References [

14,

17,

36]. Rhee et al. [

17] related this effect to the influence of the spin-orbit coupling when it becomes comparable with the thermal energy at about 20 K, which results in an ‘unquenching’ of the orbital magnetic moments of Fe

${}^{2+}$ ions, thus increasing their total magnetic moment.

The magnetization curves of the LiFePO

${}_{4}$ sample were measured at temperatures 4.2 K and 60 K and are presented in

Figure 4. A linear dependence of the magnetic moment on the magnetic field typical for an AF material was observed at the liquid helium temperature. A larger value of the magnetic moment at 60 K is due to the transition from the AF to the paramagnetic state, which is clearly seen in the temperature dependence of the magnetization in a field of 1 T (

Figure 3). We can state that the AF character of the magnetic order in the studied LiFePO

${}_{4}$ sample is confirmed by the magnetic measurements discussed above, and we can take it into account in the subsequent analysis of the Mössbauer spectra.

#### 3.2. Mössbauer Spectroscopy of LiFePO${}_{4}$

Mössbauer spectra were collected and evaluated for three different experimental conditions varying in temperature and external magnetic field: (i)

$T=60$ K (

$>{T}_{\mathrm{N}}$),

${B}_{\mathrm{ext}}=0$ T; (ii)

$T=4.2$ K (

$<{T}_{\mathrm{N}}$),

${B}_{\mathrm{ext}}=0$ T; and (iii)

$T=4.2$ K (

$<{T}_{\mathrm{N}}$),

${B}_{\mathrm{ext}}=6$ T. The fitted parameters of the spectra are shown in

Table 1 and discussed in detail below. The overall observation is that theory—as discussed in more detail the subsequent section—underestimates the

${V}_{\mathrm{zz}}$ and

$\eta $ parameters and overestimates

${B}_{\mathrm{hf}}$ (when only the Fermi contact interaction contributes to

${B}_{\mathrm{hf}}$).

The

${}^{57}$Fe Mössbauer spectrum of LiFePO

${}_{4}$ above the Néel temperature at temperature

$T=60$ K, see

Figure 5, was fitted with a symmetrical, Lorentzian-shaped quadrupole doublet D. The doublet D with reasonably narrow line widths

$\Gamma =0.30\left(1\right)$ mm/s, quadrupole splitting

$QS=3.02\left(2\right)$ mm/s and isomer shift

$IS=1.33\left(2\right)$ mm/s (in agreement with work [

16]), see

Table 1, was assigned to ferrous Fe

${}^{2+}$ ions in the high spin state

$S=2$ [

15,

37].

At temperature

$T=4.2$ K the

${}^{57}$Fe Mössbauer spectrum was fitted with one octet O (eight absorption lines [

38]) with a narrow distribution of

${B}_{\mathrm{hf}}$ in the static Hamiltonian (see

Figure 6). The octet component O with

${B}_{\mathrm{hf}}=12.4\left(2\right)$ T,

$QS=3.05\left(2\right)$ mm/s, and

$IS=1.35\left(2\right)$ mm/s were ascribed, similarly to the doublet D at

$T=60$ K, to Fe

${}^{2+}$ ions with the high spin state

$S=2$. The orientation of

${\mathit{b}}_{\mathrm{hf}}$ on

${}^{57}$Fe nuclei is antiparallel to the orientation of the magnetic moment of Fe

${}^{2+}$ ions (like in pure iron), which is parallel to the crystallographic direction [010] [

10]. The principal axis of the EFG main component

${V}_{\mathrm{zz}}=16.7\left(1\right)\times {10}^{21}$ V/m

${}^{2}$, with respect to the zero polar angle

$\vartheta \sim {0}^{\circ}$, see

Table 1, is collinear with the direction [010], which is expectable in some respect (because of the Fe site symmetry, one principal EFG axis should be parallel with this direction). At the Fe

${}^{2+}$ sites, due to a non-zero value of the asymmetry parameter

$\eta \sim 0.8$, there is no ordinary local symmetry axis, except the normal to the mirror plane (i.e.,

${C}_{1\mathrm{h}}$ point symmetry).

The in-field Mössbauer spectrum of LiFePO

${}_{4}$ at

$T=4.2$ K and

${B}_{\mathrm{ext}}=6$ T is presented in

Figure 7. The application of the external field provokes a broadening of the octet O lines due to the vectorial sum of randomly oriented local

${\mathit{B}}_{\mathrm{hf}}$ and the applied external field

${\mathit{B}}_{\mathrm{ext}}$ perpendicular to the

$\gamma $-ray direction. The average value of the polar angle

$\vartheta \sim {13}^{\circ}$, see

Table 1, and a higher probability above the mean value of

${B}_{\mathrm{hf}}$, see inset in

Figure 7, indicates the canting of the Fe

${}^{2+}$ moments arranged originally antiferromagnetically. In fact, there are four nonequivalent Fe ion sites with respect to hyperfine interactions when an external magnetic field is applied. Since the single Fe ion magnetic anisotropy is strong, which is explained below, the external field does not cause significant deviation of the Fe ion MM from its original direction along [010], and in the fit we can consider just one Fe site where its MM is slightly deviated from [010], while

${B}_{\mathrm{eff}}$ possesses a broad distribution.

#### 3.3. Theoretical Exploration of Magnetism and Hyperfine Interactions

The magnetic structure of LFP was deduced to be antiferromagnetic (AF)—as also confirmed here for the sample studied—with the easy magnetization axis along the [010] direction, based on neutron scattering experiments [

9,

10]. Other experiments [

11,

12,

13] confirmed this finding, except small deviations from this direction due to anisotropic exchange, which is not taken into account in the following considerations. In addition to the easy magnetization direction, another question arises. Namely, what is the arrangement of the magnetic moments in the LiFePO

${}_{4}$ unit cell? There are three possible MM orders which result in an AF order. These are: Fe1 and Fe2 MMs are parallel (

$\uparrow \uparrow $) being antiparallel (

$\uparrow \downarrow $) to

$\downarrow \downarrow $ Fe3 and Fe4 (AF1), (Fe1

$\uparrow \uparrow $ Fe3)

$\uparrow \downarrow $ (Fe2

$\downarrow \downarrow $ Fe4) (AF2), and (Fe1

$\uparrow \uparrow $ Fe4)

$\uparrow \downarrow $ (Fe2

$\downarrow \downarrow $ Fe3) (AF3). Our DFT PBE spin-polarized calculations (without SOC) show that the AF2 order exhibits the lowest energy, followed by AF1 and the highest energy occurs for AF3. In all calculations the lattice parameters from Reference [

3], i.e.,

$a=10.332$ Å,

$b=6.010$ Å and

$c=4.692$ Å, were employed.

Table 2 collects results of these calculations (see the second column). The AF2 order is observed in reality (see, e.g., Reference [

10]) and was also confirmed in other computations [

8,

18].

Table 2 also contains results of hyperfine parameter calculations. One can see that

${B}_{\mathrm{hf}}$ 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

${V}_{\mathrm{zz}}$ 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 (

$\eta $) 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

${B}_{\mathrm{hf}}$ 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

$\mathit{a}$,

$\mathit{b}$, and

$\mathit{c}$. 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

${V}_{\mathrm{zz}}$ axis is always along

$\mathit{b}$, and

${V}_{\mathrm{yy}}$ and

${V}_{\mathrm{xx}}$ axes are always in the

a-

c plane, though they are tilted from the

$\mathbf{a}$ and

$\mathbf{c}$ directions. For

${B}_{\mathrm{ext}}=0$, all Fe ions are equivalent with respect to hyperfine interactions. When the external magnetic field is applied along

$\mathit{b}$, EFG principal axis orientations are unimportant since

$\vartheta ={0}^{\circ}$ [

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

${\mathit{B}}_{\mathrm{ext}}$ is deviated from the easy magnetization axis

$\mathit{b}$, results in the outcome that all Fe sites are nonequivalent regarding the hyperfine interactions.

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

$\mathit{a}$,

$\mathit{b}$, and

$\mathit{c}$ 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

$\mathit{b}$ ([010]), whereas

$\mathit{c}$ ([001]) appears to be the hard direction. The

$\mathit{a}$ ([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

${\mu}_{\mathrm{B}}$, 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 (

${m}_{\mathrm{l}}$) to the magnetic moment are also given in

Table 3 for all three investigated cases (magnitude is shown only). The largest contribution 0.11

${\mu}_{\mathrm{B}}$ 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 Fe

${}^{2+}$ ion with a magnetic moment of 4

${\mu}_{\mathrm{B}}$ in the magnetic field

${B}_{\mathrm{ext}}$, one gets that these energies match at the field

${B}_{\mathrm{ext}}\simeq 40$ T. Then, a significant deviation from the easy magnetization direction may happen only when

${B}_{\mathrm{ext}}$ approaches 40 T, demonstrating a large single ion magnetic anisotropy of Fe ions in LiFePO

${}_{4}$. This is consistent with the in-field MS results discussed above where a relatively small (average) deviation of the Fe ion MMs from the easy axis direction is observed at the field

${B}_{\mathrm{ext}}=6$ T. Moreover, the above considerations about distinct hyperfine interactions at four Fe ion sites confirm the approach regarding how the corresponding MS spectrum was fitted.

When coming to the hyperfine parameters, the EFG—characterized by

${V}_{\mathrm{zz}}$ and

$\eta $—is almost not affected by switching on the SOC (cf.

Table 2). On the other hand,

${B}_{\mathrm{hf}}$ is strongly affected and depends on the magnetization direction.

${B}_{\mathrm{hf}}$ values (see

Table 3) now include the orbital magnetic moment and dipolar spin contributions [

33] (whereas previous numbers originated from the Fermi contact interaction only). The value

${B}_{\mathrm{hf}}=11.7$ T calculated for the magnetization direction collinear with the

$\mathit{b}$ axis ([010]) is now very close to the corresponding experimental value 12.4 T (see

Table 1).

From these considerations, it becomes obvious that for an adequate description of magnetic hyperfine interactions it is necessary to include their anisotropy. A sufficiently general expression for the nuclear Hamiltonian could be written as

where

$\widehat{\mathit{I}}$ is the operator of the nuclear spin,

$\langle \widehat{\mathit{S}}\rangle $ is the quantum-mechanical and thermodynamic mean value of the Fe ion spin operator, and

$\mathcal{A}$ is the second order tensor (matrix) describing such type of hyperfine interaction. In principle, each Fe ionic site may have different tensor

$\mathcal{A}$ in the global coordinate system, which is important in the case when an external magnetic field is applied. A possibly anisotropic (dipolar) contribution from other Fe sites to the hyperfine field should also be considered. Currently, the code [

21] used to fit the Mössbauer spectra does not allow us to take into account anisotropic magnetic hyperfine interactions. In the case of an isotropic interaction, the tensor

$\mathcal{A}$ reduces to a constant

A. When the nuclear magnetic moment operator is

$\widehat{\mathbf{\mu}}={g}_{\mathrm{N}}{\mu}_{\mathrm{N}}\widehat{\mathit{I}}$ (

${g}_{\mathrm{N}}$ and

${\mu}_{\mathrm{N}}$ being, respectively, the

g-factor of the

${}^{57}$Fe nucleus and nuclear magneton),

${\mathit{B}}_{\mathrm{hf}}=-A\langle \widehat{\mathit{S}}\rangle /\left({g}_{\mathrm{N}}{\mu}_{\mathrm{N}}\right)$. Since

$\langle \widehat{\mathit{S}}\rangle $ is proportional to the ion magnetic moment, so is

${\mathit{B}}_{\mathrm{hf}}$. An example of how to handle the anisotropic magnetic hyperfine interaction can be found in Reference [

40] where iron phosphide (FeP) having an orthorhombic structure with the same space group as LiFePO

${}_{4}$ was investigated.

${B}_{\mathrm{hf}}$ values from

Table 3 could be used to estimate the tensor

$\mathcal{A}$ (supposed for simplicity to be diagonal in the coordinate system with axes parallel to orthorhomic

$\mathit{a}$,

$\mathit{b}$, and

$\mathit{c}$ translation vectors) considering that Fe ions are in the high spin state

$S=2$ regardless of the magnetic moment orientation.

When studying magnetic anisotropy, the single ion anisotropy may play an important role in addition to the exchange interaction anisotropy. The general formula to describe the magnetic anisotropy energy originating from the individual atoms of a crystal reads

with

${K}_{lm}^{\left(i\right)}$ being the anisotropy constants of

i-th atom/ion and

${Y}_{lm}$ are spherical harmonics expressed in terms of spherical coordinates (

$\theta $ polar and

$\phi $ 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

${C}_{1h}$ (mirror) point symmetry in LiFePO

${}_{4}$:

where

${K}_{lm\left[c\right|s]}$ are anisotropy constants and

$\theta $,

$\phi $ are defined with respect to a suitable coordinate system. Namely,

$\theta $ is the deviation from the axis

$\mathit{b}$, whereas

$\phi $ 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

$\theta $ and

$\phi $ 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.