research papers
Tensor description of Xray magnetic dichroism at the Fe L_{2,3}edges of Fe_{3}O_{4}
^{a}Debye Institute for Nanomaterials Science, Utrecht University, 99 Universiteitsweg, Utrecht 3584 CG, The Netherlands, ^{b}Institute of Theoretical Physics, Heidelberg University, 19 Philosophenweg, Heidelberg 69120, Germany, ^{c}Jülich Centre for Neutron Science, Forschungszentrum Juelich GmbH, Jülich 52425, Germany, ^{d}Faculty of Science, Helwan University, Cairo 11795, Egypt, and ^{e}Diamond Light Source, Harwell Science and Innovation Campus, Didcot OC11 0DE, United Kingdom
^{*}Correspondence email: h.m.e.a.elnaggar@uu.nl, f.m.f.degroot@uu.nl
A procedure to build the optical conductivity tensor that describes the full magnetooptical response of the system from experimental measurements is presented. Applied to the Fe L_{2,3}edge of a 38.85 nm Fe_{3}O_{4}/SrTiO_{3} (001) thinfilm, it is shown that the computed polarization dependence using the conductivity tensor is in excellent agreement with that experimentally measured. Furthermore, the magnetic field angular dependence is discussed using a set of fundamental spectra expanded on spherical harmonics. It is shown that the convergence of this expansion depends on the details of the ground state of the system in question and in particular on the valencestate spin–orbit coupling. While a cubic expansion up to the third order explains the angulardependent Xray magnetic linear dichroism of Fe^{3+} well, higherorder terms are required for Fe^{2+} when the orbital moment is not quenched.
1. Introduction
The determination of the electronic and magnetic structure of engineered magnetic nanostructures is essential to tailor their properties for technological applications such as information storage, spin transport and sensing technology. These devices often rely on magnetic thinfilms and nanostructures comprising multiple layers, such as for instance transition metaloxides magnetic tunnel junctions and exchange biased systems. Xray magnetic dichroism spectroscopy is a powerful tool that can provide element and sitespecific magnetic information in heteromagnetic nanostructures (Kuiper et al., 1993; Nunez Regueiro et al., 1995; Alders et al., 1998; Scholl et al., 2000; Hillebrecht et al., 2001; Haverkort et al., 2004; van der Laan, 2013; Luo et al., 2019). Xray (XMCD) can be used to determine the spin and orbital magnetic moments using sum rules (Carra et al., 1993) while Xray magnetic linear dichroism (XMLD) can be used to determine the anisotropic magnetic moments and spin–orbit interaction (Lüning et al., 2003; Csiszar et al., 2005; Arenholz et al., 2006; Finazzi et al., 2006; van der Laan et al., 2011; Chen et al., 1992, 2010; Iga et al., 2004). However, using dichroism experiments for magnetometry is far from being straightforward because it requires an understanding of the spectral shape and magnitude of the dichroism signal as well as its dependence on the relative orientation of the Xray polarization, the exchange field and the crystallographic axes.
The aim of this work is to provide a general method to construct and analyse dichroism effects in dipole transitions such as at the Fe L_{2,3}edge in magnetite (Fe_{3}O_{4}). We illustrate the procedure to build the conductivity tensor from a few well chosen experimental measurements describing all possible dichroism effects at a single magnetic field orientation. Furthermore, the angular dependence of the magnetic field is discussed using a set of fundamental spectra expanded using spherical harmonics which can describe the full magnetooptical response of the system (Haverkort et al., 2010). Such expansions have been used previously to explain the angular dependence of XMLD (Arenholz et al., 2006, 2007; van der Laan et al., 2008, 2011), yet the new aspect we provide in this work is a thorough inspection of the convergence of the expansion using a comprehensive set of XMLD data measured on Fe_{3}O_{4} in combination with theoretical calculations. Fe_{3}O_{4} serves as an adequate model system: it is a ferrimagnetic mixedvalence strongly correlated system containing two different Fe sites where Fe^{3+} ions reside in tetrahedral (T_{d}) coordinated intersites (A sites), while both Fe^{2+} and Fe^{3+} ions are in octahedral (O_{h}) coordinated intersites (B sites). This provides us with an opportunity to study the effect of the electronic structure on the quality of the expansion between the orbital singlet Fe^{3+} and the orbital triplet Fe^{2+} ions.
2. Methods
2.1. Experimental
The Fe_{3}O_{4} thinfilm was grown on a conductive 0.1% Nbdoped SrTiO_{3} (001) TiO_{2}terminated substrate using pulsed laser deposition as reported by Hamed et al. (2019). The film thickness and surface roughness were concluded to be 38.85 nm and 0.4 nm, respectively, from Xray reflectivity measurements (Fig. 10 of Appendix A). The was observed at 114.97 ± 0.29 K from the magnetization versus temperature measurements in zero field cooling mode with 500 Oe applied field (Fig. 11 of Appendix A). Hysteresis measurements were also performed along the [1,0,0] direction to inspect the saturation of the thinfilm below and above the (Fig. 11 of Appendix A). The largest coercivity is observed for the lowest temperature (H_{c} = 0.1 T) and an external magnetic field of ∼0.25 T is required to saturate the inplane magnetization (see Fig. 12 of Appendix A). On the contrary, the magnetization is not saturated along the [0,0,1] direction with a field of H = 2 T as shown by the XMCD measurement shown in (Fig. 13 of Appendix A).
Xray absorption spectroscopy (XAS) measurements were carried out on beamline I06 of Diamond Light Source, UK. The beam spot at the sample position was estimated to be ∼200 µm × 100 µm. The polarization of the beam can be controlled using an AppleII type undulator to produce linearly and circularly polarized Xrays. A vector magnet set to 1 T was used to saturate the magnetization to (nearly) any arbitrary direction. All measurements were performed at T = 200 K in a normalincidence configuration, i.e. with the incoming beam impinging at an angle of 90° with respect to the sample surface. The energy resolution was estimated to be ∼200 meV full width at halfmaximum (FWHM). The measurements were performed in total electron yield mode. All experimental spectra were first normalized to the incident The spectra were then fitted using a model consisting of two error functions to take into account the L_{2,3}edge jumps. In addition, a set of Gaussian functions were used to fit the multiplet features of the spectra (refer to Appendix B for more details). The L_{2,3}edge jumps were subtracted from the spectra and the spectra were renormalized to the spectral area.
2.2. Computational
The data treatment and the Kramers–Kronig transformation were performed using Python. Crystal field multiplet calculations were performed using the quantum manybody program Quanty (Haverkort et al., 2012). The Hamiltonian we use is of the form
The electron–electron Hamiltonian (H_{e−e}) is of the form
where F^{k} (f_{k}) and G^{k} (g_{k}) are the Slater–Condon parameter for the radial (angular operators) part of the direct and exchange Coulomb interactions, respectively. The radial integrals are obtained from atomic Hartree–Fock calculation scaled to 70% and 80% for valence and valencecore interactions, respectively, to take into account interatomic screening and mixing effects. This is in line with works in the literature such as those by Arenholz et al. (2006) and Pattrick et al. (2002). This reduction is related to two effects: (i) the 80% reduction is to correct the Hartree–Fock calculations such that they agree with atomic data, as shown by Cowan (1981) and many others; (ii) the additional reduction to 70% is to take into account the effects of charge transfer; in other words, the nephelauxetic effects.
The spin–orbit Hamiltonian (H_{SO}) is of the form
where l_{i} and s_{i} are the one electron orbital and spin operators, respectively, and the sum over i is over all electrons. The prefactor ξ is an atomdependent constant (which is to a good approximation material independent) and hence we used here tabulated data for this with ξ = 0.052 eV for 3d orbitals and ξ = 8.20 eV for 2p orbitals.
The crystal field Hamiltonian (H_{CF}) is of the form
where C_{k,m}(θ,ϕ) are the angular crystal field operators expanded on renormalized spherical harmonics and A_{k,m} are proportional to the distortion parameters used in crystal field theory, 10D_{q}, D_{s} and D_{t}. In cubic symmetry we consider only 10D_{q} which we found to be 1.25 eV and 0.5 eV for the B and A sites, respectively, by fitting to the and XMCD spectra. The optimized parameters used for the calculations can be found in Tables 4, 5 and 6. Details of the ground state for the three Fe ions in Fe_{3}O_{4} are shown in Appendix C. We note that we have not taken into account charge transfer effects explicitly in our model. The mixing of iron and oxygen orbitals gives rise to charge transfer effects in core level spectroscopies. It has been shown that neutral experiments on relatively ionic systems map very accurately to the crystal field multiplet model [refer to de Groot & Kotani (2008) for example]. This is the basis of crystal field theory, where the is effectively taken care of by the reduction of the Slater integrals from their atomic values, i.e. an extra reduction with respect to the 80% reduction of the Hartree–Fock values.
Finally, the magnetic exchange Hamiltonian is given as
where S is the spin operator, n is a unit vector giving the direction of the magnetization and J_{exch} is the magnitude of the meanfield exchange interaction which we use as 90 meV in our calculation. This value is based on previous 2p3d RIXS measurements that showed that the spinflip excitation is observed at this energy [see, for example, Huang et al. (2017) and Elnaggar et al. (2019a,b)].
3. Results and discussion
3.1. Construction of the conductivity tensor
The general where ε is the polarization vector, Im is the imaginary part of the equation and σ is the conductivity tensor describing the material properties (Haverkort et al., 2010),
can be expressed by equation (6)The conductivity tensor is a 3 × 3 matrix for a dipole transition as shown in equation (7). The matrix elements of the conductivity tensor are defined in equation (8) where ψ is the ground state wavefunction, T_{x(y)} = ε_{x(y)}·r_{x(y)} is the dipole transition operator, H is the Hamiltonian (taking into account the corehole effect) and γ is the Lorentzian broadening given by the corehole lifetime [L_{3} = 200 meV and L_{2} = 500 meV half width at halfmaximum (HWHM) (de Groot, 2005); L_{2} has a larger lifetime broadening due to the Coster–Kronig Auger decay],
In the most general case, nine independent measurements are required to fully reconstruct the conductivity tensor. However, the crystal symmetry can simplify the conductivity tensor by dictating the equivalence between matrix elements or cancelling out some of the matrix elements. For a cubic σ_{xx}, σ_{yy}, σ_{zz}; and two offdiagonal elements: σ_{yz} and σ_{zy}). The cubic crystal field implies that the x, y and z directions are equivalent by symmetry; however, if the external magnetic field is aligned to the x axis (and consequently the magnetization), it breaks the equivalency. For this reason, σ_{xx} will be different from σ_{zz} and σ_{yy}. In addition, the magnetization along x induces offdiagonal terms σ_{yz} (σ_{zy}) leading to a scenario where an electric field in the y(z) direction can produce an excitation in the z(y) direction. The offdiagonal terms cannot be directly measured; however, they can be reconstructed from linear combinations of measurements. We first focus on reconstructing the terms σ_{xx}, σ_{yy}, σ_{xy} and σ_{yx}; hence four independent measurements were performed for this purpose as shown in Table 1.
with the magnetic field aligned parallel to the high symmetry [1,0,0] direction, only five of the nine matrix elements are nonzero (three diagonal elements:

The response function is a complex quantity and one needs to compute the real part of the function. The real and the imaginary parts of the response function are related to each other through the Kramers–Kronig relation, which allows the computation of the real part from the (b) and 1(c)]. Linear combinations of these measurements can now be created according to Table 1 to give the matrix elements of the conductivity tensor. The four matrix elements (σ_{xx}, σ_{yy}, σ_{xy} and σ_{yx}) are shown in Fig. 1(d). One notices that σ_{xx}, σ_{yy} are different which results in a significant XMLD [see Fig. 1(e)]. The offdiagonal terms σ_{xy} and σ_{yx} are about 50 times smaller than the diagonal terms and are roughly equal. These symmetric offdiagonal contributions are possibly due to the presence of small noncubic distortion in the thinfilm. In contrast to the XMLD, the XMCD is negligible as can be seen in Fig. 1(f).
measurements [see Figs. 1The same procedure can be used to reconstruct the conductivity tensor with the magnetic field aligned to the [0,0,1] direction from four . A striking difference can be observed in comparison with Fig. 1(d): the offdiagonal terms σ_{xy} and σ_{yx} are nearly an order of magnitude stronger and are antisymmetric where σ_{xy} ≃ −σ_{yx}. The offdiagonal term differences seen are likely due to small misalignments in the orientation of the magnetization in particular given that the [0,0,1] direction is a magnetically hard direction and does not saturate with 1 T (see Fig. 13 of Appendix A) in combination with the presence of small noncubic crystal distortion in the thinfilm as we showed in earlier work (Elnaggar et al., 2020). The antisymmetric offdiagonal elements result in a significant XMCD as seen in Fig. 2(c). On the other hand, the XMLD signal is negligible because σ_{xx} ≃ σ_{yy} [refer to Fig. 2(b)].
measurements. These matrix elements are shown in Fig. 2The full conductivity tensor can be created by merging together the matrix elements obtained with B  [1,0,0] and B  [0,0,1] as shown in Fig. 3. This procedure assumes that the crystal field is cubic, which is an acceptable assumption given that the offdiagonal matrix elements related to the noncubic crystal field are very small. With the full conductivity tensor at hand, we can compute for any arbitrary polarization using equation (6). As such, we consider the polarization dependence as it is rotated from [1,0,0] to [0,1,0] in Fig. 4. The computed isotropic and the polarization dependence at E = 706.4 eV (red), 707.4 eV (green), 708.4 eV (magenta) and 709.4 eV (blue) are shown in Figs. 4(a) and 4(b), respectively. The measured polarization dependences at these energies are shown in the bottom row of Fig. 4(b) and agree very well with the computation using the full conductivity tensor.
3.2. Magnetic field dependence
The conductivity tensor shown in Fig. 3 gives the response function of the system at a certain magnetic field direction. It is of interest to find the full magnetooptical response of the system as it provides information about the anisotropic magnetic spin–orbit interaction and magnetic moments (van der Laan, 1998; Dhesi et al., 2001, 2002). Symmetry operations of the crystal can be used to relate the conductivity tensor with different magnetic field directions. For example, in a cubic the conductivity tensors with the magnetic field along x, y and z transform into each other through a 90° rotation. Similar symmetry arguments can be used to relate the conductivity tensor as a function of the magnetic field for different crystal symmetries. Haverkort et al. showed that the conductivity tensor can be expressed as a sum of linear independent spectra multiplied by functions depending on the local magnetization direction as given in equation (9) (Haverkort et al., 2010),
Here θ and ϕ define the direction of the local moment with θ being the polar angle, and ϕ being the azimuthal angle. Y_{k,m}(θ, ϕ) is a spherical harmonic function and σ_{i,j} is the i,j component of the conductivity tensor on a basis of linear polarized light in the coordinate system of the crystal. This expression allows one to describe the full, magnetic field directional dependent, magnetooptical response of a system by using only a few linear independent fundamental spectral functions. This expression may be simplified for certain crystal systems as we will discuss in the following.
3.2.1. Spherical field expansion
The σ^{(0)}, σ^{(1)} and σ^{(2)}) connected to the spherical harmonics Y_{0,0}(θ,ϕ), Y_{1,0}(θ,ϕ) and Y_{2,0}(θ,ϕ) are required to describe the conductivity tensor with an arbitrary magnetization direction [equation (10)],
can in some systems be small (in comparison with other interactions such as spin–orbit coupling in rareearth compounds) and the crystal symmetry can be considered to be nearly spherical. This approximation implies that the spectral shape modification is solely determined by the relative orientation between the magnetization and the polarization. Three fundamental spectra (The three fundamental spectra of the spherical expansion, σ^{(0)}, σ^{(1)} and σ^{(2)}, in Fe_{3}O_{4} are shown in Fig. 5(a) and can be used to compute spectra for any orientation of the magnetization. To evaluate the quality of the expansion, we start by comparing the measured and the computed magnetic field angular dependence of XMLD [I_{XMLD} = I(ϕ) − I(90°)] with linear horizontal polarization where the magnetic field is rotated from [1,0,0] to [0,0,1] [see Fig. 5(b)]. The expansion reproduces the measured angular dependence well and only minor discrepancies in the absolute intensities are observed. The angular dependence in this case is given by equation (11) where the two fundamental spectra σ^{(0)} and σ^{(2)} come into play,
A more interesting case can be observed when the magnetic field angular dependence is measured with the polarization rotated 30° clockwise from the [1,0,0] direction {i.e. ε  [cos(30°),− sin(30°), 0]} as shown in Fig. 5(c). Contrary to the results with linear horizontally polarized light, a strong deviation from spherical symmetry is now observed. The expected angular dependence from the spherical field expansion should follow equation (12); however, the spherical field expansion completely breaks down when the polarization is aligned to a low symmetry direction. This is not a surprising result for Fe_{3}O_{4} as its is cubic and the between the t_{2g} and e_{g} orbitals parametrized through 10D_{q} is ∼1 eV for Fe in Fe_{3}O_{4} while the spin–orbit coupling is ∼0.05 eV and the mean field exchange interaction is ∼0.09 eV. These values suggest that the crystal field cannot be neglected, and a spherical field expansion consequently cannot describe the magnetooptical response of Fe in Fe_{3}O_{4} well,
3.2.2. Cubic field expansion
The _{3}O_{4} is nearly cubic (Bragg, 1915), and therefore a more realistic treatment would be to perform a cubic field expansion. In this case, distinctly different measurements can be taken, for example along the fourfold and the threefold symmetry axes, and the fundamental spectra of order k branch according to their symmetry representations in the cubic This is shown in equation (13) where σ^{(2)} branches to for diagonal elements and for the offdiagonal elements [see Fig. 6(a)]. Furthermore, higherorder k terms such as become important (Haverkort et al., 2010),
of the Fe in FeA comparison between the measured and computed magnetic field angular dependence of XMLD with linear horizontal polarization can be seen in Fig. 6(b). The agreement between the measurements and the computed field dependence for the spherical and cubic field expansions are of similar quality. The field dependence in this case is given by equation (14) which is of the same form as the spherical field expansion [compare equation (14) with equation (11)] and the fundamental spectra involved are very similar [compare Fig. 6(a) with Fig. 5(a)],
However, contrary to the spherical expansion results, an excellent agreement between the cubic field expansion and the magnetic field angular dependence performed with rotated polarization is now observed in Fig. 6(c). The reason for this improvement is the branching of the σ^{(2)} fundamental spectrum into and which is probed when the polarization is aligned to a low symmetry direction bringing offdiagonal elements into play. The field dependence for the 30° rotated polarization is given by equation (15) which highlights the role of the fundamental spectrum. Another important conclusion is that it is essential to measure with the polarization aligned to a low symmetry direction to sensitively probe the crystal symmetry,
3.2.3. Convergence of the field expansion
In symmetries lower than spherical, the expansion of the spin (or the magnetic field) direction on spherical harmonics does not truncate at finite k. There is thus, in principle, an infinite number of linearly independent fundamental spectra. Not all of them are important and most of them will be of very low intensity. We have included in our previous analysis terms up to k = 3. The cubic field expansion showed a satisfactory agreement with the experimental data and only small discrepancies were showed. Here we investigate theoretically the origin of these discrepancies and the convergence of the field expansion. The quality of an expansion on the spin (or magnetic field direction) is foreseen to depend on the details of the ground state. This is because the magnetization direction depends on both the orbital and spin moments and hence whether the valence orbital moment is quenched or not will affect the efficiency of the expansion. Fe_{3}O_{4} contains both types of ions, Fe^{3+} and Fe^{2+}, providing us with an excellent opportunity to test the effect of the ground state for the two cases. We approach this by calculating the field dependence of XMLD in two ways:
(1) Performing a new full XMLD calculation for every magnetic field orientation.
(2) Computing once the conductivity tensor in equation (13) and then generating the field dependence of XMLD from the cubic fundamental spectra.
The first method is exact and involves no approximations. On the other hand, the accuracy of the second method depends on the order of the expansion used in the calculation. Let us first consider the magnetic field (B) angular dependence probed with the linear polarized Xrays aligned to [1,0,0] where B is rotated about [0,0,1] at ϕ = 0°. The XMLD signal [computed as XMLD = XAS(ϕ) − XAS(90°)] for Fe^{3+} and Fe^{2+} in O_{h} symmetry is shown in Figs. 7(a) and 7(b), respectively. The exact calculations (solid lines) and the cubic field expansion (dashed lines) match well which can initially suggest that the series expanded up to k = 3 is sufficient to describe XMLD in 3d transition metal oxides. Similar conclusions were reached by Arenholz et al. (2006, 2007) and van der Laan et al. (2008, 2011). However, a difference between the convergence of the series for both ions can be seen when the polarization is aligned parallel to [cos(30°),−sin(30°),0] [Figs. 7(c) and 7(d)]. Only minor discrepancies are observed for Fe^{3+} while a larger disagreement is observed for Fe^{2+}.
The reason behind the mismatch observed lies in the ground state of Fe^{2+}. In the absence of a magnetic/exchange field, the ground state of the Fe^{2+} ion in O_{h} symmetry is ^{5}T_{2g} composed of 15fold degenerate states. This degeneracy is split by exchange and spin–orbit interactions leading to a ground state characterized by the spin and orbital momenta projections s_{z} = 1.971 (±0.01) and l_{z} = 0.98 (±0.20). On the other hand, the ground state of Fe^{3+} is characterized by the spin and orbital projections s_{z} = 2.498 (±0.001) and l_{z} = 0.001 (±0.001). These values are obtained using the wavefunction calculated by solving equation (1). The reported errors are obtained from the errors in the distortion parameters that are obtained by fitting the XMCD signal using our calculations. We note that the ground state of Fe^{3+} in T_{d} symmetry is almost identical to that in O_{h} symmetry and therefore we focus here on the O_{h} sites (refer to the Appendix C for more details). As the magnetic field is rotated, the spin moment follows the field for the Fe^{3+} as shown in Fig. 8(a). In the case of Fe^{2+}, however, the coupling between the orbital and spin momenta results in a scenario where neither the spin nor the orbital moments follow the rotation of the magnetic field [see Fig. 8(b)] due to the magnetocrystalline anisotropy (Alders et al., 2001). The spin moment can be phase shifted from the direction of the magnetic field with ∼0.5° while the orbital moment can lag ∼4° in some directions. This causes the series to converge slower and hence higher orders of k are required. This is further confirmed by the calculation in Fig. 9(a) where the valence spin–orbit coupling is artificially switched off for Fe^{2+}. Now the cubic field expansion reproduces the XMLD exquisitely well and no phase shift is observed [Fig. 9(b)].
4. Conclusions
In conclusion, we illustrated the procedure to build the conductivity tensor from experimental measurements which describes the full magnetooptical response of the system. Applied to the Fe L_{2,3}edge of a 38.85 nm Fe_{3}O_{4}/SrTiO_{3} (001) thinfilm, we showed that the convergence of the cubic expansion depends on the details of the ground state. The key aspect that affects the convergence of the expansion in this work is the valence state spin–orbit interaction. While the cubic expansion explains the angular dependence of the XMLD of Fe^{3+} with terms up to the third order, higherorder terms are required for Fe^{2+}. This conclusion is expected to apply for other systems where the valence orbital moments are not quenched.
APPENDIX A
Sample characterization
A1. Xray reflectivity measurement
The Fe_{3}O_{4} (001)/SrTiO_{3} film thickness, interface and surface roughness were examined by Xray reflectivity using a Philips XPert MRD with Cu K_{α} radiation (see Fig. 10). The film thickness was concluded to be ∼38.85 nm. The surface roughness was concluded to be ∼0.4 nm on average.
A2. Magnetic measurement
Bulk magnetic properties of the Fe_{3}O_{4} (001)/SrTiO_{3} thinfilm were investigated using a quantum design dynacool physical properties measurement system. versus temperature was measured in zerofield cooling mode with 500 Oe applied field [Fig. 11(a)]. A clear peak can be observed at T = 114.97 ± 0.29 K in the derivative signal (the Verwey transition) confirming the good stoichiometry and quality of the thinfilm. A comparison between the derivative signal between the thinfilm used for this work and a single crystal of the same orientation is shown in Fig. 11(b). The is significantly broader for the thinfilm (FWHM = 13.03 ± 0.71 K, centre = 114.97 ± 0.29 K) in comparison with the single crystal (FWHM = 1.40 ± 0.01 K, centre = 127.08 ± 0.01 K). This could be related to defects and domain formations in the thinfilm which is typical for growth on SrTiO_{3}.
Hysteresis loop measurements were also performed along the [1,0,0] axis to inspect the saturation of the film below and above the ). The largest coercivity is observed for the lowest temperature (H_{c} = 0.1 T) and an external magnetic field of 0.25 T is required to saturate the inplane magnetization. On the contrary, the magnetization is not saturated along the [0,0,1] direction with a field of H_{c} = 2 T as shown by the XMCD signal shown in Fig. 13.
parallel to the inplane [1,0,0] direction (Fig. 12APPENDIX B
Data treatment
In order to compare the L_{3} and L_{2}) were fitted by two error functions positioned at 708.8 eV and 721.5 eV, respectively [refer to the grey dashed line in Fig. 14(a)]. The multiplet features of the spectra were fitted by a set of Gaussian functions. Six Gaussian functions were used to fit the L_{3} part of the spectra [red peaks in Fig. 14(a)] and four to fit the L_{2} [blue peaks in Fig. 14(a)]. Only the amplitude of the functions was allowed to float between the data while the energy positions and widths were kept constant between all the data. Table 2 (Table 3) shows the centre and width of the Gaussian peaks used for the L_{3} (L_{2}) edge. We normalized the spectra by setting the L_{2} edge jump to unity. The L_{2,3} edge jumps were subtracted from the data as shown in Fig. 14(b). Finally, the spectra were normalized to the spectral area.
results measured at different magnetic field orientations it is necessary to normalize the spectra. The edge jumps (


APPENDIX C
Ground state of Fe ions in Fe_{3}O_{4}
C1. Fe^{2+} in octahedral symmetry
The high spin Fe^{2+} ion in octahedral symmetry 15fold ^{5}T_{2g} state [shown in Fig. 15(a)] is split by exchange interaction [90 meV applied according to equation (5)] which lowers the degeneracies leading to a triplet ground state as illustrated in Fig. 15(b). The spin–orbit coupling finally lifts all the degeneracies as shown in Fig. 15(c). The ground state is 99.8% pure in terms of crystal field configuration and has an occupation (t_{2g})^{3.9997}(e_{g})^{2.0003}〉 and is 97.21% composed of the state characterized by m_{z} = −2 and l_{z} = −1. We point out that the first and second excited states are ∼22 and 53 meV higher in energy than the ground state which leads to a Boltzmann occupation of ∼75.8%, 20.6% and 3.6% of the ground, first and second excited states, respectively, at 200 K. These have been included in the calculation of the spectra. All the parameters used in the multipet calculations of Fe^{2+} are reported in Table 4.

C2. Fe^{3+} in octahedral and tetrahedral symmetry
The ground states of Fe^{3+} in O_{h} and T_{d} symmetries are almost identical with s_{z} = 2.499 and l_{z} = 0.001. Indeed, the ^{6}A_{1} splits to the doublet E_{2} and quartet G states; however, this splitting can be neglected (the splitting is less than 0.1 meV and is thermally populated). This is illustrated in Figs. 16 and 17 where the ^{6}A_{1(g)} state [panel (a)] is split mainly due to exchange interaction [panel (b)] while spin–orbit coupling has negligible effect [panel (c)] for Fe^{3+} in both T_{d} and O_{h} symmetries. We point out that the first is ∼90 meV higher in energy than the ground state which leads to a Boltzmann occupation of ∼99.6% and 0.4% of the ground and first respectively, at 200 K. We therefore only used the ground state in the calculation of the spectra. All the parameters used in the multipet calculations of Fe^{3+} O_{h} and T_{d} are reported in Tables 5 and 6, respectively.


Acknowledgements
We are thankful to R.P. Wang, M. Ghiasi and M. Delgado for helping with the synchrotron measurements. The synchrotron experiments were performed at the I06 beamline, Diamond Light Source, UK under proposal number SI17588. We are grateful for the help of the beamline staff to setup and perform the experiments.
Funding information
The following funding is acknowledged: European Research Council (grant No. 340279 to Frank M. F. de Groot; award No. SI17588 to Hebatalla Elnaggar).
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