Structure and thermal expansion of end-member olivines I: Crystal and magnetic structure, thermal expansion, and spontaneous magnetostriction of synthetic fayalite, Fe₂SiO₄, determined by high-resolution neutron powder diffraction

The thermoelastic and thermodynamic properties of the olivine group of minerals are of considerable interest, being necessary to understand the structure of the Earth's upper mantle (down to 410 km depth), where olivine is believed to account for between 30% to 60% (by volume) for typical mantle phase assemblages (e.g. Ringwood, 1969; Bass and Anderson, 1984). Olivines with compositions between forsterite (Mg₂SiO₄) and fayalite (Fe₂SiO₄) are among the most studied minerals due to their abundance and importance in a spectrum of geological and cosmic environments, their physical properties determining, amongst other things, the formation and thermal evolution of planetary bodies. For example, as pointed out recently by Béjina et al. (2021), determination of the thermoelastic properties of iron-rich olivine is likely to be crucial in understanding the Martian interior. Since the original determination of their crystal structures by Bragg and Brown (1926), there have been numerous investigations of the temperature dependence of the volume of the (Mg,Fe)₂SiO₄ olivines. The first studies of the linear thermal expansion of olivines were made by Kozu et al. (1934) and Rigby et al. (1946) using dilatometers for their measurements. The first thermal expansion study for pure forsterite was made by Skinner (1962) using powder X-ray diffraction, and for pure fayalite by Suzuki et al. (1981), using dilatometry.

In the Mg–Fe solid-solution, the Fe²⁺ and Mg²⁺ ions are sufficiently similar in size that this substitution results in very little structural change (e.g. Hazen, 1977). However, the fayalite end-member does show some very significant differences in behaviour compared to that of the isostructural forsterite. In particular, fayalite becomes magnetically ordered at low temperature and the change with temperature in the thermal expansion coefficient of its b axis takes a very different form to that of forsterite. Fayalite is orthorhombic, with space group Pbnm and cell parameters a ≈ 4.82 Å, b ≈ 10.48 Å and c ≈ 6.09 Å at room temperature. The crystal structure can be considered in terms of a layered hexagonal close-packed (HCP) oxygen network. Silicon ions occupy tetrahedral sites (in 4c positions at x, y, ¼, with x ≈ 0.43 and y ≈ 0.10 at room temperature) lying on mirror planes. Iron cations occupy two symmetrically non-equivalent octahedral sites, namely M1 (= Fe1) and M2 (= Fe2); the first of these is located in the 4a (0, 0, 0) positions and the second in 4c positions with x ≈ 0.99 and y ≈ 0.28 (Fig. 1). There are three non-symmetry-equivalent oxygen ions in the unit cell; two of these, O1 and O2, are located in 4c positions on the mirror planes, with x ≈ 0.77, y ≈ 0.09 and x ≈ 0.21, y ≈ 0.45, respectively, at room temperature. The remaining oxygen ions are in the general 8d positions, with x ≈ 0.29, y ≈ 0.17 and z ≈ 0.46. The smaller, more distorted, M1 octahedral site (at 0, 0, 0) is located on a centre of symmetry, and shares edges with the larger M2 octahedral site – which lies on the mirror planes running perpendicular to the c axis, cutting it at ¼ and ¾. The coordination polyhedron of the M1 site is formed by bonding to two O1, two O2 and two O3 atoms; that for the M2 site involves bonding to one O1, one O2 and four O3 atoms. In the SiO₄ tetrahedra the Si bonds to one O1, one O2 and two O3 atoms.

Kondo and Miyahara (1963) were the first to propose that pure fayalite exhibits a Néel point between 77 and 300 K, however Santoro et al. (1966) were the first to report detailed information about the two magnetic transitions at 23 K and 65 K by means of neutron powder diffraction. Below 65 K the Fe₂SiO₄ structure is antiferromagnetically ordered (henceforth termed AFM1) with the spins of the Fe atoms on both the M1 and M2 sites having a collinear arrangement parallel to the c axis. Below 23 K, there is a different antiferromagnetic structure (AFM2) in which the spins on the Fe atoms at the M1 sites become canted, possessing components along all three crystallographic directions (e.g. Müller et al., 1982). Suzuki et al. (1981) first identified the high-temperature anomalous behaviour of fayalite's b axis, which is completely different from that found in forsterite, with the thermal expansion coefficient for this axis showing a shallow, roughly parabolic form between 300 K and 1100 K, which they considered to be similar to the behaviour of some of the elastic moduli as reported by Sumino (1979). In subsequent studies, Kroll et al. (2012, 2014), also detected the anomalous behaviour of b, however their data are sparse below room temperatures and show significantly more scatter than those of Suzuki et al. (1981); furthermore, the behaviour of the thermal expansion coefficient of the b axis below room temperature reported by Kroll et al. (2014; as shown in figure 5b of their paper) differs markedly from that which we observe in the present study. Anomalous thermal expansion associated with a magnetic phase transition has also been reported by Sazonov et al. (2010) in the isostructural compound Co₂SiO₄. This material, however, differs markedly from Fe₂SiO₄; firstly, in that only one phase transition is observed, to the phase denoted here as AFM2 (at 50 K) and, secondly, as the b axis of Co₂SiO₄ (equivalent to the a axis in the Pnma setting of the space group used by Sazonov et al., 2010) does not show any unusual behaviour, or indeed any anomaly at the temperature of the magnetic phase transition.

To date, the effects of magnetism on the lattice parameters and the high-temperature anisotropy of the thermal expansion in fayalite have never been studied together and there have been no detailed measurements of the lattice parameters within the antiferromagnetic region. Using neutron powder diffraction, we have identified the two antiferromagnetic phase transitions and have determined their effects on the cell parameters of synthetic Fe₂SiO₄. The neutron powder data were collected with the High Resolution Powder Diffractometer (HRPD) at the STFC ISIS spallation neutron source. Measurements were made between 10 K and 1453 K, i.e. close to fayalite's melting point at 1478 K, allowing us to investigate the structural basis for the unusual anisotropy of fayalite's thermal expansion.

Polycrystalline Fe₂SiO₄ was prepared from a stoichiometric mixture of metallic Fe, Fe₂O₃ and SiO₂. The sample was synthesised using a new WC-COW solid-state oxygen buffering reaction within a ceramic crucible system (Dobson, 2021) that is effective at maintaining the oxygen fugacity at a level between iron–wüstite and quartz–iron–fayalite during solid-state synthesis of Fe-bearing silicates at 1 atm and 1273–1373 K. The sample was sintered for 6 days at 1323 K in a set of nested alumina crucibles inside a 5 L muffle furnace. In total, three sintering cycles were used, each lasting 2 days, with the fayalite being re-ground and pelleted between cycles. The recovered sample was examined by powder X-ray diffraction at room temperature; it was found to contain a very small amount of quartz as an impurity phase, with the Rietveld-refined X-ray diffraction pattern indicating a composition of 99.8 wt.% Fe₂SiO₄ and 0.2 wt.% SiO₂.

Time-of-flight neutron powder diffraction patterns were collected with the High Resolution Powder Diffractometer (HRPD) (Ibberson et al., 1992; Ibberson 2009) at the STFC ISIS spallation neutron source, Rutherford Appleton Laboratory, UK. Diffraction data, at all temperatures, were collected in HRPD's standard 100 ms wide time-of-flight window in the range 30–130 ms (d-spacings = 0.65–2.6 Å), normalised to the incident spectrum and corrected for detector efficiency by reference to a V:Nb standard, and then exported in a format suitable for analysis with GSAS/EXPGUI (Larson and Von Dreele, 2000; Toby, 2001) using the Mantid library of diffraction algorithms (Mantid, 2013; Arnold et al., 2014).

For the low-temperature data collection, between 10 and 340 K, the sample was loaded into an 18 mm × 23 mm aluminium-alloy ‘slab can’ sample holder with a depth of 5 mm. The Al sample holder is open at the front (beam-facing) and rear, and the sample is then contained by 125 μm thick vanadium foil windows, held in place with steel frames and sealed with indium wire. Exposed steel and aluminium surfaces on the front face of the holder were masked from the incident beam with Gd and Cd foils. The sample holder has holes drilled on either side of the specimen area, one to accept a RhFe resistance thermometer, and the other to take a cartridge heater; this arrangement allows rapid and accurate control of the true sample temperature provided that an appropriate time – ~10 minutes – is allowed for thermal equilibration after each change of temperature (Fortes, 2019). The assembly was then loaded into a closed-cycle refrigerator (CCR) held under a partial pressure of ~50 mbar of He exchange-gas. An initial dataset was collected at 100 K, followed by measurements in 5 K increments on cooling to 10 K. Thereafter, the sample was warmed to 110 K and measurements were obtained in 10 K steps up to 340 K. The majority of the datasets were ‘short’ acquisitions of ~10 minutes duration (8 μAh of proton beam current) intended solely for the refinement of unit-cell parameters, but several longer measurements for ~100 minutes (80 μAh) were made at 40 K and 10 K on cooling and at 300 K on warming; these provided diffraction data of excellent statistical quality suitable for us to carry out high-precision structure refinements.

To prevent sample degradation in the evacuated furnace at high temperatures, the sample was loaded under He gas at room temperature into a Ti–Zr–Mo alloy (‘TZM’) cylindrical sample can (8 mm in internal diameter with a 0.25 mm wall thickness), which was then sealed with a niobium wire gasket. This was then placed in a vacuum furnace (10^–6 mbar) and heated using vanadium-foil elements. Two K-type thermocouples were secured on either side of the sample can. After the first data collection at room temperature, ~297 K (20 μAh), a measurement was made at 373 K (40 μAh), after which data were taken in 20 K increments from 373 K to 1273 K, counting for 10 μAh at each point. As before, longer counting-times were used at 573 K (80 μAh), 873 K (80 μAh) and 1173 K (120 μAh) to provide data more suitable for high-precision structure refinements. The experiment was interrupted by a failure of the furnace elements at ~1270 K, as a result of which the sample cooled rapidly back to room temperature. After the elements were replaced, the sample was reheated, with data collected at 1073 K and 1173 K (to provide an overlap with the first high-temperature dataset), whereafter diffraction patterns were obtained in 20 K intervals from 1273 K up to 1453 K, close to the melting point of Fe₂SiO₄ which is reported to be at 1478 K. In the second high-temperature series, data were collected for 10 μAh at each temperature, except at 1353 K and 1453 K where the counting times were extended to 200 μAh and 167 μAh respectively. After heating to 1453 K, the recovered Fe₂SiO₄ sample, measured at 573 K for 53 μAh, was found to be unaltered, with no additional peaks seen in the powder pattern (the unit-cell parameters of the recovered sample are given in the footnote to Table 2; the refined values of the fractional coordinates differed by 0.0007, at most, from those originally found at 573 K).

Neutron diffraction patterns of Fe₂SiO₄ collected at 10 K, 300 K and 1453 K are presented in Fig. 2. A total of 119 data sets were collected for Fe₂SiO₄: 45 in the low- and 75 in the high-temperature sample environment. The good statistical quality of even the ‘short’ data acquisitions allowed us to carry out refinements of the nuclear and magnetic structure using all of the data collected between 10 and 1453 K. For consistency, the same set of parameters was refined for both the ‘long’ and ‘short’ counting times. As one might expect, for the short counting times, the estimated standard deviations of the refined parameters are larger than those derived from the longer measurements, however we are nonetheless able to discern trends in – for example – bond lengths, bond angles and magnetic moments with a high degree of certainty, as illustrated below. The magnetic and nuclear structures were refined as separate phases with the Rietveld method using the GSAS I suite of programmes (Larson and Von Dreele, 2000) with the EXPGUI graphical interface (Toby, 2001). Consideration of previous studies of the magnetic structure of Fe₂SiO₄ (e.g. Müller et al., 1982) indicated that the space group remains Pbnm in all phases and so the magnetic and nuclear unit cells were constrained to be the same throughout. Following Müller et al. (1982), for the canted antiferromagnetic (AFM2) region (T < 23 K), all three components of the magnetic moment on the Fe1 site were allowed to vary while for the collinear (AFM1) region, 25 ≤ T ≤ 65 K, these moments were constrained to remain parallel to the c axis; the moments on the Fe2 site were constrained to remain parallel to the c axis at all temperatures below 65 K. At and below 65 K, in the AFM1 phase, 59 variables were included in the refinement (one scale factor, six profile parameters, eight background coefficients, three cell parameters, two magnetic moments, 11 fractional coordinates, 28 displacement parameters); below the second AFM transition at 23 K, the number of refined magnetic moments increases to four. For all data collected with the sample in the CCR, the instrumental parameters DIFC and DIFA, which define the primary neutron flight path and the contribution to the path from wavelength-dependent absorption from the sample, were fixed at 48216.25 and −0.38, respectively, these being derived from an instrument calibration carried out using a NIST silicon standard, SRM640e. Above 65 K we observed no peaks from the magnetic cell and therefore only the nuclear phase was refined, including 57 variables. Bragg reflections from the TZM sample can were present in the data collected with the sample in the furnace. This TZM alloy has a composition of ~99.4% Mo, alloyed with 0.5% Ti, 0.08% Zr and 0.02% C; it was, therefore, included in the refinement on the basis that it had the body-centred cubic structure of Mo, with the intensities calculated via the Le Bail method as it was highly textured. The lattice parameters of Fe₂SiO₄ and those of the Mo sample can, the fractional coordinates, atomic displacement parameters, scale factors, phase fractions, background and profile parameters were refined with DIFC and DIFA (adjusted to take account of the different sample geometry), fixed at 48216.25 and −0.52, respectively. Due to a small offset between the two stages, as well as between the two sets of high-temperature measurements, all high-temperature cell parameter values were scaled to match the low-temperature results. This was done by fitting first-order polynomials to the data from 300–340 K (CCR) and 297–433 K (furnace) and 960–1000 K (first high-temperature run) and 900–1020 K (second high-temperature run) and then calculating the scale factors for the offsets (see Table 2) from the ratios of the values of the polynomials at 300 K and 980 K. Refined fractional coordinates, anisotropic atomic displacement parameters (and their isotropic equivalents) and magnetic moments of the ten data sets with longer collection times, from the CCR and the furnace, are shown in Table 1. The use of anisotropic displacement parameters with powder data is sometimes questionable. However, we believe that they can be justified in this case as we have neutron data extending to 0.65 Å and because the resulting equivalent isotropic displacements (Table 1) are such that the values for the same types of atoms are always very similar. In the high-temperature data, when the scattering at short d-spacings is reduced in intensity, the values are probably less reliable, with one component of the displacement ellipsoid of a few atoms for some data sets (e.g. U¹¹ for Si and O1 at 1453 K) occasionally taking values that are too small, without, however, showing a systematic trend (Supplementary Fig. 1 shows plots of the displacement ellipsoids).

The crystallographic information files have been deposited with the Principal Editor of Mineralogical Magazine and are available as Supplementary material (see below).

The evolution of the unit-cell volume of Fe₂SiO₄ over the full temperature range is plotted in Fig. 4 and listed in Table 2. Our results are in good agreement with those of earlier high-temperature X-ray diffraction and dilatometry studies (Suzuki et al., 1981; Kroll et al., 2012; Hazen, 1977). The detailed low-temperature behaviour of the unit-cell lattice parameters and volume have not, until now, been reported.

The resulting values for all data between 297 K and 1453 K, fitted in EoSFit7c (Angel et al., 2014), are V_TR = 307.73(1) ų, a₀ = 3.02(6)×10⁻⁵ K⁻¹, a₁ = 0.86(5)×10⁻⁸ K⁻², and a₂ = −0.97(8) K.

Where N is the number of atoms in the unit-cell (in this case N = 28); θ_D is the Debye temperature, and k_B is Boltzmann's constant.

The full line in Fig. 5 is obtained by differentiation of equation 6 whereas the points show the results from simple numerical differentiation by differences, point by point, of the V(T) data. Once again, the agreement between model and data is good, despite the deficiencies of the theory. In particular, the neglect of anharmonicity can lead to an underestimate of the thermal expansion coefficient at high temperatures (e.g. Wood et al., 2002), but there is little indication of this in the present case.

For T > θ_D, our volume thermal expansion has slightly higher values compared to those in the literature, with the value of α_V(T) of Suzuki et al. (1981) at 1000 K being 9.3% smaller than ours, while those of Kroll et al. (2012), when calculated on a point-by-point basis, are scattered. Kroll et al. (2012) measured the thermal expansion by single-crystal X-ray diffraction and Suzuki et al. (1981) used dilatometry. We consider our data to be more accurate than those of Kroll et al. (2012) as we have the advantages of neutron diffraction data collected at a very high Bragg angle on a time-of-flight diffractometer with an extremely long flight path. The time-of-flight method gives us a resolution in d-spacing which is effectively constant across the whole of the diffraction pattern, unlike the angle-dispersive X-ray diffraction method used by Kroll et al. (2012), in which measurements at very high Bragg angles would be required to obtain comparable resolution. The differences between our results and those of Suzuki et al. (1981) may be due to differences in the stoichiometry of the samples, although it should be remembered that dilatometry and diffraction sample fundamentally different properties of the material (see e.g. Simmons and Balluffi, 1962).

The Debye temperature of Fe₂SiO₄ obtained from the fit to equation 6, 498(9) K, is in very good agreement with that reported by Anderson and Suzuki (1983), 510 K, or Anderson and Isaak (1995), 511 K, but somewhat lower than that of Suzuki et al. (1981), 565 K. The value of V₀, 306.280(9) ų, corresponds to the volume that the unit-cell of a paramagnetic phase of Fe₂SiO₄, with disordered local magnetic moments, would occupy if such a phase persisted to limiting low temperature. An estimate of the incompressibility, K₀, can be obtained directly from the coefficient Q, provided that the Grüneisen parameter is known. Suzuki et al. (1981) found γ equal to 1.097(5). If we apply this value of γ we obtain K₀ = 143(4) GPa which is a little larger than published values for K₀, obtained directly from high-pressure studies, which lie between 128 and 136 GPa (e.g. Graham et al., 1988; Béjina et al., 2019, 2021; Speziale et al., 2004; Zhang, 1998; Zhang et al., 2017). The first derivative of the incompressibility with respect to pressure can also be estimated from the coefficient b. The resulting value, |$K_0^{\prime}$|= 12.8(2), is, however, higher than those published from high-pressure studies, which range between 4.1 and 5.3 (e.g. Graham et al., 1988; Béjina et al., 2019, 2021; Speziale et al., 2004; Zhang, 1998). This is an indication of the limitations of the model of equation 6 that are reflected in the fitted parameters. In the V_G(T) term of equation 6 the coefficients Q and b are assumed to be temperature independent, whereas, in reality, the Grüneisen parameter, the incompressibility and its first derivative with respect to pressure all have some temperature dependence (e.g. Voc̆adlo et al., 2002).

Where |$Q_{\rm X} = K_{X_0}V_0/\gamma$|⁠, with the subscript X indicating that we are considering axial rather than volumetric expansion. For an orthorhombic crystal, the expression for the parameter b_X becomes more complex than is for the case of volumetric expansion. Taking as an exemplar the a axis of Fe₂SiO₄ we have |$Q_a = K_{a_0}V_0/\gamma$| and |$b_a = [ {K_{a_0}^{\prime} \;\ndash \;2( {K_{a_0}/K_{b_o}} ) \;-\;2( {K_{a_0}/K_{c_o}} ) \;-1} ] /2$|⁠. The axial incompressibilities are related to the elastic compliances, s_ij, such that for example, |$K_{a_0} = 1/( {s_{11} + s_{12} + s_{13}} )$|⁠. Similar equations can be derived for the b and c axes. The magnetic, X_M(T), term of equation 9 takes the same form as was used in equation 6. Therefore equation 9 can be used to fit the data for a(T), b(T) and c(T). The results are plotted in Fig. 6 and Table 3 lists the values of all fitted parameters.

For an orthorhombic crystal, the bulk incompressibility, K₀, and the axial incompressibilities are related by K₀ = |$1/( {1/K_{a_0} + 1/K_{b_0} + 1/K_{c_0}} )$|⁠. Assuming that a value of γ = 1.097 (Suzuki et al., 1981) applies in all cases, we obtain axial incompressibilities of 408(17), 519(18) and 293(7) GPa for the a, b, and c axes, respectively. By combining the axial incompressibilities we obtain a value of 128(12) GPa for K₀, which is in excellent agreement with values reported in the literature (128–136 GPa – see above), but 6.3% smaller than that obtained by fitting the V(T) data.

Inspection of Fig. 6 reveals that although equation 9 adequately represents the behaviour of the a and c axes, the same is not true for the b axis. The misfits between the model and the observed data for b(T) are much greater than for either a(T) or c(T) and some of the fitted parameters for b(T) are not physically sensible; in particular, the value of the Debye temperature, 14(1) K, is extremely low, reflecting the fact that equation 9 is not able to account satisfactorily for the observed trend. This failure is, perhaps, more clearly seen by examination of the thermal expansion coefficient for the b axis, which does not show the expected fall-off below room temperature, instead remaining almost constant until the transition to the AFM1 phase is reached (Fig. 6). The expansion coefficient for the b axis that we report here corresponds very closely to that observed above room temperature by Suzuki et al. (1981) using single-crystal dilatometry, however our new data reveal that its unusual behaviour continues to much lower temperatures.

A further deficiency of the model defined by equation 9, as applied to the individual cell parameters of fayalite, is revealed by comparison of our derived axial incompressibilities with those measured directly, at room temperature and high pressure, by X-ray diffraction by Zhang (1998) and Zhang et al. (2017). Zhang (1998) reported axial incompressibilities of 741, 304 and 568 GPa for the a, b and c axes respectively; Zhang et al. (2017) adopted a value of 135 GPa for the volumetric incompressibility and obtained axial values of 682(67), 281(24) and 479(24) GPa. Although our values for the volumetric incompressibility (143(4) K from the fit to V(T) and 128(12) GPa from fitting the axes) agree well with those listed in table 2 of Zhang et al. (2017), the agreement for the individual axes is not good, even to the extent of being unable to determine correctly the relative order of incompressibility. Both Zhang (1998) and Zhang et al. (2017) have |$K_{a_0}$| > |$K_{c_0}$| > |$K_{b_0}$|⁠, whereas from equation 9 we obtain |$K_{b_0}$| > |$K_{a_0}$| > |$K_{c_0}$|⁠. Bearing in mind the unusual form of the thermal expansion of the b axis, this result is, perhaps, not surprising. As we intend to discuss in a future accompanying paper, in this respect fayalite behaves very differently from forsterite, in which the three crystallographic axes all show the expected reciprocal relationship between thermal expansion coefficient and incompressibility.

Our neutron data are consistent with previous results for fayalite (e.g. Santoro et al., 1966; Müller et al., 1982; Lottermoser et al., 1986) showing that below the antiferromagnetic phase transitions at 65.4 K (to the AFM1 structure) and ~23 K (to AFM2) the magnetic cell remains equal to the crystallographic (chemical) cell and the space group remains Pbnm. The magnetic moments of the Fe2 ions are antiferromagnetically coupled and constrained by symmetry (as the Fe2 sites are on mirror planes) to lie parallel/antiparallel to the c axis in both the AFM1 and AFM2 structures (Fig. 7). For the Fe1 sites, which lie on centres of symmetry, there is no symmetry constraint on the spin orientation, however the four Fe1 sites in the unit cell are related by symmetry such that there is no net magnetic moment. Previous neutron diffraction studies on powder samples (Santoro et al., 1966) and with single crystals (Müller et al., 1982; Lottermoser et al., 1986) have shown that in the AFM1 phase the moments of the Fe1 and Fe2 sites are collinear, but in the AFM2 phase the moments on the Fe1 sites are canted. Santoro et al. (1966; see also Lottermoser et al., 1986) described two possible models for the spin canting on the Fe1 site, only one of which (that originally proposed by Cox et al., 1965) would seem to be consistent with space group Pbnm. Given that the symmetry is unchanged on passing from the AFM1 to the AFM2 phase, there has been some discussion, on the basis of results from single-crystal neutron diffraction (Lottermoser et al., 1986; Fuess et al., 1988), as to whether there was a distinct transition between the two phases or whether there was, instead, just a gradual change of the canting angle accompanied by a decrease of the magnetic moments. However, an analysis of Fe₂SiO₄ magnetisation and its anisotropy by Ehrenberg and Fuess (1993), indicated a change in spin canting direction on the Fe1 sites below ~20 K which is consistent with Mössbauer experiments on temperature-dependent hyperfine fields (e.g. Hafner et al., 1990; Lottermoser et al., 1995, 1996).

Our refinements of the AFM2 magnetic structure were based on the model proposed by Cox et al. (1965), and later confirmed by Müller et al. (1982) and Lottermoser et al. (1986); in our refinements of the AFM1 structure, the moments of the Fe1 ions were constrained to lie along the c axis. As discussed by Cococcioni et al. (2003 – see their figure 2), there are, however, two possible configurations for the relative orientation of the moments on the Fe1 and Fe2 sites. We found that the arrangement shown in Fig. 7 gave the best fit to our data; this arrangement also corresponds to the ground state of the system, as determined by quantum-mechanical modelling (Cococcioni et al., 2003), with the magnetisation of the Fe2 ion being in the same direction to that of its closest Fe1 ion, an arrangement suggesting antiferromagnetic ordering that occurs between corner-sharing octahedra.

Our refined values for the total moments on the Fe1 and Fe2 ions and for their direction cosines are listed in Table 4. At 10 K, our total moments on the Fe1 and Fe2 ions (4.16(4) and 4.20(3) μ_B, respectively) and for the direction cosines of the Fe1 moment with respect to the a, b and c axes (0.56(4), 0.31(7), 0.76(3)) are in very good agreement with the single-crystal results of Lottermoser et al. (1986), who found values of 4.41(5) and 4.4(1) μ_B, 0.57(2), 0.31(2) and 0.77(1). With increasing temperature, we find a gradual decline, more pronounced in the AFM2 phase, in the Fe1 magnetic moments, which are considerably reduced as compared both to the spin only value of μ = 4 μ_B and to those of the Fe2 site (Fig. 8). The behaviour of the moment on the Fe2 site is different, remaining fairly constant until ~50 K and then falling steeply. Once again, our results are in agreement with the trends observed by Müller et al. (1982) and Lottermoser et al. (1986).

With X referring to the lattice parameters a, b, and c.

Initially, we determined the strains by fitting first-order Grüneisen approximations to the zero-pressure equation of state (e.g. Vočadlo et al., 2002, equivalent to setting the parameter b in equation 4 to zero) to a, b, c and V in the range 70 K ≤ T ≤ 200 K and then using these to extrapolate the paramagnetic behaviour into the AFM temperature range, but we then found that that the sum of the resulting axial magnetostrictive strains was not equal to the volumetric strain, as is required by equation 13. The reason for this is that the b axis varies essentially linearly for 65 K ≤ T ≤ 200 K and so it cannot be reliably extrapolated in this way; the correlation coefficient between b₀ and θ_D in the non-linear least-squares algorithm was found to be 99.9%, and thus the temperature at which the thermal expansivity of the b axis begins to reduce, and hence its value at 0 K, cannot be determined. It was decided, therefore, that the most robust method for extrapolation of the cell parameters of the paramagnetic phase was to fit a, c and V in the range 70 K ≤ T ≤ 200 K, as described above, and then to determine b_par from V_par/(a_parc_par). For T ≤ 65 K, equation 7 was then fitted to the differences between the measured and the extrapolated cell parameters and the volumetric and linear magnetostrictive strains were calculated from equations 11 and 12 and were normalised to their calculated values at 0 K. The fitted lattice parameters are shown in Fig. 9, with the volumetric and linear spontaneous magnetostrictive strains given in Fig. 10a and the normalised magnetostrictive strains in Fig. 10b (the fitted values of the parameters are given in Supplementary Table 1; and Supplementary Fig. 2 shows the self-consistency of ω_V and λ_a + λ_b + λ_c). It can be seen from Fig. 10a that the strain λ_a is much smaller than λ_b and λ_c, for which the strains have opposite signs, and that the transition at 23 K has a minimal effect on the length of the b axis. In this latter respect, the behaviour of Fe₂SiO₄ at the 23 K transition is similar to that shown at the antiferromagnetic phase transition by Co₂SiO₄ (Sazonov et al., 2010); however, in Co₂SiO₄, the values of λ_a and λ_c are in the opposite sense to those in Fe₂SiO₄. Comparison of the normalised values of the spontaneous magnetostriction (Fig. 10b) with the refined values of the magnetic moments on the Fe1 and Fe2 sites, Fig. 8, suggests that λ_a and λ_c show a temperature dependence that is similar to that of Fe1 moments whereas λ_b follows a trend more similar to that of the Fe2 sites.

In Table 1, we present our refined values of the atomic coordinates for Fe₂SiO₄ for the 10 well-counted data sets spanning the range from 10 K to 1453 K. However, as it was found that even the shorter counting times allowed us to obtain structure refinements of very good precision, the derived parameters characterising the Fe1O₆, Fe2O₆ and SiO₄ coordination polyhedra are shown in Fig. 11 and Fig. 12 for all data, with the numerical values listed in Supplementary Tables 2–5. None of the quantities plotted show any significant differences in behaviour associated with the onset of the paramagnetic ordering.

In Fe₂SiO₄ we find that the average Fe–O bond distances in both the Fe1O₆ and Fe2O₆ coordination octahedra show a roughly linear dependence on temperature, with Fe1–O increasing at ~6.36×10⁻⁴ Å K^–1 and Fe2–O at ~7.53×10⁻⁴ Å K^–1,while the shorter and more rigid Si–O bonds show very little, to no, expansion; the same features are visible in the volumes of the coordination polyhedra (Fig. 11). Various measures of the distortion of the polyhedra are shown in Fig. 12. The angular variance of the SiO₄ tetrahedra (defined as |$\sum ( \rm {O\widehat{{Si}}O-109.47} ) ^2/6$|⁠) is almost invariant, whereas the Fe1O₆ and Fe2O₆ octahedral angular variances (defined as |$\rm \sum ( {O\widehat{M}O-90} ) ^2/12$|⁠) both increase roughly linearly with temperature with a possible sharper increase above ~1300 K. Figure 12 also shows the polyhedral volume distortions from ideal polyhedra, as calculated by the program Ivton2 (Balić-Žunić and Vicković, 1996). These were obtained via the relationship (V_i – V_d) / V_i, where V_d is the volume of the measured coordination polyhedron and V_i the volume of the ideal polyhedron having fixed angles and an average for bond distances (taken from the centroid of the polyhedron to the coordinating ligands). The distortion of the SiO₄ tetrahedral sites remains almost constant throughout the temperature range. As was seen in the octahedral angular variances, the smaller Fe1O₆ octahedron remains more distorted throughout the temperature range and the distortion increases at a slightly faster rate (~1.4×10⁻⁴ K⁻¹) than that of the larger Fe2O₆ octahedron (~9.0×10⁻⁵ K⁻¹) in agreement with suggestions (e.g. Burns and Sung, 1978), that the larger thermal expansion of the c axis results from the tendency of the Fe1O₆ octahedra to elongate.

We have collected high-resolution neutron powder diffraction patterns from Fe₂SiO₄ over the widest range of temperatures yet reported, from 10 K to 1453 K, which is within 25 K of the melting point (1478 K), at a homologous temperature of 0.983. Our refined lattice parameters are of substantially better accuracy and precision than any preceding diffraction study, allowing us to investigate the effect of the two antiferromagnetic transitions on the lattice parameters of Fe₂SiO₄ and thereby determine the spontaneous magnetostrictive strains. Our refinements of the magnetic structures confirm that the unit cell and space group (Pbnm) remain the same in all phases of Fe₂SiO₄. The moments on the Fe2 (M2) sites remain collinear and parallel to the c axis at all temperatures below 65.4 K. Below 65.4 K, the spins on the Fe1 (M1) sites are collinear and parallel to the c axis, but below 23 K, they become canted, possessing components along all three crystallographic directions.

The behaviour of the unit-cell volume of Fe₂SiO₄ is adequately described by a Grüneisen approximation to the zero-pressure equation of state, combined with a model of the magnetostriction based on modified mean-field curves, so as to include the contributions of the two AFM transitions at 23 K and 65.4 K. However, in the paramagnetic phase, the temperature dependence of the linear thermal expansion coefficient of the b axis takes an unusual form. In contrast to the unit-cell volume and to the a and c axes, for which the thermal expansion coefficients show the expected reduction in magnitude below ~300 K, the expansion coefficient of the b axis remains almost constant between ~70 K and 1000 K. This behaviour has been reported previously by e.g. Suzuki et al. (1981) above room temperature, but we have now shown that it persists to 70 K, i.e. throughout the paramagnetic phase field. Calculation of the average interatomic distances, and the volumes, angular variances, and distortions of the coordination polyhedra show that the changes with temperature in the FeO₆ and SiO₄ polyhedra are small, with the SiO₄ tetrahedra almost invariant. The volumes and distortions of the FeO₆ octahedra increase roughly linearly with temperature, with the smaller Fe1O₆ octahedron remaining more distorted throughout. In a future accompanying paper we intend to compare the thermal expansion coefficients of Fe₂SiO₄ with those of Mg₂SiO₄ and to discuss further the effects of temperature on the structures of these two olivine end-members.

We acknowledge the support of the Natural Environment Research Council, NERC, grants NE/S007229/1 and NE/T006617/1 and the ISIS Facility for the provision of beam-time (RB1910186, doi:10.5286/ISIS.E.RB1910186). The authors are grateful to the Pressure and Furnaces technicians who supported the high-temperature neutron diffraction measurements. The ISIS Design Division, in particular Ali Mortazavi, are thanked for the development of the new TZM sample can used for the first time as part of this study.

The supplementary material for this article can be found at https://doi.org/10.1180/mgm.2023.66.

The authors declare none.