Introduction
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, Reference Ringwood1969; Bass and Anderson, Reference Bass and Anderson1984). 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. (Reference Béjina, Bystricky, Tercé, Whitaker and Chen2021), 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 (Reference Bragg and Brown1926), 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. (Reference Kozu, Ueda and Tsurumi1934) and Rigby et al. (Reference Rigby, Lovell and Green1946) using dilatometers for their measurements. The first thermal expansion study for pure forsterite was made by Skinner (Reference Skinner1962) using powder X-ray diffraction, and for pure fayalite by Suzuki et al. (Reference Suzuki, Seya, Takei and Sumino1981), 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, Reference Hazen1977). 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.
Figure 1. Polyhedral model of the Fe2SiO4 structure, consisting of SiO4 tetrahedra (blue) and Fe1 (= M1, orange) and Fe2 (= M2, green) octahedra, viewed along [100]. Image produced using VESTA (Momma and Izumi, Reference Momma and Izumi2011).
Kondo and Miyahara (Reference Kondo and Miyahara1963) were the first to propose that pure fayalite exhibits a Néel point between 77 and 300 K, however Santoro et al. (Reference Santoro, Newnham and Nomura1966) 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., Reference Müller, Fuess and Brown1982). Suzuki et al. (Reference Suzuki, Seya, Takei and Sumino1981) 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 (Reference Sumino1979). In subsequent studies, Kroll et al. (Reference Kroll, Kirfel, Heinemann and Barbier2012, Reference Kroll, Kirfel and Heinemann2014), 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. (Reference Suzuki, Seya, Takei and Sumino1981); furthermore, the behaviour of the thermal expansion coefficient of the b axis below room temperature reported by Kroll et al. (Reference Kroll, Kirfel and Heinemann2014; 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. (Reference Sazonov, Hutanu, Meven, Heger, Hansen and Senyshyn2010) 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., Reference Sazonov, Hutanu, Meven, Heger, Hansen and Senyshyn2010) 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.
Experimental method
Sample synthesis
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, Reference Dobson2021) 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.
Neutron Powder Diffraction
Time-of-flight neutron powder diffraction patterns were collected with the High Resolution Powder Diffractometer (HRPD) (Ibberson et al., Reference Ibberson, David and Knight1992; Ibberson Reference Ibberson2009) 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, Reference Larson and Von Dreele2000; Toby, Reference Toby2001) using the Mantid library of diffraction algorithms (Mantid, 2013; Arnold et al., Reference Arnold, Bilheux, Borreguero, Buts, Campbell, Chapon, Doucet, Draper, Leal, Gigg and Lynch2014).
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, Reference Fortes2019). 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).
Refinement
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, Reference Larson and Von Dreele2000) with the EXPGUI graphical interface (Toby, Reference Toby2001). Consideration of previous studies of the magnetic structure of Fe2SiO4 (e.g. Müller et al., Reference Müller, Fuess and Brown1982) 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. (Reference Müller, Fuess and Brown1982), 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. U 11 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).
Figure 2. Examples of neutron powder diffraction patterns of Fe2SiO4 collected at different temperatures. Observed data are shown as red crosses, the calculated diffraction pattern as a green line, and their differences are given by the lower pink trace. The black tick marks in each of the plots show the positions of the Bragg reflections from the nuclear Fe2SiO4 crystal structure. For the refinement at 10 K, the red tick marks show the positions of the Bragg reflections from the Fe2SiO4 magnetic structure; the contribution from magnetic scattering to the diffraction pattern is judged negligible below 0.986 Å. For the 1453 K refinement, the red tick marks show the positions of the Bragg reflections from the TZM sample can.
Table 1. Fractional coordinates, anisotropic atomic displacements, and magnetic moments at 10 temperature points (numbers in parenthesis are one standard error of the least significant digits).
* Equivalent isotropic displacement parameters calculated from the anisotropic values.
Table 2. Lattice parameters and unit-cell volumes* measured in this study. The listed values are unscaled and the numbers in parenthesis are one standard error of the least significant digit.
*Notes: The values listed in the Table are taken directly from the Rietveld refinements. In subsequent analyses of the cell parameters, the CCR data used were those shown above, with the following multiplicative scale factors then applied to correct for the small offsets between the different sample environments: (1) between the CCR and the high-temperature furnace for a, b, c axes and unit-cell volume, respectively: 1.000199, 1.000139, 1.000198 and 1.000536; (2) between the two high-temperature measurements for a, b, c axes and unit-cell volume, respectively: 0.99984, 0.999856, 0.99982 and 0.999526.
The cell parameters of the ‘recovered’ sample (i.e. after it had been heated to 1453 K) at 573 K were found to be a = 4.83072(2) Å, b = 10.50281(4) Å, c = 6.11305(2) Å and V = 310.152(1) Å3; the small differences between these values and those shown at 573 K in the Table are readily attributable to the change in sample environment.
The crystallographic information files have been deposited with the Principal Editor of Mineralogical Magazine and are available as Supplementary material (see below).
Results and Discussion
Antiferromagnetic Transition Temperatures
Although the space group of the paramagnetic, canted, and collinear AFM phases of Fe2SiO4 remains Pbnm, some reflections that are systematically absent when only the nuclear scattering is considered become visible in the collinear AFM and canted AFM phases. To determine the temperature of the transition between the paramagnetic and AFM1 phases, we considered the behaviour of the 052 reflection, as this peak, with intensity coming solely from the magnetic scattering, occurs at a d-spacing (~1.72 Å) where it is well separated from other peaks in the diffraction pattern (Fig. 3). The values of |F obs|2 for 052, obtained from the GSAS refinements, are shown as a function of temperature in Fig. 3, with those for T ≥ 50 K fitted by non-linear least squares to the expression:
where A is a scale factor, T N is the transition temperature and β is a critical exponent. Although our data are rather sparse, it can be seen that an excellent fit is obtained, with T N = 65.4(1) K and β = 0.31(3). Our calculated T N is in good agreement with previous measurements, which lie in the range 64.9–65.3 K (e.g. Lottermoser et al., Reference Lottermoser, Müller and Fuess1986; Müller et al., Reference Müller, Fuess and Brown1982; Suzuki et al., Reference Suzuki, Seya, Takei and Sumino1981; Santoro et al., Reference Santoro, Newnham and Nomura1966; Aronson et al., Reference Aronson, Stixrude, Davis, Gannon and Ahilan2007). Determination of the transition temperature between the AFM1 and AFM2 antiferromagnetic phases from our diffraction patterns is more problematic as no additional, non-overlapped, reflections appear in the AFM2 phase. Thus, although this transition is visible in the behaviour of the a and c axes, and in the refined values of the magnetic moments on the Fe atoms (see below), we do not consider that we can reliably fix the transition temperature beyond saying that it occurs at 20 ≤ T ≤ 25 K; in our analysis of the lattice parameters we have, therefore, adopted the value of 23 K quoted by previous authors (e.g. Müller et al., Reference Müller, Fuess and Brown1982).
Figure 3. (a) Stacked diffraction patterns from 10 K (bottom-most) to 70 K (top-most) showing the magnetic reflection at 1.72 Å (hkl = 052) and (b) |Fobs|2 for the 052 reflection as a function of temperature fitted to equation 1 (solid black line) giving a transition temperature T N = 65.4(1) K.
Lattice parameters and thermal expansion
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., Reference Suzuki, Seya, Takei and Sumino1981; Kroll et al., Reference Kroll, Kirfel, Heinemann and Barbier2012; Hazen, Reference Hazen1977). The detailed low-temperature behaviour of the unit-cell lattice parameters and volume have not, until now, been reported.
Figure 4. Measured unit-cell volumes of Fe2SiO4 against temperature. Experimental data are shown as open circles and the model of equation 6 as a solid black line. Unit-cell volume error bars are omitted because they are smaller than the symbols; the smaller inner panel shows the fit of equation 6 below 90 K in more detail. The lower panel shows the differences between measured and calculated unit-cell volumes as a function of temperature when employing the model of equation 6.
To describe thermal expansion at high temperatures an equation of the form suggested by Fei (Reference Fei and Ahrens1995) is commonly used. In this formulation, the unit-cell volume is given by:
where $V_{T_{\rm R}}$
is the volume at a reference temperature (in this case T R = 297 K) and a V(T) is the volumetric thermal expansion coefficient given by a polynomial expression of the form
The resulting values for all data between 297 K and 1453 K, fitted in EoSFit7c (Angel et al., Reference Angel, Alvaro and Gonzalez-Platas2014), are V T R = 307.73(1) Å3, a 0 = 3.02(6)×10−5 K−1, a 1 = 0.86(5)×10−8 K−2, and a 2 = −0.97(8) K.
A more physically meaningful interpretation of the temperature dependence of the unit-cell volume, encompassing its full temperature range, can be obtained by using Grüneisen approximations to the zero-pressure equation of state (Wallace, Reference Wallace1998). This approach, in which the effects of the thermal expansion are considered to be equivalent to the elastic strain induced by the thermal pressure, also allows estimates of the Debye temperature and Grüneisen parameters. Specifically, the second-order approximation, is more appropriate for covering a wide temperature range (e.g. Voc̆adlo et al., Reference Vočadlo, Knight, Price and Wood2002; Wood et al., Reference Wood, Knight, Price and Stuart2002; Lindsay-Scott et al., Reference Lindsay-Scott, Wood and Dobson2007; Hunt et al., Reference Hunt, Wann, Dobson, Vočadlo and Wood2017) and takes the form:
where: $Q = {\rm \;}{{V_0K_0} \over \gamma }$
and $b = {{( {K_0^{\rm^{\prime}} -1} ) } \over 2}$
, V 0 is the unit-cell volume at T = 0 K, γ is a Grüneisen parameter which is assumed to be constant and K 0 and ${\rm \; }K_0^{\rm ^{\prime}}$
are the bulk modulus and its first derivative with respect to pressure (P) at T = 0 K and P = 0 GPa. The internal energy, U(T), can be calculated using the Debye approximation to describe the energy of thermal vibrations (e.g. Cochran, Reference Cochran1973).
Where N is the number of atoms in the unit-cell (in this case N = 28); θD is the Debye temperature, and k B is Boltzmann's constant.
However, in the case of Fe2SiO4, the behaviour of the unit-cell volume and lattice parameters is more complex than that which is describable by equation 4 because of the magnetostriction resulting from the two magnetic transitions. In previous work on ferromagnetic Fe3C (Wood et al., Reference Wood, Vočadlo, Knight, Dobson, Marshall, Price and Brodholt2004) we have accurately modelled V(T) over the full temperature range by assuming that the structural, V G(T), and magnetic, V M(T), contributions to the thermal expansion may be separated, i.e., that
where V G(T) was described in a similar way to equation 4. The magnetic contribution, V M(T) = 0 above the temperature at which magnetic ordering occurs; at temperatures below the ferromagnetic phase transition, a mean-field model was used to describe the spontaneous magnetisation, M S, which was then assumed to induce a change in volume that was proportional to M S2. For ferromagnetic systems, such as Fe3C, symmetry dictates that there can be no coupling between the unit-cell dimensions and any odd power of the spontaneous magnetisation, as otherwise reversal of the magnetisation would reverse the sign of the change in the lattice parameters; thus, the lowest-order coupling is to M S2. In a simple antiferromagnet, however, this restriction does not apply as the crystal contains equal numbers of atoms with magnetic moments of opposite sign and thus linear coupling between the cell parameters and the spontaneous magnetisation is allowed, assuming that M S is here taken to be the moment of one of the sets of oppositely-magnetised atoms.
Fe2SiO4 is, though, a more complex material than either Fe3C or a simple antiferromagnet, in that there are two magnetic transitions with each of the antiferromagnetic phases having two independent sets of magnetic moments. In addition, there is evidence, from e.g. the intensity of the 052 reflection, that the behaviour close to T N for the paramagnetic to collinear AFM transition is not mean-field like. Nonetheless, we have found that a model based on a modified mean-field magnetisation curve is adequate to allow us to describe the behaviour of the unit-cell volume of Fe2SiO4 at low temperature with sufficient accuracy and thereby separate the effects of thermal vibration and of magnetic ordering on the unit-cell volume. We assume that the magnetostrictive contribution to V(T) results from an effective spontaneous magnetisation, MS, which may be described by the following equation:
where A65 and A23 are constants of proportionality, $T_{{\rm N}_{65}}$
and $T_{{\rm N}_{23}}$
the transition temperatures and β is an exponent (assumed to be the same for both phase transitions, with the factor of 2 introduced so as to ensure that a value of β = ½ would correspond to mean-field behaviour). To obtain M S(T) in a mean-field approximation (e.g. Blundell, Reference Blundell2001) it is necessary to solve the following equation:
where m is the reduced spontaneous magnetisation (i.e. M S(T)/M S(0 K)) and t = T/T N. The right side of this equation is the Brillouin function B J(y) calculated here for J = ½, as this is known to provide a good fit to the magnetisation curves of materials containing Fe and other ferromagnetic transition elements (e.g. Dekker, Reference Dekker1964). The theoretical justification for this approach to quantifying the spontaneous magnetostriction is, perhaps, somewhat scant, but equation 7 does result in a curve with the correct asymptotic behaviour as T approaches 0 K and which is also capable of describing the behaviour at temperatures close to T N. The black solid line in Fig. 4 shows the result obtained from fitting the data to equation 6 by a weighted non-linear least-squares algorithm. The resulting values of the six fitted parameters were: V 0 = 306.280(9) Å3, θD = 498(9) K, Q = 4.0(1)× 10−17 J, b = 5.90(1), A 65 = −0.183(1) Å3, A 23 = −0.009(3) Å3 and β = 0.315(3). It can be seen that the model of equation 6 provides a good description of the behaviour of the unit-cell volume over the full temperature range of the experiment and that the effect of the 23 K transition on the unit-cell volume is very small. The quality of the fit is also reflected in the volumetric thermal expansion curve, Fig. 5, calculated from
Figure 5. Volumetric thermal expansion coefficient of Fe2SiO4 as a function of temperature. Open circles were obtained by point-by-point numerical differentiation of the experimental unit-cell volume data reported in Table 2 and Fig. 3. The solid black line represents the fitted model as calculated by differentiation of equation 6. Grey and black symbols refer to experimental data from previous studies (grey, Kroll et al., Reference Kroll, Kirfel, Heinemann and Barbier2012; black, Suzuki et al., Reference Suzuki, Seya, Takei and Sumino1981).
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., Reference Wood, Knight, Price and Stuart2002), but there is little indication of this in the present case.
For T > θD, our volume thermal expansion has slightly higher values compared to those in the literature, with the value of αV(T) of Suzuki et al. (Reference Suzuki, Seya, Takei and Sumino1981) at 1000 K being 9.3% smaller than ours, while those of Kroll et al. (Reference Kroll, Kirfel, Heinemann and Barbier2012), when calculated on a point-by-point basis, are scattered. Kroll et al. (Reference Kroll, Kirfel, Heinemann and Barbier2012) measured the thermal expansion by single-crystal X-ray diffraction and Suzuki et al. (Reference Suzuki, Seya, Takei and Sumino1981) used dilatometry. We consider our data to be more accurate than those of Kroll et al. (Reference Kroll, Kirfel, Heinemann and Barbier2012) 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. (Reference Kroll, Kirfel, Heinemann and Barbier2012), 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. (Reference Suzuki, Seya, Takei and Sumino1981) 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, Reference Simmons and Balluffi1962).
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 (Reference Anderson and Suzuki1983), 510 K, or Anderson and Isaak (Reference Anderson, Isaak and Ahrens1995), 511 K, but somewhat lower than that of Suzuki et al. (Reference Suzuki, Seya, Takei and Sumino1981), 565 K. The value of V 0, 306.280(9) Å3, 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, K 0, can be obtained directly from the coefficient Q, provided that the Grüneisen parameter is known. Suzuki et al. (Reference Suzuki, Seya, Takei and Sumino1981) found γ equal to 1.097(5). If we apply this value of γ we obtain K 0 = 143(4) GPa which is a little larger than published values for K 0, obtained directly from high-pressure studies, which lie between 128 and 136 GPa (e.g. Graham et al., Reference Graham, Schwab, Sopkin and Takei1988; Béjina et al., Reference Béjina, Bystricky, Tercé, Whitaker and Chen2019, Reference Béjina, Bystricky, Tercé, Whitaker and Chen2021; Speziale et al., Reference Speziale, Duffy and Angel2004; Zhang, Reference Zhang1998; Zhang et al., Reference Zhang, Hu, Shelton, Kung and Dera2017). 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., Reference Graham, Schwab, Sopkin and Takei1988; Béjina et al., Reference Béjina, Bystricky, Tercé, Whitaker and Chen2019, Reference Béjina, Bystricky, Tercé, Whitaker and Chen2021; Speziale et al., Reference Speziale, Duffy and Angel2004; Zhang, Reference Zhang1998). 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., Reference Vočadlo, Knight, Price and Wood2002).
A modification of equation 6 can be used to model the anisotropic axial expansivities in Fe2SiO4.
For the structural term, X G(T), a modification of the Grüneisen approximation, was used (see Lindsay-Scott et al., Reference Lindsay-Scott, Wood and Dobson2007).
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.
Figure 6. (Left column) Lattice parameters of Fe2SiO4 as a function of temperature. Symbols denote the experimental data, as obtained from the Rietveld refinement, and the solid black lines show the fit of the model of equation 9 to the data. Error bars are smaller than the symbols. (Right column) Axial expansivities as a function of temperature (circles), compared to those of Suzuki et al. (Reference Suzuki, Seya, Takei and Sumino1981; red crosses). The solid black line represents the fitted model as calculated by differentiation of equation 9; the points were obtained by point-by-point numerical differentiation of the experimental data. The lower panels show the differences between the observed and calculated values.
Table 3. Fe2SiO4 fitted parameters of equation 6 and equation 9 to unit-cell axes and volume data (numbers in parenthesis are one standard error of the least significant digits).
For an orthorhombic crystal, the bulk incompressibility, K 0, and the axial incompressibilities are related by K 0 = $1/( {1/K_{a_0} + 1/K_{b_0} + 1/K_{c_0}} )$
. Assuming that a value of γ = 1.097 (Suzuki et al., Reference Suzuki, Seya, Takei and Sumino1981) 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 K 0, 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. (Reference Suzuki, Seya, Takei and Sumino1981) 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 (Reference Zhang1998) and Zhang et al. (Reference Zhang, Hu, Shelton, Kung and Dera2017). Zhang (Reference Zhang1998) reported axial incompressibilities of 741, 304 and 568 GPa for the a, b and c axes respectively; Zhang et al. (Reference Zhang, Hu, Shelton, Kung and Dera2017) 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. (Reference Zhang, Hu, Shelton, Kung and Dera2017), 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 (Reference Zhang1998) and Zhang et al. (Reference Zhang, Hu, Shelton, Kung and Dera2017) 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.
Magnetic structures and spontaneous magnetostriction
Our neutron data are consistent with previous results for fayalite (e.g. Santoro et al., Reference Santoro, Newnham and Nomura1966; Müller et al., Reference Müller, Fuess and Brown1982; Lottermoser et al., Reference Lottermoser, Müller and Fuess1986) 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., Reference Santoro, Newnham and Nomura1966) and with single crystals (Müller et al., Reference Müller, Fuess and Brown1982; Lottermoser et al., Reference Lottermoser, Müller and Fuess1986) 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. (Reference Santoro, Newnham and Nomura1966; see also Lottermoser et al., Reference Lottermoser, Müller and Fuess1986) described two possible models for the spin canting on the Fe1 site, only one of which (that originally proposed by Cox et al., Reference Cox, Frazer, Almodovar and Kay1965) 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., Reference Lottermoser, Müller and Fuess1986; Fuess et al., Reference Fuess, Ballet, Lottermoser, Ghose, Coey and Salje1988), 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 (Reference Ehrenberg and Fuess1993), 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., Reference Hafner, Stanek and Stanek1990; Lottermoser et al., Reference Lottermoser, Forcher, Amthauer and Fuess1995, Reference Lottermoser, Forcher, Amthauer, Treutmann and Hosoya1996).
Figure. 7. Model of the spin configuration at: (a) 10 K in the canted and (b) at 40 K in the collinear antiferromagnetic regions. Fe1 ions (M1 sites) are shown in gold; Fe2 (M2) ions are shown in green.
Our refinements of the AFM2 magnetic structure were based on the model proposed by Cox et al. (Reference Cox, Frazer, Almodovar and Kay1965), and later confirmed by Müller et al. (Reference Müller, Fuess and Brown1982) and Lottermoser et al. (Reference Lottermoser, Müller and Fuess1986); 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. (Reference Cococcioni, Dal Corso and de Gironcoli2003 – 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., Reference Cococcioni, Dal Corso and de Gironcoli2003), 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) μB, 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. (Reference Lottermoser, Müller and Fuess1986), who found values of 4.41(5) and 4.4(1) μB, 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 μB 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. (Reference Müller, Fuess and Brown1982) and Lottermoser et al. (Reference Lottermoser, Müller and Fuess1986).
Table 4. Fe2SiO4 Magnetic moments and direction cosines (numbers in parenthesis are one standard error of the least significant digits).
Figure 8. (a) Magnetic and (b) squared magnetic moments for the Fe1 (M1) and Fe2 (M2) sites as a function of temperature. Moments on the Fe1 site are considerably reduced as compared with Fe2 and the spin-only value of 4 μB (see Supplementary Table 1).
Our results suggest that the Fe1 and Fe2 sites make separable and complementary contributions to the evolution of the lattice parameters of the AFM1 and AFM2 phases with temperature. The transition at 65.4 K from the paramagnetic state affects all axes, producing a decrease in a and b, but an increase in c; the effect of the transition at 23 K is much less pronounced, but is clearly visible in the c axis (Fig. 9). Changes in magnetic ordering in a crystal are, in general, accompanied by a magnetostrictive deformation. A method to obtain the spontaneous volume magnetostriction, ωV, is to find the volume difference between the state in which the material is antiferromagnetically ordered, and a hypothetical state in which it is paramagnetically disordered (e.g. Kusz et al., Reference Kusz, Böhm and Talik2000).
where V afm and V par are the antiferromagnetic (observed) and hypothetical paramagnetic values of the unit-cell volume below T N, respectively. Similarly, the linear spontaneous magnetostriction can be calculated as
Figure 9. Lattice parameters of Fe2SiO4 below 120 K. Extrapolation of the paramagnetic behaviour of Fe2SiO4 below T N using the 1st-order Grüneisen-Debye approximation of the thermal expansion (equation 4, with the parameter b = 0) is shown in dotted black lines and fitted magnetostrictive components (equation 7) in solid black lines.
With X referring to the lattice parameters a, b, and c.
Clearly, the derived values of the spontaneous magnetostriction will be strongly dependent on the procedure used to extrapolate the cell parameters of the paramagnetic phase to temperatures below T N. However, a further constraint on correctness is provided by the requirement that, for small strains,
Initially, we determined the strains by fitting first-order Grüneisen approximations to the zero-pressure equation of state (e.g. Vočadlo et al., Reference Vočadlo, Knight, Price and Wood2002, 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 b 0 and θD 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 ωV and λa + λb + λc). It can be seen from Fig. 10a that the strain λa is much smaller than λb and λc, 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., Reference Sazonov, Hutanu, Meven, Heger, Hansen and Senyshyn2010); however, in Co2SiO4, the values of λa and λc 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 λa and λc show a temperature dependence that is similar to that of Fe1 moments whereas λb follows a trend more similar to that of the Fe2 sites.
Figure 10. (a) Linear and volumetric spontaneous magnetostriction of Fe2SiO4. Experimental values (shown as symbols) were obtained from equation 11 and equation 12; the lines show the values calculated from equations 4 and 7 (for the volume) and 9 and 10 (for the axes). (b) Normalised values of spontaneous magnetostriction, with symbols as for (a), with the a and c axes showing a similar temperature dependence to that of Fe1 (M1) moments while b and V follow temperature dependences more like that of the moments on the Fe2 (M2) sites.
Crystal structure
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.
Figure 11. Fe2SiO4: Average polyhedral bond distances and polyhedral volumes as a function of temperature.
Figure 12. Fe2SiO4: Angular variances (left column) and volumetric distortions (right column) from the ideal polyhedra as a function of temperature.
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, Reference Balić-Žunić and Vicković1996). 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, Reference Burns and Sung1978), that the larger thermal expansion of the c axis results from the tendency of the Fe1O6 octahedra to elongate.
Conclusions
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. (Reference Suzuki, Seya, Takei and Sumino1981) 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.
Acknowledgements
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.
Supplementary material
The supplementary material for this article can be found at https://doi.org/10.1180/mgm.2023.66.
Competing interests
The authors declare none.
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