Degree of order of the investigated samples
The Fe content at the octahedral site obtained from the structural refinement of the mfr sample yielded an inversion parameter x = 0.892(2) (Table 2). The inversion parameter for magnesioferrite can also be calculated using the previously determined linear relationship between the inversion parameter and unit-cell lattice parameter a0, using the equation x = 81.34–9.598 a0 (Å) (O’Neill et al. 1992). From this relationship, we obtain an inversion parameter x = 0.889 in excellent agreement with that obtained from the structural refinements. Therefore, the mfr sample synthesised in this study has a high degree of order. Moreover, the degree of order of the mfr sample is higher after annealing at 5 GPa and 1300°C than that of the magnesioferrite starting material (Uenver-Thiele et al. 2017a), which had x = 0.837 and was synthetized at 900°C at room pressure. This confirms that pressure favours cation ordering in these spinels, in agreement with the results reported by Turkin and Drebushchak (2005) and Antao et al. (2005a). In fact, the determined inversion parameter is in good agreement with parameters determined at a similar pressure and temperature by Antao et al. (2005a) for magnesioferrite (i.e. 0.906(8)).
For samples with mixed compositions, like Fe50, the degree of order cannot be simply determined from X-ray diffraction, as it is not possible to distinguish between Fe2+ and Fe3+ in structural refinements. However, the collection of the SMS spectra showed the presence of all ferric iron in the tetrahedral site allowing the inversion parameter of sample Fe50 to be obtained directly from the Mg and Fe site occupancies at this site. As a result, the inversion parameter of sample Fe50 is x = 0.98(3), showing a highly ordered, and almost completely inverse spinel, consistent with the high-pressure and moderate temperature (9 GPa, 1000°C) synthesis conditions.
Crystal Chemistry Of The Mgfeo – Feo Solid Solution
The room pressure crystal chemistry and state of ordering of the samples in this study can be investigated through a comparison with literature data on magnetite-magnesioferrite samples and, in particular, with those used in previous compressibility studies (Table 2). Unit-cell parameters for magnesioferrite from compressibility studies are quite varied and range from a = 8.3841 to 8.3970 Å, with one outliner sample having a = 8.3600 Å (Nakatsuka et al. 2004). These samples (Table 2) have been synthesised at temperatures between 500 to 1200 °C with some being quenched in a step-wise manner, which has thus affected the final degree of order that is probably the main cause of the unit-cell variation. The predictions by O’Neill et al. (1992) for the expected unit-cell parameters of fully ordered and disordered magnesioferrite are indicated in Fig. 2A. The sample of Nakatsuka et al. (2004) (Table 2) has a very small unit-cell parameter not consistent with the other reported values and below the predicted fully ordered end-member. This may be caused by a degree of non-stoichiometry through the substitution of a maghemite γ-Fe2O3 component, which is expected to decrease the unit-cell parameter further (O’Neill et al.1992). The mfr sample investigated in this study is at the lower end of the main cluster of values from the literature (a = 8.3821 (2) Å), and is, therefore, one of the most ordered samples examined to date.
Reported magnetite end-member unit-cell parameters show a much smaller range in values, between 8.3941 Å and 8.3967 Å, consistent with all the samples being ordered inverse spinel. As opposed to mfr, magnetite cation disorder cannot, to our knowledge, be quenched as it results only from the movement of an electron. Only the study of Finger et al. (1986) reports a significantly smaller value (a = 8.3778 (5) Å), which is interpreted to be due to the presence of a γ-Fe2O3 maghemite component (Volenik et al. 1975), consistent with the high temperature synthesis from a liquid.
In order to investigate the behavior of the magnetite-magnesioferrite solid solution, the unit- cell parameters (Table 2) are plotted as a function of the molar Mg/(Mg + Fe2+) fraction, XMg, in Fig. 2A. A line has been drawn between the value obtained for mfr in this study and an average value (a = 8.3962 Å) for magnetite. The Fe50 sample investigate in this study lies on the linear trend between these most ordered samples. It should be noted, however, that it may be possible to produce a more ordered magnesioferrite sample, particularly at high pressures (Antao et al. 2005a), which would render a non-linear unit-cell relationship with Mg content, so there are no inferences from this line in terms of Vegards law.
The Mg/Fe2+ substitution in the close packed structure of spinels influences the tetrahedral, T-O, and octahedral, M-O, bond lengths (Fig. 2B and 2C). In the ordered magnetite-magnesioferrite solid solution Mg substitutes for Fe2+ at the octahedral site, whereas only Fe3+ occupies the tetrahedral site. With increasing Mg content, the M-O bond length decreases due to the smaller radius of Mg with respect to Fe2+. This decrease causes a small increase of the tetrahedral bond distance, due to the close interconnectivity of the spinel structure. The magnetite samples reported in Table 2 have similar bond distance values, as expected due to their high degree of order. A line through the average of these values and the bond distances of the mfr sample investigated in this study can be used to describe the linear behaviour of the most ordered magnetite-magnesioferrite solid solution. The M-O bond distance of the ordered Fe50 sample lies just slightly below this trend and the T-O bond length slightly above (Fig. 2C and 2B respectively). This very slight shortening of the M-O distance compared to the linear trend may be related to the raised incompressibility of this intermediate sample, as discussed later.
Cation disorder should decrease the M-O bond distances and increase the T-O bond distances even further, as Fe2+/Mg enter the tetrahedral site and are replaced by Fe3+ at the octahedral site. This is clearly the case for the single-crystal Mg0.956Fe2.044O4 sample studied by Andreozzi et al. (2001), which is slightly more disordered (x = 0.87) than our mfr sample. Data for the other reported end-member magnesioferrite samples (Table 2), however, do not appear to follow this behaviour (Fig. 2B and 2C) and show in some cases an increase in M-O distance even though they report higher levels of disorder. These studies, however, have been performed on polycrystalline samples and it is likely that the large correlations between refined parameters during the Rietveld refinements resulted in a poorly constrained oxygen position due to its low scattering factor. This is well illustrated in the study by Antao et al. (2005b) where different values for the M-O and T-O bond distances are reported for the same sample analysed with two different X-ray sources (Table 2), in spite of the fact that the same unit-cell lattice parameter is obtained from the two techniques.
Compressibility Of The Mgfeo – Feo Solid Solution
The decreasing trends of the unit-cell volumes with pressure for the mfr and Fe50 samples are shown in Fig. 3. No evidence of a phase transition was observed in the pressure range investigated. The end-member magnesioferrite is clearly more compressible than the Fe50 sample. The normalized stress, FE, versus Eulerian finite strain, fE, plot (Angel 2000) is illustrated in Fig. 4. Both data sets are well represented within uncertainties by horizontal straight lines indicating that a second order truncation of the Birch-Murnaghan (BM) equation of state (EoS) (Birch 1947) is sufficient to describe the experimental P-V data. In this case only two EoS parameters are refined, the room pressure unit-cell volume, V0, and the bulk modulus, KT0, whereas the first pressure derivative of the bulk modulus, K´, assumes the value of four. The quality of the P-V data, however, appears adequate to constrain the value of K´, therefore, a third-order truncation of the BM EoS was also used, with three refined EoS parameters, V0, KT0, and K´. The results from fitting a BM2 and a BM3 EoS to the P-V data of both samples are reported in Table 1. The K´ values of the BM3 EoS are identical to 4 within their uncertainties. KT0 is found to decrease significantly by approximately 10 GPa between the Fe50 (188.0 ± 0.6 GPa BM2) and mfr (178.4 ± 0.5 GPa BM2) samples (Fig. 6).
Data reported in the literature (Table 1) for the end-members magnetite and magnesioferrite vary considerably between the different studies (Fig. 5). For the magnesioferrite studies, KT0 values range between 170.5 and 233 GPa and K´ between 3.3 and 6.32 (Fig. 5). The reasons for this large difference among the reported data sets are probably multiple but likely include non-hydrostatic conditions and insufficient data coverage. A comparison between the inversion parameter x for the samples used in each study and the obtained EoS terms shows no obvious correlation. The very large K´ in the study of Levy et al. (2004) probably results from non-hydrostatic conditions arising from the use of an N2 pressure medium, which has been shown to become non-hydrostatic above approximately 6 GPa (Angel et al. 2007). The results of Greenberg et al. (2009) are very similar to those of this study up to approximately 20 GPa, however, their sample appears to become softer at higher pressures, leading to a low determined value of K´ of 3.3. This can be seen in the FE- fE plot reported by Greenberg et al. (2009) which shows a kink above 20 GPa. Although somewhat speculative one possible expansion for this would be the approach or commencement of a phase transition to a post-spinel phase, such as that observed by Andrault and Bolfan-Cassanova (2001). As magnesioferrite is anyway expected to break down at ~ 10 GPa and high temperatures (Uenver-Thiele et al. 2017b) the measurements in the current study should cover a sufficient range (up to ∼19 GPa) to obtain suitable elastic properties for thermodynamic calculations of its stability field.
Amongst various studies on magnetite compressibility, the study of Gatta et al. (2007) used a very similar methodology to that employed here and the results are in good agreement with several other studies on magnetite (Nakagiri et al., 1986; Rozenberg et al., 2007; Reichmann and Jacobsen 2004). As shown in Fig. 6, the KT0 value obtained for the mfr sample (178.4 (5) GPa) is only slightly lower than the value of 180 (1) GPa for magnetite (Gatta et al. 2007) while the values of K´ obtained by using a BM3 EoS are identical within the uncertainties. That the values of KT0 for the two endmembers are similar, with just a very small increase in KT0 between the Mg and Fe2+ endmembers, is consistent with studies on the normal spinels MgAl2O4 (193 ± 1 GPa) and FeAl2O4 (193.9 ± 1.7 GPa; Nestola et al. 2007, 2015) and MgCr2O4 (182.5 ± 1.4 GPa) and FeCr2O4 (184.8 ± 1.7 GPa; Nestola et al. 2014). The sample with mixed composition (Fe50) is stiffer than both end-members (Fig. 6), indicating that a simple linear relationship of bulk moduli along the most ordered magnesioferrite-magnetite solid solution cannot describe the intermediate compositions. This behaviour is unusual, as other normal spinels such as Fe2SiO4-Mg2SiO4 (Higo et al. 2006) and MgAl2O4-FeAl2O4 (Bruschini et al. 2018) show near monotonous changes in KT0 across significant sections of the solid solutions. The difference between the KT0 of the Fe50 sample and its two endmembers is also greater than the differences between any of the studied Fe-Mg spinel endmembers.
One consideration is that although the magnetite and Fe50 samples are fully ordered inverse spinels, the mfr sample retains some level of cation disorder (x = 0.89). If the fully ordered mfr were stiffer, this might render a linear XMg-KT0 relationship among the samples. Studies at least on the normal spinel MgAl2O4, appear to indicate that there is no resolvable effect of varying cation ordering on KT0 (Nestola et al., 2007; Bruschini et al., 2018). This implies that the potential softening effect of putting Mg into the octahedral site is balanced by the hardening influence of Al entering the tetrahedral site. However, the magnesioferrite inverse spinel may behave completely different. The increase in volume at room temperature on disordering, for example, is larger for magnesioferrite compared to MgAl2O4 spinel (O’Neill et al., 1992; Nestola et al., 2007). Similarly, the softening effect of Mg entering the tetrahedral site may not be balanced by Fe3+ entering the octahedral site, as half of this site is already filled by Fe3+. Density functional theory calculations on MgAl2O4 spinel (Núñez-Valdez et al. 2018), do predict a higher bulk modulus for a theoretically fully ordered inverse spinel, compared to normal ordered spinel. This effect is not predicted to be much larger, however, and is probably not sufficient to raise the mfr KT0 significantly to render a linear relationship between Mg substitution and KT0.
The higher KT0 for the Fe50 sample implies that some type of Fe2+-Mg interaction is causing the octahedral site to be stiffer than when it is dominated by either Mg-Mg or Fe2+-Fe2+ neighbours. One way that this might occur is through an interruption of the electron hopping between Fe2+ and Fe3+, which causes the averaged Fe2.5+ valence observed in Mössbauer spectra of magnetite. If the presence of local Mg reduces the extent of hopping then the more localised Fe2+ 3d electron may increase the degree of covalency, and therefore the strength, of the Fe2+-O octahedral bond. This could decrease the compressibility. Once the XMg=0.5 composition has been reached a further increase in Mg may then dilute this effect. If this is the case, then the non-linear elastic behaviour with composition is likely to be a peculiarity of the magnetite-magnesioferrite solid solution.
A final aspect is to consider whether the change in incompressibility across the solid solution would result in a significant excess molar volume at high-pressure, which could potentially contribute to the degree of non-ideality of the solid solution. At 10 GPa, however, i.e. near the limit of high-pressure magnetite-magnesioferrite stability, the predicted excess molar volume is at most 0.07 cm3/mol, which would not have a significant influence on the thermodynamics of mixing.