br A model for the bulk modulus evaluating input data

A model for the bulk modulus: evaluating input data The isothermal bulk moduli of 12 4–2 oxide spinels at ambient P-T condition have been experimentally measured, and the corresponding experimental details are summarized in Table 1. Since our major interest here is to probe the composition dependence of the bulk modulus of the ringwoodites by extending the variation ranges of the bulk modulus and other physical properties, any data about the Rw solid solutions along the Mg2SiO4–Fe2SiO4 join would not help and are not used in the model-building process.
A model for the bulk modulus: building and examining As revealed in the previous section, 11 sets of bulk modulus-volume data are available for the calibration of our empirical model (Table 2). The range of the KT0 values, from 152 to 223 GPa, is about 4.4 times that of the KT0 values of Mg2SiO4-Sp and Fe2SiO4-Sp whereas the range of the unit-cell volume values, from 520.6 to 604.68 Å3, is about 2.6 times that of the volume values of Mg2SiO4-Sp and Fe2SiO4-Sp. A weighted multiple linear least-squares fit to the data has been conducted. The data without uncertainties are weighted by assuming (1) a 0.05% uncertainty in the unit-cell volume, (2) a 0.1 uncertainty in the EN-total, and (3) a 6 GPa uncertainty in the bulk modulus of the germanate spinels (Table 2). These germanate spinels were investigated by using the ultrasonic method in Liebermann (1975) up to 0.75 GPa and in Rigden and Jackson (1991) up to 3 GPa, and the average difference of the obtained bulk moduli by these two studies was ∼6 GPa. The derived equation is as follows:with KT0 in GPa and V0 in Å3. The reduced chi-squared value for the regression is 1.6, indicating a good fit, as shown in Fig. 1. When the unit-cell volume of the 4–2 oxide spinels increases, the bulk modulus decreases, which is in general agreement with Anderson and Anderson (1970). As the electronegativity total of the cations increases, the bulk modulus increases, presumably suggesting that the bonds between the cations and oxygens become more covalent. Fig. 2 shows the differences between the bulk moduli calculated by our model and those determined in the previous section, varying from −10 to 11 GPa and with an average of 0(7) GPa. Especially, the bulk moduli of the magnesium 4–2 spinels have been perfectly reproduced by our empirical model (difference varying from −2 to 1 GPa only; Table 2), which is in sharp purchase cisapride to the poor performance of the theoretical methods (simulated bulk modulus for the Mg2SiO4-Rw varying from 173 to 190 GPa; Table 1). On the other hand, the difference for other spinels is relatively larger, presumably reflecting that the cations (Fe, Ni, Co and Zn) in these spinels had multiple charge states, and no all cations were 2+. It follows, anyhow, that our empirical model can well reproduce all experimental data used in its calibration, with an accuracy of about 5% (Table 2). An independent examination of our model with the data of the Rw solid solutions along the Mg2SiO4–Fe2SiO4 join has been carried out (Table 2). The result shows that our model can reproduce almost all experimental data well, actually excellent for the XFe ≦ 0.5 samples. Distinctly, the bulk moduli for all iron-rich samples from Hazen (1993) can not be well reproduced, which is probably an issue related to the charge status of the Fe cation, approximately 5% iron occurring as Fe3+ in these samples. The effect of Fe3+ on the bulk modulus of spinels has not been well constrained though (Yamanaka et al., 2013; Xiong et al., 2015). Since the ringwoodites in the MTZ have XFe much lower than 0.5, it can be concluded that our empirical model is geologically meaningful. Our empirical model predicts the bulk modulus of Fe2TiO4, Mg2SnO4, Co2SnO4 and Mn2SnO4 as 162, 167, 187 and 166 GPa, respectively. As reviewed in the previous section, the bulk modulus of Fe2TiO4 was experimentally determined as 121 GPa by some ultrasonic measurements (Syono et al., 1971; Liebermann et al., 1977) and as 250.8(25) GPa by some high-P X-ray diffraction data (Yamanaka et al., 2009, 2013), both of which were very different to the predicted value. Recently, we remeasured the bulk modulus of Fe2TiO4-Sp with a diamond-anvil cell coupled with synchrotron X-ray radiation and obtained 148(4) GPa (KT0′ fixed as 4; Xiong et al. (2015)), which closely approached the predicted value of 162 GPa. Considering the iron-rich nature of the Fe2TiO4 spinel, this application of our empirical model essentially confirms its earlier application to the ringwoodites along the Mg2SiO4–Fe2SiO4 join: it has relatively low but still acceptable accuracy at Fe-rich conditions. For the stannates, the experimentally obtained bulk modulus was 125(2), 126(14) and 128(4), respectively (Liebermann et al., 1977), with the ultrasonic measurements conducted up to 0.75 GPa only, which probably resulted in an underestimate of the residual porosity in the samples. In order to resolve the discrepancy, anyhow, more experimental investigations on the stannates are necessary.