One way around this problem is to

One way around this droperidol problem is to build an empirical model which relates the bulk moduli to some other physical properties of some materials with the same crystal structure but different compositions (Anderson and Anderson, 1970; Duffy and Anderson, 1989). By extending the ranges of all the relevant properties, those uncertainties in the employed experimental techniques and variations of the physical and chemical states of the investigated samples become unimportant, and the correlation of the physical properties can be accurately established. Rw is crystallographically a cubic 4–2 oxide spinel ( with Z = 8) (Yagi et al., 1974; Sasaki et al., 1982), and shares the same crystal structure with a large number of phases such as Ni2SiO4-Sp (spinel), Co2SiO4-Sp, Mg2GeO4-Sp, Fe2GeO4-Sp, Co2GeO4-Sp, Mg2TiO4-Sp, Fe2TiO4-Sp, Co2TiO4-Sp and Zn2TiO4-Sp. Because of the similarity in the crystal structures, these spinels have been widely used as analogues to explore the high-P physical properties of Mg2SiO4-Rw (Mao et al., 1970; Liu et al., 1974; Liebermann, 1975; Liebermann et al., 1977; Sato, 1977; Finger et al., 1979; Weidner and Hamaya, 1983; Bass et al., 1984; Rigden et al., 1988; Rigden and Jackson, 1991). This study aims at evaluating the isothermal bulk moduli of the Mg2SiO4-Rw and Fe2SiO4-Rw, and those cubic 4–2 oxide spinels obtained by different experimental techniques in recent years, and building an empirical model to describe the relationship between the isothermal bulk moduli and other physical properties such as the unit-cell volume and the electronegativity of the cations. It has been demonstrated that such an empirical model can reproduce well all the data used in its calibration. In addition, this model has been examined with some extra experimental data which have not been used in its construction, and satisfactory agreement has been achieved as well. To apply this model to the Rw solid solutions in the system Mg2SiO4–Fe2SiO4, a well-established volume–composition relationship is a prerequisite. We have summarized all the Rw volume-composition data reported so far, and found a nearly linear solid solution behavior for the Rw solid solutions. Eventually a much smaller composition effect on the bulk modulus of the Rw has been revealed. Furthermore, the compositional data of the Rw reported in some high-P experimental studies with peridotitic mantle compositions have been carefully scrutinized. Contrary to the prevalent assumption of a constant composition, we have found that the XFe of the Rw at the P-T conditions of the lower part of the MTZ decreases significantly with P increase. Taking all these observations into account, finally, the sound velocity features of the Rw at the P-T conditions of the lower part of the MTZ have been explored, and their geophysical implications have been discussed in this paper.
A model for the bulk modulus: building techniques The general chemical formula for the Rw and other 4–2 oxide spinels is AB2O4, with A being a 4+ and B a 2+ cation. The first-order crystal structural feature of these compounds is a cubic close packing array made of the oxygens, which is slightly modified by the A and B cations occupying one eighth of the tetrahedral (8a) and half of the octahedral (16d) sites, respectively. The A and B cations might partially switch their positions due to changes of temperature, pressure and composition (e.g., Wechsler et al., 1984; Liu and Prewitt, 1990; Wittlinger et al., 1998; O\'Neill et al., 2003; Antao et al., 2005; Rozenberg et al., 2007; Gatta et al., 2014), and these materials become crystallographically disordered, so that a more general chemical formula can be written as [AB]tet[AB]octO4. If i = 0, the spinel has an ordered normal cation distribution ([A]tet[B2]octO4, normal spinel); if i = 1, the spinel has an ordered inverse cation distribution ([B]tet[AB]octO4, inverse spinel); if i = 2/3, the spinel has a completely random cation distribution between the tetrahedral and octahedral sites.