A semi-empirical tight-binding theory of the electronic structure of semiconductors

Advances, during the Upper Mantle Project, in investigations on phase equilibria and elastic properties of the mantle minerals in the MgO-FeO-SiO2 system are reviewed. The experimental procedure for a phase equilibrium study is described. Some advantages and disadvantages of various high-pressure, high-temperature apparatus are discussed. Necessities of establishing a pressure calibration method at high temperatures are also pointed out. A brief description on a method, recently developed for the measurements of ultrasonic wave velocities of very small samples, is given.Olivine-spinel solid solution equilibria in the system Mg2SiO4-Fe2SiO4 have been studied at several laboratories over the pressure range 40–200 kbar at 800 and 1,000°C. Ringwood and Major first discovered a peculiar mode of the high-pressure transformation in compositions close to pure Mg2SiO4, resulting from the formation of β phase. Crystal structure of the β phase was clarified as modified spinel structure through a series of studies on the high-pressure transformation of Co2SiO4 and Mn2GeO4. The appearance of the modified spinel phase in the Mg2SiO4-Fe2SiO4 system was confirmed in further investigations at the author's laboratory. The isothermal section of the phase diagram for the Mg2SiO4-Fe2SiO4 system was constructed at 800 and 1,000°C on the basis of these recent experimental results. At 800°C a continuous series of spinel solid solutions was synthesizable from Fe2SiO4 to (Mg0.9Fe0.1)2SiO4. At 1,000°C, however, all the attempts to synthesize a true spinel phase of (Mg0.9Fe0.1)2SiO4 were unsuccessful up to about 140 kbar and coexistence of the modified spinel phase with true spinel was usually identified. This suggests that the stability field of the β(Mg, Fe)2 SiO4 is highly temperature dependent. A remarkable expansion of the β(Mg, Fe)2SiO4 region is expected at the higher temperatures. Based on the cell parameters of βMg2SiO4 and the extrapolated value for γMg2SiO4 (true spinel), the density increase associated with forsterite- βMg2 SiO4 transformation, and forsterite — γMg2 SiO4 transformation was calculated to be 7.9% and 10.8% respectively.Experimental data on the phase equilibria of the MgSiO3-FeSiO3 system are presented. A highpressure disproportionation of clino-pyroxene solid solutions into stishovite plus spinel solid solutions was found. It was established that the disproportionation curve for clino-ferrosilite (FeSiO3) was well represented by the boundary curve for the coesite-stishovite transformation. A preliminary phase diagram for the MgSiO3-FeSiO3 system was constructed at 800 and 1,000°C from the data by Ringwood and Major and by the author's laboratory.Experimental data on the high-pressure phase transformations of SiO2 are summarized. The boundary curve for the coesite-stishovite transformation was determined over the temperature range 550–1,200°C in the pressure range 83–101 kbar by means of a tetrahedral anvil press. The transition curve was fitted by the linear relation P(kbar) = 67 + 0.028 T(°C). This determination was found to be in reasonable agreement with the previous data.A stoichiometric compound, Fe1,000O, was synthesized at high-pressures above 40 kbar at 775°C by a reaction between wüstite, Fe0.950O, and metallic iron. The cell dimension of Fe1,000O was determined to be 4.323 ± 0.001 Å.Compressional- and shear-wave velocities of the synthetic (Mg, Fe)2 SiO4 olivine, Fe2SiO4 spinel, (Mg, Fe)SiO3 orthopyroxene, eoesite, stishovite and Fe0.98O were measured by means of the ultra-sonic pulse transmission method. Results are represented on Birch's diagram, where the wave velocities are plotted as a function of density. It was found that the wave velocities of these ferromagnesian silicates and oxide decrease linearly with the increase of the FeO/(FeO+MgO) ratio, and that the isomorphic lines of (Mg, Fe)2 SiO4 olivine, (Mg, Fe)SiO3 orthopyroxene and (Mg, Fe)O magnesiowüstite are approximately parallel to each other. The Compressional- and shear-wave velocities of the true spinel phase of Mg2 SiO4 were estimated to be 10.0 km/sec and 5.7 km/sec respectively.Compressional- and shear-wave velocities of stishovite were determined to be 11.0 km/sec and 5.55 km/sec. The bulk modulus of stishovite was calculated from these values to be 3.43 Mbar. Compressional-wave velocity of the three polymorphs of silica, αquartz, coesite and stishovite, was found to increase regularly along Birch's mean atomic weight line of M = 21.