Equilibrium free energies for the α and ω phases of Ti are constructed. The result is a consistent picture of the ambient pressure, static high pressure, and shock data, as well as first-principles electronic structure calculations. The Hugoniot consists of three segments: a metastable α-phase region, a transition region, and an ω-phase branch. All the Hugoniot data are consistent with a transition occurring at ∼12 GPa. An early identification [R. G. McQueen et al., in High Velocity Impact Phenomena, edited by R. Kinslow (Academic, New York, 1970)] of a phase transition at 17.5 GPa appears to have been an artifact. The shock Hugoniot extends further into the metastable region than static data, indicating the existence of a relaxation process occurring on a time scale intermediate between those of the static and dynamic measurements.

An analytic model of the shear modulus applicable at temperatures up to melt and at all densities is presented. It is based in part on a relation between the melting temperature and the shear modulus at melt. Experimental data on argon are shown to agree with this relation to within 1%. The model of the shear modulus involves seven parameters, all of which can be determined from zero-pressure experimental data. We obtain the values of these parameters for 11 elemental solids. Both the experimental data on the room-temperature shear modulus of argon to compressions of $\sim 2.5,$ and theoretical calculations of the zero-temperature shear modulus of aluminum to compressions of $\sim 3.5$ are in good agreement with the model. Electronic-structure calculations of the shear moduli of copper and gold to compressions of 2, performed by us, agree with the model to within uncertainties.

The Urbach tail in crystalline GaAs and Si due to thermal vibration is evaluated in the temperature range 0–810 K. Recent experiments suggest that there is no thermally induced Urbach tail in undoped GaAs. This is most unexpected since thermally induced tails at the absorption edge having the Urbach form and widths ${\mathit{E}}_0$(T) linear in T are observed in a wide range of crystalline and amorphous semiconductors. Clear understanding of the tail is important in device applications. In our calculations, we obtain an absorption tail having a simple exponential energy dependence (Urbach form) in GaAs at all temperatures with ${\mathit{E}}_0$(T) proportional to T. However, due to a very weak electron-phonon interaction, the magnitude of ${\mathit{E}}_0$(T) is unusually small, much smaller than the temperature independent, structurally induced component ${\mathit{E}}_0$≃10 meV. As a test of the method, the tail is similarly evaluated in Si providing an ${\mathit{E}}_0$(T) value in agreement with experiment.

Elastic constants and zone-boundary phonon frequencies of gold are calculated by total energy electronic structure methods to twofold compression. A generalized force constant model is used to interpolate throughout the Brillouin zone and evaluate moments of the phonon distribution. The moments are used to calculate the volume dependence of the Grüneisen parameter in the fcc solid. Using these results with ultrasonic and shock data, we formulate the complete free energy for solid Au. This free energy is given as a set of closed-form expressions, which are valid to compressions of at least ${V/V}_0=0.65\mathrm{}$ and temperatures up to melting. Beyond this density, the Hugoniot enters the solid-liquid mixed phase region. Effects of shock melting on the Hugoniot are discussed within an approximate model. We compare with proposed standards for the equation of state to pressures of $\sim 200\mathrm{GPa}.$ Our result for the room-temperature isotherm is in very good agreement with an earlier standard of Heinz and Jeanloz.