Fourth-order quasi-harmonic equations of state for solids of cubic and tetragonal symmetry

General expressions are given for the dependence of the pressure and the effective elastic moduli on deformation and temperature in the form of a Taylor series expansion with respect to elastic and thermal strains. The temperature dependence of these expressions is derived within the quasi-harmonic approximation of lattice dynamics. The expressions are developed in terms of the Lagrangian strain and an alternative strain measure identical with the Eulerian strain for a pure deformation. They are then used to obtain the third- and fourth-order equations of state for crystals of cubic and tetragonal symmetry and to relate the parameters entering these equations to quantities which are commonly (or may be potentially) measured experimentally. It is shown that available ultrasonic data are not completely sufficient to evaluate the parameters of fourth-order equations of state. For tetragonal symmetry, this problem is still in abeyance; while in the cubic case, it is possible to estimate the fourth-order parameters from shock-wave data and so to give illustrative numerical applications of our equations. Finally, the third- and fourth-order Hugoniots and isotherms of Cu and Ag are calculated in terms of both the Lagrangian and Eulerian strain measures.