Cold shear modulus and Grüneisen parameter at all densities

We derive the relation −1/2 d ln G/d ln V−1/6=γ for the volume dependence of the cold (T=0) shear modulus, G, where γ is the cold Grüneisen parameter given by the formula γ = − 1 / 2 d ln ( B − 2 / 3 t P ) / d ln V − 1 / 6 , B and P being the cold bulk modulus and pressure, respectively. For constant t, this formula reduces to the known Slater, Dugdale–MacDonald, and Vashchenko–Zubarev relations for t=0, 1, and 2, respectively. However, as we demonstrate, in the case of a real solid under pressure, t is a variable such that t→5/2 as P→∞. This formula is the basis for the analytic model of the cold Grüneisen parameter, γ ( V ) = 1 / 2 + γ 1 V 1 / 3 + γ 2 V q , q>1, developed previously by two of the authors, and the corresponding analytic model of the cold shear modulus. The model of the shear modulus is compared to electronic-structure calculations and experimental data on rare-gas solids, iron, and cobalt, and good agreement is found in all cases.