Cold shear modulus and Grüneisen parameter at all densities

We derive the relation -1/2d 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 ̈ = -1/2 d ln(B - 2/3tP)/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 ₁V^1/3 + γ₂V^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.