Thermodynamics of Crystals and Melting

The Helmholtz free energy, A, of a rigid body is a function of temperature, and of the six homogeneous strain components. If the crystal is to be rigid, three inequalities must be satisfied for the derivatives of A with respect to the six strain components, for a regular (cubic) lattice. This enables one to limit the pressure-temperature range for which the crystal is stable. The violation of the condition c440, that the crystal resist shearing, is interpreted as leading to melting. From a knowledge of the forces between the molecules the phase integral, and therefore the free energy, may be calculated as a function of T, V, and the six strain components. The numerical calculations are carried out for a body-centered cubic lattice. The product of all the frequencies is calculated directly, so that the assumption that the Debye equation for the frequency distribution holds, is not necessary. The melting curve, pressure against temperature, is then determined.