Thermodynamics of Phase Transitions in Compressible Solid Lattices

The calculations on the Ising lattice and the order-disorder phenomenon indicate that the specific heat may become infinite at the λ temperature or transition point. The general thermodynamics of such a system in a variable magnetic field is considered, and the relation to the generalized theories of critical points of Tisza and of Semenchenko is indicated. The general import of these theories is that infinite values of the heat capacity at constant pressure Cp, or the analogous heat capacity in a system with variables other than p, V, and T, can become infinite in a one-phase system only at a critical point. The existence of a locus of points where Cp = ∞ is thermodynamically possible in the pVT system, and equations analogous to the Clapeyron and Ehrenfest equations may be derived. However, the thermodynamic requirements show that such a locus is not likely to occur. Interesting possibilities occur where there is an interaction between several sets of variables. If an Ising lattice is compressible, for example, the transition temperature will depend on the lattice distance. In this case, the infinite heat capacity is to be considered as a singularity in Cv rather than in Cp. The effect of the compressibility is to change any transition in which infinite specific heats occur to a first-order transition. If there is already a first-order transition, even for an assumed incompressible lattice, the introduction of compressibility increases the latent heat and a discontinuity in lattice distance appears. This situation appears to arise in the order-disorder transformation of an alloy like AuCu3, and the phases may also be expected to have different compositions; this latter phenomenon has also been considered from the point of view of this paper. If there is a λ point without infinite specific heats, the introduction of compressibility may or may not change it to a first-order transition; in any event, the λ point is made ``sharper.'' There may be a critical point in this case. Various thermodynamic relationships are deduced, and the nature of the isotherms in all these cases is considered.