Crystallography and phase equilibria a review: Part III—Second-order transitions and approximations

Abstract

In this third part of the review of the relevance of crystallography to phase equilibria, the distinction between firstorder and second-order transitions is the first item to be considered. In a first-order transition G/ is discontinuous; however, in a second-order transition G/ is continuous, but G/d 2 is discontinuous. Here G is the Gibbs energy and is a state variable such as temperature, pressure, or composition (T, P, or X). Plots of G versus T are shown in Fig. III-1 to illustrate the difference between firstand second-order transitions. In Fig. III-1(a), the curve for the -symmetry structure crosses the curve for the -symmetry structure at a point where the Gibbs energies are the same, but the slopes of the two curves are different, dG /dT dG /dT. The juncture is the transition temperature, Tt. It is obvious that has the lower Gibbs energy at T Tt has the lower Gibbs energy and is the stable structure. At Tt the two phases can exist in a two-phase equilibrium. Since dG/dT −S, it is also obvious that iS is not zero. Thus, the combination of iG 0 and iS 0 characterizes a first-order transition with two distinct phases in equilibrium at Tt. For comparison Fig. III-1(b) illustrates the G versus T behavior of a second-order reaction. In this figure, the and -symmetry structures have Gibbs energies that differ at low temperatures but merge at a point where both G G and dG /dT dG /dT and at temperatures above the merger point, the symmetry difference disappears and the Gibbs energies are indistinguishable. The transition temperature is the point of merger. Figure III-1(b) characterizes a second-order transition, and there is no phase change but rather a transition between two symmetries of the same single phase. The second-order transition contrasts with the first-order transition in that both tG and tS equal zero in the second-order transition. The symmetries of the two structures in a second-order transition are related in a manner that allows the possibility of the structures to continuously change from one to the other. The specific conditions that must apply before a secondorder transition can occur were first delineated by Landau and Lifshitz on the mathematical basis of group theory. Their result in the language of group theory necessitates that all of four conditions apply: • Space groups of the structures in a second-order transition must be in a group-subgroup relationship. • The difference between the particle density functions of the two structures must be a basis function, or combination of basis functions, of the irreducible representation of the higher-symmetry space group. • It must not be possible to form a totally symmetric third-order combination of such basis functions. • The low-symmetry structure must be locked in by symmetry to the high-symmetry space group (even if incommensurate).