Incommensurate phases in the statistical theory of the crystalline state

The paper is devoted to the elaboration of a mathematical apparatus for studying second-order phase transitions, both commensurate and incommensurate, and the properties of emerging phases on the basis of the approach in equilibrium statistical mechanics proposed earlier by the author. It is shown that the preliminary symmetry analysis for a concrete crystal can be performed analogously with the one in the Landau phenomenological theory of phase transitions. The analysis enables one to deduce a set of transcendental equations that describe the emerging phases and corresponding phase transitions. The treatment of an incommensurate phase is substantially complicated because the symmetry of the phase cannot be described in terms of customary space groups. For this reason, a strategy of representing the incommensurate phase as the limit of a sequence of long-period commensurate phases whose period tends to infinity is worked out.