Incommensurate phases in statistical theory of the crystalline state

TL;DR

This paper develops a mathematical framework within statistical mechanics to analyze second-order phase transitions, including incommensurate phases, by extending symmetry analysis and representing incommensurate phases as limits of long-period commensurate phases.

Contribution

It introduces a novel approach for studying incommensurate phases using a limit of long-period commensurate phases, overcoming symmetry description challenges.

Findings

Symmetry analysis can be applied similarly to Landau theory.

A method to represent incommensurate phases as limits of commensurate phases.

The approach avoids complexities related to the devil's staircase phenomenon.

Abstract

The paper continues a series of papers devoted to treatment of the crystalline state on the basis of the approach in equilibrium statistical mechanics proposed earlier by the author. This paper is concerned with elaboration of a mathematical apparatus in the approach for studying second-order phase transitions, both commensurate and incommensurate, and properties of emerging phases. It is shown that the preliminary symmetry analysis for a concrete crystal can be performed analogously with the one in the Landau theory of phase transitions. After the analysis one is able to deduce a set of 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. The strategy enables one to obviate difficulties due to the devil's staircase that occurs in this situation.