Twofold Incommensurate Phase as Analogs of Semicommensurate Phases. On the Transitions [Prototypic→Ordinarily (or Onefold) Incommensurate →Twofold Incommensurate]

Abstract

The order parameter Q(x) for incommensurate transitions can, as is well known, be expressed as a sum of harmonics Q_n e^inhx, where n takes on integers. Let α, the degree of anisotropy of the free energy function, have t divisors α₁(=α)>α₂>…>α_t−1>α_t(=1). Then, t different types of incommensurate phases can be conceived, the jth of which has Q_ns≠0 for and only for n=1+α_jν (ν=0, ±1, ±2, …). The j=1 type is ordinarily incommensurate. The j=t type is essentially the same as recently termed semicommensurate. The 2≤j≤t−1 types are new proposals; they can be recognized as twofold incommensurate (while the j=1 type as onefold incommensurate). Transitions involving twofold incommensurate phases are possible. A detailed account of this possibility is given.