Phenomenological model for type-II incommensurate phases having a soliton regime: Thiourea case

It is shown how a rather strong soliton regime as that detected experimentally in the incommensurate phase of thiourea can be described by a thermodynamic potential expanded in terms of a one-dimensional order parameter. The modulation of the order parameter admits a parametrization similar to that commonly used for incommensurate phases with Lifshitz invariants (usually termed as type I). It is demonstrated that, if this parametrization is used, the sine-Gordon equation, which is usually considered only adequate for type-I incommensurate phases, is also approximately valid for those of type II, such as thiourea. In this way, the evolution of the order-parameter modulation towards a two-stepped function can be easily modeled. By extending the thermodynamic potential to include acoustic degrees of freedom, the coexistence (experimentally detected in thiourea) of solitonic and sinusoidal structural modulations is also explained. According to the present work, the differences between some of the so-called incommensurate phases of types I and II can be essentially reduced to the fact that, while in the former ones the eigenvector of the order parameter mode can be considered constant, in the latter, the variation of the soft-mode eigenvector with the modulation wave vector cannot be neglected.