PHASE-DIAGRAMS OF CRYSTALS WITH ONE-DIMENSIONAL MODULATED PHASES INDUCED BY 2D ACTIVE REPRESENTATIONS

A one-dimensional model of particles with a displacive degree of freed om for crystals possessing incommensurate phases which arise as a resu lt of the condensation of either real two-dimensional or complex one-d imensional irreducible representations, has been proposed. For these r epresentations all invariants of the free energy expansion can be divi ded to four general forms. For the active irreducible representations for which the invariants belong to the first form a complete list of i nvariants is derived. In this case the incommensurate modulation propa gates along the symmetry axis and for such crystals a proposed one-dim ensional model may be a good approach to describe the main features of the devil's staircase curve. The particles of the model interact with harmonic and anharmonic terms. The last ones may contain an additiona l third order term provided a soft phonon branch has a symmetry tau1. The calculated phase diagrams show sequences of the incommensurate and commensurate one-dimensional phases. In the presence of the third ord er anharmonic term the incommensurate phase proves to be stable closer to the phase boundary to the normal phase.