Images, Landau expansions, and symmetry changes.

Abstract

The phase diagrams obtained from the fourth-degree Landau expansions associated with the ${C}_{4}$ and ${C}_{4v}$ images are compared. While the first image yields a Landau expansion with a four-dimensional parameter space (${\ensuremath{\alpha}}_{1}$,${\ensuremath{\alpha}}_{2}$,${\ensuremath{\beta}}_{1}$,${\ensuremath{\beta}}_{2}$), the second yields a potential which is contained in the three-dimensional subspace with ${\ensuremath{\beta}}_{2}$=0. Within this subspace the phase diagrams of both ${C}_{4}$ and ${C}_{4v}$ are identical. Two ordered phases (with the order parameter along either an axis or a diagonal, respectively) are separated by a first-order line. Analysis of symmetry changes involved lead, however, to different results. In the ${C}_{4v}$ case the two phases have distinct symmetries. On the other hand the ${C}_{4}$ image implies identical symmetry in the two ordered phases. Physical interpretation of the phase diagram is thus directly related to the actual image. Experimental consequences are discussed.