Hard Vibrational Modes Which Are, or Are Not, Induced to Condense by the Condensation of Soft Modes. Extension of the Concept of Faintness Index for Ferroelectricity and Ferroelasticity

The problem “By a condensation of a given set of degenerate soft modes what hard modes are induced to condense and what hard modes are not?” (here either soft or hard modes are not restricted to ferroelectric or ferroelastic ones) is treated theoretically in a general and unified way with an extension of the concept of faintness index for ferroelectricity and ferroelasticity. Also, this general theory is applied to more concrete cases where the prototypic phase belongs to a specified point group and the soft modes are of zero wavenumber and belong to a specified irreducible representation; for all these cases an evaluation is made of the faintness indices of each possible ferroic phase with respect to all hard modes of zero wavenumber.