Group-Theoretical Interpretation of Faintness Index for Ferroelectricicty or Ferroelasticity. I. Faintness Index of Soft Vibrational Modes

What has been called the faintness index of a degenerate set of soft vibrational modes with respect to a component y of electric polarization vector of mechanical strain tensor is group-theoretically interpreted as the smallest of the permissible values of such integer N that the symmetrized Kronecker N th power of the irreducible representation of G for the soft modes may contain as an irreducible component the representation of G for y , where G is the space group of the prototypic phase. If there is no such N , the faintness index is defined as as equal to infinity. A few examples are presented and discussed.