An analytical solution for the zero frequency hyperbolic RUM modes of distortion of SiO2-tridymite

Abstract

The constraint equations controlling the existence of soft, Rigid Unit Mode (RUM) modes of distortion of the ideal tridymite tetrahedral framework structure are derived and used to obtain an analytical solution for the potential soft phonon mode wave-vectors and their associated displacement patterns. One of two types of RUM known to exist are characterized by modulation wave-vectors perpendicular to 〈110〉 and atomic displacement patterns involving tetrahedral edge rotation while a second type of RUM is found to be soft on a hyperbolic reciprocal space distribution and to involve tetrahedral rotation about in-plane rotation axes. The analytical form of this hyperbolic diffuse distribution is derived as is an analytical expression for the associated RUM eigenvector of each point on the diffuse distribution. Show more