Group Theory of Vibrations of Symmetric Molecules, Membranes, and Plates

Abstract

It is shown that the detailed group-theoretical analysis of molecular vibrations can be extended to continuous bodies of various shapes by regarding them as the limits of discrete systems with the number of particles becoming indefinitely large. In this paper an elementary review of the method is given and its application is illustrated with 2-dimensional systems and thin systems having pyramidal symmetry (i.e., tents and shells). Both stationary and rotating systems containing any number of particles arranged with the symmetry of a regular polygon or polygonal pyramid are considered. For macroscopic systems, unlike molecules, different possible constraints must be taken into account in enumerating the genuine vibrations of various types. Otherwise, one proceeds as in the molecular case. By going to the limit of large numbers of particles, one obtains the fractional occurrence numbers of vibrations of various symmetry types for stationary or spinning membranes, plates, tents, or shells of all regular polygonal shapes.