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International Journal of Mathematics and Mathematical Sciences
Volume 6 (1983), Issue 4, Pages 783-794
http://dx.doi.org/10.1155/S0161171283000666

Two dimentional lattice vibrations from direct product representations of symmetry groups

Department of Hathematical Sciences, Virginia Commonwealth University, Richmond 23284, Virginia, USA

Received 6 October 1981

Copyright © 1983 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Arrangements of point masses and ideal harmonic springs are used to model two dimensional crystals. First, the Born cyclic condition is applied to a double chain composed of coupled linear lattices to obtain a cylindrical arrangement. Then the quadratic Lagrangian function for the system is written in matrix notation. The Lagrangian is diagonalized to yield the natural frequencies of the system. The transformation to achieve the diagonalization was obtained from group theorectic considerations. Next, the techniques developed for the double chain are applied to a square lattice. The square lattice is transformed into the toroidal Ising model. The direct product nature of the symmetry group of the torus reveals the transformation to diagonalize the Lagrangian for the Ising model, and the natural frequencies for the principal directions in the model are obtained in closed form.