Document Type
Article
Original Publication Date
1983
Journal/Book/Conference Title
International Journal of Mathematics and Mathematical Sciences
Volume
6 (1983)
Issue
4
DOI of Original Publication
10.1155/S0161171283000666
Date of Submission
August 2014
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.
Rights
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.
Is Part Of
VCU Mathematics and Applied Mathematics Publications
Comments
Originally published at http://dx.doi.org/10.1155/S0161171283000666