Two dimentional lattice vibrations from direct product representations of symmetry groups

Two dimentional lattice vibrations from direct product representations of symmetry groups

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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 princess polly dresses long sleeve 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 read more to diagonalize the Lagrangian for the Ising model, and the natural frequencies for the principal directions in the model are obtained in closed form.