A new treatment of the normal vibrations of a linear diatomic lattice

Abstract

The vibrational analysis of the diatomic linear lattice has been worked out by the method of extra forces. In this method one takes as the zero-order problem that part of forces in the complete equations of motion yielding an exactly solvable problem. The normal coordinates thus obtained form new generalized coordinates for the modified Lagrange's equations and forces that were neglected in zero-order are introduced as extra generalized forces. The diatomic chain can be regarded as a system of two uniform chains, one containing N ∗ M-particles and the other n ∗ = N ∗ − 1 m-particles, where N ∗ + n ∗ = N is the total number of particles, that are superimposed to form a symmetrical chain. The zero-order problem comes from the two independent chains and the extra forces arise from the coupling of the chains. The analysis shows that for a long diatomic chain, only those zero-order modes of the separate chains having the same index r are coupled, to high order of approximation. The frequencies of the optical and acoustical branches predicted by this method differ from those quoted in the literature by the replacement of the customary cos 2 ( rπ 2N ∗ ) by cos 4 ( rπ 2N ∗ ). The basic method of extra forces, including proper symmetry conditions, enables one to set up the vibrational analysis of more complicated diatomic arrays.