Knowledge Bank

Lattice vibrations are considered on basis of characteristic modes of a large molecule rather than as standing wave patterns in a quasi-continuous medium with arbitrary boundary conditions. Symmetry properties are used in solution of problem for uniform linear chain without recourse to the controversial Born-Von Karman cyclic end conditions. An iterative process, based on Lagrange’s equation with ""extra"" forces, is developed for treating such modifications of the uniform linear chain as: alternating masses, both nearest and next-nearest neighbor coupling, change in masses and/or force constants near ends of the chain, ""impure"" mass in midst of the chain, coupling of linear chains to form planar and three-dimensional arrays, etc. The method permits proper approximations as the number of particles becomes very large. Interesting results are obtained for two- and three-dimensional arrays on the basis of valence-type vibrational forces similar to those employed in small molecules.