Lattice Vibration of the Cayley Tree

Abstract

Vibrational properties of a Cayley-tree-type system are investigated: Normal modes and squared frequency spectral densities are calculated for infinite homogeneous monatomic and diatomic Cayley trees. Effect of an impurity is then investigated. Existence of a virtual localized mode is thereby discussed with emphasis. Eigenfrequencies and eigenfunctions of a spherical Cayley tree of finite size are also calculated. Then the spectral density of the infinite homogeneous Cayley tree and that of the spherical Cayley tree in the large limit size are compared, and the relation between these two systems is discussed. J~ 4J on the vibrational properties of the Cayley tree. In 1961, Rubin and Zwanzig2J studied lattice dynamics of a rooted Cayley tree and found that its vibrational spectral density becomes very unusual in the limit of large size: It is dense within the interval (m 112 -1, m 1/ 2 + 1), but discontinuous at every frequency for which it does not vanish. Here m is the branching number. Recently, Tsuchiya3J obtained the diagonal Green function of a homogeneous Cayley tree, or the Bethe lattice, and calculated therefrom the spectral density of squared frequency D (ul). This density is smooth and has no singular points at all, contrary to the density obtained by Rubin and Zwanzig. Tsuchiya also con­ sidered the Bethe lattice with an impurity and found that there exists an impurity frequency above the band. The purpose of the present paper is to study in more detail the vibrational properties of Cayley trees and, especially, to clarify the reason why the spectrum obtained by Rubin and Zvvanzig and that obtained by Tsuchiya are at variance with one another. The systems studied are homogeneous monatomic and diatomic Bethe lattices and spherical Cayley trees of finite and infinite size. Here a Cayley tree is defined as a uniformly branching structure which has no closed loops. And an infinite homogeneous Cayley tree with translational invariance is called the Bethe lattice. At first, vibrational spectral densities and eigenfunctions of a monatomic and a diatomic Bethe lattice are calculated in §§ 2 and 3, respec­ tively, by using lattice Green functions. The spectra obtained are smooth, except that it consists of two separate bands in the case of diatomic lattice. It is shown5J that, if the coordination number z is larger than 3, the eigenfunctions fall off in