Effect of Free Ends on the Vibration Frequencies of One-Dimensional Lattices

Abstract

Vibration frequencies have been calculated for finite one-dimensional lattices in which the point masses alternately have the values $m_{\alpha }$ and $m_{\beta }$, $m_{\alpha }<m_{\beta }$. Only nearest neighbor Hooke's law interactions are considered. The end atoms are assumed to interact only with their nearest neighbors on the interior of the lattice and are otherwise free. If the numbers of atoms having masses $m_{\alpha }$ and $m_{\beta }$ are equal, there exists a single mode whose frequency squared lies at the middle of the "forbidden" gap between the optical and acoustical branches. For this "surface" mode the displacements of the atoms from their equilibrium positions decrease roughly exponentially from the end having the lighter atom. For the case of $N$ atoms of mass $m_{\beta }$ and $N+1\mathrm{}$ atoms of mass $m_{\alpha }$ there exist two modes whose frequencies lie in the "forbidden" gap provided $\left(\frac{m_{\alpha }}{m_{\beta }}\right)<\mathrm{}\frac{N}{\mathrm{}(N+1\mathrm{})\mathrm{}}$. These modes correspond to symmetric and antisymmetric displacements. The displacements are largest for the end atoms and decrease roughly exponentially toward the center of the lattice.