Vibrational spectrum in the vicinity of a local mode in linear chains with isotopic impurities

Abstract

The vibrational spectrum in the vicinity of the frequency of a local vibration of an isolated light isotopic impurity in a one-dimensional disordered chain containing such impurities is calculated numerically. Chains consisting of 108–109 atoms, corresponding to macroscopic samples several centimeters in size, are used. A complex hierarchical structure of the spectrum is obtained for this system. The results of the numerical calculations are compared with simple analytical expressions. It is shown that the smoothed density of states in a narrow neighborhood of the local mode can be obtained to sufficient accuracy from arguments based on separating out the nearest neighbor of the chosen impurity atom. It turns out that the expression for the density of states obtained in that manner gives an acceptable result not only outside but also inside the region of concentration broadening of the local mode.