Oscillatory spectrum structure in chains with isotopes

M.A.Ivanov, V.S.Molodid, Yu.V.Skrypnyk

G.V.Kurdyumov Institute of Metal Physics of NASU

Kyiv, 03142, Ukraine

skrypnyk@i.com.ua

The oscillatory spectrum in the vicinity of the frequency of a local mode in a one-dimensional disordered chain containing light isotopic impurities is calculated numerically. Chains corresponding to macroscopic samples several centimeters long are simulated. It is shown that by systematizing impurities with respect to the distance to the next closest impurity the density of states can be estimated to a sufficient accuracy not only outside but also inside the region of concentration broadening of the local mode. A complex hierarchical structure of the spectrum which has features characteristic of a fractal is obtained for this system.

Determination of the oscillatory spectrum of a disordered system is a traditional topic in condensed matter physics. Generally speaking, the dimensionality times the number of atoms equations should be solved simultaneously in order to obtain the full information on oscillatory modes of the disordered crystal. Since translation symmetry is absent and often there are no other ways to simplify the problem, analytical approximations and numerical methods have been developed for the physically most important models of disorder.

Beginning with the pioneering works of I. M. Lifshits; different authors have proposed their own analytical approaches to the calculation of the oscillatory spectrum of disordered systems. Often it is difficult to estimate inherent limitations of the approximations. Numerical simulation provides a possibility to check directly analytical results for a given simplified model system. One of them is the linear chain with nearest-neighbor interaction and substitution disorder. Despite its basic character, it is widely used, as it is capable of expressing main features of real physical objects such as chain crystals and some photonic materials.

Back in the 1960s Dean carried out calculations of the spectra of disordered chains at comparatively high (0.1- 0.5) concentrations of impurities. However, because chains of the sufficient length could not be simulated, systems with low impurity concentration remained unstudied. In contrast, this very case is of a particular interest, since, on the one hand, even the presence of only a faint trace of impurities can result in a substantial change of the main properties of the crystal and, on the other hand, it is at the low impurity concentration that one can neglect the indirect interaction between impurities and for which numerous analytical results had been obtained.

This paper is devoted to disordered chains with low concentration of impurities. Oscillatory spectrum is analyzed in the close proximity of the local mode caused by the presence of light substitutional atoms. Results of the numerical simulation of chains comprised of 10⁸-10⁹ atoms are compared to simple estimates for the spectral distribution of frequencies.