Phenomenological model of changes in the speed of a sound wave in crystals caused by dopants

The paper examines theoretically changes in the speed of an acoustic wave in crystals caused by metallic dopants. It is well known that an acoustic wave has a wavelength larger than the typical dimension of the elementary crystallographic cell, so it is sensitive only to large-scale features. The model bases on the assumption that a crystal with uniform distributed dopants can be transformed into a multiplayer virtual structure form which some conclusions can be drawn. The approach presented is valid for longitudinal acoustic waves, where the atomic displacement can be considered as a continuum field. It provides positive or negative changes in the acoustic wave speeds for different dopants in a hypothetical lithium niobate crystal.