Acoustic axes conditions revised

TL;DR

This paper revises the conditions for acoustic axes in crystals using a minimal polynomial approach, providing a compact, qualitative, and computationally effective framework for identifying acoustic axes and their properties.

Contribution

It introduces an alternative minimal polynomial-based system for acoustic axis conditions, unifying and simplifying existing criteria and enabling practical computations for various crystal symmetries.

Findings

Derived necessary and sufficient criteria for acoustic axes using the reduced acoustic tensor.

Recast Khatkevich criteria in terms of the reduced acoustic tensor.

Applied the approach to high symmetry cases like isotropic and RTHC crystals.

Abstract

The explanation of the basic acoustic properties of crystals requires a recognition of the acoustic axes. To derive the acoustic axes in a given material, one requires both a workable method and the necessary and sufficient criteria for the existence of the acoustic axes in a partial propagation direction. We apply the reduced form of the acoustic tensor to the acoustic axis conditions in the present work. Using this tensor, we obtain in a compact form, allowing for qualitative analysis, the necessary and sufficient criteria for the existence of the acoustic axis. Furthermore, the well-known Khatkevich criteria and their variants are recast in terms of the reduced acoustic tensor. This paper's primary input is an alternate minimal polynomial-based system of acoustic axes conditions. In this approach, we derive an additional characteristic of acoustic axes: the directions in which the minimal polynomial of the third order is reduced to that of the second order. Next, we offer a general solution to the second-order minimum polynomial equation, that utilizes a scalar and a unit vector for defining the acoustic tensor along the acoustic axis. It is shown that the scalar matches the eigenvalue of the reduced acoustic tensor, and the vector corresponds to the polarization into the single eigenvalue direction. We use the minimal polynomial construction to demonstrate the equivalence of different acoustic axis criteria. We demonstrate the applicability of this approach to actual computations of the acoustic axes and their fundamental properties (phase speeds and polarizations) for high symmetry cases, such as isotropic materials and RTHC crystals.