General distance function for quantifying and comparing material anisotropies demonstrated through calculations of elastic and magnetic anisotropies of crystalline materials, A

Abstract

A new method for quantifying material anisotropy, as formulated in scalar and tensor functions, is presented. This general method is based on mathematical principles rather than bespoke models of unique physics; hence it is able to describe material anisotropy of any crystal symmetry and arbitrary physics, regardless of their dimension (i.e., scalars vs. tensors). In some simple cases for example two dimensional problems, the integrals derived in this new methodology can be evaluated analytically, otherwise they can be solved using numerical quadrature. This framework reduces any anisotropy function to a positive real number, measuring deviation from isotropy. The framework is through two specific examples: crystal elasticity and ferromagnetic crystal anisotropy. The method is verified to produce comparable results to established techniques in both fields and is shown to enable new discussions of material anisotropy were previously impossible or cumbersome to compute. It is also shown able to compare anisotropies of different physics across the same crystals. The impact of this ability is elucidated through the result that hexagonal Cobalt is more magnetically anisotropic than cubic Iron but less elastically anisotropic, resulting in understanding that “more or less” anisotropic is not a general property of a crystal symmetry, but rather a convolved property of the crystal symmetry and the physics. Furthermore, the technique can compare the anisotropies of the same physics across different crystal symmetries. In this regard, the result that cubic Potassium is more anisotropic than monoclinic Nickel Titanium shows that “more or less” anisotropic does not directly correlate to “more or less” symmetry of the crystal structure. Finally, one natural generalization of this technique to Cartesian tensors of arbitrary order is proposed.