Relations between angular and Cartesian orientational expansions

TL;DR

This paper provides explicit formulas to convert between angular and Cartesian orientational expansions, facilitating comparison of theoretical and experimental results in fields like liquid crystal physics.

Contribution

It introduces detailed formulas for converting between angular and Cartesian orientational expansions for functions of multiple angles in 2D and 3D.

Findings

Explicit conversion formulas for 1, 2, and 3 angles

Application to orientational order parameters

Enhances comparison of theory and experiment

Abstract

Orientational expansions, which are widely used in the natural sciences, exist in angular and Cartesian form. Although these expansions are orderwise equivalent, it is difficult to relate them in practice. In this article, both types of expansions and their relations are explained in detail. We give explicit formulas for the conversion between angular and Cartesian expansion coefficients for functions depending on one, two, and three angles in two and three spatial dimensions. These formulas are useful, e.g., for comparing theoretical and experimental results in liquid crystal physics. The application of the expansions in the definition of orientational order parameters is also discussed.