Classifying local anisotropy formed by rigid molecules: symmetries and tensors

TL;DR

This paper develops a tensor-based framework to classify local anisotropy in a volume of rigid molecules by analyzing molecular symmetries and their invariant tensors, aiding in understanding mesoscopic symmetry states.

Contribution

It introduces a method to classify local anisotropy using invariant tensors derived from molecular symmetries and their application to mesoscopic symmetry classification.

Findings

Explicit expressions for invariant tensors under molecular symmetries.

A classification scheme for mesoscopic symmetries based on order parameter tensors.

Analysis of three-fold, four-fold, and polyhedral symmetries.

Abstract

We consider an infinitesimal volume where there are many rigid molecules of the same kind, and discuss the description and classification of the local anisotropy in this volume by tensors. First, we examine the symmetry of a rigid molecule, which is described by a point group in $SO(3)$. For each point group in $SO(3)$, we find the tensors invariant under the rotations in the group. These tensors shall be symmetric and traceless. We write down the explicit expressions. The order parameters to describe the local anisotropy are then chosen as some of the invariant tensors averaged about the density function. Next, we discuss the classification of local anisotropy by the symmetry of the whole infinitesimal volume. This mesoscopic symmetry can be recognized by the value of the order parameter tensors in the sense of maximum entropy state. For some sets of order parameter tensors involving different molecular symmetries, we give the classification of mesoscopic symmetries, in which the three-fold, four-fold and polyhedral symmetries are examined.