Determination of the symmetry classes of orientational ordering tensors

The orientational order of nematic liquid crystals is traditionally studied by means of the second-rank ordering tensor S. When this is calculated through experiments or simulations, the symmetry group of the phase is not known a priori, but needs to be deduced from the numerical realisation of S, which is affected by numerical errors. There is no generally accepted procedure to perform this analysis. Here, we provide a new algorithm suited to identifying the symmetry group of the phase. As a by product, we prove that there are only five phase-symmetry classes of the second-rank ordering tensor and give a canonical representation of S for each class. The nearest tensor of the assigned symmetry is determined by group-projection. In order to test our procedure, we generate uniaxial and biaxial phases in a system of interacting particles, endowed with D_{\infty h} or D_{2h} symmetry, which mimic the outcome of Monte–Carlo simulations. The actual symmetry of the phases is correctly identified, along with the optimal choice of laboratory frame.