On the Cartesian definition of orientational order parameters

Abstract

Orientational order parameters can be effectively and economically defined using spherical tensors. However, their definition in terms of Cartesian tensors can sometimes provide a clearer physical intuition. We show that it is possible to build a fully Cartesian theory of the orientational order parameters which is consistent with the traditional spherical tensor approach. The key idea is to build a generalised multi-pole expansion of the orientational probability distribution function in terms of outer products of rotation matrices. Furthermore, we show that the Saupe ordering super-matrix, as found, for example, in the text by de Gennes and Prost [The Physics of Liquid Crystals, 2nd ed. (Oxford University Press, Oxford, UK, 1995)] and which is used to define the Cartesian second-rank orientational order parameters, is not consistent with its spherical tensor counterpart. We then propose a symmetric version of the Saupe super-matrix which is fully consistent with the spherical tensor definition. The proposed definition is important for a correct description of liquid crystal materials composed of low symmetry molecules. (C) 2011 American Institute of Physics. [doi:10.1063/1.3589961

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