Scalar and Tensor Order Parameters

The physics of liquid crystals is best described in terms of the so-called order parameters . If we use the long axis of the molecule as a reference and denote it as k, the microscopic scalar order parameter S is defined as follows  

On the other hand, for molecules lacking such symmetry, or in cases where such rotational symmetry is destroyed by the presence of asymmetric dopants or intramolecular material interactions, a more general tensor order parameter Sy is needed. Sa is defined as 

ORDER PARAMETER, PHASE TRANSITION, AND FREE ENERGIES 

Note that Sn+Sjj+Sa=0. Put another way, S is a traceless tensor because its diagonal elements add up to zero. 

For a more complete description of the statistical properties of the liquid crystal orientation, functions involving higher powers of cos 0 are needed. The most natural functions to use are the Legendre polynomials P (cos 0) (/ = 0,1,2.), in terms of which we can write Equation (2.1) as S = P2) which measures the average of cos 0. The next nonvanishing term is 4), which provides a measure of the dispersion of cos 0 . 

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