Orientation order parameter

The alignment of a system can be characterized by calculating the orientation order parameter, S:

${\begin{alignedat}{2}S&=\langle P_{2}(\cos \chi )\rangle \\&={\frac {3\langle \cos ^{2}\chi \rangle -1}{2}}\end{alignedat}}$

Where $\scriptstyle \chi$ is the angle with respect to the director, $\scriptstyle P_{2}$ is the 2nd Legendre polynomial, and the angle-brackets denote averaging over all entities. In scattering, if $\scriptstyle \chi$ is the angle around the scattering pattern, one can compute the order parameter for aligned systems. This quantity is often used in research of intrinsically anisotropic materials, such as liquid crystals.

Note that in two-dimensions, the order parameter has a different definition:

${\begin{alignedat}{2}S&=2\langle \cos ^{2}\chi \rangle -1\\&=\langle \cos(2\chi )\rangle \end{alignedat}}$