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.

The order parameter provides a convenient quantification of alignment:

• S = 1 denotes perfect alignment (along the director).
• 0 < S < 1 denotes partial alignment
• S = 0 denotes a completely random alignment
• S = 1/2 denotes 'anti-alignment' (against the director)

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

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

Alternatives

For a highly-aligned system, one will observe a sharp peak (in the $\scriptstyle \chi$ direction), in which case one can also characterize the orientational spread by measuring the peak FWHM, or fitting the peak to a Gaussian (to measure $\scriptstyle \sigma _{\chi }$). A more general possibility is to fit the scattering intensity along $\scriptstyle \chi$ to a function that has proper 'circular wrapping' (c.f. circular orientation distribution function).