Orientation order parameter
The alignment of a system can be characterized by calculating the orientation order parameter, S:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} S & = \langle P_2 ( \cos \chi ) \rangle \\ & = \frac{ 3 \langle \cos^2 \chi \rangle - 1} {2} \end{alignat} }
Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \chi} is the angle with respect to the director, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle P_2} is the 2nd Legendre polynomial, and the angle-brackets denote averaging over all entities. In scattering, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} S & = 2 \langle \cos ^2 \chi \rangle - 1 \\ & = \langle \cos( 2 \chi) \rangle \end{alignat} }
Alternatives
For a highly-aligned system, one will observe a sharp peak (in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \sigma_{\chi}} ). A more general possibility is to fit the scattering intensity along Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \chi} to a function that has proper 'circular wrapping' (c.f. circular orientation distribution function).
