Circular orientation distribution function

In assessing the orientation of aligned materials, one can use the orientation order parameter to quantify order. Another possibility is to fit scattering data using an equation that has 'circular wrapping' (i.e. periodic 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 2 \pi} ). Ruland et al. present such an equation:

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 I (\chi) = \frac{1 - \eta^2}{(1+\eta)^2 - 4 \eta \cos^2 \chi} }

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 along the arc of the scattering ring/feature. The single fit parameter (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 \eta } ) is convenient in that it behaves in a similar way to an order parameter: a value close to 1.0 indicates strong alignment, while progressively smaller values indicate lesser alignment. For a random sample, the scattering is isotropic and 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 \eta = 0} .

See Also

• Ruland, W.; Tompa, H., The Effect of Preferred Orientation on the Intensity Distribution of (Hk) Interferences. Acta Crystallographica Section A 1968, 24, 93-99.
• Ruland, W.; Smarsly, B., Saxs of Self-Assembled Oriented Lamellar Nanocomposite Films: An Advanced Method of Evaluation. J. Appl. Crystallogr. 2004, 37, 575-584.