Circular orientation distribution function

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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 ).


Ruland et al. present such an equation:

Where is the angle along the arc of the scattering ring/feature. The single fit parameter () 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 .

Eta func-I chi.png


The function normalized so that the maximum is always at 1 would be:


Maier-Saupe distribution parameter

Where is a parameter that can be related to the order parameter ; specifically is for an isotropic distribution (), while is for a well-aligned system ().

MaierSaupe-ODF-01.png MaierSaupe-ODF-021.png

The parameter c can be used to normalize:

MaierSaupe-ODF-01b.png MaierSaupe-ODF-02b.png