On the Calculation of the Molecular Cylindrical Distribution Function and the Order Parameter from X-Ray Diffraction Data of Liquid Crystals

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Abstract
A general formula is derived for the x-ray diffraction from an assembly of elongated molecules, and it is shown that this formula is too complex to be solved. To enable the calculation of the molecular cylindrical distribution function (mcdf) and the order parameter from the x-ray data, the approach taken in the literature has been to assume that only molecules approximately perpendicular to the scattering vector contribute significantly to the scattered radiation. Calculations on a model indicate, however, that this ``literature approximation'' is totally unjustified in the derivation of the mcdf, and, consequently, that the mcdf's obtained from x-ray data must be regarded as of no value. For the order parameter, the model calculations show that the errors made in using the ``literature approximation'' are not as large as in the case of the mcdf, and it appears that under favorable conditions the x-ray data may yield approximate values for the order parameter.