Molecular orientation distribution derived from an arbitrary reflection

Abstract

A straightforward method using Legendre series enables the orientation distribution in a specimen with uniaxial symmetry to be derived from the azimuthal profile of a single arbitrary reflection. Moreover, the moments of the distribution (P2,(cos,)) can be obtained directly from the azimuthal profile without needing to calculate the complete distribution.Pole figures derived from X-ray diffraction measurements are the standard method of quantifying orientation in crystalline materials. Similarly, the azimuthal profiles of the diffuse arcs found for liquid crystals (Leadbetter & Wrighton, 1979) and non-crystalline polymers (Wilchinsky, 1968) have been used to give a measure of orientation.In polymers and liquid crystals, it is usually the orientation distribution for the molecular axes which is required, but this is only obtained directly from a pole figure if there is a strong reflection from planes perpendicular to the molecular axes. However, Wilchinsky (1963) has shown that, provided the molecules are random about their axes, a single arbitrary reflection can give the value of (cos 2.), where, is the angle between the molecular axis and the specimen axis.In this communication we show that, for a specimen with uniaxial symmetry, the higher moments of the orientation distribution can also be obtained from the azimuthal profile of an arbitrary reflection. Hence the full orientation distribution can be calculated without recourse to solving integral equations or inverting matrices.The scattering from a distribution of independent molecules is given by a convolution of the orientation distribution of molecular axes with the scattering for a single molecule (Ruland & Tompa, 1968). If both the orientation distribution D(,) and the molecular scattering I'(-) have cylindrical symmetry, then the resultant scattering I(,) also has cylindrical symmetry (Deas, 1952) and all three functions can be expanded in series of even-order Legendre polynomials (P2,), e.g. Equatorial reflection (, 0 = zt/2):(-1)" 22"(n!) 2 (P2.)o --(P2.)r (2n)!

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