The integrals determining orientational order in liquid crystals by x-ray diffraction revisited

ABSTRACTThe orientational distribution function f(β) relative to the average orientation direction or director of liquid crystals and related materials can be obtained from the angular variation of the diffracted x-ray intensity distribution I(θ) of the wide-angle or equatorial arcs. The two quantities f(β) and I(θ) are related by an integral equation. Two different integral equations, one by Kratky and the other by Leadbetter and Norris describing the same phenomena, are the basis of the analyses. There has been discussion of which of the equations is correct; however, the form first derived by Kratky is correct. In this paper, solutions to the Kratky form of the equation are presented. Analytical, closed-form expressions for for n = 0, 1, 2 and 3 are presented. These allow calculation of the first three non-zero orientational order parameters. Experimental data are analysed using solutions to the Kratky kernel and the often used Leadbetter–Norris kernel as a series and in integral form, and the method o...