Orientational order determination in liquid crystals by x-ray diffraction

The orientational distribution function f relative to the director of elongated molecules of liquid crystals and related materials can be determined, in principle, from the intensity distribution I along equatorial arcs in the x-ray-diffraction pattern. The integral equation relating these two quantities is solved here, yielding an analytic, closed-form expression for f in terms of the measured I. Analytic expressions for several important related quantities, like the second- and fourth-order order parameters, the average tilt angle, etc. are also derived. The accuracy obtainable in calculating these quantities from measured data is discussed, and examples demonstrating the use of the method are given. These examples show the robustness and accuracy of the solution and its related quantities even for highly noisy and/or sparse experimental input data I.