Elliptical Small-Angle X-Ray Scattering Patterns from Aligned Lamellar Arrays

Semicrystalline polymers aligned by drawing into films or fibers produce small-angle X-ray scattering patterns with two or four spots. Some liquid crystalline materials, cybotactic nematics, produce extremely similar patterns when oriented in magnetic fields. A structure of stacked lamellae, tilted for four point patterns, explain the basic features. However, the peak intensity positions of the broadened reflections lie not on a layer line or on the arc of a circle, but very close to an ellipse. Specific structural explanations of this feature have been suggested, but models using an equilibrium distribution of molecular orientations and lamellar tilts can predict elliptical shapes for the reflections. The model parameters are chosen by fitting the entire 2D intensity distribution of the scattering pattern. Assumptions required for modeling make some fitted parameters uncertain, but it is clear that the elliptical form can emerge from a statistical distribution of the properties of the stacks of lamellae, without a directly assignable structural cause.