'Butterfly' small-angle X-ray scattering patterns in semicrystalline polymers are double-elliptical

We have previously shown that most small-angle scattering (SAS) patterns from a range of polymer fibers can be best fit using elliptical coordinates, where the minor axis of the ellipse is parallel to the fiber axis. Analysis of small-angle X-ray scattering patterns obtained during elongation of poly(ethylene-co-1-octene) films now shows that elliptical coordinates also provide a natural way to fit four-point radiating-out patterns, also called "butterfly" or "eyebrow" patterns. To fit this type of pattern the minor axis of the ellipse is tilted away from the fiber draw direction by some angle α, and as the films have rotational symmetry, the pattern is fitted with the sum of two ellipses, at +α and -α. As a result, the reflections appear to fall on a hyperbolic arc - the butterfly pattern. The Matlab code to carry out this analysis on full 2D data is available on request. The elliptical analysis is primarily an empirical fit, much strengthened by its ability to be applied to all observed types of SAS patterns and by its relevance to the mechanical behavior of polymers.