  When a macroscopic-sized non-crystalline sample is illuminated using coherent
x-ray radiation, a bifurcation of photon energy flow may occur. The
coarse-grained complex refractive index of the sample may be considered to
attenuate and refract the incident coherent beam, leading to a coherent
component of the transmitted beam. Spatially-unresolved sample microstructure,
associated with the fine-grained components of the complex refractive index,
introduces a diffuse component to the transmitted beam. This diffuse
photon-scattering channel may be viewed in terms of position-dependent fans of
ultra-small-angle x-ray scatter. These position-dependent fans, at the exit
surface of the object, may under certain circumstances be approximated as
having a locally-elliptical shape. By using an anisotropic-diffusion
Fokker-Planck approach to model this bifurcated x-ray energy flow, we show how
all three components (attenuation, refraction and locally-elliptical diffuse
scatter) may be recovered. This is done via x-ray speckle tracking, in which
the sample is illuminated with spatially-random x-ray fields generated by
coherent illumination of a spatially-random membrane. The theory is developed,
and then successfully applied to experimental x-ray data.
