XRD image calibration: a fluid dynamics approach.

Finding the centre of a X-ray diffraction (XRD) image is a crucial task for most of modern techniques (SWAXS,GISAXS/GIWAXS/GIUSAXS), whenever partial profiles are available. A XRD pattern is a positive definite distribution function recorded onto a 2D detector (CCD, Image Plate, etc.), resembling the snapshot of the wavefront of a perturbation propagating in a medium. This approach aims at modelling the global propagation of such a perturbation in an oversimplified picture, as follows. The dynamical evolution of intrados/extrados of a wavefront is governed by fluid mechanics: due to the axial symmetry of the sample-to-detector geometry, the wavefront intrados shall evolve towards the image centre. A number of approximations are accounted for: the edges of detector itself will be treated as perfect boundaries, and, thus, the reflection effects can be safely neglected. Moreover, without loss of generality, the wavelength of such a wave can be safely assumed to be significantly larger than the depth of the medium (shallow water approximation). Shallow water equations (SWEs) indeed are used to model waves, especially in water, in a variety of environments, from droplets in a bathtub to tsunamis in the ocean. The SWEs are derived from the Navier--Stoke equations under the main assumption consisting in neglecting the vertical velocity: while a seafloor kink does cause a vertical acceleration (beyond that of gravity), the vertical velocity is still negligible when compared to tsunamis.