Modelling of X-ray patterns using Fourier transforms: application to nanomaterials

Stucture of crystalline materials is obtained by studying the diffraction patterns using electromagnetic radiations like X-rays, electron and neutron beams. These patterns are essentially Fourier transforms of the sample space. As the diffraction pattern corresponds to reciprocal lattice of the atomic arrangement, Fourier transform can be used to convert the data from real space to reciprocal space. Thus, an image of real space when transformed to Fourier space should resemble the X-ray diffraction profile. For this, we have developed an algorithm for image processing using Fourier transform employing GNU Octave. In essence we would like to investigate the Fourier transform of several two-dimensional ordered systems which mimic two dimensional nanostructures in general. For this, we have built several two-dimensional models to study the ordered patterns with various shapes of the repeating entity like circular, rectangle, squares, benzene shape along irregular patterns. We study the variation in diffraction patterns that are in transformed space and compare them inorder to simulate with two dimensional images.