On the efficient evaluation of Fourier patterns for nanoparticles and clusters

Samples made of an isotropically oriented ensemble of atomic clusters or structures that are not large crystals (i.e. extended less than 10 periods in each direction) are at the frontier of today's material science and chemistry. Examples are nanoparticles, nanotubes, amorphous matter, polymers, and macromolecules in suspension. For such systems the computation of powder diffraction patterns (which may provide an efficient characterization) is to be performed the hard way, by summing contributions from each atom pair. This work deals with performing such computation in the most practical and efficient way. Three main points are developed: how to encode the enormous array of interatomic distances (which increase as the square or higher powers of the cluster diameter) to a much smaller array of equispaced values on a coarse grid (whose size increases linearly with the diameter); how to perform a fast computation of the diffraction pattern from this equispaced grid; how to optimize the grid step to obtain an arbitrarily small error on the computed diffraction pattern. Theory and examples are jointly developed and presented.