Simulation of small-angle scattering curves by numerical Fourier transformation

A simple numerical approach for calculating the q-dependence of the scattering intensity in small-angle X-ray or neutron scattering (SAXS/SANS) is discussed. For a user-defined scattering density on a lattice, the scattering intensity I(q) (q is the modulus of the scattering vector) is calculated by three-dimensional (or two-dimensional) numerical Fourier transformation and spherical summation in q space, with a simple smoothing algorithm. An exact and simple correction for continuous rather than discrete (lattice-point) scattering density is described. Applications to relatively densely packed particles in solids (e.g. nanocomposites) are shown, where correlation effects make single-particle (pure form-factor) calculations invalid. The algorithm can be applied to particles of any shape that can be defined on the chosen cubic lattice and with any size distribution, while those features pose difficulties to a traditional treatment in terms of form and structure factors. For particles of identical but potentially complex shapes, numerical calculation of the form factor is described. Long parallel rods and platelets of various cross-section shapes are particularly convenient to treat, since the calculation is reduced to two dimensions. The method is used to demonstrate that the scattering intensity from `randomly' parallel-packed long cylinders is not described by simple 1/q and 1/q⁴ power laws, but at cylinder volume fractions of more than ∼25% includes a correlation peak. The simulations highlight that the traditional evaluation of the peak position overestimates the cylinder thickness by a factor of ∼1.5. It is also shown that a mix of various relatively densely packed long boards can produce I(q) ≃ 1/q, usually observed for rod-shaped particles, without a correlation peak.