Abstract The diffracted intensity of an x-ray or neutron diffraction experiment is expressed as an integral over an atomic position distribution function. A generalized Debye scattering formula results. Since this distribution function is expanded into a series of spherical harmonics, an inverse Hankel transform of the intensity allows the calculation of the expansion coefficients which describe the atomic arrangement completely. The connections between the generalized Debye scattering formula and the original Debye formula as well as the Laue scattering formula are derived.
The tail of the particle swarm optimisation (PSO) position distribution at stagnation
is shown to be describable by a power law. This tail fattening is attributed to
particle bursting on all length scales. The origin of the power law is concluded to
lie in multiplicative randomness, previously encountered in the study of first-order
stochastic difference equations, and generalised here to second-order equations. It
is argued that recombinant PSO, a competitive PSO variant without multiplicative
randomness, does not experience tail fattening at stagnation.
Materials property prediction using symmetry-labeled graphs as atomic position independent descriptors