  The availability of neutron spallation-source instruments that provide total
scattering powder diffraction has led to an increased application of real-space
structure analysis using the pair distribution function. Currently, the
analytical treatment of finite size effects within pair distribution refinement
procedures is limited. To that end, an envelope function is derived which
transforms the pair distribution function of an infinite solid into that of a
spherical particle with the same crystal structure. Distributions of particle
sizes are then considered, and the associated envelope function is used to
predict the particle size distribution of an experimental sample of gold
nanoparticles from its pair distribution function alone. Finally, complementing
the wealth of existing diffraction analysis, the peak broadening for the
structure factor of spherical particles, expressed as a convolution derived
from the envelope functions, is calculated exactly for all particle size
distributions considered, and peak maxima, offsets, and asymmetries are
discussed.
