Site-occupancy factors in the Debye scattering equation. A theoretical discussion on significance and correctness

The Debye scattering equation (DSE) [Debye (1915). Ann. Phys. 351, 809-823] is widely used for analyzing total scattering data of nanocrystalline materials in reciprocal space. In its modified form (MDSE) [Cervellino et al. (2010). J. Appl. Cryst. 43, 1543-1547], it includes contributions from uncorrelated thermal agitation terms and, for defective crystalline nanoparticles (NPs), average site-occupancy factors (s.o.f.'s). The s.o.f.'s were introduced heuristically and no theoretical demonstration was provided. This paper presents in detail such a demonstration, corrects a glitch present in the original MDSE, and discusses the s.o.f.'s physical significance. Three new MDSE expressions are given that refer to distinct defective NP ensembles characterized by: (i) vacant sites with uncorrelated constant site-occupancy probability; (ii) vacant sites with a fixed number of randomly distributed atoms; (iii) self-excluding (disordered) positional sites. For all these cases, beneficial aspects and shortcomings of introducing s.o.f.'s as free refinable parameters are demonstrated. The theoretical analysis is supported by numerical simulations performed by comparing the corrected MDSE profiles and the ones based on atomistic modeling of a large number of NPs, satisfying the structural conditions described in (i)-(iii).