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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)., (open access.)

Many functional materials are today synthesized in the form of nanoparticles displaying preferred orientation effects to some small or large extent. The analysis of diffraction data of such kinds of systems is best performed in the framework of the total scattering approach that prescinds from translation symmetry assumptions. Therefore modified expressions were derived for the most common total scattering functions, in particular the Debye scattering equation (DSE) which yields the texture-averaged differential cross section as a function of atomic coordinates and texture parameters. The modified DSE encodes higher-order even spherical Bessel functions which account for the texture effect. Selection rules arising from experimental geometries and symmetries are discussed. In addition the duality of the texture effect is introduced showing the effects of texture on both the I(Q) and {\cal{G}}(r). The paper includes several definitions and appendices which are meant to be useful for those involved in the development of crystallographic computing.

Faulted face-centred cubic platinum nanocrystals, grown within a nanoporous silica matrix, have been extensively characterized by the Debye function analysis method applied to wide-angle synchrotron X-ray total scattering data. A method for building databases of sampled interatomic distances of weakly faulted materials is proposed, maintaining statistical significance and allowing complete populations of differently sized and shaped nanocrystals to be used within the DEBUSSY approach. This study suggests that anisotropic Pt nanoclusters are formed in the presence of a shape-directing (templating) agent, and tentatively describes the effects of post-synthetic temperature treatments on fault probability, size, shape and dispersion of the nanocrystal populations. Surface relaxation effects are also observed in the smallest particles.

One hundred years of the Debye scattering equation are celebrated with a series of articles arising from the DSE 2015 conference.