Particle size distribution and X-ray diffraction peak profiles in supersaturated solid solutions

An analytical approximation of the correlation function relevant to a collection of spherical particles is obtained starting from a peaked particle distribution and neglecting any interparticle correlation. The distribution depends on two parameters, Δ and D₀, respectively the total width of the peak and the position of its left end. The distribution is such that by an appropriate variable rescaling and a suitable choice of the parameter Δ/D₀, a very accurate analytical approximation is obtained for the correlation function relevant to the particle distribution predicted by Lifshitz & Slyozov [J. Phys. Chem. Solids (1961), 19, 35-50] and by Wagner [Z. Elektrochem. (1961), 65, 581-590] for advanced stages of phase-separation processes in supersaturated solid solutions. By a Fourier transformation, two simple analytical expressions are obtained for the corresponding SAXS and WAXS peak profiles. The characteristic feature of these is the presence, in the experimentally accessible region, of some oscillations in the SAXS Porod plot and in the latter generalization appropriate to WAXS peaks. The observation of these features in peak profiles, relevant to samples where a demixing process may have occurred, and the subsequent use of the functional expressions presented here should make the numerical analysis of experimental data easier and yield a more definite answer on the nature of the demixing process.