Quantifying structural disorder in spinels by X-ray diffractometry through constrained–restrained Rietveld refinements

Abstract

Accurate determination of the structural disorder (i.e., cation misplacement, anionic and cationic vacancies, and reduced cations) of spinels is crucial to understanding the properties of this technologically attractive sub-family of ceramics, but it is very difficult to do in practice. Here, constrained–restrained Rietveld refinements of the experimentally measured X-ray diffraction patterns are proposed to tackle the challenging quantification of the structural disorder in spinels. First, it is demonstrated that the constraints–restraints to be imposed during the Rietveld refinements can be formulated by mathematical modelling through the linear inverse problem whose framework is first presented generically and then particularised to the different types of possible normal, inverse, and mixed spinels, namely, stoichiometric perfect spinels, stoichiometric and non-stoichiometric imperfect spinels, and non-stoichiometric imperfect spinels with oxidation state changes. And second, by way of example this type of constrained–restrained Rietveld refinement is successfully applied to the experimental quantification of the structural disorder in two custom-made spinels (i.e., zinc-ferrite and nickel-ferrite spinels).