Structure analysis of multiphase systems by anomalous small-angle X-ray scattering

Abstract

The theory of small-angle scattering is reviewed with special attention paid to the anomalous scattering and multiphase systems. A general equation is derived that describes the scattering of a multiphase system as a sum of scattering functions of each of the phases, as if it scattered alone in a two-phase system, and interphase interference scattering functions. These scattering functions depend only on the spatial distribution of the phase boundaries, but not on the scattering density. Contrast variation techniques are most rewarding when the scattering density of only one phase can be varied. For anomalous small-angle X-ray scattering (ASAXS), this means the most favourable is the case in which resonant atoms are contained in one phase only. The general equation involves n(p − 1) unknown partial atomic number density differences, where p is the number of phases and n the number of the different atom types in the sample. These partial atomic number density differences can be found if a suitable structure model is applied to calculate the phase scattering functions. Then, the phase compositions and densities can be calculated by solving a system of linear equations incorporating the atom number conservation law. The partial structure factors formalism is also reviewed. Corresponding equations for a system of n types of atoms and p phases are derived. The number of independent partial structure factors is p(p − 1)/2 and depends on the number of phases, but not on the number of the types of the atoms in the sample, as in the case of wide-angle scattering.

Keywords:

• anomalous small-angle X-ray scattering
• multiphase systems
• partial structure factors
• scattering contrast variation

Acknowledgments

The author expresses his thanks to Prof. R. Kranold, Dr. A. Hoell and Dr. A. Heinemann for their comments on the manuscript.