The Solution of the Eigenvector Problem in Synchrotron Radiation Based Anomalous Small-Angle X-Ray Scattering

Abstract

In the last three decades Synchrotron radiation became an indispensable experimental tool for chemical and structural analysis of nano-scaled properties in solid state physics, chemistry, materials science and life science thereby rendering the explanation of the macroscopic behavior of the materials and systems under investigation. Especially the techniques known as Anomalous Small-Angle X-ray Scattering provide deep insight into the materials structural architecture according to the different chemical components on lengths scales starting just above the atomic scale (≈1 nm) up to several 100 nm. The techniques sensitivity to the different chemical components makes use of the energy dependence of the atomic scattering factors, which are different for all chemical elements, thereby disentangling the nanostructure of the different chemical components by the signature of the elemental X-ray absorption edges i.e. by employing synchrotron radiation. The paper wants to focus on the application of an algorithm from linear algebra in the field of synchrotron radiation. It provides a closer look to the algebraic prerequisites, which govern the system of linear equations established by these experimental techniques and its solution by solving the eigenvector problem. The pair correlation functions of the so-called basic scattering functions are expressed as a linear combination of eigenvectors.

Coaggregation of Two Anionic Azo Dyestuffs: A Combined Static Light Scattering and Small-Angle X-ray Scattering Study

“…These combinations can be represented by ( i = 1, j = 2, k = 3), ( i = 2, j = 3, k = 1), and ( i = 3, j = 1, k = 2). A detailed outline of this combinatorics and its statistical consequences for the determination of S res ( q ) has been presented by Goerigk. , In the present work, the final pure resonant scattering is calculated as an average of the three independent combinations. Principally, S res ( q ) refers to the scattering of all strontium ions, independent of whether they are specifically bound to dyestuff aggregates or unbound (i.e., freely dissolved in water).…”

Systematic Limitations in Concentration Analysis via Anomalous Small-Angle X-ray Scattering in the Small Structure Limit

“…A comprehensive description of the mathematical details can be found in [28]. |A Pb pqq| 2 is the Fourier transform of the pair correlation function of the resonant scattering Pb 2+ -counterions.…”

“…When solving the vector equation constituted by ASAXS measurements at a minimum of three energies by the Gaussian elimination procedure, the form factor can be determined in analytical form [ 15 , 23 , 24 , 25 , 26 , 27 ]: where F(E 1 , E 2 , E 3 ) represents a normalization factor composed of the anomalous dispersion corrections at the respective three energies E 1 , E 2 , E 3 . A comprehensive description of the mathematical details can be found in [ 28 ].…”