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

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 ac cording to the different chemical components on lengths scales starting just above the atomic scale amp; 8776;1 nm up to sev eral 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 synchro tron radiation. The paper wants to focus on the application of an algorithm from linear algebra in the field of synchro tron radiation. It provides a closer look to the algebraic prerequisites, which govern the system of linear equations es tablished 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 eigenvector

The Significance of Solutions Obtained from Ill Posed Systems of Linear Equations Constituted by Synchrotron Radiation Based Anomalous Small Angle X Ray Scattering

Synchrotron radiation based experimental techniques known as Anomalous Small Angle X ray Scattering ASAXS provide deep insight into the nanostructure of uncountable material systems in condensed matter research i.e. solid state physics, chemistry, engineering and life sciences thereby rendering the origin of the macroscopic functionalization of the various materials via correlation to its structural architecture on a nanometer length scale. The techniques constitute a system of linear equations, which can be treated by matrix theory. The study aims to analyze the significance of the solutions of the stated matrix equations by use of the so called condition numbers first introduced by A. Turing, J. von Neumann and H. Goldstine. Special attention was given for the comparison with direct methods i.e. the Gaussian elimination method. The mathematical roots of ill posed ASAXS equations preventing matrix inversion have been identified. In the framework of the theory of von Neumann and Goldstine the inversion of certain matrices constituted by ASAXS gradually becomes impossible caused by non definiteness. In Turing s theory which starts from more general prerequisites, the principal minors of the same matrices approach singularity thereby imposing large errors on inversion. In conclusion both theories recommend for extremely ill posed ASAXS problems avoiding inversion and the use of direct methods for instance Gaussian eliminatio

Machine Prepared Thermoplastic Polyurethanes of Varying Hard Segment Content Morphology and Its Evolution in Tensile Tests

Machine cast thermoplastic polyurethanes are strained and monitored by small angle X ray scattering SAXS . They are prepared from 4,40 methylene diphenyl diisocyanate, 1,4 butane diol, and polytetrahydrofuran. Upon stretching hard domains are destroyed. Most stable are the domains of materials with a hard domain content HSC of 30 . Domain stability decreases with increasing HSC and crosslinking. Most materials show stability up to a strain 0.6. At higher strain, the apparent long period decreases for the materials with HSC530 . Correlated hard domains, the strain probes relax as others are destroyed. The fraction of relaxing probes and their ultimate relaxation decrease with increasing HSC. Chord distribution functions computed from the SAXS exhibit the same sequence of static long period bands. The band positions form a Fibonacci series, related to the underlying polyaddition process. This indicates a nearly quasicrystalline arrangement of stringed hard domains, identified as the strain probes of the discrete SAXS. At strains lt;0.6, the probes experience half of the macroscopic strain, which reflects hard domain rigidity. VC 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part B Polym. Phys. 2015, 00, 000 00