Small Angle X-Ray Scattering

Quasielastic neutron scattering for the investigation of diffusive motions in solids and liquids.

Quasielastic Neutron Scattering for the Investigation of Diffusive Motions in Solids and Liquids

Optical particle sizing in the range of 10 nm up to several micrometers by means of quasi-elastic and elastic light scattering requires sophisticated data inversion techniques. We have developed an optimized regularization technique that can be used for the inversion of such light-scattering data. The technique has been successfully tested for a large number of simulated and measured data. It is easy to handle. Typical problems that arise in practical applications are discussed.

Optical sizing of small colloidal particles: an optimized regularization technique

Small-Angle Scattering and Its Applications to Materials Science

Publisher Summary This chapter discusses small-angle scattering experiments with particles in solution—i.e., the particles are nonoriented. A large number of particles contribute to the scattering and the resulting spatial average leads to a loss in information. The information contained in the three-dimensional electron density distribution is thereby reduced to the one-dimensional distance distribution function. This function is proportional to the number of lines with length, which connect any volume element i with any volume element k of the same particle. The spatial orientation of these connection lines is of no account to the function. The connection lines are weighted by the product of the number of electrons situated in the volume elements i and k , respectively. The correlation between the function and the structure of the particle is also discussed in the chapter. The connection between the distance distribution function and the measured experimental scattering curve is also shown. It is observed that the each distance between two electrons of the particle, which is part of the function, leads to an angular-dependent scattering intensity. This physical process of scattering can be mathematically expressed by a Fourier transformation, which defines the way in which the information in “real space” (distance distribution function) is transformed into “reciprocal space” (scattering function). The chapter also discusses monochromatization and the camera type developed in Graz.

[11] Small-angle x-ray scattering

The concept of spline functions is introduced as a new tool for resolution error correction in small-angle scattering. After discussing the approximating properties with respect to the estimation of experimental distributions and their derivatives from statistical data natural cubic splines are applied to the closed solution of the slit-height effect with Gaussian weighting functions including infinite slit height and to the correction of the slit-width effect based on Sauder's unfolding series. On the basis of an extended variational problem with one constraint taking adequately into account the statistical errors of the data a general procedure using cubic splines is developed for the solution of the integral equations associated with the three distinct types of resolution errors and arbitrary weighting functions. Computer simulations of small-angle scattering experiments have been performed to demonstrate the efficiency of the spline function algorithms in the different applications using scattering functions and slit-height and slit-width distributions of various kinds.

F. Hossfeld

Papers

Application of spline functions to the correction of resolution errors in small-angle scattering