Numerical analysis as applied to x-ray scattering curves.

Abstract

Various numerical methods involving polynomials were employed for both interpolation and linear least-squares curve fitting of the atomic scattering factors of X-ray diffraction. This included use of Lagrangian and orthogonal Legendre polynomials, as well as cubic spline and Stineman interpolation.

Interpolation is a method that uniquely matches known data points within small segments of an unknown curve to approximate points in between. Special emphasis was placed on establishing both the minimum grid spacing and polynomial degree that are needed to perform accurate interpolations. The grid spacing in a region of the x-argument near 1.0 A{dollar}\sp{lcub}-1{rcub}{dollar} that is commonly employed in standard tabulations of X-ray scattering factors was found to be too coarse for wholly accurate interpolation by polynomials of low degree.

Least-squares curve fitting is a procedure that approximates the entire unknown curve with an analytical function that matches the known data points as closely as possible by minimizing the sum of their squared deviations from the fitted curve. Analytical representations of X-ray scattering curves are advantageous, since otherwise the complete scattering table must be stored and the values of individual scattering factors derived by interpolation. The literature of previous nonlinear least-squares fits of Gaussian expansions and polynomial series to approximate X-ray scattering curves is reviewed exhaustively. Several transformations were tested for linear least-squares curve fitting of X-ray scattering factors. It was concluded that the best approximation is obtained from a combination of Gaussian and polynomial terms.

A special chapter on the historical impact of numerical analysis by computers in modern chemical education is also included.