Determining surface relaxation from low-energy electron diffraction via a transform method

Abstract

A method for determining the atomic structure of a clean metal surface from low-energy-electron diffraction data is presented using the Fourier transform of the beam intensity as a function of electron momentum (the Patterson function). A model Patterson function which is made up of the convolution product of a window transform function and a series of $\delta$ functions representing the crystal layer spacing is fitted in a least-squares sense to the Patterson function of the data. By varying the positions of the $\delta$ functions and examining the quality of the fit, the structure can be determined. The method is fully illustrated using model kinematic data from a surface which is both relaxed and unrelaxed. The approximations necessary to obtain a perfect fit between the Patterson function of the kinematic data and the convolution product are discussed showing that even ideal data cannot be analyzed to give an "exact" solution. Nevertheless, it is shown that the method is capable of determining structure even from data that are highly dynamic. This is illustrated by analyzing multiple-scattering data for W(110) which have been calculated by Van Hove and Tong. Four nonequivalent beams are analyzed for inward relaxation, outward relaxation, and no relaxation. The effect of using data with different energy ranges is discussed as well as different prescriptions for choosing the window transform function.