IMPROVEMENTS TO THE RADIUS OF CONVERGENCE OF DIRECT INVERSION TECHNIQUES IN HELIUM ATOM SCATTERING

Recent studies (I. G. Shuttleworth, Surf. Rev. Lett. 14(2) (2007) 321) have demonstrated a direct method of diffraction pattern inversion for the helium atom scattering (HAS) experiment. The method requires the expansion of the Patterson function as a multivariate Taylor series to form a set of simultaneous equations. Benchmark tests of the procedure show that incomplete Taylor expansions introduce inconsistency into the set of simultaneous equations, whereas larger Taylor expansions attract significant numerical errors during their solution. The current work replaces the conventional matrix inversion techniques with a multidimensional Newton–Raphson algorithm. Tests have shown that the Newton–Raphson procedure removes the Taylor series and numerical limitations from the inversion technique for any realistic surface.