An effective procedure to determine corrugation functions from atomic beam-diffraction intensities

A computational method is described, which, starting from given difraction intensities, approaches effectively the best-fit corrugation function $\text{ζ(}\text{R}\text{)}$. Because of the approximations involved, the procedure works well for smooth corrugations with amplitudes not exceeding ～10% of the lattice constant. The method rests on two crucial observations: (i) With the full knowledge of the scattering amplitudes $\text{A}{}_{\mathrm{G}}\text{= ¦A}{}_{\mathrm{G}}\text{¦}\text{exp}\text{(}\text{i}\text{?}{}_{\mathrm{G}}\text{)}$ (absolute values plus phases), the corrugation function can be calculated to a high degree of accuracy from $\text{ζ(}\text{R}\text{) = (2}\text{i}\text{k}{}_{\mathrm{i}}\text{)}{}^{\mathrm{?1}}$ In $\text{¦?ΣA}{}_{\mathrm{G}}\text{exp}\text{(}\text{i}\text{G·R}\text{)¦}$ which is derived easily from the hard corrugated wall scattering (HCWS) equation by approximating $\text{k}{}_{\mathrm{G}}$ by $\text{?}\text{k}{}_{\mathrm{i}}$ ($\text{k}{}_{\mathrm{i}}$ and $\text{k}{}_{\mathrm{G}}$ being the wavevectors of the incoming and diffracted beams, respectively), (ii) With only the $\text{¦A}{}_{\mathrm{G}}\text{¦'s}$ (or intensities) known, approximate solutions of the HCWS equation can be obtained with a rough estimate of the relative phases of only a few intense diffraction beams; the estimate is readily performed by investigating systematically a coarse mesh of phases. In this way, approximate corrugations are found with which a full set of phases can be generated, which allows the calculation of an improved ζ(R); this step is repeated in a loop, until optimum agreement between calculated and given intensities is obtained. The effectiveness of the procedure is demonstrated for three one-dimensional model corrugations described by several Fourier coefficients. The method is finally applied to the case of H₂ diffraction from the quasi-one-dimensional adsorbate corrugation Ni(110) + H(1 × 2).