Computed Fourier series representations of the interaction potential of an argon atom on an argon crystal surface

Abstract

Undoubtedly the quest for a Fourier series representation of the adatomsurface interaction potential at a crystal surface is of paramount importance. Reliable knowledge of the Fourier coefficients is a prerequisite for progress in the solution of many problems in surface physics. Such problems are for example, scattering from crystal surfaces, the evaluation of the thermodynamic properties of adsorbed atoms, phase transitions in adsorbed layers and epitaxy. The purpose of this study was to calculate the relevant Fourier coefficients of the equilibrium interaction potential øe (x,y) and for the equilibrium height zmin (x,y) of an argon atom above a (001) argon crystal surface using a Lennard-Jones (6-12) potential. The extent of the numerical computations necessitated truncation of the Fourier series. The form of the series for øe (x,y), subjected to the symmetry properties imposed by the substrate, was derived analytically using a purely mathematical approach and the more physical approach of the reciprocal lattice formalism. A computer program was written to evaluate the equilibrium interaction energy øe (x,y) at any point in a two-dimensional unit cell above the crystal substrate. Truncation of the crystal at 5a (a being the lattice parameter} was also necessitated by computational limitations and was justified by the fact that increasing the radius from Sa to 6a altered øe (x,y) at the origin by less than 0,3%. All calculations were carried out using J44 data points; using more data resulted in negligible increase of accuracy. Inclusion of higher order harmonics do not significantly influence the values obtained for the lower order coefficients. The Fourier coefficients converge rapidly to zero with harmonic order; the magnitudes of fifth order coefficients are only about 0,04% of those of the first order coefficients. Values for coefficients in a one-dimensional section, obtained by putting y=0 in the two-dimensional representation, were verified using analytical one-dimensional expressions derived by Hildebrand. The calculated desorption energy Ed and the surface migration activation energy Ea of an adsorbed atom compared well with those a calculated by Bacigalupi and Neustadter, Ed and Ea can be obtained using Fourier approximations truncated at low orders. In order to obtain fair results for the value of the lateral force constant kxy' at least third order harmonics must be included in the approximation. The coefficients in the computed Fourier series representation of the equilibrium height also converge to zero fairly rapidly, but with more variance in the magnitudes of the higher order coefficients. This is possibly partly due to rounding off errors. It was shown that the equilibrium height at an adsorption site was 0,52815a which represents a slight outward relaxation of the (001) surface planes. This agrees well with previous work by L.C.A. Stoop.