The intermediate scattering function of an interacting adlayer as a characteristic function: a closed-form theory of Ising lattice-gas surface diffusion

Abstract

The intermediate scattering function (ISF) measured by helium spin-echo is a characteristic function (CF): the Fourier transform of the distribution of adsorbate displacements, whose value at zero time is the static structure factor. We use this double role to bring lateral interactions of Ising lattice-gas type into surface diffusion. The ISF then follows in closed form in momentum transfer, time, coverage and temperature, and with it the whole linear-response hierarchy. A single correlation parameter fixes both the equilibrium structure and the interaction-renormalized hop rate; it is exact along a one-dimensional channel, and is carried onto the surface as the nearest-neighbour correlation of a pair approximation, the momentum dependence of the structure factor then coming from a collective (random-phase) treatment. The closed form is not postulated. Projecting the master equation onto the density gives an exact equation of motion whose amplitude is an identity and whose initial rate obeys a sum rule, so that the quasi-elastic weight is the layer's own structure factor and de Gennes narrowing becomes a theorem rather than an analogy; the closed form is what remains when the memory kernel of that equation is dropped. Kinetic Monte Carlo (KMC) on the same Hamiltonian reproduces the dressed hop rate to better than about one per cent and the initial collective rate to $1$--$2\%$. Four correlated, non-critical fingerprints of the interaction follow --- in the quasi-elastic amplitude, its linewidth, the Arrhenius slope, and the dependence of the fitted linewidth on the range of times over which the decay is fitted. They invert in closed form for the attempt frequency, the static barrier and the interaction. Applications to Na/Cu(111) and H/Pt(111) are analysed and discussed.