Surface diffusion with lateral interactions: a closed-form intermediate scattering function of an Ising adlayer from the memory-equation formalism

Abstract

The intermediate scattering function and its value at zero time, which is the static structure factor, are both characteristic functions. An exact equation of motion for the ISF carrying a local rate and a memory function is then obtained. To interpret this equation, the ISF is expressed in terms of relaxation modes. The local rate is fixed by a sum rule which reversibility turns into an equilibrium average. The de Gennes narrowing is establshed as a theorem. The memory function being non-negative, the closed form that results is a rigorous lower bound on the ISF at every momentum transfer and every time, and its accuracy is itself predicted. In the diffusive regime, the closed form keeps the single-exponential shape to which spin-echo data are routinely fitted, with both coefficients renormalized by the interaction: the amplitude becomes the SSF of the layer and the dephasing rate the dressed one-adsorbate rate divided by it, explicit in momentum transfer, time, coverage and temperature. When introducing the lateral interaction, a single correlation parameter fixes the equilibrium structure and the renormalized hop rate: exactly along a one-dimensional channel, and on the surface as the nearest-neighbour correlation of a pair approximation completed by a collective (random-phase) structure factor. The result contains no adjustable parameter and is exact at vanishing interaction for every coverage and momentum; the Darken relation comes with it, the correlation between successive moves separated off as a computable factor. Applications to Na/Cu(111) and H/Pt(111) are analysed and discussed.