Hop rates of H and D on Pt(111) at finite coverage: the memory equation and the collective friction

Helium scattering measures the coherent intermediate scattering function of an adlayer, the characteristic function of the separation between adparticles at two times. Published dephasing rates of H and D on Pt(111) are analysed with its memory equation and a nearest-neighbour Ising lattice gas. The measured rate belongs to the adlayer. The hop rate of a single adsorbate is extracted from it by dividing out an interaction factor, without a model of the barrier. This factor lies between $0.97$ and $1.17$ at $0.1$~ML, so that the published single-adsorbate analyses hold there. At $0.4$~ML and $80$~K it lies between $1.2$ and $4.7$, depending on how the occupied neighbours modify a tunnelling jump. The memory function is interpreted as a friction, the collective friction, which the adlayer exerts on its own density fluctuations. The lateral coupling, fixed by the coverage series, is the free energy of a neighbouring pair, with a vibrational pair entropy. The small deformation of the measured curves from the Chudley--Elliott form, the tilt, is proposed as an observable of the coupling. The tilts at $0.15$ and $0.2$~ML are consistent with it, although they are hard to extract from the published curves. The memory equation provides a transition-state theory for the adlayer, with the collective friction in the role of the friction of the substrate and without divergence at the crossover. Combined with the standard quantum transition-state theory of a single adsorbate, the extracted hop rates give the friction of the substrate, $124$ and $75$~ps$^{-1}$ for H and D, independent of the coverage.