Dynamical theory of migration of an adsorbed atom on solid surfaces

<jats:p>A dynamical theory of an adsorbed atom on solid surfaces is developed starting from a Hamiltonian which includes: (1) motion of the adsorbed atom parallel to the surface; (2) the vibration of the adsorbed atom perpendicular to the surface; (3) the lattice vibrations (phonons) of the solid; (4) the coupling between the vibration of the adsorbed atom and its parallel motion; (5) the coupling between the lattice vibrations and the parallel motion of the adsorbed atom; (6) the concerted interaction among the phonons, the vibration, and the parallel motion. Using a canonical transformation, we describe the motion of the adsorbed atom as one of a pseudomolecule formed from the adsorbed atom and the distorted lattice. An equation for the probability of finding the adsorbed atom at a lattice site is derived and the mean square displacement of the motion of the adsorbed atom is calculated. At low temperature, the migration of the adsorbed atom is coherent and the mean square displacement is proportional to the square of time. The adsorbed atom behaves like a free particle with a certain velocity. At high temperature, the migration has diffusional character and the mean square displacement is proportional to time. This is due to the interaction between the parallel motion of the adsorbed atom and phonons as well as to the one between the vibration of the adsorbed atom and phonons. At intermediate temperature, after a certain time, which becomes shorter as the temperature increases, the migration becomes diffusional. The resulting diffusion coefficient consists of two parts: one represents the diffusional character of the migration, and the other its coherent nature. The temperature dependence of the diffusional part is of the Arrhenius type, while that of the coherent part increases as temperature decreases. The coherent part is small when the interaction between the adsorbed atom and the lattice atoms is very strong. Thus the theory gives reasonable trends with respect to the variation of parameters taken into account. Comparison with existing experiments on tungsten and rhodium is briefly made to estimate the order of magnitude of the lattice distortion due to the strong interaction between the adsorbed atom and the lattice atoms.</jats:p>