The interaction of atoms and molecules with solid surfaces - XI—The dispersal of energy from an activated link

One of the problems which has been considered in this series of papers (Lennard-Jones, Strachan, Devonshire, Goodwin 1935-7) is the activation of the vibrational states of an atom adsorbed on a surface and the converse problem of the deactivation from one vibrational state to another. Various mechanisms have been examined, such as the interaction of the adsorbed atom with the thermal vibrations of the solid and also with the conduction electrons of a metal, and in each case it has been found that deactivation occurs in a time which is of the order of a few vibrations of the adsorbed atom. In this connexion it is instructive to examine a simple problem which is of a similar type and can be solved by classical methods. Let the positions of a number of similar atoms forming a one-dimensional chain when in equilibrium be denoted by x = na, n = 0, ±1, ±2, . . . , and let ξn denote the lateral displacement of the nth atom from its equilibrium position. When the relative displacements are small, the potential energy due to a relative shift of two neighbouring particles can be written as ½k(ξn - ξn-1)2, and the energy of the system when in a state of vibration is ∑½mξn2 + ∑½k(ξn - ξn-1)2.