Frequency of collision between adatoms

Following Holland [1, 2], a localised atom is assigned to one of the localised quantum states, with energy less than E, associated with an adsorption site (see Fig. 1). Quantum states, whose energies are in excess of, embrace the whole surface and are approximated to be the states of a particle in a two-dimensional box of uniform potential. An adatom in one of these quantum states is able to move about the surface and encounter other adatoms, both localised and mobile, the effective collision diameter always being o,. 

Let us focus our attention on one such mobile adatom. In unit time, it will travel on average a distance c, and it will collide with all the adatoms whose centres are within a distance o, of the line defining its path. Since there is a total of (A/a), adatoms per unit area, each mobile adatom will have a collision frequency of 2 o, c, (Na),. Further, since it takes two adatoms to make a collision and there are (m A/a), mobile adatoms eligible per unit area, the total collision frequency per unit area is [2o, c, (Na) ] 

To estimate (m JVa), Holland supposes that the same vibrational motion 

The frequency (vp) of the two degenerate vibrational modes parallel to the surface of an adatom confined to an adsorption site is given to a first approximation by the relation 

A similar expression has been given by Ehrlich [3] who said that the frequency of hopping for each adatom would be proportional to exp (— Em /RT) and the chance that an adjacent site was already occupied would be (ATa) t /Ns, so 

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