Two-dimensional random walks of diatomic clusters
Authors: Wrigley John D. , Ehrlich Gert 
Published: July 2002

Diffusion on bcc (110) of dimers made up of two chemically different atoms is examined for a model in which motion occurs by transitions from a ground state, in which two atoms are in nearest-neighbor sites along 〈111〉, to an intermediate state with the atoms in adjacent sites along either 〈100〉 (horizontal) or 〈110〉 (vertical transition). In this representation, diffusion can be viewed as a random walk of the center of positions (COP) of the dimer atoms over a plane grid of stable and intermediate sites. Starting from the Kolmogoroff equation, a detailed stochastic analysis is carried out for horizontal transitions. This yields the mean-square displacement in terms of four elementary jump processes of the dimer; in addition, the moment generating
function and higher moments of the dimer displacements along the Cartesian coordinates x and y are also obtained. These quantities are found to differ for the two axes quite apart from differences in length scales: relaxation to the long-time limits occurs at different rates. For systems in which the population of dimers in the intermediate state is negligible, only the rates a and c at which the two different dimer atoms jump to the intermediate state affect the moments significantly; however, these rates are not separately accessible from observations of the mean-square displacement. The probability distribution for dimer displacements over (110) is therefore worked out explicitly. Displacements are referenced to coordinate axes χ and η along the close-packed 〈111〉 direction, along which COP motion is assumed to
occur. On this χ–η grid, the dimer executes a novel random walk, in which transitions from one axis to another are allowed only at intermediate-state positions; at ground-state positions, motion is restricted to one dimension. For short diffusion intervals, the probability distribution is found to be sensitively dependent on the jump rates a and c. The asymmetry of the distribution of displacements about 〈100〉 provides an immediate qualitative indication of any difference in the rates at which the chemically different dimer atoms jump out of the stable configuration. The actual value of these rates can be obtained from a quantitative comparison with experimentally measured distributions. Dimer diffusion involving vertical transitions, through an intermediate oriented along 〈110〉, is also considered, but more briefly. It is shown that the sum of the two rate constants for jumps into this
intermediate can be derived from observations of the rate at which flips occur from one orientation to another. The mean-square displacement under steady-state conditions is demonstrated to have the same general form as for horizontal transitions alone, but with an additional term for the effective jump rate out of the ground state via this new channel. Finally, the general formalism is adapted to the diffusion of simple dimers, of chemically identical atoms, both on bcc (110) and (100). For simple dimers which are not found in the intermediate state to any perceptible extent, a single jump rate suffices to define the mean-square displacement as well as the distance distribution function.