In recent observations of Brownian motion of islands of adsorbed atoms and of vacancies with mean radius R, the cluster diffusion constant ${\mathit{D}}_{\mathit{c}}$ varies as ${\mathit{R}}^{\mathrm{-}1}$ and ${\mathit{R}}^{\mathrm{-}2}$. From an analytical continuum description of the cluster’s steplike boundary, we find a single Langevin equation for the motion of the cluster boundary, rather than three special cases. From this we determine ${\mathit{D}}_{\mathit{c}}$ and the correlation function ${\mathit{G}}_{\mathit{sh}}$ for fluctuations of the shape around an assumed equilibrium circular shape. In three limiting cases we find the scaling relations ${\mathit{D}}_{\mathit{c}}$∼${\mathit{R}}^{\mathrm{-}\mathrm{\alpha }}$ and, at early elapsed time t, ${\mathit{G}}_{\mathit{sh}}$∼${\mathit{t}}^{1/(1+\mathrm{\alpha })}$, where α=1, 2, and 3, corresponding to the three generic surface mass-transport mechanisms of straight steps. We thereby provide a unified treatment of the dynamics of steps and of clusters. To check how well the continuum results describe clusters of the size in experiments, we perform Monte Carlo simulations of simple lattice gas models. Further, we estimate atomic diffusion parameters from the available experimental data on diffusion of large clusters. © 1996 The American Physical Society.

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