Scaling of the Step Position Distribution of Stepped Crystal Surfaces

Both Monte Carlo simulations and Scanning Tunneling Microscope images of stepped crystal surfaces can only include some finite length, $\Delta y$, along average direction of steps. This has important consequences, because the variance of the Step Position Distribution (SPD), $\sigma^2_Q(\Delta y)$, calculated from these ``snapshots'' depends on $\Delta y$. For $\Delta y \! < \! \xi_Q$, where $\xi_Q$ is the correlation length of the steps, $\sigma^2_Q(\Delta y) \! \propto \! (\Delta y)^{0.8}$; for $\Delta y \! > \! 4 \xi_Q$, $\sigma^2_Q(\Delta y)/\sigma^2_{Q, {\rm W}} \! \approx \! 0.158+ 0.318 ln(\Delta y)$, where $\sigma^2_{Q, {\rm W}}$ is the finite value of $\sigma^2_Q(\infty)$ predicted by the two-step approximation which yields the generalized Wigner distribution (GWD) for the Terrace Width Distribution (TWD). A ``Wigner length'', $L_{\rm W}$, can be defined by $\sigma^2_Q(L_{\rm W}) \! = \! \sigma^2_{Q, {\rm W}}$. It appears that $L_{\rm W} \! \approx \! 14.2 \xi_Q$ independent of the magnitude of step interaction. This is very close to a length which must be introduced to reproduce the GWD from an ensemble averge of Gruber-Mullins approximations of the TWD.