Instead of an expansion just in terms of $\exp(-|\epsilon_1|/k_BT)$, the expansion also is in terms of $\exp(\epsilon_2 /2k_BT)$. Hence, to take the NNN-expansion to the same order of magnitude as the NN-expansion, an unspecified number of terms is required, depending on the size of the ratio $\epsilon_2/\epsilon_1$.

Since the NNN Ising model cannot be solved exactly and we cannot generalize the low-$T$ expansion, we turn to an SOS model, which was used in earlier examinations of step problems, most notably in the seminal work of Burton, Cabrera, and Frank,[@BCF] and later used for steps of arbitrary orientation by Leamy, Gilmer, and Jackson.[@Leamy] It was also applied to an interface of arbitrary orientation in a square-lattice Ising model.[@Burk]

Although the SOS model can be treated exactly, the result is somewhat unwieldy. Fortunately, at low temperature—the appropriate regime for the experiments under consideration—the solution reduces to a simple expression.

Description of Model
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Consider a step edge of projected length $L$ separating an upper adatom-free region from a lower adatom-filled region (see Fig. \[fig:nnnbonds\]).

![ A finite-sized step edge whose projected length is $L$. The step has height $y_i$ at position $i$ ($0 \leq i \leq L$). The height difference $y_L - 