y_0$ is fixed; thus, the step edge makes an angle $\theta$ with the horizontal axis, and has an overall slope $m$ (shown as the top of the gray region). The energy of the step edge is found by counting the number of broken links required to form it. Here all NN and NNN broken links are shown. ](nnnbondsColor.eps){width="8"}

[\[fig:nnnbonds\]]{}

The step edge is completely described by specifying its height $y_i$ at position $i$ ($0 \leq i \leq L$). The energy of the step edge depends on the number of broken bonds required to form it. Let $V$ and $H$ represent the vertical and horizontal NN bond strengths divided by $k_B T$, and let $U$ and $D$ represent up-diagonal and down-diagonal NNN bond strengths over $k_B T$. Then the step-edge energy $E \equiv E(\{\Delta_i\})$ depends only on $\Delta_i \equiv y_i-y_{i-1}$.

For clarity, we consider two examples. First, if $\Delta_i=3$ (as is the case between columns a and b in Fig. \[fig:nnnbonds\]), then between positions $i$ and $i+1$ there are $3$ broken $H$-links, $2$ broken $U$-links, and $4$ broken $D$-links. There are also $2$ broken $V$-links, but this number is independent of $\Delta_i$, since every step-edge configuration of projected length $L$ requires 