picture. The solid curve denoted “Ising NN" corresponds to $\epsilon_2 \! =\! 0$. The dots labeled “Exp’t" are taken from Fig. 2 of Ref.  and were derived from the equilibrium shape of islands on Cu(001) at 302K, with the line segments serving as guides for the eye. To minimize clutter, we omit similar data derived from correlation functions of vicinal surfaces at various temperatures. Note that for $\epsilon_2 \! =\! \epsilon_1/4$ a maximum has developed near $\tan \theta \! =\! 1/2$ that is not evident in the experimental data. ](nnnplot.eps){width="9"}

[\[fig:nnnplot\]]{}

Fig. \[fig:nnnCompare\] compares Eq. (\[eq:nnnstiffness\]) to corresponding exact solutions \[found by numerically solving Eqs. (\[eq:gtostiffrho\]), (\[eq:ddgfinal\]), and (\[eq:rhotom\])\] at several temperatures when $\epsilon_2 = \epsilon_1/10$. We see that Eq. (\[eq:nnnstiffness\]) overlaps the exact solution at temperatures as high as $T_c/6$. As the temperature increases, the stiffness becomes more isotropic, and Eq. (\[eq:nnnstiffness\]) begins to overestimate the stiffness near $\theta = 0^\circ$.

Finally, in Fig. \[fig:nnnplot\] (using the experimental value[@GI] $\varepsilon_k = 128$ meV $\Rightarrow \epsilon_1 = -256$ meV), we compare Eq. (\[eq:nnnstiffness\]) to the NN-Ising model at $T=320$K, as well as to the experimental results of Ref. . For strongly attractive (negative) $\epsilon_2$, $k_B T / \tilde{\beta}a$ decreases significantly. 