the absence of NNN interactions ($\epsilon_2 \! =\! 0$).[@rottman; @Ising45; @Z00]

By considering just the lowest and second lowest energy configurations,[@ZP1; @ZP2] Zandvliet et al. obtained the result[@ZP2] (expressed with our sign convention for $\epsilon_2$) for the maximally misoriented case $m \! =\!1$

$$\begin{aligned}
  \label{eq:ZVP}
     \frac{k_B T}{\tilde{\beta}a}=\frac{\sqrt{2}}{1 + e^{-\epsilon_2/2k_BT}},\end{aligned}$$

which has, for the attractive $\epsilon_2$ of primary concern here, some qualitative similarities to Eq. (\[eq:m=1\]) (including the value $1/\sqrt{2}$ for $\epsilon_2 \! =\! 0$) but is too small by a factor of 2 for $\epsilon_2/2k_BT \ll 0$; even the coefficient of the first-order term in an expansion in $\epsilon_2/2k_BT$ is half the correct value. For the opposite limit of repulsive $\epsilon_2$, Eq. (\[eq:ZVP\]) levels off (at $\sqrt{2}$), in qualitative disagreement with the actual exponential increase seen in Eq. (\[eq:m=1\]).

![ The range of validity of Eq. (\[eq:nnnstiffness\]) is examined by comparing it to exact numerical solutions of the SOS model at several temperatures. In the legend $T_c$ refers to the NN lattice-gas (Ising) model; for $|\epsilon_1|= 256$meV, $T_c = 1685$K.](nnnCompare.eps){width="9"}

[\[fig:nnnCompare\]]{}

![ Eq. (\[eq:nnnstiffness\]) is plotted for a variety of different values of $D = -\epsilon_2/2k_B T$, where $\epsilon_1$ and $\epsilon_2$ are NN- and NNN-interaction energies, respectively, in a lattice-gas 