formation of Cu islands implies $\epsilon_1 \! <\! 0$); thus, as necessary, $\varepsilon_k \! >\! 0$. Note that for clarity we reserve the character $\epsilon$ for lattice-gas energies,[@LG] which are deduced by fitting this model to energies which can be measured, such as $\varepsilon_k$.

The goal of this paper is to compute the step line tension $\beta$ and the stiffness $\tilde{\beta}$ as functions of azimuthal misorientation $\theta$, when NNN (in addition to NN) interactions contribute. Since it is difficult to generalize the low-temperature expansion of the Ising model,[@rottman; @avron] we instead study the SOS (solid-on-solid) model, which behaves very similarly at low temperatures and at azimuthal misorientations that are not too large, but can be analyzed exactly even with NNN interactions. This derivation is described in Section II, with most of the calculational details placed in the Appendix. In Section III we derive a simple expression for the stiffness in the low-temperature limit, presented in Eq. (\[eq:nnnstiffness\]). We also make contact with parameters relevant to Cu(001), for which this limit is appropriate. In Section IV we extend the formalism to encompass the presumably-strongest trio (3-atom, non-pairwise) interaction, showing that its effect can be taken into account by shifting the pair 