form on the Cu(001) surface, it is obvious that $\epsilon_1$ is attractive. If $\epsilon_2$ is also attractive, as required for reduction of the overestimate of $k_BT/\tilde{\beta}$, then the low-temperature equilibrium shape has clipped corners (octagonal-like, with sides of alternating lengths), as noted in Ref. ; no evidence of such behavior has been seen. The lack of evidence of a decreasing stiffness near $\theta \approx 45^\circ$ suggests that $\epsilon_2/\epsilon_1$ is at most 1/5.

There is implicit experimental information for $\epsilon_2$: from island shapes[@Ising45] and fluctuations[@SGVI] $\beta(0) = 220 \pm \! 11\!$ meV. Since related measurements showed $\scriptstyle{\frac{1}{2}}\textstyle{\epsilon_1} \!=\! -128$meV, we deduce $\epsilon_2 \!=\! -92 \!$ meV if $\epsilon_{RT}$ is insignificant. These values imply that $\epsilon_2/\epsilon_1$ is somewhat larger than 1/3, which seems unlikely in light of the unobserved predictions about the shape of islands in that case (cf. the end of Section III).

To corroborate this picture, one should estimate the values of $\epsilon_1$ and $\epsilon_2$, as well as $\epsilon_{RT}$, from first-principles total-energy calculations. In contrast to Cu(111),[@Bogi; @Feibel] however, no such information even for $\epsilon_1$ has been published for Cu(001); there are, however, several semiempirical calculations which found $\varepsilon_k \approx 0.14$eV.[@semi] In such calculations based on the embedded atom method 