Problem 2 – Hint 4

So, when you consider shape deformations, you may add or remove the topmost (long flat) step, and adjust slightly the positions of the other steps so as to conserve the volume. It is clear that in the curved part of the crystal’s surface, the shape will still obey the same power law (as it was shown to be optimal), but the prefactor \lambda needs to take a new value. Also, the length of the new flat facet may differ slightly from the previous value. However, the exact length of the new flat facet is not very important here. Indeed, a small displacement \delta of any of the steps yields the same change \epsilon_n(\delta) in the interaction energy (see the previous hint), and so what defines the change of the total energy is the sum of all the step displacements, which in its turn, can be related to the volume change.