Problem 2 – Hint 2

The surface will take the shape of minimal energy by fixed volume. This means that any small volume-preserving deformation, in linear approximation (i.e. when we keep only terms with the deformation amplitude in the first power, and neglect all the terms by which the deformation amplitude has second or higher power), there is no change in the total energy of the crystal. (If the change of the total energy were to include a non-zero linear term, it would be possible to choose such sign of the deformation which would yield a decrease in the total energy implying that shape had not been optimal.) One way to deform the shape of the crystal is to change the positions of the steps in the x-directions while preserving the volume and keeping the number of steps intact; let us call it the step-shifting deformation. In the original EuPhO problem, it was shown that for such deformations to have no effect on the total energy in the linear approximation, the interaction energy between two steps must be inversely proportional to the squared distance between the two steps. Conversely, if the interaction energy between two steps is inversely proportional to the squared distance between the steps, the total energy change by step-shifting deformations is zero in the linear approximation if the distance d_n between two adjacent steps is proportional to n^{-1/3}, where n denotes the order number of the step. If you look at the solution of the EuPhO problem, you’ll see that this is the necessary and sufficient condition for the total energy change to be zero by step-shifting deformations, i.e. this is all the information what we can get by considering step-shifting deformations only! Therefore, in order to determine the factor \mu , you need to consider other deformation types.

Please note that this problem can be solved without knowing advanced mathematical techniques (like variational method or Lagrange multipliers), and as a matter of fact, solutions based on such techniques tend to be more complicated than those using simpler methods.

Be careful with the numerical prefactors, several (almost correct) solutions of the first two weeks were rejected because of a mistake by a factor of 1.5 or 2.