Statistical mechanics and the partition of numbers II. The form of crystal surfaces

Abstract

The classical theory of partition of numbers is applied to the problem of determining the equilibrium profile of a simple cubic crystal. It is concluded that it may be thermo-dynamically profitable for the surface to be ‘saw-toothed’ rather than flat, the extra entropy associated with such an arrangement compensating for the additional surface energy. For both a two- and a three-dimensional ‘saw-tooth’ the extra entropy varies, to a first approximation, in the same way as the surface energy, i.e. is proportional to or respectively, where N is the number of molecules in a ‘tooth’. For the simple cubic lattice, the entropy associated with the formation of a tooth containing N atoms is estimated to be 3.3 It is also possible to estimate the variation of the ‘equilibrium roughness’ of a crystal with temperature, if its surface energy is known.