Crystalline Symmetry and Surface Tension

The fundamental mechanisms causing crystalline symmetry in matter at low temperature and pressure are not yet understood’), though there has been some recent progress”‘4. In particular in”,“) we considered two-dimensional models of particles interacting through a two body force consisting of a hard core and short range attraction (modelling a molecular force). One expects the ground state of such a system to consist, away from boundaries, of a perfect crystal with each particle surrounded by six neighbors. The only difficulty is that for a finite system (at zero pressure) such a crystal would have a regular hexagonal boundary and a surface tension tending to make the boundary circular, while a circular boundary is incompatible with a perfectly crystalline interior. The problem then is whether or not the surface tension is strong enough to overcome the tendency to crystalline ordering. We showed it is not, but through delicate inequalities which do not give one a feeling for how the problem would turn out under other circumstances, such as ionic or metallic bonding, or in three dimensions. In fact it was conceivable that a simpler approach might have shown surface tension inherently incapable of affecting crystalline symmetry. This paper consists of an analysis of another two-dimensional model, a type common in studying antiferromagnetism though we prefer to think of it as modelling ions. We show that its ground state has a natural crystalline symmetry which, however, is indeed broken by surface tension. Thus surface tension is now proven to be an important aspect of the crystal problem.