Diophantine Approach to the Classification of Two-Dimensional Lattices: Surfaces of Face-Centered Cubic Materials

The long-range periodic order of a crystalline surface is generally represented by means of a two-dimensional Bravais lattice, of which only five symmetrically distinct types are possible. Here, we explore the circumstances under which each type may or may not be found at the surfaces of face-centered cubic materials and provide means by which the type of lattice may be determined with reference only to the Miller indices of the surface; the approach achieves formal rigor by focusing on the number theory of integer variables rather than directly upon real geometry. We prove that the {100} and {111} surfaces are, respectively, the only exemplars of square and triangular lattices. For surfaces exhibiting a single mirror plane, we not only show that rectangular and rhombic lattices are the only two possibilities, but also capture their alternation in terms of the parity of the indices. In the case of chiral surfaces, oblique lattices predominate, but rectangular and rhombic cases are also possible and arise according to well-defined rules, here partially recounted.