Crystallization in a one-dimensional periodic landscape

We consider the crystallization problem for a finite one-dimensional collection of identical hard spheres in a periodic energy landscape. This issue arises in connection with the investigation of crystalline states of ionic dimers, as well as in epitaxial growth on a crystalline substrate in presence of lattice mismatch. Depending on the commensurability of the radius of the sphere and the period of the landscape, we discuss the possible emergence of crystallized states. In particular, we prove that crystallization in arbitrarily long chains is generically not to be expected.