Locally minimal Epstein zeta functions

Extract

One of the problems of solid state physics is to explain why the atoms of certain elements (such as iron) arrange themselves in a body-centred cubic lattice, rather than in the much denser face-centred cubic lattice packing [8]. Recently, we discovered the geometric significance of these body-centred lattice packings: they are fragile in the sense that, assuming we have a sphere of fixed radius at each lattice point, any perturbation which does not alter the set of nearest neighbours of any sphere results in a denser configuration [3]. In other words, in-these packings it is the density of the interstitial void between the spheres that is being maximized. This fact might seem only to deepen the mystery of why such arrangements ever appear in nature at all. Our purpose here is to demonstrate that these lattice packings are in fact quite “stable”, provided one seeks a more subtle form of stability, one prompted by physical rather than purely geometrical considerations. More precisely, we will show for those lattices which possess a high degree of symmetry–and for the bodycentred cubic lattice in particular–that the EPSTEIN ZETA FUNCTION of the lattice, ∑ |r|^-2s (where |r| is the distance of the r-th lattice point from the origin and s is any fixed number greater than n/2), is locally minimized, in the sense that, any lattice, obtained by a perturbation which does not alter the set of nearest neighbours of any lattice point, has an Epstein zeta function of larger value, for any s > n/2.