On energy ground states among crystal lattice structures with prescribed bonds

Abstract

Abstract We consider pairwise interaction energies and we investigate their minimizers among lattices with prescribed minimal vectors (length and coordination number), i.e. the one corresponding to the crystal’s bonds. In particular, we show the universal minimality—i.e. the optimality for all completely monotone interaction potentials—of strongly eutactic lattices among these structures. This gives new optimality results for the square, triangular, simple cubic (sc), face-centred-cubic (fcc) and body-centred-cubic (bcc) lattices in dimensions 2 and 3 when points are interacting through completely monotone potentials. We also show the universal maximality of the triangular and fcc lattices among all lattices with prescribed bonds. Furthermore, we apply our results to Lennard-Jones type potentials, showing the minimality of any universal minimizer (resp. maximizer) for small (resp. large) bond lengths, where the ranges of optimality are easily computable. Finally, a numerical investigation is presented where a phase transition of type ‘square–rhombic–triangular’ (resp. ‘sc-rhombic-bcc-rhombic-fcc’) in dimension d = 2 (resp. d = 3) among lattices with more than 4 (resp. 6) bonds is observed.