Stability of crystals

Abstract

Summarizing the results of the two last parts we see there are two cases for a BraveIs lattice of given parameters, (i) the equilibrium conditions cannot be satisfied, (ii) they can be satisfied. In the latter ease there are again two cases, stability or instability as determined by the second order terms of the potential energy. The fulfilment of the equilibrium conditions depends apparently on a sufficient number of symmetry elements. The triclinlo cell leads to no equilibrium even if it is rhombic, apart from those cases which have symmetries (the three cubic lattices}: or a prism with a quadratic or rhombic base, leads to no equilibrium except for a certain ratio of the perpendicular axis to the plane ones. Moat of the equilibrium con¬ figurations are unstable. According to our result, it is very probable that the faee-centred lattice is the only Bravais lattice stable under central forces. This result seems rather trivial, as central forces of the type considered are almost equivalent to rigid sphere^, which arrange themselves of course in the densest packing. The purpose of this investigation was more to develop suitable methods which might be applied to more complicated cases.