Octupolar order in three dimensions

Octupolar order in three space dimensions is described by a real-valued, fully symmetric and traceless, third-rank tensor A. The real generalized eigenvalues of A are also the critical values of a realvalued potential Φ defined on the unit sphere S2 by A. Generalized eigenvalues of A and critical points of Φ are equivalent means to describe octupolar order in a molecular assembly according to Buckingham’s formula for the probability density distribution. Intuition suggests that Φ would generically have four maxima, corresponding to the most probable molecular orientations, so that a (possibly distorted) tetrahedron would effectively describe A. This paper shows that another generic octupolar state flanks the expected one, featuring three maxima of Φ. The two generic states are divided by a separatrix manifold, which may physically represent an intra-octupolar transition.