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 real-valued potential [Formula: see text] defined on the unit sphere [Formula: see text] by A. Generalized eigenvalues of A and critical points of [Formula: see text] are equivalent means to describe octupolar order in a molecular assembly according to Buckingham's formula for the probability density distribution. Intuition suggests that [Formula: see text] 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 [Formula: see text]. The two generic states are divided by a separatrix manifold, which may physically represent an intra-octupolar transition.