Eur. Phys. J. E (2016) 39: 113
https://doi.org/10.1140/epje/i2016-16113-7

Regular Article

Octupolar order in three dimensions

¹ Dipartimento di Matematica, Università degli Studi di Milano, via Saldini 50, I-20133, Milano, Italy
² Dipartimento di Matematica, Università di Pavia, via Ferrata 5, I-27100, Pavia, Italy

^* e-mail: giuseppe.gaeta@unimi.it
^** e-mail: eg.virga@unipv.it

Received: 22 September 2016
Accepted: 2 November 2016
Published online: 28 November 2016

Abstract

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