Introduction

The prediction that atomic nuclei can possess stable low-lying configurations with the tetrahedral symmetry has been confirmed by several independent calculations using a variety of nuclear mean-fields[1,2,3,4]. However, no experimental evidence has been reported so far, which can be put down to the lack of understanding of the excitation mechanisms in such exotic systems. We investigate here the collective rotation, since the deep minima and large barriers reported e.g. in Ref.[2] suggest tetrahedral nuclei may be amenable to sustain rotational bands. This article is a sequel to previous works where the structure and moments of inertia of tetrahedral collective bands were analyzed[5], and the consequences of the collective rotation in terms of symmetries and quantum numbers were discussed[6]. To complement these studies, we wish to focus here on the stability of the tetrahedral minimum as a function of the rotational frequency, which is one of the pre-requisites to an eventual observation of rotational bands.

Today there exists several articles in the literature about the nuclear tetrahedral symmetry (associated to the point group of symmetry $T_{d}^{D}$) and we refer the reader interested in a general introduction to this subject to e.g. Ref.[7]. For the purpose of the present study, it suffices to recall that a nucleus with a non-axial $\alpha_{32}$ octupole deformation (in the standard expansion of the nuclear radius on the basis of spherical harmonics, with deformation parameters $\alpha_{\lambda\mu}$) has all the symmetries of a regular tetrahedron. Although other realizations of the group $T_{d}^{D}$, involving higher-order multipole terms with $\lambda\geq 7$, also appear to generate stable minima in the potential energy landscape[8], they will not be considered in this article.

The cranking model was the subject of several comprehensive review articles and we refer to Refs.[9,10] for a general discussion of its main features. We just recall here how the model should be adapted in the 3-dimensional case: the rotation is described with the help of 3 Lagrange multipliers $\vec{\omega}\equiv (\omega_x, \omega_y, \omega_z)$ which are interpreted as the classical rotational frequencies along the x-, y- and z-axis of the body-fixed frame respectively. Equivalently, one can choose the spherical representation in which the rotational frequency vector $\vec{\omega}$ is parameterized by $(\omega, \theta, \varphi)$.

It is known that a collective rotation of a nucleus with prolate or oblate deformation takes place about an axis perpendicular to the symmetry axis. However, in the case of a nuclear shape with the $T_{d}^{D}$ symmetry, the quadrupole deformation is equal to zero, and consequently no simple criterion exists to determine the sorientation of the rotation axis.

A possible procedure to find optimum rotation axis consists in computing the total energy as a function of the orientation of the rotation axis, characterized by the two angles $(\theta,\varphi)$, for different rotational frequencies $\omega$, as was done in Refs.[5,6]. One thus obtains two-dimensional maps whose minima signal the energetically-favored axes of rotation. These calculations were performed using a macroscopic-microscopic technique, in which the total energy is the sum of a liquid-drop contribution parameterized as in Ref.[11] and shell-correction extracted from a Woods-Saxon potential with the form defined in Ref.[12]. The results[5,6] suggest that at low rotational frequencies (up to $\omega \sim 0.3$ MeV/$\hbar$), no particular axis of rotation is favored, while at higher frequencies, several well-defined minima emerge. However, these calculations assumed a fixed tetrahedral deformation and did not include pairing correlations: therefore, in the present study we want to see whether the tetrahedral minimum survives the increase of angular momentum.