Stability of the tetrahedral minimum

In order to study the stability of the tetrahedral minimum and to treat on the same footing the response of the nuclear system to rotation and pairing correlations, we performed the Hartree-Fock-Bogoliubov (HFB) calculations in the current implementation of the code HFODD (v2.17k)[13] for $^{110}$Zr, which shows[2] a pronounced tetrahedral minimum at $I = 0$. In our calculations we employed the SLy4[14] Skyrme force in the particle-hole channel and the density-dependent delta interaction in the particle-particle channel, whose intensity was fitted so as to reproduce the trend of experimental pairing gaps in the neutron-rich Zr isotopes, see Ref.[4] for details.

The calculations were performed for two families of rotation axes: (i) the axis passing through the middle of the edge of the tetrahedron, (ii) the axis passing through the tip of the tetrahedron. They will be referred hereafter as to edge and tip axes, respectively. These two axes are given a particular attention as they are symmetry axes of the rotating tetrahedron; a 4-fold axis with inversion (edge) and a 3-fold axis (tip), i.e. these symmetries commute with the cranking hamiltonian.

For the edge and tip axes, we performed the constrained HFB calculations. In the case of the edge axis we requested that the discrete antilinear $T$-simplexes, $S_{x}^{T}$ and $S_{y}^{T}$, be conserved. (We refer to Ref. [13] for a comprehensive review of the discrete symmetries implemented in the code HFODD). This was equivalent to conserving the $z$-signature symmetry, $R_{z}$. They were supplemented by full symmetry-unconstrained calculations, which was the only option available for the rotation about the tip axis. In the latter case, we also used the option to readjust the orientation of the rotation axis self-consistently in the course of the iterations. We did not notice (i) any change in the orientation of $\vec{\omega}$ fixed at the beginning of the iterations, at least up to the first single-particle crossing, (ii) any significative difference in energy between the various orientations of $\vec{\omega}$ and between the symmetry-constrained and symmetry-unconstrained case. This confirms a spherical-like behaviour of the rotating tetrahedron as claimed in Ref.[5].

Figure 1: Evolution of the deformation parameters in the tetrahedral minimum of $^{110}$Zr as function of the rotational frequency $\omega$ for the edge- (circles) and tip-axes (squares and diamonds). For both classes of axes, the quadrupole deformation remains equal to 0. For the tip-axis, the theoretical value of $\alpha _{33}$ in the exact $T_{d}^{D}$ limit is marked as the dashed line.

With the standard choice of the spherical harmonics for the calculation of the multipole moments, the edge-axis coincides with the z-axis of the body-fixed frame[6]. The deformations $\alpha_{\lambda\mu}$ extracted from the multipole moments can be plotted directly, cf. Fig. 1. The curve with plain circles shows the "tetrahedral" deformation $\alpha_{32}$. It is worth noticing that it is nearly constant as function of $\omega$, thus proving the resilience of the tetrahedral symmetry to the rotation.

In the case of the tip-axis, we chose to first perform a rotation of the nucleus at rest by the Euler angles $(\alpha,\beta,\gamma)$ in such a way that the new tip-axis coincides with the z-axis of the reference frame. This corresponds to an angle $\beta$ such that $\cos^{2}\beta = 1/3$, and $\alpha = \gamma = 0$. The multipole moments $Q_{\lambda\mu}$ must be transformed according to the general relation:

$$Q'_{\lambda\mu} = \sum_{\mu'} D^{*\lambda}_{\mu\mu'}(\alpha,\beta,\gamma)Q_{\lambda\mu}$$ (1)

where the $D^{*\lambda}_{\mu\mu'}(\alpha,\beta,\gamma)$ are the Wigner matrices. In our case, the initial deformation was characterized by $Q_{32} \neq 0$ and all other moments were null. After the rotation, only $Q_{30}$ and $Q_{33}$ are different from zero, and they are related by: $Q_{30}/Q_{33} = -\sqrt{10}/2$. We report in Fig. 1 the equivalent $\alpha_{30}$ and $\alpha _{33}$ deformations as function of the rotational frequency, together with the value of $\alpha _{33}$ in the exact symmetry limit. The little deviation from the exact symmetry case seems to increase slightly with $\omega$ although remaining very small.

In conclusion, we have investigated the rotation of tetrahedral nuclei using a fully-microscopic 3-dimensional cranking model. The results of self-consistent Skyrme Hartree-Fock Bogoliubov calculations show that the tetrahedral minimum remains as function of the rotational frequency and that practically no quadrupole polarization remains. The latter observation implies strongly hindered stretched-$E2$ transitions for the eventual rotational bands.

This work was supported in part by the exchange programme between IN2P3 (France) and Polish Nuclear Physics Laboratories, by the Polish Committee for Scientific Research (KBN) under Contract No. 1 P03B 059 27 and by the Foundation for Polish Science (FNP).