Microscopic study of tetrahedrally symmetric nuclei by an angular-momentum and parity projection method

Abstract

We study the properties of the nuclear rotational excitations with hypothetical tetrahedral symmetry by employing the microscopic mean-field and residual-interaction Hamiltonians with angular-momentum and parity projection method; we focus on the deformed nuclei with tetrahedral doubly-closed shell configurations. We find that for pure tetrahedral deformation the obtained excitation patterns satisfy the characteristic features predicted by group-representation theory applied to the tetrahedral symmetry group. We find that a gradual transition from the approximately linear to the characteristic rigid-rotor, parabolic energy-vs.-spin dependence occurs as a function of the tetrahedral deformation parameter. The form of this transition is compared with the similar well-known transition in the case of quadrupole deformation. PACS numbers: 21.10. Re, 21.60.Jz, 23.20.Lv, 27.70.+q Symmetries play an important role in physics, often serving as guide-lines in studying characteristic features of motion of quantum systems. In particular, in nuclear physics, the spatial symmetries of the nuclear mean-field potential are crucial in determining both the properties of the independent particle motion and of the nuclear stability. The best known, the spherical symmetry of the potential, implies the high degeneracy of the single-particle level energies, the so-called magnetic degeneracy, (2j + 1), of the orbitals characterised by the spherical-shell angular-momentum quantum-number, j, and leads to the well-known spherical magic numbers in nuclei. The spontaneous breaking of the spherical symmetry arises when the energies of the spherical configuration of the system with certain particle numbers are higher as compared to the energies of the alternative non-spherical spatial distribution of nucleons, the mechanism to which the Pauli exclusion principle contributes importantly. The spherical symmetry breaking removes the (2j + 1)-degeneracy and leads to the deformed single-particle orbital scheme [1, 2].More generally, nuclei which are not spherical may acquire the forms governed by the point-group symmetries, some being more likely than the others -depending on the actual number of nucleons. In this context, it has been suggested, cf. Ref.[3] and references therein, that each symmetry group 'sufficiently rich' in terms of symmetry elements may lead to its proper scheme of magic numbers, which in turn can be seen as characterising such a symmetry group from the point of view of realistic realisations of the nuclear mean-field theories. In particular, a series of earlier publications related to point-group symmetries focused on the tetrahedral and octahedral symmetry groups which are the only ones that lead to an extra (four-fold) degeneracy of single-nucleonic levels in deformed nuclei and to an increased nuclear stability. In other words: The tetrahedral and octahedral symmetries are the only ones within which both the fourfold degeneracy ('unusual' case) appears for some levels and the two-fold degeneracy ('usual' case) appears fo...

Are There Signatures of Harmonic Oscillator Shells Far from Stability? First Spectroscopy of Zr110

“…Furthermore, the spectral systematics along the Zr isotopic chain allow a direct mapping of our strongest transitions to the 2 + and 4 + levels in 108 Zr, which exhibits a nearly identical ground state band [45]. For these reasons, we reject the interpretation of the ground state band as being built upon a tetrahedral deformed minimum as suggested by [27], however, we can not exclude the existence of an excited tetrahedral band, as in [30]. To further quantify the influence of the e↵ective interaction on the above observations, we performed a local sensitivity test within the 5DCH approach by increasing the spin-orbit term of the Gogny D1S interaction from -130 to -140 MeV fm 5 , with no change of the other terms of the interaction.…”

“…Our data also suggest that the ground state band we observe is not of a tetrahedral character. According to calculations by Tagami et al [30,31], the lowest-lying levels of the tetrahedral band in 110 Zr are predicted to be (3 ,4 + ,6 + ). However the low energy of the transitions we measure, combined with the fact that we observe the decay in flight, excludes the possibility that our most intense gamma ray results from E3 and higher multipolarity transitions.…”

“…This exotic symmetry, hitherto unobserved, is expected to compete strongly with deformed minima and its emergence is known to be very sensitive to pairing e↵ects [29]. If a tetrahedral configuration persists in the ground state of 110 Zr however, it would manifest a unique energy spectrum [30,31], distinguishable in a first-spectroscopy measurement. Meanwhile, most predictions of the 110 Zr ground state show well deformed prolate [32,33] or shape coexistent minima [34][35][36][37][38][39], though the exact structure is highly sensitive to the details of the e↵ective interaction [22].…”

Investigation of negative-parity states in Dy156 : Search for evidence of tetrahedral symmetry

“…Ongoing theoretical work based on group representation theory and microscopic solutions of the many-body problem [28,29] suggests that a near linear energy versus spin dependence should exist within tetrahedral bands. As none of the bands in 156 Dy meet this criterion, the existence of a tetrahedral minimum could not be confirmed from the present experiment.…”

Tetrahedral shapes of neutron-rich Zr isotopes from a multidimensionally constrained relativistic Hartree-Bogoliubov model

“…The possible existence of tetrahedral symmetry in light nuclei such as 16 O [52] as well as superheavy nuclei [53,54] was also proposed. Furthermore, the rotational properties of tetrahedral nuclei have been studied theoretically [55][56][57] which would certainly be helpful for experimentalists to identify tetrahedral nuclei.…”