]>PLB34428S0370-2693(19)30090-510.1016/j.physletb.2019.02.001The AuthorsTheoryFig. 1Mean-field energy E, projected energy EPROJ=E1I=0 in function of the quadrupole moments Qˆ20 (left) and Qˆ22 (right), calculated for 16O by the SkV, SLY4, SIII, and SKP functionals.Fig. 1Fig. 2Same as Fig. 1, but for octupole moments Qˆ30, Qˆ31, Qˆ32 and Qˆ33. The quadrupole moments are set to zero during these calculations.Fig. 2Fig. 3The quadrupole moments against the octupole moments, during the HF variations for the calculation with constraints of octupole moments.Fig. 3Fig. 4Potential energy surfaces in the (Qˆ20, Qˆ32) (left) and (Qˆ22, Qˆ32) (right) plane, calculated by SkV functionals. The contour scale is 1.0 MeV for the mean-field energy E (a) and (b), and 0.2 MeV for the projected energy EPROJ (c) and (d).Fig. 4Fig. 5Same as Fig. 4, but for energy surfaces in the (Qˆ30, Qˆ32), (Qˆ31, Qˆ32), and (Qˆ33, Qˆ32) plane. The contour scale is 1.0 MeV for the mean-field energy E (a), (b) and (c), and 0.15 MeV for the angular-momentum projected energy EPROJ (d), (e) and (f).Fig. 5Fig. 6(a–c) Density distributions of 16O, with the pure moment Qˆ32=40fm3 (β32 = 0.339), calculated by SkV functional, in (x, y), (y, z), (z, x) planes, respectively. The contour scale is 0.01 fm−3, and the third axis left, i.e., z, x, y is fixed at 0.161 fm. (d) The 3D-density distribution of 16O at the density ρ = 0.15 fm−3.Fig. 6Fig. 7The experimental ground-state band of 16O, taken from the observed spectrum [34] and organized by Ref. [4], is given for a comparison with calculations (labeled as “Exp.”). Calculated ground-state band of 16O from the cranking-HF solution of tetrahedral symmetry with Q32 = 40 fm3 with cranking axis as x, y, z and the axis in x − y plane (π/4 degree between x and y axis), are labeled as “Crank(x)”, “Crank(y)”, “Crank(z)” and “Crank(xy)” respectively. The projection energies after the convergence of CHF with different cranking axis are also given (labeled as “PROJ(x)”, “PROJ(y)”, “PROJ(z)”, and “PROJ(xy)” correspondingly).Fig. 7Tetrahedral symmetry in the ground state of 16OX.B.Wanga⁎xbwang@zjhu.edu.cnG.X.DongaZ.C.GaobY.S.ChenbC.W.ShenaaSchool of Science, Huzhou University, Huzhou 313000, ChinaSchool of ScienceHuzhou UniversityHuzhou313000ChinabChina institute of atomic energy, Beijing 102413, ChinaChina institute of atomic energyBeijing102413China⁎Corresponding author.Editor: W. HaxtonAbstractBased on the Skyrme energy density functional, the self-consistent HF calculations have been performed for 16O, and the results show that the double point group tetrahedral symmetry TdD may play an important role in the configuration of many-body fermion system in the ground state of 16O. The corresponding total density distribution in the ground state, calculated by using the HF wave functions, presents the distinct 4α cluster-like tetrahedral structure with the Td symmetry. Among others, the necessary restoration of the rotational and parity symmetry, plays a crucial role for the occurrence of the tetrahedral symmetry in 16O.KeywordsTetrahedral symmetryCluster-like structureThe tetrahedral symmetry that breaks spontaneously both the spherical and spatial-reflection symmetries has been identified in molecules, fullerenes, metal clusters and many other quantum objects, all of which are governed by electromagnetic interaction. A matter of fundamental interest has been the possibility of the tetrahedral symmetry in atomic nuclei, as a strong interaction finite many-body quantum system. Recently, the low-lying tetrahedral states were predicted by the potential energy surface calculations to appear in 156Gd [1]. To test the tetrahedral symmetry the ultrahigh-resolution gamma-ray spectroscopy of 156Gd was carried out in a collaboration between France, Poland, Bulgaria, Switzerland and Italy [2]. The experimental result, however, gives a strong evidence against tetrahedral symmetry in the lowest negative-parity band of 156Gd. The same conclusion for the non-tetrahedral symmetry of this negative-parity band also was drawn by another experimental data [3]. The searching for new candidates for the tetrahedral symmetry in other nuclear mass regions becomes an important task to address the issue.Very recently, the evidence for the tetrahedral symmetry in light nucleus 16O has been identified by Bijker and Iachello with the algebraic cluster model to reproduce the rotation-vibration spectrum of an object with Td symmetry (tetrahedral) and compare it with the observed ground state rotational band in 16O [4]. This study clearly shows that the low-lying states in 16O can be described as a 4α cluster with Td symmetry for both the energies and the B(EL) values of the ground-band states. A very recent ab initio lattice calculation of the low-lying even-parity states of 16O has been carried out in the framework of nuclear lattice effective field theory, and the result also shows that the nucleons in the ground state of 16O are arranged in a tetrahedral configuration of 4α clusters [5]. The fingerprints of tetrahedral configuration in 16O is also found from the investigation of giant dipole resonance [6].In this letter, we report our investigation based on the nuclear mean-field solution for the tetrahedral symmetry in the ground state of 16O rather than the starting point of the α-cluster picture. The specific results may, therefore, provide a deeper insight into the tetrahedral symmetry in the nuclear system where the mean field approximation has been proven as an essential starting point for the nuclear modelings. We will show that the HF calculation based on the Skyrme energy density functional predicts the tetrahedral shape with double point group symmetry TdD as the major configuration in the ground state of 16O, and the beyond mean-field effect, namely, the restoration of rotational symmetry, plays a crucial role in the occurrence of the tetrahedral symmetry. We show also that the ground state of 16O with the tetrahedral TdD symmetry has a density distribution of nuclear matter presenting a 4α-cluster structure with the Td symmetry.The density functional theory (DFT) is based on theorems presenting the existence of energy functionals for many-body systems, which, in principle, include all many-body correlations [7–9]. Actually, the first nuclear energy density functionals have been presented in the context of the Hartree–Fock (HF) method with the zero range, density dependent interactions such as the Skyrme force [10–12]. The potential energy surfaces (PES) is obtained by the constraints of multipole moments Q¯λμ, using the augmented Lagrangian method, which is very robust and can give precisely the requested solutions [13]. During PES calculations, a constraint is always imposed on the center of mass of the nucleus: 〈r1Y10〉=0, to exclude the possible coupling to the spurious center of mass motion.The symmetry restoration is very important to study the “true” ground and excited states of deformed nuclei, for references of investigating tetrahedral symmetry, see Refs. [14–16]. In the deformed mean field, the angular-momentum-projection (AMP) operator PˆMKI and parity projection operator Pˆπ [17,18], can be used to obtain the angular momentum and parity conserving wave function,(1)|IMKπ〉=PˆMKIPˆπ|Φ〉≡2I+18π2∫DMKI⁎(Ω)Rˆ(Ω)|Φπ〉dΩ, where, I is the angular momentum, and M and K are its projections along the laboratory and intrinsic z axes, respectively. Pˆπ=1/2(1+πPˆ), where Pˆ is the parity operator and π=±1. Ω denotes the set of three Euler angles (α,β,γ), while DMKI⁎(Ω) are Wigner functions [19]. Rˆ(Ω)=e−iαIˆze−iβIˆye−iγIˆz is the rotation operator.As the rotation symmetry is broken in the mean field, K is no longer a good quantum number, so that different K components must be mixed with the coefficients determined by minimizing the energy. The K-mixing is realized by the assumption,(2)|IMπ〉(i)=∑KgK(i)|IMKπ〉≡∑KgK(i)PˆMKI|Φπ〉, where gK(i) are the mixing coefficients of different K components. The label i=1,2,… indicates the different collective states with the same angular momentum I. Then the Hill–Wheeler (HW) [20] equation is solved to obtain the eigen energies EiIπ and mixing coefficients g(i)K,(3)HIπg¯(i)=EiIπNIπg¯(i), where the matrix elements HK′KIπ=〈Φ|HˆPˆK′KIPˆπ|Φ〉 and NK′KI=〈Φ|PˆK′KIPˆπ|Φ〉 represent the Hamiltonian and overlap kernels, respectively. g¯(i) represents a column of the gK(i) coefficients. States with different angular momentum and parity Iπ are solved separately. When solving Eq. (3), the norm matrix is diagonalized to build the collective subspace. To avoid the numerical unstable solution, the cut-off parameter needs to be used to remove the “zero” norm eigenstates. The cut-off is chosen to satisfy the plateau condition for the corresponding state [21].The EDF calculation and its extensions in this work are performed by the computer code HFODD (v2.73y) [22], which can solve HF/HFB equations in the basis of three-dimensional Cartesian harmonic oscillators. The AMP is provided by the code HFODD already and we implemented the parity projection for the current study. Calculations were performed in the spherical basis of 12 major harmonic-oscillator shells. The harmonic oscillator frequency of the basis is chosen as 1.2×41MeV/A (the value 1.2 is based on experience of diagonalizing the Woods–Saxon Hamiltonian on the HO basis [23,24]). During the calculations, we break all intrinsic symmetries of the nuclear mean field, i.e., simplex, signature, and parity symmetry, to adopt all possible deformation freedoms. The AMP energy and overlap kernels are calculated by using 40 Gauss–Chebyshev integration points in the α and γ directions and 40 Gauss–Legendre points in the β direction.We use several Skyrme-EDFs, as the SLy4 [25], SIII [26], and SkP [27] functionals which are frequently used for DFT predictions, to initiate the calculations. The SkV [26] functional, derived from the density-independent force and free from singularity problem existed in multi-reference calculations [28,29], also serves DFT predictions commonly, especially for beyond-mean-field descriptions [30,31].We first calculate the energy curves against one deformation freedom only, while other deformations are forced to be zero. The results for quadrupole deformations and octupole deformation, are shown in Figs. 1 and 2, respectively. The mean field energies from variational calculations are shown in the upper panel of the figures. The lowest projected energies E1I=0, in function of the quadrupole moments (Fig. 1) and octupole moments (Fig. 2), are shown in the lower panel, labeled as EPROJ. As expected, in the mean-field energy curves of 16O there is a deep minimum at the spherical configuration and no other local minimum. The projected energy curves are extremely flat against quadrupole deformations, being consistent with the results in Ref. [32].However, when the necessary restoration of the rotational symmetry is considered through the angular momentum plus parity projection (AMPP), the octupole deformed states become lower in energy than the spherical state, as the well established minima seen in the lower panel of Fig. 2. For all these Skyrme functionals, non-axial octupole minima with moment Qˆ32 are slightly lower than axial deformed one with Qˆ30, and are also slightly lower than the corresponding other non-axial octupole minima. Especially, the results from the SkV functional shows that the Qˆ32-deformed minimum is explicitly lowest. In general, the HF solutions with the Skyrme functionals and AMPP indicates that the system in the ground state of 16O favors to have the Y32 shape, the tetrahedral symmetry.We also do the variations against single octupole deformation without constraints of other deformation freedom. The unconstrained moments, e.g., quadrupole and other octupole moments are initialized as zero. Thus, they can occur naturally during the HF variations (minimizing procedure). The quadrupole moments after convergence are shown in Fig. 3. The other, unconstrained, octupole moments are nearly zero (less than 10−1fm3) after the convergence. As in Fig. 3 (a), when Q30 moment increases, Q20 moment grows, without triaxial deformations. When Q31 moment increases, Q20 moment increases and triaxality kicks in (Q22 moment appears), seen in Fig. 3 (b). As in Fig. 3 (c), when Q32 moment increases, no quadrupole deformation appears at all. The results of Q33 are similar as Q30 moment, as in the panel (d) of Fig. 3.We then test the competition of the tetrahedral degree of freedom with the quadrupole and other octupole degrees of freedom. The potential energy surfaces in the (Qˆ20, Qˆ32) and (Qˆ22, Qˆ32) plane are given, calculated by using the SkV functionals and setting other multipole moments to zero during the entire computation, in Fig. 4. The Fig. 5 shows the energy surfaces in the (Qˆ30, Qˆ32), (Qˆ31, Qˆ32), and (Qˆ33, Qˆ32) plane, calculated with other multipole moments being excluded. As expected, in the mean-field energy surfaces only the spherical minimum survives. After the restoration of the symmetry, the projected energy surfaces in the lower panel of Figs. 4 and 5, gives a pure Qˆ32-deformed minimum, indicating the tetrahedral symmetry nature in the ground state of 16O. The projected energies give rise to the tetrahedral minimum at about Q32=40fm3. The above projected energy surface calculations demonstrate that the tetrahedral degree of freedom as the non-axial octupole deformation, Y32, could win against or strongly compete with the other important nuclear deformation degrees of freedom, namely, the quadrupole and other octupole deformations.The total density distribution in the ground state of 16O is calculated by using the wave function predicted by the projected energy surface of SkV functional, namely, with the tetrahedral shape at the moment Q32=40fm3 (tetrahedral minimum of the projected energy surfaces, as can be seen in Figs. 2, 4, and 5), and the results are plotted in Fig. 6. This total density distribution of 16O, as a nucleonic system, coincides with the 4α-cluster structure of 16O. Hence, the present Skyrme functional calculations could provide the microscopic support to the 4α-cluster modelings of the tetrahedral structure of 16O, for an example, as that given in Ref. [4].The tetrahedral rotational band is a very important proof for the tetrahedral symmetry [33]. In Ref. [4], the ground tetrahedral rotational band of 16O has been used as the experimental indicator of tetrahedral symmetry. However, unlike the axially deformed nuclei, the favored cranking axis of the tetrahedral deformed state has uncertainties [35,36]. In principle, the tilted axis cranking needs to be performed. To test the collective rotation on the tetrahedral deformed state, we select several different cranking axis, as x axis, y axis, z axis, and the axis in xy plane (π/4 degree between x and y axis). We do the cranking calculation with the constraints of 〈Iˆ〉=I based on the tetrahedral-deformed minimum suggested by projected energy surface, by the SkV functional. The AMPP calculations is performed after the convergence of cranking-HF (CHF) calculations. In the CHF calculation, the time-odd field is also included. The projection after the convergence of cranking mean field can improve the description of the cranking momentum of inertia at low spins for projection calculations, which is an approximated variational-after-projection (VAP) method [21,37]. Results are shown in Fig. 7. The total Routhians by different cranking axis at the same spin are quite close. Both the CHF and projection calculations can give good reproduction of the experimental tetrahedral rotational band of 16O.The present Skyrme functional HF calculations, as the microscopic mean-field study, could support, therefore, the conclusion that the tetrahedral symmetry as the 4α structure exists in the ground state of 16O from the very recent studies, the algebraic cluster model (ACM) calculation [4] and the ab initio lattice calculation [5]. The necessary restoration of the rotational symmetry is essential for the appearance of well established tetrahedral minima in the projected energy surfaces generated by the Skyrme functionals. The spectrum calculations performed with the AMPP on the CHF solutions indicate again that the tetrahedral symmetry configuration might be a major one in the ground state of 16O. As a typical quantum system, the “true” ground state 16O should be a mixing of different possible configurations, and what matters most would be the relative importances for these configurations, so that, the mixing between different configurations, i.e., spherical shape, quadrupole deformed shape, Y32−4α, C12+α structure, etc., have to be performed in the future, to understand the full picture of the ground state in 16O.AcknowledgementsThis work has been supported by the National Natural Science Foundation of China under Grant Nos. 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