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

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

Jie Zhao (赵杰) CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China Physics Department, Faculty of Science, University of Zagreb, Bijenicka 32, Zagreb 10000, Croatia    Bing-Nan Lu (吕炳楠) CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China Institut für Kernphysik (IKP-3) and Jülich Center for Hadron Physics, Forschungszentrum Jülich, D-52425 Jülich, Germany    En-Guang Zhao (赵恩广) CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou 730000, China    Shan-Gui Zhou (周善贵) sgzhou@itp.ac.cn CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China School of Physics, University of Chinese Academy of Sciences, Beijing 100049, China Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou 730000, China Synergetic Innovation Center for Quantum Effects and Application, Hunan Normal University, Changsha, 410081, China
July 14, 2019

We develop a multidimensionally constrained relativistic Hartree-Bogoliubov (MDC-RHB) model in which the pairing correlations are taken into account by making the Bogoliubov transformation. In this model, the nuclear shape is assumed to be invariant under the reversion of and axes; i.e., the intrinsic symmetry group is and all shape degrees of freedom with even are included self-consistently. The RHB equation is solved in an axially deformed harmonic oscillator basis. A separable pairing force of finite range is adopted in the MDC-RHB model. The potential energy curves of neutron-rich even-even Zr isotopes are calculated with relativistic functionals DD-PC1 and PC-PK1 and possible tetrahedral shapes in the ground and isomeric states are investigated. The ground state shape of Zr is predicted to be tetrahedral with both functionals and so is that of Zr with the functional DD-PC1. The tetrahedral ground states are caused by large energy gaps around and when deformation is included. Although the inclusion of the deformation can also reduce the energy around and lead to minima with pear-like shapes for nuclei around Zr, these minima are unstable due to their shallowness.

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I Introduction 

The intrinsic shape of most known nuclei deviates from a sphere Bohr and Mottelson (1969, 1975). The deformation of a nucleus can be described by the parametrization of the nuclear surface with a multipole expansion,


where ’s are deformation parameters. Various nuclear deformations characterized by with different are connected with many nuclear phenomena which have been predicted and some of them have also been observed.

In recent decades, the triaxiality and reflection asymmetry in nuclear shapes have become more and more attractive because several interesting theoretical predictions related to these shapes have been confirmed experimentally. The static triaxial shapes may result in the wobbling motion Bohr and Mottelson (1975) or chiral doublet bands Frauendorf and Meng (1997) in atomic nuclei; both phenomena were observed in 2001 Odegard et al. (2001); Starosta et al. (2001). In 2006, Meng et al. predicted that multiple chiral doublet (MD) bands may exist in one nucleus with different triaxial configurations Meng et al. (2006a). In recent years, MD bands were indeed found in several nuclei Ayangeakaa et al. (2013); Lieder et al. (2014); Kuti et al. (2014); Tonev et al. (2014); Liu et al. (2016). The triaxiality may also play a role in the low-spin signature inversion Bengtsson et al. (1984); Liu et al. (1995, 1996); Zhou et al. (1996); Riedinger et al. (1997); Liu et al. (1998). Moreover, the development of triaxial shapes could lead to the termination of rotational bands in nuclei with ; see Ref. Afanasjev et al. (1999) for a review. The pear-like nuclear shape, characterized by , was predicted to be very pronounced in many nuclei Butler and Nazarewicz (1996); Möller et al. (2008); Agbemava et al. (2016). Low-lying negative-parity levels which are connected with the ground state band via strong transitions in actinides and some rare-earth nuclei are related to reflection asymmetric shapes with nonzero Shneidman et al. (2003, 2006); Wang et al. (2005); Yang et al. (2009); Robledo and Bertsch (2011); Zhu et al. (2012a, b); Nomura et al. (2013, 2014, 2015). The direct evidence for a static octupole deformation has been shown experimentally in Ra Gaffney et al. (2013) and in Bucher et al. (2016).

What new features can we expect if the nonaxial and reflection asymmetric deformations are put together or combined? It has been revealed in Ref. Lu et al. (2012a) that both the triaxial and octupole shapes become important around the second saddle point in the potential energy surface (PES) of actinide nuclei and they lower the second fission barrier considerably. Recently, Liu et al. observed for the first time octupole correlations between MD bands in Br and proposed that chirality-parity quartet bands may appear in a nucleus with both a static triaxial deformation () and an octupole deformation () Liu et al. (2016).

The nonaxiality and reflection asymmetry are combined intrinsically in deformations characterized by with odd and nonzero . Among such deformations, the deformation is of particular interest and has been investigated extensively Hamamoto et al. (1991); Skalski (1991); Li and Dudek (1994); Takami et al. (1998); Yamagami et al. (2001); Dudek et al. (2002, 2006); Olbratowski et al. (2006); Zberecki et al. (2006); Dudek et al. (2010). A nucleus with a pure deformation, i.e., if and , has a tetrahedral shape with the symmetry group . The study of single-particle structure of a nucleus with tetrahedral symmetry predicted large energy gaps at , 20, 32, 40, 56–58, 70, and 90–94 and and 136/142 Li and Dudek (1994); Dudek et al. (2002, 2007, 2003); Heiss et al. (1999); Arita and Mukumoto (2014). These shell gaps are comparable to or even stronger than those at spherical shapes. Thus, a nucleus with proton and/or neutron numbers equal to these values may have a static tetrahedral shape or strong tetrahedral correlations.

Various nuclei such as Zr, Yb, Gd, and Fm were predicted to have ground or isomeric states with tetrahedral shapes from the macroscopic-microscopic (MM) model Dudek et al. (2002, 2007, 2006); Schunck et al. (2004); Dudek et al. (2014) and the Skyrme Hartree-Fock (SHF) model, the SHF plus BCS model, or the Skyrme Hartree-Fock-Bogoliubov (SHFB) model Dudek et al. (2007); Schunck et al. (2004); Dudek et al. (2006); Yamagami et al. (2001); Olbratowski et al. (2006); Zberecki et al. (2009, 2006); Takami et al. (1998). The possible existence of tetrahedral symmetry in light nuclei such as O Bijker and Iachello (2014) as well as superheavy nuclei Chen and Gao (2013, 2010) was also proposed. Furthermore, the rotational properties of tetrahedral nuclei have been studied theoretically Gao et al. (2004); Tagami et al. (2013, 2015) which would certainly be helpful for experimentalists to identify tetrahedral nuclei.

Several experiments were devoted to the study of tetrahedral shapes in Yb Bark et al. (2010), Gd Bark et al. (2010); Jentschel et al. (2010); Doan et al. (2010), U Ntshangase et al. (2010), and Zr Sumikama et al. (2011). For Yb and Gd, although the observed odd and even-spin members of the lowest energy negative-parity bands are similar to the proposed tetrahedral band of Gd, the deduced nonzero quadrupole moments existence of tetrahedral shapes in these nuclei. The possibility of a tetrahedral shape for the negative-parity bands in U was excluded in Ref. Ntshangase et al. (2010) by the observed similarity in the energies and electric dipole moments of those bands in the and isotones and the measured values for quadrupole moment in Ra. An isomeric state observed in Zr has been assigned a tetrahedral shape Sumikama et al. (2011). However, this isomeric state can also been explained by assuming an axial shape with the angular momentum projected shell model Liu et al. (2011, 2015).

Besides the tetrahedral shape which corresponds to a pure deformation, the study of nuclear shapes with a non-zero superposed on a sizable has become a hot topic too. In recent years, nuclei with have been studied extensively because such studies can not only reveal the structure for these nuclei but also give useful structure information for superheavy nuclei Afanasjev et al. (2003); Leino and Hessberger (2004); Herzberg et al. (2006); Herzberg and Greenlees (2008); Zhang et al. (2011, 2012, 2013); Liu et al. (2014); Shi et al. (2014); Wang et al. (2014); Afanasjev (2014); Agbemava et al. (2015); Afanasjev (2015); Li and He (2016). One example of such studies is the observation of very low-lying bands in several even- isotones Robinson et al. (2008). This low-lying feature was argued by Chen et al. to be a fingerprint of sizable in these well deformed nuclei Chen et al. (2008). A self-consistent and microscopic study of effects in these isotones was carried out in Ref. Zhao et al. (2012) and it was found that, for ground states of Cf and Fm, and the energy gain due to the distortion is more than 300 keV.

In this work we will study the potential energy surface and tetrahedral shape in neutron rich even-even Zr isotopes by using the covariant density function theory (CDFT). The nuclear CDFT is one of the state-of-the-art approaches for the study of ground states as well as excited states in nuclei ranging from light to superheavy regions Serot and Walecka (1986); Reinhard (1989); Ring (1996); Bender et al. (2003); Vretenar et al. (2005); Meng et al. (2006b); Paar et al. (2007); Nikšić et al. (2011); Liang et al. (2015); Meng and Zhou (2015); Meng (2016). Recently, the global performance of various relativistic functionals on the ground state properties and beyond-mean-field correlations of nuclei across the nuclear chart has also been examined Afanasjev et al. (2013); Agbemava et al. (2014); Lu et al. (2015); Agbemava et al. (2015); Afanasjev and Agbemava (2016).

Recently, we have developed multidimensionally constrained CDFT (MDC-CDFT) by breaking the reflection and axial symmetries simultaneously. In the MDC-CDFT, the nuclear shape is assumed to be invariant under the reversion of and axes; i.e., the intrinsic symmetry group is and all shape degrees of freedom with even (, , , , , ) are included self-consistently. The MDC-CDFT consists of two types of models: the multidimensionally constrained relativistic mean field (MDC-RMF) model and the multidimensionally constrained relativistic Hartree-Bogoliubov (MDC-RHB) model. In the MDC-RMF model, the BCS approach has been implemented for the particle-particle (pp) channel. This model has been used to study potential energy surfaces and fission barriers of actinides Lu et al. (2012a, 2014a, 2014b, 2014c, b); Zhao et al. (2015a), the spontaneous fission of several fermium isotopes Zhao et al. (2016), the correlations in isotones Zhao et al. (2012), and shapes of hypernuclei Lu et al. (2011, 2014d); see Refs. Li et al. (2014); Lu et al. (2016); Zhou (2016) for recent reviews. The Bogoliubov transformation generalizes the BCS quasi-particle concept and provides a unified description of particle-hole (ph) and pp correlations on a mean-field level. In the MDC-RHB model, pairing correlations are treated by making the Bogoliubov transformation and a separable pairing force of finite range Tian and Ma (2006); Tian et al. (2009a, b); Nikšić et al. (2008) is adopted. Due to its finite range nature, the dependence of the results on the pairing cutoff can be avoided. The MDC-RHB model has been used to study the spontaneous fission of fermium isotopes Zhao et al. (2015b).

The detailed formulas for the MDC-RMF model have been presented in Ref. Lu et al. (2014b) and some applications of this model were reviewed in Refs. Li et al. (2014); Lu et al. (2016); Zhou (2016). In this paper, we present the formulas of the MDC-RHB model and its application to the study of tetrahedral shapes of neutron-rich Zr isotopes. The paper is organized as follows. The formalism of the MDC-RHB model is given in Sec. II. In Sec. III we present the numerical details and the results on neutron-rich even-even Zr isotopes. A summary is given in Sec. IV. Some detailed formulas used in the MDC-RHB model are given in Appendices A and B.

Ii Formalism of the MDC-RHB Model

In the CDFT, a nucleus is treated as a system of nucleons interacting through exchanges of mesons and photons Serot and Walecka (1986); Reinhard (1989); Ring (1996); Bender et al. (2003); Vretenar et al. (2005); Meng et al. (2006b); Paar et al. (2007); Nikšić et al. (2011); Liang et al. (2015); Meng and Zhou (2015); Meng (2016). The effects of mesons are described either by mean fields or by point-like interactions between the nucleons Nikolaus et al. (1992); Burvenich et al. (2002). The nonlinear coupling terms Boguta and Bodmer (1977); Brockmann and Toki (1992); Sugahara and Toki (1994) or the density dependence of the coupling constants Fuchs et al. (1995); Nikšić et al. (2002) were introduced to give correct saturation properties of nuclear matter. Accordingly, there are four types of covariant density functionals: the meson exchange or point-coupling nucleon interactions combined with the nonlinear or density dependent couplings. All these four types of functionals have been implemented in the MDC-RHB model. In this section, we mainly present the formalism of the RHB model with density dependent point couplings.

The starting point of the RHB model with density dependent point couplings is the following Lagrangian,


where is the nucleon mass, , , and are density-dependent couplings for different channels, is the coupling constant of the derivative term, and is the electric charge. , , and are the isoscalar density, isoscalar current, and isovector current, respectively.

With the Green’s function technique, one can derive the Dirac-Hartree-Bogoliubov equation Kucharek and Ring (1991); Ring (1996),


where is the quasiparticle energy, is the chemical potential, and is the single-particle Hamiltonian,


with the scalar potential, vector potential, and rearrangement terms,


As usual we assume that the states are invariant under the time-reversal operation, which means that, in the above equations, all the currents or time-odd components vanish. In this case the single-particle Hamiltonian (4) has the time-reversal symmetry. For convenience two vector densities are defined as the time-like components of the 4-currents and ,


which are the only surviving components. The densities are related to ’s and ’s through


In the above summations, the no-sea approximation is applied.

The pairing potential reads


where is used to represent the large and small components of the Dirac spinor. is the effective pairing interaction and is the pairing tensor,


As is usually done in the RHB theory, only the large components of the spinors are used to build the pairing potential Serra and Ring (2002). In practical calculations this means


The other components , , and are omitted.

In the pp channel, we use a separable pairing force of finite range Tian and Ma (2006); Tian et al. (2009a, b); Nikšić et al. (2008),


where is the pairing strength and and are the center-of-mass and relative coordinates, respectively. denotes the Gaussian function,


where is the effective range of the pairing force. The two parameters MeVfm and fm Tian et al. (2009a, b) have been adjusted to reproduce the density dependence of the pairing gap of symmetric nuclear matter at the Fermi surface calculated with the Gogny force D1S. In the present work, the pairing strength is fine-tuned to reproduce the pairing gaps of Zr isotopes as discussed in Sec. III.1.

The RHB equation (3) is solved by expanding the large and small components of the spinors and in an axially deformed harmonic oscillator (ADHO) basis Gambhir et al. (1990),




which are eigensolutions of the Schrödinger equation with an ADHO potential,




In Eq. (14), is a constant introduced for convenience, is the collection of quantum numbers, and and are the oscillator frequencies along and perpendicular to the symmetry () axis, respectively. The deformation of the basis is defined through relations and , where is the frequency of the corresponding spherical oscillator. These bases are also eigenfunctions of the component of the angular momentum with eigenvalues . For such a basis state we can define a time-reversal state by , where is the time-reversal operator and is the complex conjugation. It is easy to see that is also an eigensolution of Eq. (15) with the same energy , while the direction of the angular momentum is reversed, . forms a complete and discrete basis set in the space of two-component spinors.

In practical calculations, the ADHO basis is truncated as Warda et al. (2002); Lu et al. (2014b),


for the large component of the Dirac spinor. is a certain integer constant and and are constants calculated from the oscillator lengths , , and . For the small component, the truncation is made up to major shells in order to avoid spurious states Gambhir et al. (1990). The convergence of the results on is discussed in Sec. III.1.

The symmetry is imposed in the MDC-CDFT Zhou (2016); i.e., the nuclear potentials and densities are invariant under the following operations


where represents nuclear potentials and densities. Thus, both axial symmetry and reflection symmetry are broken. Due to the symmetry, one can introduce a simplex operator to block-diagonalize the RHB matrix. For a fermionic system, is a Hermitian operator and satisfies . The basis states are also eigenstates of with , which means that with span a subspace with , while their time-reversal states span the one with . Due to the time-reversal symmetry, the RHB matrix is block-diagonalized into two smaller ones denoted by the quantum number , respectively. Furthermore, for a system with the time-reversal symmetry, it is only necessary to diagonalize the matrix with and the other half is obtained by making the time reversal operation on Dirac spinors.

In the subspace, solving the RHB equation (3) is equivalent to diagonalizing the matrix


where and are column matrices, and


are matrix elements of the single-particle Hamiltonian and the pairing field, where and run over all the quantum numbers satisfying the truncation condition (17).

We expand the potentials and and the densities in Eq. (II) in terms of the Fourier series,


By applying the symmetry conditions (18), it is easy to see that and for odd . Thus the expansion in Eq. (21) can be simplified as




are real functions of and . The calculation of matrix elements and densities are similar to that of the MDC-RMF model; the details can be found in Appendices of Ref. Lu et al. (2014b).

Due to the finite range nature of the paring force given in Eq. (11), it is not easy to calculate the matrix element , where the numbers , and denote the ADHO basis wave functions. In Ref. Tian et al. (2009b), the authors have shown that the anti-symmetrized matrix element can be written as a sum of separable terms in both axially deformed and anisotropic HO bases. For this purpose the first step is to represent the product of two single-particle HO wave functions as a sum of HO wave functions in their center-of-mass frame,


where and are the Talmi-Moshinski brackets which are given in Appendix B. The matrix element in the center-of-mass frame reads






More details are given in Appendix A. The pairing field and pairing energy can also be written in the separable form as




Since is conserved, we can check that the matrix element of the pairing tensor has the structure . Combined with the in , one gets


i.e., only with even survive and thus we can skip safely those with odd in the sum in Eq. (28). The sum in Eq. (25) runs over the ADHO quantum numbers , , and in the center-of-mass frame with .

The total energy of the nucleus reads


where the center-of-mass correction is calculated, depending on the relativistic functional, either phenomenologically as


or from the quasiparticle vacuum


where is the total linear momentum and is the nuclear mass number.

The intrinsic multipole moments are calculated as


where is the spherical harmonics and refers to the proton, neutron, or the whole nucleus. The deformation parameter is obtained from the corresponding multipole moment by


where fm and is the number of proton, neutron, or nucleons.

Iii Results and discussions

In this section, we study neutron rich even-even Zr isotopes with by using the MDC-RHB model. These nuclei are among those with and which show interesting nuclear structure properties, such as drastic changes in shapes with and shape coexistences Skalski et al. (1997); Lalazissis et al. (1999); Xu et al. (2002); Dudek et al. (2002); Geng et al. (2003); Schunck et al. (2004, 2004); Olbratowski et al. (2006); Verma et al. (2008); Zberecki et al. (2009); Bender et al. (2009); Rodríguez-Guzmán et al. (2010); Dong et al. (2010, 2011); Mali (2011); Liu et al. (2011); Shi et al. (2012); Xiang et al. (2012); Mei et al. (2012); Liu and Sun (2013); Agbemava et al. (2014); Liu et al. (2015); Togashi et al. (2016); Xiang et al. (2016); Kremer et al. (2016); Nomura et al. (2016). The structures of nuclei around are also of particular interest for nuclear astrophysics because neutron-rich nuclei with are around the -process path, which is determined by the equilibrium between neutron capture and photodisintegration Kratz et al. (1993); Dobaczewski et al. (1994); Chen et al. (1995); Pearson et al. (1996); Dobaczewski et al. (1996); Pfeiffer et al. (1996, 2001); Wanajo and Ishimaru (2006); Qian and Wasserburg (2007); Kratz et al. (2007); Nishimura et al. (2011).

Zr isotopes have been studied extensively with the CDFT under the assumption of reflection symmetry Lalazissis et al. (1999); Geng et al. (2003); Mali (2011); Xiang et al. (2012); Mei et al. (2012); Agbemava et al. (2014); Xiang et al. (2016). In these studies, the coexistence of prolate and oblate minima in potential energy curves of even-even Zr isotopes were discussed. By including the deformation, a pear-like ground state of Zr is predicted with the functional DD-PC1 Agbemava et al. (2016). From the experimental point of view, the analysis of rotational bands, -decay half-lives, and lifetimes of the first states reveals that some of these nuclei are well deformed Nishimura et al. (2011); Hwang et al. (2006a); Mach et al. (1990); Pereira et al. (2009); Hwang et al. (2006b); Cheifetz et al. (1970); Sumikama et al. (2011); Hua et al. (2004); Sarriguren and Pereira (2010); Sarriguren et al. (2014); Browne et al. (2015); Ni and Ren (2014); Yeoh et al. (2010); Li et al. (2008) and there exist shape coexistences in Zr Hwang et al. (2006a); Mach et al. (1990); Pereira et al. (2009); Petrovici et al. (2013).

Next we give some numerical details and illustrative calculations of the MDC-RHB model. Then we present and discuss the results for neutron-rich even-even Zr isotopes calculated with functionals DD-PC1 Nikšić et al. (2008) and PC-PK1 Zhao et al. (2010).

iii.1 Numerical details

Zr Zr
Experiment 1.10 1.55 1.08 1.55
DD-PC1 1.08 1.51 0.99 1.50
PC-PK1 1.17 1.60 1.11 1.60
Table 1: Pairing gaps (in MeV) calculated from functionals DD-PC1 Nikšić et al. (2008) and PC-PK1 Zhao et al. (2010) for Zr. The experimental values are obtained from odd-even mass differences and the mass values are taken from Ref. Wang et al. (2012).

In the MDC-RHB model, the potentials and densities are calculated in a spatial lattice in which mesh points in the and directions are designed in a way that the Gaussian quadrature can be performed and those for the azimuthal angle are equally distributed. Since we keep the mirror reflection symmetry with respect to the or planes, only mesh points with positive and are considered. For axially deformed nuclei the azimuthal degree of freedom vanishes, and for reflection symmetric nuclei mesh points with can also be omitted. The values of the localized fields and potentials in the full lattice space can be simply obtained by symmetry transformations such as rotations or the spatial reflection.

To reproduce the available empirical pairing gaps of Zr (see Table 1), the strength of the pairing force given in Eq. (11) is readjusted a little for protons compared to those originally proposed in Refs. Tian et al. (2009a, b): , with MeVfm.

Figure 1: (Color online)  The potential energy curve of Zr from MDC-RHB calculations with different symmetries imposed (a) and with different truncations of the ADHO basis [(b), (c), and (d)]. The calculations are preformed with axial symmetry and reflection symmetry (AS & RS), axial symmetry and reflection asymmetry (AS & RA), and triaxial symmetry and reflection asymmetry (TA & RA) imposed, respectively. In [(b), (c), and (d)], the results calculated with , 14, and 16 are depicted by dotted, dashed, and solid curves, respectively. The corresponding amplified figures [(e), (f), and (g)] show the detailed structure of the PES’s near ; the width of each subfigure is 0.06 and the height is 1 MeV.

The calculated physical observables should converge as the truncation . In Fig. 1, we show the one-dimensional (1D) potential energy curve of Zr around the ground state with axial symmetry and reflection symmetry, axial symmetry and reflection asymmetry, and nonaxial and reflection asymmetry imposed, respectively. The effective interaction DD-PC1 Nikšić et al. (2008) is used. Calculations with different truncations (, 14, and 16) are depicted by dotted, dashed, and solid curves, respectively. As shown in the upper panel of Fig. 1, the predicted ground state shape of Zr is oblate with when both axial and reflection symmetries are imposed. There is another minimum around . The energy of this minimum is reduced by about 1 MeV when the axial octupole deformation () is included, while the energy gain is about 2 MeV when the nonaxial octupole deformation () is included. As a result, the ground state changes to be . To see more clearly the truncation errors, we have amplified the figure near . We can see that for Zr, the energy changes about 400 keV when increases from 12 to 14; further increasing from 14 to 16, the energy changes only about 100 keV. Furthermore, from Fig. 1, one can find that the inaccuracy caused by the finite basis size is almost independent of the imposed symmetry (in this case, independent of whether the axial or nonaxial deformations, reflection symmetric or reflection asymmetric deformations are included). This means a good convergence and is sufficient in this mass region.

Figure 2: The binding energy (a) and deformation (b) of Zr from MDC-RHB calculations with different truncations made in the Fourier expansion of densities and potentials. The calculations are performed with . The functional DD-PC1 is used in the MDC-RHB calculations.

For axially symmetric nuclei where potentials and densities are independent of the azimuthal angle , the Fourier coefficient when . For nonaxial nuclei, when . Therefore, characterizes the nonaxiality of potentials and densities. In practical calculations, the Fourier series in Eq. (22) has to be truncated, i.e., . To study the dependence of the calculated results on , we show the binding energy and nonaxial deformation parameter of Zr calculated with different in Fig. 2. As shown in Fig. 2(b), if all ’s with are omitted (), the nucleus remains axial and . The convergence of the binding energy and with increasing is very fast. The results obtained with are very precise already; the energy gain when increases from 2 to 3 is only around 30 keV. In practical calculations, is determined automatically by the relation , where () is the largest (smallest) in the truncated ADHO basis. This is sufficiently large for all of results discussed here.

iii.2 Neutron-rich even-even Zr isotopes

Figure 3: (Color online) Potential energy curves () for Zr isotopes. The energy is normalized with respect to the energy minimum with for each nucleus. The dotted line denotes results with axial symmetry (AS) and reflection symmetry (RS) imposed, the dash-dotted line denotes results restricted to AS and reflection asymmetry (RA), while the solid line represents results with both AS and RS broken, i.e., with triaxial (TA) deformation and RA. The functional DD-PC1 is used in MDC-RHB calculations.

We calculate one-dimensional (1D) potential energy curves () for even-even Zr nuclei with . The functionals DD-PC1 Nikšić et al. (2008) and PC-PK1 Zhao et al. (2010) are used. To investigate different roles played by the nonaxiality and reflection asymmetry, calculations are preformed with different symmetries imposed: (i) axial and reflection symmetry, (ii) axial symmetry and reflection asymmetry, and (iii) nonaxial and reflection asymmetry, and the results are denoted by dotted, dash-dotted, and solid lines, respectively.

First we focus on 1D potential energy curves obtained with DD-PC1, which are shown in Fig. 3. If nuclear shapes are restricted to be axial and reflection symmetric (red dotted line), a prolate ground state with coexisting with an oblate minimum with is obtained for Zr. This is consistent with the experimental observations Mach et al. (1990); Hwang et al. (2006a); Hua et al. (2004); Cheifetz et al. (1970). For Zr, the ground state shapes are predicted to be oblate with . As neutron number increases, the energy of the minimum with increases. A minimum with is developed starting from Zr. This minimum becomes lower in energy and the pocket becomes deeper with the neutron number increasing.

If nuclei are allowed to be reflection asymmetric but axial symmetric, one obtains 1D potential energy curves, denoted by blue dash-dotted lines in Fig. 3. The deformation is involved around for all of the nuclei investigated here. As a result, a minimum develops for Zr around . The energy of the minimum with for Zr is lowered substantially by the distortion. Due to this lowering effect, the energy of the minimum with pear-like shape (, ) is lower than the minimum with oblate or prolate shape for Zr, Zr, and Zr. This lowering effect is also found in Skyrme HFB+BCS calculations Zberecki et al. (2009). The deformation is not important either around the oblate minimum () or around the prolate minimum (). Note that these results are different from those presented in Ref. Agbemava et al. (2016) where the octupole deformation is found only in the ground state of Zr with the functional DD-PC1. Such differences are mainly caused by different pairing strengths used in these two works. We have checked that if we use the pairing strengths given in Ref. Agbemava et al. (2016), i.e., with MeVfm, we can get the same results as Ref. Agbemava et al. (2016).

When the deformation is allowed in the calculations, both axial and reflection symmetries are broken. As seen in Fig. 3, the deformation changes very much the energy around (the black solid line). Similar to the cases discussed in the previous paragraph about the inclusion of , a minimum also develops for Zr around . However, the distortion effect is more pronounced than that of the deformation for most of these nuclei. The energy of the minimum with for Zr is lowered substantially. A tetrahedral ground state is predicted for Zr. For Zr, the predicted pear-like shape is lower in energy than the tetrahedral shape. From Fig. 3, we know that the distortion effect is the most pronounced for Zr, where the inclusion of the deformation lowers the energy of the minimum around by about 2 MeV. Similar to the situation of , the deformation is important neither around the oblate minimum () nor around the prolate minimum (). The obtained ground state deformation parameters as well as the binding energies with DD-PC1 for Zr are listed in Table 2. The corresponding available experimental values are also included for comparison. The predicted energies of various minima (with respect to the corresponding ground state energy) are presented in Table 3.

Table 2: The quadrupole deformation , axial octupole deformation , nonaxial octupole deformation , and hexadecapole deformation together with binding energies for the ground states of Zr isotopes obtained in MDC-RHB calculations with the functional DD-PC1. denotes experimental binding energies taken from Ref. Wang et al. (2012). represents experimental quadrupole deformations taken from Ref. Raman et al. (2001). The data with “*” stand for estimated values Wang et al. (2012). All energies are in MeV.
Nucleus Zr Zr Zr Zr Zr Zr Zr Zr
Table 3: The binding energy (relative to the ground state) for various energy minima (when they exist) in Zr isotopes obtained in the MDC-RHB calculations with the functional DD-PC1. All energies are in MeV.
Figure 4: The single-particle levels near the Fermi surface for protons (a) and neutrons (b) of Zr as functions of with fixed at zero. The functional DD-PC1 is used in the MDC-RHB calculations.

From Fig. 3 and Table 2, one finds that the ground states of Zr are well deformed with large quadrupole deformation. For Zr, a tetrahedral isomeric state is also predicted. The ground states of Zr are predicted to have tetrahedral shape and the most pronounced effect from the distortion is in Zr with the functional DD-PC1. The formation of the tetrahedral ground state around Zr can be traced back to the large energy gaps at and in the single-particle levels when deformation is included. In Fig. 4, we show the single-particle levels of Zr near the Fermi surface as functions of with fixed at zero. Due to the tetrahedral symmetry, the single-particle levels split into multiplets with degeneracies equal to the irreducible representations of the group. For example, the spherical levels with and are two-fold and four-fold degenerate and they can be reduced to the two-dimensional (2D) irreducible representation