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Possible shape coexistence in odd-A Ne isotopes and the impurity effects of Λ hyperons

  • In this study, shape evolution and possible shape coexistence are explored in odd-A Ne isotopes in the framework of the multidimensionally constrained relativistic-mean-field (MDC-RMF) model. By introducing sΛ and pΛ hyperons, the impurity effects on the nuclear shape, energy, size, and density distribution are investigated. For the NN interaction, the PK1 parameter set is adopted, and for the ΛN interaction, the PK1-Y1 parameter set is used. The nuclear ground state and low-lying excited states are determined by blocking the unpaired odd neutron in different orbitals around the Fermi surface. Moreover, the potential energy curves (PECs), quadrupole deformations, nuclear r.m.s. radii, binding energies, and density distributions for the core nuclei as well as the corresponding hypernuclei are analyzed. By examining the PECs, possibilities for shape coexistence in 27,29Ne and a triple shape coexistence in 31Ne are found. In terms of the impurity effects of Λ hyperons, as noted for even-even Ne hypernuclear isotopes, the sΛ hyperon exhibits a clear shrinkage effect, which reduces the nuclear size and results in a more spherical nuclear shape. The pΛ hyperon occupying the 1/2[110] orbital is prolate, which causes the nuclear shape to be more prolate, and the pΛ hyperon occupying the 3/2[101] orbital displays an oblate shape, which drives the nuclei to be more oblate.
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  • In the extensive nuclide chart, most nuclei are observed to be deformed. In these nuclei, shape evolution and possible shape coexistence, which was first introduced by Morinaga in 1956 [1], have attracted considerable attentions. Later, Heyde and Wood revealed that shape coexistence is ubiquitous and may appear across the entire nuclide chart [24]. Moreover, triple and multiple shape coexistences have been observed or predicted [5, 6]. As one evidence of the appearance of shape coexistence, exploring the delicate interplay between the single-particle and collective behavior of nucleons, which exhibit opposite trends in nuclei, could promote understanding of nuclear collectivity and the nuclear shell structure [7].

    Investigations on hypernuclear systems provide invaluable information for exploring many-body hadronic physics with "strangeness" utilized as a new degree of freedom [813]. In hypernuclei, hyperons have the advantage of being free from the Pauli exclusion principle for nucleons, and as a result, they can move deep into the nuclear interior as an impurity to probe nuclear structures and properties. Numerous studies have been performed on single-Λ hypernuclei both experimentally [1417] and theoretically [1824]. The interesting impurity effects of hyperons have been studied from various aspects, such as the nuclear size and binding energies [25, 26], nuclear cluster structures [18, 19, 27], neutron drip line [2830], nucleon and hyperon skin or halo [3033], and pseudospin symmetry of nucleons [34].

    The impurity effects of Λ hyperons may also be observed from drastic changes in nuclear shape and deformation. Various studies have shown that s-wave and p-wave hyperons have significantly different impurity effects. By including a spherical sΛ hyperon, the deformation of nuclei can be reduced, and the corresponding core nucleus becomes more spherical. For instance, as shown in Ref. [35], research using the axially deformed RMF model for 29ΛSi and 13ΛC revealed that the oblate nuclei 28Si and 12C became spherical after adding an sΛ hyperon. A similar conclusion was obtained in Ref. [36], where the triaxially-deformed RMF model was used to study the potential energy surfaces E(β,γ), which found that the additional sΛ hyperon drove the ground state of light C, Mg, and Si isotopes to a point with a small β and soft γ. In contrast with the sΛ hyperon, a pΛ hyperon exhibits strong polarization effects, which may enhance nuclear deformation [37]. In Ref. [38], the study using the deformed Skyrme-Hartree-Fock (DSHF) model showed that pΛ hyperons in the 1/2[110] and 3/2[101] states had different effects on nuclear deformation, which resulted in more prolate and oblate nuclear shapes, respectively. In recent years, shape-driven effects induced by a valence nucleon(s) in high-spin states have been extensively researched [3941]. The nucleon occupying the high-j and low-Ω orbital can cause the nucleus to be more prolate. A similar hyperon effect has been confirmed that polarizes nuclear shapes [37, 42].

    To describe nuclear deformation with various shape degrees of freedom, it is better to reduce the symmetric restrictions imposed when solving the equations of motion. In CDFT, we can do this with a harmonic oscillator (HO) basis [43, 44] or three-dimensional lattice space [4548]. Recently, Zhou et al. developed the multidimensionally constrained covariant density functional theories (MDC-CDFTs) [4952], which can accommodate various shape degrees of freedom. They applied this model to a series of investigations on, for example, the fission barriers of actinides [50, 51, 5356], nonaxial octupole Y32 correlation in N=150 isotones [57], the third minima and triple-humped barriers in light actinides [58], and potential energy curves (PECs) in the superheavy nucleus 270Hs [59]. Subsequently, by including Λ hyperons, the MDC-CDFTs have been extended to study hypernuclei. Shape evolution in the C, Mg, and Si isotopes and the possible polarization effects of the Λ hyperon [36], the superdeformed states in Ar isotopes [42], the ΛΛ pairing correlations [60], and the new effective ΛN interactions [61] have been either studied or developed. Recently, we used the MDC-CDFTs to explore shape evolution and possible shape coexistence in even-even Ne isotopes [62]. By exploring the PECs, possibilities for shape coexistence were found in nuclei 24,26,28Ne. Furthermore, the impurity effects of the sΛ and pΛ hyperons on the nuclear shape, size, and binding energies were studied.

    In this paper, following our previous study in Ref. [62], shape evolution and possible shape coexistence in odd-A Ne isotopes are explored with the MDC-RMF model by including the blocking effect for the unpaired odd neutron and the impurity effects are investigated by adding a single-Λ hyperon occupying the lowest s or p orbitals. For shape coexistence in odd-A Ne isotopes, in Ref. [63], constrained RMF+BCS calculations with the NL075 force has been performed and possible shape coexistence was predicted in 25,27Ne. The paper is organized as follows: In Sec. II, the MDC-RMF model for single-Λ hypernuclei including the blocking effect is briefly presented. In Sec. III, numerical details are provided. After the results and discussions in Sec. IV, a summary and perspectives are provided in Sec. V.

    In the meson-exchange MDC-RMF model for single-Λ hypernuclei, the covariant Lagrangian density is composed of two parts:

    L=L0+LΛ,

    (1)

    where L0 is the standard RMF Lagrangian density for nucleons. For details, see Refs. [6468]. In the case of the Lagrangian density for the Λ hyperon LΛ, considering the neutral and isoscalar particle properties, only couplings with scalar-isoscalar σ and vector-isoscalar ω mesons are included, and LΛ is expressed as

    LΛ=ˉψΛ[iγμμmΛgσΛσgωΛγμωμ]ψΛ+fωΛΛ4mΛˉψΛσμνΩμνψΛ,

    (2)

    where the mass of the Λ hyperon, mΛ=1115.6 MeV, the coupling constants of the Λ hyperon with the σ and ω meson fields, gσΛ and gωΛ, and the parameter fωΛΛ in the tensor coupling term between the Λ hyperon and ω field, which is strongly related to small single-Λ spin-orbit splitting [69], constitute the ΛN interaction. The field tensor of the ω field, Ωμν=μωννωμ.

    In the framework of the RMF model, using the variational procedure, the Dirac equations for baryons as well as the Klein-Gordon equations for mesons and photons can be obtained under the mean-field and no-sea approximations. The Λ hyperon satisfies the following Dirac equation:

    [αp+β(mΛ+SΛ)+VΛ+TΛ]ψΛ=ϵψΛ,

    (3)

    where α and β are the Dirac matrices, and SΛ, VΛ, and TΛ represent the scalar, vector, and tensor parts, respectively, of the mean-field potentials for the Λ hyperon,

    SΛ=gσΛσ,

    (4)

    VΛ=gωΛω,

    (5)

    TΛ=fωΛΛ2mΛβ(αp)ω.

    (6)

    In odd-A nuclei, the blocking effect for the unpaired nucleon should be treated, which is of a crucial importance [7072]. The ground state of an odd system is a one-quasiparticle state, and in the BCS approach, it could be described by the following wave function [70, 71]:

    ˆα+k1|BCS=ˆa+k1k>0,kk1(uk+vkˆa+kˆa+ˉk)|0,

    (7)

    where |BCS is the BCS vacuum state, ˆα+k1 and ˆa+k1 are the creation operators for the quasiparticle and single-particle, respectively, and k1 denotes the blocked orbital occupied by the unpaired particle.

    To determine the nuclear ground state and low-lying excited states in an odd-A nucleus, a variety of calculations are performed with the odd nucleon blocked in different single-particle states k around the Fermi surface. As a result, the state with the lowest total binding energy is considered the ground state, while others form the low-lying excited energy spectra.

    To obtain potential energy curves (PECs), constraint calculations [70] with a modified linear constraint method [50, 51] are performed, which has been effectively demonstrated in MDC-RMF calculations compared to the quadratic constraint method [36]. The Routhian is calculated as

    E=ˆH+λμ12CλμQλμ,

    (8)

    where ˆH is the RMF Hamiltonian, and Cλμ are variables that change their values in different iteration steps as follows:

    C(n+1)λμ=C(n)λμ+kλμ(β(n)λμβλμ),

    (9)

    where C(n+1)λμ and C(n)λμ are values obtained in the (n+1)th and nth iterations, respectively, βλμ is the desired nuclear deformation, and β(n)λμ is the calculated value in the nth iteration. Changes in Cλμ and βλμ are linked by a constant kλμ.

    The intrinsic multipole moment Qλμ in Eq. (8) can be calculated with the vector density ρV(r) and spherical harmonics Yλμ(Ω) by

    Qλμ=d3rρV(r)rλYλμ(Ω).

    (10)

    Subsequently, with the multipole moment, the corresponding deformation parameter βλμ can also be calculated using

    βλμ=4π3BRλQλμ,

    (11)

    where R=1.2A1/3fm is the radius of the nucleus, with A as the total number of nucleons, and B represents the number of protons, neutrons, Λ hyperons, or total baryons.

    In an axially symmetric system, we only have the good quantum numbers of parity and the z component of the angular momentum. In this case, calculations with only a single-constraint on the quadrupole deformation β2 will be performed.

    Shape evolution and possible shape coexistence are explored in odd-A Ne isotopes with A=1933 using the self-consistent MDC-RMF model. Moreover, the impurity effects of Λ hyperons occupying the s or p orbitals are investigated. As a continuation of our previous study on even-even Ne isotopes [62], the same numerical details are used.

    In the RMF functional, the PK1 [73] parameter set is used for the NN interaction. For the ΛN interaction, the PK1-Y1 [74] parameter set is adopted, where the scalar-isoscalar coupling constant gσΛ=0.580gσ and vector-isoscalar coupling constant gωΛ=0.620gω, which were determined by fitting the experimentally observed single-Λ binding energies, and the tensor coupling constant fωΛΛ=gωΛ, which was determined by reproducing pΛ spin-orbit splittings in 9ΛBe and 13ΛC.

    To solve the RMF equations, an axially deformed harmonic oscillator (ADHO) basis [43, 44] is taken with the truncation parameters NF=14 for fermion shells and NB=20 for boson shells, as in Ref. [36]. With these parameters, the truncation error in the binding energy of the nucleus 26Si is less than 30 keV.

    In the mean-field approximation, the translational symmetry is broken. To remedy this, the following microscopic center-of-mass (c.m.) correction [75] is employed:

    Ec.m.=12MˆP2,

    (12)

    where M is the total nuclear mass.

    The BCS approach is used to treat the pairing effects with a finite-range separable pairing force [7678].

    V(r1r2)=Gδ(˜R˜R)P(˜r)P(˜r)1ˆPσ2,

    (13)

    where G is the strength of the pairing force, ˜R and ˜r are the center of mass and relative coordinates between the paired particles, respectively, and P(˜r) is a Gaussian shaped function.

    P(˜r)=1(4πa2)3/2e˜r2/a2,

    (14)

    where a is the effective range of the pairing force. Here, the strength G and range a are taken as

    G=728.0MeVfm3,a=0.644fm,

    (15)

    which can be used to obtain the same momentum dependency of the pairing gap in nuclear matter as that of the D1S Gogny force [76].

    In Fig. 1, the binding energy per nucleon E/A, matter radius rm, and quadrupole deformation β2 in the ground states of Ne isotopes determined by unconstrained RMF calculations with the PK1 parameter set are compared with the available experimental data [79]. In general, good consistency with the experimental results is revealed, indicating that the choice of the PK1 parameter set for the description of Ne isotopes is suitable.

    Figure 1

    Figure 1.  (color online) Binding energy per nucleon E/A, matter radius rm, and quadrupole deformation β2 as a function of the mass number A in the ground states of Ne isotopes in comparison with the available experimental data [79].

    To explore shape coexistence on the mean field level, the PECs are mainly analyzed. If two close-lying energy minima with a difference of a few hundred keV that own prolate and oblate quadrupole deformations are observed in combination with a pronounced barrier between them, possible shape coexistence is indicated because their ground states may have two competing configurations.

    In Fig. 2, the PECs for odd-A Ne isotopes from A=19 to 33 are plotted as functions of the quadrupole deformation β2, which are obtained using constrained calculations with the self-consistent MDC-RMF model. The unpaired odd neutron is blocked in different orbitals around the Fermi surface, and the state with the lowest binding energy is deemed the ground state while others form the local minima.

    Figure 2

    Figure 2.  (color online) Potential energy curves (PECs) as a function of the deformation parameter β2 in odd-A Ne isotopes, with the odd neutron blocked in different orbitals around the Fermi surface denoted by the Nilsson quantum numbers Ωπ[Nn3ml]. The open circles denote local energy minima.

    In Fig. 3, to observe the level structures in the Ne isotopes, the single-neutron levels Ω[Nn3ml] obtained using the constrained MDC-RMF calculations are plotted for the even-even nucleus 30Ne, where the solid lines represent levels with positive parity, and the dashed lines represent those with negative parity. At a spherical shape with the deformation β2=0, we observe the neutron shell closures N=8 and N=20, whereas the shell closure N=28 vanishes owing to the inversion of the 1f7/2 and 2p3/2 levels. Referring to this single-neutron level structure, the odd neutron in 1929Ne most likely occupies the 2s,1d orbitals and 2p,1f orbitals in 31,33Ne. Calculations have been performed with all these configurations, and the obtained PECs are presented in Fig. 2, where the local energy minima are marked by open circles. The values of the quadrupole deformations, binding energies, and blocked orbitals of the unpaired odd neutron corresponding to the local energy minima are listed in Table 1.

    Table 1

    Table 1.  Quadrupole deformations and binding energies (in MeV) for ground states and local energy minima (labeled with asterisks) in 1933Ne.
    Nucleus Quadrupole deformationEnergy
    β2β2n β2p
    19Ne(1/2+[220])0.3610.3030.413142.612
    19Ne(5/2+[202])0.1450.1380.151141.477
    21Ne(3/2+[211])0.5120.5150.509165.917
    21Ne(3/2+[211])0.2010.2090.193161.081
    23Ne(5/2+[202])0.3880.3700.411181.759
    23Ne(1/2+[220])0.2230.2360.205179.701
    25Ne(1/2+[211])0.2060.1930.225193.711
    25Ne(1/2+[211])0.1310.1310.130193.487
    27Ne(1/2+[200])0.1900.1770.213202.836
    27Ne(3/2+[202])0.1300.1350.120202.503
    29Ne(3/2+[202])0.0800.0680.103210.166
    29Ne(1/2+[200])0.0750.0730.079210.144
    31Ne(1/2[321])0.1910.1960.181213.707
    31Ne(7/2[303])0.0900.0890.091213.022
    31Ne(3/2+[202])0.4290.4390.408212.986
    33Ne(3/2[312])0.4190.4280.397216.533
    33Ne(3/2[312])0.1620.1820.116212.418
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    Figure 3

    Figure 3.  (color online) Single-neutron levels Ω[Nn3ml] as a function of the deformation parameter β2 in 30Ne.

    Regarding the nucleus 19Ne with a neutron number just exceeding an N=8 shell closure, it is optimal that the odd neutron occupies the 1d5/2 orbital, which can be splitted into the Ωπ[Nn3ml]=1/2+[220], 3/2+[211], and 5/2+[202] orbitals with axially symmetric quadrupole deformation. Further analysis, shown in Fig. 2(a), demonstrates that the odd neutron blocked in the 1/2+[220] orbital corresponds to the ground state with a prolate deformation of β2=0.361, whereas the odd neutron blocked in 5/2+[202] contributes to the second energy local minimum, which exhibits an oblate shape at β2=0.145. In the case of the unpaired neutron occupying other orbitals, the obtained PECs (denoted by dashed lines) are significantly higher. Hereafter, we focus our attention on the configurations for the ground state and other local energy minima that may generate shape coexistence. With two more neutrons in 21Ne, the odd neutron occupying the 3/2+[211] orbital results in both the ground state and second local energy minimum, which are located at the prolate deformation of β2=0.512 and oblate deformation of β2=0.201, respectively. As for 23Ne, the configurations of the ground state and the second local minimum are opposite to those in 19Ne, that is, the odd neutron occupying the 5/2+[202] orbital leads to the prolate ground state observed at β2=0.388, while the odd neutron blocked in the 1/2+[220] orbital corresponds to the second local energy minimum identified at the deformation β2=0.223. Because the corresponding energy differences between the ground state and second local energy minimum in 19,21,23Ne are as large as 1.135,4.836,2.058 MeV, possible shape coexistence is excluded in those nuclei. With regard to the nucleus 25Ne, the configuration of the unpaired neutron blocked in 1/2+[211] from the spherical 2s1/2 orbital contributes to the lowest PEC, on which the ground state with the prolate deformation β2=0.206 and the second local energy minimum with an oblate shape at β2=0.131 are obtained. Although the energy difference between them is as small as 0.224 MeV, the circumjacent PEC is flat, where different deformations within a limited scope correspond to similar binding energies. In this case, the occurrence of shape coexistence is believed to be difficult. Moving further to the nuclei 27Ne and 29Ne, odd neutrons have great possibilities of occupying the 1d3/2 (1/2+[200] and 3/2+[202]) orbital. For 27Ne, when the odd neutron is blocked in the 1/2+[200] and 3/2+[202] orbitals, the ground state with prolate deformation at β2=0.190 and second energy local minimum exhibiting an oblate shape at β2=0.130 are observed, respectively. A small energy difference of 0.333 MeV in combination with a barrier of 0.984 MeV in height between them are observed, providing great potential for shape coexistence in 27Ne. In the case of 29Ne, with the odd neutron blocked in 3/2+[202], the prolate ground state is identified at a deformation of β2=0.080, whereas the configuration with the odd neutron occupying 1/2+[200] leads to an oblate second local minimum with a deformation of β2=0.075. Similar to 27Ne, owing to the small energy difference of 0.022 MeV coupled with a barrier of 0.4 MeV between the two local energy minima, possible shape coexistence is expected in 29Ne. Far from the N=20 shell closure, the unpaired neutron in the nuclei 31Ne and 33Ne is likely to occupy the 2p3/2 (1/2[321], 3/2[312]) and 1f7/2 (1/2[330], 3/2[321], 5/2[312], 7/2[303]) orbitals. For the nucleus 31Ne, a possible triple shape coexistence is observed, which is comprised of the prolate ground state at β2=0.191 with the odd neutron blocked in 1/2[321], a second oblate local energy minimum at β2=0.090 with the odd neutron in 7/2[303], and a third prolate local minimum at β2=0.429 with the odd neutron in 3/2+[202]. The corresponding excitation energies are 0.685 and 0.721 MeV with respect to the ground state, which together with a proper barrier between them, guarantees the appearance of shape coexistence. In the case of 33Ne, with the odd neutron occupying 3/2[312], both the ground state and the second local minimum are observed owing to a prolate shape at β2=0.419 and an oblate shape at β2=0.162, respectively. Because of the large energy difference of ΔE=4.1 MeV, the appearance of shape coexistence is difficult.

    To explore the impurity effects of Λ hyperons, taking the single-Λ hypernucleus 28ΛNe (or denoted by 27NeΛ) as an example, PECs are plotted in Fig. 4 where the s-wave (Ωπ[Nn3ml]=1/2+[000]) and three p-wave (1/2[110], 1/2[101], and 3/2[101]) single-Λ hyperons are considered (denoted by dashed lines). In panels (a) and (b), the odd neutron is blocked in the 1/2+[200] and 3/2+[202] orbitals, respectively.

    Figure 4

    Figure 4.  (color online) PECs as a function of deformation β2 in 27Ne and the single-Λ hypernucleus 28ΛNe (27NeΛ). The odd neutron is blocked in the (a) 1/2+[200] and (b) 3/2+[202] orbitals, and the single-Λ hyperon is injected into the lowest s or p orbitals. The open circles denote the local energy minima.

    Generally, after injecting an sΛ hyperon, the PEC shapes remain almost unchanged, while the depths deepen significantly owing to the attractive ΛN interaction. Moreover, a clear "glue-like" effect of the sΛ hyperon is observed, which results in smaller nuclear deformations and a more spherical nuclear shape. For example, in Fig. 4(a), the reduced deformation of the prolate ground state changes from β2=0.190 to β2=0.165, and in Fig. 4(b), the deformation of the oblate second local minimum reduces to β2=0.115 from β2=0.130.

    Regarding single-pΛ hypernuclei, PECs obtained with the Λ hyperon injected into the 1/2[101] orbital from the 1p1/2 orbital and the 3/2[101] orbital from the 1p3/2 orbital are almost degenerate owing to small spin-orbit splitting; however, they exhibit significant differences from those obtained by injecting the pΛ hyperon into the 1/2[110] orbital. Hereafter, considering that the pΛ hyperon occupying the 1/2[101] and 3/2[101] orbitals exhibit similar effects, we only discuss the latter case. In general, after injecting a pΛ hyperon, the PEC shapes as well as the locations of the local energy minima clearly change. Furthermore, different polarization effects are exhibited by pΛ hyperons occupying the 1/2[110] and 3/2[101] orbitals. For instance, in Fig. 4(a), with the Λ hyperon blocked in the 1/2[110] orbital, the prolate deformation of β2=0.190 corresponding to the nuclear ground state in 27Ne is driven to β2=0.265, whereas it is reduced to β2=0.126 when the pΛ hyperon is blocked in the 3/2[101] orbital. Similar effects by the pΛ hyperon have been observed in Fig. 4(b), where the unpaired neutron in the core nucleus is blocked in the 3/2+[202] orbital, which corresponds to the second local minimum. In detail, the pΛ hyperon in the 1/2[110] state drives the nucleus toward a spherical shape, and the oblate deformation of β2=0.130 corresponding to the second local energy minimum decreases to β2=0.044. Meanwhile, the pΛ hyperon occupying the 3/2[101] orbital causes the hypernucleus to become more oblately deformed with β2=0.162.

    Similar investigations as those of 28ΛNe shown in Fig. 4 have also been performed for other Ne hypernuclei. The same impurity effects of single-Λ hyperons on nuclear deformations and binding energies are obtained. As a result of the introduced Λ hyperons, the nuclear PECs are significantly deepened; however, their increments are varied at different deformations β2. Therefore, the energy difference ΔE between different local energy minima might change, which may influence the possibility of shape coexistence. In Table 2, the values of ΔE in the nuclei 1933Ne are presented in comparison with values after injecting a single-Λ hyperon into the 1/2+[000], 1/2[110], and 3/2[101] orbitals. With the additional sΛ hyperon, the energy difference ΔE decreases significantly in all the hypernuclei. As a result, possible shape coexistence in 27,29,31Ne can persist well in 28,30,3228,30,3ΛNe. For example, the value of ΔE between the ground state and second local energy minimum reduces to 0.238 MeV in 28ΛNe from 0.333 MeV in 27Ne. Moreover, the hypernucleus 20ΛNe becomes a new candidate for possible shape coexistence because ΔE therein is reduced to 0.924MeV. With the addition of a pΛ hyperon, the influence on the value of ΔE becomes complex. With the addition of a single-pΛ hyperon occupying the 1/2[110] orbital, there is a significant increase in ΔE in all hypernuclei, which reduces the possibility of shape coexistence. For instance, the energy difference ΔE in 28ΛNe increases to 1.811MeV from 0.333MeV. Furthermore, one of the local energy minima disappears in 22,2622,2ΛNe with the addition of a pΛ hyperon. Similarly, by including a pΛ hyperon in the state of 3/2[101], the values of ΔE increase in most of the nuclei, and the probabilities of shape coexistence decrease. This differs from those in even-even Ne isotopes [62], where the additional pΛ occupying the 3/2[101] orbital provides a higher possibility of shape coexistence.

    Table 2

    Table 2.  Energy difference ΔE (in MeV) between the two local energy minima in 1933Ne (core nuclei) and the corresponding single-Λ hypernuclei 1933NeΛ with the Λ hyperon injected into the 1/2+[000], 1/2[110], and 3/2[101] orbitals.
    Core nucleiΛ(1/2+[000])Λ(1/2[110])Λ(3/2[101])
    19NeΛ1.1350.9242.9160.138
    21NeΛ4.8364.4343.188
    23NeΛ2.0581.8314.8040.502
    25NeΛ0.2240.113
    27NeΛ0.3330.2381.8110.443
    29NeΛ0.0220.0200.9020.356
    31NeΛ0.6850.6021.9340.019
    33NeΛ4.1153.9556.2182.834
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    To study the impurity effects of sΛ and pΛ hyperons on the nuclear quadrupole deformations β2, nuclear root mean square (r.m.s.) radii r, and binding energies E, we take the hypernucleus 28ΛNe as an example and list the corresponding values in Table 3. The single-Λ separation energies SΛ=E(A+1ΛNe)E(ANe) are also given. Configurations with the odd-neutron blocked in the 1/2+[200] and 3/2+[202] (denoted by asterisks) orbitals, which correspond to the ground state and the second local minimum of the core nucleus 27Ne, are considered. In general, the sΛ hyperon is more deeply bound than the pΛ hyperon, which results in a relatively small nuclear size rΛ and a large single-Λ separation energy SΛ. Moreover, the sΛ and pΛ hyperons induce different impurity effects. The sΛ hyperon exhibits a significant shrinkage effect, which makes the nuclei more bound with a smaller size. For example, with the addition of the sΛ hyperon, the nuclear r.m.s. radius in the ground state 27Ne(1/2+[200]) decreases from 3.083 fm to 3.074 fm, the total binding energy E becomes 16.546 MeV deeper, and the nuclear shape tends to be more spherical. For the quadrupole deformation β2 of 27NeΛ(1/2+[000]), it maintains the same sign as the core nucleus. In the case of the sΛ hyperon, its deformation β2Λ is influenced in turn by the core nucleus. For instance, in the hypernucleus 27Ne(1/2+[200])sΛ, the Λ hyperon is slightly prolate with a deformation β2Λ=0.053, the sign of which is the same as that of the core nucleus 27Ne(1/2+[200]). The same behavior is found in 27Ne(3/2+[202])sΛ. In contrast with the sΛ hyperon, the pΛ hyperon may enhance the nuclear size slightly. Additionally, pΛ hyperons in the states of 1/2[110] and 3/2[101] exhibit significantly different effects on nuclear deformation. For the pΛ hyperon occupying 1/2[110], which is a prolate shape, the core nucleus becomes more prolate. Conversely, for the hyperon occupying 3/2[101] or 1/2[101], which is oblate, the core nucleus becomes more oblate or less prolate. For instance, in 27Ne(1/2+[200])pΛ(1/2[110]), because both the pΛ hyperon and core nucleus are prolate, the pΛ hyperon enhances the nuclear prolate deformation from β2=0.190 to β2=0.265, while shape decoupling occurs in 27Ne(1/2+[200])pΛ(3/2[101]), where the pΛ hyperon and core nucleus have different shapes, which leads to a reduction in the total nuclear deformation from β2=0.190 to β2=0.126. In Refs. [80, 81], shape decoupling in the deformed halo nuclei 42,44Mg are discussed, in which the shapes of the core and outside halo are different.

    Table 3

    Table 3.  Quadrupole deformation parameters, root mean square (r.m.s.) radii, binding energies, and Λ separation energies in the odd-A nucleus 27Ne and the corresponding single-Λ hypernuclei 27NeΛ with the Λ hyperon injected into the lowest s orbital or three p orbitals. The odd neutron is blocked in the 1/2+[200] and 3/2+[202] orbitals, which correspond to the ground state and the second local energy minimum (denoted by asterisks), respectively.
    NucleusQuadrupole deformationr.m.s. radii/fmEnergy/MeV
    β2β2nβ2pβ2ΛrmrcorerΛESΛ
    27Ne(1/2+[200])0.1900.1770.2133.083202.836
    27NeΛ(1/2+[000])0.1650.1570.1880.0533.0563.0742.511219.38216.546
    27NeΛ(1/2[110])0.2650.2170.2621.0873.1053.0913.461210.6937.857
    27NeΛ(1/2[101])0.1280.1400.1620.4143.0933.0793.443209.1336.297
    27NeΛ(3/2[101])0.1260.1380.1610.4293.0923.0793.431209.3176.481
    27Ne(3/2+[202])0.1300.1350.1203.077202.503
    27NeΛ(1/2+[000])0.1150.1240.1090.0333.0513.0692.505219.14416.641
    27NeΛ(1/2[110])0.0440.0820.0610.7743.0883.0753.432208.8826.379
    27NeΛ(1/2[101])0.1620.1550.1390.5093.0943.0803.444209.6257.122
    27NeΛ(3/2[101])0.1620.1540.1390.5303.0943.0803.439209.7607.257
    DownLoad: CSV
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    In Fig. 5, the density distributions ρ(r,z) in the r-z plane with the symmetric axis along the z-axis are shown for the nucleus 27Ne (a) and the corresponding hypernucleus 28ΛNe (b,c,d). Cases with the core nucleus 27Ne in the ground state and the second local minimum (denoted by asterisks) as well as the single-Λ hyperon occupying the 1/2+[000] (b), 1/2[110] (c), and 3/2[101] (d) orbitals are studied. In the upper and lower parts of panels (b–d), densities are plotted for the total hypernucleus and single-Λ hyperon, respectively. In Fig. 5(a), a prolate shape is observed for the ground state of 27Ne, while an oblate shape is observed for the second energy minimum. After introducing an sΛ hyperon, in Fig. 5(b), the nuclear shapes remain almost unchanged but with an increment of inner density. Meanwhile, the injected sΛ hyperon is slightly deformed and has the same shape as the core nucleus. In Fig. 5(c), a "dumbbell" shape is observed for the prolate pΛ hyperon in the state of 1/2[110], the addition of which increases (reduces) the prolate (oblate) deformation of the core nucleus in the ground state (second energy minimum). At the same time, owing to the coupling of the pΛ hyperon and nuclear core, slight differences in the density distributions of the pΛ hyperon are observed in the two cases (panel c). In Fig. 5(d), a "ring" shape is observed for the oblate pΛ hyperon in the state of 3/2[101]. As a result, deformation of the prolate ground state decreases, and the oblate deformation of the second energy minimum is enhanced. A similar analysis of the impurity effects of the pΛ hyperon on the nuclear density can be found in Refs. [38, 62].

    Figure 5

    Figure 5.  (color online) Two-dimensional density distributions in the r-z plane for (a) the core nucleus 27Ne and (b), (c), (d) the corresponding single-Λ hypernuclei with the Λ hyperon injected into the (b) 1/2+[000], (c) 1/2[110], and (d) 3/2[101] orbitals, respectively. The odd neutron in the ground state and the second local minimum (denoted by asterisks) occupies the 1/2+[200] and 3/2+[202] orbitals, respectively. The upper and lower parts of panels (b), (c), and (d) are the densities of the entire hypernuclei and the single-Λ hyperon, respectively.

    To explain the mechanism behind the effects of sΛ and pΛ hyperons on nuclear densities and shapes in an intuitive way, taking 27NeΛ with the core nucleus in the ground state an example, we presentdensity distributions ρ along the r- and z-axes in Fig. 6. For the sΛ hyperon, almost the same density distributions (red dashed lines in the lower part of each panel) are found along the r- and z-axes; however, the distribution is slightly extended along the z-axis, which suggests the sΛ hyperon has a weakly prolate shape. With the addition of an sΛ hyperon to 27Ne, the nuclear density clearly increases in the inner part (r<3fm, z<2.5 fm) both in the directions of the r- and z-axes. However, for the densities on the outer nuclear side, different influences are found, that is, ρ(r) becomes slightly more extended while ρ(z) shrinks. These cause 27NeΛ to become more spherical compared to the core nucleus. In the case of the pΛ hyperon occupying 1/2[110], ρ(r) vanishes and a visible distribution is presented along the z-axis with a maximum at z=2.5 fm. As a result, a clear increment in the nuclear density along the z-axis is induced, and the nuclear shape becomes more prolate. For the pΛ hyperon in the state of 3/2[101], the nuclear density is enhanced in the direction of the r-axis and prolate deformation is weakened.

    Figure 6

    Figure 6.  (color online) Density distributions ρ(r,z) along the r-axis (a) and z-axis (b) in 27Ne with the odd neutron occupying 1/2+[200] (solid lines) and the corresponding single-Λ hypernuclei (dashed lines) with the Λ hyperon injected into the 1/2+[000] (red short dashed line), 1/2[110] (blue dash-dotted line), and 3/2[101] (olive dashed line) orbitals. In the upper and lower parts of each panel, density distributions contributed by the total nuclei and the Λ hyperon are plotted, respectively.

    In this study, following our previous research [Sci. China-Phys. Mech. Astron. 64, 282011(2021)], shape evolution and possible shape coexistence in odd-A Ne isotopes are explored using the MDC-RMF model, which has achieved great success in describing nuclear deformations. Moreover, by introducing s- or p-wave Λ hyperons, the impurity effects on nuclear shape, energy, size, and density distribution are discussed. For NN and ΛN interactions in the RMF functional, the PK1 and PK1-Y1 parameter sets are adopted, respectively.

    By blocking the unpaired odd neutron in different orbitals around the Fermi surface, the nuclear ground state and other local energy minima are determined, and by examining the PECs, possible shape coexistences in 27,29Ne and a possible triple shape coexistence in 31Ne are predicted. For 27Ne, with the odd neutron blocked in the 1/2+[200] and 3/2+[202] orbitals, the ground state is observed in a prolate shape at a deformation of β2=0.190 and the second energy minimum exhibits an oblate shape at β2=0.130, respectively. A small energy difference of 0.333 MeV and a barrier with a height of 0.984 MeV between them are observed, supporting possible shape coexistence in 27Ne. Similarly, in 29Ne, with the odd neutron blocked in the 3/2+[202] orbital, the prolate ground state is identified at β2=0.080, while the configuration with the odd neutron occupying 1/2+[200] leads to an oblate second local minimum at β2=0.075. In the case of 31Ne, possible triple shape coexistence, including the ground state at β2=0.191 with the odd neutron blocked at 1/2[321], the second local energy minimum at oblate deformation β2=0.090 with the odd neutron at 7/2[303], and the third local minimum at prolate deformation β2=0.429 with the odd neutron at 3/2+[202], is predicated.

    To discuss the impurity effects of sΛ and pΛ hyperons, nuclear quadrupole deformations, r.m.s. radii, binding energies, and density distributions are compared in detail for 27Ne and the corresponding hypernuclei 28ΛNe in cases where the odd neutron occupies the 1/2+[200] and 3/2+[202] orbitals. The sΛ hyperon exhibits clear shrinkage effects, which reduces the nuclear size and deformation. In contrast, the pΛ hyperon exhibits strong polarization effects, which may enhance the nuclear deformation. Meanwhile, the pΛ hyperons occupying different orbitals exhibit different effects: the hyperon from the 1/2[110] orbital drives the nuclear shape to become more prolate, and the hyperon in the 3/2[101] or 1/2[101] state results in a more oblate nuclear shape. These conclusions are consistent with those for even-even Ne isotopes [62]. Furthermore, with the addition of the sΛ hyperon, the energy difference ΔE between the ground state and second local energy minimum decreases, which may increase the probability of shape coexistence. However, introducing a pΛ hyperon results in a distinct disadvantage for shape coexistence in most of hypernuclei.

    In the present study, we investigate shape evolution and possible shape coexistence at the mean-field level by analyzing PECs. This is the first step in our research. Recently, in Refs. [82, 83], the angular momentum and parity projected multidimensionally constrained relativistic Hartree-Bogoliubov model was developed. In the future, we aim to go beyond the mean-field to perform studies [8487]. Quantities such as electric quadrupole transitions B(E2) will also be analyzed to study the shape coexistence and impurity effects of Λ hyperon, as conducted in Refs. [38, 88].

    The authors thank Prof. Shan-Gui Zhou for providing the MDC-RMF code and helpful discussions. The theoretical calculation was supported by the nuclear data storage system at Zhengzhou University.

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Qian-Kun Sun, Ting-Ting Sun, Wei Zhang, Shi-Sheng Zhang and Chen Chen. Possible shape coexistences in the odd-A Ne isotopes and the impurity effects of Λ hyperon[J]. Chinese Physics C. doi: 10.1088/1674-1137/ac6153
Qian-Kun Sun, Ting-Ting Sun, Wei Zhang, Shi-Sheng Zhang and Chen Chen. Possible shape coexistences in the odd-A Ne isotopes and the impurity effects of Λ hyperon[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ac6153 shu
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Possible shape coexistence in odd-A Ne isotopes and the impurity effects of Λ hyperons

  • 1. School of Physics and Microelectronics, Zhengzhou University, Zhengzhou 450001, China
  • 2. School of Physics and Nuclear Energy Engineering, Beihang University, Beijing 100191, China
  • 3. School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China

Abstract: In this study, shape evolution and possible shape coexistence are explored in odd-A Ne isotopes in the framework of the multidimensionally constrained relativistic-mean-field (MDC-RMF) model. By introducing sΛ and pΛ hyperons, the impurity effects on the nuclear shape, energy, size, and density distribution are investigated. For the NN interaction, the PK1 parameter set is adopted, and for the ΛN interaction, the PK1-Y1 parameter set is used. The nuclear ground state and low-lying excited states are determined by blocking the unpaired odd neutron in different orbitals around the Fermi surface. Moreover, the potential energy curves (PECs), quadrupole deformations, nuclear r.m.s. radii, binding energies, and density distributions for the core nuclei as well as the corresponding hypernuclei are analyzed. By examining the PECs, possibilities for shape coexistence in 27,29Ne and a triple shape coexistence in 31Ne are found. In terms of the impurity effects of Λ hyperons, as noted for even-even Ne hypernuclear isotopes, the sΛ hyperon exhibits a clear shrinkage effect, which reduces the nuclear size and results in a more spherical nuclear shape. The pΛ hyperon occupying the 1/2[110] orbital is prolate, which causes the nuclear shape to be more prolate, and the pΛ hyperon occupying the 3/2[101] orbital displays an oblate shape, which drives the nuclei to be more oblate.

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    I.   INTRODUCTION
    • In the extensive nuclide chart, most nuclei are observed to be deformed. In these nuclei, shape evolution and possible shape coexistence, which was first introduced by Morinaga in 1956 [1], have attracted considerable attentions. Later, Heyde and Wood revealed that shape coexistence is ubiquitous and may appear across the entire nuclide chart [24]. Moreover, triple and multiple shape coexistences have been observed or predicted [5, 6]. As one evidence of the appearance of shape coexistence, exploring the delicate interplay between the single-particle and collective behavior of nucleons, which exhibit opposite trends in nuclei, could promote understanding of nuclear collectivity and the nuclear shell structure [7].

      Investigations on hypernuclear systems provide invaluable information for exploring many-body hadronic physics with "strangeness" utilized as a new degree of freedom [813]. In hypernuclei, hyperons have the advantage of being free from the Pauli exclusion principle for nucleons, and as a result, they can move deep into the nuclear interior as an impurity to probe nuclear structures and properties. Numerous studies have been performed on single-Λ hypernuclei both experimentally [1417] and theoretically [1824]. The interesting impurity effects of hyperons have been studied from various aspects, such as the nuclear size and binding energies [25, 26], nuclear cluster structures [18, 19, 27], neutron drip line [2830], nucleon and hyperon skin or halo [3033], and pseudospin symmetry of nucleons [34].

      The impurity effects of Λ hyperons may also be observed from drastic changes in nuclear shape and deformation. Various studies have shown that s-wave and p-wave hyperons have significantly different impurity effects. By including a spherical sΛ hyperon, the deformation of nuclei can be reduced, and the corresponding core nucleus becomes more spherical. For instance, as shown in Ref. [35], research using the axially deformed RMF model for 29ΛSi and 13ΛC revealed that the oblate nuclei 28Si and 12C became spherical after adding an sΛ hyperon. A similar conclusion was obtained in Ref. [36], where the triaxially-deformed RMF model was used to study the potential energy surfaces E(β,γ), which found that the additional sΛ hyperon drove the ground state of light C, Mg, and Si isotopes to a point with a small β and soft γ. In contrast with the sΛ hyperon, a pΛ hyperon exhibits strong polarization effects, which may enhance nuclear deformation [37]. In Ref. [38], the study using the deformed Skyrme-Hartree-Fock (DSHF) model showed that pΛ hyperons in the 1/2[110] and 3/2[101] states had different effects on nuclear deformation, which resulted in more prolate and oblate nuclear shapes, respectively. In recent years, shape-driven effects induced by a valence nucleon(s) in high-spin states have been extensively researched [3941]. The nucleon occupying the high-j and low-Ω orbital can cause the nucleus to be more prolate. A similar hyperon effect has been confirmed that polarizes nuclear shapes [37, 42].

      To describe nuclear deformation with various shape degrees of freedom, it is better to reduce the symmetric restrictions imposed when solving the equations of motion. In CDFT, we can do this with a harmonic oscillator (HO) basis [43, 44] or three-dimensional lattice space [4548]. Recently, Zhou et al. developed the multidimensionally constrained covariant density functional theories (MDC-CDFTs) [4952], which can accommodate various shape degrees of freedom. They applied this model to a series of investigations on, for example, the fission barriers of actinides [50, 51, 5356], nonaxial octupole Y32 correlation in N=150 isotones [57], the third minima and triple-humped barriers in light actinides [58], and potential energy curves (PECs) in the superheavy nucleus 270Hs [59]. Subsequently, by including Λ hyperons, the MDC-CDFTs have been extended to study hypernuclei. Shape evolution in the C, Mg, and Si isotopes and the possible polarization effects of the Λ hyperon [36], the superdeformed states in Ar isotopes [42], the ΛΛ pairing correlations [60], and the new effective ΛN interactions [61] have been either studied or developed. Recently, we used the MDC-CDFTs to explore shape evolution and possible shape coexistence in even-even Ne isotopes [62]. By exploring the PECs, possibilities for shape coexistence were found in nuclei 24,26,28Ne. Furthermore, the impurity effects of the sΛ and pΛ hyperons on the nuclear shape, size, and binding energies were studied.

      In this paper, following our previous study in Ref. [62], shape evolution and possible shape coexistence in odd-A Ne isotopes are explored with the MDC-RMF model by including the blocking effect for the unpaired odd neutron and the impurity effects are investigated by adding a single-Λ hyperon occupying the lowest s or p orbitals. For shape coexistence in odd-A Ne isotopes, in Ref. [63], constrained RMF+BCS calculations with the NL075 force has been performed and possible shape coexistence was predicted in 25,27Ne. The paper is organized as follows: In Sec. II, the MDC-RMF model for single-Λ hypernuclei including the blocking effect is briefly presented. In Sec. III, numerical details are provided. After the results and discussions in Sec. IV, a summary and perspectives are provided in Sec. V.

    II.   MODEL
    • In the meson-exchange MDC-RMF model for single-Λ hypernuclei, the covariant Lagrangian density is composed of two parts:

      L=L0+LΛ,

      (1)

      where L0 is the standard RMF Lagrangian density for nucleons. For details, see Refs. [6468]. In the case of the Lagrangian density for the Λ hyperon LΛ, considering the neutral and isoscalar particle properties, only couplings with scalar-isoscalar σ and vector-isoscalar ω mesons are included, and LΛ is expressed as

      LΛ=ˉψΛ[iγμμmΛgσΛσgωΛγμωμ]ψΛ+fωΛΛ4mΛˉψΛσμνΩμνψΛ,

      (2)

      where the mass of the Λ hyperon, mΛ=1115.6 MeV, the coupling constants of the Λ hyperon with the σ and ω meson fields, gσΛ and gωΛ, and the parameter fωΛΛ in the tensor coupling term between the Λ hyperon and ω field, which is strongly related to small single-Λ spin-orbit splitting [69], constitute the ΛN interaction. The field tensor of the ω field, Ωμν=μωννωμ.

      In the framework of the RMF model, using the variational procedure, the Dirac equations for baryons as well as the Klein-Gordon equations for mesons and photons can be obtained under the mean-field and no-sea approximations. The Λ hyperon satisfies the following Dirac equation:

      [αp+β(mΛ+SΛ)+VΛ+TΛ]ψΛ=ϵψΛ,

      (3)

      where α and β are the Dirac matrices, and SΛ, VΛ, and TΛ represent the scalar, vector, and tensor parts, respectively, of the mean-field potentials for the Λ hyperon,

      SΛ=gσΛσ,

      (4)

      VΛ=gωΛω,

      (5)

      TΛ=fωΛΛ2mΛβ(αp)ω.

      (6)

      In odd-A nuclei, the blocking effect for the unpaired nucleon should be treated, which is of a crucial importance [7072]. The ground state of an odd system is a one-quasiparticle state, and in the BCS approach, it could be described by the following wave function [70, 71]:

      ˆα+k1|BCS=ˆa+k1k>0,kk1(uk+vkˆa+kˆa+ˉk)|0,

      (7)

      where |BCS is the BCS vacuum state, ˆα+k1 and ˆa+k1 are the creation operators for the quasiparticle and single-particle, respectively, and k1 denotes the blocked orbital occupied by the unpaired particle.

      To determine the nuclear ground state and low-lying excited states in an odd-A nucleus, a variety of calculations are performed with the odd nucleon blocked in different single-particle states k around the Fermi surface. As a result, the state with the lowest total binding energy is considered the ground state, while others form the low-lying excited energy spectra.

      To obtain potential energy curves (PECs), constraint calculations [70] with a modified linear constraint method [50, 51] are performed, which has been effectively demonstrated in MDC-RMF calculations compared to the quadratic constraint method [36]. The Routhian is calculated as

      E=ˆH+λμ12CλμQλμ,

      (8)

      where ˆH is the RMF Hamiltonian, and Cλμ are variables that change their values in different iteration steps as follows:

      C(n+1)λμ=C(n)λμ+kλμ(β(n)λμβλμ),

      (9)

      where C(n+1)λμ and C(n)λμ are values obtained in the (n+1)th and nth iterations, respectively, βλμ is the desired nuclear deformation, and β(n)λμ is the calculated value in the nth iteration. Changes in Cλμ and βλμ are linked by a constant kλμ.

      The intrinsic multipole moment Qλμ in Eq. (8) can be calculated with the vector density ρV(r) and spherical harmonics Yλμ(Ω) by

      Qλμ=d3rρV(r)rλYλμ(Ω).

      (10)

      Subsequently, with the multipole moment, the corresponding deformation parameter βλμ can also be calculated using

      βλμ=4π3BRλQλμ,

      (11)

      where R=1.2A1/3fm is the radius of the nucleus, with A as the total number of nucleons, and B represents the number of protons, neutrons, Λ hyperons, or total baryons.

      In an axially symmetric system, we only have the good quantum numbers of parity and the z component of the angular momentum. In this case, calculations with only a single-constraint on the quadrupole deformation β2 will be performed.

    III.   NUMERICAL DETAILS
    • Shape evolution and possible shape coexistence are explored in odd-A Ne isotopes with A=1933 using the self-consistent MDC-RMF model. Moreover, the impurity effects of Λ hyperons occupying the s or p orbitals are investigated. As a continuation of our previous study on even-even Ne isotopes [62], the same numerical details are used.

      In the RMF functional, the PK1 [73] parameter set is used for the NN interaction. For the ΛN interaction, the PK1-Y1 [74] parameter set is adopted, where the scalar-isoscalar coupling constant gσΛ=0.580gσ and vector-isoscalar coupling constant gωΛ=0.620gω, which were determined by fitting the experimentally observed single-Λ binding energies, and the tensor coupling constant fωΛΛ=gωΛ, which was determined by reproducing pΛ spin-orbit splittings in 9ΛBe and 13ΛC.

      To solve the RMF equations, an axially deformed harmonic oscillator (ADHO) basis [43, 44] is taken with the truncation parameters NF=14 for fermion shells and NB=20 for boson shells, as in Ref. [36]. With these parameters, the truncation error in the binding energy of the nucleus 26Si is less than 30 keV.

      In the mean-field approximation, the translational symmetry is broken. To remedy this, the following microscopic center-of-mass (c.m.) correction [75] is employed:

      Ec.m.=12MˆP2,

      (12)

      where M is the total nuclear mass.

      The BCS approach is used to treat the pairing effects with a finite-range separable pairing force [7678].

      V(r1r2)=Gδ(˜R˜R)P(˜r)P(˜r)1ˆPσ2,

      (13)

      where G is the strength of the pairing force, ˜R and ˜r are the center of mass and relative coordinates between the paired particles, respectively, and P(˜r) is a Gaussian shaped function.

      P(˜r)=1(4πa2)3/2e˜r2/a2,

      (14)

      where a is the effective range of the pairing force. Here, the strength G and range a are taken as

      G=728.0MeVfm3,a=0.644fm,

      (15)

      which can be used to obtain the same momentum dependency of the pairing gap in nuclear matter as that of the D1S Gogny force [76].

      In Fig. 1, the binding energy per nucleon E/A, matter radius rm, and quadrupole deformation β2 in the ground states of Ne isotopes determined by unconstrained RMF calculations with the PK1 parameter set are compared with the available experimental data [79]. In general, good consistency with the experimental results is revealed, indicating that the choice of the PK1 parameter set for the description of Ne isotopes is suitable.

      Figure 1.  (color online) Binding energy per nucleon E/A, matter radius rm, and quadrupole deformation β2 as a function of the mass number A in the ground states of Ne isotopes in comparison with the available experimental data [79].

    IV.   RESULTS AND DISCUSSIONS
    • To explore shape coexistence on the mean field level, the PECs are mainly analyzed. If two close-lying energy minima with a difference of a few hundred keV that own prolate and oblate quadrupole deformations are observed in combination with a pronounced barrier between them, possible shape coexistence is indicated because their ground states may have two competing configurations.

      In Fig. 2, the PECs for odd-A Ne isotopes from A=19 to 33 are plotted as functions of the quadrupole deformation β2, which are obtained using constrained calculations with the self-consistent MDC-RMF model. The unpaired odd neutron is blocked in different orbitals around the Fermi surface, and the state with the lowest binding energy is deemed the ground state while others form the local minima.

      Figure 2.  (color online) Potential energy curves (PECs) as a function of the deformation parameter β2 in odd-A Ne isotopes, with the odd neutron blocked in different orbitals around the Fermi surface denoted by the Nilsson quantum numbers Ωπ[Nn3ml]. The open circles denote local energy minima.

      In Fig. 3, to observe the level structures in the Ne isotopes, the single-neutron levels Ω[Nn3ml] obtained using the constrained MDC-RMF calculations are plotted for the even-even nucleus 30Ne, where the solid lines represent levels with positive parity, and the dashed lines represent those with negative parity. At a spherical shape with the deformation β2=0, we observe the neutron shell closures N=8 and N=20, whereas the shell closure N=28 vanishes owing to the inversion of the 1f7/2 and 2p3/2 levels. Referring to this single-neutron level structure, the odd neutron in 1929Ne most likely occupies the 2s,1d orbitals and 2p,1f orbitals in 31,33Ne. Calculations have been performed with all these configurations, and the obtained PECs are presented in Fig. 2, where the local energy minima are marked by open circles. The values of the quadrupole deformations, binding energies, and blocked orbitals of the unpaired odd neutron corresponding to the local energy minima are listed in Table 1.

      Nucleus Quadrupole deformationEnergy
      β2β2n β2p
      19Ne(1/2+[220])0.3610.3030.413142.612
      19Ne(5/2+[202])0.1450.1380.151141.477
      21Ne(3/2+[211])0.5120.5150.509165.917
      21Ne(3/2+[211])0.2010.2090.193161.081
      23Ne(5/2+[202])0.3880.3700.411181.759
      23Ne(1/2+[220])0.2230.2360.205179.701
      25Ne(1/2+[211])0.2060.1930.225193.711
      25Ne(1/2+[211])0.1310.1310.130193.487
      27Ne(1/2+[200])0.1900.1770.213202.836
      27Ne(3/2+[202])0.1300.1350.120202.503
      29Ne(3/2+[202])0.0800.0680.103210.166
      29Ne(1/2+[200])0.0750.0730.079210.144
      31Ne(1/2[321])0.1910.1960.181213.707
      31Ne(7/2[303])0.0900.0890.091213.022
      31Ne(3/2+[202])0.4290.4390.408212.986
      33Ne(3/2[312])0.4190.4280.397216.533
      33Ne(3/2[312])0.1620.1820.116212.418

      Table 1.  Quadrupole deformations and binding energies (in MeV) for ground states and local energy minima (labeled with asterisks) in 1933Ne.

      Figure 3.  (color online) Single-neutron levels Ω[Nn3ml] as a function of the deformation parameter β2 in 30Ne.

      Regarding the nucleus 19Ne with a neutron number just exceeding an N=8 shell closure, it is optimal that the odd neutron occupies the 1d5/2 orbital, which can be splitted into the Ωπ[Nn3ml]=1/2+[220], 3/2+[211], and 5/2+[202] orbitals with axially symmetric quadrupole deformation. Further analysis, shown in Fig. 2(a), demonstrates that the odd neutron blocked in the 1/2+[220] orbital corresponds to the ground state with a prolate deformation of β2=0.361, whereas the odd neutron blocked in 5/2+[202] contributes to the second energy local minimum, which exhibits an oblate shape at β2=0.145. In the case of the unpaired neutron occupying other orbitals, the obtained PECs (denoted by dashed lines) are significantly higher. Hereafter, we focus our attention on the configurations for the ground state and other local energy minima that may generate shape coexistence. With two more neutrons in 21Ne, the odd neutron occupying the 3/2+[211] orbital results in both the ground state and second local energy minimum, which are located at the prolate deformation of β2=0.512 and oblate deformation of β2=0.201, respectively. As for 23Ne, the configurations of the ground state and the second local minimum are opposite to those in 19Ne, that is, the odd neutron occupying the 5/2+[202] orbital leads to the prolate ground state observed at β2=0.388, while the odd neutron blocked in the 1/2+[220] orbital corresponds to the second local energy minimum identified at the deformation β2=0.223. Because the corresponding energy differences between the ground state and second local energy minimum in 19,21,23Ne are as large as 1.135,4.836,2.058 MeV, possible shape coexistence is excluded in those nuclei. With regard to the nucleus 25Ne, the configuration of the unpaired neutron blocked in 1/2+[211] from the spherical 2s1/2 orbital contributes to the lowest PEC, on which the ground state with the prolate deformation β2=0.206 and the second local energy minimum with an oblate shape at β2=0.131 are obtained. Although the energy difference between them is as small as 0.224 MeV, the circumjacent PEC is flat, where different deformations within a limited scope correspond to similar binding energies. In this case, the occurrence of shape coexistence is believed to be difficult. Moving further to the nuclei 27Ne and 29Ne, odd neutrons have great possibilities of occupying the 1d3/2 (1/2+[200] and 3/2+[202]) orbital. For 27Ne, when the odd neutron is blocked in the 1/2+[200] and 3/2+[202] orbitals, the ground state with prolate deformation at β2=0.190 and second energy local minimum exhibiting an oblate shape at β2=0.130 are observed, respectively. A small energy difference of 0.333 MeV in combination with a barrier of 0.984 MeV in height between them are observed, providing great potential for shape coexistence in 27Ne. In the case of 29Ne, with the odd neutron blocked in 3/2+[202], the prolate ground state is identified at a deformation of β2=0.080, whereas the configuration with the odd neutron occupying 1/2+[200] leads to an oblate second local minimum with a deformation of β2=0.075. Similar to 27Ne, owing to the small energy difference of 0.022 MeV coupled with a barrier of 0.4 MeV between the two local energy minima, possible shape coexistence is expected in 29Ne. Far from the N=20 shell closure, the unpaired neutron in the nuclei 31Ne and 33Ne is likely to occupy the 2p3/2 (1/2[321], 3/2[312]) and 1f7/2 (1/2[330], 3/2[321], 5/2[312], 7/2[303]) orbitals. For the nucleus 31Ne, a possible triple shape coexistence is observed, which is comprised of the prolate ground state at β2=0.191 with the odd neutron blocked in 1/2[321], a second oblate local energy minimum at β2=0.090 with the odd neutron in 7/2[303], and a third prolate local minimum at β2=0.429 with the odd neutron in 3/2+[202]. The corresponding excitation energies are 0.685 and 0.721 MeV with respect to the ground state, which together with a proper barrier between them, guarantees the appearance of shape coexistence. In the case of 33Ne, with the odd neutron occupying 3/2[312], both the ground state and the second local minimum are observed owing to a prolate shape at β2=0.419 and an oblate shape at β2=0.162, respectively. Because of the large energy difference of ΔE=4.1 MeV, the appearance of shape coexistence is difficult.

      To explore the impurity effects of Λ hyperons, taking the single-Λ hypernucleus 28ΛNe (or denoted by 27NeΛ) as an example, PECs are plotted in Fig. 4 where the s-wave (Ωπ[Nn3ml]=1/2+[000]) and three p-wave (1/2[110], 1/2[101], and 3/2[101]) single-Λ hyperons are considered (denoted by dashed lines). In panels (a) and (b), the odd neutron is blocked in the 1/2+[200] and 3/2+[202] orbitals, respectively.

      Figure 4.  (color online) PECs as a function of deformation β2 in 27Ne and the single-Λ hypernucleus 28ΛNe (27NeΛ). The odd neutron is blocked in the (a) 1/2+[200] and (b) 3/2+[202] orbitals, and the single-Λ hyperon is injected into the lowest s or p orbitals. The open circles denote the local energy minima.

      Generally, after injecting an sΛ hyperon, the PEC shapes remain almost unchanged, while the depths deepen significantly owing to the attractive ΛN interaction. Moreover, a clear "glue-like" effect of the sΛ hyperon is observed, which results in smaller nuclear deformations and a more spherical nuclear shape. For example, in Fig. 4(a), the reduced deformation of the prolate ground state changes from β2=0.190 to β2=0.165, and in Fig. 4(b), the deformation of the oblate second local minimum reduces to β2=0.115 from β2=0.130.

      Regarding single-pΛ hypernuclei, PECs obtained with the Λ hyperon injected into the 1/2[101] orbital from the 1p1/2 orbital and the 3/2[101] orbital from the 1p3/2 orbital are almost degenerate owing to small spin-orbit splitting; however, they exhibit significant differences from those obtained by injecting the pΛ hyperon into the 1/2[110] orbital. Hereafter, considering that the pΛ hyperon occupying the 1/2[101] and 3/2[101] orbitals exhibit similar effects, we only discuss the latter case. In general, after injecting a pΛ hyperon, the PEC shapes as well as the locations of the local energy minima clearly change. Furthermore, different polarization effects are exhibited by pΛ hyperons occupying the 1/2[110] and 3/2[101] orbitals. For instance, in Fig. 4(a), with the Λ hyperon blocked in the 1/2[110] orbital, the prolate deformation of β2=0.190 corresponding to the nuclear ground state in 27Ne is driven to β2=0.265, whereas it is reduced to β2=0.126 when the pΛ hyperon is blocked in the 3/2[101] orbital. Similar effects by the pΛ hyperon have been observed in Fig. 4(b), where the unpaired neutron in the core nucleus is blocked in the 3/2+[202] orbital, which corresponds to the second local minimum. In detail, the pΛ hyperon in the 1/2[110] state drives the nucleus toward a spherical shape, and the oblate deformation of β2=0.130 corresponding to the second local energy minimum decreases to β2=0.044. Meanwhile, the pΛ hyperon occupying the 3/2[101] orbital causes the hypernucleus to become more oblately deformed with β2=0.162.

      Similar investigations as those of 28ΛNe shown in Fig. 4 have also been performed for other Ne hypernuclei. The same impurity effects of single-Λ hyperons on nuclear deformations and binding energies are obtained. As a result of the introduced Λ hyperons, the nuclear PECs are significantly deepened; however, their increments are varied at different deformations β2. Therefore, the energy difference ΔE between different local energy minima might change, which may influence the possibility of shape coexistence. In Table 2, the values of ΔE in the nuclei 1933Ne are presented in comparison with values after injecting a single-Λ hyperon into the 1/2+[000], 1/2[110], and 3/2[101] orbitals. With the additional sΛ hyperon, the energy difference ΔE decreases significantly in all the hypernuclei. As a result, possible shape coexistence in 27,29,31Ne can persist well in 28,30,3228,30,3ΛNe. For example, the value of ΔE between the ground state and second local energy minimum reduces to 0.238 MeV in 28ΛNe from 0.333 MeV in 27Ne. Moreover, the hypernucleus 20ΛNe becomes a new candidate for possible shape coexistence because ΔE therein is reduced to 0.924MeV. With the addition of a pΛ hyperon, the influence on the value of ΔE becomes complex. With the addition of a single-pΛ hyperon occupying the 1/2[110] orbital, there is a significant increase in ΔE in all hypernuclei, which reduces the possibility of shape coexistence. For instance, the energy difference ΔE in 28ΛNe increases to 1.811MeV from 0.333MeV. Furthermore, one of the local energy minima disappears in 22,2622,2ΛNe with the addition of a pΛ hyperon. Similarly, by including a pΛ hyperon in the state of 3/2[101], the values of ΔE increase in most of the nuclei, and the probabilities of shape coexistence decrease. This differs from those in even-even Ne isotopes [62], where the additional pΛ occupying the 3/2[101] orbital provides a higher possibility of shape coexistence.

      Core nucleiΛ(1/2+[000])Λ(1/2[110])Λ(3/2[101])
      19NeΛ1.1350.9242.9160.138
      21NeΛ4.8364.4343.188
      23NeΛ2.0581.8314.8040.502
      25NeΛ0.2240.113
      27NeΛ0.3330.2381.8110.443
      29NeΛ0.0220.0200.9020.356
      31NeΛ0.6850.6021.9340.019
      33NeΛ4.1153.9556.2182.834

      Table 2.  Energy difference ΔE (in MeV) between the two local energy minima in 1933Ne (core nuclei) and the corresponding single-Λ hypernuclei 1933NeΛ with the Λ hyperon injected into the 1/2+[000], 1/2[110], and 3/2[101] orbitals.

      To study the impurity effects of sΛ and pΛ hyperons on the nuclear quadrupole deformations β2, nuclear root mean square (r.m.s.) radii r, and binding energies E, we take the hypernucleus 28ΛNe as an example and list the corresponding values in Table 3. The single-Λ separation energies SΛ=E(A+1ΛNe)E(ANe) are also given. Configurations with the odd-neutron blocked in the 1/2+[200] and 3/2+[202] (denoted by asterisks) orbitals, which correspond to the ground state and the second local minimum of the core nucleus 27Ne, are considered. In general, the sΛ hyperon is more deeply bound than the pΛ hyperon, which results in a relatively small nuclear size rΛ and a large single-Λ separation energy SΛ. Moreover, the sΛ and pΛ hyperons induce different impurity effects. The sΛ hyperon exhibits a significant shrinkage effect, which makes the nuclei more bound with a smaller size. For example, with the addition of the sΛ hyperon, the nuclear r.m.s. radius in the ground state 27Ne(1/2+[200]) decreases from 3.083 fm to 3.074 fm, the total binding energy E becomes 16.546 MeV deeper, and the nuclear shape tends to be more spherical. For the quadrupole deformation β2 of 27NeΛ(1/2+[000]), it maintains the same sign as the core nucleus. In the case of the sΛ hyperon, its deformation β2Λ is influenced in turn by the core nucleus. For instance, in the hypernucleus 27Ne(1/2+[200])sΛ, the Λ hyperon is slightly prolate with a deformation β2Λ=0.053, the sign of which is the same as that of the core nucleus 27Ne(1/2+[200]). The same behavior is found in 27Ne(3/2+[202])sΛ. In contrast with the sΛ hyperon, the pΛ hyperon may enhance the nuclear size slightly. Additionally, pΛ hyperons in the states of 1/2[110] and 3/2[101] exhibit significantly different effects on nuclear deformation. For the pΛ hyperon occupying 1/2[110], which is a prolate shape, the core nucleus becomes more prolate. Conversely, for the hyperon occupying 3/2[101] or 1/2[101], which is oblate, the core nucleus becomes more oblate or less prolate. For instance, in 27Ne(1/2+[200])pΛ(1/2[110]), because both the pΛ hyperon and core nucleus are prolate, the pΛ hyperon enhances the nuclear prolate deformation from β2=0.190 to β2=0.265, while shape decoupling occurs in 27Ne(1/2+[200])pΛ(3/2[101]), where the pΛ hyperon and core nucleus have different shapes, which leads to a reduction in the total nuclear deformation from β2=0.190 to β2=0.126. In Refs. [80, 81], shape decoupling in the deformed halo nuclei 42,44Mg are discussed, in which the shapes of the core and outside halo are different.

      NucleusQuadrupole deformationr.m.s. radii/fmEnergy/MeV
      β2β2nβ2pβ2ΛrmrcorerΛESΛ
      27Ne(1/2+[200])0.1900.1770.2133.083202.836
      27NeΛ(1/2+[000])0.1650.1570.1880.0533.0563.0742.511219.38216.546
      27NeΛ(1/2[110])0.2650.2170.2621.0873.1053.0913.461210.6937.857
      27NeΛ(1/2[101])0.1280.1400.1620.4143.0933.0793.443209.1336.297
      27NeΛ(3/2[101])0.1260.1380.1610.4293.0923.0793.431209.3176.481
      27Ne(3/2+[202])0.1300.1350.1203.077202.503
      27NeΛ(1/2+[000])0.1150.1240.1090.0333.0513.0692.505219.14416.641
      27NeΛ(1/2[110])0.0440.0820.0610.7743.0883.0753.432208.8826.379
      27NeΛ(1/2[101])0.1620.1550.1390.5093.0943.0803.444209.6257.122
      27NeΛ(3/2[101])0.1620.1540.1390.5303.0943.0803.439209.7607.257

      Table 3.  Quadrupole deformation parameters, root mean square (r.m.s.) radii, binding energies, and Λ separation energies in the odd-A nucleus 27Ne and the corresponding single-Λ hypernuclei 27NeΛ with the Λ hyperon injected into the lowest s orbital or three p orbitals. The odd neutron is blocked in the 1/2+[200] and 3/2+[202] orbitals, which correspond to the ground state and the second local energy minimum (denoted by asterisks), respectively.

      In Fig. 5, the density distributions ρ(r,z) in the r-z plane with the symmetric axis along the z-axis are shown for the nucleus 27Ne (a) and the corresponding hypernucleus 28ΛNe (b,c,d). Cases with the core nucleus 27Ne in the ground state and the second local minimum (denoted by asterisks) as well as the single-Λ hyperon occupying the 1/2+[000] (b), 1/2[110] (c), and 3/2[101] (d) orbitals are studied. In the upper and lower parts of panels (b–d), densities are plotted for the total hypernucleus and single-Λ hyperon, respectively. In Fig. 5(a), a prolate shape is observed for the ground state of 27Ne, while an oblate shape is observed for the second energy minimum. After introducing an sΛ hyperon, in Fig. 5(b), the nuclear shapes remain almost unchanged but with an increment of inner density. Meanwhile, the injected sΛ hyperon is slightly deformed and has the same shape as the core nucleus. In Fig. 5(c), a "dumbbell" shape is observed for the prolate pΛ hyperon in the state of 1/2[110], the addition of which increases (reduces) the prolate (oblate) deformation of the core nucleus in the ground state (second energy minimum). At the same time, owing to the coupling of the pΛ hyperon and nuclear core, slight differences in the density distributions of the pΛ hyperon are observed in the two cases (panel c). In Fig. 5(d), a "ring" shape is observed for the oblate pΛ hyperon in the state of 3/2[101]. As a result, deformation of the prolate ground state decreases, and the oblate deformation of the second energy minimum is enhanced. A similar analysis of the impurity effects of the pΛ hyperon on the nuclear density can be found in Refs. [38, 62].

      Figure 5.  (color online) Two-dimensional density distributions in the r-z plane for (a) the core nucleus 27Ne and (b), (c), (d) the corresponding single-Λ hypernuclei with the Λ hyperon injected into the (b) 1/2+[000], (c) 1/2[110], and (d) 3/2[101] orbitals, respectively. The odd neutron in the ground state and the second local minimum (denoted by asterisks) occupies the 1/2+[200] and 3/2+[202] orbitals, respectively. The upper and lower parts of panels (b), (c), and (d) are the densities of the entire hypernuclei and the single-Λ hyperon, respectively.

      To explain the mechanism behind the effects of sΛ and pΛ hyperons on nuclear densities and shapes in an intuitive way, taking 27NeΛ with the core nucleus in the ground state an example, we presentdensity distributions ρ along the r- and z-axes in Fig. 6. For the sΛ hyperon, almost the same density distributions (red dashed lines in the lower part of each panel) are found along the r- and z-axes; however, the distribution is slightly extended along the z-axis, which suggests the sΛ hyperon has a weakly prolate shape. With the addition of an sΛ hyperon to 27Ne, the nuclear density clearly increases in the inner part (r<3fm, z<2.5 fm) both in the directions of the r- and z-axes. However, for the densities on the outer nuclear side, different influences are found, that is, ρ(r) becomes slightly more extended while ρ(z) shrinks. These cause 27NeΛ to become more spherical compared to the core nucleus. In the case of the pΛ hyperon occupying 1/2[110], ρ(r) vanishes and a visible distribution is presented along the z-axis with a maximum at z=2.5 fm. As a result, a clear increment in the nuclear density along the z-axis is induced, and the nuclear shape becomes more prolate. For the pΛ hyperon in the state of 3/2[101], the nuclear density is enhanced in the direction of the r-axis and prolate deformation is weakened.

      Figure 6.  (color online) Density distributions ρ(r,z) along the r-axis (a) and z-axis (b) in 27Ne with the odd neutron occupying 1/2+[200] (solid lines) and the corresponding single-Λ hypernuclei (dashed lines) with the Λ hyperon injected into the 1/2+[000] (red short dashed line), 1/2[110] (blue dash-dotted line), and 3/2[101] (olive dashed line) orbitals. In the upper and lower parts of each panel, density distributions contributed by the total nuclei and the Λ hyperon are plotted, respectively.

    V.   SUMMARY AND PERSPECTIVES
    • In this study, following our previous research [Sci. China-Phys. Mech. Astron. 64, 282011(2021)], shape evolution and possible shape coexistence in odd-A Ne isotopes are explored using the MDC-RMF model, which has achieved great success in describing nuclear deformations. Moreover, by introducing s- or p-wave Λ hyperons, the impurity effects on nuclear shape, energy, size, and density distribution are discussed. For NN and ΛN interactions in the RMF functional, the PK1 and PK1-Y1 parameter sets are adopted, respectively.

      By blocking the unpaired odd neutron in different orbitals around the Fermi surface, the nuclear ground state and other local energy minima are determined, and by examining the PECs, possible shape coexistences in 27,29Ne and a possible triple shape coexistence in 31Ne are predicted. For 27Ne, with the odd neutron blocked in the 1/2+[200] and 3/2+[202] orbitals, the ground state is observed in a prolate shape at a deformation of β2=0.190 and the second energy minimum exhibits an oblate shape at β2=0.130, respectively. A small energy difference of 0.333 MeV and a barrier with a height of 0.984 MeV between them are observed, supporting possible shape coexistence in 27Ne. Similarly, in 29Ne, with the odd neutron blocked in the 3/2+[202] orbital, the prolate ground state is identified at β2=0.080, while the configuration with the odd neutron occupying 1/2+[200] leads to an oblate second local minimum at β2=0.075. In the case of 31Ne, possible triple shape coexistence, including the ground state at β2=0.191 with the odd neutron blocked at 1/2[321], the second local energy minimum at oblate deformation β2=0.090 with the odd neutron at 7/2[303], and the third local minimum at prolate deformation β2=0.429 with the odd neutron at 3/2+[202], is predicated.

      To discuss the impurity effects of sΛ and pΛ hyperons, nuclear quadrupole deformations, r.m.s. radii, binding energies, and density distributions are compared in detail for 27Ne and the corresponding hypernuclei 28ΛNe in cases where the odd neutron occupies the 1/2+[200] and 3/2+[202] orbitals. The sΛ hyperon exhibits clear shrinkage effects, which reduces the nuclear size and deformation. In contrast, the pΛ hyperon exhibits strong polarization effects, which may enhance the nuclear deformation. Meanwhile, the pΛ hyperons occupying different orbitals exhibit different effects: the hyperon from the 1/2[110] orbital drives the nuclear shape to become more prolate, and the hyperon in the 3/2[101] or 1/2[101] state results in a more oblate nuclear shape. These conclusions are consistent with those for even-even Ne isotopes [62]. Furthermore, with the addition of the sΛ hyperon, the energy difference ΔE between the ground state and second local energy minimum decreases, which may increase the probability of shape coexistence. However, introducing a pΛ hyperon results in a distinct disadvantage for shape coexistence in most of hypernuclei.

      In the present study, we investigate shape evolution and possible shape coexistence at the mean-field level by analyzing PECs. This is the first step in our research. Recently, in Refs. [82, 83], the angular momentum and parity projected multidimensionally constrained relativistic Hartree-Bogoliubov model was developed. In the future, we aim to go beyond the mean-field to perform studies [8487]. Quantities such as electric quadrupole transitions B(E2) will also be analyzed to study the shape coexistence and impurity effects of Λ hyperon, as conducted in Refs. [38, 88].

    ACKNOWLEDGMENTS
    • The authors thank Prof. Shan-Gui Zhou for providing the MDC-RMF code and helpful discussions. The theoretical calculation was supported by the nuclear data storage system at Zhengzhou University.

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