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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 [2–4]. 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 [8–13]. 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 [14–17] and theoretically [18–24]. 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 [28–30], nucleon and hyperon skin or halo [30–33], 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 for29Λ Si and13Λ C revealed that the oblate nuclei 28Si and 12C became spherical after adding ansΛ hyperon. A similar conclusion was obtained in Ref. [36], where the triaxially-deformed RMF model was used to study the potential energy surfacesE∼(β,γ) , which found that the additionalsΛ 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 thesΛ hyperon, apΛ 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 thatpΛ hyperons in the1/2−[110] and3/2−[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 [39–41]. 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 [45–48]. Recently, Zhou et al. developed the multidimensionally constrained covariant density functional theories (MDC-CDFTs) [49–52], 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, 53–56], nonaxial octupole
Y32 correlation inN=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], theΛΛ pairing correlations [60], and the new effectiveΛN 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 nuclei24,26,28 Ne. Furthermore, the impurity effects of thesΛ andpΛ 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
25,27 Ne. 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. [64–68]. In the case of the Lagrangian density for the Λ hyperonLΛ , considering the neutral and isoscalar particle properties, only couplings with scalar-isoscalar σ and vector-isoscalar ω mesons are included, andLΛ is expressed asLΛ=ˉψΛ[iγμ∂μ−mΛ−gσΛσ−gωΛγμωμ]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 andgωΛ , and the parameterfωΛΛ 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 theΛN interaction. The field tensor of the ω field,Ωμν=∂μων−∂νωμ .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:
[α⋅p+β(mΛ+SΛ)+VΛ+TΛ]ψΛ=ϵψΛ,
(3) where
α and β are the Dirac matrices, andSΛ ,VΛ , andTΛ 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(α⋅p)ω.
(6) In odd-A nuclei, the blocking effect for the unpaired nucleon should be treated, which is of a crucial importance [70–72]. 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+k1∏k>0,k≠k1(uk+vkˆa+kˆa+ˉk)|0⟩,
(7) where
|BCS⟩ is the BCS vacuum state,ˆα+k1 andˆa+k1 are the creation operators for the quasiparticle and single-particle, respectively, andk1 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, andCλμ are variables that change their values in different iteration steps as follows:C(n+1)λμ=C(n)λμ+kλμ(β(n)λμ−βλμ),
(9) where
C(n+1)λμ andC(n)λμ are values obtained in the(n+1) th and nth iterations, respectively,βλμ is the desired nuclear deformation, andβ(n)λμ is the calculated value in the nth iteration. Changes inCλμ andβλμ are linked by a constantkλμ .The intrinsic multipole moment
Qλμ in Eq. (8) can be calculated with the vector densityρV(r) and spherical harmonicsYλμ(Ω) byQλμ=∫d3rρV(r)rλYλμ(Ω).
(10) Subsequently, with the multipole moment, the corresponding deformation parameter
βλμ can also be calculated usingβλμ=4π3BRλQλμ,
(11) where
R=1.2A1/3 fm 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=19−33 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 theΛN interaction, the PK1-Y1 [74] parameter set is adopted, where the scalar-isoscalar coupling constantgσΛ=0.580gσ and vector-isoscalar coupling constantgωΛ=0.620gω , which were determined by fitting the experimentally observed single-Λ binding energies, and the tensor coupling constantfωΛΛ=−gωΛ , which was determined by reproducingpΛ spin-orbit splittings in9Λ Be and13Λ C.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 andNB=20 for boson shells, as in Ref. [36]. With these parameters, the truncation error in the binding energy of the nucleus 26Si is less than30 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.=−12M⟨ˆP2⟩,
(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 [76–78].
V(r1−r2)=−Gδ(˜R−˜R′)P(˜r)P(˜r′)1−ˆPσ2,
(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, andP(˜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=728.0MeV⋅fm3,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 radiusrm , and quadrupole deformationβ2 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 radiusrm , and quadrupole deformationβ2 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 to33 are plotted as functions of the quadrupole deformationβ2 , 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 numbersΩπ[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 deformationβ2=0 , we observe the neutron shell closuresN=8 andN=20 , whereas the shell closureN=28 vanishes owing to the inversion of the1f7/2 and2p3/2 levels. Referring to this single-neutron level structure, the odd neutron in19−29 Ne most likely occupies the2s,1d orbitals and2p,1f orbitals in31,33 Ne. 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 deformation Energy β2 β2n β2p 19Ne(1/2+[220]) 0.361 0.303 0.413 − 142.61219Ne∗(5/2+[202]) − 0.145− 0.138− 0.151− 141.47721Ne(3/2+[211]) 0.512 0.515 0.509 − 165.91721Ne∗(3/2+[211]) − 0.201− 0.209− 0.193− 161.08123Ne(5/2+[202]) 0.388 0.370 0.411 − 181.75923Ne∗(1/2+[220]) − 0.223− 0.236− 0.205− 179.70125Ne(1/2+[211]) 0.206 0.193 0.225 − 193.71125Ne∗(1/2+[211]) − 0.131− 0.131− 0.130− 193.48727Ne(1/2+[200]) 0.190 0.177 0.213 − 202.83627Ne∗(3/2+[202]) − 0.130− 0.135− 0.120− 202.50329Ne(3/2+[202]) 0.080 0.068 0.103 − 210.16629Ne∗(1/2+[200]) − 0.075− 0.073− 0.079− 210.14431Ne(1/2−[321]) 0.191 0.196 0.181 − 213.70731Ne∗(7/2−[303]) − 0.090− 0.089− 0.091− 213.02231Ne∗(3/2+[202]) 0.429 0.439 0.408 − 212.98633Ne(3/2−[312]) 0.419 0.428 0.397 − 216.53333Ne∗(3/2−[312]) − 0.162− 0.182− 0.116− 212.418Table 1. Quadrupole deformations and binding energies (in MeV) for ground states and local energy minima (labeled with asterisks) in
19−33Ne .Figure 3. (color online) Single-neutron levels
Ω[Nn3ml] as a function of the deformation parameterβ2 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 the1d5/2 orbital, which can be splitted into theΩπ[Nn3ml]=1/2+[220] ,3/2+[211] , and5/2+[202] orbitals with axially symmetric quadrupole deformation. Further analysis, shown in Fig. 2(a), demonstrates that the odd neutron blocked in the1/2+[220] orbital corresponds to the ground state with a prolate deformation ofβ2=0.361 , whereas the odd neutron blocked in5/2+[202] contributes to the second energy local minimum, which exhibits an oblate shape atβ2=−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 the3/2+[211] orbital results in both the ground state and second local energy minimum, which are located at the prolate deformation ofβ2=0.512 and oblate deformation ofβ2=−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 the5/2+[202] orbital leads to the prolate ground state observed atβ2=0.388 , while the odd neutron blocked in the1/2+[220] orbital corresponds to the second local energy minimum identified at the deformationβ2=−0.223 . Because the corresponding energy differences between the ground state and second local energy minimum in19,21,23 Ne are as large as1.135,4.836,2.058 MeV, possible shape coexistence is excluded in those nuclei. With regard to the nucleus 25Ne, the configuration of the unpaired neutron blocked in1/2+[211] from the spherical2s1/2 orbital contributes to the lowest PEC, on which the ground state with the prolate deformationβ2=0.206 and the second local energy minimum with an oblate shape atβ2=−0.131 are obtained. Although the energy difference between them is as small as0.224 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 the1d3/2 (1/2+[200] and3/2+[202] ) orbital. For 27Ne, when the odd neutron is blocked in the1/2+[200] and3/2+[202] orbitals, the ground state with prolate deformation atβ2=0.190 and second energy local minimum exhibiting an oblate shape atβ2=−0.130 are observed, respectively. A small energy difference of0.333 MeV in combination with a barrier of0.984 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 in3/2+[202] , the prolate ground state is identified at a deformation ofβ2=0.080 , whereas the configuration with the odd neutron occupying1/2+[200] leads to an oblate second local minimum with a deformation ofβ2=−0.075 . Similar to 27Ne, owing to the small energy difference of0.022 MeV coupled with a barrier of0.4 MeV between the two local energy minima, possible shape coexistence is expected in 29Ne. Far from theN=20 shell closure, the unpaired neutron in the nuclei 31Ne and 33Ne is likely to occupy the2p3/2 (1/2−[321] ,3/2−[312] ) and1f7/2 (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 atβ2=0.191 with the odd neutron blocked in1/2−[321] , a second oblate local energy minimum atβ2=−0.090 with the odd neutron in7/2−[303] , and a third prolate local minimum atβ2=0.429 with the odd neutron in3/2+[202] . The corresponding excitation energies are0.685 and0.721 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 occupying3/2−[312] , both the ground state and the second local minimum are observed owing to a prolate shape atβ2=0.419 and an oblate shape atβ2=−0.162 , respectively. Because of the large energy difference ofΔE=4.1 MeV, the appearance of shape coexistence is difficult.To explore the impurity effects of Λ hyperons, taking the single-Λ hypernucleus
28Λ Ne (or denoted by27Ne⊗Λ ) 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] , and3/2−[101] ) single-Λ hyperons are considered (denoted by dashed lines). In panels (a) and (b), the odd neutron is blocked in the1/2+[200] and3/2+[202] orbitals, respectively.Figure 4. (color online) PECs as a function of deformation
β2 in27Ne and the single-Λ hypernucleus28Λ Ne (27Ne⊗Λ ). 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 attractiveΛN interaction. Moreover, a clear "glue-like" effect of thesΛ 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 fromβ2=0.190 toβ2=0.165 , and in Fig. 4(b), the deformation of the oblate second local minimum reduces toβ2=−0.115 fromβ2=−0.130 .Regarding single-
pΛ hypernuclei, PECs obtained with the Λ hyperon injected into the1/2−[101] orbital from the1p1/2 orbital and the3/2−[101] orbital from the1p3/2 orbital are almost degenerate owing to small spin-orbit splitting; however, they exhibit significant differences from those obtained by injecting thepΛ hyperon into the1/2−[110] orbital. Hereafter, considering that thepΛ hyperon occupying the1/2−[101] and3/2−[101] orbitals exhibit similar effects, we only discuss the latter case. In general, after injecting apΛ hyperon, the PEC shapes as well as the locations of the local energy minima clearly change. Furthermore, different polarization effects are exhibited bypΛ hyperons occupying the1/2−[110] and3/2−[101] orbitals. For instance, in Fig. 4(a), with the Λ hyperon blocked in the1/2−[110] orbital, the prolate deformation ofβ2=0.190 corresponding to the nuclear ground state in 27Ne is driven toβ2=0.265 , whereas it is reduced toβ2=0.126 when thepΛ hyperon is blocked in the3/2−[101] orbital. Similar effects by thepΛ hyperon have been observed in Fig. 4(b), where the unpaired neutron in the core nucleus is blocked in the3/2+[202] orbital, which corresponds to the second local minimum. In detail, thepΛ hyperon in the1/2−[110] state drives the nucleus toward a spherical shape, and the oblate deformation ofβ2=−0.130 corresponding to the second local energy minimum decreases toβ2=−0.044 . Meanwhile, thepΛ hyperon occupying the3/2−[101] orbital causes the hypernucleus to become more oblately deformed withβ2=−0.162 .Similar investigations as those of
28Λ Ne 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 deformationsβ2 . Therefore, the energy differenceΔE between different local energy minima might change, which may influence the possibility of shape coexistence. In Table 2, the values ofΔE in the nuclei19−33 Ne are presented in comparison with values after injecting a single-Λ hyperon into the1/2+[000] ,1/2−[110] , and3/2−[101] orbitals. With the additionalsΛ hyperon, the energy differenceΔE decreases significantly in all the hypernuclei. As a result, possible shape coexistence in27,29,31 Ne can persist well in28,30,3228,30,3Λ Ne. For example, the value ofΔE between the ground state and second local energy minimum reduces to0.238 MeV in28Λ Ne from0.333 MeV in 27Ne. Moreover, the hypernucleus20Λ Ne becomes a new candidate for possible shape coexistence becauseΔE therein is reduced to0.924 MeV. With the addition of apΛ hyperon, the influence on the value ofΔE becomes complex. With the addition of a single-pΛ hyperon occupying the1/2−[110] orbital, there is a significant increase inΔE in all hypernuclei, which reduces the possibility of shape coexistence. For instance, the energy differenceΔE in28ΛNe increases to1.811MeV from0.333 MeV. Furthermore, one of the local energy minima disappears in22,2622,2Λ Ne with the addition of apΛ hyperon. Similarly, by including apΛ hyperon in the state of3/2−[101] , the values ofΔE 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 additionalpΛ occupying the3/2−[101] orbital provides a higher possibility of shape coexistence.Core nuclei Λ(1/2+[000]) Λ(1/2−[110]) Λ(3/2−[101]) 19Ne ⊗Λ 1.135 0.924 2.916 0.138 21Ne ⊗Λ 4.836 4.434 − 3.188 23Ne ⊗Λ 2.058 1.831 4.804 0.502 25Ne ⊗Λ 0.224 0.113 − − 27Ne ⊗Λ 0.333 0.238 1.811 − 0.44329Ne ⊗Λ 0.022 0.020 0.902 − 0.35631Ne ⊗Λ 0.685 0.602 1.934 − 0.01933Ne ⊗Λ 4.115 3.955 6.218 2.834 Table 2. Energy difference
ΔE (in MeV) between the two local energy minima in19−33 Ne (core nuclei) and the corresponding single-Λ hypernuclei19−33 Ne⊗Λ with the Λ hyperon injected into the1/2+[000] ,1/2−[110] , and3/2−[101] orbitals.To study the impurity effects of
sΛ andpΛ hyperons on the nuclear quadrupole deformationsβ2 , nuclear root mean square (r.m.s.) radii r, and binding energies E, we take the hypernucleus28Λ Ne as an example and list the corresponding values in Table 3. The single-Λ separation energiesSΛ=E(A+1ΛNe)−E(ANe) are also given. Configurations with the odd-neutron blocked in the1/2+[200] and3/2+[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, thesΛ hyperon is more deeply bound than thepΛ hyperon, which results in a relatively small nuclear sizerΛ and a large single-Λ separation energySΛ . Moreover, thesΛ andpΛ hyperons induce different impurity effects. ThesΛ hyperon exhibits a significant shrinkage effect, which makes the nuclei more bound with a smaller size. For example, with the addition of thesΛ hyperon, the nuclear r.m.s. radius in the ground state 27Ne(1/2+[200]) decreases from3.083 fm to3.074 fm, the total binding energy E becomes16.546 MeV deeper, and the nuclear shape tends to be more spherical. For the quadrupole deformationβ2 of 27Ne⊗Λ(1/2+[000]) , it maintains the same sign as the core nucleus. In the case of thesΛ hyperon, its deformationβ2Λ 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 deformationβ2Λ=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 thesΛ hyperon, thepΛ hyperon may enhance the nuclear size slightly. Additionally,pΛ hyperons in the states of1/2−[110] and3/2−[101] exhibit significantly different effects on nuclear deformation. For thepΛ hyperon occupying1/2−[110] , which is a prolate shape, the core nucleus becomes more prolate. Conversely, for the hyperon occupying3/2−[101] or1/2−[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 thepΛ hyperon and core nucleus are prolate, thepΛ hyperon enhances the nuclear prolate deformation fromβ2=0.190 toβ2=0.265 , while shape decoupling occurs in 27Ne(1/2+[200])⊗pΛ(3/2−[101]) , where thepΛ hyperon and core nucleus have different shapes, which leads to a reduction in the total nuclear deformation fromβ2=0.190 toβ2=0.126 . In Refs. [80, 81], shape decoupling in the deformed halo nuclei42,44 Mg are discussed, in which the shapes of the core and outside halo are different.Nucleus Quadrupole deformation r.m.s. radii/fm Energy/MeV β2 β2n β2p β2Λ rm rcore rΛ E SΛ 27Ne(1/2+[200]) 0.190 0.177 0.213 3.083 − 202.83627Ne⊗Λ(1/2+[000]) 0.165 0.157 0.188 0.053 3.056 3.074 2.511 − 219.38216.546 27Ne⊗Λ(1/2−[110]) 0.265 0.217 0.262 1.087 3.105 3.091 3.461 − 210.6937.857 27Ne⊗Λ(1/2−[101]) 0.128 0.140 0.162 − 0.4143.093 3.079 3.443 − 209.1336.297 27Ne⊗Λ(3/2−[101]) 0.126 0.138 0.161 − 0.4293.092 3.079 3.431 − 209.3176.481 27Ne∗(3/2+[202]) − 0.130− 0.135− 0.1203.077 − 202.50327Ne∗⊗Λ(1/2+[000]) − 0.115− 0.124− 0.109− 0.0333.051 3.069 2.505 − 219.14416.641 27Ne∗⊗Λ(1/2−[110]) − 0.044− 0.082− 0.0610.774 3.088 3.075 3.432 − 208.8826.379 27Ne∗⊗Λ(1/2−[101]) − 0.162− 0.155− 0.139− 0.5093.094 3.080 3.444 − 209.6257.122 27Ne∗⊗Λ(3/2−[101]) − 0.162− 0.154− 0.139− 0.5303.094 3.080 3.439 − 209.7607.257 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 hypernuclei27Ne⊗Λ with the Λ hyperon injected into the lowest s orbital or three p orbitals. The odd neutron is blocked in the1/2+[200] and3/2+[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 hypernucleus28ΛNe (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 the1/2+[000] (b),1/2−[110] (c), and3/2−[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 ansΛ hyperon, in Fig. 5(b), the nuclear shapes remain almost unchanged but with an increment of inner density. Meanwhile, the injectedsΛ hyperon is slightly deformed and has the same shape as the core nucleus. In Fig. 5(c), a "dumbbell" shape is observed for the prolatepΛ hyperon in the state of1/2−[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 thepΛ hyperon and nuclear core, slight differences in the density distributions of thepΛ hyperon are observed in the two cases (panel c). In Fig. 5(d), a "ring" shape is observed for the oblatepΛ hyperon in the state of3/2−[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 thepΛ 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 the1/2+[200] and3/2+[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 andpΛ hyperons on nuclear densities and shapes in an intuitive way, taking27Ne⊗Λ with the core nucleus in the ground state an example, we presentdensity distributions ρ along the r- and z-axes in Fig. 6. For thesΛ 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 thesΛ hyperon has a weakly prolate shape. With the addition of ansΛ hyperon to 27Ne, the nuclear density clearly increases in the inner part (r<3 fm,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 whileρ(z) shrinks. These cause 27Ne⊗Λ to become more spherical compared to the core nucleus. In the case of thepΛ hyperon occupying1/2−[110] ,ρ(r) vanishes and a visible distribution is presented along the z-axis with a maximum atz=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 thepΛ hyperon in the state of3/2−[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 occupying1/2+[200] (solid lines) and the corresponding single-Λ hypernuclei (dashed lines) with the Λ hyperon injected into the1/2+[000] (red short dashed line),1/2−[110] (blue dash-dotted line), and3/2−[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 andΛN 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
27,29 Ne and a possible triple shape coexistence in 31Ne are predicted. For 27Ne, with the odd neutron blocked in the1/2+[200] and3/2+[202] orbitals, the ground state is observed in a prolate shape at a deformation ofβ2=0.190 and the second energy minimum exhibits an oblate shape atβ2=−0.130 , respectively. A small energy difference of0.333 MeV and a barrier with a height of0.984 MeV between them are observed, supporting possible shape coexistence in 27Ne. Similarly, in 29Ne, with the odd neutron blocked in the3/2+[202] orbital, the prolate ground state is identified atβ2=0.080 , while the configuration with the odd neutron occupying1/2+[200] leads to an oblate second local minimum atβ2=−0.075 . In the case of 31Ne, possible triple shape coexistence, including the ground state atβ2=0.191 with the odd neutron blocked at1/2−[321] , the second local energy minimum at oblate deformationβ2=−0.090 with the odd neutron at7/2−[303] , and the third local minimum at prolate deformationβ2=0.429 with the odd neutron at3/2+[202] , is predicated.To discuss the impurity effects of
sΛ andpΛ hyperons, nuclear quadrupole deformations, r.m.s. radii, binding energies, and density distributions are compared in detail for 27Ne and the corresponding hypernuclei28Λ Ne in cases where the odd neutron occupies the1/2+[200] and3/2+[202] orbitals. ThesΛ hyperon exhibits clear shrinkage effects, which reduces the nuclear size and deformation. In contrast, thepΛ hyperon exhibits strong polarization effects, which may enhance the nuclear deformation. Meanwhile, thepΛ hyperons occupying different orbitals exhibit different effects: the hyperon from the1/2−[110] orbital drives the nuclear shape to become more prolate, and the hyperon in the3/2−[101] or1/2−[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 thesΛ hyperon, the energy differenceΔE between the ground state and second local energy minimum decreases, which may increase the probability of shape coexistence. However, introducing apΛ 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 [84–87]. 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.
