-
Since the first discovery of a hypernucleus in cosmic rays [1], the study of hypernuclei has become one of the most interesting topics in nuclear physics on both the experimental and theoretical sides [2–8]. In particular,
Λ hypernuclei have been studied by many experiments and related theoretical analyses. Due to the limitation of current experimental conditions, the experimental data of hypernuclei are mainly for single-Λ hypernuclei [2, 3, 6, 7, 9–14], with only a few data available for double-Λ hypernuclei [4, 6, 8, 15–21]. Of particular interest is the fact that the addition ofΛ hyperons can lead to the appearance of the so-called impurity effect [2, 7], since the hyperon(s) can enter deeply into the center of a hypernucleus regardless of the restriction of the Pauli exclusion principle, and thus can be used as a good probe to study the nuclear environment. The impurity effects of single-Λ hypernuclei have been investigated extensively in the past decades, such as the shrinkage effect [22–24], the modification of the drip lines [25, 26], and the modification of the deformation [27–33].Since the experimental discovery of double-
Λ hypernuclei [15–21], several kinds of nuclear models have been extended to study theΛΛ hypernuclei sector. For example, the microscopic cluster model was used to study theΛ binding energies of light hypernuclei and reproduced well the observation of the ground state of11ΛΛ Be [34, 35]. The Faddeev calculations with the Nijmegen soft-core potential NSC97 described well the binding energies of lightΛΛ hypernuclei [36]. The shell-model calculation showed how theΛN spin-dependent interaction terms influence theΛΛ hypernuclei across the nuclearp shell [37]. The beyond-mean-field approach was applied to study the evolution of nuclear deformation inΛΛ hypernuclei and the hyperon impurity effect in hypernuclei with shape coexistence [38, 39]. Extensive research of binding energies and deformation effects has been carried out by the self-consistent mean-field model on the shape of hypernuclei [26, 27, 29, 40–44].Very recently, the impurity effects of multiple
Λ hyperons on the deformations in the hyperisotope chains8+n nΛ Be (n=2 , 4),20+n nΛ Ne (n=2 , 8), and28+n nΛ Si (n=2 , 8) have been studied using relativistic mean field (RMF) theory in Ref. [24]. It was pointed out that in the Ne hyperisotopes, the deformation is slightly reduced by the additionalΛ hyperons, whereas it is significantly reduced or even disappears in the Si hyperisotopes. Studies on multi-Λ hyperisotopes have theoretical significance, although the corresponding experiments are currently unfeasible. First of all, the impurity effects in multi-Λ hyperisotopes are evidently stronger than those of single-Λ ones. Moreover, a multi-Λ system can provide important information on theΛΛ interaction, and the effects of the core nucleus on theΛ hyperons can be studied.The aforementioned work in Ref. [24] studied only a few nuclei and did not address the impurity effects caused by six
Λ hyperons. Therefore, further studies on the impurity effect ofn= 2, 4, 6, 8Λ hyperons on the properties of even–even nuclei ranging from8 Be to40 Ca will be carried out in this work. In contrast to the RMF model adopted in Ref. [24], we will employ the deformed Skyrme–Hartree–Fock (SHF) approach [45–47], which is one of the widely used models for hypernuclei [26, 43, 48]. -
In the framework of the SHF approach, the energy of a hypernucleus is given by an energy-density functional,
E=∫d3rε(r) ,ε=εNN+εΛN+εΛΛ,
(1) where
εNN ,εΛN , andεΛΛ account for the nucleon–nucleon (NN ) interaction, the hyperon–nucleon (YN ) interaction, and the hyperon–hyperon (YY ) interaction, respectively. The energy-density functional depends on the one-body densitiesρq , kinetic densitiesτq , and spin–orbit currentsJq ,[ρq,τq,Jq]=Nq∑i=1niq[|ϕiq|2,|∇ϕiq|2,ϕiq∗(∇ϕiq×σ)/i],
(2) where
ϕiq (i=1,…,Nq ) are the self-consistently calculated single-particle (s.p.) wave functions of theNq occupied states for the speciesq=n,p,Λ in a hypernucleus. They satisfy the Schrödinger equation, obtained by the minimization of the total energy functional (1) according to the variational principle,[∇⋅12m∗q(r)∇−Vq(r)+iWq(r)⋅(∇×σ)]ϕiq(r)=eiqϕiq(r),
(3) in which
Wq(r) is the spin–orbit interaction part for the nucleons as given in Refs. [45, 49]. The central mean fieldsVq(r) , corrected by the effective-mass terms following the procedure described in Refs. [40, 42, 50] areVN=VSHFN+∂εNΛ∂ρN+∂∂ρN(mΛm∗Λ(ρN))(τΛ2mΛ−35ρΛ(3π2ρΛ)2/32mΛ),
(4) VΛ=∂(εNΛ+εΛΛ)∂ρΛ−(mΛm∗Λ(ρN)−1)(3π2ρΛ)2/32mΛ.
(5) For the nucleonic part
εNN , we use the Skyrme force SLy5 [47, 51], which has been fitted in a wide nuclear region. The energy-density contributionsεNΛ [41, 42] andεΛΛ [50] are parameterized as (densitiesρ given in units of fm−3 , energy densityε in MeVfm−3 ):εNΛ(ρN,ρΛ)=−(ε1−ε2ρN+ε3ρN2)ρNρΛ+(ε4−ε5ρN+ε6ρN2)ρNρΛ5/3,
(6) εΛΛ(ρΛ)=−ε7ρΛ2Θ(NΛ>1),
(7) together with
m∗ΛmΛ(ρN)=μ1−μ2ρN+μ3ρN2.
(8) The parameters
ε1, …, ε6 in Eq. (6) and theΛ effective-mass parametersμi were determined in Brueckner–Hartree–Fock calculations of hypernuclear bulk matter with the Nijmegen potential NSC97f [41, 42], while the empirical expression involving the parameterε7 in Eq. (7) has been proposed by fitting the bond energy of6ΛΛ He in Ref. [50]. All parameters are listed in Table 1. This procedure gives a good description of the binding energies of single- and double-Λ hypernuclei [40, 42, 50].ε1 ε2 ε3 ε4 ε5 ε6 ε7 μ1 μ2 μ3 384 1473 1933 635 1829 4100 33.25 0.93 2.19 3.89 The occupation probabilities
niq (for nucleons only) in Eq. (2) are calculated by taking into account pairing within a BCS approximation. In this work, the pairing interaction is taken as a density-dependentδ interaction [52],Vq(r1,r2)=V0[1−ρN((r1+r2)/2)0.16fm−3]δ(r1−r2).
(9) For the
p -shell nuclei and their corresponding hypernuclei, the strength of the pairing force is set toV0= −410 MeVfm3 for both neutrons and protons, which gives reasonable binding energies for12 C and13 Λ C [29, 53]. For the heavier (hyper)nuclei,V0 is taken as−1000 MeVfm3 for both neutrons and protons as in Ref. [29].Regarding
ΛΛ pairing, currently theΛΛ pairing interaction is basically unknown, see e.g. Ref. [54], and there are no experimental data on hyperon pairing phenomena to date. Nevertheless some theoretical studies have been performed [55–57] with results depending on the theoretical assumptions made. It is thus premature to include this aspect in this work. Nevertheless, ifΛΛ pairing is strong, it might affect the deformation properties, especially the shape-coexistence features observed in C, O, S, and Ar hyperisotopes discussed later.In this work, we focus mainly on the impurity effects of multi-
Λ hyperons on the deformation of nuclei. The deformed SHF Schrödinger equation is solved in cylindrical coordinates(r,z) under the assumption of axial symmetry of the mean field [46, 47]. The optimal quadrupole deformation parameterβ2=√π5⟨2z2−r2⟩⟨z2+r2⟩
(10) is determined by minimizing the energy-density functional.
-
Due to the lack of experimental data for multi-
Λ hypernuclei, we compare in Fig. 1 the averageΛ binding energy of multi-Λ hypernuclei,⟨BΛ⟩≡BnΛ/n , with that of experimental single-sΛ hypernuclei, double-sΛΛ hypernuclei, and single-pΛ hypernuclei. Both theoretical and experimental results show that⟨BΛ⟩ decreases withA−2/3 . Due to the weakΛΛ interaction, the⟨BΛ⟩ values of double-sΛΛ hypernuclei are very close to those of single-sΛ hypernuclei. For a given isotope,⟨BΛ⟩ decreases with increasing hyperon number, since the higherΛ s.p. orbits are being filled. As a consequence, the⟨BΛ⟩ of 8Λ hypernuclei are close to those of the single-pΛ hypernuclei. These comparisons are rather qualitative, and experimental binding energies of multi-Λ hypernuclei are necessary to perform a strict evaluation of the current theoretical calculations. Nevertheless, as the energies of single- and double-Λ hypernuclei are reasonably well reproduced, we continue the analysis for other quantities based on the current model.Figure 1. (color online) Average
Λ binding energies in multi-Λ hypernuclei⟨BΛ⟩≡BnΛ/n as function ofA−2/3 calculated by SHF in comparison with the experimental data of single-sΛ hypernuclei, double-sΛΛ hypernuclei, and single-pΛ hypernuclei. The experimental data are taken from Ref. [7] and references therein.The impurity effect of additional hyperons in single-
Λ or double-Λ hypernuclei is usually reflected by the shape shrinkage or deformation reduction of the nuclear core. To study the impurity effect of multi-Λ systems, we show in Fig. 2 the calculated potential energy surfaces as functions of the quadrupole deformationβ2 for even–even nuclei ranging from8 Be to40 Ca and their corresponding multi-Λ (nΛ=2,4,6,8 ) hypernuclei. All energies are normalized with respect to the binding energy of the absolute minimum for a given isotope. Apart from12 C,32 S,36 Ar, and the doubly-magic nuclei16+n nΛ O and40+n nΛ Ca (n=0,2,4,6,8 ), all other (hyper)nuclei are well deformed.186Λ C,306Λ Mg,28+n nΛ Si (n=0,2,4,6,8 ),386Λ S, and426Λ Ar are oblately deformed, while the others are prolately deformed.8+n nΛ Be (n=4,6,8 ) and208Λ C are unbound systems, i.e., theΛ dripline is reached before [40].Figure 2. (color online) Potential energy surfaces as functions of quadrupole deformation
β2 calculated by the self-consistent deformed SHF method for even–even nuclei ranging from8 Be to40 Ca and their corresponding multi-Λ (n=2 , 4, 6, 8) hypernuclei. Energies are normalized with respect to the binding energy of the absolute minimum for a given isotope. Positive (negative) values ofβ2 correspond to prolate (oblate) deformation.We note that in this work we use the unmodified SLy5 Skyrme force, which predicts spherical core nuclei
12 C,32 S, and36 Ar. In some other works [28, 48, 53, 58] the spin–orbit component of the Skyrme force is reduced in order to enforce a deformation for these nuclei at the cost of not reproducing the binding energy correctly.One observes that the impurity effects become stronger with more hyperons involved, but the dependence is not regular: For
2Λ and8Λ hypernuclei, the impurity effect gives similar results of deformation reduction as in the case of single-Λ hypernuclei. This observation is the same as that obtained by the RMF model in Ref. [24]. However, for4Λ and6Λ hypernuclei, the opposite impurity effect can be seen, namely the deformations of the hypernuclei become larger than those of the core nuclei. The energy differences between the prolate and oblate local minima in Ne, Mg, and Si isotopes are smaller than 2 MeV, which characterize them as typical nuclei with shape-coexistence phenomenon [59]. With the addition of hyperons, not only these nuclei retain their shape coexistence, but also other hypernuclei, such as C, O, S, and Ar, can develop the shape-coexistence phenomenon.Before analyzing the results in detail, we first test the robustness of the above findings with respect to other choices of the
NN andNY forces. In Fig. 3, taking24 Mg and its corresponding multi-Λ hypernuclei as examples, the potential energy surfaces calculated with SLy5 [47, 51], SGII [60], or SIII [61] parameters for theNN interaction and NSC97f+EmpC or SLL4 [62] parameters for theYN interaction are shown. It is seen that the deformations of the core nuclei are always reduced by 2 or 8Λ hyperons while enhanced by adding 4 or 6Λ hyperons regardless of the parametrizations. Therefore, this feature is robust with respect to the choice of (realistic) interactions.Figure 3. (color online) Potential energy surfaces of
24 Mg and its multi-Λ hypernuclei calculated with SLy5, SGII, or SIII parameters for theNN interaction and NSC97f+EmpC or SLL4 parameters for theYN interaction.As pointed out before, for the light Be and C cores not all
Λ p -states can be filled, because theΛ dripline is reached first [40]. For heavier cores this is not an issue, but another mechanism that could limit the maximum number of stably boundΛ 's is the fact thatΛ -rich hypernuclei might be unstable due to theΛΛ→Ξ−p strong reaction [50, 63], which occurs whenS2Λ+28.6MeV< SΞ−+Sp , whereS2Λ ,SΞ− , andSp are the relevant separation energies. We examine this possibility using a recent determination of theΞ− -nucleus interaction, SLX3 [64]. Taking the mostΛ -rich hypernucleus186Λ C as an example, and neglecting the unknownΛΞ interaction, one estimates the separation energies asS2Λ=E(164ΛC)−E(186ΛC)=1.6MeV,
(11) Sp=E(164ΛC)−E(174ΛN)=7.7MeV,
(12) SΞ−=E(174ΛN)−E(184ΛΞ−C)
(13) ≈E(13N)−E(14Ξ−C)=7.6MeV.
(14) This leaves the above reaction blocked by an energy gap of about 15 MeV. Hence,
186Λ C as well as all otherp -state hypernuclei in the current study are stable with respect to theΛΛ→Ξ−p reaction.In order to achieve a microscopic understanding of the behavior of
⟨BΛ⟩ in Fig. 1 and the impurity effects of multiΛ 's on the deformation in Figs. 2 and 3, we take488Λ Ca as example, and show in Fig. 4 the s.p. energies ofΛ hyperons as a function ofβ2 , and in Fig. 5 the density distributions atβ2=0 as functions ofr (z=0 ) andz (r=0 ) for the occupieds andp Λ s.p. orbits. Note that thez axis is the symmetry axis.Figure 4. (color online) Calculated
Λ hyperons,p,d s.p. energy levels as function of quadrupole deformationβ2 in488Λ Ca.Figure 5. (color online) Density distributions for the occupied
s andp s.p. orbits ofΛ hyperons in488Λ Ca atβ2=0 as functions ofr (z=0 ) andz (r=0 ). Thez axis is the symmetry axis.The figures show that the [000]1/2
+ s orbit is the lowestΛ s.p. energy level with a spherical density distribution concentrated at the center. As two hyperons can occupy this level, and their mutual interaction is small, the⟨BΛ⟩ values of double-Λ hypernuclei are very close to those of single-Λ hypernuclei.Regarding the three negative-parity
p states, Fig. 4 shows that the [101]1/2− and [101]3/2− orbits are degenerate as the spin–orbit interaction is neglected in theΛΛ channel. Their s.p. energies are lower than those of the [110]1/2− orbit on the oblate side, but higher on the prolate side. Therefore a partial filling of thep states (4Λ and 6Λ hypernuclei) allows a reduction of the total energy by increasing the magnitude of the deformation, whereas a complete filling (8Λ hypernuclei) does not exhibit this feature. As shown in Fig. 5, the [110]1/2− orbit is prolate with zero density atz=0 , while the degenerate [101]1/2− and [101]3/2− orbits are both oblate with zero densities atr=0 . When the 8 hyperons occupy fully the threep orbits, their density distribution becomes spherical.A more detailed visualization of the effects of hyperons on the deformation of hyperisotopes is given in Fig. 6, which shows the
Λ density distribution in the(r,z) plane. We choose the prolate Ne, the oblate Si, and the spherical Ca hyperisotopes as examples. It can be seen that the density distribution of double-Λ hypernuclei changes in accordance with the deformation of the core nuclei, since the additional twoΛ s occupy the spherical [000]1/2+ orbital.Figure 6. (color online) Density distribution of hyperons in the
(r,z) plane in Ne, Si, and Ca hyperisotopes. Thez axis is the symmetry axis.When 4
Λ hyperons are filled in, the shape of the firstp orbit occupied by the hyperons is the same as that of the core nuclei. For example, the hyperons of244Λ Ne with prolate deformation first fill into the [110]1/2− orbit, which is also prolate, and then gradually fill into the [101]1/2− and [101]3/2− orbits, which are oblate. Thus the deformation of244Λ Ne reaches the largest value due to the maximal distribution of 4Λ hyperons in the prolate orbit. When the hyperons begin to fill into the oblate orbits, a reduction of the deformation occurs in266Λ Ne. Finally, the spherical distribution of 8Λ 's renders also the core nucleus more spherical.The hyperons in
28 Si hyperisotopes with oblate core nucleus first fill into the degenerate oblate orbits [101]1/2− and [101]3/2− . Therefore, the deformation increase can last up to 6Λ hypernuclei, when the deformation reaches the maximum. Then hyperons will fill into the prolate [110]1/2− orbit, and cause a reduction of the deformation. This also explains the different trends of deformation of prolate and oblate hyperisotopes with the increasing hyperon number as shown in Fig. 2.However, for spherical-core nuclei, such as
16 O and40 Ca, the 4Λ hypernuclei have no preference for oblate or prolate orbits; therefore their deformation trends have the characteristics of both oblate and prolate hypernuclei. As a consequence, they show a more or less soft potential energy surface around the spherical shape, in particular in the16 O hypernuclei as shown in Fig. 2. -
We study within the deformed SHF formalism the impurity effects of
Λ hyperons on the deformation of even–even nuclei ranging from8 Be to40 Ca, employing an effectiveΛN interaction that reproduces well the experimental binding energies of single-Λ hypernuclei.The deformations of the nuclear core are generally reduced by adding 2 or 8
Λ hyperons, but enhanced by adding 4 or 6. These behaviors are interpreted in a microscopic manner by analyzing theΛ s.p. orbits and the density distributions for the occupied ones. It is demonstrated that the order of filling the hyperons into thep orbits is determined by the shape of the core nucleus. When the core nucleus is oblate, the hyperons are filled first into the degenerate oblate [101]1/2− and [101]3/2− orbits, so that the core deformation increases to the oblate side, and can reach a maximum when 6 hyperons are added. When the core nucleus is prolate, the hyperons are filled first into the prolate [110]1/2− orbit, which leads to an increase of the prolate core deformation, reaching a maximum when 4 hyperons are added. When the core nucleus is spherical, hyperons have no preferences for the shape of thep orbits and result in a soft potential energy surface.Future experimental data of multi-
Λ hypernuclei are necessary to examine these effects and refine the assumptions and ingredients of the model calculations presented here. While we think the qualitative phenomena reported here are robust, a more realistic quantitative evaluation will require in particular knowledge of theΛΛ interaction including the residual pairing force and a beyond-mean-field formalism appropriate for reliably modelling shape-coexistence and weakly-bound hyperon states.
Effects of Λ hyperons on the deformations of even–even nuclei
- Received Date: 2022-01-25
- Available Online: 2022-06-15
Abstract: The deformations of multi-