-
In the framework of the SHF approach, the energy of a hypernucleus is given by an energy-density functional,
$ \begin{array}{l} E = \int {\rm d}^3{\boldsymbol{r}}\;{\varepsilon}({\boldsymbol{r}}) \ , \quad {\varepsilon} = {\varepsilon}_{NN} + {\varepsilon}_{{\Lambda} N} + {\varepsilon}_{{\Lambda} {\Lambda}} \;, \end{array} $
(1) where
$ {\varepsilon}_{NN} $ ,$ {\varepsilon}_{{\Lambda} N} $ , and$ {\varepsilon}_{{\Lambda}{\Lambda}} $ 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 densities$ \rho_q $ , kinetic densities$ \tau_q $ , and spin–orbit currents$ {\boldsymbol{J}}_q $ ,$ \begin{array}{l} \Big[ \rho_q, \; \tau_q, \; {\boldsymbol{J}}_q \Big] = \displaystyle\sum\limits_{i = 1}^{N_q} {n_q^i} \Big[ |\phi_q^i|^2 , \; |\nabla\phi_q^i|^2 , \; {\phi_q^i}^* (\nabla \phi_q^i \times \mathit{\boldsymbol{\sigma}})/i \Big] \:, \end{array} $
(2) where
$ \phi_q^i $ ($ i = 1, \; \ldots, \; N_q $ ) are the self-consistently calculated single-particle (s.p.) wave functions of the$ N_q $ occupied states for the species$ q = n, \; p, \; {\Lambda} $ in a hypernucleus. They satisfy the Schrödinger equation, obtained by the minimization of the total energy functional (1) according to the variational principle,$ \begin{array}{l} \bigg[ \nabla \cdot \dfrac{1}{2m_q^*({\boldsymbol{r}})}\nabla - V_q({\boldsymbol{r}}) + {i} {\boldsymbol{W}}_q({\boldsymbol{r}}) \cdot \left(\nabla\times\mathit{\boldsymbol{\sigma}}\right) \bigg] \phi_q^i({\boldsymbol{r}}) = e_q^i \, \phi_q^i({\boldsymbol{r}}) \:, \end{array} $
(3) in which
$ {\boldsymbol{W}}_q({\boldsymbol{r}}) $ is the spin–orbit interaction part for the nucleons as given in Refs. [45, 49]. The central mean fields$ V_q({\boldsymbol{r}}) $ , corrected by the effective-mass terms following the procedure described in Refs. [40, 42, 50] are$\begin{aligned}[b] V_N =& V_N^{\text{SHF}} + \frac{\partial{\varepsilon}_{N{\Lambda}}}{\partial{\rho_N}} \\ &+ \frac{\partial}{\partial{\rho_N}} \left(\frac{m_{\Lambda}}{m_{\Lambda}^*({\rho_N})}\right) \left( \frac{\tau_{\Lambda}}{2m_{\Lambda}} - \frac{3}{5}\frac{{\rho_\Lambda}(3\pi^2{\rho_\Lambda})^{2/3}}{2m_{\Lambda}} \right) \:, \end{aligned} $
(4) $\begin{aligned}[b] V_{\Lambda} = \frac{\partial ({\varepsilon}_{N{\Lambda}}+{\varepsilon}_{{\Lambda}{\Lambda}})}{\partial{\rho_\Lambda}} - \left( \frac{m_{\Lambda}}{m_{\Lambda}^*({\rho_N})} - 1 \right) \frac{(3\pi^2{\rho_\Lambda})^{2/3}}{2m_{\Lambda}} . \end{aligned}$
(5) For the nucleonic part
$ {\varepsilon}_{NN} $ , we use the Skyrme force SLy5 [47, 51], which has been fitted in a wide nuclear region. The energy-density contributions$ {\varepsilon}_{N{\Lambda}} $ [41, 42] and$ {\varepsilon}_{{\Lambda}{\Lambda}} $ [50] are parameterized as (densities$ \rho $ given in units of fm$ ^{-3} $ , energy density$ {\varepsilon} $ in MeVfm$ ^{-3} $ ):$\begin{aligned}[b] {\varepsilon}_{N{\Lambda}}({\rho_N}, {\rho_\Lambda}) =& -( {\varepsilon}_1 - {\varepsilon}_2{\rho_N} + {\varepsilon}_3{\rho_N}^2) {\rho_N}{\rho_\Lambda} \\& +( {\varepsilon}_4 - {\varepsilon}_5{\rho_N} + {\varepsilon}_6{\rho_N}^2) {\rho_N}{\rho_\Lambda}^{5/3} \:, \end{aligned}$
(6) $ {\varepsilon}_{{\Lambda}{\Lambda}}({\rho_\Lambda}) = -{\varepsilon}_7 {\rho_\Lambda}^2 \Theta(N_{\Lambda}>1)\:, $
(7) together with
$ \begin{array}{l} \dfrac{m_{\Lambda}^*}{m_{\Lambda}}({\rho_N}) = \mu_1 - \mu_2{\rho_N} + \mu_3{\rho_N}^2 \:. \end{array} $
(8) The parameters
$ {\varepsilon}_1, $ $ \ldots, $ $ {\varepsilon}_6 $ in Eq. (6) and the$ {\Lambda} $ effective-mass parameters$ \mu_i $ were determined in Brueckner–Hartree–Fock calculations of hypernuclear bulk matter with the Nijmegen potential NSC97f [41, 42], while the empirical expression involving the parameter$ {\varepsilon}_7 $ in Eq. (7) has been proposed by fitting the bond energy of$ ^{\ \ 6}_{{\Lambda}{\Lambda}} $ 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-$ {\Lambda} $ hypernuclei [40, 42, 50].$ {\varepsilon}_1 $ $ {\varepsilon}_2 $ $ {\varepsilon}_3 $ $ {\varepsilon}_4 $ $ {\varepsilon}_5 $ $ {\varepsilon}_6 $ $ {\varepsilon}_7 $ $ \mu_1 $ $ \mu_2 $ $ \mu_3 $ 384 1473 1933 635 1829 4100 33.25 0.93 2.19 3.89 The occupation probabilities
$ n^i_q $ (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-dependent$ \delta $ interaction [52],$ \begin{array}{l} V_{q}({\boldsymbol{r}}_1, {\boldsymbol{r}}_2) = V_0 \left[ 1 - \dfrac{{\rho_N}(({\boldsymbol{r}}_1+{\boldsymbol{r}}_2)/2)}{0.16\;\rm{fm}^{-3}} \right] \delta({\boldsymbol{r}}_1-{\boldsymbol{r}}_2) \:. \end{array} $
(9) For the
$ p $ -shell nuclei and their corresponding hypernuclei, the strength of the pairing force is set to$V_0 = $ $ -410\;$ MeVfm$ ^3 $ for both neutrons and protons, which gives reasonable binding energies for$ ^{12} $ C and$ ^{13}_{\ {\Lambda}} $ C [29, 53]. For the heavier (hyper)nuclei,$ V_0 $ is taken as$ -1000\; $ MeVfm$ ^3 $ for both neutrons and protons as in Ref. [29].Regarding
$ {\Lambda}{\Lambda} $ pairing, currently the$ {\Lambda}{\Lambda} $ 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, if$ {\Lambda}{\Lambda} $ 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-
$ {\Lambda} $ 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 parameter$ \beta_2 = \sqrt{\frac{\pi}{5}} \frac{\langle 2z^2-r^2 \rangle}{\langle z^2+r^2 \rangle} $
(10) is determined by minimizing the energy-density functional.
-
Due to the lack of experimental data for multi-
$ {\Lambda} $ hypernuclei, we compare in Fig. 1 the average$ {\Lambda} $ binding energy of multi-$ {\Lambda} $ hypernuclei,$ {\langle B_{\Lambda} \rangle} \equiv B_{n{\Lambda}}/n $ , with that of experimental single-$ s_{\Lambda} $ hypernuclei, double-$ s_{{\Lambda}{\Lambda}} $ hypernuclei, and single-$ p_{\Lambda} $ hypernuclei. Both theoretical and experimental results show that$ {\langle B_{\Lambda} \rangle} $ decreases with$ A^{-2/3} $ . Due to the weak$ {\Lambda}{\Lambda} $ interaction, the$ {\langle B_{\Lambda} \rangle} $ values of double-$ s_{{\Lambda}{\Lambda}} $ hypernuclei are very close to those of single-$ s_{\Lambda} $ hypernuclei. For a given isotope,$ {\langle B_{\Lambda} \rangle} $ decreases with increasing hyperon number, since the higher$ {\Lambda} $ s.p. orbits are being filled. As a consequence, the$ {\langle B_{\Lambda} \rangle} $ of 8$ {\Lambda} $ hypernuclei are close to those of the single-$ p_{\Lambda} $ hypernuclei. These comparisons are rather qualitative, and experimental binding energies of multi-$ {\Lambda} $ hypernuclei are necessary to perform a strict evaluation of the current theoretical calculations. Nevertheless, as the energies of single- and double-$ {\Lambda} $ hypernuclei are reasonably well reproduced, we continue the analysis for other quantities based on the current model.Figure 1. (color online) Average
$ {\Lambda} $ binding energies in multi-$ {\Lambda} $ hypernuclei$ {\langle B_{\Lambda} \rangle} \equiv B_{n{\Lambda}}/n $ as function of$ A^{-2/3} $ calculated by SHF in comparison with the experimental data of single-$ s_{\Lambda} $ hypernuclei, double-$ s_{{\Lambda}{\Lambda}} $ hypernuclei, and single-$ p_{\Lambda} $ hypernuclei. The experimental data are taken from Ref. [7] and references therein.The impurity effect of additional hyperons in single-
$ {\Lambda} $ or double-$ {\Lambda} $ hypernuclei is usually reflected by the shape shrinkage or deformation reduction of the nuclear core. To study the impurity effect of multi-$ {\Lambda} $ systems, we show in Fig. 2 the calculated potential energy surfaces as functions of the quadrupole deformation$ \beta_2 $ for even–even nuclei ranging from$ ^8 $ Be to$ ^{40} $ Ca and their corresponding multi-$ {\Lambda} $ ($ n_{\Lambda} = 2, \; 4, \; 6, \; 8 $ ) hypernuclei. All energies are normalized with respect to the binding energy of the absolute minimum for a given isotope. Apart from$ ^{12} $ C,$ ^{32} $ S,$ ^{36} $ Ar, and the doubly-magic nuclei$ ^{16+n}_{\ \ \ n{\Lambda}} $ O and$ ^{40+n}_{\ \ \ n{\Lambda}} $ Ca ($ n = 0, \; 2, \; 4, \; 6, \; 8 $ ), all other (hyper)nuclei are well deformed.$ ^{18}_{6{\Lambda}} $ C,$ ^{30}_{6{\Lambda}} $ Mg,$ ^{28+n}_{\ \ \ n{\Lambda}} $ Si ($ n = 0, \; 2, \; 4, \; 6, \; 8 $ ),$ ^{38}_{6{\Lambda}} $ S, and$ ^{42}_{6{\Lambda}} $ Ar are oblately deformed, while the others are prolately deformed.$ ^{8+n}_{\ \ n{\Lambda}} $ Be ($ n = 4, \; 6, \; 8 $ ) and$ ^{20}_{8{\Lambda}} $ C are unbound systems, i.e., the$ {\Lambda} $ dripline is reached before [40].Figure 2. (color online) Potential energy surfaces as functions of quadrupole deformation
$ \beta_2 $ calculated by the self-consistent deformed SHF method for even–even nuclei ranging from$ ^8 $ Be to$ ^{40} $ Ca and their corresponding multi-$ {\Lambda} $ ($ 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 of$ \beta_2 $ 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, and$ ^{36} $ 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
$ 2{\Lambda} $ and$ 8{\Lambda} $ hypernuclei, the impurity effect gives similar results of deformation reduction as in the case of single-$ {\Lambda} $ hypernuclei. This observation is the same as that obtained by the RMF model in Ref. [24]. However, for$ 4{\Lambda} $ and$ 6{\Lambda} $ 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 $ and$ NY $ forces. In Fig. 3, taking$ ^{24} $ Mg and its corresponding multi-$ {\Lambda} $ hypernuclei as examples, the potential energy surfaces calculated with SLy5 [47, 51], SGII [60], or SIII [61] parameters for the$ NN $ interaction and NSC97f+EmpC or SLL4 [62] parameters for the$ YN $ interaction are shown. It is seen that the deformations of the core nuclei are always reduced by 2 or 8$ {\Lambda} $ hyperons while enhanced by adding 4 or 6$ {\Lambda} $ 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-$ {\Lambda} $ hypernuclei calculated with SLy5, SGII, or SIII parameters for the$ NN $ interaction and NSC97f+EmpC or SLL4 parameters for the$ YN $ interaction.As pointed out before, for the light Be and C cores not all
$ {\Lambda} $ $ p $ -states can be filled, because the$ {\Lambda} $ dripline is reached first [40]. For heavier cores this is not an issue, but another mechanism that could limit the maximum number of stably bound$ {\Lambda} $ 's is the fact that$ {\Lambda} $ -rich hypernuclei might be unstable due to the$ {\Lambda}{\Lambda} \rightarrow \Xi^-p $ strong reaction [50, 63], which occurs when$S_{2{\Lambda}}+28.6{\;\rm{MeV}} < $ $ S_{\Xi^-}+S_p$ , where$ S_{2{\Lambda}} $ ,$ S_{\Xi^-} $ , and$ S_p $ are the relevant separation energies. We examine this possibility using a recent determination of the$ \Xi^- $ -nucleus interaction, SLX3 [64]. Taking the most$ {\Lambda} $ -rich hypernucleus$ ^{18}_{6{\Lambda}} $ C as an example, and neglecting the unknown$ {\Lambda}\Xi $ interaction, one estimates the separation energies as$ S_{2{\Lambda}} = E(^{16}_{4{\Lambda}}\text{C}) - E(^{18}_{6{\Lambda}}\text{C}) = 1.6{\;\text{MeV}} \:, $
(11) $ S_p = E(^{16}_{4{\Lambda}}\text{C}) - E(^{17}_{4{\Lambda}}\text{N}) = 7.7{\;\text{MeV}} \:, $
(12) $ S_{\Xi^-} = E(^{17}_{4{\Lambda}}\text{N}) - E(^{18}_{4{\Lambda}\Xi^-}\text{C}) $
(13) $ \quad\;\; \approx E(^{13}\text{N}) - E(^{14}_{\Xi^-}\text{C}) = 7.6{\;\text{MeV}} \:. $
(14) This leaves the above reaction blocked by an energy gap of about 15 MeV. Hence,
$ ^{18}_{6{\Lambda}} $ C as well as all other$ p $ -state hypernuclei in the current study are stable with respect to the$ {\Lambda}{\Lambda}\rightarrow \Xi^-p $ reaction.In order to achieve a microscopic understanding of the behavior of
$ {\langle B_{\Lambda} \rangle} $ in Fig. 1 and the impurity effects of multi$ {\Lambda} $ 's on the deformation in Figs. 2 and 3, we take$ ^{48}_{8{\Lambda}} $ Ca as example, and show in Fig. 4 the s.p. energies of$ {\Lambda} $ hyperons as a function of$ \beta_2 $ , and in Fig. 5 the density distributions at$ \beta_2 = 0 $ as functions of$ r $ ($ z = 0 $ ) and$ z $ ($ r = 0 $ ) for the occupied$ s $ and$ p $ $ {\Lambda} $ s.p. orbits. Note that the$ z $ axis is the symmetry axis.Figure 4. (color online) Calculated
$ {\Lambda} $ hyperon$ s, p, d $ s.p. energy levels as function of quadrupole deformation$ \beta_2 $ in$ ^{48}_{8{\Lambda}} $ Ca.Figure 5. (color online) Density distributions for the occupied
$ s $ and$ p $ s.p. orbits of$ {\Lambda} $ hyperons in$ ^{48}_{8{\Lambda}} $ Ca at$ \beta_2 = 0 $ as functions of$ r $ ($ z = 0 $ ) and$ z $ ($ r = 0 $ ). The$ z $ axis is the symmetry axis.The figures show that the [000]1/2
$ ^+ $ $ s $ orbit is the lowest$ {\Lambda} $ 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$ {\langle B_{\Lambda} \rangle} $ values of double-$ {\Lambda} $ hypernuclei are very close to those of single-$ {\Lambda} $ 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 the$ {\Lambda}{\Lambda} $ 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 the$ p $ states (4$ {\Lambda} $ and 6$ {\Lambda} $ hypernuclei) allows a reduction of the total energy by increasing the magnitude of the deformation, whereas a complete filling (8$ {\Lambda} $ hypernuclei) does not exhibit this feature. As shown in Fig. 5, the [110]1/2$ ^- $ orbit is prolate with zero density at$ z = 0 $ , while the degenerate [101]1/2$ ^- $ and [101]3/2$ ^- $ orbits are both oblate with zero densities at$ r = 0 $ . When the 8 hyperons occupy fully the three$ p $ 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
$ {\Lambda} $ 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-$ {\Lambda} $ hypernuclei changes in accordance with the deformation of the core nuclei, since the additional two$ {\Lambda} $ 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. The$ z $ axis is the symmetry axis.When 4
$ {\Lambda} $ hyperons are filled in, the shape of the first$ p $ orbit occupied by the hyperons is the same as that of the core nuclei. For example, the hyperons of$ ^{24}_{4{\Lambda}} $ 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 of$ ^{24}_{4{\Lambda}} $ Ne reaches the largest value due to the maximal distribution of 4$ {\Lambda} $ hyperons in the prolate orbit. When the hyperons begin to fill into the oblate orbits, a reduction of the deformation occurs in$ ^{26}_{6{\Lambda}} $ Ne. Finally, the spherical distribution of 8$ {\Lambda} $ '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 6$ {\Lambda} $ 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 and$ ^{40} $ Ca, the 4$ {\Lambda} $ 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 the$ ^{16} $ O hypernuclei as shown in Fig. 2.
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-