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The purpose of the present work is to qualitatively discuss the effect of an additional
$ K^- $ meson on the ground-state properties of nuclei in the SHF approach combined with a simple density-dependent Skyrme-type$ K^-N $ interaction. In this approach the total energy of the nucleus is expressed as [47, 49, 63–69]$ E = \int {\rm d}^3{\boldsymbol{r}}\, {\varepsilon}({\boldsymbol{r}}) \:,\quad {\varepsilon} = {\varepsilon}_{NN} + {\varepsilon}_{KN} + {\varepsilon}_{\rm C} \:, $
(1) where
$ {\varepsilon}_{NN} $ denotes the energy density of the nucleon-nucleon part,$ {\varepsilon}_{KN} $ is the energy density due to the kaon-nucleon strong interaction, and$ {\varepsilon}_{\rm C} $ is the Coulomb contribution of protons and kaons.For each single-particle (s.p.) state
$ \phi_q^i $ $ (q=n,p,K) $ , the minimization of the total energy E in Eq. (2) implies the SHF Schrödinger equation$ \left[ -\nabla \cdot \frac{1}{2m^*_q({\boldsymbol{r}})} \nabla + V_q({\boldsymbol{r}}) - {i}{\boldsymbol{W}}_q({\boldsymbol{r}}) \cdot (\nabla\times{\boldsymbol{\sigma}}) \right] \phi_q^i({\boldsymbol{r}}) = e_q^i \phi_q^i({\boldsymbol{r}}) \:, $
(2) with the mean fields
${V_K} = \frac{{\partial {\varepsilon _{KN}}}}{{\partial {\rho _K}}} - {V_{\rm C}} , $
(3) ${V_q} = V_q^{{\rm{SHF}}} + V_q^{(K)}\,, $
(4) $ V_q^{(K)} = \frac{{\partial {\varepsilon _{KN}}}}{{\partial {\rho _q}}},\;(q = n,p) , $
(5) where
$ V_{\rm C} $ denotes the Coulomb field,$ V^{\rm{SHF}}_q $ the standard nucleonic Skyrme mean field,$ {\boldsymbol{W}}_q $ the nucleonic spin-orbit mean field, and$ V_q^{(K)} $ the change of the nucleonic mean fields by the$ K^-N $ interaction.For the nucleonic part, we use the standard Skyrme force SLy4 [70]. For the kaonic energy-density contribution, a simple linear energy density functional is assumed as in Ref. [49],
$ {\varepsilon}_{KN} = -a_0 \rho_K [(1+x_0)\rho_p + (1-x_0)\rho_n] \:, $
(6) where
$ a_0 $ and$ x_0 $ are the$ K^-N $ interaction strength parameters. Under this assumption, the mean fields in Eqs. (3) and (4) are [49]$V_K = -a_0 [(1+x_0)\rho_p + (1-x_0)\rho_n] - V_{\rm C} \:, $
(7) $V_{p,n}^{(K)} = -a_0(1 \pm x_0)\rho_K . $
(8) In the following calculations, we use a (p,n)-symmetric
$ K^-N $ interaction, i.e.,$ x_0=0 $ and$ a_0=500{\;{\rm{MeV}}\,{\rm{fm}}^{3}} $ , which were justified as reasonable values in Ref. [49], to which we refer for more information. This choice yields a mean field$ V_K\approx130{\;\rm{MeV}} $ for the$ ^{12}_{K^-}{\rm{C}} $ nucleus. In that reference we also studied the extreme case of$ x_0=1 $ , neglecting the$ K^-n $ interaction completely, which required smaller values of$ a_0 $ to cause similar effects as$ x_0=0 $ . We delay the decision on more realistic values of$ a_0 $ and$ x_0 $ (and eventual further nonlinear interaction parameters) to the future when experimental results will hopefully allow a more realistic analysis.The pairing interaction of the nucleonic part employs a density-dependent δ pairing force [71],
$ V_q({\boldsymbol{r}}_1,{\boldsymbol{r}}_2) = -V_0 \left[1-\frac{\rho_N(({\boldsymbol{r}}_1+{\boldsymbol{r}}_2)/2)}{0.16 \; {\,{\rm{fm}}^{-3}}}\right] \delta({\boldsymbol{r}}_1-{\boldsymbol{r}}_2), \: $
(9) with a pairing strength
$ V_0=410{\;{\rm{MeV}}\,{\rm{fm}}^{3}} $ for both neutrons and protons [72–76]. A smooth energy cutoff is included in the BCS calculation [76]. In the case of a nucleus with an odd number of nucleons, the orbit occupied by the last odd nucleon is blocked, as described in Ref. [77]. Dobaczewski et al. [78] stated that the SHF+BCS method was not well suited to describe nuclei close to the neutron-drip line due to the neutron gas effect. However, Anguiano et al. [79] studied the importance of this effect in the description of nuclei with large neutron excess within the BCS approach and concluded that from the quantitative point of view, the neutron gas problem was irrelevant and SHF+BCS calculations were reliable in all regions of the nuclear chart. In the present work, we thus use this approach in a pragmatic way.We assume axially-symmetric mean fields and the properties of axially-deformed nuclei are studied in cylindrical coordinates. The coupled SHF+BCS equations for nucleons and kaon are solved self-consistently by iteration within a coordinate-space representation, imposing the quadrupole deformation parameter
$ \beta_2 = \sqrt{\frac{\pi}{5}} \frac{\langle 2z^2-r^2 \rangle}{\langle z^2+r^2 \rangle} $
(10) as additional constraint. The physical value of
$ \beta_2 $ is taken as the one minimizing the total energy [64, 67, 68, 80]. Specifically, the r-space box sizes for nucleons and antikaons are the same and depend on the mass of the nucleus and the range of input deformation. The results are converged and confirmed to be independent of the box and step sizes.At this point, we also comment on the imaginary part of the
$ K^-N $ interaction (optical potential), due to the decay channels$ KN \rightarrow \pi Y $ and$ KNN \rightarrow YN $ ($ Y=\Lambda,\Sigma $ ) [32, 35, 36]. This is a difficult and intensely studied theoretical problem, also due to the presence of the$ \Lambda(1405) $ resonance [81] as a possible intermediate state,$ KN \rightarrow $ $ \Lambda(1405) \rightarrow \pi\Sigma $ [18, 34]. In this work, we neglect the imaginary part in attendance of reliable data. It has been found that the effect of a moderate$ {\rm{Im}}\,V_K\lesssim 20 $ MeV (neglecting the kaon multinucleon absorption) on the real part is negligible, whereas too large widths might make kaon bound states unobservable [35, 36]. We consider this feature an open problem that can only be solved by future confrontation with accurate data.However, we estimated the qualitative effect in our formalism in Ref. [49] by solving the SHF Schrödinger Eq. (2) incorporating a complex kaon potential
$ V_K({\boldsymbol{r}}) = V_R({\boldsymbol{r}}) + {i}V_I({\boldsymbol{r}}) $ . The imaginary part modified the kaon wave function, single-particle energy, density distribution, and the kaon removal energy$ B_K $ . In Ref. [49], we found that the change in$ B_K $ is small, even up to a large magnitude of the imaginary part. Furthermore, a given value of$ B_K $ could always be restored by adjusting the value of the interaction parameter$ a_0 $ in this model. This demonstrated that the imaginary part of the kaon mean field did not play a key role in the SHF model, at least regarding its effect on the real part and the kaon removal energy. The treatment of real and imaginary parts can be fairly well separated. Equivalent results have been found in the RMF model [42]. Of course, more experimental information is required for a final quantitative determination of this feature. -
To study in detail the effects of an additional
$ K^- $ meson on the ground-state properties of nuclei (comprising unstable nuclei), we examine the one-nucleon separation energies$S_n \equiv E[^AZ] - E[^{A-1}Z] , $
(11) $S_p \equiv E[^AZ] - E[^{A-1}(Z-1)] , $
(12) which provide the location of unstable nuclei, and we compare the results obtained for normal and kaonic nuclei. In addition, we focus on the highest-occupied (valence) nucleon s.p. levels (the last level with occupation probability
$ v^2>0.5 $ ), which become weakly bound for unstable nuclei. If the s.p. energy of the highest-occupied nucleon levels is still negative in the minimum of the total energy, the nucleus is supposed to exist [60]. -
In Fig. 1, the energies of the highest-occupied nucleon s.p. levels
$ -e_q $ (a), the one-nucleon separation energies$ S_q $ (b), the binding energies E (c), and the quadrupole deformations$ \beta_2 $ (d) of Be isotopes and their corresponding$ K^- $ nuclei are presented in comparison with the available experimental results from Ref. [82]. As indicated by the theoretical$ e_q $ and$ S_q $ values, the nuclei with$ N=2 – 8 $ neutrons and their corresponding$ K^- $ nuclei exist. However, 13,17,19Be ($ N=9,13,15 $ ) are unbound systems due to pair breaking. Experimentally, the nuclei from 6Be ($ N=2 $ ) to 16Be ($ N=12 $ ) have been observed [82], but the experimental values of$ S_n $ for 13Be ($ N=9 $ ) and$ S_p $ for 7B ($ N=2 $ ) are negative. Therefore, the proton and neutron drip lines locate at 6Be ($ N=2 $ ) and 12Be ($ N=8 $ ), respectively. Note that the SHF mean-field approach makes rather good predictions for the binding and separation energies even for the lightest isotopes 6,7Be.Figure 1. (color online) (a) Energies of the highest-occupied nucleon s.p. levels
$ -e_q $ ($ q=n,p $ ) (the neutron level is indicated on the top; the proton level is$ 1p_{3/2} $ ), (b) one-nucleon separation energies$ S_q $ , (c) binding energies E, (d) quadrupole deformations$ \beta_2 $ of Be isotopes (dashed black lines) and their corresponding$ K^- $ nuclei (dotted red lines), obtained with$ a_0=500{\;{\rm{MeV}}\,{\rm{fm}}^{3}} $ and$ x_0=0 $ in Eqs. (6), (7). The experimental$ S_q $ and E values of normal nuclei obtained from Ref. [82] are also included for comparison (solid black lines). The spherical shells on the top are the approximate quantum numbers in case of deformation.The additional
$ K^- $ meson clearly shifts down the energies of the highest-occupied nucleon s.p. levels$ e_q $ (most notable for those in strongly-bound inner orbits) and thus increases the total binding energies. The shift of the proton levels is larger than that of the neutron levels, even for weakly-bound states, which is due to the additional$ K^-p $ Coulomb attraction. The decrease in quadrupole deformations, as shown in Fig. 1 (d), is due to the attractive$ K^-N $ interaction [49]. With an additional$ K^- $ meson,$ ^{17}_{K^-} {\rm{Be}}$ ($ N=13 $ ) and$ ^{19}_{K^-} {\rm{Be}}$ ($ N=15 $ ) remain unbound, while$ ^{13}_{K^-} {\rm{Be}}$ ($ N=9 $ ) becomes marginally bound, but its$ S_n $ is still negative. Moreover, due to the major impact on the proton levels, a bound nucleus$ ^{\ \ 5}_{K^-} {\rm{Be}}$ ($ N=1 $ ) is found, but its one-proton separation energy$ S_p $ remains negative. An extension effect of the additional$ K^- $ meson on Be isotopes is thus found, which can be attributed to the strong$ K^-p $ attraction including the Coulomb interaction. A similar phenomenon was also found by additional Λ hyperons due to the attractive$ \Lambda N $ interaction in [60].Fig. 2 shows the preceding results for 11-28O (
$ N=3-20 $ ) as well as the corresponding$ K^- $ nuclei. In this case all isotopes are nearly spherical as illustrated in panel (d). The theoretical results of the total energies are in good agreement with the experimental values. All isotopes from 12O ($ N=4 $ ) to 28O ($ N=20 $ ) exist with negative$ e_n $ and$ e_p $ . The separation energies$ S_n $ and$ S_p $ of all nuclei with$ N=5-16 $ are positive, whereas$ S_n $ becomes negative at$ N=17 $ theoretically and experimentally [82], and thus the neutron drip line locates at 24O ($ N=16 $ ). The theoretical proton drip line is reached at$ N=4 $ with 13F ($ Z=9,\;N=4 $ ) being unbound, whereas experimentally it lies at$ N=5 $ , where the experimental one-proton separation energy$ S_p $ of 14F ($ Z=9,\;N=5 $ ) is negative. 11O ($ N=3 $ ) does not exist theoretically and experimentally.Figure 2. (color online) Same as Fig. 1, but for O isotopes.
In contrast to the results of Be isotopes in Fig. 1, the additional
$ K^- $ meson increases slightly the energies of the highest-occupied neutron s.p. levels of$ ^{11-24}_{\hskip1em K^-} {\rm{O}}$ ($ N= $ $ 3 - 16 $ ), but decreases those of$ ^{25-28}_{\hskip1em K^-} {\rm{O}}$ (valence neutron levels$ 1d_{3/2} $ ). Therefore,$ ^{25}_{K^-} {\rm{O}}$ becomes unbound and the neutron removal energies of$ ^{27-28}_{\hskip1em K^-} {\rm{O}}$ are reduced. Thus, a reducing effect of the$ K^- $ on neutron-rich O isotopes is found. This is an interesting result and is analyzed in detail in the following. However, a bound$ ^{11}_{K^-} {\rm{O}}$ and increasing one-proton separation energies of$ ^{12-24}_{\hskip1em K^-} {\rm{O}}$ are found. Thus, an extension effect of the$ K^- $ meson on the proton-rich O isotopes is evident.The same phenomenon occurs for Ne isotopes, as observed in Fig. 3. All isotopes 16-34Ne (
$ N=6-24 $ ) exist, as experimentally [82]. 35,36Ne are unbound systems. For 17-30Ne ($ N=7 - 20 $ ),$ S_n $ and$ S_p $ are positive both experimentally and theoretically, while$ S_n $ of 31Ne ($ N=21 $ ) is negative theoretically but marginally positive experimentally. Thus, the neutron drip line locates at 30Ne ($ N=20 $ ) theoretically and at 32Ne ($ N=22 $ ) experimentally. The proton drip line is not reached for nuclei with$ N>6 $ experimentally and theoretically. A weakly bound 16Ne ($ Z=10,\;N=6 $ ) and an unbound 15F ($ Z=9,\;N=6 $ ) are found on the theoretical side, while the experimental$ S_p $ of 16Ne is slightly negative [82].Figure 3. (color online) Same as Fig. 1, but for Ne isotopes.
However,
$ ^{16}_{K^-} {\rm{Ne}}$ with positive$ S_p $ and$ ^{15}_{K^-} \;{\rm{F}}$ with negative$ S_p $ are bound nuclei and thus the additional$ K^- $ meson firmly establishes the proton drip line at$ ^{16}_{K^-} {\rm{Ne}}$ for$ N=6 $ nuclei, and in fact also$ ^{15}_{K^-} {\rm{Ne}}$ becomes bound. All kaonic nuclei$ ^{15-34}_{\hskip1em K^-} {\rm{Ne}}$ ($ N=5 - 24 $ ) exist. Thus, the additional$ K^- $ meson does not affect the existence of neutron-rich Ne isotopes, although the energies of the highest-occupied neutron s.p. level ($ 1d_{3/2} $ ,$ 1f_{7/2} $ ) of$ ^{27-32}_{\hskip1em K^-} {\rm{Ne}}$ ($ N=17-22 $ ) decrease due to the additional$ K^- $ meson. This is like the case of O isotopes as displayed in Fig. 2. In addition, in these larger nuclei, the deformation changes due to the kaon are much smaller than those of Be isotopes in Fig. 1(d). -
As mentioned in Sec. II, the results of
$ K^- $ nuclei shown in the previous figures are obtained with the$ K^-N $ interaction strength$ a_0=500{\;{\rm{MeV}}\,{\rm{fm}}^{3}} $ . Because of the uncertainty of this value, we explore the energies of the highest-occupied neutron s.p. levels of$ ^{27-32}_{\hskip1em K^-} {\rm{Ne}}$ ($ N= $ $ 17 - 22 $ ) with$ K^-N $ interaction strengths$ a_0= 100, $ $ 200,\; \ldots,\; 600{\;{\rm{MeV}}\,{\rm{fm}}^{3}} $ in Fig. 4. It is interesting to note that the$ K^- $ meson does not always increase the energies of the highest-occupied neutron s.p. levels as increasing the$ K^-N $ interaction strength. For the weakly bound$ 1f_{7/2} $ levels, a weak$ K^-N $ interaction ($ a_0=100 $ and$ 200{\;{\rm{MeV}}\,{\rm{fm}}^{3}} $ ) causes a slight increase of binding, whereas repulsion only set in for$ a_0 \gtrsim 300 $ $ {{\rm{MeV}}\,{\rm{fm}}^{3}} $ . The reason will be analyzed later.Figure 4. (color online) Energies of the highest-occupied neutron s.p. levels of
$ ^{27-32}_{\hskip1em K^-} {\rm{Ne}}$ and their corresponding normal nuclei$ (a_0=0) $ obtained with different$ K^-N $ interaction strengths$ a_0 $ (in units of$ {\;{\rm{MeV}}\,{\rm{fm}}^{3}} $ ).To illustrate the above features, we show in Fig. 5 the effect of an additional
$ K^- $ meson on the neutron s.p. levels of the spherical drip-line nuclei 12Be, 28O, and 30Ne with three different$ K^-N $ interaction strengths$ a_0=100,\; 300,\; 500{\;{\rm{MeV}}\;{\rm{fm}}^{3}} $ . The spin-orbit splitting of the orbitals$ 1p_{1/2,\; 3/2} $ and$ 1d_{3/2,\; 5/2} $ in$ K^- $ nuclei is larger than that in the corresponding normal nuclei. This effect reduces the binding of the$ 1d_{3/2} $ valence levels in$ ^{28}_{K^-} {\rm{O}}$ and$ ^{30}_{K^-} {\rm{Ne}}$ . Here we also note that the level inversion between orbitals$ 2s_{1/2} $ and$ 1d_{5/2} $ in light kaonic nuclei obtained by the RMF model in Ref. [58] is not found in our SHF calculations.Figure 5. (color online) Partial neutron s.p. levels of spherical nuclei 12Be, 28O, 30Ne (solid bars), and their corresponding
$ K^- $ nuclei (dashed bars) with$ a_0=100 , \;300, \; 500{\;{\rm{MeV}}\;{\rm{fm}}^{3}} $ .To understand the shift of the neutron s.p. levels by the addition of a kaon, we analyze the Schrödinger Eq. (2). Note that both the central potential
$ V_n $ and the spin-orbit potential$ W_n $ are modified when including a kaon [47, 49]. We note that the spin-orbit potential of nucleons in the SHF approach with SLy4 force is [63, 70]$ W_q = \frac{W_0}{2}(\nabla\rho+\nabla\rho_q) \:, $
(13) with
$ q=n $ or p. Qualitatively, the strong$ K^-N $ attraction shrinks the nucleus. Similar effects have been discussed in the case of Λ hyperons [80, 83–85] or antiprotons [86, 87] bound in nuclei. This leads to a deepening of the mean fields in the core region of the nucleus, but a weakening in the peripheral part that is essential for weakly-bound valence neutrons. Moreover, there is always a delicate competition between$ V_n $ and$ W_n $ for some levels. To visualize these effects, we compare in Fig. 6 the potentials$ V_n(r) $ and$ W_n(r) $ of the drip-line nuclei 12Be, 30Ne, and their corresponding$ K^- $ nuclei, together with the partial densities$ \rho_i = 4\pi r^2 v_i^2 |\phi_i(r)|^2 $ of the various occupied neutron s.p. levels. Note that indeed the attractive$ K^-N $ interaction contracts the density distributions and thus enhances self-consistently the mean fields in the core region.Figure 6. (color online) Mean fields
$ V_n(r) $ and spin-orbit potentials$ W_n(r) $ , and the partial densities$ \rho_i(r)= 4\pi r^2 v_i^2 |\phi_i(r)|^2 $ normalized to the actual occupation numbers of all occupied neutron s.p. levels in the normal nuclei 12Be and 30Ne and their corresponding$ K^- $ nuclei with$ a_0=500{\;{\rm{MeV}}\,{\rm{fm}}^{3}} $ . The vertical dotted lines label the crossings between the$ V_n(r) $ and$ W_n(r) $ of normal and$ K^- $ nuclei.Note that
$ V_n(r) $ and$ W_n(r) $ of$ ^{12}_{K^-} {\rm{Be}}$ and$ ^{30}_{K^-} {\rm{Ne}}$ are deeper than those of their normal nuclei for$ r\lesssim 2.9\; {\rm{fm}}$ ,$ r\lesssim 3.0\; {\rm{fm}}$ , and$ r\lesssim 3.2\; {\rm{fm}}$ ,$ r\lesssim 3.8\; {\rm{fm}}$ , respectively (dotted vertical lines). The strongly-bound neutron levels$ 1p_{1/2,3/2} $ ,$ 1d_{5/2} $ , and$ 2s_{1/2} $ of 30Ne are concentrated well within the core region$ r\lesssim3\; {\rm{fm}}$ , which indicates their gain of energy and the larger splitting of$ 1p_{1/2} $ and$ 1p_{3/2} $ in$ K^- $ nuclei, see Fig. 5. Similar phenomena are found in 12Be. On the contrary, a large amount of the$ 1d_{3/2,5/2} - $ state neutrons locate in the range of$ 3.2-3.8\; {\rm{fm}}$ , where the central potentials are smaller and the spin-orbit potentials are larger in$ K^- $ nuclei than in the normal nuclei. Therefore, the splitting of$ 1d_{5/2} $ and$ 1d_{3/2} $ is still enhanced in$ K^- $ nuclei. However, the most peripheral$ 1d_{3/2} $ neutrons are embedded in weaker both central and spin-orbit mean fields at$ r\gtrsim3.8\; {\rm{fm}}$ , which accounts for the upward shift of that level.These considerations explain the possible reduction of the neutron drip line by an added kaon. One might wonder whether a similar effect is possible for the proton dripline of heavier nuclei in spite of the additional
$ K^-p $ Coulomb attraction in this case, which causes an extension of the proton dripline in light nuclei, as observed before for$ ^{\ 5}_{K^-} {\rm{Be}}$ and$ ^{11}_{K^-} {\rm{O}}$ . This is addressed in Fig. 7 for the 36Ca nucleus, comparing calculations with and without Coulomb interaction, and note that even with Coulomb interaction the additional$ K^- $ meson still decreases the energy of the highest-occupied proton s.p. levels$ 1d_{3/2} $ . Thus, both an extension or reduction of the proton drip line are in principle possible, depending on the balance between the core shrinking effect and the Coulomb attraction. The quantitative realization of these effects depends in our current model on the values of the interaction parameters$ a_0 $ and$ x_0 $ , which can hopefully be fixed better with the aid of future experimental data.Figure 7. (color online) Partial proton s.p. levels of the spherical nucleus 36Ca (solid bars) and corresponding
$ K^- $ nucleus (dashed bars) with or without$ K^-p $ Coulomb interaction with$ a_0=500{\;{\rm{MeV}}\,{\rm{fm}}^{3}} $ .In conclusion, the
$ K^- $ meson increases the total binding energies for all nuclei, whereas it shows obvious impact on their deformation only for the lighter nuclei without shell closure. Moreover, the effect of an additional$ K^- $ meson on the nuclei near the neutron drip line depends on the highest-occupied neutron s.p. level. If this level is an orbit without spin-orbit splitting (e.g.,$ 2s_{1/2} $ ) or the lower orbit with splitting (e.g.,$ 1p_{3/2} $ and$ 1d_{5/2} $ ), the additional$ K^- $ may allow new unstable isotopes or only make the isotopes more stable, and thus extend or not shift the neutron drip line. If instead, this level is an upper orbit with splitting (e.g.,$ 1d_{3/2} $ ), the additional$ K^- $ can make some weakly bound nuclei unbound and reduce the neutron drip line, such as for O isotopes. The same mechanism is found for the proton dripline because of the small$ K^-p $ Coulomb interaction. This effect is caused by the shrinking of the nucleon wave functions due to a particularly strong attractive$ K^-N $ interaction. A similar role of an additional$ K^- $ in influencing s.p. levels was pointed out in the RMF approach [58].
Effects of a kaonic meson on the ground-state properties of nuclei
- Received Date: 2021-12-20
- Available Online: 2022-06-15
Abstract: The effects of an additional