Quadrupole deformation in Λ-hypernuclei

EPJ manuscript No. (will be inserted by the editor) Quadrupole deformation in Λ-hypernuclei Bipasha Bhowmick a , Abhijit Bhattacharyya b , and G. Gangopadhyay c arXiv:1403.6261v1 [nucl-th] 25 Mar 2014 Department of Physics, University of Calcutta 92, Acharya Prafulla Chandra Road, Kolkata-700 009, India the date of receipt and acceptance should be inserted later Abstract. Shapes of light normal nuclei and Λ-hypernuclei are investigated using relativistic mean field approach. The FSUGold parametrization is used for this purpose. The addition of a Λ is found to change the shape of the energy surface towards prolate. The deformation in a Λ-hypernucleus, when the hyperon is in the first excited state, is also discussed. The effect of the inclusion of the hyperon on the nuclear radius is generally small with one exception. PACS. 21.80.+a Hypernuclei 1 Introduction One of the unique and interesting aspects of hypernuclei is the structural change caused by the hyperon. As an impurity in normal nuclei, a hyperon may be expected to induce many effects on the core nucleus, such as change in size[1], shape change, modification of its cluster structure[2], occurrence of nucleon and hyperon skin or halo[2,3], shift of neutron drip line to more neutron-rich side[3,4,5] etc. Owing to recent experimental developments some of those have been observed in light p-shell hypernuclei. As examples, we can refer to the reduction of B(E2) in 7Λ Li[6] and the identification of the super-symmetric hypernuclear state in 9Λ Be[7]. We can expect that a new experimental facility of Japan Proton Accelerator Research Complex (J-PARC) will reveal new spectral information on p and sd shell as well as neutron-rich hypernuclei. As the shape of nuclei plays a decisive role in determining their properties, such as quadrupole moments and radii, mean field calculations have been performed in recent years to investigate the change of nuclear shape due to the addition of a Λ hyperon. Deformed Skyrme- HartreeFock(SkHF) studies in Ref. [8] have shown that the deformation of any hypernucleus is slightly less than the corresponding core nucleus. On the other hand, relativistic mean field (RMF) study in Ref. [9] found that the 29 deformation completely disappears in 13 Λ C and Λ Si hypernuclei though the corresponding normal core nuclei are deformed. Recently, a study with anti-symmetrized molecular dynamics by Isaka et al.[10] have found that the Λ in p-wave enhances the nuclear deformation, while that in s-wave reduces it. a b c [email protected] [email protected] [email protected] To perform a systematic and quantitative study of the structure change of the p and sd-shell nuclei, caused by the hyperon, we study the binding energy, quadrupole deformation, and the root mean square radii of a number of hypernuclei within a RMF model. The relativistic mean field theory of nucleus has been fairly successful in reproducing the properties of finite nuclear systems[11]. It has also been extensively applied to the study of hypernuclei[12]. 2 Results There have been a number of RMF parametrizations for prediction of the nuclear ground state properties. In the present work the FSUGold Lagrangian density has been employed[13]. This parametrization has already been extended to include hyperons and to study the properties of hypernuclear systems[14]. In this work it is used to study the change in quadrupole deformation due to addition of hyperon. The parameters of the ΛN interaction have been determined by fitting the experimental separation energies of 12 hypernuclei in the mass region 16 to 208[14]. The masses of the two physical mesons have been taken from experiment. The nucleon mass is taken as 939 MeV. The mass of the Λ has been fixed at 1115.6 MeV. In the present work, we have chosen to limit the type of deformation to azimuthally symmetric and reflection symmetric systems which corresponds to prolate and oblate ellipsoids for quadrupole deformation. This eliminates all the three-vector components of the boson fields. Therefore, the limitation on the type of deformation simplifies the calculation with contributions from only the scalar meson, the zero components of the isoscalar vector meson, the photon, and the neutral ρ meson fields. This is the same set of boson fields that was required for spherical 2 Bipasha Bhowmick et al.: Quadrupole deformation in Λ-hypernuclei nuclei; however, these boson fields now have an additional Table 1. Calculated Binding energy/nucleon(-E/A) and Λ separation energy (EΛ ) in MeV, quadrupole deformation paangular dependence. Solving the field equations with no further simplifica- rameter β, and the root mean square radius in fm at the mintion can be very involved due to the difficulties encoun- imum energy for the core nucleus and the corresponding hytered in obtaining solutions for coupled partial differential pernucleus in different Λ states. p equations. We expand the boson fields in terms of LegenNucleus -E/A β rp rn hr 2 i dre polynomials and The nucleonic orbital wave functions (Λ-state) in terms of spherical angle functions. The method of solu8 Be 5.668 0.38 2.41 2.31 2.36 tion has been explained in detail in Ref. [15]. Terms up to 5.533 -0.37 2.39 2.28 2.33 9 angular momentum L = 6 have been taken into account. 5.837 7. 189∗ 0.34 2.40 2.56 2.34 Λ Be (s) 10 As we concentrate on very light nuclei, pairing is not exB 6.201 0.27 2.41 2.26 2.33 pected to play a very important role. The grid size for 6.116 -0.20 2.39 2.24 2.32 11 solving the differential equations is taken to be 0.1 fm. 6.501 9.501 0.33 2.54 2.49 2.40 Λ B (s) 12 We calculate the quadrupole deformation parameter C 7.175 -0.21 2.36 2.18 2.27 and the root mean square radii for a number of Λ-hypernuclei 13 7.534 11.842† 0.13 2.59 2.39 2.39 Λ C (s) 13 up to A < 30 putting the Λ in its deformed ground state 6.683 0.779 0.14 2.61 3.41 2.92 Λ C (p) 18 and also in the first excited state, which have opposite F 7.562 0.14 2.70 2.45 2.58 parities. The results of our calculation are presented in 7.561 -0.11 2.69 2.44 2.57 19 7.883 13.661 0.13 2.71 2.55 2.56 Table 1. Λ F (s) 19 7.399 4.465 0.14 2.71 2.69 2.63 Λ F (p) In our calculation, the ground state of Λ is a pure s22 Na 7.565 0.22 2.85 2.57 2.72 state as the higher positive parity states are too high in 7.475 -0.13 2.83 2.55 2.69 the continuum. The first excited state is a negative parity 23 Na (s) 7.904 15.362 0.21 2.86 2.65 2.70 Λ state which has contributions from both the p orbitals. 23 Na (p) 7.537 6.921 0.22 2.86 2.74 2.74 Λ We thus indicate the solutions for hypernuclei with the 24 Mg 7.814 0.25 2.91 2.60 2.76 hyperon in the ground or the first excited states with ‘s’ 7.640 -0.15 2.89 2.59 2.74 and ‘p’ in parentheses in the table and in the relevant fig25 Mg (s) 8.142 16.014 0.23 2.91 2.67 2.74 Λ ures. We have looked for both prolate as well as oblate 25 Mg (p) 7.823 8.039 0.24 2.91 2.75 2.78 Λ minima in normal nuclei and hypernuclei. It is observed 25 Mg 7.872 0.20 2.88 2.64 2.76 that in all the cases, the deformed hypernuclear minimum 7.793 -0.13 2.87 2.63 2.75 corresponds only to prolate deformation. Thus, in case of 26 8.248 17.648 0.22 2.92 2.70 2.75 π − Λ Mg (s) the excited state, the hyperon occupies the K = 1/2 26 7.927 9.30 0.23 2.92 2.78 2.79 Λ Mg (p) state. No oblate minima are observed in the hypernuclear 25 Al 7.657 0.15 2.95 2.56 2.77 systems. This observation will be elaborated later in this 7.565 -0.11 2.94 2.55 2.76 work. Very little information is available for the Λ sepa26 8.001 16.601 0.14 2.95 2.63 2.75 Λ Al (s) ration energy in the ground state of the hypernucleus; the 26 7.676 12.103 0.15 2.95 2.71 2.78 Λ Al (p) existing experimental values agree reasonably well with 26 Al 7.828 0.11 2.92 2.60 2.76 our calculation. 7.795 -0.09 2.92 2.59 2.76 27 We see that when the Λ is placed in the ground state, 8.200 17.872 0.13 2.96 2.66 2.76 Λ Al (s) 27 the deformation changes slightly on inclusion of the hy7.874 9.07 0.14 2.96 2.74 2.80 Λ Al (p) 28 peron, with only one exception. The deformation may inSi 8.080 0.00 2.95 2.59 2.77 29 crease as well as decrease, but in general the shape remains 8.453 18.897 0.06 3.00 2.65 2.78 Λ Si (s) 29 similar to that of the core nucleus. 8.118 9.182 0.06 3.00 2.72 2.86 Λ Si (p) However, in the case of 13 Λ C, the shape appears to ∗ change drastically on the inclusion of the Λ. The ground † Exp. value is 6.84 ± 0.05 MeV[16] Exp. values are 11.69±0.12 MeV[16], 11.38 MeV[7] 12 state of C shows oblate deformation, whereas the corre13 sponding hypernucleus Λ C is prolate. This necessitated further investigation where we study 13 the total energy of the system against deformation in a and Λ C to be oblate and prolate, respectively, actually a flat minimum is being replaced by a more sharp prolate constrained calculation. In Fig. 1 we have represented the one. Our result is different from the previous HF calculatotal binding energy with respect to the minimum (Etot 12 13 tions by Zhou et al.[8] or the SkHF+BCS calculations by C as a function of deformation paramEmin ) for C and tot Λ eter β. We see that the energy surface of 12 C is almost flat Win et al.[17], which have reported similar deformations 12 13 with a oblate minimum around β = −0.2. Though there for C and Λ C. Some calculations have also reported a spherical configuration for 13 is no minimum for prolate deformation, around β = 0.1 Λ C[9,10,18]. As a case where deformation appears in hypernuclei there exists a region which might have formed a minimum. Inclusion of the Λ makes the oblate minimum disappear while the core is spherical, we study 29 Λ Si. Lu et al.[18] and the prolate minimum is found to be formed in the concluded that the prediction of the shape of 29 Λ Si is pa29 Si is slightly prolate rameter dependent. We see that C. One can see that although the unenergy surface of 13 Λ Λ constrained calculations predict the ground state of 12 C deformed with the FSUGold parameter whereas 28 Si is Bipasha Bhowmick et al.: Quadrupole deformation in Λ-hypernuclei it slightly increases the deformation excluding the case of 29 Λ Si, where the deformation remains the same. Calculation for the deformation of hypernuclei for an excited Λ state reported by Isaka et al.[10] found similar results. In 9Λ Be and 11 Λ B, the excited Λ state is unbound. Etot - Emin tot (MeV) 6 12 C 13 5 ΛC 4 3 2 1 0 -1 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 β Fig. 1. Total energy with respect to the minimum (Etot -Emin tot ) as function of deformation β for 12 C and the corresponding hypernucleus 13 Λ C. We have also investigated the changes in the root mean square radii of the hypernuclei (Table-1). The last column, in case of hypernuclei, includes the effect of the hyperon also. It is seen that the inclusion of a Λ hyperon, causes either a slight increase or no modification in neutron and proton radii (rn and rp , respectively) when the Λ particle is in the s-state. When the hyperon is excited to the pstate the size of the hypernucleus increases only slightly in almost all cases. The only exception is 13 Λ C, where this increase in radius is significant, so that the hypernucleus is much larger than the corresponding core nucleus. spherical. In Fig. 2 we have represented the total binding min energy with respect to the minimum (Etot − Etot ) for 28 29 Si and Λ Si as a function of deformation. From Fig. 2, 28 29Si Λ Si Etot - Emin tot (MeV) 10 8 6 4 2 0 0 0.1 0.2 0.3 0.4 β Fig. 2. Total energy with respect to the minimum as function of deformation β for 28 Si and the corresponding hypernucleus 29 Λ Si. we see that the rather flat energy minima at the spherical configuration in case of 28 Si, becomes sharper and shifts slightly towards positive side in case of 29 Λ Si. We also note that our results for 25,27 Λ Mg agree with the SkHF+BCS calculations by Win et al.[17]. However, our results, as a whole, are different from the previous RMF and SkHF calculations[9,17,4,8,18,3] in the sense that all the previous calculations report a decrease in deformation on addition of a Λ particle. Our calculations show that the deformation may increase in some cases. In general, the inclusion of the Λ hyperon makes the energy surface sharper, the oblate minima disappears and the prolate one becomes prominent, thus producing a shift in deformation. Similar observations have been found for the other nuclei investigated in Table 1. In the case of excited Λ states, our calculations show that when the Λ goes from the ground-state to the p-state 1 monopole density (fm-3) 12 -2 -0.4 -0.3 -0.2 -0.1 3 1x10 -1 1x10 -2 1x10 -3 1x10 -4 1x10 -5 1x10 -6 13 ΛC 0 2 4 6 8 radius (fm) ρ0(s) ρ0(p) ρh0(s) ρh0(p) 10 12 14 Fig. 3. The monopole nucleon and hyperon density profile in 13 Λ C when the Λ is in different single particle state (s and p, given in parentheses). Here ρo refer to monopole nucleon density and ρh0 refer to monopole hyperon density. . The reason for the large increase in size in 13 Λ C can be seen in the distribution of densities of the nuclei. We 25 plot the monopole densities in 13 Λ C and Λ Mg in Figs. 3 and 4, respectively, to see the effect of the hyperon. Both the cases with the hyperon in its ground state and excited state are shown. Fig. 3 shows that in 13 Λ C, when the hyperon is placed in the p-state, the densities extend to larger distances. Actually, the hyperon is very loosely bound in the excited state. Hence, its wave function extends to a very large distance. As the hyperon interacts strongly with the nucleons, the nucleon density also shows a large tail, reminiscent of halo nuclei. However, this is not a general phenomena as evident from Table I. As an example, we present the density profile for 25 Λ Mg in Fig. 4. It is clear that in this hypernucleus, both the nucleon and the hyperon densities remain practically unchanged whether the hyperon is in the s-state or in the p-state. The p-shell hyperon in this case is strongly bound and does not show any halo-like behaviour. 4 Bipasha Bhowmick et al.: Quadrupole deformation in Λ-hypernuclei monopole density (fm-3) 1 1x10 -1 1x10 -2 1x10 -3 1x10 -4 1x10 -5 1x10 -6 1x10 -7 ρ0(s) ρ0(p) ρh0(s) ρh0(p) 25 Λ Mg 0 2 4 6 radius (fm) 8 10 12 Fig. 4. The monopole nucleon and hyperon density profile in 25 Λ Mg when the Λ is in different single particle state (s and p). See caption of Fig. 3 for details. 3 Summary We have studied the deformation of core and hypernuclei in the RMF approach using the FSUGold parameter set. The calculated Λ binding energies agree reasonably well with the experimentally observed values. We see that the inclusion of a Λ hyperon changes the energy surface making it steeper. There exists no oblate minimum in any light hypernucleus. Results of the present calculation differ from the previous ones. The latter usually predict that inclusion of a Λ tends to drive the shape to spherical, while our results show that the change in β is usually small and may go either way. In general, the nucleon density profile changes to a small extent on inclusion of the hyperon, whether in the ground state or the first excited state. When the Λ goes to the p-state, both the deformation and radius increases by a small amount. The only exception is 13 Λ C where, the hyperon being very loosely bound, create a halo-like structure. 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