Deformation of \Lambda hypernuclei

Deformation of hypernuclei

Myaing Thi Win Department of Physics, Tohoku University, Sendai 980-8578, Japan    K. Hagino Department of Physics, Tohoku University, Sendai 980-8578, Japan
July 27, 2019

We study the deformation property of hypernuclei using the relativistic mean field (RMF) method. We find that Si and C hypernuclei have spherical shape as a consequence of the additional particle, whereas the corresponding core nuclei, Si and C, are oblately deformed. Most of other hypernuclei have a similar deformation parameter to the core nucleus, in accordance with the previous study with the non-relativistic Skyrme-Hartree-Fock method. We discuss the sensitivity of our results to the choice of pairing interaction and to the parameter set of the RMF Lagrangian.


I Introduction

It has been well known that many open-shell nuclei are deformed in the ground state. The nuclear deformation generates the collective rotational motion, which is characterized by a pronounced rotational spectrum as well as strongly enhanced quadrupole transition probabilities. Theoretically, a standard way to discuss nuclear deformation is a self-consistent mean-field theoryBHR03 (). By allowing the rotational symmetry to be broken in the mean-field potential, the mean-field theory provides an intuitive and transparent view of the nuclear deformation. See e.g., Ref. SDNPD03 () for a recent systematic study of nuclear deformation based on the Skyrme-Hartree-Fock-Bogoliubov method. The state-of-the-art mean-field approach also takes into account the effects beyond the mean-field approximation, such as the angular momentum projection and the configuration mixing BH08 ().

The self-consistent mean-field method has been extensively applied also to hypernuclei R81 (); YBZ88 (); YB90 (); LY97 (); L98 (); CLS00 (); VPRS01 (); ZSSWZ07 (); MZ89 (); RSM90 (); GVH93 (); MJ94 (); ST94 (); VPLR98 (); LMZZ03 (); SYT06 () (see Ref. HT06 () for a recent experimental review on hypernuclei). These calculations have successfully reproduced the mass number dependence of binding energy, from a light nucleus C to a heavy nucleus Pb. We notice that most of these calculations have assumed spherical symmetry. Only recently, deformed calculations have been carried out in a broad mass region using the non-relativistic Skyrme Hartree-Fock method ZSSWZ07 (). The authors of Ref. ZSSWZ07 () have reported that the hypernuclei which they studied have a similar deformation parameter to the corresponding core nuclei with the same sign.

The aim of this paper is to study the deformation property of hypernuclei using the relativistic mean field (RMF) method, as an alternative choice of effective and interactions. The RMF method has been as successful as the Skyrme-Hartree-Fock method in describing stable nuclei as well as nuclei far from the stability lineRGL90 (); R96 (). Vretenar et al. have argued VPLR98 () that the change in the nucleon spin-orbit potential due to the presence of particle is much more emphasized in the RMF approach as compared to the non-relativistic approach, since the spin-orbit potential in the former approach is actually given as a sum of scalar and vector potentials. That is, even if the change in the mean-field potential (given as a difference of scalar and vector potentials) is small, the change in the spin-orbit potential may not necessarily be small. Therefore, a slightly different conclusion from that with the non-relativistic approach may result concerning the structure of hypernuclei. In fact, we will demonstrate below that the shape of C and Si nuclei are drastically changed when a particle is added to them.

The paper is organised as follows. In Sec. II, we briefly summarize the RMF approach for hypernuclei. In Sec. III, we apply the RMF method to Ne and Si isotopes, and discuss the influence of particle on the deformation of the hypernuclei. We also discuss the deformation of C and C nuclei. We summarize the paper in Sec. IV.

Ii RMF for hypernuclei

In the RMF approach, nucleons and a particle are treated as structureless Dirac particles, interacting through the exchange of virtual mesons, that is, the isoscalar scalar meson, the isoscalar vector meson, and the isovector vector meson. The photon field is also taken into account to describe the Coulomb interaction between protons. The effective Lagrangian for hypernuclei may be given as MZ89 (); RSM90 (); GVH93 (); MJ94 (); ST94 (); VPLR98 (); LMZZ03 (); SYT06 ()


where and are the Dirac spinor and the mass for the particle, respectively. Notice that the particle couples only to the and mesons, as it is neutral and isoscalar. Those coupling constants are denoted as and , respectively. For simplicity, we neglect the tensor - interaction. This is justified since we consider only the ground state configuration, in which the particle occupies the lowest single-particle stateVPLR98 (); ST94 (), being the projection of the single-particle angular momentum onto the symmetry axis. in Eq. (1) is the standard RMF Lagrangian for the nucleons. See e.g., Refs. VPLR98 (); RGL90 (); R96 () for its explicit form.

We solve the RMF Lagrangian (1) in the mean field approximation. The variational principle leads to the Dirac equation for the particle,


where is the single-particle energy for the particle state, and the Klein-Gordon equation for the mesons,


To derive these equations, we have used the time-reversal symmetry and retained only the time-like component of RGL90 (). and are the scalar and vector densities for the nucleons, which are constructed with the spinor for the nucleons using the so called no-sea approximation, i.e., neglecting the contribution from the antiparticles. and are the coupling constants of the nucleons to the sigma and the omega mesons, respectively, and and are the coefficients in the non-linear sigma terms in .

We solve these equations, together with the Dirac equation for the nucleons and the Klein-Gordon equations for the meson and the photon field, iteratively until the self-consistency condition is achieved. For this purpose, we modify the computer code RMFAXIAL RGL97 () to include the particle. In this code, the RMF equations for the nucleons are solved with the harmonic oscillator expansion methodRGL90 (), assuming the axial symmetry. The pairing correlation among the nucleons is also taken into account in the BCS approximation.

With the self-consistent solution of the RMF equations, we compute the intrinsic quadrupole moment of the hypernucleus,


The quadrupole deformation parameter is then estimated with the intrinsic quadrupole moment as RGL90 (); RS80 (); HLY06 (),


where is the mass number of the core nucleus for the hypernucleus. We use fm for the radius of the hypernucleus.

Iii Quadrupole Deformation of hypernuclei

We now numerically solve the RMF equations and discuss the quadrupole deformation parameter of hypernuclei. For this purpose, we use the NL3 parameter setnl3 () for the RMF Lagrangian for the nucleons, . For the -meson coupling constants, we follow Refs. MJ94 (); VPLR98 () and take and . The value for was determined from the naive quark model ST94 (), while the value for was slightly fine-tuned in order to reproduce the binding energy of O MJ94 (). For the pairing correlation among the nucleons, we employ the constant gap approximation with the pairing gap given in Ref. MN92 (), that is, and MeV for the neutron and the proton pairing gaps, respectively. (It has been known that these pairing gaps underestimate the deformation parameter of Ne nucleus when it is calculated with the NL3 parameter setBG03 (); LRR99 (). We therefore arbitrarily switch off the pairing interaction when we calculate the Ne and Ne nuclei.)

Figure 1: (Color online) Quadrupole deformation parameter for Ne isotopes obtained with the RMF method with the NL3 parameter set. The dashed line is the deformation parameter for the core nucleus, while the solid line is for the corresponding hypernucleus.
Figure 2: (Color online) Same as Fig.1, but for Si isotopes.

Figures 1 and 2 show the deformation parameter for the ground state of Ne and Si isotopes, respectively. The dashed line is the deformation parameter for the even-even core nuclei, while the solid line is for the corresponding hypernuclei. For the Ne isotopes, the deformation parameter is always similar between the core nucleus and the corresponding hypernucleus, although the deformation parameter for the hypernucleus is slightly smaller than that for the core nucleus. This is consistent with the previous results with the non-relativistic Skyrme-Hartree-Fock method ZSSWZ07 (). On the other hand, for the Si isotopes, the deformation parameter for the Si nuclei is drastically changed when a particle is added, although the change for the other Si isotopes is small. That is, the Si nuclei have oblate shape in the ground state. When a particle is added to these nuclei, remarkably they turn to be spherical.

Figure 3: (Color online) The potential energy surface for the Ne (the dashed line) and Ne (the solid line) nuclei obtained with the constrained RMF method with the NL3 parameter set. The energy surface for Ne is shifted by a constant amount as indicated in the inset.
Figure 4: (Color online) Same as Fig. 3, but for the Si and Si nuclei.

The potential energy surfaces for the Ne and Si nuclei are shown in Figs. 3 and 4, respectively. These are obtained with the constrained RMF method with quadrupole constraint RS80 (); FQKV73 (). The meaning of each line is the same as in Figs. 1 and 2. In order to facilitate the comparison, we shift the energy surface for the hypernuclei by a constant amount as indicated in the inset of the figures. For the Ne nucleus, the prolate minimum in the energy surface is relatively deep (the energy difference between the oblate and the prolate minima is 3.04 MeV, and that between the spherical and prolate configurations is 3.63 MeV), and it is affected little by the addition of the particle. On the other hand, the energy surface for the Si nucleus shows a relatively shallow oblate minimum, with a shoulder at the spherical configuration. The energy difference between the oblate and the spherical configurations is 0.754 MeV, and could be easily inverted when a particle is added.

Figure 5: (Color online) Same as Fig. 4, but obtained with the NLSH parameter set.

In order to check the parameter set dependence of the results, we repeat the same calculation with the NLSH parameter set SNR93 (). The potential energy surface for the Si nuclei obtained with the NLSH set is shown in Fig. 5. One sees that the potential energy surface is qualitatively almost the same between the NL3 and NLSH parameter sets, although the binding energy is slightly different. Namely, the oblate Si nucleus becomes spherical in the presence of particle, again with the NLSH parameter set. We also check the dependence of the results on the treatment of pairing correlation. For this purpose, we perform the calculations i) without taking into account the paring correlation and ii) with the constant force approach for the strength of the seniority pairing force. For the latter approach, we determine and so that they lead to and MeV for the ground state of each nucleus. (For instance, and MeV for the NL3 calculation of Si nucleus.) We confirm that our conclusion remains the same for both the treatments of the pairing correlation, due to the fact that or =14 is an oblate magic numberRNS78 (). We therefore conclude that the particle significantly changes the deformation of Si nucleus, at least for the two parameter sets of the RMF Lagrangian and irrespective of the treatment of pairing correlations.

nucleus const.- const.- no-pairing const.- const.- no-pairing
Si 0.31 0.29 0.33 0.29 0.25 0.32
Si 0.00 0.01 0.00 0.00 0.00 0.00
Si 0.19 0.19 0.15 0.19 0.19 0.19
Si 0.00 0.00 0.00 0.06 0.06 0.18
Si 0.14 0.14 0.20 0.15 0.15 0.20
Si 0.00 0.00 0.18 0.11 0.12 0.18
C 0.25 0.25 0.286
C 0.00 0.00 0.00
Table 1: Comparison of deformation parameter for the Si, Si, and C nuclei obtained with the NL3 and NLSH parameter sets of RMF Lagrangian. The pairing correlation is taken into account either in the constant (i.e., constant pairing gap) or in the constant (i.e., constant pairing force) approximations for a seniority pairing interaction. The results without the pairing correlation are also shown. The calculation with the NL3 set did not converge for C and the results are not shown in the table.

In contrast, for the Si nuclei, the dependence on the parameter set and the treatment of pairing is much stronger. For instance, with the NLSH parameter set, the Si is slightly oblate and the deformation parameter is similar between Si and Si. Without the pairing correlation, the deformation is similar between Si and Si for both NL3 and NLSH. Apparently more careful investigations will be necessary for these nuclei before we can draw a definite conclusion on their deformation parameter. We summarize our results for Si and Si + in Table I.

Figure 6: (Color online) Same as Fig. 5, but for the C and C nuclei.

As another example which shows a large effect of particle on nuclear deformation, we next discuss the C nucleus. For this nucleus, the calculation with the NL3 parameter set did not converge, due to the instability of the scalar meson field R88 (); FBR89 (), and we here show only the results with the NLSH set. Fig. 6 shows the potential energy surface obtained with the NLSH parameter set together with the constant gap approximation for the pairing correlation. The behaviour of energy surface of C is similar to that of Si shown in Figs. 4 and 5. That is, the energy surface has a shallow oblate minimum and a shoulder at the spherical configuration. For this nucleus, the energy difference between the oblate and the spherical configurations is as small as 0.13 MeV. By adding a particle, the oblate minimum disappears and the ground state becomes spherical. This is exactly the same effect of particle as that in the Si nucleus. For this light nucleus, the pairing correlation does not play an essential role, and we confirm that our conclusion remains the same even if we do not include the pairing correlation (see Table I).

Iv Summary

We have used the relativistic mean field (RMF) theory to investigate quadrupole deformation of hypernuclei. We have shown that, while an addition of particle does not influence much the shape of many nuclei, C and Si make important exceptions. That is, we have demonstrated that the particle makes the shape of these nuclei change from oblate to spherical. For the Si nucleus, this conclusion was achieved both with the NL3 and NLSH parameter sets of the RMF Lagrangian, although the calculation with NL3 was not converged for the C nucleus due to the instability of sigma field. We have also confirmed that the conclusion is independent of the treatment of pairing correlation among the nucleons.

An important next question will be how to observe experimentally the drastic structure change of the hypernuclei found in this paper. For this purpose, a measurement of the energy of the first 4 state, and thus a deviation from a rotational spectrum, will be extremely useful. On the other hand, the potential energy surface for the C and Si nuclei is somewhat soft and a large anharmonic effect of collective vibration might be expected. One may thus need to perform e.g., a generator coordinate method (GCM) calculation BH08 (), on top of the mean field calculation presented in this paper, and calculate the excitation spectra before one can compare the theoretical results with experimental data.

We thank H. Tamura, H. Sagawa, Nyein Wink Lwin, and Khin Nyan Linn for useful discussions. This work was supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology by Grant-in-Aid for Scientific Research under the program number 19740115.


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