Shape of Lambda Hypernuclei within the Relativistic Mean-Field Approach

Transcription

1 Universities Research Journal 2011, Vol. 4, No. 4 Shape of Lambda Hypernuclei within the Relativistic Mean-Field Approach Myaing Thi Win 1 and Kouichi Hagino 2 Abstract Self-consistent mean-field theory was used to discuss the ground state properties of lambda() hypernuclei. The deformation of lambda hypernuclei was discussed using the relativistic mean-field (RMF) approach. Although most of hypernuclei have a similar deformation parameter to the core nucleus, the shapes of 28 Si and 12 C are drastically changed, from oblately deformed to the spherical, if a lambda particle is added to this nucleus. The behavior of potential energy surfaces in these nuclei was discussed. Introduction One of the main interests in hypernuclear physics is to investigate how an addition of hyperon influences the properties of atomic nuclei. With the presence of hyperon (having strangeness number S= 1) as impurity, some bulk properties of nuclei such as shape, size and collective motion may be changed. It has been shown that hyperon makes a nucleus more stable and thus extends the neutron drip line. It also makes the nucleus shrink. The shape of nuclei plays a crucial role in determining their properties such as nuclear size and quadrupole moment. 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 transition probabilities. Theoretically, a standard way to discuss nuclear deformation is a self-consistent mean-field theory. It has been extensively applied also to hypernuclei in which mass number dependence of binding energy has been successfully reproduced, from a light nucleus 12 C to a heavy nucleus 208 Pb [1-3]. The deformation property of hypernuclei has been explored in a broad mass region using the non- 1. Assistant Lecturer, Department of Physics, Lashio University 2. Associate Professor, Department of Physics, Tohoku University, Japan

2 220 Universities Research Journal 2011, Vol. 4, No. 4 relativistic Skyrme Hartree-Fock method [2]. In this contribution, similar study was carried out using the the self-consistent Relativistic Mean-Field (RMF) approach. The RMF approach for hypernuclei was briefly summarized. The RMF method to Si isotopes was applied and the influence of particle on the deformation of the hypernuclei was discussed. The deformation of 12 C and 12+ C nuclei was also discussed. 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 σ meson, the isoscalar vector ω meson, and the isovector vector ρ 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 [3,4] μ μ L = LN + ψ [ γ μ( i gωω ) m gσσ] ψ, (1) where ψ and m are the Dirac spinor and the mass for the particle, respectively. Notice that the particle couples only to the σ and ω mesons, as it is neutral and isoscalar. L N in Eq.(1) is the standard RMF Lagrangian for the nucleons [5,6]. The RMF Lagrangian (Eq-1) is solved in the mean field approximation. The variational principle leads to the Dirac equations for the particle, r r r r [ iα. + β( m + gσσ( )) + gωω 0 ( )] ψ = ε ψ, (2) where ε is the single-particle energy for the particle state, and to the Klein-Gordon equation for the mesons, r 2 2 r s r 2 r 3 [ + mσ ] σ( ) = g σρ s ( r ) g 2σ( ) g 3σ( ) + r 0 r (3) g σψ ( ) γ ψ ( ). r 2 2 r s + r 0 r [ + m ] ω( ) = g ρ ( r) g ψ ( ) γ ψ ( ). (4) ω σ ν ω To derive these equations, the time-reversal symmetry has been used and retained only the time-like component of ω μ, ρ s and ρ v are the scalar and vector densities for the nucleons, which are constructed with the spinor for

3 Universities Research Journal 2011, Vol. 4, No the nucleons using the so-called no-sea approximation, i.e., neglecting the contribution from antiparticles. g s and gw are the coupling constants of the nucleons to the σ and ω mesons, respectively, and g 2 and g 3 are the coefficients in the nonlinear σ terms in L N. These equations (2),(3) and (4) are solved iteratively until the selfconsistency condition is achieved. For this purpose, The computer code RMFAXIAL [7] is modified to include the particle. In this code, RMF equations for the nucleons are solved with the harmonic oscillator expansion method, 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, the intrinsic quadrupole moment of the hypernucleus is computed, 16π r r + r r 2 Q = [ ( ) ( ) ( )] 20 ( ˆ). 5 dr ρv + ψ ψ r Y r (5) The quadrupole deformation parameter β 2 is then estimated with the intrinsic quadrupole moment as 16π 3 2 Q = ( AC + 1) R. (6) 0 β2 5 4π where A C = A-1 is the mass number of the core nucleus for the hypernucleus. We use R 0 = 1.2 A 1/3 C fm for the radius of the hypernucleus. Quadrupole Deformation of Hypernuclei The RMF equations are numerically solved and the quadrupole deformation parameter of hypernuclei is discussed. For this purpose, the NL3 and NLSH parameter sets for the RMF Lagrangian for the nucleons, L N are used. For the -meson coupling constants [8], g ω =2/3 g ω is taken and determined from the quark model and g σ =0.621g σ slightly fine-tuned to reproduce the binding energy of 17 O. For the pairing correlation among the nucleons, the constant gap approximation with the pairing gap values, Δ n = 4.8/N 1/3 and Δ p =4.8/Z 1/3 MeV is employed for the neutron and the proton pairing gaps, respectively. Figure 1 shows the deformation parameter for the ground state of Si isotopes obtained with the NL3 parameter set. The dashed line is the

4 222 Universities Research Journal 2011, Vol. 4, No. 4 deformation parameter for the even-even core nuclei, while the solid line is for the corresponding hypernuclei. The deformation parameter for the 28,30,32 Si nuclei is drastically changed when a particle is added [9], although the change for the other Si isotopes is small. That is, the 28,30,32 Si Fig.1. Quadrupole deformation parameter for Si isotopes obtained with the RMF method with the NL3 parameter set [9]. The dashed line is the deformation parameter for the core nucleus, while the solid line is for the corresponding hypernucleus Fig.2. The density profile for the 28 Si (the left panel) and for the 28+ Si (the right panel) obtained with the RMF method. It is plotted with the cylindrical coordinates (z,ρ), where the z axis is the symmetry axis. The unit for the density is fm -3 nuclei have oblate shape in the ground state. When a particle is added to these nuclei, remarkably they turn to be spherical. The corresponding density profile for 28,28+ Si is shown in Fig.2. The potential energy surfaces

5 Universities Research Journal 2011, Vol. 4, No for the 28,28+ Si nuclei are shown in Fig.3. The energy surface for the 28 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 MeV, and could be easily inverted when a particle is added. It has been confirmed that this conclusion remain the same for the NLSH parameter set of RMF Langrangian. Fig.3. The potential energy surface for the 28 Si (the dashed line) and 28+ Si (the solid line) nuclei obtained with the constrained RMF method with the NL3 parameter set. The energy surface for 28+ Si is shifted by a constant amount as indicated in the figure Fig.4. The same as in Fig.3, but for the 12 C and 12+ C nuclei Another example that shows a large effect of particle on nuclear deformation is the 12 C nucleus. Fig.4 shows the potential energy surfaces

6 224 Universities Research Journal 2011, Vol. 4, No. 4 for 12 C and 12+ C obtained with the NLSH parameter set. For these nuclei, the calculation with the NL3 parameter set did not converge, due to the instability of the scalar meson field [10]. The behavior of the energy surface of 12 C is similar to that of 28 Si shown in Fig.3. That is, the energy surface has a shallow oblate minimum and a shoulder at spherical configuration. For this nucleus, the energy difference between the oblate and the spherical configuration is as small as 0.13MeV. By adding a particle, the oblate minimum disappears and the ground state becomes spherical. For this light nucleus, the pairing interaction does not play an essential role and we confirm that our conclusion remain the same even if we do not include the pairing correlation. Summary The relativistic mean field (RMF) theory was used to investigate quadrupole deformation of hypernuclei [11]. While an addition of particle does not influence much the shape of many nuclei, 28 Si and 12 C make an important exception. It has been demonstrated that the particle makes the shape of these nuclei change from oblate to spherical. An important question will be how to observe experimentally the drastic structure change of the hypernuclei found in this research paper. Therefore, it will be an interesting subject to experimentally measure the deformation properties of hypernuclei. Acknowledgments We thank H. Tamura (Professor, Tohoku university) and H. Sagawa (Professor, Aizu university) for useful discussions. We thank the Japanese Ministry of Education, Culture, Sports, Science and Technology for its support for this research. We also thank Dr. Kay Thi Thin (Professor, Physics Department, Lashio University) for permiting me to write this paper.

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