Collective excitations of Lambda hypernuclei
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1 Collective excitations of Lambda hypernuclei Kouichi Hagino (Tohoku Univ.) Myaing Thi Win (Lashio Univ.) J.M. Yao (Southwest Univ.) Z.P. Li (Southwest Univ.) 1. Introduction 2. Deformation of Lambda hypernuclei 3. Collective rotational excitations of hypernuclei 4. Summary
2 Introduction Impurity effects: one of the main interests of hypernuclear physics how does Λ affect several properties of atomic nuclei? size, shape, density distribution, single-particle energy, shell effect, fission barrier the most prominent example: the reduction of B(E2) in 7 ΛLi 9.3 +/- 2.1 e 2 fm / /- 0.5 e 2 fm 4 about 19% reduction of nuclear size (shrinkage effect) K. Tanida et al., PRL86( 01)1982
3 how about heavier nuclei? sd-shell nuclei Ikeda diagram excitation energy clusterization ground state series (shell model structure) the g.s. has a shell model-like structure for nuclei heavier than Be (cluster-like structure appears in the excited states:threshold rule)
4 Shell model (mean-field) structure and nuclear deformation many open-shell nuclei are deformed in the ground state characterstic feature of finite many-body systems spontaneous symmetry breaking of (rotational) symmetry B(E2) for deformed nuclei A change in B(E2) can be interpreted as a change in β
5 sd-shell nuclei:prominent nuclear deformation an evidence for deformation rotational spectrum (MeV) Mg How is the deformation altered due to an addition of Λ particle?
6 Self-consistent mean-field method: optimized shape can be automatically determined = suitable for discussion of shape of hypernuclei First application to deformed hypernucleus J. Zofka, Czech. J. Phys. B30( 80)95 Hartree-Fock calculations with V NN : 3 range Gauss V ΛN : 2 range Gauss Λ changes the Q-moment (deformation) at most by 5% e.g., β = 0.5 β=0.475
7 Shape of hypernuclei Recent Skyrme-Hartree-Fock +BCS calculation by Zhou et al. (with assumption of axial symmetry for simplicity) similar deformation between the hypernuclei and the core nuclei hypernuclei: slightly smaller deformation than the core X.-R. Zhou et al., PRC76( 07)
8 Deformation of Λ hypernuclei Recent Skyrme-Hartree-Fock calculations by Zhou et al. How about Relativistic Mean-Field (RMF) approach? non-relativistic reduction (strong cancellation between V and S) cf. D. Vretenar et al., PRC57( 98)R1060 changes in V and S due to a Λ particle are emphasized (only in RMF)
9 RMF for deformed hypernuclei quark model 17 Λ O cf. D. Vretenar et al., PRC57( 98)R1060 Λσ and Λω couplings variational principle etc. self-consistent solution (iteration)
10 RMF for deformed hypernuclei self-consistent solution (iteration) (intrinsic) Quadrupole moment Application to hypernuclei parameter sets: NL3 and NLSH Axial symmetry pairing among nucleons: Const. gap approach (MeV) Λ particle: the lowest s.p. level (K π =1/2 + ) Basis expansion with deformed H.O. wf Deformation parameter: (fm)
11 Ne isotopes Si isotopes in most cases, similar deformation between the core and the hypernuclei hypernuclei: slightly smaller deformation than the core conclusions similar to Skyrme-Hartree-Fock (Zhou et al.) Exception: 29 ΛSi oblate ( 28 Λ Si) spherical ( 29 ΛSi) Myaing Thi Win and K.H., PRC78( 08)054311
12 Potential energy surface (constraint Hartree-Fock) If the energy curve is relatively flat, a large change in nuclear deformation can occur due to an addition of Λ particle the same conclusion also with NLSH and/or with another treatment of pairing correlation (constant G approach) Myaing Thi Win and K.H., PRC78( 08)054311
13 Another example: 13 ΛC cf. recent AMD calculations oblate spherical M. Isaka, K. Kimura, A. Dote, and A. Ohnishi, PRC83( 11) Myaing Thi Win and K.H., PRC78( 08)054311
14 Comparison between RMF and SHF E 30+Λ E 30 Gain of binding energy= in spherical configuration E = E( β = 0. 2) E( β = 0) E = E( β = 0. 2) E( β = Larger effect of NΛ force in RMF Si Si 0. 14MeV ( SHF) 0) 0. 3MeV ( RMF) SHF RMF RM F H.-J.Schulze, M.Thi Win, K.Hagino,H.Sagawa Prog.Theo.Phys123( 10) 569
15 Systematic comparison with Skyrme-Hartree-Fock method: Stronger influence of Λ in RMF than in SHF Disappearance of deformation can happen also with SHF if the energy curve is very flat H.-J. Schulze, Myaing Thi Win, K.H., H. Sagawa, PTP123( 10)569 A key point is a flatness of potential energy curve
16 3D Hartree-Fock calculation for hypernuclei So far, axial symmetric shape has been assumed for simplicity Effect of Λ particle on triaxial deformation? θ,φ z Courtesy: Takeshi Koike
17 Skyrme-Hartree-Fock calculations for hypernuclei 3D calcaulations with non-relativistic Skyrme-Hartree-Fock: the most convenient and the easiest way 3D mesh calculation ( lattice Hartree-Fock ) Imaginary time evolution of single-particle wave functions computer code ev8 available P. Bonche, H. Flocard, and P.-H. Heenen, NPA467( 87)115, CPC171( 05)49 (note)
18 M. Rayet, NPA367( 81)381 *Interaction No.1 of Yamamoto et al. + SGII (NN) (Y. Yamamoto, H. Bando, and J. Zofka, PTP80( 88)757) *Pairing among nucleons: BCS approximation with d.d. contact force *Λ particle: the lowest energy state Skyrme-Hartree-Fock calculations for hypernuclei 3D calcaulations with non-relativistic Skyrme-Hartree-Fock: the most convenient and the easiest way 3D mesh calculation ( lattice Hartree-Fock ) Imaginary time evolution of single-particle wave functions computer code ev8 available P. Bonche, H. Flocard, and P.-H. Heenen, NPA467( 87)115, CPC171( 05)49 extension to hypernuclei
19 24 Mg, 25 ΛMg +Λ Myaing Thi Win, K.H., T. Koike, Phys. Rev. C83( 11)014301
20 26 Mg, 27 ΛMg +Λ Myaing Thi Win, K.H., T. Koike, Phys. Rev. C83( 11)014301
21 Discussions Deformation is driven to speherical when Λ is in the lowest state Prolate configuration is prefered for the same value of β All of 24 Mg, 26 Mg, 26 Si, 28 Si show that Λ makes the curvature along the γ direction somewhat smaller Experiment? (the energy of 2 2+ state) quantitative estimat:rpa or GCM or Bohr Hamiltonian
22 Rotational Excitation of hypernuclei Collective spectrum of a hypernucleus: half-integer spin Bohr Hamiltonian for the core part: mass inertias: cranking approximation (Inglis-Belyaev formula for the rotational inertia)
23 Solution of Coll. H fluctuation of deformation parameters J.M. Yao, Z.P. Li, K.H. et al., arxiv: much smaller change
24 reduction of B(E2) from 2 + to Mg: B(E2) = 62.0 e 2 fm 4 J.M. Yao, Z.P. Li, K.H. et al., arxiv: Λ Mg: B(E2) = 56.4 e 2 fm 4 (about 9% reduction)
25 Summary Shape of Λ hypernuclei: from the view point of mean-field theory deformation: in important key word in the sd-shell region RMF: stronger influence of Λ particle Shape of 28 Si : drastically changed due to Λ SHF: weaker influence of Λ, but large def. change if PES is very flat 3D calcaulations softening of γ-vibration? Rotational excitations of Λ hypernuclei about 9% reduction of B(E2) value A challenging problem full spectrum of a hypernucleus odd mass, broken time reversal symmetry, half-integer spins - spectrum of a double Λ hypernucleus?
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