National Natural Science Foundation of China [10605018]; New Century Excellent Talents in University [NCET-07-0730]Properties of the even-N Zr isotopes and their corresponding single-Lambda and double-Lambda hypernuclei are studied in the deformed Skyrme-Hartree-Fock approach. Binding energy, the two-neutron separation energies, radii are studied from beta-stable zone to the neutron drip line in this paper. The neutron drip line nuclei predicted with SLy4 and SkI4 interactions are (122)Zr and (138)Zr, respectively. The neutron drip line of single-Lambda and double-Lambda hypernuclei are (139)(Lambda)Zr and (142)(2 Lambda)Zr with SkI4 interaction, respectively. The predicted hyperon drip line hypernuclei with (80)Zr and (138)Zr cores ads (100)(20 Lambda)Zr and (188)(50 Lambda)Zr, respectively

Cheng, F. Q.

Wei, N.

Zhou, X. R.

周先荣

Chinese Physics C

English

Xiamen University Institutional Repository

CPC(HEP & NP), 2009, 33(Suppl.?): 116—118 Chinese Physics C Vol. 33, Suppl.?, Mar., 2009Properties of Zr hypernuclei indeformed Skyrme Hartree-Fock approach*WEI Ning(??)1) ZHOU Xian-Rong(???)2) CHEN Fang-Qi(???)(Department of Physics and Institute of Theoretical Physics and Astrophysics,Xiamen University, Xiamen 361005, China)Abstract Properties of the even-N Zr isotopes and their corresponding single-Λ and double-Λ hypernuclei arestudied in the deformed Skyrme-Hartree-Fock approach. Binding energy, the two-neutron separation energies,radii are studied from beta-stable zone to the neutron drip line in this paper. The neutron drip line nucleipredicted with SLy4 and SkI4 interactions are 122Zr and 138Zr, respectively. The neutron drip line of single-Λand double-Λ hypernuclei are 139Λ Zr and1422Λ Zr with SkI4 interaction, respectively. The predicted hyperon dripline hypernuclei with 80Zr and 138Zr cores are 10020ΛZr and18850ΛZr, respectively.Key words Deformed Skyrme-Hartree-Fock, neutron drip line, two-neutron separation energy, neutron haloPACS 21.80.+a, 21.10.Dr, 21.10.Gv1 IntroductionThe developments of radioactive nuclear beamsare changing the field of nuclear research from thebeta-stable region towards the drip line. Exoticphenomena such as neutron skin and neutron halohave been found. The discovery of neutron halonucleus, 11Li in 1985[1], has brought much excite-ment to the atomic and nuclear physics communi-ties. It can be described briefly as a very looselybound particle tunneling into the space surround-ing a potential well in a process represented by anextended dilute wave-function tail[2]. In Ref. [3],the neutron halo was reproduced in a self-consistentway, without further modification, using the scatter-ing of cooper pairs to the 2s1/2 level in the contin-uum, excellent agreement with recent experimentaldata is observed. Moreover,a giant halo which com-posed by as many as six neutrons in Zr was pre-dicted by this model[4]. However, the existence of halostructure in Zr is model dependent in the Hartree-Fock-Bogoliubov calculation[5]. The Skyrme param-eter SkI4 and Sly4 were used in the calculation, andonly the former predicted a halo in Zr isotopes withA>122.The hyperon can be free from the Pauli blockingby nucleons, which made it possible to probe deepinto the nucleus, and thus become an ideal tool tostudy the properties of the nuclei. Besides, the hy-peron can provide attractive Λ−N interaction whichwould be helpful in the formation of the neutron halo.Therefore the properties of neutron halos and gianthalos in hypernuclei would be very interesting. Inthis paper, the properties of Zr isotopes and its cor-responding hypernuclei were studied in the deformedSkyrme-Hartree-Fock approach[6].This paper is organized as follows. In sec. 2 webriefly describe the theoretical framework. In sec. 3the calculation result are presented . Finally, a discus-sion of the results and our conclusions are containedin Sec. 4.2 Theoretical frameworkOur model is based on the deformed self-consistent Skyrme-Hartree-Fock methods[6] solved inReceived 3 September 2008* Supported by National Natural Science Foundation of China (10605018) and program for New Century Excellent Talents inUniversity(NCET-07-0730)1)E-mail:nwei@xmu.edu.cn2)E-mail:xrzhou@xmu.edu.cn© 2009 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Instituteof Modern Physics of the Chinese Academy of Sciences and IOP Publishing LtdSuppl.? WEI Ning et al?Properties of Zr hypernuclei in deformed Skyrme Hartree-Fock approach 117the coordinate space with axially symmetric shape[7].More details are given in Ref. [8] and the Refs therein.The total energy of a hypernucleus in the extendedDeformed Skyrme-Hartree-Fock model is expressed asE=∫d3rε(r) (1)with the energy density functionalε= εN[ρn,ρp, τn, τp,Jn,Jp]+εΛ[ρn,ρp,ρΛ, τΛ], (2)where εN is the total energy density of neutrons andprotons[6, 9], and the local density ρq, kinetic densityJq, and spin-orbit current Jq readρq=Nq∑i=1niq|φiq|2 , (3)τq=Nq∑i=1niq|∇φiq|2 , (4)Jq =Nq∑i=1niqφiq∗(∇φiq×σ)/i , (5)where φiq (i=1,Nq) are the single-particle wave func-tions of the Nq occupied states for particles of kindq=n,p,Λ. The occupation probabilities niq (for nu-cleons only) are calculated by taking a residual pair-ing interaction into account, which is taken to be adensity-dependent delta-force[10],Vq(r1, r2)=V ′q(1− ρN(r)ρ0)δ(r1−r2), (6)where ρN(r) is the HF density at r=(r1+r2)/2 andρ0 = 0.16 fm−3. The pairing strength is taken to beV ′p =−1146 MeV fm3, V ′n =−999 MeV fm3.The minimization of the total energy Eq.(1) im-plies the SHF Schro¨dinger equation[−∇· 12m∗q(r)∇+Vq(r)− i∇Wq(r) ·(∇×σ)]φiq(r)=eiqφiq(r). (7)3 ResultsThe two-neutron separation energy is crucial tojudge the neutron drip line, and expose the structureof the nucleus. It is defined asS2n(N,Z)=E(N,Z)−E(N−2,Z), (8)where E(N,Z) is the total energy of the nucleus withN neutrons and Z protons. The two-neutron separa-tion energy for Zr isotopes and its corresponding hy-pernuclei with single-Λ and double-Λ hyperons werecalculated by SHF with SkI4 interaction and Zr iso-topes with SLy4 interaction as a comparison.The binding energies of Zr isotopes from 80Zr to110Zr, which the experiment can reached until now,with SLy4, SkI4 interactions and experimental dataare shown in Fig. 1. The binding energies with SLy4and SkI4 interactions were generally 2 MeV differ-ent from the experimental data. The maximal withSLy4 interaction is 4.8582 MeV while 3.9388 MeVwith SkI4 interaction at 110Zr. It is not possible totell which Skyrme parameters is better for Zr iso-topes from the experimental data. Then we studiedthe radii, two-neutron separation energies with SLy4and SkI4 interactions, respectively. The neutron dripline nuclei predicted with SLy4 and SkI4 interactionsare 122Zr and 138Zr, respectively, which agrees withRef. [5].Fig. 1. Results from DSHF calculation withSLy4, SkI4 interactions and experimental dataFig. 2. The two-neutron separation energies forZr isotopes with SLy4 and SKI4 interactions,respectively.In Fig. 2 the neutron drip line of the correspond-ing hypernuclei with SkI4 are 139Λ Zr and1422Λ Zr, re-spectively. There are two additional neutrons inthe last bound hypernucleus, 1422Λ Zr, compared to thecorresponding ordinary nuclei, because of the addi-tional attractive force provided by the Λ-N interac-tion. The two-neutron separation energies for the118 Chinese Physics C (HEP & NP) Vol. 33last bound nucleus with SkI4 and SLy4 interactions,are 0.2295 MeV for 138Zr and 4.4229 MeV for 122Zr,respectively. It decreased sharply when the massnumber beyond 122, while 3.5913 MeV for 122Zr,0.9414 MeV for 124Zr, 2.2443 MeV for 126Zr with SkI4interaction. Because the small two-neutron separa-tion energies in the continuum nuclei could be a sig-nature for the existence of neutron halo, the formerforce predicts a neutron halo with mass number be-yond 122. We get the same results as vividly shownin Fig. 3.Fig. 3. Neutron radii for Zr isotopes with SLy4,SkI4 interactions and corresponding hypernu-clei as a function of neutron numbers.Fig. 4. The hyperon separation energy as afunction of hyperon numbers. The solid cir-cle indicate multi-Λ hypernuclei with a 80Zrcore and solid square with a 138Zr core, re-spectively.The neutron radii with SLy4 and SkI4 interac-tions are almost identical when A<122. They in-crease rapidly when A beyond 122 with SkI4 inter-action. Then we studied the neutron halos proper-ties in hypernuclei with single-Λ and double-Λ in Zrisotopes using SkI4 interaction. The radii becomesmall as the Λ numbers increase. The shrink effectin medium and heavy hypernuclei is not obvious aslight ones. The neutron radii of single-Λ and double-Λ hypernuclei are almost identical with the even-N Zrisotopes. A neutron halo is predicted when neutronnumbers is beyond 82. Because the last bound hy-pernucleus with double-Λ, 1422Λ Zr, has two additionalneutrons than 138Zr, that will be helpful to form thegiant neutron halo. Finally, we studied the Λ separa-tion energies in our model with SkI4 interaction. Wedid not take the interaction between hyperons intoconsideration. The results are shown in Fig.4. Formulti-Λ hypernuclei with 80Zr and 138Zr cores, thehyperon drip line hypernuclei are 10020ΛZr and18850ΛZr,respectively.4 ConclusionWe studied the properties of even-N Zr isotopesand their corresponding single-Λ and double-Λ hyper-nuclei are studied in the deformed Skyrme-Hartree-Fock approach. The prediction of neutron drip linedepends on the Skyrme parameter used. The neutrondrip line nucleus is 122Zr with SLy4 interaction, andit will become 138Zr if the interaction is replaced bySkI4. We compared the binding energy given by thetwo sets of parameters with the experimental datain Fig. 1. It is not possible to tell which Skyrmeparameter is better for even-N Zr isotopes from theexperimental data until now. The neutron halo struc-ture is model dependent. The neutron drip line nucleipredicted with SkI4 and SLy4 interactions are 122Zrand 138Zr, and only the former force predict a neu-tron halo structure. The predicted hyperon drip linehypernuclei with the 80Zr and 138Zr cores are 10020ΛZrand 18850ΛZr, respectively.References1 Tanihata I, Hamagaki H, Hashimoto O et al. Phys. Rev.Lett., 1985, 55: 2676—26792 Riisager K. Rev. Mod. Phys., 1994, 66: 1105—11163 MENG J, RING P. Phys. Rev. Lett., 1996, 77: 3963—39664 MENG J, RING P. Phys. Rev. Lett., 1998, 80: 460—4635 Grasso M, Yoshida S, Sndulescu N et al. Phys. Rev. C,2006, 74: 064317(7)6 Vautherin D. Phys. Rev. C, 1973, 7: 296—3167 Blum V, Lauritsch G, Maruhn J A et al. J. Comput. Phys.,1992, 100: 364—3768 ZHOU X R, Schulze H J, Sagawa H et al. Phys. Rev. C,2007, 76: 034312(9)9 Bender M, Rutz K, Reinhard P G et al. Phys. Rev. C, 1999,60: 03430410 Tajima N, Bonche P, Flocard H et al. Nucl. Phys. A, 1993,551: 434—450

Properties of Zr hypernuclei in deformed Skyrme Hartree-Fock approach

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Properties of Zr hypernuclei in deformed Skyrme Hartree-Fock approach

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