aa r X i v : . [ nu c l - t h ] J un ,, Two neutron correlations in exotic nuclei
H. Sagawa and K. Hagino Center for Mathematics and Physics, University of Aizu, Aizu-Wakamatsu, 965-8580 Fukushima, Japan Department of Physics, Tohoku University, Sendai, 980-8578, Japan (Dated:)We study the correlations between two neutrons in borromian nuclei Li and He by using athree-body model with a density-dependent contact two-body interaction. It is shown that thetwo neutrons show a compact bound feature at the nuclear surface due to the mixing of singleparticle states of different parity. We study the Coulomb breakup cross sections of Li and Heusing the same three-body model. We show that the concentration of the B(E1) strength near thethreshold can be well reproduced with this model as a typical nature of the halo nuclei. The energydistributions of two emitted neutrons from dipole excitations are also studied using the correlatedwave functions of dipole excitations.
It has been well recognized by now that the borromian nuclei such as Li and He show a strong di-neutroncorrelations in the ground states and also in the excited states. Recently, Nakamura et al. have remeasured thelow-lying dipole excitations in Li nucleus and have confirmed the strong concentration of the dipole strength nearthe threshold in the 2-neutron (2n) halo nucleus [1]. The low-lying dipole strength for another 2n halo nucleus, He,has also been measured by Aumann et al. [2]. The two neutron correlations are further measured very recently inthe Coulomb breakup process of dipole excitaitons in Li [3].The aim of this paper is to study the correlations between di-neutrons and also neutron-core correlations inborromian nuclei He and Li by using a three-body model [4, 5, 6]. In Ref. [7], the behavior of the two valenceneutrons in Li is studied at various positions from the center to the surface of the nucleus. It was found that thetwo-neutron wave function oscillates near the center whereas it becomes similar to that for a compact bound statearound the nuclear surface, and the mean distance between the valence neutrons has a well pronounced minimumaround the nuclear surface. We have pointed out that these are qualitatively the same behaviors as found in neutronmatter [8]. To elucidate these points, we show in Fig. 1 the mean distance of the valence neutrons in Li as afunction of the nuclear radius R (the distance between the core and the center of two neutrons) obtained with andwithout the neutron-neutron ( nn ) interaction. For the uncorrelated calculations, we consider both the [(1p / ) ]and [(2s / ) ] configurations. One can see that, in the non-interacting case, the neutron pair almost monotonouslyexpands, as it gets further away from the center of the nucleus. On the other hand, in the interacting case it firstbecomes smaller going from inside to the surface before expanding again into the free space configuration. Theseresults confirm the strong and predominant influence of the pairing force in the nuclear surface of Li. We alsoshow the local neutron-neutron correlation energy as a function of the radius R in the lower panel of Fig. 1. It isclearly seen that the energy gain is the maximum at the surface where the correlaiton length is the minimum. Thetwo panels in Fig. 1 confirm that the kink of the correlation length is induced by the strong pairing correlationsat the surface.Two particle densities of the correlated pair and the uncorrelated [(1p / ) ] configuration are shown in Fig. 2.The reference particle is located at ( z, x ) = (3 . , r r m s (f m ) nn correlationuncorr. [(1p ) ]uncorr. [(2s ) ]0 5 10 15R (fm)-15-10-50 E p a i r ( R ) ( M e V ) FIG. 1: (Upper panel)The root mean square distance r rms between the valence neutrons in Li as a function of thedistance R between the core and the center of two neutrons. The solid line is obtained by taking into account the neutron-neutron correlations, while the dashed and the dotted lines are obtained by switching off the neutron-neutron interaction andassuming the [(1p / ) ] and [(2s / ) ] configurations, respectively. (Lower panel) The neutron-neutron correlation energy asa function of the distance R . x ( f m ) x ( f m ) FIG. 2: (Color online) Two dimensitional (2D) plots for the two particle density of the correlated pair (left panel) anduncorrelated [(1p / ) ] configuration (right panel) in Li. It represents the probability distributions for the spin-up neutronplacing the spin-down neutron at ( z, x )=(3.4,0)fm. The core nucleus is located at the origin ( z, x )=(0,0)fm. d σ / d E ( m b / M e V ) full recoilno recoil (i)no recoil (ii) He + Pb0 1 2 3 4E rel (MeV)0123 d σ / d E ( b / M e V ) full recoilno recoil (i)no recoil (ii) Li + Pb
FIG. 3: Coulomb breakup cross sections for He+Pb at 240 MeV /nucleon and for the Li +Pb at 70 MeV/nucleon. Thesolid line is the result of the full calculations, while the dashed line is obtained by neglecting the off-diagonal component ofthe recoil kinetic energy in the excited states. The dotted line is obtained by neglecting the off-diagonal recoil term both inthe ground and the excited states. These results are smeared with an energy dependent width of Γ = 0 . · √ E rel MeV for He and Γ = 0 . ·√ E rel MeV for Li. The experimental data are taken from Refs. [2] and [1] for He and Li, respectively. symmetric two peaks in ( z, x ) plane with respect to the center of the core nucleus at ( z, x ) = (0 , / ) ] configuration. On the contrary, thepeak appears only around the position of the reference particle when the two neutron correlations are taken intoaccount in the wave functions. To compare two panels in Fig. 2, we can see a clear manifestation of the strong twoneutron correlations in the wave function of the borromian nucleus Li.Figs. 3 compare the Coulomb breakup cross sections calculated by taking into account the recoil term exactly(the solid curves) with those calculated approximately (the dashed and dotted curves). For the dashed curves,the off-diagonal component of the recoil kinetic energy is neglected in the excited J π = 1 − states, while it is fullytaken into account in the ground state. It is interesting to notice that these calculations lead to similar results tothe one in which the recoil term is treated exactly (the solid curves). The dotted curves, on the other hand, areobtained by neglecting the off-diagonal part of the recoil term both for the ground and the J π = 1 − states. Byneglecting the recoil term in the ground state, the value for h r c − n i decreases, from 13.2 fm to 9.46 fm for Heand from 26.3 fm to 20.58 fm for Li. Consequently, the B(E1) distribution as well as the breakup cross sectionsare largely underestimated. These results clearly indicate that the recoil term is important for the ground state,while it has a rather small effect on the excited states. (MeV) 0 0.2 0.4 0.6 0.8 1 e ( M e V ) (MeV) 0 0.5 1 1.5 2 2.5 e ( M e V ) FIG. 4: (Color online) The dipole strength distributions, d B ( E /de de , of Li (left panel) and He (right panel) as afunction of the energies of the two emitted neutrons relative to the core nucleus. They are plotted in units of e fm /MeV .Figures show the correlated response, which fully takes into account the ground state and final state interactions betweenthe two neutrons. Figures 4 show the dipole strength distribution, d B ( E /de de , as a function of the energies of the two emittedneutrons for the Li and He nuclei, respectively [9]. One immediately notices that the strength distribution isconsiderably different between Li and He. For Li, a large concentration of the strength appears at about e =0.375 MeV and e =0.075 MeV (and at e =0.075 MeV and e =0.375 MeV), with a small ridge at an energy ofabout 0.5 MeV. On the other hand, for He, the strength is largely concentrated at one peak around e = e = 0 . Li and He is due to theexistence of virtual s-state in the residual Li, but not in He. Thus, if the interaction between the neutron andthe core nucleus is switched off, the distribution become similar between the two nuclei because the virtual s-stateis diappeared in Li.In summary, we have studied the di-neutron correlations and the neutron-core correlations in the borromiannuclei He and Li by using the three-body model with a density dependent contact interaction. It is shown thatthe two neutron wave functions show a strong di-nuetron correlation at the nuclear surface due to the mixing ofdifferent parity single particle states. The same model is used to analyze the dipole strength distributions as wellas the Coulomb breakup cross sections of the He and Li nuclei. We have shown that the strong concentrationof the B(E1) strength near the continuum threshold can be well reproduced as a nature of the halo nuclei withthe present model. It is shown that the recoil effect plays an important role in the ground state while it may beneglected in the excited states. We have carried out the calculations of the energy and angular distributions ofthe two emitted neutron from E1 excitations in Li and He nuclei. We have shown that these distributions arestrongly affected by the existence of the virtual s-state in the residual Li nucleus. Thus, the properties of theneutron-core potential is crucial to describe the energy distributions of two emitted neutrons, rather than the twoneutron correlaions in the excited states.This work was supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology byGrant-in-Aid for Scientific Research under the program numbers (C) 19740115 and 20540277. [1] T. Nakamura et al. , Phys. Rev. Lett. , 252502 (2006).[2] T. Aumann et al. , Phys. Rev. C , 1252 (1999).[3] T. Nakamura et al. , to be published.[4] G.F. Bertsch and H. Esbensen, Ann. Phys. (N.Y.) , 327 (1991).[5] H. Esbensen and G.F. Bertsch, Nucl. Phys. A542 , 310 (1992).[6] K. Hagino and H. Sagawa, Phys. Rev. C , 044321 (2005).[7] K. Hagino, H. Sagawa, J. Carbonell, and P. Schuck, Phys. Rev. Lett. , 022506 (2007).[8] M. Matsuo, Phys. Rev. C73