Helmholtz International Summer School "Nuclear Theory and Astrophysical Application" Halo Nuclei S. N. Ershov JointInstitute for Nuclear Research new structuraldripline phenomenon with clusterization into an ordinary core nucleus and a veil of halo nucleons – forming very dilute neutron matter HALO:
Chains of the lightest isotopes (He, Li, Be, B,…) end up with two neutron halo nuclei Two neutron halo nuclei (6He , 11Li , 14Be ,… ) break into three fragments and are all Borromean nuclei One neutron halo nuclei (11Be , 19C , …) break into two fragments Nuclear scale ~10-22 sec Width 1 ev ~ 10-16sec Life time for radioactive nuclei > 10-12 sec bdecays define the life time of the most radioactive nuclei
Separation energies of last neutron (s) : 11Li n n ~ 7 fm “Residence in forbidden regions” halostable < 1 6 - 8 MeV ~ 5 fm e (11Li)= 0. 3 MeV 2.3 fm Appreciable probability for dilute nuclear matter extending far out into classically forbidden region e (11Be)= 0. 5 MeV 9Li e ( 6He)= 0. 97 MeV < r2(11Li)>1/2~ 3.5 fm ( r.m.s. for A ~ 48 ) Large sizeof halo nuclei Two-neutronhalo nuclei ( 11Li, 6He, 14Be,17B, . . .) Borromean systems Neutron halo nuclei weakly bound systems with large extension and space granularity Halo ( 6He, 11Li, 11Be, 14Be, 17B, … ) none of the constituent two-body subsystemsare bound Borromeansystem is bound
( naive picture ) narrow momentum distributions large spatial extensions Peculiarities of halo nuclei: the example of 11Li (i) weakly bound: the two-neutron separation energy (~300 KeV) is about 10 times less than the energy of the first excited state in 9Li . (ii) large size: interaction cross section of 11Li is about 30%larger than for 9Li This is very unusual for strongly interacting systems held together by short-range interactions Interaction radii : E / A = 790 MeV, light targets I. Tanihata et al., Phys. Rev. Lett., 55 (1985) 2676 (iii) very narrow momentum distributions, compared to stable nuclei, of both neutrons and9Li measured in high energy fragmentation reactions of 11Li . Nonarrow fragment distributions in breakup on other fragments, say 8Li or8He
A + 12C sIs-2ns-4n 9Li 796 m 6 11Li 1060m 10220m 40 4He503m 5 6He 722m 5 189m14 8He 817m 6 202m1795m 5 Evidence for a rather large difference between charge and masscenters in a body fixed frame concentration of the dipole strength at low excitation energies (iv) Relations between interaction and neutron removal cross sections ( mb ) at 790 MeV/A sI(A=C+xn) = sI (C) + s-xn Strong evidence for the well defined clusterization into the core and two neutrons Tanihata I. et al. PRL, 55 (1987) 2670; PL, B289 (1992) 263 (v)Electromagnetic dissociation cross sections per unit chargeare orders of magnitude larger than for stable nuclei halo stable T. Kobayashi, Proc. 1st Int. Conf. On Radiactive Nuclear Beams, 1990.
soft DR normal GDR Ex ~ 1MeV ~ 20 MeV 11Li 0.6 dB(E1) / dEx ( e 2 fm 2 / MeV ) 0.2 0 1 2 3 4 Ex ( MeV ) Soft Excitation Modes (peculiarities of low energy halo continuum) specific nuclear property of extremely neutron-rich nuclei Large EMD cross sections M. Zinser et al., Nucl. Phys. A619 (1997) 151 excitations of soft modes with different multipolarity collectiveexcitations versus direct transition from weakly bound to continuum states
9Li11Li Spin Jp:3/2- 3/2- quadrupole moments :-27.4 m 1.0 mb -31.2 m 4.5 m magnetic moments : 3.4391 m 0.0006 n.m. 3.6678 m 0.0025 n.m. 1 2 C (vi)Ground stateproperties of11Li and9Li : Schmidt limit : 3.71 n.m. Previous peculiarities cannotarise from large deformations core is not significantly perturbed by the two valence neutrons Nuclear charge radii by laser spectroccopy R. Sanches et al., PRL 96 (2006) 033002 L.B. Wang et al., PRL 93 (2004) 142501
(vii) The three-body system 11Li(9Li + n + n) is Borromean : neither the two neutron nor the core-neutron subsytems are bound Three-body correlations are the most important: due to them the system becomes bound. The Borromean rings, the heraldic symbol of the Princes of Borromeo, are carved in the stone of their castle in Lake Maggiore in northern Italy.
p n p n 1s - intruder level parity inversion of g.s. g.s. : Stable nuclei Unstable nuclei N / Z ~ 0.6 - 4 N / Z ~ 1 - 1.5 eS~ 0 - 40 MeV eS~ 6 - 8 MeV decoupling of proton and neutron distributions r0~ 0.16 fm -3 proton and neutrons homogeneously mixed, no decoupling of proton and neutron distributions neutron halos and neutron skins Prerequisite of the halo formation : lowangular momentum motion for halo particles and few-body dynamics
in groundstate in low-energy continuum weakly bound, with large extension and space granularity concentration of the transition strength near break up threshold - soft modes Decoupling of halo and nuclear core degrees of freedom BASICdynamics of halo nuclei Peculiarities of halo elastic scattering some inclusive observables (reaction cross sections, … ) nuclear reactions (transition properties)
1 2 r1C r2C C 1 2 1 2 V - basis C C The T-set of Jacobin coordinates ( ) The hyperspherical coordnates :r, T - basis ris therotation, translation and permutationinvariant variable Y - basis Volume element in the 6-dimensional space
are the spherical harmonics The kinetic energy operatorT has the separable form is a square of the 6-dimensional hyperorbital momentum Eigenfunctions of are the homogeneous harmonic polynomials are hyperspherical harmonics or K-harmonics. They give a completeset of orthogonal functions in the 6-dimensional space on unit hypersphere are the Jacobi polynomials,
The functions with fixed total orbital moment a normalizing coefficient is defined by the relation + (positive), ifK – even - (negative), ifK - odd The parity of HH depends only on The three equivalent sets of Jacobi coordinates are connected by transformation (kinematic rotation) Quantum numbers K, L, M don’t change under a kinematic rotation. HH are transformed in a simple way and the parity is also conserved. Reynal-Revai coefficients
The three-body bound-state and continuum wave functions (within cluster representation) The Schrodinger 3-body equation : where the kinetic energy operator : and the interaction : The bound state wave function (E < 0 ) - spin function of two nucleons The continuum wave function (E > 0 ) is the hypermomentum conjugated tor
the boundary conditions: where and partial-wave coupling interactions The HH expansion of the 6-dimensional plane wave Normalization condition for bound state wave function Normalization condition for continuum wave function After projecting onto the hyperangular part of the wave function the Schrodinger equation is reduced to a set of coupled equations
The asymptotic hyperradial behaviour of The simplest case : two-body potentials : a square well, radius R Atrthe system of differential equations is decoupled since effective potentials can be neglected a general behaviour of three-body effective potential if the two-body potentials are short-range potentials if E < 0 if E > 0
rnn n n R(nn)-C C Correlation density for the ground state of 6He cigar-like configurationdineutron configuration
1 3 2 a A A* Study ofhalo structure 1 2 3 events withundestroyedcore peripheralreactions complexconstituents 1 + 2 + 3 + Agr , elastic ( 4-body ) 1 + 2 + 3 + A* , inelastic ( ³ 4-bodies) a + A Three-body halo fragmentation reactions Cross section
Reaction amplitude Tfi(prior representation) a 1 halo ground state wave function target ground state wave function distorted wave for relative projectile-target motion exact scattering wave function NN - interaction between projectileand target nucleons optical potential in initial channel ky kx 2 C kf A
DW:low-energyhaloexcitationsÞsmallkx & ky ; large ki & kf (no spectators, three-body continuum, full scaleFSI ) 1 a FSI 2 3 A A* Reaction amplitude Tfi(prior representation) • Kinematically complete experiments • sensitivity to 3-body correlations (halo) • selection of halo excitation energy • variety of observables • elastic & inelastic breakup
Nuclear structure Three-body models Model assumptions Transition densities effective interactions ( NN&N-core ) Method ofhyperspherical harmonics: 3-bodyboundandcontinuumstates binding energy electromagnetic moments electromagnetic formfactors geometrical properties density distributions . . . . . . .
Reaction mechanism Distorted wave approach effective NN interactionsVpt distorted waves c (k) self-consistency complex, energy, density dependent optical potential nucleus-nucleus elastic scattering and reaction cross sections free NN scattering Model assumptions One-step process noconsistency with nuclear structure interactions
n n n n x Two-body breakup Three-body breakup y y x C n x C θx C Continuum Spectroscopy Y-system T-system Excitation Energy Orbital angular momenta Spin of the fragments Hypermoment
n n n n x y y T-system Y-system V-system x C where C no recoil: n n y x C 1. Angular correlation →no effect 2. Energy correlation is approximately symmetric relativeε= 1/2 ABC of the three-body correlations Permutation of identical particles must not producean effect on observables. Manifestation depends on the coordinate system where correlations are defined. Neutron permutation changes 1. Angular correlation is symmetric relative 2. Energy correlation →no effect
n n n n x y y x C C ABC of the three-body correlations Energy correlations of elementary modes : 2+ excitations with thehypermoment K = 2
y C n n y x C Realistic calculations: mixing of configurations T system n n x For low-lying states : only a few elementary modes Y system Energy correlations for 2+ resonance in 6He
n n n n x y y x C C Energy and angular fragment correlations
n n n n x y y x C C ■ L.V. Chulkov et al., Nucl. Phys. A759, 23 (2005)
n n n n n n n n x x y y y y x x C C C C
? Theoretical calculationsExperimental data detector V1 V2 VCM VC Inverse kinematics:forwardfocusing assume 4 -measurements of fragments T. Aumann, Eur. Phys. J. A 26 (2005) 441 "The angular range for fragments and neutrons covered by the detectors corresponds to a 4measurement of the breakup in the rest frame of the projectile for fragment neutron relative energies up to 5.5 MeV (at 500 MeV/nucleon beam energy)".
CONCLUSIONS • The remarkable discovery of new type of nuclear structure at driplines, HALO, have been made with radioactive nuclear beams. • The theoretical description of dripline nuclei is an exciting challenge. The coupling between bound states and the continuum asks for a strong interplay between various aspects of nuclear structure and reaction theory. • Development of new experimental techniques for production and /or detection of radioactive beams is the way to unexplored “ TERRA INCOGNITA “