1. Halo Nuclei S. N. Ershov Joint Institute for Nuclear Research new structural dripline phenomenon with clusterization into an ordinary core nucleus and a veil of halo nucleons into an ordinary core nucleus and a veil of halo nucleons – forming very dilute neutron matter – forming very dilute neutron matter HALO:

2. Life time for radioactive nuclei > 10 -12 sec Chains of the lightest isotopes ( He, Li, Be, B, …) end up with two neutron halo nuclei Two neutron halo nuclei ( 6 He, 11 Li, 14 Be, … ) break into three fragments and are all Borromean nuclei One neutron halo nuclei ( 11 Be, 19 C, … ) break into two fragments  decays define the life time of the most radioactive nuclei Nuclear scale ~ 10 -22 sec Width 1 ev ~ 10 -16 sec

3. weakly bound systems with large extension and space granularity “Residence in forbidden regions” Appreciable probability for dilute nuclear matter extending far out into classically forbidden region ( 6 He, 11 Li, 11 Be, 14 Be, 17 B, … )  11 Li  = 0. 3 MeV  11 Be  = 0. 5 MeV  6 He  = 0. 97 MeV none of the constituent two-body subsystems are bound Borromean systems Two-neutron halo nuclei ( 11 Li, 6 He, 14 Be, 17 B,... ) Borromean system is bound 2.3 fm ~ 5 fm ~ 7 fm nn 11 Li 9 Li Separation energies of last neutron (s) : halo stable < 1 6 - 8 MeV Large size of halo nuclei 1/2 ~ 3.5 fm ( r.m.s. for A ~ 48 )

4. Peculiarities of halo nuclei: the example of 11 Li (i) weakly bound: the two-neutron separation energy (~300 KeV) is about 10 times less than the energy of the first excited state in 9 Li. (ii) large size: interaction cross section of 11 Li is about 30 % larger than for 9 Li E / A = 790 MeV, light targets Interaction radii : I. Tanihata et al., Phys. Rev. Lett., 55 (1985) 2676 This is very unusual for strongly interacting systems held together by short-range interactions ( iii) very narrow momentum distributions, compared to stable nuclei, of both neutrons and 9 Li measured in high energy fragmentation reactions of 11 Li. ( naive picture ) narrow momentum distributions large spatial extensions No narrow fragment distributions in breakup on other fragments, say 8 Li or 8 He

5. A + 12 C  I  -2n  -4n 9 Li 796  6 11 Li 1060  10 220  40 4 He 503  5 6 He 722  5 189  14 8 He 817  6 202  17  95  5 (iv) Relations between interaction and neutron removal cross sections ( mb ) at 790 MeV/A  I (A=C+xn) =  I (C) +  -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 charge are orders of magnitude larger than for stable nuclei T. Kobayashi, Proc. 1st Int. Conf. On Radiactive Nuclear Beams, 1990. stable halo Evidence for a rather large difference between charge and mass centers in a body fixed frame concentration of the dipole strength at low excitation energies

6. (peculiarities of low energy halo continuum) soft DR normal GDR E x ~ 1 MeV ~ 20 MeV 0 1 2 3 4 0.6 0.2 dB(E1) / dE x ( e 2 fm 2 / MeV ) E x ( MeV ) 11 Li M. Zinser et al., Nucl. Phys. A619 (1997) 151 excitations of soft modes with different multipolarity collective excitations versus direct transition from weakly bound to continuum states Large EMD cross sections specific nuclear property of extremely neutron-rich nuclei

7. (vi) Ground state properties of 11 Li and 9 Li : Spin J     3/2   3/2  quadrupole moments :  27.4  1.0 mb  31.2  4.5 m magnetic moments : 3.4391  0.0006 n.m. 3.6678  0.0025 n.m. 9 Li 11 Li Schmidt limit : 3.71 n.m. Previous peculiarities cannot arise from large deformations core is not significantly perturbed by the two valence neutrons R. Sanches et al., PRL 96 (2006) 033002 L.B. Wang et al., PRL 93 (2004) 142501 Nuclear charge radii by laser spectroccopy 12 C 

8. ( vii) The three-body system 11 Li ( 9 Li + 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.

9. pnpn low angular momentum motion for halo particles and few-body dynamics Prerequisite of the halo formation : 1s - intruder level parity inversion of g.s. g.s. :  0  ~ 0.16 fm -3 proton and neutrons homogeneously mixed, no decoupling of proton and neutron distributions N / Z ~ 1 - 1.5  S ~ 6 - 8 MeV decoupling of proton and neutron distributions neutron halos and neutron skins N / Z ~ 0.6 - 4  S ~ 0 - 40 MeV

10. Peculiarities of halo in low-energy continuumin ground state weakly bound, with large extension and space granularity concentration of the transition strength near break up threshold - soft modes BASIC dynamics of halo nuclei of halo nuclei Decoupling of halo and nuclear core degrees of freedom elastic scattering some inclusive observables (reaction cross sections, … ) nuclear reactions (transition properties)

11. The hyperspherical coordnates :  Volume element in the 6-dimensional space 12 C  12 C  Y - basis T - basis 12 C  V - basis r 1C r 2C The T-set of Jacobin coordinates ( )   is the rotation, translation and permutation invariant variable

12. The kinetic energy operator T has the separable form is a square of the 6-dimensional hyperorbital momentum are hyperspherical harmonics or K-harmonics. are the Jacobi polynomials, are the spherical harmonics They give a complete set of orthogonal functions in the 6-dimensional space on unit hypersphere Eigenfunctions of are the homogeneous harmonic polynomials

13. The functions with fixed total orbital moment a normalizing coefficient is defined by the relation 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 parity of HH depends only on + (positive), if K – even - (negative), if K - odd

14. 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 : is the hypermomentum conjugated to  The bound state wave function (E < 0 ) - spin function of two nucleons The continuum wave function (E > 0 )

15. Normalization condition for bound state wave function The HH expansion of the 6-dimensional plane wave After projecting onto the hyperangular part of the wave function the Schrodinger equation is reduced to a set of coupled equations Normalization condition for continuum wave function where and partial-wave coupling interactions the boundary conditions:

16. At  the system of differential equations is decoupled since effective potentials can be neglected The asymptotic hyperradial behaviour of The simplest case : two-body potentials : a square well, radius R a general behaviour of three-body effective potential if the two-body potentials are short-range potentials if E < 0 if E > 0

17. Correlation density for the ground state of 6 He cigar-like configuration dineutron configuration nn C r nn R (nn)-C

18. 3 1 2 A* 3 12 A a 1 + 2 + 3 + A gr, elastic ( 4-body ) 1 + 2 + 3 + A*, inelastic (  4-bodies) a + A complex constituents Cross section events with undestroyed core peripheral reactions Study of halo structure

19. Reaction amplitude T fi (prior representation) 1 2 C A kxkx kyky kfkf a halo ground state wave function target ground state wave function distorted wave for relative projectile-target motion exact scattering wave function NN - interaction between projectile and target nucleons optical potential in initial channel

20. DW: low-energy halo excitations   small k x & k y ; large k i & k f (no spectators, three-body continuum, full scale FSI ) Kinematically complete experiments sensitivity to 3-body correlations (halo) selection of halo excitation energy variety of observables elastic & inelastic breakup a FSI A 1 2 3 A* Reaction amplitude T fi (prior representation)

21. Method of hyperspherical harmonics: Method of hyperspherical harmonics: 3-body bound and continuum states 3-body bound and continuum states effective interactions ( NN & N-core ) ( NN & N-core ) binding energy electromagnetic moments electromagnetic formfactors geometrical properties density distributions.............. Transition densities Nuclear structure Three-body models

22. Reaction mechanism no consistency with nuclear structure interactions free NN scattering complex, energy, density dependent effective NN interactions V pt Distorted wave approach nucleus-nucleus elastic scattering and reaction cross sections optical potential distorted waves  (k) self-consistency One-step process

23. Halo scattering on nuclei

24. Continuum Spectroscopy Two-body breakupThree-body breakup n C x θxθx nn x C y nn C x y T-system Y-system Excitation Energy Orbital angular momenta Spin of the fragments Hypermoment

25. ABC of the three-body correlations Permutation of identical particles must not produce an effect on observables. Manifestation depends on the coordinate system where correlations are defined. nn x C y T-system Neutron permutation changes 1. Angular correlation is symmetric relative 2. Energy correlation → no effect nn C x y Y-system nn C x y V-system no recoil: 1. Angular correlation → no effect 2. Energy correlation is approximately symmetric relative ε = 1/2 where

26. Energy correlations of elementary modes : 2+ excitations with the hypermoment K = 2 nn C x y nn x C y ABC of the three-body correlations

27. Energy correlations for 2+ resonance in 6 He nn x C y nn C x y T system Y system Realistic calculations: mixing of configurations For low-lying states : only a few elementary modes

28. nn x C y nn C x y Energy and angular fragment correlations

29. L.V. Chulkov et al., Nucl. Phys. A759, 23 (2005) ■ nn C x y nn x C y

30. nn C x y nn x C y nn x C y nn C x y

31. assume 4  -measurements of fragments V CM V1V1 V2V2 VCVC detector Inverse kinematics: forward focusing "The angular range for fragments and neutrons covered by the detectors corresponds to a 4  measurement 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)". T. Aumann, Eur. Phys. J. A 26 (2005) 441 Theoretical calculations Experimental data ?

32.  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 “ CONCLUSIONS

33. Special thanks to B.V. Danilin & J.S. Vaagen