Few-Body Models of Light Nuclei The 8th APCTP-BLTP JINR Joint Workshop June 29 – July 4, 2014, Jeju, Korea S. N. Ershov

22 N 23 N 18 O 19 O 20 O 21 O 22 O 23 O 24 O 18 F 19 F 20 F 21 F 22 F 23 F 24 F 25 F 26 F 27 F 31 F 29 F 19 F proton-neutron arrangement clustering of nucleons in nuclei Nucleus low-energy domain Few-Body models

23 N : 24 O : two-neutron halo nuclei K. Tanaka et al. “Observation of a Large Reaction Cross Section in the Drip-Line Nucleus 22 C “, Phys. Rev. Lett., 104, (2010)

N Charge radii

Ab initio models Interactions: NN + NNN + … Many-Body Methods: Green’s Function Monte Carlo No-Core Shell Model Coupled Cluster …. the shell models effective interactions fitted to experimental data Many-Body Methods: mean-fields (self-consistent) + residual interactions shell structure evolution when nuclei are approaching driplines Large-scale shell model calculations for medium and heavy nuclei very complicated wave functions How do they consistent with independent nucleon motion ? ( essence of the shell model ) Understand the nuclear structure with realistic nuclear forces microscopic mean-field models and beyond continuum ( reactions ) ? nuclear density functional theory quasiparticle random-phase approximation quasiparticle phonon model ….. theoretical calculations of nuclear properties throughout the whole nuclear chart

exact MICROSCOPIC cluster models: exact treatment of the antisymmetrization is the antisymmetrization operator Nucleons are separated into clusters and exchanged between them as if nucleons resonate between each group Resonating Group Method (J.A. Wheeler, 1937) Generator Coordinate Method (D.L. Hill and J.A. Wheeler, 1953) Auxiliary parameters (generator coordinates ) are introduced. Slater determinant many-body wave functions are generated for parameter set

projected out: 3-body approximate SEMI-MICROSCOPIC cluster models: approximate treatment of the antisymmetrization Orthogonality condition model (S. Saito, 1968) The inter-cluster wave function is ORTHOGONAL to the Pauli forbidden states the deep core-neutron potential (popular approach) permits forbidden orbits are less universal the shallow core-neutron potential does not have forbidden orbits disadvantages Application in three-body models does not lead to additional difficulties attractive features: potentials are more universal for treating adjacent systems, similar to the ones used for calculations in usual shell models attractive features: disadvantages additional efforts have to be applied to exclude forbidden states from dynamics in three-body models neglected: 2-body the deeply bound states are obtained as the lowest levels (forbidden orbits dominate in the structure of these states)

Few-Body cluster models, general formulation Three-body ( ) : Two-Body ( ) : wave function is factorized into a sum of products from two parts (no -operator) The sum includes core excitations : The Schrodinger equation C r1r1r1r1 r2r2r2r2 x y C n A coupling with excited core states involves additional partial waves.

Total hamiltonian (two and three-body cluster models) The wave function is a solution of the coupled Schrodinger equations matrix elements describe the core-nucleon binary interactions These matrix elements define the dynamics of the system Calculations of the bound states and continuum wave functions

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

L. Grigorenko et al., Phys. Lett. B 677(2009) 30 A.S. Fomichev et al., Phys. Lett. B708 (2012) 6 Isobaric partners: J  =0 +, T = 1 (the same nuclear structure within isobaric symmetry) 6 He(0 +,g.s.) 6 Li(0 +,3.56 MeV) 6 Be(0 +,g.s.)

Two-neutron halo nucleus 22 C S.N. Ershov et al., Phys. Rev. C 86, (2012)

Soft dipole mode Electromagnetic dissociation cross sections 10 keV 50 keV S 2n 400 keV 200 keV 100 keV 50 keV S 2n C r1r1r1r1 r2r2r2r2 x y RCRC {

simple shell – model description

exp. data : R. Palit et al., Phys. Rev. C68 (2003) the 3/2 - resonance at 2.9 MeV the 3/2 - resonance at 2.1 MeV Dipole excitations from ground to continuum states X C n

Conclusion Clustering is a very widespread phenomenon in light nuclei. The few-body cluster models present a natural and most transparent way to describe specific features of nuclear structure specified by the cluster degrees of freedom. In many nuclei the cluster degrees of freedom are most important and define the basic properties of the nuclear systems. Still in many circumstances the core can not be considered as inert system and additional degrees of freedom connected to excited core states have to be taken into account. This leads to extension of few-body cluster models and increases the applicability of a cluster approach. Investigations of the nuclear structure toward to the limits of existence of bound nuclei reveal variety of new phenomena. The challenge to the microscopic theory is to understand and reliably calculate the properties of radioactive nuclei.

W. Horiuchi, Y. Suzuki Phys. Rev. C 74, (2006) the stochastic variational method forbidden states were projected out different n - 20 C deep interactions Important findings: non-central forces give negligible contributions the (s 1/2 ) 2 configuration is predominant other components add only small admixtures total spin of halo neutrons S = 0 (weight more then 98%) S 2n = MeV 22 C : r m = fm 20 C : r m = 2.98 fm

The bound state wave function ( ) The continuum wave function at the positive energy C r1r1r1r1 r2r2r2r2 x y Set of coupled Schrodinger equations for radial wave functions