Shell Evolution and Structure of Two-Neutron Halo in Exotic Nuclei

Shell evolution in neutron-rich carbon isotopes is studied with repulsive contributions from three-nucleon forces among valence neutrons. Three-nucleon forces are shown to determine the dripline and improve the energy levels of the isotopes. A three-body model with low-energy neutron-neutron interaction is used to study structure of two-neutron halos of exotic nuclei. Observed halo radii in 22C and 29F are found to be well explained by the three-body model. A possible existence of an excited 2+ state in the halo of 29F and its consequences are discussed. The halo in 17B is found to be d-wave dominant, while the halo in 19B has s-wave component with about 40% probability.


Introduction
We first discuss shell evolution in carbon isotopes in Sect.2, where important roles of three-body forces are pointed out.Then, in Sect.3, we study structure of two-neutron halo by a three-body model, where neutron-neutron interaction is shown to play essential roles.A unique relation between the neutron halo radius and the two-neutron separation energy, S 2n , is obtained.The three-body model is applied to 22 C, 29 F, 17 and 19 B, and their halo structures are investigated.

Shell evolution in neutron-rich carbon isotopes
Shell evolution in neutron-rich carbon isotopes is discussed based on monopole terms of neutronneutron interaction.For realistic shell-model Hamiltonians, monopole terms in isospin T =1 channel are more repulsive than those in microscopic G-matrix elements [1].This repulsion can be attributed to three-nucleon (3N) forces Δ-hole excitations give rise to an attraction to single-particle energies, which are blocked when there is another nucleon in the same orbit.This contribution can be described by the exchange term of Fujita-Miyazawa (FM) 3N forces [2], which is definitely positive and induces repulsion among valence neutrons [3].
This repulsion is important to reproduce the ground state energies of carbon isotopes as shown in Fig. 1 (left panel).The experimental values are reproduced well by an interaction with the 3N forces added to the G-matrix as well as by SFO-tls [4] with the inclusion of proper repulsion in T =1 channels. 22C is found to be the dripline, consistent with the observation.The halo in 22 C is naturally s-wave dominant.The repulsion between neutron 1d 5/2 and 2s 1/2 orbits gives rise to a level inversion in neutron effective single-particle energies at around A = 16-17 [5].In 12 C the s-orbit is lower, while the d-orbit becomes lower at higher mass region: A >16.The two orbits become almost degenerate at A =16-17.
The effects of 3N forces on the low-lying levels in the carbon isotopes with A = 16-18 are shown in Fig. 1 (right panel).Energies of first 2 + states in 16 C and 18 C come closer to the experimental values with 3N, and the low-lying levels in 17 C are well reproduced with the 3N.

Structure of two-neutron halo by a three-body model
Structure of two-neutron halo is studied by a three-body model, where for the neutron-neutron (n-n) interaction the one with a Gaussian form which reproduces the scattering length and the effective range is used.For the core-neutron interaction, one-body Woods-Saxon (WS) potential with the standard Bohr-Mottelson parameters is used.The strength and the range of the Gaussian for the n-n interaction and the WS parameters are the only parameters in the present model without any other extra fixing.
We first apply the model to 22 C.The halo in 22 C is s-wave dominant, while the 20 C-core is known to have appreciable s-orbit components as confirmed experimentally [6] as well as by shell-model calculations [7].Here, we adopt the following model [8].The halo is taken to be a pure s-wave, and the orthogonality condition between the halo s-orbit and the core s-orbit is set to satisfy the Pauli principle.The core s-orbit has halo components while the harmonic oscillator 2s 1/2 component is reduced by the blocking effect.Energy of the 20 C-core in 22 C is pushed up, that is, the S 2n is reduced.For such a correlated 20 C-core, the halo radius is obtained to be 6-7 fm almost independent of S 2n .This value is consistent with the experimental one, 6.79+0.70/-0.66fm, derived from the matter radius of 22 C [9].
Next, we discuss two-neutron halo in 29 F. The present three-body model gives the halo radius of 5.80 fm when a pure 2p 2  3/2 configuration is assumed and the experimental value for S 2n = 1.44 MeV is used.This value is consistent with the one, 6.6±0.8 fm derived from matter radii of 29 F and 27 F [10].For a three-body model with three configurations, 2p 2  3/2 , 1f 2 7/2 and 1d 2 3/2 , the dominance of the p-orbit component is shown to be necessary to get a halo radius as large as 5.7 fm [11].There is also a shell-model calculation with an effective interaction EEdf1, obtained from chiral N 3 LO [12] and FM 3N forces by an extended Kuo (EKK) method [13,5].The halo radius is obtained to be 5.85 fm, while an appreciable admixture of 1f 7/2 components is noticed [10].
The existence of an excited p 2 3/2 (2 + ) state is possible for the halo in 29 F for the present three-body model.The excitation energy of the 2 + state is calculated to be 0.41 MeV.It is obtained to be about 1.3 MeV in another three-body model [14].
Possible consequences of the 2 + state is investigated by a weak-coupling model, where 29 F is constructed by coupling 27 F-core and the two-neutron halo with p 2 3/2 .
The 27 F-core has the ground 5/2 + state and the excited 1/2 + state, whose energy is 0.915 MeV from observation [15] and 1.48 MeV from shell-model calculation with the SDPF-MU [16].The halo has 0 + and 2 + states.The 1/2+ and 5/2 + states in 29 F has two and three components, respectively.There appear 3/2 + , 7/2 + and 9/2 + states also by the couplings.Non-diagonal terms are evaluated by using the matrix elements of shell-model interactions, EEdf1 or SDPF-MU.Reduction of matrix elements are taken to account for the neutron halo effects.IOP Publishing doi:10.1088/1742-6596/2586/1/0120284 1.5 for proton and 0.35-0.25 for neutron.For neutron, effects of halo are considered to adopt a small effective charge compared to the standard one.In case with 0 + only for the halo, a large E2 transition occurs for 1/2 + → 5/2 + .With the inclusion of 2 + state, levels are split to more states due to the core-halo coupling and many E2 transitions take place.Here, we show two examples, (a) one with high excitation energies (1.48 MeV for 1/2 + in 27 F and 1.3 MeV for 2 + in the halo) and (b) the other with low excitation energies (0.915 MeV for 1/2 + in 27 F and 0.41 MeV for 2 + in the halo) for both 27 F and the halo.In both cases, there is one dominant E2 transition corresponding to the one in case of 0 + only.Rates for the E2 transitions are determined by the B(E2) and the phase space factors, E 5  fi where E fi is the energy difference between the initial and final states..Because of small phase space factors and E2 strengths, the rates are suppressed below 10 −2 of the dominant one except for the 5/2 + 3 → 5/2 + 1 transition in case of (b); the rate is about 0.04 times the dominant rate.There is only one γ transition observed [15].Transitions with rates suppressed below 10 −2 might not be observed.If it is true, the existence of the 2 + state is not excluded completely yet.The halo radius is enhanced by 0.06 fm when the 2 + state exists.

F
We finally discuss halo structure in 17 B and 19 B by the three-body model with mixed configurations of 2s 2  1/2 and 1d 2 5/2 [17].A relation between the halo radius and S 2n is obtained by the three-body model.The probability for the d-orbit component in the halo of 17 B is obtained to be 0.83±0.02from constraints on the halo radius and S 2n from experimental data.This is consistent with the recent quasifree (p, pn) reaction, which suggests the s-orbit probability to be 9±2 % [18].
The relations between the halo radius and S 2n obtained for pure d-orbit, pure s-orbit and mixed configurations are obtained for 19 B. If small matter and halo radii are taken [19], the halo in 19 B is d-wave dominant.However, a large E1 strength was found in a recent Coulomb break-up reaction experiment [20], which suggests an appreciable s-wave mixing in the halo and a larger halo radius than in Ref. [19].When S 2n is taken to be ∼0.5 MeV as suggested in Ref. [20], the probability of s-orbit component is obtained to be 40% with the halo radius of 7.2 fm for S 2n = 0.52 MeV for our three-body model.Detailed discussion is given in Ref. [17].A similar probability for the s-orbit, 52-57%, was reported in another three-body model [21].

Figure 1 .
Figure 1.(Left) Ground state energies of carbon isotopes obtained with microscopic G-matrix and G-matrix + FM 3N forces as well as with the SFO-tls Hamiltonian.(Right) Energy levels of 16 C, 17 C and 18 C obtained with G-matrix and G-matrix +FM 3N forces as well as the experimental data.

Figure 2 .
Figure 2. Energy levels below 1.5 MeV and B(E2) values in 29 F obtained by the weak-coupling model.B(E2) values are denoted in the parentheses in units of e 2 fm 4 .Cases with 0 + state only and with 0 + and 2 + states in the halo are shown as well as the experimental data [15].Cases (a) and (b) correspond to high and low excitation energies for the 1/2 + state in 27 F and the 2 + state in the halo, respectively.