Faddeev Calculation for Neutron-Rich Nuclei

Faddeev Calculationfor Neutron-Rich Nuclei Focus on He isotopes Eizo Uzu (Tokyo Univ. of Science) Collaborators Masahiro Yamaguchi(RCNP) Hiroyuki Kamada(Kyusyu Inst. Tech.) Yasuro Koike(Hosei Univ., CNS)

TUNL Nuclear Data Evaluation Project (unit: MeV) http://www.tunl.duke.edu/nucldata/ (extraction) unbound bound

He isotopes (even nuclei) Borromean-nuclei n unbound n n unbound n unbound unbound unbound unbound 4He 6He 6He 8He 3-cluster model

TUNL Nuclear Data Evaluation Project (unit: MeV) http://www.tunl.duke.edu/nucldata/ (extraction) Borromean

He isotopes (even nuclei) Borromean-nuclei series n n n n n n 4He 6He 8He 6He 8He 10He Faddeev equations 3-cluster model

6He 6Li • A. Eskandarian and I. R. Afnan, PRC46, 1992 (2344). • T = 1: • Faddeev eqs. • Yamaguchi-type potentials • 1S0 for n-n • S1/2, P1/2, and P3/2 for 4He-n

Checking on 6Li T = 1 (in MeV)

6He 6Li • A. Eskandarian and I. R. Afnan, PRC46, 1992 (2344). • T =1: • Faddeev eqs. • Yamaguchi-type potentials • 1S0 for n-n • S1/2, P1/2, and P3/2 for 4He-n • Apply to 8He and 10He with our extention

TUNL Nuclear Data Evaluation Project (unit: MeV) http://www.tunl.duke.edu/nucldata/ (extraction)

Odd Nuclei

Even Nuclei

Interesting Nature “He” isotopes are getting stable with increasing the number of neutrons … ? Remembering the three-nucleon forces… N N N The binding energy of 3H becomes deeper D N N N

TUNL Nuclear Data Evaluation Project (unit: MeV) http://www.tunl.duke.edu/nucldata/ (extraction) Core excitation is important!

Brief Summary • 6He, 8He, 10He … Borromean-nuclei series • 4He, 6He, 8He … Core: Ground states • 6He*, 8He* … Core: Excited states • 5He, 7He, 9He … resonances • Faddeev eqs. • employ Yamaguchi-type potentials for n-n and (core)-n subsystems

Potentials between (core)-n • 8He, 10He … Borromean-nuclei series • 6He, 8He … Core: Ground states • 6He*, 8He* … Core: Excited states n n n n couple couple 6He 7He 6He* 8He 9He 8He*

Potentials between (core)-n • 7He, 9He … resonances • employ Yamaguchi-type potentials for n-n and (core)-n systems • Fix the parameters to reproduce energy levels and decay widths for 7He and 9He n n n n couple couple 6He 7He 6He* 8He 9He 8He*

Potentials between 8He-n • for the study of 10He Separable Potential n n 8He 8He* 1 2 Yamaguchi-type form factor ci: mixing parameter (state probability) bi : “width” parameter `i : angular momentum

Potentials between 8He-n • parameters: c1, c2, b1, b2 b1, b2 = 1.5166 fm-1 fixed (fit to 5He data) c1, c2 Fit! 8He* determined uniquly 8He 10He 8He 9He

Ground (0+) State of 10He Jp of 9He Jp of 9He* E (MeV) G (MeV) -0.187 3/2- 1/2- 0.591 0.131 1/2+ -0.534 1/2- 3/2- 1/2+ 3/2+ -1.862 1/2+ 1/2- 0.306 0.034 1/2- 1/2+ 0.657 0.782 exp. 1.07 0.3

Potentials between 6He-n • parameters: c1, c2, b1, b2 b1, b2 = 1.5166 fm-1 fixed (fit to 5He data) c1, c2 Fit! determined uniquly 6He* 6He 6He 7He 8He

Ground (0+) State of 8He

0+ 66.2% 6He* 33.8% n n n n n n n n n n n n n n 2+ 6He 6He 6He 6He 6He 6He State Probability + 66.2% 8He + 33.8%

Summary • Study the He isotopes (Borromean nuclei) • within the 3-cluster model … core + 2n • The three-body Faddeev eqs. • Yamaguchi-type potentials • parameters are fit to the smaller next odd nuclei • taking into account the core excitation • states of 9He(1/2-) and 9He*(1/2+) (for 10He) • probability of 6He(66.2%) and 6He*(33.8%) in 8He

ti R(E) E Outlook For improvement the potentials … • Response function of the Coulomb break-up reactions (e.g. 8He + Pb) • compare directly with the experimental data

simulation (integrate ±0.3MeV) 0.02 0.01 0.00 0 1 2 3 4 c.m. energy (MeV) Response Function (preliminary) for 8He + Pb reaction only include 0+ state of (6He + 2n) our prediction 8He 2nd 0.02 8He 1st 7He 1st R(E) (MeV-1) 7He g.s. 0.01 0.00 0 1 2 3 4 c.m. energy (MeV)

Summary • Study the He isotopes (Borromean nuclei) • within the 3-cluster model … core + 2n • The three-body Faddeev eqs. • Yamaguchi-type potentials • parameters are fit to the smaller next odd nuclei • taking into account the core excitation • states of 9He(1/2-) and 9He*(1/2+) (for 10He) • probability of 6He(66.2%) and 6He*(33.8%) in 8He

n n n n n n 6He 6He 6He Reaction • Low energy reaction of (core + 2n) + (heavy ion) e.g. (8He + Pb) Coulomb interaction Pb

n n 6He Cross Section Coulomb scattering response function : W.F. of the bound state : W.F of the final (continuum) state ( ) : Momentum shift op. projection op. Pb

n n 6He Brief Summary Faddeev Eqs. (Faddeev-like eqs.) v3 v1 v2

n n 6He Brief Summary fitting potential parameters to peaks of the response function ti v3 R(E) v1 v2 E

n n 6He Response Function v3 v1 v2

n n n n n n 6He 6He 6He Three-Cluster System 2 2 1 2 1 1 3 3 3 channel 1 channel 2 channel 3 (Faddeev theory) ti : T-matrix for two-body subsystem

0+ 66.2% 6He* 33.8% n n n n n n n n n n n n 2+ 6He 6He 6He State Probability (preliminary) + 66.2% 8He + 33.8% 8He(0+) 48.8% 10He 51.2% 8He*(2+)

Application : He isotopes • Binding Energy :: 6He < 8He • Resonance Levels :: 5He ¼7He ¼9He • Decay Width :: 5He > 7He ¼9He • Is nucleus stable with increasing neutron?

TUNL Nuclear Data Evaluation Project (unit: MeV) http://www.tunl.duke.edu/nucldata/ (extraction) not well established

8He 8He* 8He n n n n 8He-n coupling channel 1 channel 2 TUNL Nuclear Data Evaluation Project http://www.tunl.duke.edu/nucldata/ 6He+2n (extraction) unit: MeV 8He 9He 10He

Neutron-rich Nuclei 3-cluster model Faddeev Equations n n Exact treatment of the break-up thresholds (or boundary conditions) core Focus on “He” isotopes

Yamaguchi-type Pot. (for the first attempt) Separable Potential Yamaguchi-type form factor • ci: mixing parameter • (state probability) • : coupling constant bi : “width” parameter `i : angular momentum ti R(E) E

Content • Motivation • Potential Model • Calculations

Faddeev Calculation for Neutron-Rich Nuclei