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

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

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

6 He 6 Li A. Eskandarian and I. R. Afnan, PRC46, 1992 (2344). T = 1: Faddeev eqs. Yamaguchi-type potentials 1 S 0 for n - n S 1/2, P 1/2, and P 3/2 for 4 He- n

JJ Eskandarian and AfnanOurs E  E  T = 1 (in MeV) Checking on 6 Li

6 He 6 Li A. Eskandarian and I. R. Afnan, PRC46, 1992 (2344). T =1: Faddeev eqs. Yamaguchi-type potentials 1 S 0 for n - n S 1/2, P 1/2, and P 3/2 for 4 He- n Apply to 8 He and 10 He with our extention

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Odd Nuclei

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Even Nuclei

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

(extraction) TUNL Nuclear Data Evaluation Project (unit: MeV) Core excitation is important!

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

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

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

for the study of 10 He Potentials between 8 He- n nn 8 He 8 He * 12 Separable Potential Yamaguchi-type form factor c i : mixing parameter (state probability) b i : “width” parameter ` i : angular momentum

parameters: c 1, c 2,  1,  2 Potentials between 8 He- n 8 He 9 He 10 He Fit! (fit to 5 He data) c 1, c 2  1,  2 = fm  1 fixed determined uniquly 8 He 8 He *

Ground (0 + ) State of 10 He J p of 9 He J p of 9 He * 3/2  1/2  1/2 + 1/2  1/2 + 1/2  3/2  3/2 + 1/2  1/2  E (MeV)  (MeV)    exp

parameters: c 1, c 2,  1,  2 Potentials between 6 He- n 6 He 7 He 8 He Fit! (fit to 5 He data) c 1, c 2  1,  2 = fm  1 fixed determined uniquly 6 He 6 He *

Ground (0 + ) State of 8 He J  of 7 HeJ  of 7 He * E (MeV) 3/2   /2  1/2   /2  5/2   exp.  2.140

State Probability 8 He 6 He n n + n n % 33.8% 6 He * n n n n 6 He 66.2% 33.8% He nn nn nn

Summary Study the He isotopes (Borromean nuclei) –within the 3-cluster model … core + 2 n 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 9 He(1/2  ) and 9 He * (1/2 + ) (for 10 He) probability of 6 He(66.2%) and 6 He * (33.8%) in 8 He

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

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

Summary Study the He isotopes (Borromean nuclei) –within the 3-cluster model … core + 2 n 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 9 He(1/2  ) and 9 He * (1/2 + ) (for 10 He) probability of 6 He(66.2%) and 6 He * (33.8%) in 8 He

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

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

Brief Summary (Faddeev-like eqs.) 6 He nn v1v1 v2v2 v3v3 Faddeev Eqs.

Brief Summary titi fitting potential parameters to peaks of the response function E 6 He nn v1v1 v2v2 R(E)R(E) v3v3

Response Function 6 He nn v1v1 v2v2 v3v3

Three-Cluster System 6 He nn nn nn channel 1channel 2channel t i : T-matrix for two-body subsystem (Faddeev theory)

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

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Application : He isotopes Binding Energy :: 6 He < 8 He Resonance Levels :: 5 He ¼ 7 He ¼ 9 He Decay Width :: 5 He > 7 He ¼ 9 He Is nucleus stable with increasing neutron?

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8 He- n 8 He nn 8 He * coupling channel 1 channel 2 8 He nn 9 He 10 He 6 He+2 n (extraction) TUNL Nuclear Data Evaluation Project unit: MeV

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

Yamaguchi-type Pot. Separable Potential Yamaguchi-type form factor c i : mixing parameter (state probability) l: coupling constant b i : “width” parameter ` i : angular momentum titi E R(E)R(E) (for the first attempt)

Content Motivation Potential Model Calculations