Faddeev Calculationfor Neutron-Rich Nuclei

Eizo Uzu (Tokyo Univ. of Science)

CollaboratorsMasahiro Yamaguchi (RCNP)

Hiroyuki Kamada (Kyusyu Inst. Tech.)

Yasuro Koike (Hosei Univ., CNS)

Focus on He isotopes

(extraction)

TUNL Nuclear Data Evaluation Projecthttp://www.tunl.duke.edu/nucldata/

(unit: MeV)

unboundbound

He isotopes (even nuclei)

6He

n n

3-cluster model

4He

n n

6He 8He

Borromean-nuclei

unbound unbound

unbound unbound

unbound unbound

(extraction)

TUNL Nuclear Data Evaluation Projecthttp://www.tunl.duke.edu/nucldata/

(unit: MeV)

Borromean

He isotopes (even nuclei)

6He

n n

3-cluster model

4He

n n

8He

n n

6He 8He 10He

Borromean-nuclei series

Faddeev equations

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

J

Eskandarian and Afnan OursE E

0+ -0.56 0.00 -0.565 0.0002+ +0.95 0.30 +0.947 0.287

T = 1

(in MeV)

Checking on 6Li

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

(extraction)

TUNL Nuclear Data Evaluation Projecthttp://www.tunl.duke.edu/nucldata/

(unit: MeV)

Odd Nuclei

Odd Nuclei

Odd Nuclei

(extraction)

TUNL Nuclear Data Evaluation Projecthttp://www.tunl.duke.edu/nucldata/

(unit: MeV)

Even Nuclei

Even Nuclei

Interesting Nature

“He” isotopes are getting stable with increasingthe number of neutrons … ?

Remembering the three-nucleon forces…

N N N

N N N

The binding energy of 3H

becomes deeper

(extraction)

TUNL Nuclear Data Evaluation Projecthttp://www.tunl.duke.edu/nucldata/

(unit: MeV)

Core excitation is important!

• 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

Brief Summary

• 8He, 10He … Borromean-nuclei series• 6He, 8He … Core: Ground states• 6He*, 8He* … Core: Excited states

Potentials between (core)-n

n n

8He 8He*

n n

6He 6He*

couple couple

7He 9He

• 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

Potentials between (core)-n

n n

8He 8He*

n n

6He 6He*

couple couple

7He 9He

• for the study of 10He

Potentials between 8He-n

n n

8He 8He*

1 2

Separable Potential

Yamaguchi-type form factorci: mixing parameter (state probability) i : “width” parameter`i : angular momentum

• parameters: c1, c2, 1, 2

Potentials between 8He-n

8He 9He 10He

Fit!

(fit to 5He data)

c1, c2

1, 2 = 1.5166 fm1 fixed

determineduniquly

8He

8He*

Ground (0+) State of 10He

J of He J of He*

3/2

1/2

1/2+

1/2

1/2+

1/2+

1/2

3/2

3/2+

1/2

1/2

E (MeV) (MeV)

0.187

0.5910.5341.8620.3060.782

0.131

0.0340.657

exp. 1.07 0.3

• parameters: c1, c2, 1, 2

Potentials between 6He-n

6He 7He 8He

Fit!

(fit to 5He data)

c1, c2

1, 2 = 1.5166 fm1 fixed

determineduniquly

6He

6He*

Ground (0+) State of 8He

J of 7He J of 7He* E (MeV)3/2 1.3173/2 1/2 2.1753/2 5/2 1.318

exp. 2.140

State Probability

8He

6Hen

n+

6He nn+

66.2%

33.8% 6He*

nn

nn

6He 66.2%

33.8%

0+

2+

6He

n n6He

n n6He

n n

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

OutlookFor improvement the potentials …

• Response function of the Coulomb break-up reactions (e.g. 8He + Pb)– compare directly with the experimental data

ti

E

R(E)

Response Function (preliminary)

only include 0+ state of (6He + 2n)

0 1 2 3 4

0.01

0.02

0.00

c.m. energy (MeV)

R(E)

(MeV

-1)

our prediction

7He g.s.

0 1 2 3 4

0.01

0.02

0.00

c.m. energy (MeV)

simulation (integrate ±0.3MeV)

for 8He + Pb reaction

7He 1st

8He 1st

8He 2nd

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

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

6Henn

Pb

6Henn

6He

Coulomb interaction

nn

Cross Section

response functionCoulomb scattering

: W.F. of the bound state : W.F of the final (continuum) state

: Momentum shift op.

Pb

n6He

( ) n

projection op.

Brief Summary

(Faddeev-like eqs.)

6He

n nv1 v2

v3

Faddeev Eqs.

Brief Summary

ti

fitting potential parametersto peaks of the response function

E

6He

n nv1 v2

R(E)v3

Response Function

6He

n nv1 v2

v3

Three-Cluster System

6He

n n

6He

n n

6He

n n

channel 1 channel 2 channel 3

12

3

12

3

12

3

ti : T-matrix for two-body subsystem

(Faddeev theory)

State Probability (preliminary)

8He

6Hen

n+

6He nn+

66.2%

33.8% 6He*

nn

nn

6He 66.2%

33.8%

0+

2+

10He8He(0+)

8He*(2+)

nnnn

48.8%

51.2%

(extraction)

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

Application : He isotopes

• Binding Energy :: 6He < 8He　• Resonance Levels :: 5He ¼ 7He ¼ 9He　• Decay Width :: 5He > 7He ¼ 9He

• Is nucleus stable with increasing neutron?

(extraction)

TUNL Nuclear Data Evaluation Projecthttp://www.tunl.duke.edu/nucldata/

not wellestablished

(unit: MeV)

8He-n

8He

n n

8He*

coupling

channel 1 channel 28He

n n

8He 9He 10He

6He+2n(extraction)

TUNL Nuclear DataEvaluation Project

http://www.tunl.duke.edu/nucldata/

unit: MeV

Neutron-rich Nuclei

core

n n

Focus on“He” isotopes

Faddeev Equations

Exact treatment ofthe break-up thresholds

(or boundary conditions)

3-cluster model

Yamaguchi-type Pot.Separable Potential

Yamaguchi-type form factorci: mixing parameter (state probability): coupling constanti : “width” parameter`i : angular momentumti

E

R(E)

(for the first attempt)

Content

• Motivation• Potential Model• Calculations