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