A Three-Body Faddeev Calculation of the Double Polarized 3He(d,p)4He Reaction in the Super Low-Energy Region

1S. Gojuki, K. Sonoda, Y. Hiratsuka and S. OryuDepartment of Physics, Tokyo University of Science

1SGI Japan Ltd.

Agenda

• Introduction– What’s interesting?– What’s our purpose?

• How to calculate the 3He(d,p)4He reaction?– Three body Faddeev theory– Potentials

• Results• Summary

IntroductionWhat’s Interesting?• What’s interesting for the 3He(d,p)4He in super low-

energy region?– Nucleosynthesis in Universe – Nuclear-Fusion Power Generation

• Mirror Reaction of the 3H(d,n)4He• Neutronless reaction• Polarization effects

http://grin.hq.nasa.gov/

TOKAMAK

Nucleosynthesis

http://www.fusionscience.org

IntroductionWhat’s our Purpose?

T.W.Bonner et al., Phys.Rev.88,473 (1952), W.H.Geist et al., Phys.Rev.C60,054003-1 (1999)

3/2+ R

eson

ance

2

11 1

2

1

S-wave S-wave

np

p

np

np

pn

p

3He 3Hed d

Jπ=1/2+ Jπ=3/2+

The 3/2+ state can be set by the double parallel polarization.

Get the cross section enhancement !?

Double Parallel Polarization

How to calculate the 3He(d,p)4He reaction?

Five nucleon Problem(Big degree of freedom)

Select three clusters (3He, p, and n)(Because of super low energy)

Three cluster Faddeev calculation(Reduce the degree of freedom)

Potentials(p-n, p-3He, and n-3He)

p-n p-3Hen-3He

Three Cluster Faddeev Equation

Faddeev Equation

Amado-Lovelace-Mitra Equation

Separable Expansion (reduce degree of freedom)

We calculate this equation on the each energy.

Potential p-n• Paris Potential (EST expanded)

– One of the most popular nucleon-nucleon potential

1S0

3S1

3D1

M.Lacombeet al., Phys. Rev. C21 (1980) 861

0

20

40

60

80

100

120

140

160

180

0 100 200 300 400 500

Lab. Energy [MeV]

Phas

e Sh

ift [d

egre

e] rank=1rank=4rank=6rank=8Exp. AExp. B

-40

-30

-20

-10

0

10

20

30

40

50

0 100 200 300 400 500

Lab. Energy [MeV]

Phas

e Sh

ift [d

egre

e] rank=1rank=4rank=6rank=8Exp. AExp. B

-40

-30

-20

-10

0

10

20

30

40

50

60

70

0 100 200 300 400 500

Lab. Energy [MeV]

Phas

e Sh

ift [d

gere

e]

rank=1rank=3rank=5Exp. AExp. B

Exp. A :R.A.Arndt, L.D.Roper, R.A.Bryan, R.B.Clark, B.J.VerWest, and P.Signell, Phys. Rev. D28, 97 (1983)Exp. B : R.A.Arndt, J.S.Hyslop III, and L.D.Roper, Phys. Rev. D35, 128 (1987)

Potentials p-3He, n-3He

Base TheoryResonating Group Method(RGM)

Pauli PrincipleOrthogonal Condition Model

Separable PotentialEST Expansion

I.Reichstein,P.R.Thompson,and Y.C.Tang., Phys. Rev. C3, 2139 (1971)H.Kanad and T.Kaneko., Phys. Rev. C34, 22 (1986)

S.Saito, Prog. Theor. Phys. 40, 893 (1968)S.Saito, Prog. Theor. Phys. 41, 705 (1969)

D.J.Ernst,C.M.Shakin,and R.M.Thaler, Phys. Rev. C8, 46 (1973)

Just theory!

Potential p-3He

1S0

EST Expansion

Resonating Group Method & Orthogonal Condition Model

-160

-140

-120

-100

-80

-60

-40

-20

0

0 5 10 15 20 25 30

Lab. Energy [MeV]

Phas

e Sh

ift [d

egre

e]

rank=1rank=3

○;T.A.Tombrello, Phys.Rev.138,B40(1965)□;D.H.Mc Sherry and S.D.Baker, Phys.RevC1,888(1970)△;J.R. Morales, T.A. Cahill, and D.J. Shadoan, Phys.Rev..C11,1905(1975)◊;D.Müller, R.Beckmann, and U. Holm, Nucl.Phys.A311,1.(1978)+;L.Beltrmin, R.del Frate, and G. Pisent, Nucl.Phys.A442,266(1985)●;Y.Yoshino, V.Limkaisang, J.Nagata, H.Yoshino, and M.Matsuda, Prog. Theor.Phys.103,107(2000)

Potential n-3He

1S0

EST Expansion

Resonating Group Method & Orthogonal Condition Model

-100

-80

-60

-40

-20

0

0 5 10 15 20

Lab. Energy [MeV]

Phas

e Sh

ift [d

egre

e]

rank=1rank=3

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

340 350 360 370 380 390 400 410

Lab. Energy [keV]

Tota

l Cro

ssSe

ctio

n [m

b]

Total Cross Sectionp-n: 1S0, 3S1-3D1 ,p-3He: 1S0 ,n-3He: 1S0

Total Jπ=1/2(+-) – 9/2(+-)

p-n: 1S0(rank=3 or 5), 3S1-3D1(rank=4 or 6 or 8)p-3He: 1S0(rank=3) ,n-3He: 1S0(rank=3)

p-n: 1S0(rank=1), 3S1-3D1(rank=1)p-3He: 1S0(rank=3) ,n-3He: 1S0(rank=3)

p-n: 1S0(rank=1 or 3), 3S1-3D1(rank=1 or 4)p-3He: 1S0(rank=1) ,n-3He: 1S0(rank=1)

Converged!

Polarized Total Cross Section

Unpolarized Total Cross Section

x2.2

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

10

100

1000

10000

100000

340 350 360 370 380 390 400 410

Lab. Energy [keV]

Tota

l Cro

ss S

ectio

n [m

b]Total Cross Sectionp-n: 1S0, 3S1-3D1 ,p-3He: 1S0 ,n-3He: 1S0

Jπ=3/2+ Polarized

Jπ=3/2+ Unpolarized

Jπ=1/2+ Polarized

The 375keV peak is made from the 3/2+ state!

Jπ=3/2- Polarized

Jπ=1/2- Polarized

Jπ=1/2- UnpolarizedJπ=3/2- UnpolarizedJπ=1/2+ Polarized

Jπ=5/2+ Polarized

Jπ=5/2+ Unpolarized

Jπ=5/2-PolarizedJπ=5/2- Unpolarized

Summary

• The double parallel polarization effects– The total cross section in the 375 keV grows up to 2.2 times by th

e double parallel polarization effects.• The 3/2+ peak is found by the 1S0 rank=3 of the N-3He poten

tial. – The more realistic 4He structure is important.– But the peak is not broad…(experiment is broad. )

• Future– More exact two-body potential (higer rank and partial wave)– Internal Coulomb effect (Now: only initial and final states)