Faddeev three-body calculation of triple-alpha reaction

Souichi IshikawaHosei University, Japan

1

The Fifth Asia-Pacific Conference on Few-Body Problems in Physics 2011 (APFB2011)22~26 August 2011, Seoul, Republic of Korea

• Triple-alpha reaction

-Resonant process (T>108 K) 8Be, 12C* Resonance formula

-Non-resonant process (T<108 K) Nuclear Astrophysics Compilation of Reaction Rates (NACRE) [1] based on Nomoto et al. [2]: Extension of the resonance formula with energy dependent width at low energies

1. INTRODUCTION

2

4 4 4 12He He He C

4 4 8

8 4 12 * 12 122 1 1

He He Be

Be He C 0 C 2 C 0

Refs. [1] C. Angulo et al., NPA656 (1999) 3. [2] K. Nomoto et al., A&A149 (1985) 239.

3

Resonant

Non-resonant

=T/(107 K)

Astrophysical input: 3 reaction rate <> [cm6/s]

n12 (n4): Number density of 12C (4He) 34

12 6

nn

(2) Ogata et al. (OKK rate)[3] Quantum 3-body calculations by the method of Continuum-Discretized

Coupled-Channel (CDCC):

*Normalized to the NACRE rate at T7=100 ~1026 larger at T7=1

~106 larger at T7=10 compared to the NACRE rate

“Severe inconsistency with the current understanding of the observations.” [4]

[3] K. Ogata et al., PTP122 (2009) 1055.[4] T. Suda et al., arXiv:1107.4984, and references therein.

NACRE

OKK

~1026

~106

In the present talk:(3) Faddeev method, which was successfully applied to

three-nucleon scattering systems in a sufficient accuracy with Coulomb force [5].

CONTENTS (1. Introduction) 2. Formalism 3. - and -- potentials 4. Results 5. Summary

[5] S. Ishikawa, PRC80, 054002 (2009); MPL A 24, 855 (2009) (APFB 2008); Proc. of INPC 2011 (to be published).

2. Formalism• Consider 12C as an -- system. • The inverse process: (E2-)photodisintegration of 12C(2+).

12C(2+) + (E2) + + (L=0)

• Define a wave function for the disintegration process and apply the Faddeev 3-body formalism to calculate it.

12 2 121

3 23/ 2 /

3 (2 ) 3 (2 )0

240 3 BE k T

C C

dEEc e E E E E

mc kT

12 5/ 23

2 2

/ 222 2 2

0

1;

4 3, cos , sin

3 4

~ cos sin ;

iKR

C

eE H B E

E i H R

R x y x R y R

E E d B E

yyxx

Faddeev eq.:Multiple scattering with rearrangements

1 2

3

1 2

3

1 2

3

Problem in the presence of long-range Coulomb forces (1)Rearrangement at long distance Severe singularity in kernel(2)Spectator particle should be distorted by Coulomb force

Channel-1Channel-3

Channel-2

Sasakawa-Sawada method [6]:Auxiliary Coulomb potential

(23)1 (12)3Distortion of the spectator particle(Partial) cancellation of long-range - Coulomb force

The cancellation is not perfect for breakup channels. treat this problem approximately by a (mandatory) cutoff procedure.

[6] T. Sasakawa and T. Sawada, PRC20 (1979) 1954.

cut

43 cut( /

3 1

)

17 fm

1 1 x R

R

ex y

1

1

y

3

1

x

• Shallow -potential (no forbidden state)[7]

• -- Potential to reproduce the resonance energy (continuum 0+ state) and the binding energy (2+ state).

2

2

/1.530 2

/ 2.85

ˆ ˆ125 20

30.18

xL L

x

V x P P e

e

[7] D.V. Fedorov and A. S. Jensen, PLB 389 (1996) 631

3. - and -- Potentials

23

/ 3.9 2 20 2

1

ˆ ˆ168 56 3.97L L ii

V P P e r

1. E2-photodisintegration cross section of 12C(2+):

2. 3 reaction rate <>

4. Results

10

12 2 121

3 23/ 2 /

3 (2 ) 3 (2 )0

240 3 BE k T

C C

dEEc e E E E E

mc kT

12 21(2 ) 3C

Photodisintegration cross section

12 21

12

(2 ) 3

(2 )

C

C

E

E E E

Er=0.383MeV [Exp.=0.379MeV]

=11.7eV [Exp.=8.3(1.0)eV]

B(E2,0+22+

1) = 9.4 e2fm4

[Exp=13.3 e2fm4]

• is normalized with respect to the E2 transition strength, B(E2,0+

22+1),

(effective charge ~ 0.2).

Normalization of

12

12 21(2 ) 3C

213.31.41 1 0.19

9.4

12 21(2 ) 3C

W3

(MeV)

Er

(keV)

(eV)

B(0+2+)

(e2fm4)

-168 383.2 11.7 9.4

Exp. 379.8 8.3(1.0) 13.3(1.3)

reaction rate

OKK

This work

NACRE

~1026 for OKK

~0.98

5. SUMMARY• Calculations of the 3-reaction as a quantum mechanical three-body

problem

• A wave function corresponding to the inverse process: 12C(2+) + + + applying the Faddeev three-body theory with accommodating long-range Coulomb force effect, which has been successfully applied for three-nucleon systems.

• Present calculations of <> : ~1000 times larger than the NACRE rate at T7=1.

• The result is not consistent with the CDCC calculations for T7 < 20 (Why ?)

• Three-body Coulomb problem is still tough one.

15

Photodisintegration cross section

12 21

12

(2 ) 3

(2 )

C

C

E

E E E

• OKK’s insist:Due to a reduction of Coulomb barrier of - subsystem between the incoming particle for non-resonant - system.

Enhancement at low temperature

17

Cancellation of the long-range character in - Coulomb force