faddeev three-body calculation of triple- alpha reaction souichi ishikawa hosei university, japan 1...
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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
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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.
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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