Transcript Slide 1

ANC Techniques and r-matrix analysis
Santa Fe, April 2008
ANC Techniques and r-matrix analysis
Grigory Rogachev
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
Outline
 Sub-Coulomb a-transfer for astrophysics.

13C(a,n)
reaction rate from sub-Coulomb
13C(6Li,d)

14C(a)
reaction.
reaction rate from sub-Coulomb
14C(6Li,d), 14C(7Li,t)

14O(a,p)
reactions.
reaction rate from sub-Coulomb
14C(6Li,d), 14C(7Li,t)
reactions.
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
 Rates of some (a,n), (a,p) and (a) reactions are important
input parameters for various astrophysical processes.
 S-process neutron sources.
 rp-process in X-ray binaries and novae.
 In many cases cross section is prohibitively small for direct
measurements at energies of interest. Needs to be
extrapolated.
 Low energy resonances often dominate the cross section.
 One needs to know properties of these resonances to make
reliable extrapolation.
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
 In some cases resonances that are crucial for the
specific reaction rate, are known and most of their
properties determined, except for a
13C(a,n); 1/2+ at 6.356 MeV in 17O.
14C(a); 3- at 6.404 MeV in 18O.
14O(a,p); 1- at 6.15 MeV in 18Ne.
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
For these “known resonance” cases...
 a transfer reactions (6Li,d) or (7Li,t) can be used to
measure Sa spectroscopic factor and deduce the partial a
widths.
 However, final result depends on:
✔Optical potentials used for entrance and exit channels.
✔Shape of binding potentials for core-a and a-d(t)
formfactors.
✔Number of nodes assumed in the core-a wavefunction.
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
 ALL uncertainties can be avoided if:
✔ a transfer reaction is performed at sub-Coulomb energy.
This eliminates dependence of the calculated cross section
on optical potentials.
✔ ANCs are extracted from experimental data. This eliminates
dependence of the final result on the shape of form-factor
binding potentials and number of wavefunction nodes.
 This approach was used by [C.R. Brune, et al., PRL 83 (1999) 4025] in
pioneering 12C(6Li,d) atransfer at sub-Coulomb energy experiment, in which
the contributions from 16O sub-threshold resonances to the 12C(a,) reaction
rate were determined.
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
ANC approach
a
dA

a
A
d
I ab 
S ab  ab  C ab
W
r
 S ab b ab2
 ab  b ab
C
2
ab
aA
Bn
W
Model ab cluster
r wavefunction
Single-particle ab cluster wavefunction
Definition of ANC through single-particle ANC
d
d
 Sad SaA
d  exp
d  DWBA
d
~ ba2d ba2A X X depends only on entrance and
exit channel optical potentials
d  DWBA
1
2
2
2
2 d
C a d C a A ~ 2 2 ba d ba A
ba d ba A X
d  exp
~
C
P~ a
A
nB
M
~
W


r
a
A

E


it
a
A
2
A.M. Mukhamedzhanov, R.E. Tribble,
Phys. Rev. C59, 3418 (1999)
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
The 13C(a,n) reaction
The 13C(a,n) reaction is considered to be the main
source of neutrons for s-process in
Asymptotic Giant Branch stars.
The 13C(a,n) reaction rate was identified as
“necessary ingredient” for better models of
AGB stars in NSAC 2002 Long Range Plan
(p. 68).
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
The 13C(a,n) reaction
Partial a width of the ½+ state at 6.356 MeV in 17O is the main
source of the 13C(a,n) reaction rate uncertainty.
an ~

a
n
E2 tot
2
 2
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
The 13C(a,n) reaction
 The Sa factor of the ½+ 6.356 MeV state in 17O was
measured using the 13C(6Li,d) reaction at 60 MeV of
6Li
[S. Kubono, et al., PRL 90 (2003) 062501].
 Result – Sa = 0.011
 However, it was shown by [N. Keeley, K.W. Kemper
and D.T. Khoa, Nucl. Phys. A726 (2003) 159] that the
data is consistent with Sa ranging from 0.15 to 0.5,
depending on the DWBA parameters.
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
Sub-Coulomb 13C(6Li,d)17O experiment
at FSU
 In order to avoid influence of optical potentials the reaction
has to be sub-Coulomb for both entrance and exit channels.
Therefore very low energy (<3.0 MeV in c.m.) has to be
used.
 Inverse kinematics was used to provide additional “boost” for
deuterons and eliminate of 12C background.
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
Sub-Coulomb 13C(6Li,d) experiment
[S.Kubono, et al., PRL 90 (2003) 062501]
Spectrum of deuterons from 6Li(13C,d)
reaction, measured at 8.5 MeV of 13C.
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
13C(6Li,d)
angular distribution
 Coulomb modified
8.5 MeV of
13
13CC
8.0 MeV of 13C
ANC of ½+ resonance is
0.89+/-0.23 fm-1.
 S(0) factor of ½+
resonance is
2.5+/-0.7*106 MeV*b.
 This is a factor of ten
smalled than adopted in
NACRE [1] compilation
and a factor of ~5 larger
than in [2].
Angular distribution of deuterons from
sub-Coulomb 13C(6Li,d)17O(1/2+; 6.356 MeV)
reaction at 8.5 and 8.0 MeV.
[1] C.Angulo, et al., Nucl, Phys. A656 (1999) 3
[2] S.Kubono, et al., PRL 90 (2003) 062501
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
13C(a,n)
s-factor and reaction rate
[4] H.W. Drotleff et al., AJ 414 (1993) 735
[15] C.R. Brune, et al., PRC 48 (1993) 3119
r-matrix fit to the direct
measurements [4,15]
combined with coherent
contribution from ½+
6.356 MeV state,
determined using the
measured ANC.
 
2
r
C2
2 R
2 dS

C
W 2 ( R)
dE
B  S ( Ei )
A.M. Mukhamedzhanov, R.E. Tribble,
Phys. Rev. C59, 3418 (1999)
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
r-matrix fit to the 13C(a,n) and 13C(n,n) data
S-factor (MeV*b)
Total CS (b)
13C(a,n)
13C(n,n)
Eex - 4.16 (MeV)
Eex – 6.36 (MeV)
 Two channels were included into the r-matrix fit
13C(n,n)
and 13C(a,n).
 18 known resonances from 4.6 to 8.0 MeV in 17O.
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
13C(a,n)
reaction rate
 The final reaction rate
is a factor of 3 lower
than in NARCE
compilation.
 Uncertainty at
temperatures relevant
for s-process was
reduced to 25 %
E. Johnson, et al., PRL 97 (2006) 192701
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
Abundance of 19F in AGB stars.
There is experimental evidence that
19F
is produced within the interior of
AGB stars.
The major uncertainties in
abundance of 19F are associated
with 14C(a) and 19F(a,p) reaction
rates [M. Lugaro, et al.,
Comparison of the observed
Astro. J., 615 (2004) 934.]
and predicted fluorine
abundances. [M. Lugaro ApJ,
615 (2004)]
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
14C(a)
reaction rate.
States of interest at 0.1 GK:
3- at 6.40 MeV
1- at 6.20 MeV
a 
2J 1
 
(2Sa  1)(2S A  1) 
  
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
The sub-Coulomb 14C(6Li,d) and 14C(7Li,t) a-transfer
experiment at FSU.
Radioactive
14C
beam at energies 8.8, 10.5
and 11.5 MeV was delivered using the special
14C SNICS source.
Both the 14C(6Li,d) and 14C(7Li,t) reactions at
sub-Coulomb energies were used to measure
the ANCs of the 6.4 MeV 3- and 6.2 MeV 1states.
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
The sub-Coulomb 14C(7Li,t) a-transfer
Spectra of tritons from 7Li(14C,t) reaction at 11.5 MeV of 14C
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
The sub-Coulomb 14C(7Li,t) a-transfer
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
The sub-Coulomb 14C(7Li,t) a-transfer
kR
C2
a  2 2
F (kR)  G 2 (kR) 2R  C 2 dS
W 2 ( R)
dE
a3 at 6.4 MeV = (1.05+/-0.25)x10-13 eV
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
Contribution of the compound nucleus.
States with unnatural parity (0-,1+,2-,etc.) cannot be populated in direct alpha
transfer reaction, however they are populated through compound nucleus.
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
The 14C(a) reaction rate.
At temperatures
relevant for 19F
nucleosynthesys
in AGB stars the
14C(a) reaction
rate is totally
determined by
the strength of
the 3- state.
The Direct Capture (DC) and resonance capture due to
4+ at 7.11 MeV are from J. Gorres, et al. Nucl. Phys. A548 (1992)
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Santa Fe, April 2008
14
The O(a,p)
reaction.
 14O(a,p) reaction rate is an important input parameter for rpprocess in X-ray burst models [H. Schatz, K.E. Rehm, NP A777
(2006) 601].
 Two near threshold resonances are considered to be the main
contributors to the 14O(a,p) reaction rate at X-ray burst energies,
1- at 6.15 MeV and 3- at 6.30 MeV.
 Partial a width for these resonances is uncertain. There is a
significant disagreement between direct measurements [M.
Notani, et al., Nucl. Phys. A746 (2004) 113c] and indirect (time
inverse reaction) measurements [J.C. Blackmon, et al., NP A688
(2001) 142; B. Harss, et al., PRC].
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
14
The O(a,p)
reaction.
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
The 14O(a,p) reaction.
Reduced width of the 6.15 MeV resonance in 18Ne and
6.2 MeV resonance in 18O is assumed to be the same.


2

 k Ne R  
CO


a  2 2
2 
R
2 dS 
 FNe  GNe   2

C
O
 WO 2 ( R)
dE O 
a(1- at 6.15 MeV in 18Ne) = 1.4+/-0.3 eV
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
The 14O(a,p) reaction.
B. Harss, et al. PRC, 65 (2002)
M. Notani, NPA 746 (2006)
Direct 14O(a,p) measurement
Time reverse 17F(p,a) measurement
a = 3.2+5-2 eV from [B. Harss, et al. PRC, 65 (2002)]
Our value is 1.4 +/- 0.3 eV
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
14
The O(a,p)
S-factor
MeV*b
reaction.
1- at 6.15 MeV
Strong cluster
1- at 8.9 MeV
1- at
4+ at
7.05 MeV 7.6 MeV
Ecm (MeV)
Effects of constructive and destructive interference on 1- state
at 6.15 MeV are estimated to be ~20% at resonance energy.
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
14
The O(a,p)
reaction.
Based on the results of this work a to proton
decay branching ratio for this 1- resonance at
6.15 MeV in 18Ne is ~3*10-5 - not too bad and it
is possible to design an experiment which can
test this branching ratio directly.
Example: 16O(3He,n)18Ne(1-)
17F+p
14O+a
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
Conclusion
 Sub-Coulomb alpha transfer can be used to extract ANCs of sub and near
threshold resonances and calculate their contribution to corresponding low
energy reactions on parameterless basis.
 Mirror symmetry allows to apply knowledge of ANCs in one nucleus to
evaluate width of the corresponding resonances in it’s harder to excess mirror.
 ANC of the 1/2+ state at 6.36 MeV in
17O
was measured and the 13C(a,n)
reaction rate uncertainty was reduced from 300% to 25%.
 ANCs of the 1- and 3- states at 6.2 and 6.4 MeV in
18O
were measured. The 3-
state provides dominant contribution to the 14C(a) reaction rate at ~0.1 GK
and the 1- state is the mirror of the 6.15 MeV state in 18Ne which is the
dominant state for the 14O(a,p) reaction in explosive environment of x-ray
binaries. Its partial alpha width was evaluated with an accuracy of ~30%.
ANC Techniques and r-matrix analysis
Santa Fe, April 2008
Acknowledgements
Florida State University
E. Johnson
Texas A&M University
A. Mukhamedzhanov
J. Mitchell
V.Z. Goldberg
L. Miller
R.E. Tribble
S. Brown
B. Green
B. Roeder
A. Momotyuk
K. Kemper