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


Grigory Rogachev

Outline Sub-Coulomb -transfer for astrophysics.

13C(,n) reaction rate from sub-Coulomb

13C(6Li,d) reaction.

14C() reaction rate from sub-Coulomb

14C(6Li,d), 14C(7Li,t) reactions.

14O(,p) reaction rate from sub-Coulomb

14C(6Li,d), 14C(7Li,t) reactions.



Rates of some (,n), (,p) and () 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.



In some cases resonances that are crucial for the

specific reaction rate, are known and most of their

properties determined, except for

13C(,n); 1/2+ at 6.356 MeV in 17O.

14C(); 3- at 6.404 MeV in 18O.

14O(,p); 1- at 6.15 MeV in 18Ne.



transfer reactions (6Li,d) or (7Li,t) can be used to

measure S spectroscopic factor and deduce the partial

widths.

However, final result depends on:

✔ Optical potentials used for entrance and exit channels.

✔ Shape of binding potentials for core- and -d(t)

formfactors.

✔ Number of nodes assumed in the core- wavefunction.

For these “known resonance” cases...



ALL uncertainties can be avoided if:

✔ 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) transfer at sub-Coulomb energy experiment, in which

the contributions from 16O sub-threshold resonances to the 12C(,) reaction

rate were determined.



ANC approach

exp

2222

22

22

exp

22

1~

~

d

dbb

XbbCC

Xbbd

d

d

dSS

d

d

bSC

r

Wb

r

WCSI

AdAd

Ad

AdDWBA

DWBAAd

ababab

abab

abababab

dAAd nBA Model ab cluster wavefunction

Single-particle ab cluster wavefunction

Definition of ANC through single-particle ANC

X depends only on entrance and exit channel optical potentials

2

~~

~t

A

nBA

A iE

CWr

PM

A.M. Mukhamedzhanov, R.E. Tribble,

Phys. Rev. C59, 3418 (1999)



The 13C(,n) reaction rate was identified as

“necessary ingredient” for better models of

AGB stars in NSAC 2002 Long Range Plan

(p. 68).

The 13C(,n) reactionThe 13C(,n) reaction is considered to be the main

source of neutrons for s-process in

Asymptotic Giant Branch stars.



The 13C(,n) reactionPartial width of the ½+ state at 6.356 MeV in 17O is the main

source of the 13C(,n) reaction rate uncertainty.

2

2

2

~

tot

nn

E



The S 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 – S = 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 S ranging from 0.15 to 0.5,

depending on the DWBA parameters.

The 13C(,n) reaction



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.



Sub-Coulomb 13C(6Li,d) experiment

Spectrum of deuterons from 6Li(13C,d)

reaction, measured at 8.5 MeV of 13C.

[S.Kubono, et al., PRL 90 (2003) 062501]



8.0 MeV of 13C

13C8.5 MeV of 13C

Angular distribution of deuterons from

sub-Coulomb 13C(6Li,d)17O(1/2+; 6.356 MeV)

reaction at 8.5 and 8.0 MeV.

Coulomb modified 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].

13C(6Li,d) angular distribution

[1] C.Angulo, et al., Nucl, Phys. A656 (1999) 3[2] S.Kubono, et al., PRL 90 (2003) 062501



13C(,n) s-factor and reaction rate

r-matrix fit to the direct measurements [4,15] combined with coherent contribution from ½+ 6.356 MeV state, determined using the measured ANC.

[4] H.W. Drotleff et al., AJ 414 (1993) 735[15] C.R. Brune, et al., PRC 48 (1993) 3119



A.M. Mukhamedzhanov, R.E. Tribble,

Phys. Rev. C59, 3418 (1999)

)( iESB

dEdS

CRWRC

r2

2

22

)(2

r-matrix fit to the 13C(,n) and 13C(n,n) dataS-factor (MeV*b)

Eex – 6.36 (MeV)

Total CS (b)

Eex - 4.16 (MeV)

13C(,n) 13C(n,n)

Two channels were included into the r-matrix fit 13C(n,n) and 13C(,n). 18 known resonances from 4.6 to 8.0 MeV in 17O.



13C(,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



Abundance of 19F in AGB stars.

There is experimental evidence that

19F is produced within the interior of

AGB stars.

Comparison of the observed and predicted fluorine abundances. [M. Lugaro ApJ, 615 (2004)]



The major uncertainties in abundance of 19F are associated with 14C() and 19F(,p) reaction rates [M. Lugaro, et al.,Astro. J., 615 (2004) 934.]

14C() reaction rate.

States of interest at 0.1 GK:

3- at 6.40 MeV

1- at 6.20 MeV



a

Aa SS

J

)12)(12(

12

The sub-Coulomb 14C(6Li,d) and 14C(7Li,t) -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 1-

states.



Spectra of tritons from 7Li(14C,t) reaction at 11.5 MeV of 14C

The sub-Coulomb 14C(7Li,t) -transfer







at 6.4 MeV = (1.05+/-0.25)x10-13 eV



dEdS

CRWRC

kRGkRF

kR

22

2

22

)(2)()(

2

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.



The 14C() reaction rate.

The Direct Capture (DC) and resonance capture due to 4+ at 7.11 MeV are from J. Gorres, et al. Nucl. Phys. A548 (1992)



At temperatures relevant for 19F nucleosynthesys in AGB stars the 14C() reaction rate is totally determined by the strength of the 3- state.

The 14O(,p) reaction. 14O(,p) reaction rate is an important input parameter for rp-

process 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(,p) reaction rate at X-ray burst energies, 1- at 6.15 MeV and 3- at 6.30 MeV.

Partial 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].



The 14O(,p) reaction.



(1- at 6.15 MeV in 18Ne) = 1.4+/-0.3 eV




OO

O

O

NeNe

Ne

dEdS

CRW

RC

GF

Rk2

2

2

22

)(

22

Reduced width of the 6.15 MeV resonance in 18Ne and 6.2 MeV resonance in 18O is assumed to be the same.


= 3.2+5-2 eV from [B. Harss, et al. PRC, 65 (2002)]

Our value is 1.4 +/- 0.3 eV



B. Harss, et al. PRC, 65 (2002)

M. Notani, NPA 746 (2006)

Direct 14O(,p) measurement

Time reverse 17F(p,) measurement




S-factor MeV*b

Ecm (MeV)

1- at 6.15 MeV

4+ at7.05 MeV

1- at

7.6 MeV

Strong cluster 1- at 8.9 MeV

Effects of constructive and destructive interference on 1- state at 6.15 MeV are estimated to be ~20% at resonance energy.


Based on the results of this work 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+



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(,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() 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(,p) reaction in explosive environment of x-ray

binaries. Its partial alpha width was evaluated with an accuracy of ~30%.

Conclusion



Acknowledgements

A. Mukhamedzhanov

V.Z. Goldberg

R.E. Tribble

Texas A&M University

E. Johnson

J. Mitchell

L. Miller

S. Brown

B. Green

B. Roeder

A. Momotyuk

K. Kemper

Florida State University



Documents

ANC Techniques and r-matrix analysis Santa Fe, April 2008 ANC Techniques and r-matrix analysis Grigory Rogachev