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

2. Outline  Sub-Coulomb  -transfer for astrophysics.  13 C( ,n) reaction rate from sub-Coulomb 13 C( 6 Li,d) reaction.  14 C(  ) reaction rate from sub-Coulomb 14 C( 6 Li,d), 14 C( 7 Li,t) reactions.  14 O( ,p) reaction rate from sub-Coulomb 14 C( 6 Li,d), 14 C( 7 Li,t) reactions. ANC Techniques and r-matrix analysis Santa Fe, April 2008

3.  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. ANC Techniques and r-matrix analysis Santa Fe, April 2008

4.  In some cases resonances that are crucial for the specific reaction rate, are known and most of their properties determined, except for    13 C( ,n); 1/2 + at 6.356 MeV in 17 O. 14 C(  ); 3 - at 6.404 MeV in 18 O. 14 O( ,p); 1 - at 6.15 MeV in 18 Ne. ANC Techniques and r-matrix analysis Santa Fe, April 2008

5.   transfer reactions ( 6 Li,d) or ( 7 Li,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... ANC Techniques and r-matrix analysis Santa Fe, April 2008

6.  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 12 C( 6 Li,d)  transfer at sub-Coulomb energy experiment, in which the contributions from 16 O sub-threshold resonances to the 12 C( ,  ) reaction rate were determined. ANC Techniques and r-matrix analysis Santa Fe, April 2008

7. ANC approach 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 A.M. Mukhamedzhanov, R.E. Tribble, Phys. Rev. C59, 3418 (1999) ANC Techniques and r-matrix analysis Santa Fe, April 2008

8. The 13 C( ,n) reaction rate was identified as “necessary ingredient” for better models of AGB stars in NSAC 2002 Long Range Plan (p. 68). The 13 C( ,n) reaction The 13 C( ,n) reaction is considered to be the main source of neutrons for s-process in Asymptotic Giant Branch stars. ANC Techniques and r-matrix analysis Santa Fe, April 2008

9. The 13 C( ,n) reaction Partial   width of the ½ + state at 6.356 MeV in 17 O is the main source of the 13 C( ,n) reaction rate uncertainty. ANC Techniques and r-matrix analysis Santa Fe, April 2008

10.  The S  factor of the ½ + 6.356 MeV state in 17 O was measured using the 13 C( 6 Li,d) reaction at 60 MeV of 6 Li [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 13 C( ,n) reaction ANC Techniques and r-matrix analysis Santa Fe, April 2008

11. Sub-Coulomb 13 C( 6 Li,d) 17 O 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 12 C background. ANC Techniques and r-matrix analysis Santa Fe, April 2008

12. Sub-Coulomb 13 C( 6 Li,d) experiment Spectrum of deuterons from 6 Li( 13 C,d) reaction, measured at 8.5 MeV of 13 C. [S.Kubono, et al., PRL 90 (2003) 062501] ANC Techniques and r-matrix analysis Santa Fe, April 2008

13. 8.0 MeV of 13 C 13 C 8.5 MeV of 13 C Angular distribution of deuterons from sub-Coulomb 13 C( 6 Li,d) 17 O(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*10 6 MeV*b.  This is a factor of ten smalled than adopted in NACRE [1] compilation and a factor of ~5 larger than in [2]. 13 C( 6 Li,d) angular distribution [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

14. 13 C( ,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 ANC Techniques and r-matrix analysis Santa Fe, April 2008 A.M. Mukhamedzhanov, R.E. Tribble, Phys. Rev. C59, 3418 (1999)

15. r-matrix fit to the 13 C( ,n) and 13 C(n,n) data S-factor (MeV*b) E ex – 6.36 (MeV) Total CS (b) E ex - 4.16 (MeV) 13 C( ,n) 13 C(n,n)  Two channels were included into the r-matrix fit 13 C(n,n) and 13 C( ,n).  18 known resonances from 4.6 to 8.0 MeV in 17 O. ANC Techniques and r-matrix analysis Santa Fe, April 2008

16. 13 C( ,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

17. Abundance of 19 F in AGB stars. There is experimental evidence that 19 F is produced within the interior of AGB stars. Comparison of the observed and predicted fluorine abundances. [M. Lugaro ApJ, 615 (2004)] ANC Techniques and r-matrix analysis Santa Fe, April 2008 The major uncertainties in abundance of 19 F are associated with 14 C(  ) and 19 F( ,p) reaction rates [M. Lugaro, et al., Astro. J., 615 (2004) 934.]

18. 14 C(  ) reaction rate. States of interest at 0.1 GK: 3 - at 6.40 MeV 1 - at 6.20 MeV ANC Techniques and r-matrix analysis Santa Fe, April 2008

19. The sub-Coulomb 14 C( 6 Li,d) and 14 C( 7 Li,t)  -transfer experiment at FSU.  Radioactive 14 C beam at energies 8.8, 10.5 and 11.5 MeV was delivered using the special 14 C SNICS source.  Both the 14 C( 6 Li,d) and 14 C( 7 Li,t) reactions at sub-Coulomb energies were used to measure the ANCs of the 6.4 MeV 3 - and 6.2 MeV 1 - states. ANC Techniques and r-matrix analysis Santa Fe, April 2008

20. Spectra of tritons from 7 Li( 14 C,t) reaction at 11.5 MeV of 14 C The sub-Coulomb 14 C( 7 Li,t)  -transfer ANC Techniques and r-matrix analysis Santa Fe, April 2008

21. The sub-Coulomb 14 C( 7 Li,t)  -transfer ANC Techniques and r-matrix analysis Santa Fe, April 2008

22. The sub-Coulomb 14 C( 7 Li,t)  -transfer      at 6.4 MeV  = (1.05+/-0.25)x10 -13 eV ANC Techniques and r-matrix analysis Santa Fe, April 2008

23. 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

24. The 14 C(  ) 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) ANC Techniques and r-matrix analysis Santa Fe, April 2008 At temperatures relevant for 19 F nucleosynthesys in AGB stars the 14 C(  ) reaction rate is totally determined by the strength of the 3 - state.

25. The 14 O( ,p) reaction.  14 O( ,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 14 O( ,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]. ANC Techniques and r-matrix analysis Santa Fe, April 2008

26. The 14 O( ,p) reaction. ANC Techniques and r-matrix analysis Santa Fe, April 2008

27.   (1 - at 6.15 MeV in 18 Ne) = 1.4+/-0.3 eV The 14 O( ,p) reaction. ANC Techniques and r-matrix analysis Santa Fe, April 2008 Reduced width of the 6.15 MeV resonance in 18 Ne and 6.2 MeV resonance in 18 O is assumed to be the same.

28. The 14 O( ,p) reaction.   = 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 B. Harss, et al. PRC, 65 (2002) M. Notani, NPA 746 (2006) Direct 14 O( ,p) measurement Time reverse 17 F(p,  ) measurement

29. The 14 O( ,p) reaction. ANC Techniques and r-matrix analysis Santa Fe, April 2008 S-factor MeV*b E cm (MeV) 1 - at 6.15 MeV 4 + at 7.05 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.

30. The 14 O( ,p) reaction. Based on the results of this work  to proton decay branching ratio for this 1 - resonance at 6.15 MeV in 18 Ne is ~3*10 -5 - not too bad and it is possible to design an experiment which can test this branching ratio directly. Example: 16 O( 3 He,n) 18 Ne(1 - ) 17 F+p 14 O+  ANC Techniques and r-matrix analysis Santa Fe, April 2008

31.  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 17 O was measured and the 13 C( ,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 18 O were measured. The 3 - state provides dominant contribution to the 14 C(  ) reaction rate at ~0.1 GK and the 1 - state is the mirror of the 6.15 MeV state in 18 Ne which is the dominant state for the 14 O( ,p) reaction in explosive environment of x-ray binaries. Its partial alpha width was evaluated with an accuracy of ~30%. Conclusion ANC Techniques and r-matrix analysis Santa Fe, April 2008

32. 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 ANC Techniques and r-matrix analysis Santa Fe, April 2008