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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) atransfer 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 dA a A d I ab S ab ab C ab W r S ab b ab2 ab b ab C 2 ab aA Bn 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 E2 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) 2R C 2 dS W 2 ( R) dE a3 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) ANC Techniques and r-matrix analysis 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