1. 15 N Zone 8 Zone 1 Zone 28 p Zone 1 Zone 28 16 O Zone 1 Zone 4 Zone 8 Zone 28 15 N 16 O p Reaction rates are used to determine relative abundance of elements in the CNO cycle. Because the theoretical S(E) factor is lower than experimental data at lower energies, theoretical reaction rates (RRTotal) are also lower than experimental (NACRE). At 0.3 T 9, reaction rates increase due to a narrow resonance. More resonances contribute to the hot CNO cycle—above 0.1 T 9. New measurements at low energies and new calculations of reaction rates should consider all resonances. (NACRE) C. Angulo et al., Nucl. Phys. A 656 (1999) 3-183. C. Rolfs and W. S. Rodney, Nucl. Phys. A235 (1974) 450. Rolfs and Rodney (RR) measured at higher energies and extrapolated data to zero energy. From their direct measurements, the S(E) factor is contributed by two J  =1 - resonances at E cm =317 keV and 964 keV and nonresonant captures. Their result was very sensitive to direct capture; the direct capture amplitude was varied arbitrarily to fit the data. They conclude 1 leak per 880 CN cycles. S(0 keV)=64 ± 6 keV b D.F. Hebbard, Nucl. Phys. 15 (1960) 289. Hebbard measured an S(E) factor contributed by two resonances at E cm =317 keV and 947 keV. The experimental data were renormalized at the second resonance with 42% error, contributing 18% uncertainty at lower energies. Hebbard concludes 1 leak per 2200 CN cycles. S(25 keV)=32 ± 5.76 keV b Why the stars shine Energy production and nucleosynthesis in stars Hydrogen burning in the stars produces energy by two processes, with the same net result 4p  4 He + 2e + + 2 + Q (Q=26.73 MeV). The fusion of hydrogen into helium fuels the luminosity of the stars, and produces energy to make the stars shine. In very old first-generation stars, energy is produced by burning hydrogen via the proton-proton (p-p) chain. In second-generation stars, energy is produced by hydrogen burning of heavier elements in the CN cycle. Carbon and nitrogen are heavier elements than helium with relatively small Coulomb barriers and high abundance, compared to heavier elements. Catalytic material is lost from the process via the leak reaction 15 N(p,  ) 16 O. Subsequent reactions restore the catalytic material to the cycle, creating oxygen-16 and heavier elements. The nucleosynthesis of heavier elements can be explained by the relative abundance and reaction rates of CN elements. The important leak reaction 15 N(p,  ) 16 O is the subject of this research. Rolfs, C., and Rodney, W.S., Cauldrons in the Cosmos (The University of Chicago Press, Chicago, 1988). Interference Interference between resonances and direct capture amplitudes depends on selection rules – angular momentum and parity conservation laws. Total angular momentum is the sum of channel spins and relative orbital angular momentum. Resonances and direct capture amplitudes interfere if they have the same quantum numbers in the initial state. Resonances For the reaction 15 N(p,  ) 16 O: 15 N has spin ½, proton has spin ½, and relative orbital angular momentum l i is 0. The angular momentum J f of the resonant state 16 O is 1. 15 N(1/2) + p(1/2) + l i (0) = J f (1) 15 N has parity -, proton has parity +, and parity of l i is +. Parity of 16 O is -. (-) (+) (+) = (-) Mechanics The leak reaction involves proton capture. An impinging proton experiences two forces: the long-range electromagnetic force (Coulomb potential) outside the nucleus, and the short-range strong force (nuclear potential) inside the nucleus. In classical mechanics, E=T+V, and thus r cl is the classical closest approach distance. In quantum mechanics, due to the tunnel effect, a proton can penetrate the Coulomb barrier with energy E p <E c. At astrophysical energies, the Coulomb barrier is the main obstacle for the reaction to occur, and the probability to penetrate the barrier is small. Proton Capture Nonresonant reactions: In nonresonant (direct-capture) reactions, a proton goes directly to the state in the final compound nucleus, and  -radiation is emitted. This process can occur for all proton energies. Resonant reactions: In resonant reactions, an excited state of the compound nucleus is first formed, and then  -decays to the final compound state. This is a two- step process which occurs at fixed proton energies. p(p,e +,ν)d d(p,γ) 3 He 3 He( 3 He,2p) 4 He 3 He(α,γ) 7 Be 7 Be(e -,ν) 7 Li 7 Li(p, α) 4 He 7 Be(p, γ) 8 B 8 B(e +, ν) 8 Be 8 Be(α) 4 He 86%14%.02% Proton-proton chain CNO Cycles (p,γ)(p,α) (e+,ν) (p,γ) (p,α) (p,γ) (e+,ν) I II III (p,α) IV (p,γ) (e+,ν) (p,γ) (e+,ν) IMPORTANT LEAK REACTION Astrophysical Factor for the CNO Cycle Reaction 15 N(p,γ) 16 O Adele Plunkett, Middlebury College REU 2007, Cyclotron Institute, Texas A&M University Mentor: A.M. Mukhamedzhanov Radius Energy p Coulomb Barrier Nuclear Potential Direct Capture Resonant Capture Experimental S(E) factor for 15 N(p,γ) 16 O Energy Levels 16 O Our Analysis In our analysis, the astrophysical S(E) factor is contributed by two resonances AND direct captures. The S(E) factor is determined by asymptotic normalization coefficients (ANCs) and resonant parameters. The ANCs were recently measured in Prague by Nuclear Physics Institute, Czech Republic; Catania National Lab, Italy; and Cyclotron Institute, TAMU, USA. The resonant parameters were fit for both channels 15 N(p,  ) 12 C and 15 N(p,  ) 16 O. J i =J f Astrophysical S(E) Factor The cross section  is the effective geometrical area of a projectile and target nucleus interaction, and it is directly related to the probability of a reaction. For projectile energies below the Coulomb barrier, such as astrophysical energies, the cross section drops by many orders of magnitude with decreasing energy. We use the astrophysical S(E) factor in our theory to extrapolate to lower astrophysical energies, essentially to zero energy. The astrophysical S(E) factor has units [EnergyArea]. Fit of Direct Data Resonant Parameters To find the resonant parameters, we fit direct data for 15 N(p,  ) 12 C with two interfering J  =1 - resonances at E cm =312 keV and 960 keV. Direct data is from A. Redder et al., Z. Phys. A305, 325 (1982). keV eV Rolfs1.1100984512±232 ±5 Hebbard1.1100934012.888 Theory1.19590458.740  R-Matrix Approach In the R-Matrix approach, the configuration space is divided into an internal and an external region, divided at a radius r 0. The nuclear parameters inside combine resonant and nonresonant capture. The collision matrix outside includes only nonresonant capture from r 0  ∞. The advantage of the R-Matrix approach is the essential independence of astrophysical results on the radius r 0 division of configuration space. S(E) Factor Expression for 15 N(p,γ) 16 O capture amplitude (through resonance i) proton partial resonant width radiative width total width remove the Gamow factor from the cross section  direct capture amplitude to bound state i kinematical factor Fitting the Data for 15 N(p,γ) 16 O  are fixed by the resonant peaks - - - nonresonant part ■ Rolfs, C., and Rodney, W.S., Nucl. Phys. A235 (1974) 450. ▲ D.F. Hebbard, Nucl. Phys. 15 (1960) 289. Considerations We consider factors which may contribute to discrepancies with experimental and theoretical data. Stopping power and straggling has little effect on proton energy when a thin target is used. The Rolfs & Rodney (RR) analysis contains inconsistent use of lab and center of mass systems. Systematic uncertainties of resonant S(E) factors in the RR analysis are not given. We estimate 15% uncertainties. The RR analysis theory is not specified, and a significant overestimation of the nonresonant capture may be due to incorrect division of configuration space; channel radius is not specified. Conclusions The calculated astrophysical factor S(0 keV)=38 keV b. The result does not depend on channel radius. The ratio of S(E) factors for 15 N(p,  ) 12 C and 15 N(p,  ) 16 O is 1436:1. Our analysis calls for new measurements of 15 N(p,  ) 16 O. Reaction Rates for 15 N(p,γ) 16 O DC Element abundance in the Sun www.nucastrodata.org C. Angulo, Nucl. Phys. A656 (1999) 3-183. from 15 N(p,  ) 12 Cfrom 15 N(p,  ) 16 O