Astrophysical S(E)-factor of the 15 N(p,α) 12 C reaction at sub-Coulomb energies via the Trojan-horse method Daniel Schmidt, Liberty University Cyclotron Institute —Texas A&M University Mentor: Dr. Akram Zhanov

The CNO Cycle in Stellar Nucleosynthesis 12 C + p → 13 N + γ 13 N → 13 C + e + + ν 13 C + p → 14 N + γ 14 N + p → 15 O + γ 15 O → 15 N + e + + ν 15 N + p → 12 C + α In this branch of the CNO (carbon, nitrogen, oxygen) cycle, 12 C acts as a nuclear catalyst which allows the fusion of four protons into a 4 He nucleus (or α particle).

The CNO Cycle in Stellar Nucleosynthesis The final step of the CNO cycle, 15 N + p → 12 C + α (also written 15 N(p,α) 12 C) is the subject of this research. When a proton impinges on the 15 N nucleus, it is repelled by the Coulomb force from the seven other protons, and thus it must have sufficient energy to overcome this “Coulomb barrier” of 2.14 MeV if it is to react. At energies typical of stellar interiors, protons have energies less than 100 keV. The reaction still occurs, but it depends on quantum tunneling. Accelerator experiments can reproduce the 15 N(p,α) 12 C reaction at energies E> 70 keV, but at astrophysically relevant energies the reaction proceeds too slowly to be measured directly in the lab.

The Trojan horse method An indirect technique, the Trojan horse method, can be used at low energies. In this method, a deuteron impinges on a 15 N nucleus. After penetrating the Coulomb barrier, the deuteron breaks, dividing its energy between the proton and neutron. In some cases, the neutron will exit the scene with most of the deuteron’s energy, leaving a low-energy proton inside the Coulomb barrier to react with the 15 N nucleus. By selecting those events in which the neutron leaves with high energy, we can study the reaction at low energies without suppression from the Coulomb barrier.

The Trojan horse method α 12 C d n 15 N 16 O p

Comparison of Data Unfortunately, the data from these two experimental methods (direct and Trojan horse) do not agree.

Correcting the Trojan Horse Data The next task is to see if we can find a theoretical correction to the Trojan horse data that will bring the two into agreement. Differences between the two methods include the following: –The direct data is affected by the Coulomb barrier, while the Trojan horse data is not. –The direct data is also affected by a centrifugal barrier, so that final states with angular momentum l = 0 are preferred. Again, this does not affect the Trojan horse data. –In the direct method, the proton is a real particle, whereas in the Trojan horse method, the proton transferred from d to 15 N is an “off energy shell” or virtual particle (E ≠ p 2 /2m). –The most significant differences are the off-shell effects.

Correcting the Trojan Horse Data The cross section for a resonant reaction is given by the well - known Breit-Wigner formula. πλ 2 represents the geometrical cross section. (2J+1)/ (2J 1 +1)(2J 2 +1) is a statistical factor, where J 1, J 2 and J, represent the angular momenta of the projectile, target and compound nucleus, respectively. Γ a, Γ b are the partial widths of formation and decay, respectively, of the compound nucleus, and Γ = Γ a + Γ b. E R is the energy of the resonance.

Correcting the Trojan Horse Data The R-matrix predictions for the direct method are shown below for reference. Note that the agreement with the data is very good except in the high-energy tail.

Correcting the Trojan Horse Data The Breit-Wigner formula cannot correctly reproduce the Trojan horse data, since off-shell effects are involved. We can modify the formula by replacing Γ a with a form factor that takes these other effects into account.

Correcting the Trojan Horse Data In the case of 15 N(p,α) 12 C there are multiple resonances involved, as well as subthreshold states, so we have:

Correcting the Trojan Horse Data The predictions for the Trojan horse method are shown here. The data indicates a significantly wider resonance, with a peak at slightly higher energy, than was predicted. (This is a graph of S(E) = E e 2πη σ(E), the “astrophysical S-factor”)

Checking the Method on a Simpler Reaction It is beneficial to check a second reaction to see if a similar discrepancy exists there. The reaction 3 He(d,p) 4 He is a good case study, since there is only one significant resonance in this case, and interference effects can be ignored. We first see that the direct and indirect data do not agree. We can apply theoretical corrections to the data as follows: –Find a polynomial fit to the direct data. –Divide this function by the width of the resonance, and multiply by the form factor discussed above. –This gives a prediction for the indirect method.

Checking the Method on a Simpler Reaction Interestingly, the data still do not fit the predictions and the difference is similar to the case of 15 N(p,α) 12 C. (Shown here are both linear an logarithmic plots.)

Conclusions More work is needed to determine the source of this error. In the mean time, however, the Trojan horse method may still be used fairly accurately in astrophysical studies, since it is reasonably consistent with the direct method at low energies. It is this region that is of interest in stellar evolution and nucleosynthesis. Diagram courtesy M. La Cognata et. al. Published in: Phys. Rev. C (2005)

This research was supported by NSF grant PHY , the Texas A&M Cyclotron Institute, and the Department of Energy. Special thanks to Dr. Akram Zhanov and the Cyclotron Institute staff.