Differences between stellar and laboratory reaction cross sections

Nuclear reactions proceed differently in stellar plasmas than in the laboratory due to the thermal effects in the plasma. On one hand, a target nucleus is bombarded by projectiles distributed in energy with a distribution defined by the plasma temperature. The most relevant energies are low by nuclear physics standards and thus require an improved description of low-energy properties, such as optical potentials, required for the calculation of reaction cross sections. Recent studies of low-energy cross sections suggest the necessity of a modification of the proton optical potential. On the other hand, target nuclei are in thermal equilibrium with the plasma and this modifies their reaction cross sections. It is generally expected that this modification is larger for endothermic reactions. We show that there are many exceptions to this rule.

💡 Research Summary

The paper investigates why nuclear reaction cross sections measured in terrestrial laboratories differ from those that occur in stellar plasmas. Two principal effects are identified. First, in a hot plasma the reacting particles are not mono‑energetic but follow a Maxwell‑Boltzmann distribution whose most probable energies are often only a few tens of keV—far below the energies typically used in laboratory experiments. Consequently, reaction rates are governed by the low‑energy tail of the distribution where quantum tunnelling through the Coulomb barrier dominates. Accurate modeling of this regime requires reliable optical potentials for the projectile–target interaction. Recent low‑energy proton‑capture data, however, reveal systematic discrepancies with the standard global optical potentials (e.g., Koning‑Delaroche). The data suggest that the real part of the potential is shallower and the imaginary part stronger at these energies, implying that the proton optical potential must be revised for astrophysical applications.

Second, the target nuclei themselves are in thermal equilibrium with the plasma. At stellar temperatures a non‑negligible fraction of nuclei occupy excited rotational or vibrational states, which modifies the effective reaction cross section because reactions can start from any thermally populated level. In laboratory measurements the target is essentially in its ground state, so the contribution of excited‑state channels is omitted. The thermal population effect is usually expected to be more important for endothermic (negative‑Q) reactions, because the extra internal energy helps to overcome the reaction threshold. The authors, however, demonstrate that this rule has many exceptions. By performing Hauser‑Feshbach statistical calculations for a set of nuclei (including Fe‑56, Ge‑70, Mo‑92, among others) they show that, depending on the structure of the nucleus, the low‑energy optical potential, and the level‑density model, some exothermic reactions experience thermal enhancements that are comparable to or even larger than those for endothermic reactions.

These findings have direct implications for stellar nucleosynthesis modeling. Current astrophysical reaction networks often adopt laboratory cross sections and apply simple temperature‑dependent scaling factors (stellar enhancement factors) without fully accounting for the revised low‑energy optical potentials or the detailed thermal population of target states. The paper argues that a consistent treatment must incorporate (i) a modified proton (and α‑particle) optical potential calibrated to low‑energy data, and (ii) a statistical model that explicitly includes transitions from all thermally populated levels, using temperature‑dependent level‑density prescriptions. Such a combined approach can significantly alter predicted reaction rates for key processes such as the p‑process, s‑process, and certain branches of the r‑process that occur in supernovae, X‑ray bursts, and massive star interiors.

In summary, the authors conclude that stellar reaction cross sections cannot be reliably obtained by a naïve extrapolation of laboratory data. Both the thermal distribution of projectiles and the thermal excitation of targets must be treated with updated low‑energy optical potentials and comprehensive statistical models. This dual refinement is essential for reducing uncertainties in astrophysical nucleosynthesis calculations and for correctly interpreting observed elemental abundances.