electron capture, these reactions are experimentally studied through charge-exchange experiments. One way to study neutrino reactions directly is the production of 12C from the rare isotope 13C using reactor neutrinos. Theoretical studies, however, are needed for most of the nuclei of interest, and their neutrino interactions, building on the shell model treatments that have proven successful in studies of β decays.

In this context, the process of neutrino oscillations may alter the neutrino flavors between the neutrino source and the region where interactions with nuclei occur. This has been recognized and studied in recent years (e.g. Ko et al, 2020) (see recent review by Fischer et al, 2024).

2.6 Nuclear theory complements

The β decay, and even β-delayed neutron emission rates can be measured in detail for a subset of specific nuclei (Kratz, 2001; M¨oller et al, 2003; Kratz et al, 2017). Yet the number of nuclei involved in heavy-isotope reaction paths is large, and many of these nuclei are inaccessible to experiments (see Figure 2). Also in terms of nuclear properties as needed for neutron binding and β decays, there remains considerable uncertainty (Arnould et al, 2007; Lewitowicz and Widmann, 2024). Nuclear fission becomes important at the high-mass end, setting a limit to the reaction flow of the r process beyond masses of 260, but difficult to constrain by experiment (Giuliani et al, 2020). Theoretical models are needed to describe nuclear properties in the wider region of thousands of nuclei that are expected to exist, and to extrapolate beyond measured nuclei (Cowan et al, 2021).

FRIB reach

r-process

Fig. 5 Comparison of nuclear binding models away from the region of stable nuclei and towards the

Figure 9. Two-neutron separation energies (S2n) of Gd isotopes obtained in various

neutron drip line. Two-neutron separation energies are shown for Gd isotopes, comparing different mass models often used in r-process simulations shown relative to DZ mass formula

mass models (see text) with experimental values as represented by AME2016 (from Horowitz et al predictions. The experimental values from AME2016 [399, 400] are also shown. The

The original description of nuclear binding energies (through determining their masses)ofwasknownmademasses.anAnotherensemblemeasure(macroscopic)of mass model’seffectperformancethrough theis itsliquidab lity-dropto model

reproduce important bindingerencesenergysuchdi as neutron separation energies or

Q-values for β-decay. For such quantities, the model performance is often significantly better compared to masses as many systematic22 theoretical errors are expected to cancel out.

Figure 9 shows a typical example of the current theoretical situation, using the two-neutron separation energies S2n of the Gd isotopes and mass models often used for r-process simulations. In general, mass predictions are consistent where experimental data are available.

A few mass units past the last measured point, as one enters the region of interest for the r-process,erencesdi between models increase significantly. Nevertheless, discrepancies stay within about 800 keV for about 24 mass units across the r-process path. Around the N = 126 shell closure and towards the neutronerencesdripline, di between S2n values become much larger and can reach up to 2 MeV. While this may provide some indication of the expected uncertainties, it is not necessarily a reliable measure. Uncertainties may be overestimated because of one low-performing model, or uncertainties may be underestimated because not all models shown in Fig. 9 are based on similar phenomenology near stability. An indication for the latter is the good agreement between mass models near stability, despite large discrepancies with data. Such model

(Weizs¨acker, 1935). Since then, there had been an impressive evolution of theories (see M¨oller, 2023, for a personal review). A bottom-up approach starting from the nucleonnucleon forces (microscopic approach) was pursued, in parallel to efforts aiming at an establishment of an empiric energy density functional (macroscopic approach) (Warbinek et al, 2024; Ryssens et al, 2023). Nuclear mass models are a compromise between empirical descriptions of macroscopic (and more-easily measurable) components and microscopic descriptions of nucleon-nucleon interactions deriving from fundamental theory or from empirical prescriptions. Nuclear theory attempts to advance models in specific regions of interest, while astrophysical nucleosynthesis models require consistent treatment of neutron captures and β decays across the entire path of reactions; therefore, the FRDM baseline from 1995 is still widely used, as β decays have been evaluated for a large number of nuclei here. Nuclear mass models evolved from the macroscopic finite range liquid drop model (FRDM) (M¨oller et al, 2016) to Hartree-Fock Bogoliubov (HFB) model versions of different sophistication levels, adding corrections to nucleon-nucleon forces with shell and deformation effects to a Fermi gas model for nuclei, among others (see Goriely et al, 2022, for testing model alternatives). Towards the higher range of nuclear masses, nuclear deformations play a large role. The additional degrees of freedom for nuclear rotation and oscillations provide a larger phase space, thus making predictions of effective nuclear binding quite uncertain. Experimental results are analyzed and fitted to descriptions of mass models, thus determining model parameters; often models have a large number ( 30) of parameters determined in this way. As a measure, the 2-neutron separation energies S2n between models and measurements (where available) still show an RMS difference of 800 keV (see Figure 5), while reliable astrophysical r-process calculations would require a precision of order 100 keV (Horowitz et al, 2019). Weak-interaction rates can be estimated theoretically in the QRPA approach (Sarriguren, 2017). But correspondingly, also the weak-interaction processes in atomic nuclei (β-decay rates, radioactive lifetimes) cannot be reliably calculated in theoretical models, if the details of shells and nuclear structure are not covered; shell models have proven to provide more realistic rates, in better agreement with experimental values (see Langanke and Mart´ınez-Pinedo, 2003; Grawe et al, 2007, for discussions of weak interactions in cosmic environments). As agreements with experimental values are unsatisfactory, phenomenological models are widely used. This also extends to the determination of fission barriers, correspondingly (Erler et al, 2012; M¨oller and Schmitt, 2024).

Due to the large number of nuclei and nuclear-interaction parameters, also machine learning has been employed. For extrapolations into regions of unknown nuclei this may be inappropriate, as the learning algorithms can be trained and thus constrained on available data only (Mumpower et al, 2023). Further advancement of the underlying model and its understanding remains a promising road in any case.

For a better understanding of nucleon-nucleon interactions and the properties of nuclear matter, also astronomical observations are exploited (Sun et al, 2008; Lattimer, 2021) (see discussions of neutron star observations below).

Our limitations of nuclear-physics knowledge incur uncertainties in astrophysical calculations of nucleosynthesis processes whenever nuclear reaction rates need to be

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