The accurate nuclear data of deuteron-induced reactions are most important for selecting and validating the structural materials and the major technologies to be taken into consideration within the nuclear design of ITER, DEMO, and IFMIF. Requested deuteron activation cross-sections are of significant interest for shielding design as well as the radiation damage estimation including the hydrogen, tritium, and helium emission. They are related to “gas bubble accumulation,” which goes through the surface swelling effects and finally deteriorates the properties of the material [18].

However, the systematics of deuteron activation cross-sections related to gas accumulation, e.g., (d, p), (d, 2p), (d, t), and (d, α), is modest compared to the case of neutrons. At the same time, evident discrepancies have been found even between the newest evaluation predictions, e.g., TENDL-2021 [10], in respect to the existing data. They pointed out neglected peculiarities of the deuteron interaction process, as well as the need to complete the theoretical frame of deuteron–nucleus interaction analysis. Thus, the non-compound processes of direct interactions (DIs), namely, breakup (BU) and direct reactions (DRs), should be considered in addition to the pre-equilibrium emission (PE) and evaporation from the compound nucleus (CN). Neglect of these reaction-mechanism contributions to the deuteron activation cross-section estimation triggers the current discrepancies of the evaluation predictions.

The distinct reaction mechanism of the deuteron interactions is the deuteron breakup into its nucleon constituents, the so-called breakup nucleons. Its complexity is increased by the addition to the primary deuteron–target nucleus interaction of the nuclear reactions induced by the breakup nucleons ([19] and Refs. therein). Moreover, the larger the target nucleus mass and charge, the higher the importance of the deuteron BU, which becomes dominant for heavy target nuclei at deuteron incident energies, particularly around the Coulomb barrier [20].

The physical picture of deuteron breakup in the Coulomb and nuclear fields of the target nucleus concerns (i) the elastic breakup (EB), in which the target nucleus remains in its ground state and none of the deuteron nucleons interacts with it, and (ii) the inelastic breakup or breakup fusion (BF), where one of these deuteron constituents interacts non-elastically with this nucleus. On the other hand, the deuteron breakup processes have two opposite effects on the deuteron activation cross-sections that should be considered equally.

First, the deuteron total-reaction cross-section σ_R , which is shared among different outgoing channels, is reduced by the value of the total (EB+BF) breakup cross-section σ_BU . On the other hand, the BF component contributes to different reaction channels through breakup-nucleon interactions with the target nucleus. Thus, the absorbed proton or neutron following the deuteron breakup contributes to the enhancement of the corresponding (d, xn) or (d, xp) reaction cross-sections, respectively. The importance of the BF enhancement has been particularly found for the case of the second and third chance-emitted particle channels [19–23].

Empirical parametrization ([19] and Refs. therein) of both the total breakup-nucleon emission f B U n / p = σ B U n / p / σ R and EB f_EB =σ_EB /σ_R fractions finally provides the BU cross-sections under the assumption of equal neutron- and proton-emission breakup cross-sections. The difference f B F n / p = f B U n / p -f_EB yields the BF fraction as well. A former 2016 implementation of the IFIN-HH deuteron BU formalism in TALYS (as keyword value breakupmodel = 2) [11] has been updated, also including the enhancement of deuteron-induced reaction cross-sections due to the breakup-nucleon-induced reactions [19]. The proper account of this enhancement has been obtained using the evaluated data of TENDL-2019 [10] for the nucleon-induced reactions.

Thus, improvements in the nuclear model code capabilities to extend the evaluated deuteron-induced reaction cross-sections of Mn [22] and all stable isotopes of Zr [21] and ^95−98,100Mo [23], in view of a final work concerning natural Mo, have been considered by analyzing all data up to 50 MeV. A sample case is shown in Figure 1 for deuterons on the heaviest ¹⁰⁰Mo isotope. One should note that DI and PE+CN fractions oscillate around half of σ_R (gray-shaded area). Thus, the key role of the direct interactions [19, 21, 22], neglected so far, is emphasized.

The deuteron interaction with low- and medium-mass target nuclei, at incident energies below and around the Coulomb barrier, proceeds largely through stripping (d, p) and (d, n), pickup (d, t), and (d, α) DR mechanisms. Then, PE and CN become important at higher energies ([21] and Refs. therein). The assessment of the (d, p) and other direct reactions’ contributions not included within TALYS output is carried out using the distorted-wave Born approximation (DWBA) method within the code FRESCO [25] and available data for spectroscopic factor establishment. There have been involved, in this respect, the post/prior form distorted-wave transition amplitudes for stripping and pickup reactions, respectively, as well as the finite-range interaction. Moreover, the population of the discrete levels available within the ENSDF library [26] was considered, while the analysis of the available proton angular distributions provided the corresponding spectroscopic factors ([19] and Refs. therein).

The DR contributions are essential for describing the measured excitation functions corresponding to the first chance-emitted particles. A comparative analysis of the experimental (d, p) and (d, 2p) excitation functions, the model calculations, and the evaluation predictions has emphasized the role of deuteron BU and stripping reactions for the hydrogen gas accumulation process [23]. Thus, the deuteron-related enhanced version TALYS-1.96, along with this continuous code development (see [27], pag. 20 for the most important updates), could be used to obtain deuteron reaction data of increased accuracy [19]. Furthermore, requirements within the corresponding validation activities at ENS/EUROfusion [7, 28] could be also met. However, an additional increase due to the stripping, (d, p) and (d, n), pickup, (d, t) and (d, α), direct reactions should be carried out using FRESCO, anymore [27].

Nevertheless, a specific point of consistent model calculations is the involvement of the same model parameters within different reaction mechanisms, e.g., the same OMP parameters for the calculation of breakup cross-sections, distorted waves in the ingoing/outgoing channels of DR, PE transition rates, and transmission coefficients of various ingoing/outgoing CN channels. The overall agreement between the measured data and model calculations [19] sustains the theoretical frame of reaction mechanisms taken into account for the deuteron–nucleus interaction, emphasizing the key role of direct interactions, i.e., the breakup and direct reactions. Finally, the best point of the consistent theoretical frame associated with the analysis of the deuteron–nucleus interactions through advanced codes is the improved predictability where no data exist. On the other hand, requirements for new measurements for the completion of the large gaps of (d, 2p) data on specific nuclei along the priority list of candidate materials for ITER/IFMIF [1] are obvious.

The α-particle optical model potential (OMP) that was obtained previously by analyzing α-particle elastic scattering and induced reactions on A ≈ 45–209 nuclei, at energies ≤50 MeV [29], received another confirmation [30] by considering the α-induced reaction data published within the last decade. The use of consistent parameter sets, i.e., already validated by the analysis of other independent data that are distinct from α-induced reaction or α-emission cross-sections, has made possible the suitable data account in terms of the CN and PE models. An obvious correlation exists, however, between the accuracy of the independent data, the input parameters determined by their fit, and final uncertainties of the calculated reaction cross-sections.

On the other hand, no further empirical rescaling factors of the γ and nucleon widths have been used, in order to avoid compensation effects of less-accurate model parameters, while the former are still mandatory within large-scale nuclear data evaluations. Moreover, a suitable description of all competitive reaction channels, in addition to the α-emission of interest, should also be aimed at the validation of consistent parameter sets. At the same time, the analysis should also concern the available data not only for one of the stable isotopes of the elements other than the mono-isotopic ones in nature but also for whole isotopic chains and neighboring elements.

Then, the α-emission from excited nuclei in nucleon-induced reactions on A ∼60 [31, 32] and A ∼90 [33] nuclei has also been described by the same α-particle OMP. Similar usage of consistent parameter sets, considering all competitive reaction channels, and eventually isotopic chains of neighboring elements, has been considered for the α-emission and for α-particle-induced reaction analysis. The main drawback that had to be overcome was the underestimation of the α-emission cross-sections [34, 35] by taking into account OMPs settled by the suitable account of α-particle elastic scattering and induced reactions [29, 36–38].

The additional consideration that should be given to the pickup DR, leading to an increase in the α-emission beyond the CN+PE predictions, has been shown more recently [31, 32], while its suitable account at the Giant Quadrupole Resonance (GQR) energies of ^55,57,58Fe and ^59,61,62,65Ni (Figures 2A–C) has been attributed to a like-GQR component. An alternate solution has been, thus, provided to the so-called α-potential mystery [39] to describe the absorption and emission of α-particles of large interest for astrophysics and fusion technology.

The same analysis for the stable isotopes of Zr, Nb, and Mo becomes challenging, with Zr nuclei within both incident and emergent reaction channels. Actually, the α-emission in neutron-induced reactions on stable Mo isotopes made the object of an earlier systematic investigation up to 20 MeV [40] using α-particle OMP [41] describing A ∼60 compound-nuclei α-particle decay. A significant underestimation at incident energies around ≤10 MeV for ^92,98Mo was obtained at that time. This OMP makes predictions that differ significantly from potentials for incident α particles [42] including a double-folding model (DFM) of microscopic real potential [29, 34, 36–38]. The discrepancy in the results corresponding to OMPs [36, 41] even led to the assumption that nuclear density distribution is dependent on nuclear temperature [34, 35].

The α-emission cross-sections larger than CN+PE predictions, distinct from the data error bars, have finally been described using additional pickup DR and like-GQR decay of excited nuclei by neutrons on Zr, Nb, and Mo stable isotopes [33]. The pickup contributions to (n, α) reactions have been calculated through the DWBA method using the code FRESCO [25]. The one-step reaction takes place through the pickup of the ³He cluster, while the “spectator model” [43] for the two transferred protons in the (n, α) reaction has been taken into account. However, the lack of a measured α-particle angular distribution for the (n, α) reactions within this work made only straightforward DWBA calculations of related pickup cross-sections possible. Thus, the spectroscopic factors (SFs) of Glendenning [44] have been used for the spectator proton pair [43], in addition to SFs for the picked neutron. Thus, the latter become responsible for the angular momentum transfer. They were obtained through the analysis of α-particle angular distributions of one-nucleon pickup reactions ( H e 3 , α ) or (t, α) toward the residual nucleus of interest.

A description of both the absorption and emission of α particles, thus, becomes possible by the same potential [29], in support of its use for large-scale nuclear data evaluations as the TALYS corresponding default option. In fact, this conclusion could be found within the previous analysis for Zr [35] and Mo [34], before DR and like-GQR decay consideration, due to outstanding cross-section measurements for ⁹¹Zr(n,nα)⁸⁷Sr ^m [45] and ⁹⁵Mo(n,α)⁹²Zr [46, 47] reactions. At the same time, it is suggested that the simple OMP [41] could include, beyond the CN contribution, the additional ones that only now receive full consideration. Nonetheless, further measurements at incident energies corresponding to GQR energies of excited nuclei, as well as heavier isotopes of elements, may shed light on the eventual like-GQR α-emission.

In the meantime, a former assumed overestimation of a first direct measurement for ⁵⁹Cu(p,α)⁵⁶Ni reaction cross-sections by a factor of 2 at an energy of ∼6 MeV [48] was also solved. Due consideration of proton OMP anomalies at sub-Coulomb energies for medium-weight nuclei has been shown to be critical for the analysis of this unprecedented measurement [49]. So, the usefulness of using consistent parameter sets established by a former analysis of independent data has been shown once more.

Thus, we have shown [50] that variation in predicted cross-sections from standard statistical model calculations and cross-section range related to the anomalous proton imaginary potential depth, for target nuclei off the line of stability, is distinct and well separated (Figure 2D). Consequently, the new measurement [48] provides, under unique conditions, tests of proton isoscalar and isovector real-potential components, the anomalous imaginary potential, and the alpha-particle OMP [29], for nuclei off the line of stability. Moreover, it has, thus, been completed for the similar α-emission account by this OMP [29] for Cu stable isotopes [31] at once with all α-induced reactions on Ni stable isotopes [51].