Evaluation of light-element reactions in the resolved resonance region

. Light-element reactions at low energies in the resolved resonance region are important for a range of applications in basic and applied sciences including nuclear reactors, nonproliferation, cultural heritage, forensics and environmental control, rare event investigations and nuclear astrophysics. In this paper, we report on an e ff ort to evaluate charged-particle cross sections in the resolved resonance region and produce evaluated nuclear data files for further processing and inclusion in evaluated data libraries. We discuss the open issues in R-matrix calculations as we extend to higher energies, such as dealing with the rapidly growing number of open channels and merging with the regime of smooth cross sections described by the statistical model, and present attempts to address these issues in neutron-induced reactions relevant to nuclear reactor applications.


Introduction
Charged-particle-induced reactions at low energies in the resolved resonance region are important for a range of applications in basic and applied sciences. Ion Beam Analysis of materials for cultural heritage, environment and climate control, and forensics, depends on the knowledge of proton-, deuteron-and alpha-induced reactions at energies of a few MeV. Management of fuel in nuclear reactors involves the control of neutrons produced after the reactor operation is shutdown. For the most widely used fuel materials, UO2, UF6, PuF4 and PuO2, the dominant neutron producing reactions are (α,xn) reactions on isotopes of O and F, at incident energies above the neutron emission threshold. In nuclear astrophysics, main stellar processes producing the energy of the stars and leading to the synthesis of the light and medium-mass elements up to the iron nuclei, are fuelled by thermonuclear reactions at temperatures of tens of millions of degrees Kelvin. These conditions simulated in the earth laboratories translate into charged-particle induced reactions on light and mediummass nuclei at energies of a few hundreds of keV. Alphanucleus reactions up to 10 MeV are a potential source of low-energy neutron background for rare event research in astroparticle physics (dark matter search, neutrino, etc.) * e-mail: P.Dimitriou@iaea.org while they are also considered important in studies of energy retainment by the plasma produced in the fusion reactors.
Significant effort has been made over the past decades to measure some of these cross sections. The Ion Beam Analysis Data Library (IBANDL) [1] maintained by the IAEA contains over 6000 data sets of differential and total experimental cross sections for chargedparticle-induced reactions in the energy region below several MeV. In nuclear astrophysics, several dedicated compilations (NACRE-I and II) and theoretical reaction rate databases (REACLIB [2], BRUSLIB [3], KADONIS [4], NUCASTRODATA.ORG [5]) have been made available to meet the needs of the stellar nucleosynthesis calculations. On the other hand, the evaluated nuclear data libraries maintained by national or international coordinated efforts (ENDF, JEFF, JENDL, CENDL) are to date, incomplete as far as charged-particle-induced reactions in the resolved resonance region are concerned. For example, (α,xn) evaluated cross-section data are provided in two evaluated libraries only, JENDL-AN [6] which was produced in 2005, and the TENDL libraries (version 2021) [7].
Neutron-induced reactions on light elements in the resolved resonance region are also studied because of their relevance in nuclear criticality and safety studies. Four light systems have been identified as priorities for these studies: 9 Be, 14,15 N, 23 Na. Some of the outstanding issues in the evaluation of light elements in the energy range from a few keV to 20 MeV include: (i) lack of experimental data or discrepancies in experimental data, (ii) implementation of R-matrix algorithms at higher energies where many channels open up and/or three-body decays occur, (iii) connecting the resolved-resonance region with the unresolved resonance region and statistical model regime, (iv) treating uncertainties and producing covariance matrices, and (iv) processing of these evaluated data by the currently available processing codes.
In the following sections, we discuss our efforts to (i) evaluate charged-particle reactions in the resolved resonance region with the existing widely used R-matrix codes, (ii) extend the R-matrix algorithm to higher incident energies with a rapidly growing number of open channels, and (iii) connect the resolved and unresolved or statistical regime in neutron-induced reactions on 14 N.

Charged-particle-induced reactions
The capabilities of existing R-matrix codes to perform charged-particle R-matrix analysis has been assessed by means of an inter-comparison of the R-matrix algorithms implemented in the codes [8]. The following R-matrix codes have been used in these exercises: AMUR [9], AZURE [10], CONRAD [11], EDA (LANL), FRESCO [12], GECCCOS (TUW), RAC (Tsinghua), SAMMY [13]. Furthermore, the Ferdinand code was developed to ensure translatability of the corresponding Rmatrix calculations. Two additional exercises have been performed, one on minimization techniques used in the different codes and the other one on the evaluation of the 7 Be system. The former evaluation includes the two incident channels 3 He+ 4 He and p+ 6 Li and excitation energies up to 20 MeV. The details of the exercises and comparisons of the results can be found in Refs. [14][15][16][17]. To date, a complete evaluation up to incident energies of 30 MeV, taking into account all the available experimental data for all open channels including photon channels, has been obtained with the code RAC using the reduced Rmatrix formalism as implemented in the code [18]. Figure 1 shows comparisons of the RAC evaluation for the two above-mentioned channels with experimental excitation functions at two given angles.

Reduced R-matrix method
With increasing excitation energies, the number of open channels increases rapidly leading to very large dimensions of the R-matrix that needs to be solved, and hence to an exceedingly large number of parameters that needs to be determined, probably in the thousands or tens of thousands. Most of these parameters cannot be determined due to the limitations of the available experimental data, therefore, one needs to find ways to reduce the dimensions of the R-matrix. A practical solution to the problem was proposed by Lane and Thomas in their seminal paper [19]. The method consists of eliminating those channels that cannot be determined due to lack of experimental data. The R-matrix is replaced by a reduced R-matrix whose dimensions are comparable to the number of parameters that can be determined from the available data.
The reduced R-matrix formalism proposed by [19] has been implemented in the RAC code [18] and has been applied to the evaluation of light composite systems with remarkably good results as shown in [18]. The eliminated channels are chosen on the basis of non-available experimental data and are lumped together in an imaginary pole whose parameters are determined by adjusting to all available experimental data using a generalised least squares method. Figure 1 in the previous section demonstrates the goodness of the R-matrix fit.
An alternative approach to implementing the reduced R-matrix idea has been proposed, whereby again the reduced R-matrix is complex and the imaginary part describes the flux loss into the eliminated channels. A simplified expression derived for the one-pole approximation suggests that the essential ingredient, which is the Lfunction, allows one to account for the thresholds of the eliminated channels properly. It is also possible to estimate the cross section associated with the eliminated channels. The approach has been applied to the n+ 6 Li system and in Fig. 2 one can see the calculated breakup channel which is the eliminated one: n + 6 Li → α + n + d. Fair agreement with experiments at neutron energies between 0.1 and 6 MeV was obtained for the explicitly considered channels, i.e. elastic scattering and n + 6 Li → α + t. The results are preliminary.

Linking resolved-resonance and statistical regimes
Linking the resolved with the unresolved resonance region and eventually with the region of overlapping resonances described by the statistical model is an overarching goal in nuclear data evaluation as it ensures continuity, cohesion, and consistency of the evaluated data over the whole energy region from a few keV to [20][21][22][23][24][25][26][27][28][29][30] MeV. An attempt is made to step-by-step arrive at the Hauser-Feshbach (HF) [20] statistical theory results starting from the R-matrix theory in a transition region of unresolved resonances where both theories are applicable. This implies that the R-matrix parameters will by some manipulation be able to reproduce the optical model results. The approach is applied to n+ 14 N as the compound system has known levels and level densities, and adequate experimental data for the two channels that are open at incident energies up to 20 MeV: inelastic (n,n ′ ) and (n,α). In practice, a full R-matrix calculation including all open channels up to the excitation energy is required.
The reaction cross section for channel α to channel α ′ can be expressed in terms of the scattering matrix S αα ′ and also in terms of the average widths <Γ α >. Starting from a full R-matrix fit to the experimental data, the R-matrix parameters are used to calculate the S αα ′ terms and the average width < Γ >. These are then compared with the corresponding terms obtained from HF calculations using the Koning-Delaroche optical potential [21], and known levels from the Reference Input Parameter Library (RIPL) [22]. The results are compared in the region where the cross sections from the R-matrix fit and HF calculations are similar. In these energy regions, width fluctuation corrections are non-negligible and therefore, they have been included in the calculations. Different methods for estimating the partial widths <Γ α > with respect to the scattering amplitude |S αα ′ | have been tested.
The HF calculations including width fluctuation corrections are in reasonable agreement with the R-matrix calculations based on the <Γ> ≃ log(|S| 2 ) method as shown in Fig. 3. In this figure, HF and full R-matrix calculations are compared for the two open channels, (n,α) and (n,n 2 ). Width fluctuation corrections are non-negligible only in the former channel. More details can be found in [23].The results show that it is possible to compare full R-matrix and HF models in the transition region of unresolved resonances. It is however, necessary to include all excited residual states up to the incident energy. This is straightforward for HF calculations, but not for R-matrix analysis due to the large dimensions.

Summary
In this paper we presented the latest developments in the evaluation of light-element reactions in the resolved reso-  nance region. The effort to produce complete and reliable evaluations for charged-particle reactions is ongoing. At the same time, there is progress in extending the R-matrix theory to deal with the rapidly growing number of channels as we go to higher excitation energies as well as in linking the resolved-resonance region with the unresolved resonance and the statistical regime. We hope to be able to implement the theoretical developments in the evaluation of both charged-particle and neutron-induced reactions in the resolved resonance region as we go to higher energies and include breakup channels in the R-matrix analysis as well.

Acknowledgements
This work is performed under the auspices of the International Atomic Energy Agency within the Interna-