Present status of an R-matrix analysis code AMUR for cross-section evaluation in resolved resonance region

. An R-matrix analysis code AMUR is being progressed in terms of the correction on the experimental conditions to the theoretical calculations. In this work, new broadening options are presented both for cross-sections and angular distribution with given energy resolution. The code is also under development to analyze the J-PARC / ANNRI measurement with the double-bunch mode. In the ﬁnal part, let me focus on understanding of the R-matrix theory itself, in which a role of distant poles is discussed in the simultaneous analysis of the same compound nucleus.


Introduction
The R-matrix theory [1] is a framework which is strictly based on the quantum mechanics. Since the S-matrix elements are deduced from the observables such as measured cross sections, the theory is quite useful for the evaluation of cross-section in the resolved resonance energy region. Indeed, a number of the R-matrix codes have been developed in the world and they have been applied to the estimation of cross sections for the national nuclear data libraries over the decades. AMUR is a multi-level, multi-channel R-matrix code which is underdevelopment in JAEA. The code is organized by "theoretical" and "experimental" classes based on the object-oriented framework, in which the resonance parameters are searched/optimized with the Kalman filtering method [2]. The theoretical part is based on the Wigner and Eisenbud formalism [3] except for the radiative neutron capture channels, which is calculated by the Rich-Moore approximation [4]. Physical quantities to be calculated are the cross sections, di↵erential cross sections and analyzing power both for the neutron and chargedparticle reactions. The experimental class is designed to correct theoretical calculations so as to accommodate experimental conditions. The correction is indispensable for an elaborate resonance analysis because measured data are always a↵ected by characters of the experimental facilities, detector system and so forth. However, only a few capabilities (such as renormalization, etc) have been incorporated in this part of AMUR, unfortunately.
The main objective of this study is to enhance capability of the experimental corrections. Let me focus on the correction of resolution for neutron cross sections both for the light and heavier nuclei. For an example on the light nuclei, one of the analyses for JENDL-5 [5] is presented. For the heavier nuclei, a preliminary analysis is presented ⇤ e-mail: kunieda.satoshi@jaea.go.jp  [12] for experimental data of J-PARC/ANNRI, which usually su↵er from the "double bunch" problem. I also aims at understanding the resonant theory itself to explore general/ultimate method of resonance analysis. In this regard, a test analysis is performed for 7 Be compound system by assuming distant poles with an unprecedented way to solve a curious issue on the simultaneous analysis of the di↵erent particle reactions.

Example analysis on light nuclei : n+ 19 F
AMUR was applied to the evaluation on a number of light nuclei for development of the neutron sub-library JENDL-5. Among those isotopes, let me focus on the evaluation of n+ 19 F cross sections as an example. The measured cross sections analyzed are taken from the EXFOR database [6] and they are listed in Table 1. Those experimental data were fitted simultaneously to obtain reasonable values of the resonance parameters. In this paper, let me focus on the analysis of the angular distribution data of Elwyn et al. [10] which is given in a low energy resolution (FWHM=100 keV) of incident neutrons. Indeed, it was impossible to fit the experimental data without cor- Figure 1. Di↵erential cross sections of the 19 F(n, n 0 2 ) reaction from di↵erent libraries at E n = 600 keV which are compared with measured data [10] where the present values were broadened by experimental resolution of 100 keV.
rection of resolution to the R-matrix calculation since contributions of neighboring resonances usually overlap each other due to poor resolution.
In this work, AMUR was updated so as to reconstruct di↵erential cross sections in the two-dimensional space, viz., in terms of the incident energy and the scattering angle. Then, the code performs the broadening of excitation functions with the normal distribution for each angle. Those procedures eventually allow us to obtain angular distribution of cross sections with energy resolution of incident particle. Figure 1 illustrates an example result of the present analysis with a new option together with experimental data of Elwyn et al. Again, it is impossible to fit the measured data without the new option as the shape is totally di↵erent from each other. The data from evaluated libraries such as ENDF/B-VIII.0 and JENDL-4.0 are also shown for the reference purpose. Since the optical model had been applied to their evaluations, both curves reproduce only the average behavior of experimental data. The option allows us to obtain resonance parameters from experimental data of angular distribution, even if the data are given in a large resolution.

Extension of basic capabilities
The AMUR code was initially designed for the analysis of the light-nuclei to narrow large di↵erences of cross sections among evaluations in the world, e.g., on the 16 O(n,↵) cross-section [13,14]. Recently, the author extended capability of the code toward the analysis of heavier nuclei to fully cover range of isotopes for nuclear science and engineering. In the theoretical part, AMUR is now able to read ENDF-6 format (MF=2) file and then perform reconstruction for LRF=1,2,3 and 7, which means the code is able to calculate fission cross-sections with a reduced R-matrix formula. The code is also able to calculate the Doppler broadening with the free-gas approximation. Therefore, AMUR is now equivalent to the other resonance analysis codes (or processing codes) in terms of basic capabilities for the cross-sections calculation/reconstruction. Figure 2 illustrates the ratio of the 35 Cl(n,tot) cross-section reconstructed from ENDF/B-VIII.0 (MF=2, LRF=7) where the calculated results are compared between PREPRO [15] and AMUR. It was confirmed that the di↵erences of the reconstructed data come only from the accuracy of interpolation (0.1% was assumed in this case).

Resolution function
In general, the measured data are di↵erent from the true values of cross sections because they always su↵er from experimental conditions such as in the facility, detection system, purity of sample, and so forth. Therefore, it is necessary to simulate such experimental factors as much as possible in the resonance analysis for the nuclear data evaluation. In this regard, structure of incident particles is one of the important factors in the neutron cross sections measurements.
The cross-section measurements are underway in the J-PARC/MLF facility with the ANNRI detection system to measure neutron cross sections for minor actinoides and structural material in the resolved resonance energy region. The facility shows its own structure of neutron beam, where the operation of the accelerator is usually in the double-bunch mode, which results in the degeneration of resonant structure.
The resolution function of J-PARC/MLF had been estimated by Kino et al. [16]. The function had been also applied to the resonance analysis code REFIT [17,18]. Recently, those works were followed by myself and the function was incorporated in to AMUR. Figure 3 illustrates TOF spectrum of the 93 Nb(n, ) reaction measured by Endo et al. [19] where the corresponding calculations by AMUR (with resonance parameters of JENDL-5) are plotted for those with (single/double-bunch modes) and without resolution functions. It is well understood again that the consideration of resolution function is indispensable, and simulation of the double-bunch mode is necessary to analyze experimental data. Since the self-shielding and multiple-scattering e↵ects in the target sample could be corrected by the Mote Carlo simulation before the resonance analysis, AMUR is now ready to obtain the resonance parameters and their covariances from the J-PARC/ANNRI measurements.

Independent distant poles
Over the past decades, with the R-matrix theory, a number of works had been devoted to understanding the level structure of the nucleus and the evaluation of resonant cross sections. However, in the resonance region, both the resonant and non-resonant processes exist in which the later process had not been highlighted yet. The following test analysis and discussion may give us a guide to understand the R-matrix theory itself in terms of the non-resonant process. Here, let me focus on a very simple case of the R-matrix analysis in which di↵erent parti-cle+nucleus pairs form the same compound nucleus. Figure 4 shows an illustration of the test case in which four reaction channels, 6 Li(p, p 0 ) 6 Li, 6 Li(p, ↵ 0 ) 3 He, 3 He(↵, ↵ 0 ) 3 He and 3 He(↵, p 0 ) 6 Li, are to be analyzed. Those reactions form the same compound nucleus 7 Be ⇤ , in which they share the resonance parameters such as the energy eigenvalues and the reduced-width amplitudes. The test case was prepared in the IAEA collaborative works on the R-matrix analysis for the charged-particle reactions [20]. Since the maximum excitation energy of the compound was set to ⇠ 7.5 MeV in this study, only two and three resonances are observed for proton-and ↵-particle induced reactions, respectively. Also, the final states of the residual nuclei are always at their ground states. Although the all the reaction channels are intended to be analyzed simultaneously, the present situation is quite simple in terms of the number of the resonances and the explicit channels involved. The experimental data of the corresponding reaction channels are those for the absolute cross sections and differential cross sections where 10 sets of measurements were taken from the EXFOR database [6]. In the first step, the R-matrix analysis was carried out by a traditional way -the resonance parameters were searched for those three levels, where the pseudo levels were also assumed outside the energy range of excited levels observed to mimic the contribution of the bound states and the distant levels of 7 Be ⇤ . This approach is nearly equal to the standard in the world, as it gives a successful analysis in the neutron crosssections evaluation over the decades. However, at least by AMUR, it was di cult to obtain resonance parameter sets converged during the iterative fitting process. It was curious since all the experimental data seem not to have a serious problem, and the present case is quite simple (I was not likely to end up with a technical mistake of the analysis).

Possible solution and preliminary discussion
Since AMUR had never encountered such a problem ever before, I thought it was not a technical problem in the code. The second analysis I performed was the same as in the first analysis described above, except that additional distant (pseudo) levels were assumed for incident protons and ↵-particles, "independently". Figure 5 illustrates example results of the present R-matrix analysis with the new option. The fitting itself was quite successful for all the experimental data as it converged in the iterative process. Using resonance parameters obtained, the R-matrix calculations were also performed without the new option as illustrated by the dashed curves in the same figure, where the comparison of two curves certainly show an impact of the independent distant poles.
Through such comparisons, I have noticed that the di↵erences were seen only in the elastic scattering Figure 5. Example results of the present R-matrix analysis for the 7 Be ⇤ compound system with AMUR in which the independent distant poles were assumed as a new option (The dashed curves indicate calculations without the new option). 6 Li(p, p 0 ) 6 Li and 3 He(↵, ↵ 0 ) 3 He, while the di↵erences were not observed in the 6 Li(p, ↵ 0 ) 3 He and 3 He(↵, p 0 ) 6 Li reactions. It is well known that the elastic scattering process occurs through interference between the shape-elastic (direct) and the compound reaction processes. On the other hand, according to a number of experience in theoretical studies (e.g., Ref [21]), the (p, ↵ 0 ) and (↵, p 0 ) reactions occurs through virtually the compound process in this energy region where the direct process is less important. Those facts and our experiences suggest that the independent distant poles work as contribution of the shapeelastic (or the direct) process. Since the shape-elastic process is designed to be calculated by the hard-sphere phase shift at an arbitrary channel radius in R-matrix, the independent distant poles should correct the hard-sphere model, which does not necessarily describe real picture of the nuclear surface.

Summary
The AMUR code is being progressed in terms of the correction on the experimental conditions to the R-matrix calculations. In this work, new broadening options were presented, in which both the cross sections and angular distribution were calculated with a given energy resolution. The code was also ready to simulate neutron structure in the double-bunch mode for analysis of the J-PARC/ANNRI measurement. Finally, a role of independent distant poles was discussed, where the poles are likely to correct/mimic the direct process of nuclear reaction.