In Search of the Best Nuclear Data File for Proton Induced Reactions

EPJ Web of Conferences 239, 13005 (2020) https://doi.org/10.1051/epjconf/202023913005 ND2019

In search of the best nuclear data ﬁle for proton induced reactions: Varying both models and their parameters

1,2, 1, 1 1 2 3,4 1 E. Alhassan ⇤, D. Rochman ⇤⇤, A. Vasiliev , R.M. Bergmann , M. Wohlmuther , A.J. Koning , and H. Ferroukhi 1Laboratory for Reactor Physics and Thermal-Hydraulics, Paul Scherrer Institute, 5232 Villigen, Switzerland 2Division Large Research Facilities (GFA), Paul Scherrer Institute, Villigen, Switzerland 3Nuclear Data Section, International Atomic Energy Commission (IAEA), Vienna, Austria 4Division of Applied Nuclear Physics, Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden

Abstract. A lot of research work has been carried out in fine tuning model parameters to reproduce experimental data for neutron induced reactions. This however is not the case for proton induced reactions where large deviations still exist between model calculations and experiments for some cross sections. In this work, we present a method for searching both the model and model parameter space in order to identify the ’best’ nuclear reaction models with their parameter sets that reproduces carefully selected experimental data. Three sets of experimental data from EXFOR are used in this work: (1) cross sections of the target nucleus (2) cross sections of the residual nuclei and (3) angular distributions. Selected models and their parameters were varied simultaneously to produce a large set of random nuclear data files. The goodness of fit between our adjustments and experimental data was achieved by computing a global reduced chi square which took into consideration the above listed experimental data. The method has been applied for the adjustment of proton induced reactions on 59Co between 1 to 100 MeV. The adjusted files obtained are compared with available experimental data and evaluations from other nuclear data libraries.

1 Introduction much e↵ort driven largely by the reactor community has been put into improving the neutron-sub library through High quality proton nuclear data are important for a wide the identification and fine tuning of model parameters for range of applications, e.g., in proton therapy, medical ra- a large number of isotopes in the case of the TENDL li- dioisotope production, accelerator physics as well as in as- brary [4] for example. The identified models have been trophysics, for a better understanding of stellar nucleosyn- used over the years for evaluations without necessarily go- thesis, among others. Similar to neutrons, the evaluation of ing back to the model selection step. In Ref. [5] how- proton induced reactions normally involves a combination ever, it was demonstrated that, the simultaneous variation of nuclear reaction modelling and carefully selected exper- of models and their parameters induces prior correlations imental data. Despite the progress made in nuclear reac- and therefore could have significant impact on nuclear data tion theory over the past decade, comparison of model cal- adjustments. In Ref. [5], model selection and the adjust- culations with experimental data usually reveals discrep- ment of models (and their parameters) in order to fit dif- ancies between the two. A common solution is to adjust ferential experimental data was not emphasized. Until re- or fine tune parameters to nuclear reaction models in or- cently, much e↵ort was not devoted to the evaluation of der to fit di↵erential experimental data obtained from the proton induced reactions which is evident by the num- EXFOR database [1]. ber of evaluations available in the proton sub-library in A single nuclear reaction calculation involves several the major nuclear data libraries compared with the neutron models with several parameters, linked together in a nu- sub-library: 49 isotopes in the ENDF/B-VIII.0 library and clear reaction code such as TALYS [2] or EMPIRE [3]. In 106 in the JENDL/HE-2007 (JENDL High Energy file) the TALYS code for example, there are six level density library compared with 557 isotopes in the neutron sub- models, three optical models, four pre-equilibrium mod- library for ENDF/B-VIII.0, 406 for JENDL-4.0 and 562 els and eight gamma-strength models, among others im- for the JEFF-3.3 library. In the case of the proton induced plemented. A combination of these models usually gives reactions, the TENDL-2017 and JEFF-3.3 libraries both di↵erent TALYS outputs. One often overlooked but im- contain evaluated nuclear data for 2804 isotopes, however, portant step in the evaluation process is the identification the evaluations were all carried out with default TALYS of model combinations that can reproduce experimental models and parameters [5]. Furthermore, both the model data. This is in part due to the fact that, for several decades, and parameter space in the case of proton induced reactions have been left largely unexplored, necessitating for ⇤e-mail: [email protected] ⇤⇤e-mail: [email protected]

© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). EPJ Web of Conferences 239, 13005 (2020) https://doi.org/10.1051/epjconf/202023913005 ND2019

the simultaneous variation of both models and their pa- account by computing a global χ2 given as follows: rameters as proposed in this work. 2 2 2 2 χG,k = χk(xs) + χk(rp) + χk(DA) (1) 2 Method χ2 A total of 200 random model combinations were generated where G,k is the global chi square for the random nu- 2 2 by varying a selected number of nuclear reaction mod- clear data k, χk(xs) and χk(rp) are the chi squares com- els implemented within the TALYS code. These model puted using the reaction cross sections and the residual 2 combinations were run with the TALYS code (version 1.9) production cross sections respectively, and χk(DA) is the to produce a large set of random physical observables re- chi square computed for the elastic angular distributions. ferred here as the parent generation. A total of 682 ran- For Eq. 1 to hold, it was assume that the di↵erent experi- dom nuclear data were produced for the parent generation. mental categories as presented were uncorrelated and were The parent generation as used in this work refers to the of equal importance in the adjustment. Further, similar to initial random nuclear data (ND) files generated from the Refs.[7, 8], the experimental data points were assumed to variation of models. be uncorrelated. The reason being that, experimental cor- The random nuclear data files in the ENDF format relations especially for proton induced reactions were not χ2 were processed into XY tables for comparison with se- readily available. Our reduced c(k) for the channel c and lected experimental data from the EXFOR database using nuclear data (ND) file k, can be given as: a reduced χ2. Based on the χ2, the model combination 2 Np i i 2 with the minimum χ was chosen as the ’best’ model set. 1 σT(k) σE χ2 = − (2) The selected model combination (also referred to as the c(k) N ∆σi p i=1 E parent file) was used as the nominal file for re-sampling X ✓ ◆ i of model parameters to produce the next generation of where σT(k) is a vector of TALYS calculated observ- TALYS outputs (referred to as the 1st generation). The ables found in the kth random ND file for the channel c output of the 1st generation were again compared with ex- i and σE is a vector of experimental observables as a func- perimental data from the EXFOR database and new ’best’ i tion of incident neutron energy (i) for channel c, ∆σE is the file was selected. experimental uncertainty at an incident energy i of channel c, and N is the total number of experimental points per re- 2.1 Experimental data used p action channel considered. In cases where no matches in Three experimental categories were used: (1) cross energy (i) were observed between the TALYS output ob- sections of the target nucleus (2) cross sections of the tained and the experimental data for the cth channel, we residual nuclei (also called the residual production cross carry out a linear interpolation in order to fill in the miss- sections) and (3) angular distributions. In the case of ing TALYS values. In the case of angular distributions, cross sections of the target nucleus (also referred to as only the missing values in angle were filled through linear the reaction cross sections in this work), the following interpolation. In order to obtain perfect matches in energy eight channels were considered in the adjustments: for the elastic angular distributions, the energies at which (p,non-el), (p,n), (p,3n), (p,4n), (p,2np)g, (p,2np)m, (p,γ) angular distributions where measured where given to the and (p,xn) and for the residual production cross sec- TALYS code as input. From Eq. 2, the reduced chi square 59 46 59 48 59 52 2 tions: Co(p,x) Sc, Co(p,x) V, Co(p,x) Mn, for the reaction cross section (χk(xs)) for example, can be 59Co(p,x)55Fe, 59Co(p,x)55Co, 59Co(p,x)56Co, given as: 59 57 59 58 59 57 Co(p,x) Co, Co(p,x) Co, Co(p,x) Ni. In the Nc 2 1 2 case of angular distributions, only the elastic angular χk(xs) = χc(k) (3) Nc distributions were considered. Xc=1 A total of 169, 141 and 185 experimental data points where Nc is the number of considered channels. In were used for the reaction cross sections, the residual pro- Ref. [7], a weighted χ2 where channel weights propor- duction cross sections and the elastic angular distributions tional to the average channel cross section, was presented. respectively. Similar to Ref. [7], experiments that were ob- The idea was to assign channels with large average cross served to be inconsistent with other experimental sets and sections higher weights and those with lower relatively deviate from the trend of our model calculations as well as smaller average cross sections, lower weights. However, other evaluations (when available), were not considered. since the goal of this work is to produce a TENDL based Also, for the cases where the only experimental data avail- evaluation for a general purpose library, all channels were able for a particular energy range has no uncertainties re- assumed to carry equal weights. The file with the mini- ported, we assume a 10% uncertainty for that experimental mum global χ2 (with its set of models) was selected as our set. best file and used as the nominal file around which model parameters were varied. Because of computational re- 2.2 Optimization of models and their parameters to source constraints, the final ’best’ file produced was based experimental data on the results of the 1st generation. Also, for the selec- In this work, the reduced χ2 was used as the goodness of fit tion of models, the Bayesian approach for model selection estimator. Since three experimental categories were used could have been used. This approach is presented in a ded- in the adjustment, we take all these experimental data into icated paper [9].

2 EPJ Web of Conferences 239, 13005 (2020) https://doi.org/10.1051/epjconf/202023913005 ND2019 the simultaneous variation of both models and their pa- account by computing a global χ2 given as follows: 3 Results and Discussion for the (p,non-el) and (p,n) cross sections of 59Co. In cases rameters as proposed in this work. 2 2 where evaluations are available, comparisons are made 2 2 2 2 In Fig. 1, the global χ distribution as well as the χ dis- χ = χ (xs) + χ (rp) + χ (DA) (1) also with the JENDL-2007/He library. From the figure, G,k k k k tributions for the reaction cross sections (xs), the resid- 2 Method it can be observed that, the evaluation from the 1st gen- ual production cross sections (rp) and the angular distribu- where χ2 is the global chi square for the random nu- eration performed better than the TEND-2017 library for A total of 200 random model combinations were generated G,k tions (DA) for the 1st generation are presented and com- clear data k, χ2(xs) and χ2(rp) are the chi squares com- the (p,non-el) and (p,n) cross sections. The TENDL-2017 by varying a selected number of nuclear reaction mod- k k pared with χ2 values computed for the TENDL-2017 li- evaluation over estimates the (p,non-el) cross section from els implemented within the TALYS code. These model puted using the reaction cross sections and the residual brary and the ’best’ file from parent generation (referred production cross sections respectively, and χ2(DA) is the about 20 to 100 MeV while this evaluation is within the ex- combinations were run with the TALYS code (version 1.9) k to as the ’parent file’), using the same experimental data. perimental uncertainties over the entire incident energies. to produce a large set of random physical observables re- chi square computed for the elastic angular distributions. From Fig. 1, it can be seen that, the adjustment from the For Eq. 1 to hold, it was assume that the di↵erent experi- ferred here as the parent generation. A total of 682 ran- 1st Gen out performed the TENDL-2017 evaluation for Fig. 3 presents the comparison of file performance be- mental categories as presented were uncorrelated and were dom nuclear data were produced for the parent generation. the reaction cross sections and the angular distributions tween our evaluation, the TENDL-2017 and JENDL/He- of equal importance in the adjustment. Further, similar to The parent generation as used in this work refers to the but performed quite poorly with respect with to the resid- 2007 evaluations for the 59Co(p,x)56Co and 59Co(p,x)55Co Refs.[7, 8], the experimental data points were assumed to initial random nuclear data (ND) files generated from the ual production cross sections. Also, it can be observed residual production cross sections. In the case of the be uncorrelated. The reason being that, experimental cor- variation of models. that, the results from the 1st generation is an improvement 59Co(p,x)56Co for example, our evaluation (i.e. the 1st relations especially for proton induced reactions were not The random nuclear data files in the ENDF format over the parent file as expected: χ2 values of 22.17, 21.65, Gen), under predicts the data at incident energies below 60 readily available. Our reduced χ2 for the channel c and were processed into XY tables for comparison with se- c(k) 22.43, 22.42 for the global, reaction cross sections, resid- MeV. Our evaluation however describes the experimental nuclear data (ND) file k, can be given as: lected experimental data from the EXFOR database using ual production cross sections and angular distributions re- data reasonably well between 60 to 100 MeV. Similarly a reduced χ2. Based on the χ2, the model combination spectively, were obtained for the 1st Gen compared with in the case of the 59Co(p,x)55Co, our evaluation is unable 2 Np σi σi 2 with the minimum χ was chosen as the ’best’ model set. 1 T(k) E 35.60, 51.07, 24.43, and 22.42 for the parent file. To im- to fit satisfactorily to experimental data. This explains the χ2 = − (2) The selected model combination (also referred to as the c(k) N ∆σi prove on the 1st generation, the new ’best’ file obtained relatively large χ2 value of 22.43 obtained for this evalua- p i=1 E parent file) was used as the nominal file for re-sampling X ✓ ◆ could have been used as the nominal for re-sampling of tion (1st Gen) compared with 20.88 obtained for TENDL- i of model parameters to produce the next generation of where σT(k) is a vector of TALYS calculated observ- model parameters in an iterative fashion. This however, 2017 with respect to the residual production cross sections. TALYS outputs (referred to as the 1st generation). The ables found in the kth random ND file for the channel c can be computationally expensive and therefore not car- In order to improve the residual cross sections, the ’best’ output of the 1st generation were again compared with ex- i ried out in this work. file from the 1st Gen can be utilized as the new nominal and σE is a vector of experimental observables as a func- perimental data from the EXFOR database and new ’best’ i In Fig. 2, a comparison of file performance between file for parameter variation in an iterative fashion. This is tion of incident neutron energy (i) for channel c, ∆σE is the file was selected. experimental uncertainty at an incident energy i of channel our evaluations and the TENDL-2017 library are presented however planned for future work. c, and N is the total number of experimental points per re- 2.1 Experimental data used p action channel considered. In cases where no matches in Global @ 2 (xs+rp+DA) @ 2 (cross sections (xs)) Three experimental categories were used: (1) cross energy (i) were observed between the TALYS output ob- 100 60 sections of the target nucleus (2) cross sections of the tained and the experimental data for the cth channel, we TENDL-2017 ( @2 = 25.76) TENDL-2017 ( @2 = 25.03) Parent file ( @2 = 35.60) 50 Parent file ( @2 = 51.07) residual nuclei (also called the residual production cross carry out a linear interpolation in order to fill in the miss- 80 1st Gen ( @2 = 22.17) 1st Gen ( @2 = 21.65) sections) and (3) angular distributions. In the case of ing TALYS values. In the case of angular distributions, 40 cross sections of the target nucleus (also referred to as only the missing values in angle were filled through linear 60 the reaction cross sections in this work), the following interpolation. In order to obtain perfect matches in energy 30 eight channels were considered in the adjustments: for the elastic angular distributions, the energies at which 40 (p,non-el), (p,n), (p,3n), (p,4n), (p,2np)g, (p,2np)m, (p,γ) angular distributions where measured where given to the Counts/bin Counts/bin 20 and (p,xn) and for the residual production cross sec- 20 TALYS code as input. From Eq. 2, the reduced chi square 10 59 46 59 48 59 52 2 tions: Co(p,x) Sc, Co(p,x) V, Co(p,x) Mn, for the reaction cross section (χk(xs)) for example, can be 59 55 59 55 59 56 Co(p,x) Fe, Co(p,x) Co, Co(p,x) Co, given as: 0 0 59 57 59 58 59 57 0 100 200 300 0 50 100 150 200 Co(p,x) Co, Co(p,x) Co, Co(p,x) Ni. In the Nc 1 @2 @2 case of angular distributions, only the elastic angular χ2(xs) = χ2 (3) values values k N c(k) distributions were considered. c c=1 X @ 2 residuals (rp)) @ 2 (Ang. Dist. (DA)) 50 200 A total of 169, 141 and 185 experimental data points where Nc is the number of considered channels. In were used for the reaction cross sections, the residual pro- 2 TENDL-2017 ( @2 = 20.88) TENDL-2017 ( @2 = 31.38) Ref. [7], a weighted χ where channel weights propor- Parent file ( @2 = 24.43) Parent file ( @2 = 31.41) duction cross sections and the elastic angular distributions 40 1st Gen ( @2 = 22.43) 1st Gen ( @2 = 22.42) tional to the average channel cross section, was presented. 150 respectively. Similar to Ref. [7], experiments that were ob- The idea was to assign channels with large average cross served to be inconsistent with other experimental sets and sections higher weights and those with lower relatively 30 deviate from the trend of our model calculations as well as smaller average cross sections, lower weights. However, 100 other evaluations (when available), were not considered. 20

since the goal of this work is to produce a TENDL based Counts/bin Counts/bin Also, for the cases where the only experimental data avail- evaluation for a general purpose library, all channels were 50 able for a particular energy range has no uncertainties re- assumed to carry equal weights. The file with the mini- 10 ported, we assume a 10% uncertainty for that experimental mum global χ2 (with its set of models) was selected as our set. 0 0 best file and used as the nominal file around which model 20 30 40 50 0 200 400 600 800 parameters were varied. Because of computational re- 2 2 2.2 Optimization of models and their parameters to @ values @ values source constraints, the final ’best’ file produced was based experimental data on the results of the 1st generation. Also, for the selec- Figure 1. χ2 distributions for the 1st generation for the three experimental data categories as well as the global χ2 are presented. xs In this work, the reduced χ2 was used as the goodness of fit tion of models, the Bayesian approach for model selection denotes reaction cross sections, rp – residual production cross sections and DA – angular distributions. A total of 682 random samples estimator. Since three experimental categories were used could have been used. This approach is presented in a ded- were used for each plot. in the adjustment, we take all these experimental data into icated paper [9].

3 EPJ Web of Conferences 239, 13005 (2020) https://doi.org/10.1051/epjconf/202023913005 ND2019

59 Co(p,non-el) cross section 59Co(p,n)59Ni cross section 1400 800 Random files Best file (1st Gen) 1200 700 Parent file TENDL-2017 600 JENDL-2007/He 1000 Chodil(1967) 500 Johnson(1964) 800 400 600 Random files 300 Best file (1st Gen) 400 Parent file 200 Cross section (mb) Cross section (mb) TENDL-2017 Mccamis(1986) 200 Kirkby(1966) 100 Bearpark(1965) Makino(1964) 0 0 0 20 40 60 80 100 0 5 10 15 20 25 30 Proton Energy (MeV) Proton Energy (MeV)

Figure 2. Comparison of ﬁle performance between the evaluations from this work and the TENDL-2017 library for the (p,non-el), (p,n) cross sections of 59Co. Comparisons are made with the JENDL/He-2007 library in cases where evaluations are available. Only the experimental data sets used in the adjustment have been presented.

59Co(p,x)56Co cross section 59Co(p,x)55Co cross section 120 10 Random files Random files Best file (2nd Gen) Best file (1st Gen) Best file (1st Gen) Parent file 100 TENDL-2017 8 TENDL-2017 JENDL/He-2007 JENDL/He-2007 Ditroi (2011) Michel (1985) 80 Ditroi (2013) Michel (1997) Michel (1997) 6 60 4 40

Cross section (mb) 20 Cross section (mb) 2

0 0 0 20 40 60 80 100 0 20 40 60 80 100 Proton Energy (MeV) Proton Energy (MeV)

Figure 3. Comparison of ﬁle performance between our evaluation and the TENDL-2017 evaluation as well as the JENDL/He-2007 for the 59Co(p,x)56Co and 59Co(p,x)55Co residual production cross sections.

4 Conclusion France, April, 22-27 (2007). [3] M. Herman, R. Capote, B.V. Carlson, P. Obložinsky,` A method was presented for searching the model and pa- M. Sin, A. Trkov, H. Wienke, and V. Zerkin, Nuclear rameter space through the simultaneous variation of many Data Sheets 108, 2655-2715 (2007). TALYS models (and their parameters). By computing [4] D. Rochman, A.J. Koning, J.C. Sublet, M. Fleming, E. a reduced global χ2 which takes into consideration ex- Bauge, S. Hilaire, P. Romain, B. Morillon, H. Duarte, perimental information from reaction and residual pro- S. Goriely, S.C. van der Marck, H. Sjöstrand, S. Pomp, duction cross sections as well as the elastic angular dis- N. Dzysiuk, O. Cabellos, H. Ferroukhi, and A. Vasiliev, tributions, we were able to identify a ﬁle that performs EPJ Web of Conferences 146, 02006 (2017). favourably globally when compared with the TENDL- 2017 evaluation. The method has been applied for the [5] A.J. Koning, D. Rochman, J. C. Sublet, N. Dzysiuk, adjustment of proton induced reactions on 59Co from 1 to M. Fleming, and S.C van der Marck, Nuclear Data 100 MeV.It was observed that, by exploring a larger model Sheets 155, 1-55 (2019). space, model combinations that reproduce di↵erential ex- [6] F. Ditrói, S. Takács, F. Tárkányi, R.W. Smith, and M. perimental data can be identiﬁed for the model parameter Baba, J. Korean Phys Society 59, 1697-1700 (2011). variation step. The study also shows that there is a poten- [7] E. Alhassan, D. Rochman, H. Sjöstrand, A. Vasiliev, tial for improvement of evaluations (within the limit of the A.J. Koning, and H. Ferroukhi, Bayesian updating for models), through an iterative process. data adjustments and multi-level uncertainty propaga- tion within Total Monte Carlo, Under review in Annals References of Nuclear Energy (2019). [1] H. Henriksson, O. Schwerer, D. Rochman, M. [8] A.J. Koning, The European Physical Journal A 51, 1- Mikhaylyukova, and N. Otuka. International Nuclear 16 (2015). Data Conference for Science and Technology, Nice, [9] E. Alhassan, D. Rochman, A. Vasiliev, M. France, April, 22-27 (2007). Wohlmuther, A.J. Koning, and H. Ferroukhi, Model [2] A.J. Koning, S. Hilaire, and M.C. Duijvestijn. Nuclear selection for nuclear data adjustments and evaluation, Data Conference for Science and Technology, Nice, In Manuscript (2019).