Evaluation of cross sections for fast ion reactions with beryllium in helium and hydrogen fusion plasmas

To computationally support hydrogen and helium plasma discharges in the early stages of tokamak operation and to support the commissioning of the neutron detectors during these operational phases, creation of a realistic neutron and gamma ray particle source for Monte Carlo simulations will be needed. One of the most important parts of creating the particle source is calculating the reaction rates of the particle-emitting reactions to determine the emission profile in the plasma and the energy spectra of the emitted particles. In this paper the analysis and evaluation of cross sections for important neutron-emitting reactions, namely, 9Be(p,nγ)9B, 9Be(3He,nγ)11C, and charged-particle emission reactions 9Be(p,d)2α and 9Be(p,α)6Li that cause neutron emission in the next step of interactions are presented. The reaction cross sections were evaluated based on experimental measurements and empirical models describing the interaction of two charged particles. Evaluation of the associated uncertainties was also performed. The main goal of the work is to propose the newly evaluated cross sections for inclusion in the FENDL nuclear data library, thus making the cross section available to other researchers studying the above listed reactions.


Introduction
In the initial phase of fusion reactor operation, non-neutron emitting plasmas such as the hydrogen plasma, are envisaged to be used to test and commission various systems before power operation with deuterium or deuterium-tritium plasma, Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.e.g.ITER's pre-fusion power operation phase (PFPO) [1].At the start of the present research the ITER first wall was planned to consist of metallic tungsten and beryllium, some of which would be eroded by the interaction of the plasma ions with the wall and could be present in the ITER plasma as an impurity [2].Although the final decision on the first wall material can change, the study is nevertheless relevant to compare the relative merits of different first-wall materials.Since some proposed ITER PFPO scenarios with H and He plasmas employ a combination of high-power neutral beam injection and radiofrequency settings, the accelerated ions with energies of several MeV (depending on the temperature of the plasma) are produced and they can trigger reactions with Be impurities in the plasma.This will lead to the emission of neutrons and gammas at non-negligible rates in what are otherwise considered non-neutronic plasmas.
The neutron yield is expected to be small compared to the fusion power operation phase in a deuterium-tritium plasma (10 15 n s −1 in ITER PFPO compared to 10 21 n s −1 in ITER DT plasma) [2,3].Nevertheless, radiation field induced as a result of fast ion and Be needs to be characterized through modeling for safety reasons, and also offers the possibility of commissioning diagnostic systems before power operation begins.
Preliminary results of our study were presented at the NENE-2022 conference [4].The text with minor editorial corrections is reproduced below for clarity, quote: 'To computationally support measurements and the commissioning of diagnostic equipment, reaction cross sections are crucial for the calculation of neutron emission rates and neutron energy spectra [5,6].Unlike the major fusion reactions, such as the deuterium-deuterium and deuteriumtritium reactions, for which the cross sections are well known and available in evaluated nuclear data libraries, reaction cross sections for plasma ion reactions involving beryllium are often poorly characterized and unavailable in evaluated nuclear data libraries.Some of them (such as the reaction 9 Be(p,nγ) 9 B) are available in the TENDL-2019 and TENDL-2021 libraries [7], but the cross sections are of poor quality [2,3].' These reactions were not extensively studied in current fusion experiments because the emitted neutron rates are small compared to the main fusion reactions.However, planned ITER PFPO scenarios will be unique in that perspective since the emission rates from fast ion and Be fusion are expected to be comparable to DD neutron rates in devices such as JET and ASDEX Upgrade.Reactions of interest for neutron emission in ITER PFPO [4] are listed in table 1.
Of the above listed reactions some directly produce neutrons while some produce fast deuterium and 4 He ions (reactions 9 Be(p,d)2α and 9 Be(p,α) 6 Li) which are confined in the magnetic field and produce secondary reactions, which emit neutrons.
Cross sections for reactions 9 Be(d,nγ) 10 B are available in the TENDL-2019 library [7] and the more recent library release TENDL-2021, which contains minor improvements for the reactions of interest.Cross sections for 9 Be(α,nγ) 12 C are found in TENDL-2021 and JENDL-5 libraries [8].For the rest of the reactions listed above, the cross sections vary significantly between nuclear data libraries (if available at all) and often disagree with available experimental data.For this reason in this paper, the cross sections for the above listed missing reactions are analyzed based on cross section measurements collected in the EXFOR database [9] and empirical models describing the cross sections for interactions between ions.In addition, the uncertainty evaluation of the generated cross sections was performed and is presented in this paper.
The paper is organized as follows.In section 2, the empirical model used to determine the total cross section from experimental measurements is presented.In section 3, the methodology for the evaluation of uncertainty of the generated total cross sections is presented, based on the uncertainty of the experimental measurements and the uncertainty of the empirical model used.In sections 4-9 the cross sections and the uncertainties for each of the above listed reactions are presented.At present, the evaluated data files cannot support full transport calculations, but they conform to the ENDF-6 format, similarly like Dosimetry libraries and can be used to calculate reaction rates.

Reaction cross section model
The description of the reaction cross section model is repeated as in NENE-2022 conference [4] for clarity, quote: 'The total cross sections of a reaction between two charged particles can be described using a physics model divided into two parts which are approximately independent of each other.First is the physics of two nuclei approaching each other without a collision and the second is the physics, in case the two nuclei approach each other close enough to undergo a nuclear interaction.The dominant physics of two nuclei approaching each other is the repulsive Coulomb force and the cross section is proportional to the tunneling probability if the energy in the center of mass system is smaller than the Coulomb barrier.In the case of interaction between two nuclei the interaction is described by quantum mechanics and is always proportional to πλ 2 ∝ 1/E, where λ is the de Broglie wavelength; this term is sometimes referred to as the 'geometrical factor' [10].At low energies both parts are rapidly varying functions of energy.For this reason the cross section definition at low energy is the product of three separate energydependent factors:' 'Here, S(E) is the astrophysical S-function and B g is the Gamow factor.The motivation for this definition is that the two strong energy dependent factors (describing the Coulomb and the interaction parts of the incident nuclei, respectively) are separated, leaving the S-function to represent the nuclear part of the probability for the occurrence of the nuclear reaction.The S-function is a simple function when the interaction energy of the two nuclei is not near the resonant energy of the reaction and can be approximated by the ratio of two arbitrary polynomials [11]:' 'The A i and B i coefficients are free parameters which are used for fitting the cross section to experimental data and the polynomial degree is chosen to best represent the experimental cross section data with as low as possible degree [12].' For some of the reactions in addition to the total reaction cross section measurements, differential cross section measurements with respect to angle are also available.To use these measurements in the analysis of reaction total scattering cross section, Legendre polynomials were fitted to experimental measurements at different angles at the same energy of the incident particle.The P 0 polynomial coefficient represents the average angular cross section at given particle energy and the total cross section needed for the analysis is obtained by multiplying the P 0 term with 4π.

Cross section uncertainty evaluation
The reaction cross sections generated using the above model can be used to determine neutron production rates in the future ITER PFPO plasma with hydrogen and helium ions interacting with the beryllium impurity.An important parameter of the calculated neutron production rates is also the uncertainty of the calculated values due to the uncertainty of the generated cross sections.In order to estimate the cross-section uncertainty, the uncertainty estimation was part of the evaluation process.
The cross section covariance matrix and the uncertainties were evaluated by the Monte Carlo technique [13].
Using the values of the fitted cross section function parameters and their standard deviations, 10 000 sets of cross section were generated by random sampling the parameters within their estimated standard deviations.In some cases, the combination of the sampled cross section function parameters resulted in negative cross sections, or the perturbed parameters could give rise to singularities.However the fraction of such cases was below 0.1%.These samples were discarded because they result in unphysical shapes of the cross sections.The covariance matrix of the generated cross sections was as determined from the distribution of the generated cross sections.

Reaction 9 Be(p,nγ) 9 B
The total cross sections for the reaction 9 Be(p,nγ) 9 B do exist in the nuclear data libraries ENDF/B-VIII.0[7,14,15] (giving identical cross sections in all three nuclear data libraries), but significantly differ from measured data in EXFOR [9], as can be seen in figure 1. Due to this, the total reaction cross sections were evaluated based on equation ( 1) and experimental data from the EXFOR database.Two sets of experimental data were used for the analysis, namely measurements by E. Teranishi in 1964 (EXFOR number F0177002 [16]) and J.H. Gibbons in 1959 (EXFOR number T0010009 [17]) (shown in figure 1).It should be noted here that both cross section measurements were reported without specifying the uncertainties.
Cross section evaluation procedure is essentially the same as in [4], except that an extended set of experimental data was used.As can be seen from figure 1, the cross sections for the reaction 9 Be(p,nγ) 9 B exhibit a resonance at about 2.5 MeV.For this reason, the fitting process was divided into two parts, namely for energies up to 3 MeV and for energies above 3 MeV.The method used for the fitting process was the non-linear least square method, and the best fit function was determined from the covariance matrix of the fitted parameters.For both parts, the best fit of the cross section function to the experimental data with the lowest order of S(E) polynomial was: Both fitted cross sections were merged into a single set and the resulting total reaction cross section plot for the 9 Be(p,nγ) 9 B reaction is presented in figure 2.
The reaction has a threshold energy of about 2 MeV with a resonance peak at about 2.5 MeV and a second peak at about 5 MeV.At the time of writing, no measurements are available for energies above 5.5 MeV, thus the shape of the obtained cross sections for energies between 5 MeV and 5.5 MeV is subject to larger uncertainties.Despite these uncertainties, the obtained cross sections represent the best cross sections for the reaction 9 Be(p,nγ) 9 B based on the experimental measurements.
Based on the generated cross section from perturbed polynomial parameters, the uncertainties of the generated total reaction cross sections were evaluated using the methodology described in section 3 and are presented in figure 3. The cross section uncertainties vary from 2% to 40%.The largest uncertainties are at the ends of the intervals of the evaluation, i.e. at the reaction threshold energy and at the energy where the cross section functions were merged.Using the fitted cross section parameters the covariance matrix of the cross section uncertainties was prepared.The uncertainty in figure 3 represents only the fitting uncertainty.To account for the inherent systematic uncertainty in experimental values, a 2% uncertainty was added and the plot of the covariance matrix is presented in figure 4.

Reaction 9 Be( 3 He,nγ) 11 C
The cross sections for the reaction 9 Be( 3 He,nγ) 11 C are only available in the TENDL-2019 and TENDL-2021 nuclear data libraries [7].Three sets of experimental measurements of the cross sections are present in the EXFOR database, performed by S.N.Abramovich in 1984 (EXFOR number A0166002 [18]), B. Anders in 1981 (EXFOR number A0330006 [19]), and R.L. Harn in 1966 (EXFOR number F0205002 [20]).Experimental values are presented together with the cross sections from the TENDL-2019 and TENDL-2021 libraries in figure 5.In contrast to the cross sections presented in the previous section, the cross section measurements for reaction of 3 He with 9 Be are given with experimental uncertainties.Cross section evaluation procedure is essentially the same as in [4] except for a small refinement that had negligible impact on the results.Visual inspection of the measurements does not indicate any resonance structure in the cross sections.For this reason, the methodology presented in section 2 was used on the whole energy range of the measurements.The best fitted S(E) function was: and the generated total reaction cross sections are presented figure 6.
There are no measurements for energies below 1.95 MeV, therefore the energy dependence of the cross section below this energy is unknown.In addition, the lack of measurements means that the effective threshold energy for the reaction cannot be determined accurately.To perform computational analysis to support hydrogen and helium plasma experiments, the fitted cross sections were extrapolated to lower energies.The zero-intercept at 1.32 MeV was defined as the effective threshold and is shown in figure 6.
The uncertainty evaluation of the generated cross sections was performed using the methodology described in section 3 and the results are presented in figure 7. The uncertainties of the generated cross sections vary from a few percent to 15% at the ends of the cross section evaluation range.In addition, the covariance matrix of the generated cross section uncertainties was evaluated and is presented in figure 8.

Reaction 9 Be(p,d)2α
Reaction 9 Be(p,d)2α produces a fast deuteron and a 8 Be residual, which breaks almost instantly into two alpha particles that can interact with beryllium in plasma and produce neutrons and gamma rays.The cross sections for charged-particle production are present in the JENDL-5.0,ENDF/B-VIII.0 and TENDL-2021 nuclear data libraries; the data in TENDL-2021 are adopted from ENDF/B-VIII.0.The data in the libraries can be easily missed since they are stored implicitly as deuteronproduction data in MF6/MT5 in ENDF terminology [21].Furthermore, the cross sections represent total deuteron emission, mainly at energies above 1 MeV.At lower energies they deviate severely from the measured 9 Be(p,d) cross sections and are not considered further in the present analysis.
There are measurements included in the EXFOR database, performed by Sierk in 1973 (EXFOR number F0169005 [22]), Bertrand in 1968 (EXFOR number F0107006 [23]) and  Neuendorffer in 1951 (EXFOR number C0632002 [24]).In addition to the total reaction cross section measurements, differential cross section measurements with respect to angle were also used to determine the total cross sections.The measurements used in the analysis were performed by Bertrand in 1968 (EXFOR number F0107003 and F0107004 [23]) and Weber in 1956 (EXFOR number F0158002 [25]).The measurements of the reaction cross section are presented in figures 9 and 10.
Visual inspection of the measurements indicates the presence of two resonance peaks in the cross sections.For this reason, the fitting process was divided into three parts, namely for energies below 0.7 MeV, from 0.7 MeV up to 1.1 MeV and for energies above 1.1 MeV up to 4.5 MeV.The 'best fit' functional forms of S(E) for the respective energy intervals are: where µ(E) is the linear translation of the energy range into the interval [−1:1].The generated total cross sections are presented in figure 9.The uncertainty evaluation of the generated cross sections was performed using the methodology described in section 3 and the results are presented in figure 11.The uncertainty of the generated cross section varies from a fraction of a percent to over 100% near threshold of the fitted cross section.The largest uncertainty is at the ends of the evaluation ranges of the cross sections, i.e. at the reaction threshold energy and the point where the cross section functions were merged.However, the small uncertainties represent mainly the statistical uncertainty from the fitting process.To account for unidentified sources of uncertainty in the measurements, the uncertainty was limited to 2% when preparing the ENDF files, as shown in figure 12, together with the covariance matrix.

Reaction 9 Be(p,α) 6 Li
Reaction 9 Be(p,α) 6 Li produces a fast alpha and a lithium ion, the former of which can interact with beryllium in plasma and produce neutrons and gamma rays.The cross sections for the alpha-production are present in the JENDL-5.0,ENDF/B-VIII.0 and TENDL-2021 nuclear data libraries; the data in TENDL-2021 are just a copy of the data in ENDF/B-VIII.0.The data are easily missed since they are stored implicitly as alpha-production data in MF6/MT5 in ENDF terminology [21].Furthermore, the cross sections represent total alpha emission, mainly at energies above 1 MeV.At lower energies they deviate severely from the measured 9 Be(p,α) cross sections and are not considered further in the present analysis.
The experimental measurements present in the EXFOR database were performed by Qun-Gang Wen in 2008 (EXFOR number D0552003 [26]), Romano in 2006 (EXFOR number O1652003 [27]), Sierk in 1973 (EXFOR number F0169005 [22]), Bertrand in 1968 (EXFOR number F0107006 [23]) and Neuendorffer in 1951 (EXFOR number C0632002 [24]).In addition to the total reaction cross section measurements, differential cross section measurements with respect to angle were also used to determine the total cross sections.The measurements used in the analysis were performed by Bertrand in 1968 (EXFOR number F0107002 [23]) and Morita in 1965 (EXFOR number F0168002 [28]).The experimental measurements of the reaction cross section are presented in figures 13 and 14.From the figure it can be observed, that experimental measurements performed by Romano in 2006 deviate significantly compared to other measurements in the energy range from 0.5 MeV to 1 MeV.We do not understand exactly what was measured, so this data set was excluded from the evaluation of the reaction cross sections.Visual inspection of the measurements indicates the presence of two resonance peaks in the cross sections.For this reason, the fitting process was divided into three parts, namely for energies below 0.8 MeV, from 0.8 MeV up to 1.4 MeV and for energies above 1.4 MeV.The best fitted S(E) functions were: The plot of the generated total reaction cross section is presented in figure 13.The uncertainty evaluation of the generated cross section was performed using the methodology described in section 3 and the results are presented in figure 15.The uncertainty of the generated cross section varies from several percent to over 100% near threshold of the fitted cross section.The covariance matrix of the generated cross section uncertainty was prepared and is presented in figure 16.Reactions presented in previous two section produce fast deuterons and alpha particles, which can interact with beryllium via reactions 9 Be(d,nγ) 10 B and 9 Be(α,nγ) 12 C, producing neutrons and gamma rays.Both of the reactions are present in evaluated nuclear data libraries TENDL-2021 and JEDNL-5 respectively.
Comparison between experimental measurement and the cross section are presented in figures 17 and 18.For reaction 9 Be(d,nγ) 10 B few experimental measurements are available in the region of the peak and the measurements deviate compared to the TENDL-2021 cross sections.However, due to the low number of experimental points, the cross sections in the TENDL library currently present the best available cross sections for this reaction.
For reaction 9 Be(α,nγ) 12 C there is a big discrepancy between the cross section in the TENDL-2021 and JENDL-5 libraries.The cross sections in the JENDL-5 library are in good agreement with measurements up to around 5 MeV while the cross sections from the TENDL-2021 library do not describe any of the measured resonances.For this reason the cross sections from the JENDL-5 library are at this point the best cross sections for the reaction 9 Be(α,nγ) 12 C.However, there is a deviation of the JENDL-5 cross sections for energies above 5 MeV compared to the measurements performed by Gibbons.Covariance matrix of the generated cross section for reaction 9 Be(p,α) 6 Li.This is because the experimentalists actually measured the alpha-emission cross section, which also includes the contribution from the 9 Be(α,nα) 8 Be reaction, for which the cross sections start at 5 MeV.The reaction designation in EXFOR is incorrect.

Conclusion
The neutrons and gamma rays produced in the reactions presented in this paper can be used to commission diagnostic systems in the beginning phases of tokamak operation.Cross section are a crucial part in the computational support of the diagnostic commissioning by calculating neutron and gamma ray emission profiles for Monte Carlo particle simulations.The computational methodology being developed to support detector commissioning in future tokamaks, such as ITER and DEMO, can be validated on specialized experiments performed at currently operating tokamaks, such as the He-3 experimental campaign performed at JET, where neutrons from the reactions studied in this paper were detected.Majority of experimental cross section measurements used in this analysis were performed in the 1950 s and 1960 s with  few measurements performed in the last twenty years.Due to the importance of the studied reactions for initial operation of large tokamaks with beryllium first wall, such as ITER and DEMO, the authors are of the opinion that new cross section measurements of the studied reactions are needed to validate the generated cross sections and to expand the energy range of the evaluated cross sections to higher energies.

Figure 1 .
Figure 1.Comparison of cross section 9 Be(p,nγ) 9 B from the ENDF/B-VIII.0nuclear data library and experimental measurements in EXFOR data base.

Figure 2 .
Figure 2. Experimental measurements and fitted cross section function for the 9 Be(p,nγ) 9 B reaction.

Figure 3 .
Figure 3. Uncertainty of generated total cross section calculated by the Monte Carlo technique from the uncertainties of the fitted polynomial coefficients.

Figure 4 .
Figure 4. Covariance matrix of the generated cross sections for the neutron emission in reaction 9 Be(p,nγ) 9 B.

Figure 5 .
Figure 5.Comparison of the measured and evaluated cross sections from the TENDL-2021 nuclear data library for reaction 9 Be( 3 He,nγ) 11 C.

Figure 6 .
Figure 6.Experimental measurements and fitted cross section function (this work) for the 9 Be( 3 He,nγ) 11 C reaction.

Figure 7 .
Figure 7. Uncertainty of generated total cross section for reaction 9 Be( 3 He,nγ) 11 C calculated by the Monte Carlo technique from the uncertainties of the fitted polynomial coefficients.

Figure 8 .
Figure 8. Covariance matrix of the generated cross sections for the neutron emission from the reaction 9 Be( 3 He,nγ) 11 C.

Figure 9 .
Figure 9. Experimental measurements and fitted cross section function for the reaction 9 Be(p, d)2α.

Figure 10 .
Figure 10.Differential cross section measurements with respect to the emission angle of deuterons.The red curve at zero degrees is the average cosine of scattering.

Figure 11 .
Figure 11.Uncertainty of generated total cross section for reaction 9 Be(p, d)2α calculated by the Monte Carlo technique from the uncertainties of the fitted polynomial coefficients.

Figure 12 .
Figure 12.Covariance matrix of the generated cross section for reaction 9 Be(p, d)2α.

Figure 14 . 8 .
Figure 14.Differential cross section measurements with respect to angle.The red curve at zero degrees is the average cosine of scattering.

Figure 15 .
Figure15.Uncertainty of generated total cross section for reaction9 Be(p,α)6 Li calculated by the Monte Carlo technique from the uncertainties of the fitted polynomial coefficients.

Figure 17 .
Figure 17.Experimental measurements and cross sections from nuclear data library TENDL-2021 for the reaction 9 Be(d,nγ) 10 B.

Table 1 .
Reactions of interest for neutron emission from interactions between plasma ions and beryllium impurity in tokamak plasmas.