PHWT

The efficiency of the C₆D₆ detectors in detecting neutron capture events depends on the de-excitation paths of the compound nucleus, which are too complex to be calculated [30]. The PHWT is required in the measurements to manipulate the response function of the C₆D₆ detector for γ-rays. Thus, the detection efficiency (${ϵ}_{\gamma }$) for γ-rays with energy E_γ satisfies ${ϵ}_{\gamma }=c\cdot E_{\gamma }\mathrm{,}$ (1) where c is the proportionality coefficient. When ${ϵ}_{\gamma }$ is sufficiently low (${ϵ}_{\gamma }\ll 1$), the efficiency of detecting a capture event (ϵ_c) can be described as ${ϵ}_{\text{c}}\simeq {\displaystyle \sum _{i=1}^N{ϵ}_{\gamma i}}=c{\displaystyle \sum _{i=1}^N{E}_{\gamma i}}=c\cdot \left(S_{\text{n}}+E_{\text{n}}\right)\mathrm{,}$ (2) where S_n is the neutron binding energy of the compound nuclei, $E_{\gamma i}$ is the ith cascading γ-ray energy, and En is the kinetic energy of the incident neutron at the center of the mass system. In this study, S_n=7.89 MeV and 7.59 MeV for the compound nuclei of ⁷⁹Br and ⁸¹Br, respectively. However, only a single Sn value can be used in Eq. 2. The method in ref. [31] was adopted in this study, S_n of the most abundant isotope in the sample (⁷⁹Br) was chosen, and the abundance of ⁸¹Br isotopes in the sample was scaled according to its S_n value. In this case, the deviation caused by the selected S_n was less than 3%. The manipulation mentioned above was implemented using a weighting function (WF) [30]. The detector count of each depositional energy was multiplied by the WF to correct for the original γ-ray efficiency of the detectors and hence achieve the relationship expressed in Eq. 1. In this study, energy and resolution calibrations for each individual C₆D₆ detector were performed using standard γ-ray sources, including ⁶⁰Co and ¹³⁷Cs. A realistic Monte Carlo simulation of the experimental conditions and target setup was performed using the Geant4 [32] toolkit and used to obtain an accurate response function. Finally, a least-squares method minimizing ${\chi }^2={\displaystyle \sum _i{\left(c{E}_i-{\displaystyle {\int }_{E_{\text{L}}}^{\infty }}R\left(E_{\text{d}}\mathrm{,}{E}_i\right)WF\left(E_{\text{d}}\right)\text{d}{E}_{\text{d}}\right)}^2}$ (3) was used to determine the WF, where i is the group of monoenergetic γ-rays used in the simulation, E_L is the low-deposited energy threshold of the C₆D₆ detector, and $R\left(E_{\text{d}}\mathrm{,}{E}_{\gamma i}\right)$ is the detector response function. The WF of the natural KBr and gold samples were determined independently using this methodFig. 4. A detailed study of the possible sources of systematic uncertainties revealed that the PHWT had an accuracy of 3% [33].

Fig. 4Full-screen View

(Color online) (a) Original efficiency of the C₆D₆ detector; (b) Efficiency of the detector weighted by the WF; (c) Coefficient of the weighted efficiency vs incident γ-ray energy; (d) WF of the C₆D₆ detector system with KBr and Au samples.

Double-bunch unfolding method

At present, the accelerator complex of the CSNS operates in the normal mode, where each pulse contains two proton bunches, and the interval between the two bunches is 410 ns [13]. The neutrons generated by the two proton bunches overlap with each other. Thus, the neutron energy resolution of TOF measurement is reduced, particularly in the higher-neutron-energy region (above hundreds of eV). An unfolding method based on Bayes’ theorem was developed by Yi et al. [34] to obtain better time and energy resolutions.

In the normal mode of the CSNS, the two bunches in each single-beam pulse are essentially identical, and the temporal structure is extremely reproducible from pulse to pulse, as shown in Refs. [13, 34]. For the statistical time count spectrum measured in the normal mode, the counts of each time bin Ei measured in such a mode should theoretically depend on the counts Ci measured in the single-bunch mode. The transformation from the single-bunch mode spectrum to the normal mode spectrum can be written in the form of a matrix as $E=R\cdot C,$ (4) where the vectors E=(E₀, …, E_i, …, E_n) and C=(C₀, …, C_i, …, C_n), and the transformation matrix R can be expressed as $R=\left[\begin{array}{cccccc}{R}_{11}& & & & & \\ ⋮& R_{22}& & & & \\ R_{\left(1+\text{Δ}\right)1}& ⋮& \ddots & & & \\ & R_{\left(2+\text{Δ}\right)2}& ⋮& R_{ii}& & \\ & & \ddots & ⋮& \ddots & \\ & & & R_{n\left(n-\text{Δ}\right)}& \dots & R_{nn}\end{array}\right]\mathrm{,}$ (5) where Δ is the number of bins corresponding to the offset of the double-bunch interval (410 ns), and $R_{ij}=\frac{1}{2}$ if i=j or j=i-Δ; otherwise, Rij=0. Using Eq. 5, the unfolding problem can be treated as an inverse matrix problem. However, the inverse matrix method occasionally yields several negative-value bins and oscillations caused by statistical uncertainty in the measurements. Thus, an iterative algorithm using Bayesian estimation was developed, and the inverse matrix problem can be replaced by an iterative process [35, 34]: ${\overline{C}}_i^{k+1}=E_i\frac{C_i^{\left(k\right)}}{C_{i-\text{Δ}}^{\left(k\right)}+C_i^{\left(k\right)}}+E_{i+\text{Δ}}\frac{C_i^{\left(k\right)}}{C_i^{\left(k\right)}+C_{i+\text{Δ}}^{\left(k\right)}}\mathrm{,}$ (6) where $C_i^{\left(k\right)}$ indicates the result of the kth iteration, and the initial $C_i^{\left(0\right)}$ is set as Ei measured in the double-bunch distribution. Some unfolding results for a KBr target are shown in Fig. 5

Fig. 5Full-screen View

(Color online) (a) Comparison between the weighted original spectrum of the Br sample (black curve) and the weighted original spectrum obtained from the unfolding process (red curve); (b) ENDF/B-VIII.0 database of Br. In panel (a), the black curve shows the presence of clear double peaks caused by the double-bunch beam structure, whereas the resonance structures are restored when using the unfolding procedure, as indicated by the red curve.

The original spectra preprocessed using the PHWT and double-bunch unfolding methods were normalized using the proton beam number, as shown in Fig. 6.

Fig. 6Full-screen View

(Color online) Preprocessed and normalized (according to the proton beam number) original spectra of KBr, ^natC, ^natPb, and the empty target.

Background

There are two types of components contributing to the background level in captured cross-section measurements with C₆D₆ detectors [36]: sample-dependent background B_sample(t_n) and sample-independent background B_empty(t_n); that is, $B\left(t_{\text{n}}\right)=B_{\text{empty}}\left(t_{\text{n}}\right)+B_{\text{sample}}\left(t_{\text{n}}\right)\mathrm{.}$ (7) The contribution of B_empty(t_n) can be measured directly using an empty sample under the same experimental conditions. On the other hand, the sample-dependent background B_sample(t_n) is caused by interactions between the sample and all types of in-beam particles, including the scattered-neutron-induced background B_sn(t_n), scattered in-beam γ-rays background $B_{\text{s}}\left(t_{\text{n}}\right)$, and sample activation background B_ac.. Thus, the sample-dependent background can be expressed as [36] $B_{\text{sample}}\left(t_{\text{n}}\right)=B_{\text{sn}}\left(t_{\text{n}}\right)+B_{\text{s}}\left(t_{\text{n}}\right)+B_{\text{ac}\text{.}}\mathrm{.}$ (8) The scattered-neutron-induced background B_sn(t_n) can be determined using carbon sample measurement [37], $B_{\text{sn}}\left(t_{\text{n}}\right)=\frac{Y_{\text{Br,el}}}{Y_{\text{C,el}}}\left(W\mathrm{.}{C}_{\text{C}}\left(t_{\text{n}}\right)-W\mathrm{.}{C}_{\text{empty}}\left(t_{\text{n}}\right)\right)\mathrm{,}$ (9) where $Y_{\text{C,el}}$ and $Y_{\text{Br,el}}$ are the neutron scattering yields of the carbon and bromine targets obtained from the database, and $W\mathrm{.}{C}_{\text{C}}\left(t_{\text{n}}\right)$ and $W\mathrm{.}{C}_{\text{empty}}\left(t_{\text{n}}\right)$ are the normalized weighted counts of the carbon and empty samples, respectively. The in-beam γ-rays originated from neutron capture in the water moderator of the spallation source. Indeed, these γ-rays could be scattered by the sample. The target and energy dependence of in-beam γ-ray background components were determined from a measurement of a lead sample and the absorption valleys of 4.28 eV, 5.18 eV, 132 eV, and 5.02 keV of the Ta–Ag–Co filter, as shown in Fig. 7. As shown in this figure, the empty background B_empty(t_n) was subtracted from all the spectra, and the background due to the scattered neutrons from the lead sample was subtracted using Eq. 9. The figure also shows the activation background that was determined by fitting the spectral platform above 11 ms ($E_{\text{n}}\approx 0.2$ eV) [36]. In this region, the neutrons were absorbed by the cadmium absorber, and the in-beam γ-rays could be ignored; the counts in the residual TOF spectrum were attributed to the activation of the sample and surrounding materials. In Fig. 7 (a), the absorption valley at ~5 keV did not match the in-beam gamma-ray background shape determined by the Pb target. We believe that this was caused by delayed gamma-rays from the inelastic scattering reaction channel of ⁸¹Br. ⁸¹Br contained a 536.2 keV level with a lifetime of 34.6 μs, and the inelastic scattering reaction channel of ⁸¹Br opened with a neutron energy greater than ~500 keV (flight time of 15 μs for the Back-n facility). The flight time of the 5-keV absorption valley was approximately 80 μs. In the region of the 5 keV absorption valley, the delayed gamma-ray from ⁸¹Br contributed to some additional gamma-ray counts. However, the flight time for the 132 eV absorption valley was 460 μs, which is far greater than the lifetime of the 536.2 keV level. Therefore, we used the 132 eV black resonance to determine the in-beam gamma-ray background for the KBr crystal. The 536.2 keV level will deexcite through gamma-rays of 260.2 keV and 276 keV. Thus, a threshold of 300 keV was used in the experimental data to verify this hypothesis; the results are shown in Fig. 7 (b). In this figure, the absorption valleys at both 132 eV and 5.02 keV agree with the in-beam gamma-ray background shape determined by the Pb target. However, the majority of the gamma-ray energy released by the composite nuclear decay generated by neutron capture was also below 300 keV; therefore, this threshold was not used in the experimental data processing.

Fig. 7Full-screen View

(Color online) (a) Measured in-beam γ-ray backgrounds are normalized to match the values of the energies of the black resonances, namely, 4.28 eV (Ta filter), 5.18 eV (Ag filter), and 132 eV, and 5.016 keV (Co filter). (b) A threshold of 300 keV is used to avoid the influence of the 536 keV level.

Experimental corrections and absolute neutron flux normalization

For the RRR, sample-related corrections were included in the SAMMY [38] analysis. In the URR, multiple neutron scattering events and self-shielding corrections in the sample were determined through Geant4 simulations, as shown in Fig. 8. ${\sigma }_{\gamma }=\frac{N_{\text{w}}}{I{S}_{\text{n}}}\times \frac{{\sigma }_{\text{t}}}{1-\exp \left(N{f}_{\text{c}}t{\sigma }_{\text{t}}\right)}\mathrm{,}$ (10) where f_c is the correction factor for multiple scattering events and self-shielding effects, and N is the area density in atoms/barn of the sample, which was 0.00339 atoms/barn for the bromine sample and 0.00117 atoms/barn for the gold sample. The uncertainty of the f_c factor was considered to be 1%. The relative normalization of the well-defined energy dependence of the neutron flux could be obtained from various runs using the Li–Si neutron detectors in ES#1. The absolute flux was determined using the gold reference sample with the (n,γ) cross section of ¹⁹⁷Au as a standard.

Fig. 8Full-screen View

Correction factor for the multiple scattering events and self-shielding calculated using the Geant4 toolkit. The black solid dots represent natural Bromine, whereas the red solid dots represent the gold sample.

The first gold resonance at 4.9 eV was used to define the flux in the RRR using the saturated-resonance method, as shown in Fig. 9. The absolute yield normalization was determined through a fit of the gold target data using the R-matrix code SAMMY [38, 39] and by adopting the resonance parameters of Ref. [37]. A systematic uncertainty of 1.5% was adopted for absolute flux normalization. In the keV region, the average (n,γ) cross sections were obtained relative to gold. The background of the Au spectrum was determined using the same method as that used for the bromine spectra.

Fig. 9Full-screen View

Capture yield of the first resonance of ¹⁹⁷Au (4.9 eV) measured at the Back-n facility and normalized by the SAMMY [38] fit.

Discussion of the uncertainties

The total uncertainties, including the statistical and systematic, are discussed. The statistical uncertainty originated from the raw counts in an energy bin of four samples and was estimated to be <2.0%. In fact, because raw counts change depending on the width of the energy bins and the value of the (n,γ) cross sections, wider energy bins help to increase the counts and reduce the statistical error for energy >2.0 keV. However, energy bins that are too wide cannot exhibit a fine resonance structure.

The systematic uncertainty was mainly due to the uncertainty of the experimental conditions and data analysis method. There were several types of uncertainties in the experimental conditions, including the uncertainty of the sample parameter, beam profile of the sample, neutron energy spectrum, and proton beam power. The uncertainty of the data analysis method was mainly caused by the PHWT method, double-bunch unfolding process, normalization, background subtraction, and correction in the URR. Finally, according to error propagation, the overall experimental uncertainty was less than 10.60%, as shown in Table 2. This large error was primarily due to the uncertainty of the neutron spectrum (<8%). Therefore, a good neutron energy spectrum with a lower uncertainty would significantly improve the accuracy of this experiment.

R-matrix fits

In the region of 1 to 2000 eV, the capture yields were analyzed using the R-matrix analysis code SAMMY [38]. The yield was parameterized via the Reich-Moore approximation to the R-matrix formalism. A scattering radius of 6.85 fm and a temperature of 293 K were adopted for the correction of the Doppler effect. Other experimental effects, that is, multiple neutron scattering in the sample and neutron self-shielding, are properly taken into account within the SAMMY code. Resonance broadening owing to the neutron energy resolution function was also considered in the SAMMY fit through the implemented RPI parameterization [40].

The fitting procedure allowed us to extract the resonance parameters (radiation width ${\Gamma }_{}$, neutron width ${\Gamma }_{\text{n}}$, and orbital angular momentum L, etc.) from the measured capture yields. However, in many cases, only the resonance energy E_R and total capture kernel k should be considered as real measurable quantities. The capture kernel k is proportional to the area under an isolated resonance and is given by $k=g{\Gamma }_{\text{n}}{\Gamma }_{}/\Gamma \mathrm{,}$ (11) where g is a statistical factor defined as $g=\frac{2J+1}{\left(2s+1\right)\left(2I+1\right)}\mathrm{,}$ (12) where J is the total angular momentum, 𝒔 is the spin of the incident particles ($s=\frac{1}{2}$ for neutrons), and I is the spin of the target particles.

The resonance parameters of bromine in the Evaluated Nuclear Data Files (ENDF/B-VIII.0) database were adopted as the initial parameters for the SAMMY fitting procedure. The resonance parameters from the ENDF/B-VIII.0 database are consistent with the measurements of Macklin (1988) [12], but it contains resonances that are smaller than those of Ohkubo et al (1981) [11]. The final SAMMY-fitted results for the KBr crystal capture yield are shown in Fig. 10. The black data represent the experimental capture yield measured in this study, and the red solid curve is the actual SAMMY fit to the present data. The fitted resonance parameters and radiative kernels derived using Eq. 11 are listed in Table 5. The contributions of ^39,41K were considered using the resonance parameters from the ENDF/B-VIII.0 database. And the ratio of the capture kernels obtained from the ENDF/B-VIII.0 database and the fitted resonance parameters is shown in Fig. 11.

Fig. 10Full-screen View

(Color online) Analysis of the resonance parameters of the experimental data fitted by the R-matrix code SAMMY.

Fig. 11Full-screen View

(Color online) (a) Ratio of the capture kernels obtained from the ENDF/B-VIII.0 database and the present C₆D₆ results as a function of the resonance energy. (b) Corresponding distribution of the ratios.

For comparison with the evaluated databases, we calculated the neutron capture cross section ${\sigma }_{\text{exp}\text{.}}\left(\text{n},\gamma \right)$ for Br with our experimental yield $Y_{\text{exp}\text{.}}\left(\text{n}\mathrm{,}\lambda \right)$, ${\sigma }_{\text{exp}\text{.}}\left(\text{n}\mathrm{,}\gamma \right)=\frac{Y_{\text{exp}\text{.}}\left(\text{n}\mathrm{,}\gamma \right){\sigma }_{\text{tot}}}{\left(1-e^{-N{\sigma }_{\text{tot}}}\right)}$ (13) where σ_tot is the total cross section calculated from the ENDF/B-VIII.0 database, and N is the atomic number density of Br in the KBr crystal. The calculated ${\sigma }_{\text{exp}\text{.}}\left(\text{n},\gamma \right)$ are plotted as blue dots in Fig. 12.

Fig. 12Full-screen View

(Color online) Capture cross section ${\sigma }_{\text{exp}\text{.}}(\text{n}\mathrm{,})$ calculated from the experimental yields obtained in this study (black dots). The SAMMY fitted yield (red solid line) and evaluated databases are plotted for comparison.

Statistical analysis

The present set of resonance parameters was used for statistical analysis to determine the nuclear properties required for the cross-section calculations [39]. The cumulative number of resonances as a function of the neutron energy is shown in Fig. 13 (a) and (b) for ⁷⁹Br and ⁸¹Br, respectively. This figure provides an efficient method to investigate the population and missing levels. The average s-wave level spacings D₀ are directly related to the inverse slope of these plots and can be obtained from the linear least-squares fits indicated by the straight lines as 57.397 eV and 27.855 eV for ⁷⁹Br and ⁸¹Br, respectively. The points fall below the fitted straight line, indicating that the levels were missed in the resonance analysis. In addition, the average radiative widths $\langle {\Gamma }_{\gamma }\rangle$ were calculated using the SAMMY fitted resonance parameters, which were 287.3±10.3 meV and 295.5±12.5 meV for ⁷⁹Br and ⁸¹Br, respectively.

Fig. 13Full-screen View

(Color online) Staircase plots of the cumulative numbers of resonances in the investigated bromine isotopes.

MACSs

In the continuum region below 370 keV, average cross sections were obtained with a resolution of 20 bins per decade. The averaged cross section relative to the gold sample σ_Br(E_n) is given by ${\sigma }_{\text{Br}}\left(E_{\text{n}}\right)=\frac{{\sigma }_{\text{Au}}\left(E_{\text{n}}\right)}{\langle {\sigma }_{\text{Au}}\left(E_{\text{n}}\right)\rangle }\langle {\sigma }_{\text{Br}}\left(E_{\text{n}}\right)\rangle \mathrm{,}$ (14) where $\langle {\sigma }_{\text{Au}}\left(E_{\text{n}}\right)\rangle$ and $\langle {\sigma }_{\text{Br}}(E_{\text{n}})\rangle$ are the experimental values measured in this study and σ_Au(E_n) is the evaluated value obtained from the ENDF/B-VIII.0 database. The uncertainty of the standard ¹⁹⁷Au cross section recommended by the ENDF/B-VIII.0 library was estimated to be of the order of 6.0% below 200 keV and 4.0% between 200 and 300 keV. In the URR, the average cross section of K was lower than that of Br by two orders of magnitude, and the neutron capture events of K atoms could be ignored in this region. Finally, the results of the cross section of Br in the continuum region from 10 to 370 keV are given in 20 bins per decade in Table 3.

As shown in Fig. 14, the average cross sections obtained in this study in the continuum region (red dots) were compared with previous experimental results and the evaluated database. Our measurements are consistent with the results of Gibbons et al. (1961) and the results of Popov et al. (1961), but higher than those of all evaluated databases. For comparison, in this figure, we also plot black dots, which were calculated using the neutron flux determined by the Li–Si detector. In the region over 200 keV, the Li–Si detector-determined results were obviously higher than the results measured with the gold sample. In addition, fluctuations near 30 keV in the results determined by the Li–Si detector were caused by aluminum material in the neutron beam pipe.

Fig. 14Full-screen View

(Color online) Capture cross sections of bromine in this study relative to the ¹⁹⁷Au sample (red square dots). The cross sections calculated using the neutron flux determined by the Li–Si detector are also plotted for comparison.

The TALYS code was used to describe the isotopic average cross sections in the URR. The calculations were based on the Hauser–Feshbach statistical emission model, which assumes that the capture reactions occur by means of a compound nuclear system that reaches statistical equilibrium. The previously obtained statistical average level space D₀ average radiation width $\langle {\Gamma }_{\gamma }\rangle$ was used as the input parameter for the TALYS code calculations. In addition, the global neutron optical model potential in Ref. [41, 42] was used in the calculations. Other parameters were chosen using the method reported in Chen et al. (2020) [43], the photon strength function was given by Kopecky and Uhl [45, 46], and the level density a and nuclear temperature T were given by the Gilbert-Cameron model with adjusted parameters. In addition, $\langle {\Gamma }_{\gamma }\rangle$ was systematically multiplied by a factor of 0.9 for both isotopes to obtain the γ transmission coefficient. The calculated capture cross sections effectively reproduced the available experimental average cross sections of ⁷⁹Br and ⁸¹Br, as illustrated in Fig. 15.

Fig. 15Full-screen View

(Color online) Average capture cross sections of ⁷⁹Br and ⁸¹Br calculated using the TALYS code. The theoretical average cross sections of ⁷⁹Br are essentially consistent with the previous measurements and evaluated values, whereas those of ⁸¹Br roughly agree with the previous measurements but are higher than those of the evaluated database.

From these average cross-section values in Fig. 15, the Maxwell average cross sections (MACSs) of ⁷⁹Br and ⁸¹Br were calculated for thermal energies kT ranging from 5 to 100 keV according to ${\langle \sigma \rangle }_{kT}=\frac{2}{\sqrt{\pi }}\frac{{\displaystyle {\int }_0^{\infty }}\sigma \left(E_n\right)E_n{\text{e}}^{-E_n/kT}\text{d}{E}_n}{{\displaystyle {\int }_0^{\infty }}{E}_n{\text{e}}^{-E_n/kT}\text{d}{E}_n}\mathrm{,}$ (15) and the corresponding results are listed in Table 4, respectively.

Figure 16 shows a comparison of our results, the evaluated databases, and the recommended values compiled in the Karlsruhe Astrophysical Database of Nucleosynthesis in Stars (KADoNiS)[47]: (a) the MACS values of ⁷⁹Br obtained in this study, which were essentially located between the values of the JEFF-3.3[48], TENDL-2021[49], ENDF/B-VIII.0[50], and JENDL-5[51] databases, are in good agreement with the KADoNiS v1.0 values; (b) for ⁸¹Br, the calculated values were considerably higher than those of the evaluated database and the KADoNiS v1.0 recommended values.

Fig. 16Full-screen View

(Color online) MACSs calculated using Eq. 15 for ⁷⁹Br and ⁸¹Br, respectively.

Figure 17 shows a comparison of our results with the previously recommended MACSs from the experimental results[52], evaluated databases and theoretical values[53-55] in nuclear astrophysics concerned with a thermal energy of kT=30 keV. The value of 682±68 mb for ⁷⁹Br shown in Fig. 17 (a) is in good agreement with the KADoNiS v1.0 recommended value of 661±44 mb within the uncertainty range. However, the MACS value of 293±29 mb for ⁸¹Br shows a clear discrepancy from the KADoNiS v1.0 recommend value of 248±10 mb.

Fig. 17Full-screen View

MACSs at a thermal energy of kT=30 keV for ⁷⁹Br and ⁸¹Br. The blue-triangles represent previous experimental results: Heil et al. [3, 5], Macklin [12], Walter et al. [44], and Allen et al. [52], whereas the three dots labeled KADoNiS v0.0, KADoNiS v0.3, and KADoNiS v1.0 are the recommended values from three version of the KADoNiS database. The red dots labeled ENDF/B-VIII.0, JEFF-3.3, TENDL-2021, and JENDL-5 are calculated from the evaluated databases according to Eq. 15. The black dots represent the theoretical values taken from Goriely [54], Rauscher et al. [55], Harris [53], and Woosley et al. [4].