Method of neutron-induced total reaction cross section determination

The neutron-induced total reaction cross section was determined in a transmission experiment by attenuating the neutron beam through the target [13]. The transmission rate is the ratio of the neutron counting rate through a sample to the neutron counting rate without a sample. This method has high measurement accuracy because it does not require the absolute measurement of the neutron fluence rate and does not consider the detection efficiency. The transmission T at incident neutron energy E_i is given by Eq. (1). $T\left(E_{\text{i}}\right)=\frac{C\left(E_{\text{i}}\right)}{C_0\left(E_{\text{i}}\right)},$ (1) where C and C₀ are the neutron-counting rates of the detector with the sample inside and outside the beam, respectively.

The relationship between the neutron total cross section σ and the transmission is given by Eq. (2). $\sigma \left(E_{\text{i}}\right)=-\frac{1}{nd}\ln T\left(E_{\text{i}}\right),$ (2) where nd is the areal density given by the product of the number density of atoms n and the sample thickness d.

The time-of-flight (TOF) technique is typically used to determine incident neutron energy [14]. It determines the neutron energy by measuring the time at which the neutron flies over a fixed distance. In TOF measurements, the incident neutron energy is expressed by Eq. (3). $E_{\text{i}}=m_{\text{n}}{c}^2\left(\frac{1}{\sqrt{1-{\left(\frac{L}{c\cdot {\text{TOF}}_{\text{n}}}\right)}^2}}-1\right),$ (3) where m_n is the mass of the neutron, c is the speed of light, L is the neutron flight length, and TOF_n is the neutron flight time. The relativistic effect considered as the neutron energy was relatively high.

Experimental setup

The Back-n white neutron source and its supporting spectrometer system provide a powerful platform for measuring and researching nuclear data. On the Back-n, neutrons are produced by 1.6 GeV proton-induced spallation reactions on a tungsten target. It can provide high-throughput, wide-energy-range pulsed neutrons. The acquisition of the complete total neutron cross section throughout the entire energy spectrum via a solitary measurement has the potential to significantly enhance both the experimental efficiency and accuracy [15-17]. In the experiment, the accelerator was operated with a proton beam power of approximately 125 kW and a pulse repetition rate of 25 Hz. The double-bunch mode was used. The incident proton pulse was in two bunches, approximately 70 ns (FWHM) wide and 410 ns apart. The experimental setup is shown in Fig. 1. The double-bunch mode results in the superposition of the event distributions for two bunches. To extract the single-bunch distribution from the smeared double-bunch distribution, it was necessary to perform double-bunch unfolding, as described in Sect. 3.4.

Fig. 1Full-screen View

Layout of the experimental setup

Charged particles emitted from the opposite angle to the incident proton beam were deflected by a magnet to obtain a neutron beam. Subsequently, the neutron beam was collimated using a neutron beam switch and collimator before being directed into the experimental area. The experimental setup based on the NTOX spectrometer [18] consisted of neutron flight tubes, an automatic sample changer, and a multi-cell fission chamber [19]. A neutron collimator was installed on the inner wall of the neutron flight tubes. During the experimental measurements, the entire tube remained in a vacuum state. The automated sample changer can be remotely controlled. The multi-cell fission chamber can be filled with similar or different 8 pieces of samples, and four ²³⁵U (²³⁵U-1,²³⁵U-2,²³⁵U-4,²³⁵U-5) and four ²³⁸U (²³⁸U-1,²³⁸U-2,²³⁸U-4,²³⁸U-7) high-purity samples were used in this experiment. The chamber was filled with a gas mixture of 90% Ar and 10% CF₄ at 800 mbar. The detector was connected to the MesytecMSI-8 preamplifier. It was then connected to the electronics and the Back-n general DAQ system, which was developed by the University of Science and Technology of China, and was used to digitize and store signals from the preamplifiers [20, 21].

Selecting excessively thick samples for the neutron total cross section experiment increases the contribution of neutron scattering, whereas excessively thin samples will make the transmission significantly large, resulting in an increase in the uncertainty of the total cross section. Therefore, a suitable sample thickness should be selected such that the transmission is between 0.5 and 0.7. In the present study, a high-purity cylindrical natural lead with a thickness of 10 mm was used for the transmission measurements. The characteristic parameters of the sample are listed in Table 1. The uncertainty in the sample thickness determination was within 0.07%. The cumulative irradiation time for the sample was 142 hours and 50 hours for the measurement without the sample.

Neutron flight time determination and flight path calibration

According to Eq. (3), to obtain fission events as a function of the neutron TOF and energy, it is necessary to know the exact time-zero and the length of the neutron flight path. The spallation reaction of the proton beam bombarding the tungsten target to produce neutrons produces a large number of high-intensity γ-flash promptly. The γ-flash at the speed of light arrived at the detector earlier than the neutrons; thus, the time difference between the detected γ-flash and the measured neutron signal can be used to calculate the flight time of neutrons TOF_n. The calculation formula is shown in Eq. (4). ${\text{TOF}}_{\text{n}}=T_{\text{n}}-T_{\text{n0}}=T_{\text{n}}-\left(T_{\gamma }-{\text{TOF}}_{\gamma }\right),$ (4) where T_n is the neutron arrival time determined by the fission signal, T_n0 is the generation time of the neutron, Tγ is the time at which the γ-flash is detected,and TOFγ is the flight time of γ-flash from the target to the fission chamber.

The neutron flight path calibrations were performed with resonance peaks of 8.774 eV, 12.385 eV, and 19.288 eV for ²³⁵U. T_n of the resonance peak can be obtained by fitting the fission time spectrum with a Gaussian function [23]. The neutron flight distance can be obtained by fitting the relationship between the resonance peak energy and the neutron arrival time T_n in Eq. (5). $T_{\text{n}}-T_{\gamma }=\frac{L}{c}\left[\frac{1}{\sqrt{1-{\left(\frac{E_{\text{n}}}{m_{\text{n}}{c}^2}+1\right)}^{-2}}}-1\right],$ (5) The fitting results for ²³⁵U fission cells are listed in Table 2. The flight paths of the four ²³⁸U fission cells were not calibrated because of the small response of the reaction in low-energy regions. The fission chamber was set up with a fission cell spacing of approximately 19 mm, and the flight path corresponding to the ²³⁸U fission cells was consequently increased by 19 mm.

Background subtraction

For the experimental measurement of the neutron-induced total reaction cross section, it is important to deduct the background. The presence of an experimental background increases the measured neutron transmission, resulting in a smaller cross section.

The background in a time-of-flight measurement consists of two components [14]. The first is the background, which is independent of the neutron beam intensity, and its time structure. This background can be determined by measuring the time at which the proton beam is switched off. In this study, this type of background mainly originates from the alpha decay of the fission cells. Thirty-five hours of beam-off measurements were performed, and it was found that the background signal amplitude was significantly low compared to the neutron signal amplitude. This can be subtracted by setting a threshold, as shown in Fig. 2. The second part is related to the generation and transmission of the neutron beam, which varies with time. It includes the γ-flash produced by the scattering target and the neutrons scattered by the surrounding walls, floor, and instruments.

Fig. 2Full-screen View

(Color online) Alpha signal time-amplitude 2D distribution and sample-out TOF-amplitude 2D distribution of the ²³⁵U-1 and ²³⁸U-1 cells

The TOF and amplitude statistics of the signals from each fission cell were obtained, resulting in a sample-out TOF-amplitude 2D distribution and amplitude distribution of ²³⁵U-1 and ²³⁸U-1 cells, as shown in Fig. 2. A clear boundary area between the small-amplitude background signals and the neutron signals is shown. The distribution of the amplitudes was derived through a statistical analysis of the signal amplitudes on each fission cell, as shown in Fig. 3. The solid blue and dashed black lines in the figure corresponds to the sampled-out signal amplitude distribution of the ²³⁵U-1 fission cell and the alpha background signal amplitude distribution, respectively, shown in Fig. 2. The lowest value of the valley was selected as the signal discrimination threshold. The background signal below the threshold was subtracted from subsequent analysis. After the analysis, there was no influence of alpha particles at the threshold, and alpha particles only accounted for 0.0007% of the neutron signal.

Fig. 3Full-screen View

(Color online) Sample-out amplitude distribution of the ²³⁵U-1 cell

To estimate the on-beam neutron background more accurately, we measured the Co, In, Ag, and Cd filters. The filters were inserted in front of the ^natCr samples. The measurement of the neutron-induced total reaction cross sections of ^natCr will be published in the future. The fission events as a function of neutron energy were measured under the same experimental conditions, and different filters had different resonance peaks to maximize the absorption of neutrons in the beam. The characteristic filter parameters are listed in Table 3. There are many depressions in the low-energy region, each corresponding to the resonance of one characteristic energy. The fission events as a function of the neutron energy of the filters and the corresponding neutron-induced total cross sections are shown in Fig. 4.

Fig. 4Full-screen View

(Color online) Fission events as a function of neutron energy of the filters and the corresponding neutron-induced total cross-sections from ENDF/B-VIII.0

During the measurement time of 22 hours, the neutron counts within the energy ranges from 1.34 eV to 1.55 eV, 4.84 eV to 5.37 eV, and 123 eV to 140 eV were 81, 83, and 73, respectively. Compared with the neutron count of the beam, the neutron counts of the background were significantly small. Therefore, this portion of the background was ignored.

Neutron transmission analysis

The fission events as a function of neutron energy were obtained using Eq. (3) after subtracting the background. During the experiment, the proton beam intensity was simultaneously recorded. These data were proportional to the neutron yield and could be used as a normalization parameter for the fission events as a function of neutron energy. To increase the neutron counts and reduce the statistical error, the neutron counts counted by individual fission cells can be combined [24]. The normalized fission events of the superimposed ²³⁵U and ²³⁸U fission cells as a function of neutron energy are shown in Fig. 5 and Fig. 6. Because the multi-cell fission chamber has a serious resonance region in its fission material, with the ²³⁵U resonance peak in the 1 eV–1 keV energy region being clearly visible in the figure. The statistical uncertainties in the neutron counts are mostly within 1%. Starting from Eq. (1), the neutron transmission after considering the background and normalization factors can be expressed by Eq. (6). $T\left(E_i\right)=\frac{\left(M\left(E_i\right)-B\left(E_i\right)\right)/P}{\left(M_0\left(E_i\right)-B_0\left(E_i\right)\right)/P_0},$ (6) where M and M₀ are the signal counts of the beam-on measurements with and without the sample placement, respectively. B and B₀ are the background counts of the beam-on measurements with and without the sample placement, respectively. P and P₀ are the normalized constants based on the monitored counts of the proton beam.

Fig. 5Full-screen View

(Color online) Normalized fission events as a function of neutron energy and the corresponding transmission measured by ²³⁵U fission cells

Fig. 6Full-screen View

(Color online) Normalized fission events as a function of neutron energy and the corresponding transmission measured by ²³⁸U fission cells

The transmission spectra corresponding to the fission events of ²³⁵U and ²³⁸U as a function of neutron energy are shown in Figs. 5 and 6. The sample transmission was approximately 0.7 in the low-energy region. Due to the small number of counts in the low-energy region, the transmission fluctuates significantly in this energy region. The neutron transmissions obtained from the ²³⁵U and ²³⁸U fission cells were compared, as shown in Fig. 7. The result indicates that the data analysis and experimental results are consistent.

Fig. 7Full-screen View

(Color online) Comparison of the neutron transmissions obtained from the ²³⁵U and ²³⁸U fission cells

Double-bunch unfolding

The CSNS accelerator usually operates in a double-bunch mode to increase beam power and neutron intensity. This mode involves the presence of two identical proton bunches within a single pulse, separated by a fixed interval of 410 ns. This double-bunch effect results in the superposition of the neutron TOF spectrum, which introduces nonnegligible uncertainty in the neutron TOF measurement, particularly in the high-energy region. To recover the original fission events as a function of neutron TOF corresponding to the single-bunch mode, an unfolding algorithm based on the iterative Bayesian method was used to solve this problem. The detailed principle and algorithm derivations can be found in [25]. The fission events of ²³⁵U and ²³⁸U as a function of neutron TOF and energy obtained in the experiment were performed using the DemoUnfolding_v3.4 code in the energy ranges from 10 keV to 20 MeV and 1 MeV to 20 MeV, respectively. The fission events as a function of neutron TOF in the double-bunch mode were set with a linear uniform distribution of 41 ns per channel, and the fission events as a function of the neutron energy were set to 100 bins per decade, with five iterations.

Result and uncertainty analysis

Based on the neutron transmission spectrum, the energy-dependent neutron total cross sections can be obtained using Eq. (2). Owing to the large amount of data in the evaluation databases, the evaluation data were compared separately. Figure 8 shows a comparison of the evaluated data of the ^natPb neutron-induced total reaction cross sections from the five main nuclear databases (i.e., ENDF/B-VIII.0, JEFF-3.3, JENDL-5, CENDL-3.2, and BROND-3.1) and the ratio of other evaluation databases over CENDL-3.2. Significant differences were observed among the evaluation databases in the resonance-energy region.

Fig. 8Full-screen View

(Color online) Comparison of the evaluated data of the ^natPb neutron-induced total reaction cross sections from the five main nuclear databases and the ratio of other evaluation databases over the CENDL-3.2 library

Figure 9 shows the comparison of the neutron-induced total reaction cross sections of ^natPb measured by ²³⁵U fission cells with the experimental data available in the EXFOR library and the CENDL-3.2 evaluation data in the energy range from 0.3 eV to 1 keV. The experimental results were generally in good agreement with the CENDL-3.2 library. In this energy region, the neutron-induced total reaction cross section is basically a straight line. Experimental measurements show significant fluctuations around 4 eV due to low neutron counts. The Ta-181 material wrapped around the spallation target exhibits high neutron absorption near 4.28 eV. Previous experimental results for this energy region were relatively few, and the measurement results of Muhrer et al. [26] were consistent with our results; however, their uncertainties were relatively high. The measurement results of W. W. Havens Jr et al. (data from the EXFOR library) were lower than ours.

Fig. 9Full-screen View

(Color online) Neutron-induced total reaction cross sections of ^natPb compared with previous experimental data and the CENDL-3.2 library in the energy range from 0.3 eV to 1 keV

Figure 10 shows the neutron-induced total reaction cross sections of ^natPb after unfolding, as measured by ²³⁵U and ²³⁸U fission cells. These measurements were compared with previous experimental data and the CENDL-3.2 evaluation data in the energy range from 1 keV to 20 MeV. Good agreement was is found between the present and previous experimental results as well as with the CENDL-3.2 evaluation data. In the energy range from 1 keV to 200 keV, multiple resonance peaks were measured experimentally, which showed good agreement with the CENDL-3.2 library. However, some of the resonance peaks were not obvious, and the experimental data were generally smaller than the CENDL-3.2 data. This is mainly because the incident proton pulse had a spreading width of approximately 70 ns, whereas the resonance peak of Pb is extremely narrow and requires an energy resolution of 0.1%. There were also few experimental measurements in this energy region, and the measured resonance peaks remained inconsistent. In the energy range from 200 keV to 20 MeV, the experimental results in the high-energy region are consistent with CENDL-3.2 evaluation data. Furthermore, there are many experimental results in the EXFOR library, and the measurement data of Pruitt et al. [27] and Larson et al. (data from the EXFOR library) agree well with our measurement results, except for the results of Laptev et al. [28].

Fig. 10Full-screen View

(Color online) Neutron-induced total reaction cross sections of ^natPb compared with previous experimental data, the CENDL-3.2 library, and the theoretical calculation curve by TALYS-1.96 in the energy range from 200 keV to 20 MeV

As shown in Eq. (2), the uncertainty of the total cross section is mainly due to the uncertainties of the atomic density of the sample and the transmission. The uncertainty in transmission is related to the statistical uncertainty in the neutron counts. Uncertainty can also arise during double-bunch unfolding. The unfolding code uses an iterative method to calculate the error of each channel, including the error of the fission events as a function of neutron TOF and energy. The uncertainty of the neutron total cross section is given by Eq. (7). $\frac{\text{Δ}\sigma }{\sigma }=\frac{\sqrt{{\left(\frac{\text{Δ}T}{\text{T}}\right)}^2+{\left(\ln T\right)}^2\left[{\left(\frac{\text{Δ}d}{\text{d}}\right)}^2+{\left(\frac{\text{Δ}n}{\text{n}}\right)}^2\right]+{\left(e_{\text{in}}\right)}^2+{\left(e_0\right)}^2}}{-\ln T}$ (7) where e_in and e₀ are the uncertainties in unfolding with and without placing a sample, respectively.

For the measurement of the ^natPb sample of 10 mm thickness, the energy points (bins) were 779 and 89 with the ²³⁵U and ²³⁸U fission cells, respectively. The experimental uncertainties of the total neutron cross section were within the ranges of 0.74%–10% and 2.8%–12% with the ²³⁵U and ²³⁸U fission cells, respectively. Uncertainties below 2% were found at 32.6% of the energy points, and uncertainties below 5% were found at 92.3% of the energy points with ²³⁵U fission cells. Uncertainties below 4% were found at 49.4% of the energy points and uncertainties below 6% were found at 91% of the energy points with ²³⁸U fission cells.

Theoretical calculation

TALYS is a computer program for the analysis and prediction of nuclear reactions, developed by Koning et al. [29] of the Nuclear Research and Consultancy Group (NRG) in the Netherlands. It is used to calculate nuclear reactions involving neutrons, photons, protons, deuterons, tritons, ³He, and alpha particles with energies ranging from 1 keV to 200 MeV through an optimal combination of reliable nuclear models, flexibility, and user-friendliness. The TALYS program considers various nuclear models, usually categorized as optical, direct-reaction, pre-equilibrium, compound, and fission models, all driven by a comprehensive library of nuclear structures and model parameters. The details of the theory and application of the TALYS program are described in Ref. [29]. The theoretical calculations of the excitation function of the neutron total cross sections of ^natPb were performed using the TALYS-1.96 program, for which the excitation curve in the neutron energy range from 200 keV to 20 MeV is shown in Fig. 10 for comparison. The code was used with the default options, using the optical model parameters from Koning and Delaroche [30]. Overall, the trend of the total cross section calculated using TALYS-1.96 shows good agreement with the experimental data. The two results are in high agreement in the nonresonant region but do not reflect the resonant region well. Further studies on the theoretical model of the nuclear reactions of lead are necessary.