Uncertainty estimation in neutron TOF measurements with ANNRI

. To improve data accuracy of neutron-capture and total cross sections of minor actinides and long-lived ﬁssion products, the time-of-ﬂight experimental instrument named ”Accurate Neutron-Nucleus Reaction measurement Instrument” was constructed, and neutron time-of-ﬂight experiments have been performed with an intense pulsed-neutron beam at MLF in J-PARC. Detailed analysis methods and deduced uncertainties for capture and total cross-section measurements are presented using an example of the experiments of 243 Am. The deduced uncertainties are categorized into ﬁve types.


Introduction
For detailed engineering designs and safety evaluations of nuclear reactor systems, accurate cross-section data are crucial. Especially, neutron-capture cross sections of minor actinides (MAs) and long-lived fission products (LLFPs) are significant for evaluating and transmuting radioactive wastes and designing various innovative reactor systems [1][2][3]. However, accurate measurements of these cross sections are particularly challenging because of their high specific radioactivity. As a result, highly accurate data are not available for all materials.
To address this issue, Hokkaido University, Tokyo Institute of Technology, and JAEA collaborated to build the Accurate Neutron-Nucleus Reaction measurement Instrument (ANNRI). The ANNRI is installed at Beam Line No. 04 of the Materials and Life science experimental Facility (MLF) of the Japan Proton Accelerator Research Complex (J-PARC), which provides the world's most intense pulsed neutron source. A high flux neutrons supplied by the 1-MW pulsed spallation neutron source in the J-PARC allows better conditions for radioactive sample background reduction. Since 2008, high-intensity pulsed neutron measurements of the neutron capture and total cross sections of MAs, LLFPs, and some stable isotopes have been conducted [4]. This paper presents detailed information about the analysis methods and estimated uncertainties in TOF experiments in the ANNRI by using capture, and total crosssection measurements of 243 Am as examples [5].

ANNRI Beamline
Pulsed neutrons are produced by the spallation reaction in the mercury target in MLF, by injecting 3-GeV pulsed protons from the rapid cycling synchrotron in the J-PARC [6]. * e-mail: kimura.atsushi04@jaea.go.jp Produced neutrons are slowed down in three types of supercritical hydrogen moderators and transported to beamlines in MLF. A Coupled Moderator, which produces the most intense neutron beam, supplies neutrons to the AN-NRI. Figure 1 shows the vertical cross-sectional view of ANNRI. To provide a high-quality neutron beam to detector systems, neutron collimators, neutron resonance filters, and disk choppers have been installed.

Detector systems in ANNRI
The ANNRI has three detector systems. An array of large Ge detectors and a NaI(Tl) spectrometer were installed at flight lengths of 21.5 and 28-m, respectively. Measurements of capture cross-section are performed using these two gamma-ray spectrometers. Li-glass detectors are installed at a flight length of 28.5 m and are used for total cross-section measurements.
The array of Ge detectors consists of two cluster-type Ge detectors, eight coaxial-type Ge detectors, and anticoincidence shields around each Ge detector [7]. There are seven Ge crystals in each cluster-type Ge detector. The array of Ge detectors consequently consists of 22 Ge crystals. The photo-peak efficiency of the array of Ge detectors is 2.28 ± 0.11 % for 1.33 MeV rays. Signals from the Ge detectors are fed into CAEN V1724 (14 bit, 100 MHz) ADC boards [8]. The TOF and pulse height(PH) are recorded in an event-by-event mode.
The NaI(Tl) spectrometer comprises two NaI(Tl) detectors with neutron and γ-ray shields [4]. The cylindrical NaI(Tl) detectors, 330 mm in diameter and 203 mm in length and 203 mm in diameter and 203 mm in length, are located at 90 degrees and 125 degrees to the neutron-beam line, respectively. If necessary, the detector at 125 degree is used to reduce the effects of angular distributions of capture γ rays from resonances with l > 0. Signals from NaI(Tl) detectors are analyzed in the DAQ with a CAEN 1720 (12bit 250MHz), and TOF and PH are recorded in the same manner as the array of Ge detectors.
Two types of Li-glass detectors are installed in AN-NRI [9]. A 6 Li-glass detector is used to obtain the neutron TOF transmission spectra. The scintillator of the 6 Liglass detector is GS-20 produced by Saint-Gobain with a dimension of 50 mm × 50 mm and a thickness of 1 mm using enriched 6 Li (> 95%). Neutrons are detected via the 6 Li(n,α) 3 H reaction. A 7 Li-glass detector is used to determine the background shape due to γ rays. The scintillator of the 7 Li-glass detector is GS-30 ( 7 Li enrichment higher than 99.99%) produced by Saint-Gobain with the same size and chemical component as the GS-20 scintillator. Thus, the neutron sensitivity of the 7 Li-glass detector is significantly smaller than that of the 6 Li-glass detector due to the small neutron absorption cross section. However, both detectors have almost the same sensitivities to gamma rays. The 7 Li-glass detector is located on the upstream side of the 6 Li-glass detector. Signals from the detectors are digitized in a CAEN 1720, and TOF and PH are recorded.

Neutron beam at ANNRI
The neutron intensity at the 21.5 m sample position of the ANNRI under proton beam power of 17.5 kW is compared to those of DANCE at LANSCE and n TOF at CERN in Figiure 2 [10]. The current proton beam power is 800 kW in July 2022. The deduced present neutron intensity and the expected neutron intensity under the future 1-MW operation are also presented in Figiure 2. One finds that the present and future neutron intensities of ANNRI are much higher than those of the other facilities.

Categolization
In an analysis of TOF experiments, uncertainties are deduced step by step. Here, the deduced uncertainties are categorized into the following five types: • Statistical uncertainty • Type-1: proportional to the deduced cross section (normalization uncertainty), • Type-2: proportional to other cross sections (impurities), • Type-3: constant on the deduced cross section, • Type-4: the combination of some uncertainties.
One is statistical uncertainty. Uncertainties due to dead time correction and background subtraction include statistical uncertainty. The others are systematic uncertainties categorized into four types. Type-1 is the uncertainty proportional to the deduced cross section; for example, uncertainties due to normalization and pulse height weighting technique are categorized in Type-1 uncertainties. Type-2 is the uncertainty proportional to other cross sections; uncertainty due to impurity subtraction (sample composition) is categorized in this uncertainty. Type-3 is the uncertainty proportional to the transmission ratio and makes constant uncertainty in the deduced cross section; only neutron intensity normalization at the total cross-section measurement belongs to this type. Type-4 is the uncertainty due to a combination of some uncertainties; for example, the uncertainty due to self-shielding and multiple scattering belongs to this type. This uncertainty is deduced from sample shape uncertainty with Monte Carlo simulations.

Uncertainties in Resonance analysis
In the resonance region, resonance analysis is typically applied with resonance analysis codes, e.g., REFIT and SAMMY codes. Statistical uncertainty and Type-2 (impurities) are considered in the code. For Type-1 (normalization) and Type-3 uncertainties, some methods were proposed by P. Schillebeeckx et al. [11], and a new approach was presented by S.

Analysis methods and uncertainties in TOF experiments in ANNRI
This section presents the analysis and uncertainty estimation methods in TOF experiments in ANNRI using the 243 Am capture and total cross-section measurements as examples. Detailed information about the 243 Am experiments is described in Ref. [5].

Setup of 243 Am experiments
An 243 Am sample with a nominal activity of 240 MBq was prepared for the experiments. Americium dioxide (AmO 2 ) powder was mixed with Y 2 O 3 powder and pressed into a pellet with a diameter of 10.0 ± 0.1 mm and a thickness of 0.5 ± 0.1 mm. The pellet was encapsulated in an Al container with a diameter of 22 mm and a thickness of 0.1 mm. Similarly, the dummy sample was prepared, which contained only a Y 2 O 3 pellet. The calorimetric method was applied to determine the activity of the 243 Am sample. The determined activity was 281.8 ± 0.3 MBq. The isotopic composition of the Am sample was analyzed by thermal ionization mass spectrometry and alpharay spectroscopy. The isotopic purity of 243 Am was determined to be 97.67 ± 0.02 %. The array of Ge detectors was used for the neutron capture cross-section measurement. The 243 Am sample was put in a bag of fluorinated ethylene propylene (FEP) films and attached to a sample holder. The measurement time was about 13 hours. For the background estimation, measurements were carried out with the dummy sample, the FEP bag without any sample (Blank), and an enriched 208 Pb sample. Measurements with an enriched 10 B sample with a diameter of 5.0 ± 0.1 mm and a weight of 170.2 ± 0.1 mg were also made to determine the energy dependence of the incident neutron flux.
The 6 Li-glass and 7 Li-glass detectors were utilized for transmission measurements. The 7 Li-glass detector was installed on the upstream side of the 6 Li-glass detector. A lead filter with a thickness of 87.5 mm was installed in the neutron beamline to reduce the background gamma ray on the beamline and to control the event rate of the Li-glass detectors. The background was estimated with the black resonance technique using neutron notch filters of nat Mn, nat Co, nat In and nat Ag. The measuring times of the 243 Am and the dummy samples were 16 h and 8 h, respectively. Figure 3 shows the data analysis procedure for the capture cross-section measurements in ANNRI. In the analysis of the 243 Am experiment, uncertainties were deduced step by step.

Capture cross-section measurement
The Pulse Height Weighting Technique (PHWT) was usually applied to capture measurements [12]. The response functions of the array of Ge detectors were calculated by Monte Carlo simulation. The weighting functions for the sample were calculated from the response functions. Figure 4 shows weighted TOF spectra of the 243 Am and dummy samples. In PHWT, the most significant uncertainty originated from correcting undetected events below the threshold level. The correction was done by extrapolating the PH spectrum to under the threshold. In the analysis, an exponential function fitted the spectrum and the extrapolation of the spectrum below the threshold level. The extrapolation's uncertainty was deduced from the difference between the flat and exponential extrapolation. Consequently, the deduced uncertainty was about 3%. The deduced uncertainties are proportional to the final cross section. In short, this uncertainty belongs to Type-1. The second step is the dead-time correction. In the analysis, the dead-time correction was applied with the extended dead-time model using a dead time of 5.67 ± 0.02 µs per event. This dead time uncertainty was small enough to be negligible. The remaining uncertainty was statistical uncertainty. Figure 5 shows the weighted The third step is overlap and constant B.G. estimation and subtraction. In the experiment, the first 58 of every 62 shots were provided to MLF, and the last four were delivered to the other facilities. Thus, every 2.48 s, a broad TOF spectrum to 200 ms was obtained. The frame overlap background and constant background in the normal TOF range earlier than 25 ms were deduced by fitting an exponential decay plus constant function to this broad range TOF spectrum in the TOF region from 40 to 65 ms [13]. In this step, uncertainties were deduced from the uncertainties of the fitted parameters. The deduced uncertainties are energy-dependent, and the value of each TOF bin is correlated. Therefore, the uncertainty belongs to Type-4.
The background due to the aluminum container and Y 2 O 3 included in the sample was derived from the spectrum of the dummy sample. The background due to scattered neutrons by 243 Am was estimated by the spectrum of the 208 Pb sample by considering the differences in the elastic-scattering cross-section of 243 Am and 208 Pb. The elastic-scattering cross-section was taken from the JENDL-4.0 [14]. Figure 6 shows the TOF spectra of 243 Am sample before and after the background subtraction. In this step, uncertainties were deduced from the uncertainties of the evaluated elastic cross-sections in JENDL-4.0 and the statistical uncertainties of the blank, dummy, and lead sample measurements. Therefore, the uncertainty is a sum of statistical uncertainty and Type-4 uncertainty.
Self-shielding and multiple scattering correction factors were calculated with the Monte Carlo simulation code. The uncertainties of the areal densities mainly come from the uncertainty of the pellet diameter. Therefore, uncertainties of the correction factors were derived from the uncertainty of the pellet diameter. To derive the uncertainties, the ideal sample with area densities larger by the 1-σ uncertainty than those in the actual sample was considered. The difference in the correction factors between the actual and ideal samples was used as the uncertainty. The uncertainty depends on the uncertainty of the areal density of 243 Am and the evaluated cross section. The uncertainty is categorized into Type-4.
The relative incident neutron energy spectrum was determined by measuring 478-keV gamma-rays from the 10 B(n, αγ) 7 Li reaction. The 10 B(n, αγ) cross section was taken from JENDL-4.0. The dead-time, neutron selfshielding, and multiple scattering corrections were applied. The uncertainty of the dead-time correction was negligible. The uncertainty of the neutron self-shielding and multiple scattering correction was deduced from the uncertainties of the 10 B sample mass and diameter. The obtained relative neutron flux and uncertainties are shown in Figure 7. The uncertainty is a sum of statistical uncertainty and Type-4 uncertainty. The relative capture cross section of the 243 Am sample was calculated from the deduced net spectrum of the 243 Am sample and the relative neutron flux. In the analysis, the absolute capture cross section at the 1.356 eV resonance was derived with another analysis method [5]. The relative capture cross section of the 243 Am sample was normalized to the derived absolute cross section. The normalization uncertainty was derived from the uncertainty of the absolute cross section. The deduced uncertainty was proportional to the capture cross section of the 243 Am. The uncertainty is categorized into Type-1.
In the last step, contributions from the influences of the impurities and fission events were subtracted. The contributions were deduced using the resolution function of AN-NRI, the capture and fission cross sections in JENDL-4.0, and the sensitivity of fission events to the capture events (2.3 ± 1.0). The capture cross sections of 243 Am were derived by subtracting the contributions. The uncertainties were derived from the uncertainties of the abundances of the impurities, JENDL-4.0, and the ratio of the sensitivity. The deduced uncertainties were proportional to the crosssections of other reactions. The uncertainty is categorized into Type-2. Figure 8 shows a part of the deduced capture cross section of 243 Am and the uncertainties in the neutron energy range from 1 to 10 eV.

Total cross-section measurement
The analysis procedure for the total cross-section measurements in ANNRI is shown in Figure 9. Figure 10 shows TOF spectra of the 243 Am and dummy samples measured with the 6 Li-glass and 7 Li-glass detectors. The gamma-ray backgrounds deduced with the 7 Liglass detector were much smaller than the neutron TOF spectra with the 6 Li-glass detector.
The dead-time correction was applied to the TOF spectra using a fixed dead time of 1264 ± 4 ns (316 ± 1 ch.). In this experiment, the event rate is less than 1.6 events per 100 µs and the dead time ratio was less than 2 %. The uncertainty of the dead time correction depends not on the  uncertainty of the dead time (1264 ± 4ns) but on the statistical uncertainty.
Frame overlap and constant B.G. estimation are the same as those in the capture measurements. The deduced background is shown in Figure 11 in comparison with the broad and normal range TOF spectra. In this step, uncertainties were deduced from the uncertainties of the fitted parameters. Therefore, the uncertainty belongs to Type-4.
The TOF spectra obtained using the 7 Li-glass detector were used to determine the observed background events due to gamma rays. Gamma-ray efficiencies were slightly different between the 7 Li-glass and 6 Li-glass detectors. To correct the difference, the black resonance technique was applied. The TOF spectrum measured with the 7 Li-glass detector was normalized to that of the 6 Li-glass detector at the four dips by 55 Mn, 59 Co, 115 In and 109 Ag. The deduced normalization factor was 1.07 ± 0.03. Uncertainties due to this background subtraction were deduced from the uncertainties of the normalization factor (Type-4) and the statistical uncertainty.
In the experiment, an Al foil was set at the sample position of the array of large Ge detectors. Prompt gammarays from the 27 Al(n,γ) reaction were measured simultaneously with the Ge detector array to monitor the relative neutron intensities. The relative neutron intensity per proton beam shot for each sample was derived by dividing the number of gamma rays by the number of proton beam shots in the measurement. The factor was 1.0068 ± 0.0004 for the Am sample. Uncertainties were calculated from the uncertainty of this factor. The deduced uncertainties were proportional to the transmission ratio, i.e., constant uncertainty in the cross section (Type-3). The transmission ratio was obtained by dividing the TOF spectrum of the sample by that of the dummy. The deduced transmission ratio is shown in Figure 12. The total cross section of the 243 Am sample was derived using the transmission ratios and the areal density of 243 Am. Uncertainty was calculated from the uncertainty of the areal density. The deduced uncertainty was proportional to the cross section, and this uncertainty belongs to Type-1.
In the last step, contributions from the influences of the impurities were subtracted. This procedure is the same as that in the capture measurements. The uncertainties were derived from the uncertainties of the abundances, JENDL-4.0, and the ratio of the sensitivity. The deduced uncertainties were proportional to the cross-sections of other reactions. The uncertainties are categorized into Type-2.
Deduced total cross section of 243 Am in the neutron energy range from 0.01 to 10 eV is shown in Figure 13 with a comparison to the deduced uncertainties.

Conclusion
In the analysis of TOF experiments in ANNRI, uncertainties are deduced step by step. In experiments in ANNRI, each deduced uncertainty is reported separately. The deduced uncertainties are categorized into the five types.
Experimentalists have been trying to estimate all of the uncertainties. However, there is a possibility of missing some other uncertainties. The missing always results in an underestimation.