- Email: [email protected]
Uncertainty Quantification in Fission Cross Section Measurements at LANSCE
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Nuclear Data Sheets 123 (2015) 124–129 www.elsevier.com/locate/nds
Uncertainty Quantification in Fission Cross Section Measurements at LANSCE F. Tovesson1, ∗ 1
Los Alamos National Laboratory, Los Alamos, NM 87545, USA (Received 15 June 2014; revised received 25 August 2014; accepted 2 September 2014) Neutron-induced fission cross sections have been measured for several isotopes of uranium and plutonium at the Los Alamos Neutron Science Center (LANSCE) over a wide range of incident neutron energies. The total uncertainties in these measurements are in the range 3-5% above 100 keV of incident neutron energy, which results from uncertainties in the target, neutron source, and detector system. The individual sources of uncertainties are assumed to be uncorrelated, however correlation in the cross section across neutron energy bins are considered. The quantification of the uncertainty contributions will be described here.
I.
INTRODUCTION
Nuclear fission cross sections are essential for predicting the performance of energy and defense applications, and designers rely on the evaluated nuclear data libraries as a source of accurate cross section data. While these cross sections have been extensively measured for the major actinides there are still discrepancies and relatively large uncertainties in the evaluations of the minor actinides. In order to reduce these uncertainties, as well as extend the neutron energy range, a measurement program was carried out at the Los Alamos Neutron Science Center (LANSCE) [1–5]. The target accuracy of these measurements was approximately 3%, which is the limit of conventional fission cross section measurement techniques.
II.
EXPERIMENT
Fission cross section measurements were performed at the Weapons Neutron Science (WNR) facility (see Fig. 1) at LANSCE [6]. The neutrons were produced when a 800MeV proton beam impinges on a water-cooled tungsten spallation target, and several beam flight paths viewed the target from different angles relative to the proton beam. The basic repetition rate of the LANSCE accelerator is 120 Hz, with 100 Hz of proton pulses delivered to WNR and 20 Hz to Lujan Center. Each proton bunch delivered to WNR has a duration of approximately 625 μs, and consists of micro-pulses with 1.8 μs spacings. The micro pulses are approximately 250 ps wide. The experiment was located at the 90L flight path, which viewed the target from 90 degrees relative to the proton beam.
∗
Corresponding author: [email protected]
http://dx.doi.org/10.1016/j.nds.2014.12.022 0090-3752/© 2014 Elsevier Inc. All rights reserved.
The high energy neutron background is lower here relative to the more forward flight paths. The neutron flux at 10 meters at the 90L flight path is shown in Fig. 2 and compared to the flux at 8 meters at Lujan Center FP5, which has been used for fission cross sections at energies below 100 keV. The incident neutron energy was determined using the time-of-flight (TOF) technique, in which the time between when the proton beam hit the spallation target and the detection of a fission reaction in the detector was measured (see an example of a TOF histogram in Fig. 3). With the distance from the actinide target to the spallation target to the detector known the incident neutron energy can be determined. The actinide targets were placed inside a parallel-plate ionization chamber [7] which measures the amount of energy deposited by charged particles in the counting gas of the detector. A particle energy threshold was used to identify fission events and distinguish them from other reactions such as (n,α). The fission rates from the actinide targets are measured as a function of neutron energy and compared to the rate from a reference target (235 U), which was used to determine a relative energy differential fission cross section.
III.
UNCERTAINTY QUANTIFICATION
As already mentioned the measurements essentially consisted of counting the number of fission reactions in a target relative to a standard target. The number of fission events C measured in a sample foil with N number of atoms irradiated by a neutron beam of flux Φ is given by C = N w σ Φ + CBg ,
(1)
Uncertainty Quantification . . .
NUCLEAR DATA SHEETS
F. Tovesson
FIG. 1. (Color online) Schematic view of the neutron facilities at LANSCE. The two fligth paths used for neutron-induced fission cross section measurements are labeled: 4FP90L at WNR, and 1FP5 at Lujan Center.
8
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FIG. 2. Neutron flux at nominal beam current at 10 meters at WNR flight path 90L (4FP90L), and at 8 meters at Lujan Center flight path 5 (1FP5).
FIG. 3. Time-of-flight measurement at WNR of 235 U(n,f) in an ionization chamber (solid line). Also shown is background due to frame-overlap neutrons (dashed line).
where w is the relative time the detector system is “live” (accepting events), is the fission detection efficiency, σ is the fission cross section, and CBg are background events. The later is given in this experiment by
cross section (U 5) can be calculated as
CBg = CDC + CW A + CRS + CSC + CSF ,
σx (E) NU 5 = σU 5 (E) Nx w−1 (E)−1 x [Cx (E) − CBg,x ] × −1x wU 5 (E)−1 U 5 [CU 5 (E) − CBg,U 5 ] ΦU 5 (E) × . Φx (E)
(2)
where CDC is the number of events caused by accelerator dark current, CW A the number of events induced by frame-overlap (or “wrap-around”) neutrons, CRS is the number of events induced by room-scattered neutrons, CSC the number of events that are due to fission of other isotopes in the sample, and CSF is the number of spontaneous fission events. Using Eq. (1) the ratio between the fission cross section to be measured (x) and the 235 U(n,f)
(3)
Many of the quantities that enter into in Eq. (3) have uncertainties associated with them. When combined these determine the total uncertainty of the measurement. In the following sections the estimation of those uncertainties is discussed. 125
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NUCLEAR DATA SHEETS
The algorithm that corrects the data loops over the time-of-flight histograms, from early time to late, and corrects the bin contents using the above expression. The algorithm is exact under the assumption that the event rate is Poisson distributed, so the correction is done for every 10 seconds of data, which ensures a constant event rate over each time interval. The only input to Eq. (5) is the system recovery time D, which is directly measured. This is done by observing the length of the busy signal sent out by the data acquisition system while it is processing an event. The measured value is about 17.5 μs. The accuracy of this measurement is not very high; it has an estimated uncertainty of 1-10%. A more accurate value is obtained by using the scaler values for the total number of events and the number of events occurring when the system was live. Since the scalers measures E pi (E) and E Ni (E) exactly, these numbers are compared to the calculated ones. The algorithm input parameter D is then fine-tuned until there is prefect agreement between the measured and calculated values within the uncertainties. After optimizing the dead-time correction, the deadtime in one experiment was determined to be D = 17.711 μs. The residual systematic uncertainty is below 0.1% in the full energy range. The uncertainty could be even further reduced by continuing to fine-tune D. However, this is very time consuming, and the uncertainty due to the dead-time correction is already an order of magnitude lower than all other uncertainties.
Statistical Uncertainties
The statistical uncertainties arise from the counting of fission events in the experiment. The number of events measured in a fixed period of time follows Poisson statistics, such that the standard deviation is given √ by the square of the mean of the distribution, σ = λ. The number of events counted in each energy bin are typically large (> 100) such that the Poisson distribution is well approximated by a normal (Gaussian) distribution √ with standard deviation λ. Assuming that the number of events counted (C) is a good representation of the mean of the probability function, we have the well-known estimate of the√statistical uncertainty in a counting experiment: σ = C. B.
Dead-time Correction
When an event is registered by the data acquisition system, the system goes “dead” for a fixed amount of time. It does not register any additional event information until after the “dead-time” has elapsed. While the information about the events, such as time-of-flight and energy deposition, is lost during the dead-time period, a scaler module still counts the lost events. Additionally, there is a scaler module counting the number of events that occur while the system is “live”, so the total livetime in each measurement is measured exactly. However, the measured live-time is an integral value over all neutron times-of-flight (that is, energies). Since the count rates vary significantly with TOF in this measurement, it is important to make time-dependent deadtime corrections. The energy-dependent dead-time correction is calculated using the procedure described in Ref. [8]. A short summary of the algorithm will be given here. The collected events are sorted in a histogram according to their time-of-flight, where the content of the ith bin is denoted by Ni . The probability per neutron pulse of an event taking place in bin i is approximately given by pi
Ni , Np
C.
(4)
Np −
Ni i−1 j=i−D−1
= w−1 Ni .
Dark Current Neutrons
There is some leakage current between the nominal proton pulses that are delivered to the spallation targets by the accelerator. This proton current, commonly referred to as accelerator dark current, produces a neutron background throughout each accelerator macro-pulse. The level of dark current is monitored using facility beam monitors, and the accelerator is tuned such that the dark current is kept at below 1% of the total proton current. The amount of dark current is relatively sensitive to the accelerator tuning, and in some instances there is no detectable dark current present. In order to correct for background event caused by dark current neutrons a non-fissile target (238 U) is present in the fission chamber. The fission cross section in 238 U is several orders of magnitude lower below reaction threshold than it is above threshold. Therefore, at longer times of flight (corresponding to energies below 1 MeV) very few events are expected. The events detected at these times-of-flight are therefore attributed to accelerator dark current, and are used as a measure of the relative level of dark current. Once the level of dark current is estimated using the 238 U sample this information is used to subtract a fission event background rate from all samples that are studied. The dark current background is nearly constant on the
where Np is the number of neutron pulses. However, this expression is only an approximation since Ni is affected by dead-time. It is therefore necessary to add a factor that corrects for the dead-time caused by events that occur earlier. If the measurement system is dead for D histogram bins after an event occurs, it can be shown that an exact expression for pi is given by pi =
F. Tovesson
(5)
Nj
Note that the weight w in the above expression is the same dead-time correction factor appearing in Eq. (3). 126
Uncertainty Quantification . . .
NUCLEAR DATA SHEETS
time scale of the proton micro pulse spacing (1.8 μs), but can sometime be observed to increase during the macro pulse (625μs long). The accuracy to which the dark current levels can be determined depends on what the levels are; for a 1% dark current level it can be measure to about 10%. In this example the uncertainty introduced in the cross section measurement is less than 0.5% from reaction threshold up to 20 MeV. At 200 MeV the uncertainty is a few percent, and the same is true for the sub-threshold fission cross section of a non-fissile isotope.
D.
sion rates from enriched samples of the contamination isotope and assay information from the target manufacturer. The assay information from the manufacturer does have rather large uncertainties in some cases, but since the levels often are relatively small the large uncertainties in these numbers still translate to <1% uncertainties on the measured cross section. In some cases the assay information is verified using fission resonance analysis of TOF measurements with a moderated neutron source, which is a very sensitive method to determine the presence of contaminants.
Wrap-around Neutrons F.
At WNR the spacing between proton pulses is only 1.8 or 3.6 μs. Due to this short pulse spacing there are going to be some low energy neutrons from one pulse arriving at the detector position at the same time as high energy neutrons from a later pulse. The result is a frame-overlap background in the time-of-flight spectrum. For non-fissile isotopes the frame-overlap background is very low, since the cross sections below the energy of the frame-overlap neutrons are orders of magnitude lower compared to fast neutron cross sections. For fissile isotopes, however, the wrap-around background can be significant and needs to be quantified and corrected for. The frame overlap is quantified by measuring the event rate between macro-pulses, for about a millisecond after the last micro-pulse. Throughout the macro pulse several pulses will overlap on each other, and thus the effect will be largest at the end of a macro pulse which is where it is measured. The resulting spectrum is well described by three exponentials, and the data are fit assuming that each micro-pulse contributes equally to the frame-overlap background and can be described by three exponentials. The procedure is described in some detail in Ref. [1]. The level of frame-overlap background in the time-of-flight spectrum for 235 U obtained from the fit is shown in Fig. 3. Since the amount of frame overlap background changes throughout a macro pulse the average correction for one micro pulse is shown here. The background-to-foreground ratio is clearly highest at the two extremes, hence affecting the energies below 0.5 MeV and above 100 MeV the most. In the region between 0.5 and 100 MeV the added uncertainty due to this correction is below 0.5%.
E.
F. Tovesson
Normalization Uncertainties
FIG. 4. (Color online) The measured pulse height distribution in the ionization chamber from a 243 Am target. The arrow indicates the lower limit of the threshold used to identify fission. The peak below the threshold is due to induced and spontaneous alpha emission, as well as electronic noise.
The overall normalization of the measurement depends on the number of atoms in the samples, the uniformity of the sample deposits and neutron beam profile, and the efficiency of detecting fission. The number of atoms in the samples are known from assay information, and the uncertainty on that number can vary depending on the assay method used. The uniformity depends on the sample deposition method. The targets used for fission cross section measurements at LANSCE were either electro-deposition or evaporation. The electro-deposited targets were measured using alpha counting with a mask to have variation in thickness of about 5%, while the evaporated targets vary 1-3% in thickness across the surface. The beam profile is measured using image plates, with about 5% uncertainty. When a fission fragment is emitted at large angles relative to the normal of a fission sample there is some probability that the fragment is absorbed in the deposit and thus not counted in the detector. This was studied for
Sample Contamination
The targets used in the measurements are typically highly enriched (>99%). Moreover, the fission cross sections for the different isotopes that might occur as contaminants in the targets are comparable to the cross sections that are being measured. Hence, the uncertainty introduced due to sample contamination is small (<1%) in the fast neutron region. The correction for fission of contamination isotopes is performed using measured fis127
Uncertainty Quantification . . .
NUCLEAR DATA SHEETS
dividual sources of uncertainties and how they are estimated have been addressed. By reporting the individual contributions to the total uncertainties to the nuclear data libraries the measured data can be incorporated into the evaluation process with a fair representation of the uncertainties. This makes the comparison of several measurements more accurate than when only total uncertainties are reported. In the evaluations of fission cross sections for the major actinides there are large libraries of energy differential measurements that are incorporated, as well as integral measurements that constrain the evaluations. As a result the evaluations have significantly lower uncertainties than the individual measurements. However, in some cases measurements have discrepancies that are significantly outside their individual uncertainties, or there are other reasons to suspect problems with the evaluations. In these cases it would be beneficial to perform a measurement with an accuracy that is comparable to that of the evaluated cross section. A new promising approach to improve accuracies in energy-differential cross section measurements is to replace the conventional parallel plate ionization chamber with a Time Projection Chamber (TPC) [11]. This type of detector can be used for particle tracking, which allows for a reduction of several systematic uncertainties such as the beam and target non-uniformity as well as mis-identification of alpha particles as fission fragments. The tracking capability can also be used to employ the method developed by Budtz-Jorgensen and Knitter [9] to directly measure the loss of fission fragments due to absorption in the sample. The expectation is that this detector will be used to reach 1% uncertainty in fission cross section measurement, and would hence help verify (or identify issues with) the current evaluations of 239 Pu and 235 U.
for different deposition methods by Budtz-Jorgensen and Knitter [9] using a gridded ionization chamber, and they show that there are significant differences in the fraction of fragments absorbed between the different sample types. In the measurement reported here the cross section is always determined in a ratio between two samples of the same deposition method and similar areal density. The difference in fragment absorption of the fragments under these conditions is estimated to introduce an uncertainty in the cross section of less than 1%. The efficiency of the ionization chamber is assumed to be the same for the two samples, since they are in identical detector geometries and have similar thicknesses. However, the efficiency also depends on the pulse height threshold that is used to identify fission events. Some of the fission events will produce detector signals that are lower than the threshold, and the detection efficiency is decreased. To have the detection efficiency ratio as close as possible to unity, the pulse heights are calibrated to ensure that the thresholds are the same for all samples. In order to estimate the uncertainty introduced by setting a particle energy threshold a technique similar to one used for example in Ref. [10] was employed. The fissionfragment energy distribution was extrapolated below the threshold assuming either a constant value in this region, or a linear decreasing fission rate with decreasing particle energy. The difference in measured cross section between the two assumptions is a 1% effect, and thus this is the uncertainty assigned to the threshold.
G.
Uncertainty Reporting
The results of these measurements are reported to the experimental nuclear data library (EXFOR), in a given neutron energy bin, as the cross section, the statistical uncertainty, and all the individual systematic uncertainties. This format is compatible with EXFOR, and can be used to generate the covariance matrix for the measurement using some assumption such as no correlation between uncertainty sources and full correlation between energy bins for systematic uncertainties.
IV.
F. Tovesson
Acknowledgements: This work has benefited from the use of the Los Alamos Neutron Science Center at the Los Alamos National Laboratory. This facility is funded by the US Department of Energy and operated by Los Alamos National Security, LLC under contract DEAC52-06NA25396.
CONCLUSIONS AND OUTLOOK
The total uncertainties in conventional fission cross section measurements are typically about 3% and the in-
[1] F. Tovesson and T. S. Hill, “Neutron induced fission cross section of 237 Np from 100 keV to 200 MeV,” Phys. Rev. C 75, 34610 (2007). [2] F. Tovesson and T. S. Hill, “Subthreshold Fission Cross Section of 237 Np,” Nucl. Sci. Eng. 159, 94 (2008). [3] F. Tovesson et al., “Neutron induced fission of 240,242 Pu
from 1 eV to 200 MeV,” Phys. Rev. C 79, 14613 (2009). [4] F. Tovesson and T. S. Hill, “Cross section for 239,241 Pu(n,f) in the range En = 0.01 eV to 200 MeV,” Nucl. Sci. Eng. 165, 224 (2010). [5] F. Tovesson, A. B. Laptev and T. S. Hill, “ Fast neutroninduced fission cross sections of 233,234,236,238 U up to 200
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MeV,” Nucl. Sci Eng. 178, 57 (2014). [6] P. W. Lisowski and K. Schoenberg, “The Los Alamos Neutron Science Center,” Nucl. Instrum. Methods Phys. Res. A 562, 910 (2006). [7] S. A. Wender et al. “A fission ionization detector for neutron flux measurements at a spallation source,” Nucl. Instrum. Methods Phys. Res. A 336, 226 (1993). [8] P. B. Coates “Analytical corrections for dead time effects in the measurement of time-interval distributions,” Rev. Sci. Instrum. 63, 2084 (1992). [9] C. Budtz-Jorgensen and H.-H. Knitter, “Investigation
F. Tovesson
of fission layers for precise fission cross-section measurements with a gridded ionization chamber,” Nucl. Sci. Eng. 86, 10 (1984). [10] J. W. Meadows, “The fission cross section of 237 Np relative to 235 U from 0.1 to 9.4 MeV”, Nucl. Sci. Eng. 85, 271 (1983). [11] M. Heffner et al. “A Time Projection Chamber for High Accuracy and Precision Fission Cross Section Measurements,” Nucl. Instrum. Methods Phys. Res. A 759, 50 (2014).
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Nuclear Data Sheets 123 (2015) 124–129 www.elsevier.com/locate/nds
Uncertainty Quantification in Fission Cross Section Measurements at LANSCE F. Tovesson1, ∗ 1
Los Alamos National Laboratory, Los Alamos, NM 87545, USA (Received 15 June 2014; revised received 25 August 2014; accepted 2 September 2014) Neutron-induced fission cross sections have been measured for several isotopes of uranium and plutonium at the Los Alamos Neutron Science Center (LANSCE) over a wide range of incident neutron energies. The total uncertainties in these measurements are in the range 3-5% above 100 keV of incident neutron energy, which results from uncertainties in the target, neutron source, and detector system. The individual sources of uncertainties are assumed to be uncorrelated, however correlation in the cross section across neutron energy bins are considered. The quantification of the uncertainty contributions will be described here.
I.
INTRODUCTION
Nuclear fission cross sections are essential for predicting the performance of energy and defense applications, and designers rely on the evaluated nuclear data libraries as a source of accurate cross section data. While these cross sections have been extensively measured for the major actinides there are still discrepancies and relatively large uncertainties in the evaluations of the minor actinides. In order to reduce these uncertainties, as well as extend the neutron energy range, a measurement program was carried out at the Los Alamos Neutron Science Center (LANSCE) [1–5]. The target accuracy of these measurements was approximately 3%, which is the limit of conventional fission cross section measurement techniques.
II.
EXPERIMENT
Fission cross section measurements were performed at the Weapons Neutron Science (WNR) facility (see Fig. 1) at LANSCE [6]. The neutrons were produced when a 800MeV proton beam impinges on a water-cooled tungsten spallation target, and several beam flight paths viewed the target from different angles relative to the proton beam. The basic repetition rate of the LANSCE accelerator is 120 Hz, with 100 Hz of proton pulses delivered to WNR and 20 Hz to Lujan Center. Each proton bunch delivered to WNR has a duration of approximately 625 μs, and consists of micro-pulses with 1.8 μs spacings. The micro pulses are approximately 250 ps wide. The experiment was located at the 90L flight path, which viewed the target from 90 degrees relative to the proton beam.
∗
Corresponding author: [email protected]
http://dx.doi.org/10.1016/j.nds.2014.12.022 0090-3752/© 2014 Elsevier Inc. All rights reserved.
The high energy neutron background is lower here relative to the more forward flight paths. The neutron flux at 10 meters at the 90L flight path is shown in Fig. 2 and compared to the flux at 8 meters at Lujan Center FP5, which has been used for fission cross sections at energies below 100 keV. The incident neutron energy was determined using the time-of-flight (TOF) technique, in which the time between when the proton beam hit the spallation target and the detection of a fission reaction in the detector was measured (see an example of a TOF histogram in Fig. 3). With the distance from the actinide target to the spallation target to the detector known the incident neutron energy can be determined. The actinide targets were placed inside a parallel-plate ionization chamber [7] which measures the amount of energy deposited by charged particles in the counting gas of the detector. A particle energy threshold was used to identify fission events and distinguish them from other reactions such as (n,α). The fission rates from the actinide targets are measured as a function of neutron energy and compared to the rate from a reference target (235 U), which was used to determine a relative energy differential fission cross section.
III.
UNCERTAINTY QUANTIFICATION
As already mentioned the measurements essentially consisted of counting the number of fission reactions in a target relative to a standard target. The number of fission events C measured in a sample foil with N number of atoms irradiated by a neutron beam of flux Φ is given by C = N w σ Φ + CBg ,
(1)
Uncertainty Quantification . . .
NUCLEAR DATA SHEETS
F. Tovesson
FIG. 1. (Color online) Schematic view of the neutron facilities at LANSCE. The two fligth paths used for neutron-induced fission cross section measurements are labeled: 4FP90L at WNR, and 1FP5 at Lujan Center.
8
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FIG. 2. Neutron flux at nominal beam current at 10 meters at WNR flight path 90L (4FP90L), and at 8 meters at Lujan Center flight path 5 (1FP5).
FIG. 3. Time-of-flight measurement at WNR of 235 U(n,f) in an ionization chamber (solid line). Also shown is background due to frame-overlap neutrons (dashed line).
where w is the relative time the detector system is “live” (accepting events), is the fission detection efficiency, σ is the fission cross section, and CBg are background events. The later is given in this experiment by
cross section (U 5) can be calculated as
CBg = CDC + CW A + CRS + CSC + CSF ,
σx (E) NU 5 = σU 5 (E) Nx w−1 (E)−1 x [Cx (E) − CBg,x ] × −1x wU 5 (E)−1 U 5 [CU 5 (E) − CBg,U 5 ] ΦU 5 (E) × . Φx (E)
(2)
where CDC is the number of events caused by accelerator dark current, CW A the number of events induced by frame-overlap (or “wrap-around”) neutrons, CRS is the number of events induced by room-scattered neutrons, CSC the number of events that are due to fission of other isotopes in the sample, and CSF is the number of spontaneous fission events. Using Eq. (1) the ratio between the fission cross section to be measured (x) and the 235 U(n,f)
(3)
Many of the quantities that enter into in Eq. (3) have uncertainties associated with them. When combined these determine the total uncertainty of the measurement. In the following sections the estimation of those uncertainties is discussed. 125
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NUCLEAR DATA SHEETS
The algorithm that corrects the data loops over the time-of-flight histograms, from early time to late, and corrects the bin contents using the above expression. The algorithm is exact under the assumption that the event rate is Poisson distributed, so the correction is done for every 10 seconds of data, which ensures a constant event rate over each time interval. The only input to Eq. (5) is the system recovery time D, which is directly measured. This is done by observing the length of the busy signal sent out by the data acquisition system while it is processing an event. The measured value is about 17.5 μs. The accuracy of this measurement is not very high; it has an estimated uncertainty of 1-10%. A more accurate value is obtained by using the scaler values for the total number of events and the number of events occurring when the system was live. Since the scalers measures E pi (E) and E Ni (E) exactly, these numbers are compared to the calculated ones. The algorithm input parameter D is then fine-tuned until there is prefect agreement between the measured and calculated values within the uncertainties. After optimizing the dead-time correction, the deadtime in one experiment was determined to be D = 17.711 μs. The residual systematic uncertainty is below 0.1% in the full energy range. The uncertainty could be even further reduced by continuing to fine-tune D. However, this is very time consuming, and the uncertainty due to the dead-time correction is already an order of magnitude lower than all other uncertainties.
Statistical Uncertainties
The statistical uncertainties arise from the counting of fission events in the experiment. The number of events measured in a fixed period of time follows Poisson statistics, such that the standard deviation is given √ by the square of the mean of the distribution, σ = λ. The number of events counted in each energy bin are typically large (> 100) such that the Poisson distribution is well approximated by a normal (Gaussian) distribution √ with standard deviation λ. Assuming that the number of events counted (C) is a good representation of the mean of the probability function, we have the well-known estimate of the√statistical uncertainty in a counting experiment: σ = C. B.
Dead-time Correction
When an event is registered by the data acquisition system, the system goes “dead” for a fixed amount of time. It does not register any additional event information until after the “dead-time” has elapsed. While the information about the events, such as time-of-flight and energy deposition, is lost during the dead-time period, a scaler module still counts the lost events. Additionally, there is a scaler module counting the number of events that occur while the system is “live”, so the total livetime in each measurement is measured exactly. However, the measured live-time is an integral value over all neutron times-of-flight (that is, energies). Since the count rates vary significantly with TOF in this measurement, it is important to make time-dependent deadtime corrections. The energy-dependent dead-time correction is calculated using the procedure described in Ref. [8]. A short summary of the algorithm will be given here. The collected events are sorted in a histogram according to their time-of-flight, where the content of the ith bin is denoted by Ni . The probability per neutron pulse of an event taking place in bin i is approximately given by pi
Ni , Np
C.
(4)
Np −
Ni i−1 j=i−D−1
= w−1 Ni .
Dark Current Neutrons
There is some leakage current between the nominal proton pulses that are delivered to the spallation targets by the accelerator. This proton current, commonly referred to as accelerator dark current, produces a neutron background throughout each accelerator macro-pulse. The level of dark current is monitored using facility beam monitors, and the accelerator is tuned such that the dark current is kept at below 1% of the total proton current. The amount of dark current is relatively sensitive to the accelerator tuning, and in some instances there is no detectable dark current present. In order to correct for background event caused by dark current neutrons a non-fissile target (238 U) is present in the fission chamber. The fission cross section in 238 U is several orders of magnitude lower below reaction threshold than it is above threshold. Therefore, at longer times of flight (corresponding to energies below 1 MeV) very few events are expected. The events detected at these times-of-flight are therefore attributed to accelerator dark current, and are used as a measure of the relative level of dark current. Once the level of dark current is estimated using the 238 U sample this information is used to subtract a fission event background rate from all samples that are studied. The dark current background is nearly constant on the
where Np is the number of neutron pulses. However, this expression is only an approximation since Ni is affected by dead-time. It is therefore necessary to add a factor that corrects for the dead-time caused by events that occur earlier. If the measurement system is dead for D histogram bins after an event occurs, it can be shown that an exact expression for pi is given by pi =
F. Tovesson
(5)
Nj
Note that the weight w in the above expression is the same dead-time correction factor appearing in Eq. (3). 126
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NUCLEAR DATA SHEETS
time scale of the proton micro pulse spacing (1.8 μs), but can sometime be observed to increase during the macro pulse (625μs long). The accuracy to which the dark current levels can be determined depends on what the levels are; for a 1% dark current level it can be measure to about 10%. In this example the uncertainty introduced in the cross section measurement is less than 0.5% from reaction threshold up to 20 MeV. At 200 MeV the uncertainty is a few percent, and the same is true for the sub-threshold fission cross section of a non-fissile isotope.
D.
sion rates from enriched samples of the contamination isotope and assay information from the target manufacturer. The assay information from the manufacturer does have rather large uncertainties in some cases, but since the levels often are relatively small the large uncertainties in these numbers still translate to <1% uncertainties on the measured cross section. In some cases the assay information is verified using fission resonance analysis of TOF measurements with a moderated neutron source, which is a very sensitive method to determine the presence of contaminants.
Wrap-around Neutrons F.
At WNR the spacing between proton pulses is only 1.8 or 3.6 μs. Due to this short pulse spacing there are going to be some low energy neutrons from one pulse arriving at the detector position at the same time as high energy neutrons from a later pulse. The result is a frame-overlap background in the time-of-flight spectrum. For non-fissile isotopes the frame-overlap background is very low, since the cross sections below the energy of the frame-overlap neutrons are orders of magnitude lower compared to fast neutron cross sections. For fissile isotopes, however, the wrap-around background can be significant and needs to be quantified and corrected for. The frame overlap is quantified by measuring the event rate between macro-pulses, for about a millisecond after the last micro-pulse. Throughout the macro pulse several pulses will overlap on each other, and thus the effect will be largest at the end of a macro pulse which is where it is measured. The resulting spectrum is well described by three exponentials, and the data are fit assuming that each micro-pulse contributes equally to the frame-overlap background and can be described by three exponentials. The procedure is described in some detail in Ref. [1]. The level of frame-overlap background in the time-of-flight spectrum for 235 U obtained from the fit is shown in Fig. 3. Since the amount of frame overlap background changes throughout a macro pulse the average correction for one micro pulse is shown here. The background-to-foreground ratio is clearly highest at the two extremes, hence affecting the energies below 0.5 MeV and above 100 MeV the most. In the region between 0.5 and 100 MeV the added uncertainty due to this correction is below 0.5%.
E.
F. Tovesson
Normalization Uncertainties
FIG. 4. (Color online) The measured pulse height distribution in the ionization chamber from a 243 Am target. The arrow indicates the lower limit of the threshold used to identify fission. The peak below the threshold is due to induced and spontaneous alpha emission, as well as electronic noise.
The overall normalization of the measurement depends on the number of atoms in the samples, the uniformity of the sample deposits and neutron beam profile, and the efficiency of detecting fission. The number of atoms in the samples are known from assay information, and the uncertainty on that number can vary depending on the assay method used. The uniformity depends on the sample deposition method. The targets used for fission cross section measurements at LANSCE were either electro-deposition or evaporation. The electro-deposited targets were measured using alpha counting with a mask to have variation in thickness of about 5%, while the evaporated targets vary 1-3% in thickness across the surface. The beam profile is measured using image plates, with about 5% uncertainty. When a fission fragment is emitted at large angles relative to the normal of a fission sample there is some probability that the fragment is absorbed in the deposit and thus not counted in the detector. This was studied for
Sample Contamination
The targets used in the measurements are typically highly enriched (>99%). Moreover, the fission cross sections for the different isotopes that might occur as contaminants in the targets are comparable to the cross sections that are being measured. Hence, the uncertainty introduced due to sample contamination is small (<1%) in the fast neutron region. The correction for fission of contamination isotopes is performed using measured fis127
Uncertainty Quantification . . .
NUCLEAR DATA SHEETS
dividual sources of uncertainties and how they are estimated have been addressed. By reporting the individual contributions to the total uncertainties to the nuclear data libraries the measured data can be incorporated into the evaluation process with a fair representation of the uncertainties. This makes the comparison of several measurements more accurate than when only total uncertainties are reported. In the evaluations of fission cross sections for the major actinides there are large libraries of energy differential measurements that are incorporated, as well as integral measurements that constrain the evaluations. As a result the evaluations have significantly lower uncertainties than the individual measurements. However, in some cases measurements have discrepancies that are significantly outside their individual uncertainties, or there are other reasons to suspect problems with the evaluations. In these cases it would be beneficial to perform a measurement with an accuracy that is comparable to that of the evaluated cross section. A new promising approach to improve accuracies in energy-differential cross section measurements is to replace the conventional parallel plate ionization chamber with a Time Projection Chamber (TPC) [11]. This type of detector can be used for particle tracking, which allows for a reduction of several systematic uncertainties such as the beam and target non-uniformity as well as mis-identification of alpha particles as fission fragments. The tracking capability can also be used to employ the method developed by Budtz-Jorgensen and Knitter [9] to directly measure the loss of fission fragments due to absorption in the sample. The expectation is that this detector will be used to reach 1% uncertainty in fission cross section measurement, and would hence help verify (or identify issues with) the current evaluations of 239 Pu and 235 U.
for different deposition methods by Budtz-Jorgensen and Knitter [9] using a gridded ionization chamber, and they show that there are significant differences in the fraction of fragments absorbed between the different sample types. In the measurement reported here the cross section is always determined in a ratio between two samples of the same deposition method and similar areal density. The difference in fragment absorption of the fragments under these conditions is estimated to introduce an uncertainty in the cross section of less than 1%. The efficiency of the ionization chamber is assumed to be the same for the two samples, since they are in identical detector geometries and have similar thicknesses. However, the efficiency also depends on the pulse height threshold that is used to identify fission events. Some of the fission events will produce detector signals that are lower than the threshold, and the detection efficiency is decreased. To have the detection efficiency ratio as close as possible to unity, the pulse heights are calibrated to ensure that the thresholds are the same for all samples. In order to estimate the uncertainty introduced by setting a particle energy threshold a technique similar to one used for example in Ref. [10] was employed. The fissionfragment energy distribution was extrapolated below the threshold assuming either a constant value in this region, or a linear decreasing fission rate with decreasing particle energy. The difference in measured cross section between the two assumptions is a 1% effect, and thus this is the uncertainty assigned to the threshold.
G.
Uncertainty Reporting
The results of these measurements are reported to the experimental nuclear data library (EXFOR), in a given neutron energy bin, as the cross section, the statistical uncertainty, and all the individual systematic uncertainties. This format is compatible with EXFOR, and can be used to generate the covariance matrix for the measurement using some assumption such as no correlation between uncertainty sources and full correlation between energy bins for systematic uncertainties.
IV.
F. Tovesson
Acknowledgements: This work has benefited from the use of the Los Alamos Neutron Science Center at the Los Alamos National Laboratory. This facility is funded by the US Department of Energy and operated by Los Alamos National Security, LLC under contract DEAC52-06NA25396.
CONCLUSIONS AND OUTLOOK
The total uncertainties in conventional fission cross section measurements are typically about 3% and the in-
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F. Tovesson
of fission layers for precise fission cross-section measurements with a gridded ionization chamber,” Nucl. Sci. Eng. 86, 10 (1984). [10] J. W. Meadows, “The fission cross section of 237 Np relative to 235 U from 0.1 to 9.4 MeV”, Nucl. Sci. Eng. 85, 271 (1983). [11] M. Heffner et al. “A Time Projection Chamber for High Accuracy and Precision Fission Cross Section Measurements,” Nucl. Instrum. Methods Phys. Res. A 759, 50 (2014).
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