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All issues Volume 9 (2023) EPJ Nuclear Sci. Technol., 9 (2023) 32 Full HTML
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EPJ Nuclear Sci. Technol.
Volume 9, 2023
Templates of Expected Measurement Uncertainties: a CSEWG Effort
Article Number 32
Number of page(s) 12
DOI https://doi.org/10.1051/epjn/2023013
Published online 09 November 2023

© D. Neudecker et al., Published by EDP Sciences, 2023

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

A prompt-fission neutron spectrum (PFNS) describes the energy distribution of neutrons emitted from fission fragments from an actinide in a time window promptly after fission and before the onset of beta decay [1]. The fission reaction can be induced by neutrons, as e.g., for neutron-induced 235U(n, f) PFNS, or be spontaneous, as, e.g., for the case of the PFNS standard, namely the 252Cf spontaneous fission spectrum [2–4] that is shown in Figure 1. A PFNS gives a probability distribution of neutrons to be emitted at a specific outgoing-neutron energy, Eout. Thus, the evaluated data are normalized such that the integral over the PFNS gives unity [5]. The maximum of this distribution is usually in the low-MeV range–dependent on the isotope at hand. At higher Eout, the spectrum falls off, such that there can be a few orders of magnitude difference in the PFNS between its peak and, e.g., at Eout = 10 MeV (see left-hand side of Fig. 1). It is very difficult to detect neutrons of the PFNS to high statistical precision for Eout >  10 MeV due to the paucity of emitted neutrons in this energy range and cosmic backgrounds. Special background-reduced environments must be sought for measurements at Eout >  20 MeV (a former lead mine under a 1000 m mountain was used, for instance, for Ref. [6]). It is equally difficult to precisely measure the PFNS at low Eout, e.g., below 10 keV in even favorable experimental conditions, because of the prevalence of background and multiple-scattered neutrons that can be larger in numbers than the desired prompt-fission neutrons. This limitation in measuring the PFNS at high and low Eout can be observed by the higher uncertainties and scatter among data points in the right-hand side of Figure 1. In these energy ranges, below 10 keV and above 10 MeV, nuclear model forms or integral information help to define evaluated PFNS, even if sufficient experimental data are available for other Eout ranges.

thumbnail Fig. 1.

The evaluated 252Cf spontaneous fission spectrum by Mannhart [2, 3] is compared to the experimental data used for that evaluation and as shown in Figures 1–10 of reference [2]. (a) The PFNS in units of (1/MeV). (b) The PFNS as a ratio to a Maxwellian function.

Below, a template of expected measurement uncertainties is provided for PFNS. These templates were developed for obtaining PFNS via the time-of-flight (TOF) technique to determine Eout of the PFNS as this is the most frequently employed technique. If other techniques are used (like measuring the Eout directly, using n–p scattering and a proton recoil spectrometer) recourse should be taken to other templates that describe these techniques.

The template described here was established based on (a) the uncertainty procedure developed within the IAEA Coordinated Research Project on PFNS documented in Chapter III.M of reference [1], (b) the detailed uncertainty analysis [7–9] of several 235, 238U and 239Pu PFNS experimental data using their respective literature [10–34] and EXFOR entries [35–37], as well as (c) uncertainty studies undertaken by the Chi-Nu team as part of their measurement campaign [13, 38–43]. After the initial establishment of the templates, a sub-set of the authors reviewed 252Cf PFNS measurements; some of the knowledge, gained from references [6, 44–46], was added to this manuscript.

2. Measurement types

PFNS are typically experimentally determined from time-of-flight (TOF) measurements of outgoing neutrons. PFNS can be measured for spontaneous fission or as a function of incident-neutron energy, Einc. In the latter case, one can use mono-energetic neutron beams, where Einc is well-defined, see, e.g., references [16, 47]. PFNS can also be measured with white-neutron sources. In this case, the TOF technique is also applied to measure Einc as, for instance, in references [40, 48].

The techniques for measuring a PFNS are distinguished according to how the neutron detector efficiency, ε, was derived. This specific partition was chosen because different uncertainty-quantification methods have to be applied to estimate total PFNS covariances depending on how ε was determined. This is described in detail in the appendix of reference [49].

2.1. PFNS shape measurements

In a PFNS shape measurement1 (illustrated schematically in the upper panel of Fig. 2), ε is determined directly. It is either measured by a non-PFNS experiment or simulated/calculated. For instance, Knitter [17] measured ε relative to neutron-production cross sections, while Staples et al. [47] obtained it via SCINFUL calculations [51]. Due to the direct determination of ε, the PFNS, χs, is reported explicitly in the case of shape measurements. It is derived by an analysis,

(1)

thumbnail Fig. 2.

Schematic drawing of PFNS-measurement types. The blue rectangle in the orange fission chamber is the isotope investigated; the blue circles indicate neutrons, including the incident neutron, and scattered or background neutrons; the green arrow denotes fission neutrons headed to the neutron detector; other fission neutrons are given by the black arrows. Figure (a) is for shape measurements. Figures (b) show the main measurement (top) and the monitor measurement (bottom) for a clean-ratio shape measurements. Figures (c) show the main measurement (top) and an associated ratio-calibration measurement (bottom) made in a different experimental environment to quantify the neutron detector efficiency. Figure 3 (c-bottom) can also illustrate calibration with a spontaneous fission source such as 252Cf.

where a background, bs, is subtracted from the measured counts, cs, corrected for εs, multiple scattering, and attenuation, ms, the angular distribution of the outgoing neutrons, as, and deadtime, τs. The sub-script s indicates the sample of interest. To be exact, the measured counts, cs − bs, in equation (1) are given by:

(2)

following references [9, 43, 52]. The key difference here is that the multiple-scattering and attenuation correction merge with the neutron-detection efficiency to an environmental detector-response matrix, R, of the TOF measurement, while the variable encompasses all other corrections mentioned above and is not written out explicitly. It is important to stress that while one believes that the counts are measured with an energy Eout, the response matrix actually distorts the true PFNS, χ, and one measure counts at due to the down-scattering of outgoing neutrons. The symbol “” signifies that the outgoing-neutron energy changed due to scattering. The same is true for all other types of measurements mentioned in this section, but we follow the simplified notation in equation (1) as this is the way the analyses were usually undertaken in the past. This leads to a bias in the reported χs which will be described in the template section on ε.

2.2. PFNS clean-ratio shape measurements

In a clean-ratio shape measurement, the PFNS of the investigated isotope, χs, is measured as a ratio to a monitor spectrum, χm, in exactly the same set-up and environment as χs [18, 24, 48]. Hence, two measurements are carried out, one of χs and one of χm, as is exemplified in the middle panel of Figure 2. Due to that, ε is expected to cancel. However, reference [43] shows that differing χs and χm create a difference in the total distortion (from down scattering mainly) observed for each Eout. As the response contains ε and m, this difference is accounted for here in m in order to remain with the simplified notation. In addition to m, some other correction factors–background among them–reduce the residual difference of that correction between isotope s and m. For instance, many neutron-induced experiments are measured relative to the 252Cf(sf) PFNS, e.g., [18, 24]. While some background contributions might be similar for the measurement of the isotope of interest and the monitor one, some part of it (e.g., neutron-beam-related background contributions) applies to only the neutron-induced measurement. Hence, bs and bm do not fully cancel but the ratio of them, bs/bm, is usually reduced compared to bs. However, it is important to note that some backgrounds are subtracted before the ratio of measurement s to m is taken, minimizing the residual effects. For instance, the neutron-beam-related background contribution can be measured with a fake fission trigger and can then be subtracted from cs [48]. If the statistics are sufficient in the beam-background measurement, its uncertainty can be reduced to be small compared to the statistical uncertainty of the background measurement; then, the differences in the background are small from the main to the monitor spectrum. Clean-ratio shape measurements provide PFNS ratios χs/χm that then need to be converted to χs by using nuclear data representing χm. This ratio χs/χm is usually obtained by

(3)

where the sub-script s/m reminds the reader that this factor is a residual correction effect.

2.3. PFNS ratio-calibration shape measurements

The neutron-detector efficiency is derived in a PFNS ratio-calibration shape experiment from measuring χm in another environment and set-up than χs, namely:

(4)

making use of nuclear data values, χm ND , to represent χm. Again, two measurements are undertaken; the distinguishing factor from PFNS clean-ratio shape measurements is the different environment. Possible differences are highlighted in the lower panel of Figure 2: the measurement might be undertaken in a different room (highlighted by red, differently sized boundaries instead of black ones) or with a different fission detector. Due to that, some correction factors do not reduce as systematically as for clean-ratio shape measurements. For instance, the room-return background depends strongly on the room and the material surrounding the sample (including potentially dissimilar fission detectors). Hence, this part of the background will not cancel for χs versus χm but needs to be determined separately as was for instance done in reference [19].

In some cases, ratio data χs/χm are reported for PFNS ratio-calibration shape measurements. But, sometimes χs is also listed in cases where the nuclear data χm ND is accounted for in ε.

3. Information needed for evaluations

Sufficient experimental data are available for only a few isotopes and selected incident-neutron energies that allow to evaluate of PFNS mostly based on experimental information. The 252Cf(sf) and 235U(nth,f) PFNS [2, 3, 53] are examples of such evaluations. Even in these cases, functional forms approximating the PFNS are needed to extrapolate the evaluation based on experimental data over the whole Eout range. As mentioned in the introduction, Section 1, it is difficult to measure the PFNS below 10 keV and above 10 MeV (see also right-hand side of Fig. 1). Existing experimental data often have such high uncertainties that they are not informative for the evaluation or are not available at all. However, the PFNS needs to be given over a sufficient Eout range (usually 1.0e−5 eV to 20 or 30 MeV) to satisfy the normalization constraint and nuclear-data-format requirements [5]. Hence, functional forms approximating the PFNS–usually a Maxwellian or Watt spectrum–extending the PFNS evaluations in the tails of the spectrum are needed.

Usually, PFNS evaluations are undertaken by statistically combining model values and experimental data using associated covariances [1, 8] or fitting model parameters [54–56] as was, for instance, done in 239Pu(n,f) PFNS evaluations of JENDL-4.0 and JEFF-3.2 [57, 58]. Many of these evaluations rely on the original or extensions of the Los Alamos model by Madland and Nix [59], for instance, all 239Pu(n,f) PFNS evaluations in ENDF/B-VII.0 to ENDF/B-VIII.0, JENDL-4.0 and JEFF-3.2 [57, 58, 60–62]. More modern models are currently under development, e.g., [1, 63–65]. Models are needed for the evaluation as the experimental data of many isotopes do not cover the incident- and outgoing-neutron energy range needed for application calculations.

For an evaluation, χs or χs/χm is needed for the full range of incident- and outgoing-neutron energies measured. In addition to that, the temperature of a Maxwellian fitted to each χs (and χm if applicable) at a specific Einc is needed to convert time resolution and TOF-length uncertainties given in time and length units into covariances relative to the PFNS. A fine binning of Einc would be preferable (of 1 MeV or less), especially around second or third-chance fission thresholds to clearly resolve structures expected at these incident-neutron energies in the PFNS that are illustrated in Figure 3. The bin size of Einc should be quantified such that one can assign a weight to how much the measured PFNS contributes to an evaluation of a PFNS at a specific Einc. Partial uncertainties relative to the PFNS should be provided for counting statistics, δc, background corrections, δb, multiple scattering and attenuation corrections, δm, detector-response determination, δε, nuclear data used in the analysis process, δχm ND , neutron-angular-distribution corrections, δa, and deadtime, δτ. The time resolution, Δt, and TOF-length uncertainties, ΔL, are often provided either relative to energy, e.g., [17, 47], or in time and length units, e.g., [18, 24]. In order to convert Δt and ΔL to uncertainties relative to the PFNS, the TOF length, L, is also needed. It would be very helpful if information on the correlation coefficients of each of these partial uncertainties could be provided. However, this is rarely done, with the notable exceptions of [13, 21, 40, 41].

thumbnail Fig. 3.

ENDF/B-VIII.0 239Pu PFNS [8, 60] are used as examples to illustrate structures commonly observed around second- and third-chance fission thresholds: from thermal Einc to the on-set of second chance fission, curves similar in shape to the thermal PFNS are expected. In a window of Einc = 1–2 MeV around second-chance fission (red dashed line), one sees a characteristic structure of the PFNS appear in the 100 s of keV stemming from a second-chance fission contribution to the PFNS that is only non-zero up to these energies due to energy conservation. Around the third-chance fission threshold (blue solid line), this structure is like a broader shoulder in Eout above 100 keV. One also sees a sharp structure at high Eout (several MeV) coming from the pre-equilibrium neutron emission from the compound nucleus before fission happens.

It is also very important to mention whether the measurement is a PFNS shape, clean-ratio, or ratio-calibration shape measurement. While it is usually evident that measurement is of the type “shape”, it is not as clear to differentiate between the two ratio cases. Time resolution uncertainties in PFNS space can differ substantially at high Eout depending on whether a data set is interpreted as clean-ratio or ratio-calibration shape measurement [49]. Hence, it is very important to clearly point out in a journal article whether the ratio measurement was undertaken in the same or a very different surrounding. Related to that, if a specific nuclear data set, e.g., the 252Cf(sf) PFNS by [2, 3], was used to convert the ratio to shape data, this data set should be clearly cited. If not readily available nuclear data but rather a custom-reference data set was used for the conversion, this should be provided explicitly in the journal publication, as was done in reference [24], in order to be able to convert to shape data with the newest available reference data.

Table 1.

Typical uncertainty sources encountered in PFNS measurements are listed dependent on their specific measurement type. Also, estimates of typical uncertainty ranges are provided in case this information cannot be derived otherwise for uncertainty-quantification purposes. “ND” was used as an abbreviation for nuclear data.

Auxiliary information on the experiment (such as set-up information, isotopic content of sample, etc.) is rarely used explicitly in the evaluation process. However, its importance should not be underestimated. If auxiliary information describing the experiment in detail is provided, the data can be simulated and re-analyzed at a later time, often providing key input to understand discrepancies between existing data sets [9] that can have a significant impact on application calculations [7]. This kind of information comprises all details that describe the experiment and allows one to simulate it. For instance, MCNP input decks, tabulated values of all pertinent correction factors, raw data, schematic drawings, sample thickness, isotopic composition, and dimension of all materials in the beam that could potentially produce neutrons.

4. Template

A listing of all uncertainty sources relevant to PFNS TOF measurements is provided below, along with the template values in Tables 1 and 2. All uncertainties are given relative to the PFNS in % unless otherwise noted.

Table 2.

Typical uncertainty sources encountered in PFNS measurements are tabulated with special emphasis on shapes of correlations. “ND” was used as an abbreviation for nuclear data.

4.1. Counting-statistics uncertainties

No counting-statistic uncertainties, δc, are provided in the template. These values depend strongly on the measurement time, neutron flux, detector response and set-up. Hence, δc is hard to estimate. In addition to that, we take the approach of recommending to reject a data set if this very basic uncertainty information cannot be obtained for it. The correlation matrix associated with δc is usually diagonal for PFNS data measured within the same experiment (independent of incident- and outgoing-neutron energy) and not correlated to other measurements.

4.2. Background uncertainties

Careful background corrections are an essential part of PFNS measurements. The background, b, can assume values at some Eout that are even larger than the signal one wants to measure [9]. This is especially true at Eout <  100 keV [13, 38, 42, 52, 66]. Background can arise from γs in γ-sensitive neutron detectors, beam contaminations (e.g., secondary-neutron group, protons or charged particles in the beam), wrap-around neutrons (dependent on spacing between source-neutron pulses), from pile-up of incident neutrons, room-return, random coincidences, α-particles from the sample, etc. Adequate shielding and background analysis can effectively reduce b and δb. Also, coincidence backgrounds can be measured directly with the insertion of a trigger or signal that is random in time (i.e., random when comparing the time difference between the trigger and real fission events) [48, 67]. Typical background uncertainties are in the range of 0.2–3% from 0.1–10 MeV and can rise up to δb ∼ 50% at 10 keV in shape measurements [14, 15, 18, 25]. These values are recommended to be used in the template along with a Gaussian correlation shape to account for correlations between δb. A Gaussian shape across outgoing- and incident-neutron energies was chosen because sometimes nuclear data are used in the background-correction process or separate measurements were performed that bring a combination of statistical and systematic effects [13, 19]. This approximate functional form is chosen in lieu of knowing the more complicated correlation matrices that can be obtained by experimentalists for their own measurement. It should be highlighted that background uncertainties can change with Eout and Einc given that incident-neutron background can also contribute. Correlations between δb of different experiments can be non-zero if the same methods were used to correct for b or both measurements were undertaken at the same facility.

In clean-ratio shape measurements, the correction of b is partially reduced as some background constituents (e.g., room return) are the same for both measurements for each individual outgoing-neutron energy, though the collection of counts at each TOF may be differently impacted by room return based on differences in the outgoing neutron spectrum. However, if one of the measurements was undertaken with the neutron beam on and the other off, the beam-related contributions to b will not reduce unless corrected separately as mentioned above. This case is often applicable to ratio measurements as many neutron-induced PFNS are measured relative to the spontaneously fissioning 252Cf spectrum. Hence, a smaller δb of 0.2–2% is recommended for clean-ratio shape measurements. The uncertainty for δb of shape measurements applies twice to δb of ratio-calibration shape measurements, due to the different environmental conditions of the two experiments.

4.3. Multiple-scattering and attenuation uncertainties

Corrections for multiple scattering and attenuation are two key factors that must be diligently addressed in each PFNS measurement. The reason is that one measures outgoing neutrons, which commonly down-scatter in surrounding material, leading to a sizable correction. The correction itself depends on the geometry and isotopic composition of the target and of the experimental environment. If samples are thick, as for instance in [17, 47], the multiple-scattering correction can amount to 10–20% [9]. Also, scattering in surrounding material can be significant [7, 9], leading to combined corrections in the maximum range of 20%. The absolute size depends on the geometry of and isotopes in the surrounding material. The larger the amount of material present, the larger m is expected to be. However, some isotopes have larger neutron-scattering and neutron-capture cross sections than others leading possibly to larger corrections than are necessary for isotopes that are more abundant but have small neutron-scattering or capture cross sections. If this scattering effect is corrected by using neutron-transport simulations, e.g., as done in [13, 18], the uncertainties of underlying nuclear data dependent on Eout and Einc of neutrons will contribute to δm. The multiple-scattering correction shows a clear Eout dependence. It is usually largest at low Eout where the PFNS is small, to begin with and many neutrons down-scatter over time, while it becomes negligibly small around the peak region, and above, of the PFNS [7, 9]. To be clear, neutrons down-scatter from higher outgoing-neutron energies, but their percentage is small compared to the non-scattered neutrons making the correction small around the peak. The PFNS at low outgoing-neutron energies is small, leading to a larger relative percentage of down-scattered neutrons, and therefore a larger uncertainty, δm, in the correction factor. Hence, we recommend for the template 20% for shape measurements at 10 keV linearly decreasing to 1% at the peak region of the PFNS, if this correction has not been applied. However, at the same time, we caution to critically assess whether this missing correction factor might not adversely bias the evaluated results. If that is the case, it might be better to attempt a correction for m based on detailed simulations or reject the data altogether before assigning such a large missing uncertainty. If m was corrected but δm is missing, δm could be estimated to be 10–20% of the correction factor which is given here with 0.1–3% for shape data. As the shapes change with Einc, so does m. However, these differences are secondary effects leading to no incident-neutron-dependent template values.

In clean-ratio shape measurements, m reduces to the different response of χs compared to χm to this effect. If χs and χm are very similar, m and, hence, δm can become small [23, 24, 43]. If χs and χm differ significantly, a systematic bias can amount to a value of a few percent across the majority of the PFNS range, with corrections on the order of 10% or more possible near pre-equilibrium neutron emission regions [43]. If an m ≠ 1 due to systematic differences of χs and χm is not corrected for, δm would assume this value. This is, for instance, the case for a PFNS measured with incident neutrons of an energy right at the second-chance-fission threshold as a ratio to the 252Cf(sf) spectrum. The 252Cf spectrum is smooth in Figure 1, while clear structures in PFN spectra at the second-chance fission thresholds have been observed [1, 13, 48] which are illustrated in Figure 3. Marini et al. estimated the maximum of this effect to be 2% in an analysis of their measurement [48], while Kelly et al. showed this effect to be up to 5% for distinctly different χs and χm at the second-chance fission threshold using a different fission detector [40, 43]. The latter publication also showed that this effect is isotope dependent and does not vanish at any studied Einc if χs and χm differ. Marini et al. measured a 239Pu PFNS in ratio to 252Cf(sf), which is closer in mean energy than 235U PFNS for many Einc, hence, the effect observed was smaller than the maximum of 5% observed in reference [43]. We take the latter value, 5%, as the upper template value for clean-ratio shape measurements for PFNS not corrected by m. The discussion above also illustrates that m depends on the outgoing-neutron energy along with the incident-neutron energy as structures in the PFNS, and, therefore, differences to the 252Cf(sf) spectrum, arise contingent upon the incident-neutron energy. Again, 15% of that effect is assumed as an uncertainty value if the effect was corrected but no other information was supplied to estimate the uncertainties.

In ratio-calibration shape measurements, m needs to be determined for both measurements, because χm and χs were determined in dissimilar surroundings, and, hence, the correction is different. A dissimilar surrounding could be, for instance, significantly different sample sizes, different fission-detector dimensions and materials, different room and shielding configurations. Of course, using a 252Cf sample instead of 235U is in itself dissimilar as both have different PFNS, but this case falls under the clean-ratio shape category if all other measurement components are the same. For the ratio-calibration shape technique, δm in the template is estimated by using the respective value for shape measurements and applying it to provide an uncertainty for both, δms and δmm, leading to an overall larger uncertainty contribution. Of course, this is a strong approximation taken because of missing information.

Usually, these multiple-scattering effects are simulated or calculated with neutron-transport codes of varying sophistication [12, 13, 68]. Attempts were undertaken in only a few rare cases to measure this effect [24]. The accuracy of m depends on uncertainties of nuclear data used in the codes, the accuracy of the set-up description used in the simulation, the accuracy of the physics model used for the simulation, and finally Monte Carlo counting statistics. The latter uncertainties are expected to be small compared to the other components which would lead to strong correlations. Therefore, a Gaussian shape is assumed across outgoing-neutron energies that are strongly correlated with incident-neutron energies. An anti-correlated shape around the turning point of the PFNS (around the maximum of χs) is assumed given the fact that neutrons down-scatter (i.e., are lost) in the higher energy range and are added to a lower one. The Chi-Nu team investigated the location of this turning point for the scattering-correction factors in their measurement and showed that it appears between 3–4 MeV for 239Pu PFNS depending on the exact detector array used [13, 52], supporting a value of about two times the Maxwellian. Their multiple-scattering correction is a convoluted correction containing also detector efficiency, yielding a response function R as given in equation (2).

Correlations between δm of different experiments can arise if the same codes/ methods, nuclear data, and similar materials have been used for correcting m. Then this correlation matrix can be assumed to be Gaussian with an overall smaller correlation factor compared to correlations for δm between the same experiment.

4.4. Detector-response uncertainties

The detector-response uncertainties, ε, should be explicitly given for shape measurements. It is often termed “detector-efficiency” uncertainty [14–17, 47]. Here, the term “response” is used to highlight that, indeed, there is no one-to-one correspondence of TOF to Eout as also shown in equation (2) with the response function R. To be more specific, multiple-scattering effects lead to a response function that couples one specific TOF with several (“true” initial) Eout-bins [9, 43]. Taddeucci et al. showed in Figures 4 and 5 of reference [9] that multiple scattering distorts a simulated PFNS at low and high Eout; the large distortion at low Eout is driven by down-scattering from a much larger peak. At higher Eout, neutrons scatter down and the response gets increasingly more uncertain as the finite time resolution plays an increasing effect. This increase above the peak seen in Figures 4 and 5 of reference [9] is used to estimate a detector-response uncertainty to be 0% below Eout = 2 MeV, linearly rising to 10% at 10 MeV [7, 8]. This uncertainty applies to shape measurements, which did not take this effect into account. It can be corrected by neutron-transport simulations along with scattering corrections. This detector-response uncertainty is in addition to those uncertainties usually reported as related to detector “efficiency”. The latter uncertainties are often in the range of 2–7% [14–17, 47] across relevant measurements. These values are used here as a range for the template. This uncertainty stems from reference nuclear data used to measure or simulate ε. The uncertainties in ε become large where either these nuclear data are not well-known or at the threshold of the respective detector. As δε is related to transformed nuclear-data uncertainties, a Gaussian shape is used for correlations between δε of the same experiment. If the same detector-response function is assumed for different Einc [47], δε is fully correlated for PFNS at different Einc but the same Eout values. Non-zero correlations between δε of different experiments arise from the common usage of the same nuclear data. If the same nuclear data were likely used, a Gaussian shape can be employed. If ε determinations of two experiments are related to obviously different and uncorrelated nuclear data, the correlations between δε are zero.

One would expect ε to drop out in clean-ratio shape measurements. The ratio procedure–dividing the measured main isotope through 252Cf outgoing-neutron counts and then multiplying by PFNS nuclear data for the latter isotope–applies scattering and efficiency (intrinsic detector response) at the same time. ε cancels only then if both χm and χs are measured in exactly the same environment and the two PFNS (and thus ms and mm) are identical. If χm and χs are substantially different, a sizable difference can be observed [43] in the detector responses to both PFNS. The uncertainty related to this effect is accounted for in the template in δm for clean-ratio shape measurements.

Ratio-calibration measurements aim at determining ε via measuring χm. Hence, the uncertainties of ε for this measurement type are made up of all uncertainties determining χm, including the related nuclear-data uncertainties. It is preferable that all uncertainties related to measuring χm are provided instead of δε. Hence, no separate template value is given for that. However, it should be mentioned that the detector response measured via χm might actually not be the same as for the χs measurement. A difference can arise due to the different surroundings, fission detector, and sample size (252Cf samples are often point-source). This difference should be quantified.

4.5. Angular-distribution uncertainties

The uncertainties related to correcting for the angular distribution of neutrons, δa, are expected to be small in many measurements. An angular bias can be introduced in measurements if the fission-fragment detector has a restricted angular coverage (e.g., in the case of a parallel-plate avalanche counter) and is coupled with neutron detectors that cover only part of the angular range. Fission fragments were shown to have a non-isotropic angular distribution even below 1 MeV for many actinides [69–71]. However, the resulting neutrons in the lab frame are less strongly-forward peaked until the onset of the pre-equilibrium neutron-emission contribution to the PFNS [39]. This pre-equilibrium neutron-emission forward peak (at a few MeV in the blue solid line of Fig. 3) starts to emerge around Einc = 10 MeV for most actinides. If the neutron detectors do not cover 2π, a substantial bias in χs might be introduced in this Einc range due to the restricted angular coverage. This bias has the potential to render data unusable for evaluations [8] as was the case for [10, 12]. However, one can expect the bias introduced and δa to be small for lower Einc [39, 48] unless an unfavorable set-up leads to angular bias (see Refs. [1, 25] for examples of those). One possibility to minimize a potential angular bias is to measure neutrons across a large angular coverage.

It is expected that 252Cf(sf) prompt-fission neutrons are isotropically distributed. Hence, even if one measures as a ratio to the 252Cf(sf) spectrum, a potential bias will not drop out. The only issue that could lead to an-isotropic prompt-fission neutrons from 252Cf(sf) is that fission fragments emitted at shallow angles in the sample might remain in the sample, which could distort the start signal of the PFNS. This effect could cancel out if the 252Cf and main measurements are undertaken in the same experimental set-up (i.e., same design of fission chamber and dimensions of targets).

A small uncertainty value of 0.1% is recommended for δa in Table 1. This should serve as a reminder to question whether an unfavorable set-up could potentially bias PFNS measurements. It should be understood that this effect can be very different at thermal energies compared to, e.g., 20 MeV incident-neutron energies. This effect would be usually corrected by simulating it via neutron-transport codes or extrapolating experimental angular distributions, if a sufficient number of angles are covered. Correlations across uncertainties at different Einc, Eout, and even different experiments could result from the common usage of nuclear data and physics models. Hence, similarly to δm, Gaussian correlation shapes are assumed.

4.6. Deadtime uncertainties

Dead-time effects matter for measurements with high count rates compared to the detector or data-acquisition systems integration times. No signal is measured, i.e., the detector is “dead”, from the time it triggers to the end of the signal-integration window. Deadtime can be minimized in measurements by reducing the count rate per detector using appropriately small samples or detectors. Also, the choice of detector type can help suppress deadtime effects to a negligible amount. Liquid scintillators and Li-glass scintillators should, for instance, be fast enough to control deadtime effectively. In addition, the use of waveform digitizers, now common in many laboratories, allows a significant reduction in deadtime corrections. In past measurements, deadtime was also reduced by making the neutron detector the start signal, while making the fission counter the stop signal (that fires more often) by delaying the latter [6, 44–46]. Hence, a small uncertainty, δτ of 0.1%, is recommended in the template for all measurement types with a full correlation between δτ assuming that the effect would affect all PFNS similarly. No correlation is assumed for δτ between experiments given that often different equipment, sample sizes, etc., are used.

4.7. Nuclear-data uncertainties

There are two sources of possible nuclear-data uncertainties. The first one appears in ratio measurements. χm needs to be known either to determine ε (ratio-calibration shape measurements, see Eq. (4)) or to convert ratio data to χs (clean-ratio shape data). These uncertainties are usually straightforward to estimate if one knows what data were used for χm in equation (4) or if ratio data are explicitly provided rather than derived χs. The latter case is actually preferred as it gives evaluators the possibility to use the currently best available nuclear data for χm.

Once the exact data that were used to represent χm are known, their associated covariances can be retrieved from the relevant database and used directly in the uncertainty procedure. Hence, it is of high importance to clearly state which specific nuclear data (e.g., library reference) were used for the conversion of ratio to shape data. If ratio data were converted to shape data by using a simple functional form representing the reference PFNS, such as for instance a Maxwellian, the parameters of this function need to be documented. This specific uncertainty source can lead to non-negligible correlations between PFNS covariances of different experiments and across different Einc if they were all measured relative to the same observable. However, these are again straightforward to estimate.

The second source of nuclear data uncertainties relates to the fact that nuclear data are frequently used for many corrections to χs. One example is that nuclear data are used in neutron-transport codes to calculate m. This nuclear-data-related uncertainty source was estimated in only rare cases [40, 41, 72]. This uncertainty amounted to values of 0.1–5% for Chi-Nu data. These values were calculated by forward-propagating the 6Li(n,t) cross-section covariances used in the MCNP simulations of the Chi-Nu assembly to PFNS uncertainties by using the implicit-capture capability of MCNP-6.2 [72, 73]. These uncertainties are listed in the template but are only applicable to measurements using Li-glass scintillators. These values serve as a reminder for the reader that additional nuclear-data uncertainties should apply if neutron-transport simulations based on nuclear data are used in the measurement analysis. However, more detailed studies are needed across many measurements and nuclear-data observables. It is expected that these uncertainties reduce in clean-ratio shape measurements as the same materials are used in both experiments. The only difference is related to the response of χm compared to χs to the surrounding material. One could assume that this residual effect is a tenth of the original effect. In ratio-calibration shape measurements, this effect and related uncertainty would need to be estimated twice because of the dissimilar materials.

4.8. Time-resolution uncertainties

Time-resolution values, Δt, from 0.5 to 5 ns can be found across many TOF measurements, e.g., [10, 12, 13, 18, 19, 24, 25], with many experiments reporting 1–2 ns. If no value is provided, a value of 2.5 ns is recommended. If the detector type is known, one can compare it to the typical time resolutions of the specific detector type. It is chosen slightly more conservative than the mean of these bounding values. This value applies to shape measurements and can be used for both, χs and χm, for the two ratio types.

The correlation matrix shape for this uncertainty for the same measurement arises from the conversion of TOF to Eout and then to an uncertainty relative to the PFNS as is described in [49]. These correlation factors can be derived across incident- and outgoing-neutron energy for Δt. The absolute size of uncertainties related to Δt in PFNS space depends strongly on the measurement type used to obtain χs. A Maxwellian temperature T or Watt parameters fitted to χs (and χm if applicable) are needed for this conversion along with the TOF length L. The TOF length needs to be measured between the sample and the neutron detector(s). The spectrum parameters can be fitted by the evaluators themselves if need be. However, L must be provided by the experimentalists. This is especially the case for shape and ratio-calibration shape measurements, as δt can be a dominant uncertainty source at Eout >  8 MeV.

Usually, the time resolution is estimated from the width of the peak caused by prompt gamma rays created in the fission process. This triggers the start signal for the TOF measurement of the outgoing neutrons. This determination is assumed to be independent from one measurement to another. Hence, zero correlation is assumed between δt of two experiments unless they were performed by the same team in the same facility. An example of that is the measurement series encompassing [11, 22–24].

4.9. TOF-length uncertainties

The TOF-length uncertainty, ΔL, usually leads to a minor uncertainty source contributing to the total PFNS covariances. Its values range from 1–5 mm across the literature [11, 13, 18, 19, 22–25] with most values closer to 1 mm. Hence, a value of 2 mm was chosen for the template. It needs to be considered once for the shape measurement and twice, for χs and χm, in the case of ratio measurements. Again, a Maxwellian temperature or Watt parameters fitted to the measured PFNS should be and a TOF length must be provided in order to transform ΔL into an uncertainty relative to the PFNS. The correlation-matrix shape between the transformed ΔL uncertainties of the same experiment across Eout and Einc arises again from this conversion process. Similarly to Δt, we assume that correlations are zero between uncertainties stemming from ΔL between different experiments. This assumption is made as the determination of L is usually independent between different experiments and a bias would be randomly distributed across them.

4.10. Impurity uncertainties

The impurity uncertainties depend on the actual level of impurities in the sample. If the samples are of very high purity, > 99.9% of fissionable material, the impurity uncertainty on the PFNS is negligibly small. For subthreshold measurements of PFNS (uncommon), impurities that have a lower fission threshold can be more important. Also, impurity uncertainties can become sizable if the sample contaminations are not well known. This is also true when samples are re-used. In this case, the current composition needs to be calculated starting from the date the sample was last chemically cleaned.

A correction of the sample impurity would depend on the level of the contamination as well as nuclear data of the impurity versus the main isotope. To be more precise, one needs to know how likely it is that the impurity fissions, how many neutrons are emitted, and how much its PFNS differs from nuclear-data values of the main constituent of the sample. The differences in fission cross-section and neutron multiplicity would be only dependent on the incident-neutron energy, while the PFNS difference would lead to a correction factor dependent also on the outgoing-neutron energy. Hence, the correlation shape across Eout and Einc could be estimated as strongly correlated but not fully because of the dependence on the difference between the PFNS of the main and impurity isotope.

Correlations between impurity uncertainties of different measurements can arise if the same sample was used. Even if different samples with similar contaminants were used, the impurity uncertainty can be strongly correlated (correlation coefficient of 0.5–0.75) between experiments if the same nuclear data are used for the correction and the same method was used to measure the impurity level.

5. Conclusions

Templates of expected measurement uncertainties were established here for prompt fission neutron spectra (PFNS). These templates list expected uncertainty sources for each distinct PFNS measurement type (shape, clean-ratio, and indirect ratio measurements). They also give ranges of uncertainties for most sources and estimates of correlation coefficients between uncertainties of each source. The uncertainty values were estimated conservatively, unless otherwise noted, based on information found for a broad range of experiments in their literature or respective EXFOR entries as well as expert judgment from experimenters. In addition to these templates, a section is provided that lists what evaluators need to include experimental PFNS faithfully in the evaluation process.

1

PFNS experimental data are recommended in reference [1] to be treated as shapes for evaluation purposes, as their normalization (the average prompt-fission neutron multiplicity) can be often determined more precisely in dedicated measurements [50].

Conflict of interests

The authors declare that they have no competing interests to report.

Funding

We gratefully acknowledge the partial support of the Advanced Simulation and Computing program at LANL. This work was supported by the US Department of Energy through the Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of the US Department of Energy under Contract No. 89233218CNA000001. This material benefited from work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under the Nuclear Data InterAgency Working Group Research Program.

Data availability statement

The data that were created associated with the manuscript are all within its main text and tables.

Author contribution statement

DN wrote the original draft, MD and PM contributed text to that. All authors reviewed and proposed edits to the manuscripts. All authors were involved in investigations and data curation associated with the article and discussions on uncertainty quantification for PFNS.

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Cite this article as: Denise Neudecker, Matthew Devlin, Robert C. Haight, Keegan J. Kelly, Paola Marini, Allan D. Carlson, Julien Taieb, and Morgan C. White. Templates of Expected Measurement Uncertainties for Prompt Fission Neutron Spectra, EPJ Nuclear Sci. Technol. 9, 32 (2023)

All Tables

Table 1.

Typical uncertainty sources encountered in PFNS measurements are listed dependent on their specific measurement type. Also, estimates of typical uncertainty ranges are provided in case this information cannot be derived otherwise for uncertainty-quantification purposes. “ND” was used as an abbreviation for nuclear data.

In the text
Table 2.

Typical uncertainty sources encountered in PFNS measurements are tabulated with special emphasis on shapes of correlations. “ND” was used as an abbreviation for nuclear data.

In the text

All Figures

thumbnail Fig. 1.

The evaluated 252Cf spontaneous fission spectrum by Mannhart [2, 3] is compared to the experimental data used for that evaluation and as shown in Figures 1–10 of reference [2]. (a) The PFNS in units of (1/MeV). (b) The PFNS as a ratio to a Maxwellian function.
In the text
thumbnail Fig. 2.

Schematic drawing of PFNS-measurement types. The blue rectangle in the orange fission chamber is the isotope investigated; the blue circles indicate neutrons, including the incident neutron, and scattered or background neutrons; the green arrow denotes fission neutrons headed to the neutron detector; other fission neutrons are given by the black arrows. Figure (a) is for shape measurements. Figures (b) show the main measurement (top) and the monitor measurement (bottom) for a clean-ratio shape measurements. Figures (c) show the main measurement (top) and an associated ratio-calibration measurement (bottom) made in a different experimental environment to quantify the neutron detector efficiency. Figure 3 (c-bottom) can also illustrate calibration with a spontaneous fission source such as 252Cf.
In the text
thumbnail Fig. 3.

ENDF/B-VIII.0 239Pu PFNS [8, 60] are used as examples to illustrate structures commonly observed around second- and third-chance fission thresholds: from thermal Einc to the on-set of second chance fission, curves similar in shape to the thermal PFNS are expected. In a window of Einc = 1–2 MeV around second-chance fission (red dashed line), one sees a characteristic structure of the PFNS appear in the 100 s of keV stemming from a second-chance fission contribution to the PFNS that is only non-zero up to these energies due to energy conservation. Around the third-chance fission threshold (blue solid line), this structure is like a broader shoulder in Eout above 100 keV. One also sees a sharp structure at high Eout (several MeV) coming from the pre-equilibrium neutron emission from the compound nucleus before fission happens.
In the text

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