The Covariance of PFNS Results from the Chi-Nu Experiment

. The prompt fission neutron spectrum (PFNS) from neutron-induced fission is a fundamental quantity for the behavior of nuclear reactors, and has been measured many times on a wide variety of nuclei and covering di ff erent ranges of incident and emitted neutron energies. However, results from past measurements are frequently called into question in modern nuclear data evaluations because of a lack of thorough experimental documentation and incomplete uncertainty analyses. The Chi-Nu experiment at Los Alamos National Laboratory was designed to produce high-precision measurements of the PFNS of major actinides over a wide range of incident and emitted neutron energies, and with the documentation and covariance analysis required to ensure that the results of this experiment maintain their impact long into the future, thereby avoiding this pitfall of past measurements. In this work we describe the Chi-Nu experiment along with summaries of the treatment of and methods developed to address two important components of the analysis of Chi-Nu data: random-coincidence backgrounds and MCNP simulations. Furthermore, we describe the first results for correlations not just be-tween all data points collected on a single target nucleus, but also between all data points from separate Chi-Nu measurements on 235 U and 239 Pu. These correlations are important for accurately calculating ratios of the PFNS from one actinide to another, which are rare and can be informative for nuclear data evaluation e ff orts.


Introduction
Measurements of the prompt fission neutron spectrum (PFNS) began near the discovery of nuclear fission [1,2], spanning a wide range of nuclei and coverages in incident and emitted neutron energy from fission. Despite the multitude of impressive PFNS measurements carried out over the last seven decades, the lack of documentation, reliability, and detailed uncertainty analyses has caused many of these expensive and potentially useful data sets to either have uncertainties increased to reflect potential errors in the results or to be discarded entirely in nuclear data evaluations [3,4]. The Chi-Nu experiment at Los Alamos National Laboratory (LANL) was designed to measure the PFNS for major actindes over a wide range of incident and emitted neutron energies from fission, including a detailed documentation and covariance analysis and thorough documentation. In this way, results from Chi-Nu experiments are in position to avoid the same pitfalls rendering many past measurements either less impactful or altogether irrelevant.
In this manuscript, we describe the Chi-Nu experimental environment in Sec. 2, focusing on efforts to reduce the potential for bias in the experimental results. Two important components of the analysis and sources of systematic uncertainty for results from a Chi-Nu experiment are discussed in Secs. 3 and 4: random-coincidence background measurements and MCNP ® -based corrections for neutron interactions in the environment, respectively [5,6]. Methods developed to define covariances from each of these * e-mail: kkelly@lanl.gov sources of uncertainty are described in the subsections within Secs. 3 and 4. Correlation definitions not just between different incident and emitted energies of a single target nucleus, but between results from entirely different target nuclei in separate experiments are described in Sec. 5. Finally, concluding remarks are given in Sec. 6.

Experimental Setup
The incident neutron beam for Chi-Nu experiments is the result of 800 MeV proton beam bursts approximately 150 ps wide and typically separated by 1.78889 µs from neighboring beam pulses incident on a tungsten target. Proton-induced spallation reactions in the tungsten target yield, among other products of the reaction, neutrons with a continuous energy spectrum spanning the practical energy range of roughly 0.7-20 MeV or more. The Chi-Nu experiment at flight path 15L of the Weapons Neutron Research (WNR) facility allows neutrons to travel a 21.5 m path length before impinging on a parallel-plate avalanche counter (PPAC) target [7] containing a series of deposits of the actinide of interest, typically totaling approximately 100 mg of target material with roughly 400 µg/cm 2 target thickness on each deposit. Fission events are detected in the PPAC targets, and neutrons from fission are detected in one of two neutron detector arrays in separate experiments, both of which can be seen in Fig. 1. A 22-detector Li-glass array is used to detect neutrons with energies below approximately 1.5 MeV, and a 54-detector liquid scintillator array is used to detect neutrons with energies above 0.89 MeV. Incident neutron energies are assigned based on the time of flight from the proton spallation reaction and a fission detection. Similarly, outgoing neutron energies are assigned based on the difference between neutron detection and fission detection, though it should be noted that the outgoing neutron energy from time of flight does not necessarily correspond to the neutron energy as it was emitted from the fission event (see Sec. 4). From these detector arrays and this white incident neutron beam, Chi-Nu experiments have published results for the PFNS of 239 Pu and 235 U [8][9][10], with 238 U and 240 Pu results forthcoming.
The Chi-Nu experimental environment was specifically designed to reduce potential bias on results from common sources of spectrum distortion, specifically neutron scattering in the environment. Perhaps the most impactful effort to reduce scattering was the creation of a "get-lost basement" beneath the neutron detection area. This consists of a thin Al floor on which the PPAC and neutron detectors sit, with 2 m of empty space below that. The walls of the experimental environment were also placed at least 2-3 m from the detection environment. Both of these features place hydrogenous (and therefore highly scattering) concrete walls far enough away from neutron detectors that a neutron scattering from the concrete walls and floors will be outside the fission-neutron time coincidence window by the time it gets back to the experimental environment [11]. In terms of analysis development, the multi-year effort to include in the MCNP simulation all materials within the experimental environment that have any impact on environmental neutron scattering prior to detection has been of fundamental importance for properly interpreting data from Chi-Nu experiments. For more details of the Chi-Nu experimental setup, see Refs. [9][10][11].

Random-Coincidence Background Measurements and Uncertainties
The majority of the data observed in terms of fissionneutron time coincidences is from random coincidences, i.e., neutrons and fissions that are measured within a coincidence time window of each other, but that are not necessarily correlated with one another. These random coincidences could come from fission coincidence with neutrons from another neutron-emitting reaction on a target, epithermal neutrons in the environment, or another source of background neutrons. Alternatively, if the target actinide of interest also emits a substantial amount of α particles, which typically can not be completely separated from fission detections, then neutrons measured to be in coincidence with these α particles would also create a substantial random-coincidence background. The processes among others contribute to the data observed within the 1.8 µs and 500 ns coincidence windows defined for measurements of incident and outgoing neutrons, subject to the ≈1 ns at 1σ time resolution of the full detection setup.
To account for this background, and method was developed to use the singles (pre-coincidence) neutron and fission data directly from the experiment to determine the rate of random-coincidence detection [12]. In short, the Poisson-distributed fission and neutron detection rates as a function of time relative to the creation of the incident neutron beam allows for the calculation of the probability of (1) detecting a fission even at a time t f , (2) not detecting a neutron over the fission-neutron coincidence time difference, and (3) detecting a coincident neutron at a time t n . This method produces background spectra with negligible statistical uncertainty compared with the statistics of the foreground coincidence data.
While the statistical uncertainty of results from this random-coincidence background measurement method is negligible compared with the statistics of the coincidence data themselves, it is possible for the background to be subject to a source of systematic error rendering the background spectrum less accurate. However, it can be difficult to determine the accuracy of the background based on comparisons with the coincidence data alone (e.g., if the background is lower than the data, are there really more counts above background than expected, or is there a deficiency in the background?). Thus, understanding how experimental conditions impact the accuracy of the randomcoincidence background is fundamentally important. Most recently, the impact of detection rate changes and correlations between fission and neutron detection rates were investigated as a potential source of systematic biases in this background spectrum [13]. From this work, a few main conclusions were reached: 1. The rates of all detectors of the same type can be summed prior to calculation of the background spectrum without introducing an error 2. A single detection rate can change in any way without introducing an error in the background as long as the other relevant rates remain constant 3. If two detection rates are changing, then it is almost always true that an error in the random-coincidence background will be generated 4. Each rate can follow a Poisson distribution without introducing an error in the background provided that the entire experiment is long enough to obtain the Poisson mean value of each rate 5. A positive correlation between detection rates from, e.g., an external particle beam, causes the calculated background to be systematically lower than the correct background spectrum, and the shape is distorted as well Conclusion (1) is potentially extremely useful for time spectra because it reduces the total time required for background spectrum calculation by a factor of the product of the number of each detector type. However, when energy spectra are needed for the final result, this is less useful if the distance from each detector of either type is different (implying the need to calculate the time-dependent background for each combination before conversion to energy). Conclusions (2) and (3) above can be calculated analytically, and are not too surprising, though it is useful that variations in a single detector rate can be handled without issue. Conclusion (4) is comforting in that it is an inevitable reality of virtually all experiments. On the other hand, conclusion (5) is concerning, and is believed to be a dominant source of systematic errors (as opposed to uncertainties) in the random-coincidence background spectra calculated for Chi-Nu experiments.
In Ref. [9], a method to correct the randomcoincidence background spectrum was developed and applied. This method was later described in more detail in Ref. [13]. In short, when a time shift is introduced into the coincidence analysis for one detector type (typically the fission detection time here) such that the possibility of true coincidence detection is negligibly small but the rate correlations between detector types are approximately maintained, then the difference between the measured data with the time shift and the random-coincidence background calculated for the same analysis can be used as a correction to the random-coincidence background. This approach reduces the potential for systematic error in the background to a negligible level, but at the cost of increasing the statistical uncertainty in accordance with the statistics of the coincidence data in the time-shifted analysis. However, multiple time shifts can be averaged together to reduce the statistical uncertainty to the desired level. For more details on this method and the study of sources of systematic error from this method, see Refs. [12,13].

MCNP ® -based Corrections
Nearly all PFNS experiments rely on measurements of the 252 Cf spontaneous fission PFNS for results on any other nucleus [1]. Some measurements use 252 Cf in a direct ratio to the data from another actinide to obtain the desired PFNS result, and others use 252 Cf to measure an efficiency that is then used in the calculation of the desired PFNS, but these and all similar methods reduce down to the same method of measuring the PFNS as a ratio to the 252 Cf PFNS. Results from the Chi-Nu experiment are some of the only data sets to not measure the PFNS as a ratio to 252 Cf. Instead, the Chi-Nu team relies on highlydetailed and vetted MCNP simulations of the experimental environment [3,14]. While the use of MCNP simulations allows for a detailed understanding of how neutrons interact with the experimental environment, simulations themselves invite a series of other systematic uncertainties from, e.g., uncertainties in post-processing parameters required to match the simulation to the data, and uncertainties in the nuclear data libraries used as input to the simulation. This latter source of uncertainty, generally termed MCNP input nuclear physics uncertainty here, is a particularly challenging source of uncertainty to quantify, and is the focus of the remainder of this section.
A simple approach to the problem of quantifying the impact of this uncertainty source on a result utilizing MCNP simulations would be to vary the relevant cross sections, and run a series of sequential simulations. However, given that there are potentially a large number of cross sections to vary, each cross section may require hundreds of variations to acquire sufficient statistics, and each simulation can take hours or more, this approach quickly becomes intractable. There are also a variety of methods available within MCNP to test the sensitivity of a simulation to input nuclear data, but none of these methods allow the use to vary data according to the covariance of the input nuclear data, or allow for total freedom of choice of the user to vary different numbers of various cross sections in selected cells, for example. As such, the Chi-Nu team has attempted to develop some flexible methods of quantifying this source of uncertainty with reduced calculation time. The three most useful methods are described below.

Calculation of a Bayes-like Factor
For the purposes of Chi-Nu experiments, the output of a simulation can be summarized in a matrix of neutron energy as it was emitted from the fission event (i.e., the initial emitted neutron energy) versus the neutron energy measured using time of flight. An example of this response matrix for the Li-glass detector array is shown in Fig. 2, which is reproduced from Fig. 10 of Ref. [9]. The structures seen in this matrix are representative of nuclear cross sections relevant to the neutron transport in the environment, as well as physical objects in the experimental area.
Using only two simulations, one with the original nuclear data input and another with varied nuclear data input, one can calculate a quantity, B, that resembles a term known as the Bayes factor [15]. In short, the factor calculated here quantifies the relative likelihood between two different models describing the data, calculated separately for each bin of the response matrix. For the case of Chi-Nu PFNS measurements, the two models are (a) the change in input nuclear data did not alter the contents of the bin of the response matrix, and (b) the change in input nuclear data did alter the contents of the bin of the response matrix. It might initially seem that this information could be  obtained in a simpler way by, for example, just dividing or subtracting the response matrices before and after altering the input nuclear data, but methods like this don't work well for bins with low statistics because variations are expected given the Poisson distributions from which the counts observed in each bin are drawn.
In short, the desired information is whether the counts in each bin of the response matrix were changed enough to support a model where the Poisson mean value, λ, has changed because of the change in the input nuclear data. Assuming unity for the prior probability for the model, the Bayesian posterior probability for the case where the Poisson mean value λ is assumed to have not changed can be expressed as where P λ (c) is a Poisson distribution with mean value λ and observed counts c, G λ (α, β) is a gamma distribution with input parameters α and β defined as in the equation above, and c 1 and c 2 are the counts from the simulations before and after changing the input nuclear data. Note that the gamma distribution is included as the conjugate prior for the Poisson distribution. For the case where the λ is not constant, we instead define two Poisson mean values before and after the simulation, λ 1 and λ 2 and calculate the posterior probability as In this scenario, the ratio of the evidence (denominator) in the posterior probabilities can be calculated to yield, i.e., This factor, while similar to a Bayes factor, is not a formally correct derivation of a Bayes factor because, in part, the denominator of Bayesian posterior probabilities is not generally integrable. The reader is referred to Ref. [15] for formal discussions of the Bayes factor. Nevertheless, this factor B fulfills the needs of the Chi-Nu experiment in that it exposes which bins of the response matrix are impacted by variations in the input nuclear data. An example matrix of B factors for increasing the 27 Al(n,n) elastic neutron scattering cross section by 5% is shown in Fig. 3. As can be seen, two horizontal bands in a region of counts in the matrix representing high levels of neutron downscattering (i.e., energy measured via time of flight is less than the initially emitted neutron energy) appear to be the dominant region impacted by this Al cross section. Similar studies have been done for the Chi-Nu environment on a suite of nuclei present in the experimental environment. While this method is extremely useful for determining to which nuclear data an experiment is sensitive and has been applied successfully in that respect to the Chi-Nu experiment, unfortunately no clear method of directly relating this factor to an rigorous uncertainty on input nuclear data was developed, nor is it possible to accurately reflect input nuclear data covariances with only one variation of input nuclear data. Thus, other methods are required for the final quantification of this uncertainty source.

Implicit Capture Simulation Manipulation
In MCNP, the user can choose, among many other options, between analog and implicit capture simulation techniques. Analog simulations proceed in a step-by-step process similar to how one might imagine a real experiment occurring. Alternatively, implicit capture simulations force a capture reaction to occur whenever it can, but with a weight corresponding to the probability that the capture reaction would have occurred. Thus, a portion of the neutron, in this case, is captured and the surviving portion of the neutron carries on with a reduced weight. In Ref. [16] a method was developed to utilize the output of implicit capture simulations in order to vary input nuclear data cross sections in a post-processing calculation using the result from only the original simulation without no nuclear data variations. This is done by varying the weights of the surviving nuclei according to the altered probabilities for capture reactions occur if a cross section is changed. This method is powerful enough to incorporate covariances for input nuclear data and fast enough to allow for hundreds or thousands of nuclear data variations, but has a limited range of applicability to nuclei with reasonably-high capture cross sections. The 6 Li(n,t) cross section, which is the neutron detection mechanism for 6 Li-enriched Li-glass detectors, is ideally suited for this purpose. The reader is referred to Ref. [16] for more details on this method.

Direct Track Weight Variations
As opposed to relying on implicit capture cross simulations from MCNP, it is possible to simply write out all data from every interaction and surface crossing event from an MCNP simulation using the PTRAC output format [17]. With this information, one can again calculate varied neutron weights by altering three probabilities calculated during the course of an analog simulation: (1) the probability for a neutron to travel a path length l in a material, (2) the probability for interacting with a chosen nucleus in the material, and (3) the probability for choosing the nuclear physics reaction that occurred in the simulation. These probability variations can be described in terms of three factors altering the weight of the neutron at each step in the simulation, f l , f i , and f σ , respectively. For a given isotope, i, with a total macroscopic cross section, Σ t,i , in a material with integrated macroscopic total cross section Σ T , we otbain and where l is the path length traveled by the neutron, and a prime (as in Σ ′ t,i ) indicates a quantity calculated after a cross section was changed in the isotope of interest. The value of the factor f σ depends on the cross section being varied, but is a straightforward ratio before and after variation. These factors reflect the altered probabilities for the neutron to travel along the path taken during the simulation. This method, while it has not been applied yet to Chi-Nu PFNS results, has the potential to be used for variations of any nucleus and including covariance input. Among other potential applications, it could also be a method for carrying out optimizations for a forward analysis approach for extracting cross sections from an experiment, though this possibility has not yet been explored fully.

Correlations between PFNS Data Points Across Target Nuclei
Results from Chi-Nu PFNS measurements are unique among any other PFNS measurement in that careful covariance calculations were carried out to describe correlations between not just emitted neutron energy data points within a single incident energy range, but between all emitted energy points of all incident energies. This information then yields a single covariance matrix describing the entire data set. Correlations between data points of different incident energy ranges arise primarily from the similar analysis methods and corrections applied to each incident energy range, most of which contain correlations between different emitted energy points as well. In order to properly report the covariance of the PFNS shape, the PFNS results for each incident energy range are also normalized with the Li-glass and liquid scintillator data scaled to obtain the same data integral in the overlap region of approximately 0.89-1.5 MeV, which also has the effect of creating an anticorrelation between high and low emitted neutron energies. See Refs. [9] and [10] for more details on these covariances. In Fig. 4, we report for the first time a calculation of not just a single covariance matrix for all data points of a single PFNS measurement, but a single matrix describing correlations between all data points of all published Chi-Nu PFNS measurements [9,10]. Data point number, along both the x and y axes, is an index for outgoing neutron energy points of a PFNS result. Each PFNS result from a Chi-Nu experiment contains 65 data points, and we typically report 20 incident energies yielding 1300 total data points for the complete data set of the 239 Pu and 235 U PFNS. Data points 1-1300 on each axis correspond to the 239 Pu results, and data points 1301-2600 correspond to 235 U. Thus, while the matrix from x = 1-1300 and y = 1-1300 corresponds to the published 239 Pu correlation matrix [9] and the matrix from x = 1301-2600 and y = 1302-2600 corresponds to 235 U [10], the off-diagonal blocks (i.e., x = 1-1300 and y = 1301-2600, and its reflection over the line x = y) describes the correlations between 239 Pu and 235 U PFNS results from the Chi-Nu experiment. These correlations arise primarily from the similarities in the MCNP simulations used for analysis, since the simulations differ only in target nucleus identity and orientation of the PPAC target chamber. Thus these cross-nuclide correlations are estimated based on the magnitude of uncertainty and correlation created by MCNP-based aspects of the analysis between experiments for each nucleus. These correlations are necessary for properly calculating ratios of, e.g., the 239 Pu to the 235 U PFNS for various incident energy ranges.

Conclusions
The loss of reliability of results from PFNS measurements over the last seven decades is an unfortunate side effect of experiments that are not well documented and did not have a complete or reliable uncertainty and covariance analysis at the time analysis was being completed. The Chi-Nu experiment to measure the PFNS for major actinides was designed to produce results that stand the test of time and preserve reliability of PFNS results through thorough documentation and detailed covariance analysis. In this manuscript we describe details of two of the most important aspects of data analysis for Chi-Nu experiments: random-coincidence backgrounds and MCNP simulations. For random-coincidence backgrounds, a summary of recent work investigating the potential for systematic errors and uncertainties in the random-coincidence background measurement based on changes in detection rates one might encounter in an experiment, with important conclusions regarding correlations between detection rates provided. The uncertainty on the use of MCNP from uncertainties in input nuclear data to the simulation was also discussed in terms of three methods developed for during the course of the development of the Chi-Nu experiment: a Bayes-like factor, implicit-capture simulation output variations, and direct track weight variations. While the factor calculated for the Chi-Nu experiment is not a formally rigorous formulation of a true Bayes factor, the concept and application described in this work to nuclear physics experiments is not common, and the other methods for varying input nuclear data cross sections to MCNP were developed by the Chi-Nu team for this experiment.
Lastly, for the first time, a single correlation matrix describing the relation of each emitted energy point in each incident energy range for each target nucleus in different Chi-Nu experiments was shown. This represents an advancement in PFNS experiments beyond what has been done in previous measurements, and is essential for accurately calculating informative ratios of the PFNS of one nucleus to another. In addition to 239 Pu and 235 U, Chi-Nu measurements of the 238 U and 240 Pu PFNS have been completed as of writing this manuscript, and thus these measurements will follow the same detailed analysis methods employed for previous Chi-Nu PFNS measurements.