# Measurement of the Prompt Neutron Decay Constant in a Zero-Power Reactor Using a Novel 3D Detector System

Michel Saliba, Vincent Lamirand, Wilfried Monange, Fanny Vitullo, Mathieu Hursin, Andreas Pautz, Oskari Pakari

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Michel Saliba, Vincent Lamirand, Wilfried Monange, Fanny Vitullo, Mathieu Hursin, et al.. Measurement of the Prompt Neutron Decay Constant in a Zero-Power Reactor Using a Novel 3D Detector System. Nuclear Science and Engineering, 2025, pp.1-9. 10.1080/00295639.2025.2456411. hal-05076782

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# Measurement of the Prompt Neutron Decay Constant in a Zero-Power Reactor Using a Novel 3D Detector System

Michel Saliba, Vincent Lamirand, Wilfried Monange, Fanny Vitullo, Mathieu Hursin, Andreas Pautz & Oskari Pakari

**To cite this article:** Michel Saliba, Vincent Lamirand, Wilfried Monange, Fanny Vitullo, Mathieu Hursin, Andreas Pautz & Oskari Pakari (03 Mar 2025): Measurement of the Prompt Neutron Decay Constant in a Zero-Power Reactor Using a Novel 3D Detector System, Nuclear Science and Engineering, DOI: 10.1080/00295639.2025.2456411

**To link to this article:** https://doi.org/10.1080/00295639.2025.2456411

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# Measurement of the Prompt Neutron Decay Constant in a Zero-Power Reactor Using a Novel 3D Detector System

Michel Saliba,\* a* Vincent Lamirand,\* a,b Wilfried Monange,\* Fanny Vitullo,\* a Mathieu Hursin,\* a,b

Andreas Pautz,\* a,b and Oskari Pakari\* a,b

Received August 16, 2024

Accepted for Publication January 10, 2025

**Abstract** — Analyzing neutron noise in zero-power reactors provides information about integral parameters, such as the prompt decay constant α. This information can be used to validate codes that predict said parameters, or time-dependent behavior, particularly reactor transients. The CROCUS zero-power research reactor has been widely used for conducting noise experiments and has recently been enhanced with a new neutron detection system SAFFRON. This system includes an array of about 150 miniature neutron detectors evenly dispersed throughout the core, allowing for three-dimensional (3D) spatial detection capabilities. We conducted a neutron noise experiment using the 3D array to test this new system to determine the prompt decay constant α. Our results, derived from the Rossi-α method, aligned with the predictions from the Serpent 2 Monte Carlo code, which utilized the Lawrence Livermore National Laboratory FREYA library for fission. The agreement was within 1σ of the estimated uncertainties. We notably observed that based on previous experiments, using all 150 detectors does not achieve the levels of detection efficiency (in counts per fission) expected to be necessary to observe the fission chain decay, yet we still observe the decay with comparatively high statistical significance. We therefore show experimentally that the spatial extent of the detection system is an important parameter in the prediction of the success of the neutron noise technique. We also explored the cross-spectral density method for different combinations of SAFFRON detectors, which gave consistent results with the Rossi-α method. In addition, uncertainty estimation methods, including the bootstrap method, were applied to SAFFRON, addressing both temporal and spatial aspects. We discuss the benefits of using bootstrap methods for estimating the uncertainty of the prompt decay constant, which we found often underestimated when using the covariance matrix from a standard nonlinear fit. The results also indicate that SAFFRON is a powerful measurement tool that can be expanded for use in active perturbation experiments, 3D flux maps, and other high-resolution reactor physics experiments.

**Keywords** — Neutron noise, prompt neutron decay constant, experimental reactor physics, Monte Carlo code.

**Note** — Some figures may be in color only in the electronic version.

## I. INTRODUCTION

In nuclear reactors, all signals from monitoring detectors inherently contain neutron noise. It refers to fluctuations that occur in real signals around a mean value, and it contains information about integral parameters derived from the point-kinetics model,

<footnote><sup>a</sup>École Polytechnique Fédérale de Lausanne, Laboratory of Reactor Physics and System Behavior, Lausanne 1015, Switzerland</footnote>

<footnote><sup>b</sup>Paul Scherrer Institute, Nuclear Energy and Safety Research Division, 5232 Villigen PSI, Switzerland</footnote>

<footnote><sup>c</sup>Institut de Radioprotection et de Sûreté Nucléaire, Pôle Sûreté Nucléaire, Laboratoire de Neutronique, Fontenay-aux-Roses, France</footnote>

<footnote>*E-mail: michel.saliba@epfl.ch</footnote>

<footnote>This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The terms on which this article has been published allow the posting of the Accepted Manuscript in a repository by the author(s) or with their consent.</footnote>
such as the prompt decay constant $\alpha$.[1] Measuring the
prompt decay constant during steady-state operation of
a reactor can offer a noninvasive approach to measure
point-kinetics parameters utilized for code validation
and predicting reactor transients.

In zero-power systems, neutron noise emerges
solely from intrinsic fluctuations in the neutron popu-
lation, as all other fluctuations arising from tempera-
ture, mechanical, and hydraulic effects are considered
negligible. This is in contrast to power reactors, where
the fluctuations are dominated by thermal-hydraulic
vibrations,[2] which entirely overpower the effects of
neutron noise. Various experiments related to neutron
noise[3] have been conducted at the CROCUS zero-
power research reactor at École Polytechnique
Fédérale de Lausanne (EPFL). Moreover, experiments
have been carried out for code validation.[4,5]

The Laboratory for Reactor Physics and Systems Behavior at EPFL recently developed an advanced detection system called SAFFRON.[6–9] This detector comprises 150 evenly distributed thermal neutron detectors within the core of the CROCUS reactor. With this unique three-dimensional (3D) detection capability, the SAFFRON detector enables flux map reconstruction and 3D spatial kinetics measurements.

The outline of the paper is as follows. Section II details the CROCUS reactor and the SAFFRON detector system, the methodology used to determine the prompt decay constant, and the Monte Carlo simulations that we used for comparison. In Sec. III, we showcase, analyze, and interpret the results. Last, in Sec. IV, we provide a brief conclusion summarizing the overall experiment's key findings while providing future research directions.

## II. METHODS AND MATERIALS

### II.A. The Zero-Power Reactor CROCUS

The EPFL CROCUS reactor is a critical assembly that utilizes uranium fuel and light water as a moderator. The CROCUS reactor is primarily used for educational and research purposes[10–13] and has a maximum power of 100 W. One of the notable features of this reactor is that it uses two different fuel types arranged in concentric fuel zones: the inner zone composed of uranium oxide fuel and the outer zone consisting of metallic uranium fuel, as shown in Fig. 1. The reactor geometry is well characterized and is part of a Nuclear Energy Agency International Reactor Physics Evaluation benchmark.[14]

CROCUS has four main operation monitor detectors. Two compensated ionization chambers are located north and south, serving as basic core monitoring systems. Additionally, two fission chambers (FCs) are calibrated as power monitors in the eastern and western periphery.[15] In addition to these, CROCUS can employ various neutron detectors, including several types of proportional counters (PCs) such as $^3$He or scintillation-based neutron detectors like the SAFFRON detector system.[9] The latter is the primary detector system used for the investigations presented herein.

### II.B. A 3D Neutron Detector Array: SAFFRON

The SAFFRON[8] detector array is a new specialized system that consists of 149 small and compact neutron detectors named MiMi detectors,[6] evenly distributed in the CROCUS core, as well as 11 mobile detectors that can be positioned as required. Each detector is made of a small scintillation sheet (on the order of a tenth of mm³) composed of ZnS doped with $^6$LiF. The $^6$Li captures thermal neutrons inducing a (n,$\alpha$) reaction. The products of the reaction will excite the ZnS molecules and cause them to scintillate. This mechanism is illustrated in Fig. 2. The produced light is then collected via an optical fiber of 30-m length to individual silicon photomultipliers (one per detector and fiber) that are located outside the reactor shielding.

The main objective of SAFFRON is to provide time-resolved data for 3D space-dependent neutronics that can be used to validate high-fidelity neutronics codes in steady-state and transient scenarios. Moreover, SAFFRON is envisioned to be used for absorber oscillation experiments[16] to facilitate the local versus global reactivity components.

The SAFFRON detector is distributed axially in-core in five levels. Three primary levels at 150, 500, and 850 mm each hold 47 detectors as shown in Fig. 2. Two intermediate levels at 300 and 700 mm have four detectors. The reactivity worth of SAFFRON has been estimated at (44 ± 4) pcm using the Serpent 2 code using the ENDF/B-VII.1 cross-section library. To compensate for this effect, four metallic uranium rods were added to the periphery of the reactor core.

### II.C. Neutron Noise Analysis

Neutron noise methods such as Rossi-$\alpha$ and the power spectral density (PSD), detailed in Secs. II.C.1 and II.C.2, respectively, rely principally on assumption of the point-kinetics model. For both methods, we fitted
<figure><img src="image_8.png" /><figcaption>Fig. 1. (Left) The isometric view of the CROCUS reactor vessel and its internal components is depicted. (Right) The top grid is shown.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 2. (Left) The MiMi system utilizes a small volume of ZnS scintillator doped with ⁶LiF to detect thermal neutrons. (Right) Picture of the CROCUS core with a view of the SAFFRON system during unloading. Optical fibers carry the scintillation light out of the cavity, shielding the acquisition system from the radiation field.</figcaption></figure>

their equation using the Levenberg-Marquardt algorithm,[17] which is used to resolve nonlinear least-squared optimization problems.

### II.C.1. The Rossi-α Method

The Rossi-α method[18] relies on the direct measurement of prompt fission chains as they propagate through the reactor. To achieve this, the timestamps of detection events are recorded, and the differences between them are calculated, Eq. (1):

$$ \tau_{i,j} = \text{Timestamp}(i) - \text{Timestamp}(j) \text{ with } i < j. \quad (1) $$

For CROCUS, these differences are then binned between 0 and 0.1 s, which is the typical range taken for thermal reactors. Finally, an exponential function is fitted to the binned data, Eq. (2):

$$ f(\tau) = Ae^{-\alpha|\tau|} + B. \quad (2) $$

By utilizing a nonlinear least squared algorithm, we determine the value of the prompt decay constant α.

Furthermore, the Rossi-α method, explicitly defined in Eq. (3), incorporates additional information such as the detector's efficiency, defined as counts per fission, within its parameters:
$$P(\tau) = \frac{\varepsilon D_v}{2\alpha\Lambda^2} e^{-\alpha|\tau|} + \varepsilon F, \quad (3)$$

where $\varepsilon$ = detector's efficiency; $F$ = fission rate in the reactor; $D_v$ = Diven factor; $\Lambda$ = prompt neutron generation time.

The minimum efficiency required for a detector to successfully observe the prompt decay constant was empirically determined to be around $4 \times 10^{-5}$.[19,20] Under this limit, it was argued, the detector cannot detect enough neutrons per fission such that the Rossi-$\alpha$ exponential decay would be discernible above background noise. The efficiency of SAFFRON was determined to be $8 \times 10^{-6}$. We therefore did not expect to be able to successfully observe the prompt decay constant in CROCUS with SAFFRON.

## II.C.2. Power Spectral Density Method

This method, also known as the Cohn-$\alpha$ method,[21] utilizes frequency domain analysis to analyze the detector time series. The PSD of a signal can be estimated by computing the periodogram, which is the squared magnitude of its Fourier transform. The corresponding analytical expression is given by Eq. (4):

$$G_{xx}(\omega) = \int_{-\infty}^{\infty} e^{-i\omega t} p(t) dt = eF + \frac{\varepsilon^2 F D_v}{(\rho - \beta_{\text{eff}})^2} \frac{1}{1 + \frac{\omega^2}{\alpha^2}} \quad (4)$$

The PSD curve generally displays a Lorentzian profile with a cutoff at frequency $\alpha$. This method can also be used to cross correlate between two detectors via a cross power spectral density (CPSD) estimate of the frequency content of the signals. The formula for the CPSD between two detectors is shown in Eq. (5)[20]:

$$G_{xy}(\omega) = \int_{-\infty}^{\infty} e^{-i\omega t} p_{1,2}(t) dt = \frac{\varepsilon_x \varepsilon_y F D_v}{(\rho - \beta_{\text{eff}})^2} \frac{1}{1 + \frac{\omega^2}{\alpha^2}} \quad (5)$$

## II.D. Neutron Noise Simulation

### II.D.1. Serpent 2

The Serpent 2 code,[22] a Monte Carlo neutron transport code that is widely used for reactor physics and radiation shielding studies, was used in this study. Although Serpent 2 has been mainly used for evaluating

group cross-section libraries in diffusion codes, it can also predict kinetic parameters.[23] A detailed model of the CROCUS reactor, along with the SAFFRON detector, was created in Serpent 2.

### II.D.2. FREYA

In order to include a more physical model for neutron-induced fission events, we employed the Lawrence Livermore National Laboratory's FREYA code.[24] Specifically for Serpent 2, the number of neutrons emitted per fission is sampled from $\bar{\nu}$, giving either a rounded-up or rounded-down value (i.e., usually two or three neutrons are sampled per fission). As seen in Fig. 3 (left), the distribution of neutrons per fission using FREYA is closer to physical reality, which has been shown to be relevant for neutron noise predictions,[20] although at a higher computational cost.

To closely model the observable of the neutron noise experiment, we modified the code to save the information regarding a given interaction, such as position, energy, and timestamp in an external file, similar to the ptrac card of MCNP 6.3[25] (albeit limited by the precision of timestamp recording[26]), or the collision file of MCNPX PoliMi.[27] An example of this is shown in Fig. 3 (right).

## II.E. Uncertainty Analysis

Uncertainty in the determined parameters was estimated using the one provided by the nonlinear least squares fitting method. This method provided initial estimates of parameter uncertainties. To assess the robustness of these estimates and explore potential alternative uncertainty distributions, a bootstrap[28] analysis was conducted.

In other words, the bootstrap method is a resampling technique that generates multiple datasets by randomly selecting data points with replacement. In our application, we generated multiple datasets from the original time signal and then from the detector positions. By analyzing the distributions of the prompt decay constant obtained from these resampled datasets, a more accurate estimate of the uncertainty, potentially avoiding underestimation, is achieved.

## III. RESULTS AND DISCUSSION

### III.A. Experimental and Simulated Results of the Prompt Decay Constant $\alpha$

CROCUS was operated at steady-state power of 2 mW for 2.5 h, during which we acquired neutron
<figure><img src="image_10.png" /><figcaption>Fig. 3. (left) Probability of the number of neutrons released per fission using FREYA for $^{235}$U in CROCUS. (right) Collected neutron interaction data from a Serpent 2 simulation using FREYA for fission.</figcaption></figure>

detection event timestamps from the SAFFRON array. The timestamps were then processed with the Rossi-$\alpha$ method. The results are shown in Fig. 4.

The experimental value of the prompt decay constant $\alpha$ was estimated to be (159 ± 7) 1/s (1$\sigma$). The uncertainty around the mean value has been determined using the covariance matrix generated by the nonlinear least squares algorithm.

Simulation data from Serpent 2 (in analog mode, i.e., no implicit capture was used) coupled with the FREYA code were gathered for 600 core hours. We recorded detection events in the ZnS: $^{6}$LiF volumes of the SAFFRON detectors. The data were then processed using the Rossi-$\alpha$ method, also plotted in Fig. 4. The simulated prompt decay constant is (156 ± 3) 1/s (1$\sigma$).

The error bars in the experimental data in Fig. 4 were obtained by assuming that the occurrences in each bin follow a Poissonian distribution; i.e., the standard deviation is the square root of the occurrences in each bin. The obtained value of (156 ± 3) 1/s from the simulation is consistent with the experimentally determined interval of

[152;166]. The findings are also consistent with older similar experiments,[3,29–31] encouraging us to believe in the detector’s reliability.

### III.B. Efficiency of the Detector

The efficiency of neutron detectors is defined as the counts per second recorded by the neutron detectors divided by the total fission rate of the reactor. The efficiency of a neutron detector is an important parameter that predicts if a neutron detector is capable of detecting the prompt decay constant from a system using neutron noise methods. Traditional neutron detectors, such as FCs or $^{3}$He PCs, are typically positioned within the reflector. The efficiency of these detectors is illustrated in Fig. 5 (left).

The SAFFRON detector presents a unique case. Its efficiency, being equal to $8 \times 10^{-6}$, was observed to be lower than that of all the other neutron detectors previously used in the CROCUS reactor for noise measurements. Furthermore, when the efficiency is below the empirical threshold, $4 \times 10^{-5}$, found in Pakari et al.,[19] we should not be able to detect the prompt decay constant.

However, despite operating below the efficiency limit, the SAFFRON detector successfully measured the prompt decay constant, as detailed in Sec. III.A. In standard scenarios, such low-efficiency systems would likely lead to significant uncertainties or even failure to measure $\alpha$.

This finding suggests that the conventional understanding of efficiency in neutron detection may not fully apply to systems like SAFFRON or be a measure of predictability of detecting the prompt decay constant.

One potential explanation for this unexpected result lies in the distinctive distributed geometry of the SAFFRON detector. Unlike traditional detectors that are localized in specific regions, SAFFRON’s detectors are spread out across a broader volume. This spatial distribution could enhance the overall detection capability in a way that compensates for the lower individual detector efficiency.

<figure><img src="image_11.png" /><figcaption>Fig. 4. Comparison of the Rossi-$\alpha$ distributions of the experimental and simulation data (normalized to the first entry). The Y-axis is in log-scale.</figcaption></figure>

<figure><img src="image_12.png" /></figure>
<figure><img src="image_13.png" /><figcaption>Fig. 5. (left) Plot showing detectors with varying efficiency used in CROCUS to measure the prompt decay constant: $ ^{3}\text{He} $ in the reflector, FCs in the reflector at 4 and 10 cm from the reactor core, and SAFFRON. (right) Plot showing the connection between the efficiency used and the measurement of the prompt decay constant. The plot shows the measured prompt decay constant relative to the number of detectors used (black line). In addition, the efficiency of the number of detectors used is shown on the right-hand side of the plot (orange line). The empirical efficiency limit is also added to the plot (red dashed line).</figcaption></figure>

To explore this hypothesis further, we examined how the prompt decay constant $\alpha$ and its associated uncertainty change with the number of detectors in the SAFFRON system. As shown in Fig. 5 (right), the Rossi-$\alpha$ method was used to calculate $\alpha$ for different configurations of the SAFFRON detector, with varying numbers of detectors. The efficiency of each configuration was also plotted.

The prompt decay constant was still detectable until a system efficiency at around $2 \times 10^{-6}$. Beyond this point, our measurement becomes unreliable as the mean value and its corresponding uncertainty start to diverge.

### III.C. Cross Power Spectral Density

In this section, a CPSD analysis is conducted on two arbitrary configurations of the SAFFRON detector for exploratory purposes and for investigating the presence of spatial effects. The first configuration consists of splitting the detector ensemble into a northern group and a southern

group relative to the center of the CROCUS reactor. The second configuration separated the detectors between the detectors belonging to the uranium oxide zone and those belonging in the metallic uranium zone of the CROCUS core, i.e., inner and outer fuel zones. The prompt decay constant of the former configuration is shown in Fig. 6 (left), and the latter configuration is shown in Fig. 6 (right).

The obtained results are within $2\sigma$ from the experimentally determined prompt decay constant in Sec. III.A. These results support that there is no spatial correlation between different locations in the CROCUS reactor, solidifying that CROCUS is a point reactor.[20]

### III.D. Uncertainty Quantification with SAFFRON Using Bootstrap

In the following, we will utilize the bootstrap method to determine the prompt decay constant $\alpha$ with respect to time[3] and use the unique geometry of SAFFRON to perform this method in space.

<figure><img src="image_14.png" /><figcaption>Fig. 6. (Left) Cross power spectral density of the north and south groups of detectors. (Right) Cross power spectral density of the uranium oxide and metallic uranium group of detectors.</figcaption></figure>
### III.D.1. Time Resampling

The signal obtained from the SAFFRON detector was constructed using an integration time of 0.001 s. The signal was divided into 9000 segments.

Based on the bootstrap principles, these segments were resampled randomly with the possibility for replacement to generate 10 000 new samples. For each sample, the signal is fed to the PSD method to obtain the prompt decay constant. For each iteration, $α$ is saved to build a distribution as shown in Fig. 7.

The distribution given by the bootstrap method is similar to a normal distribution as seen in Fig. 7. The mean value of the prompt decay constant is (157 ± 7) 1/s. The uncertainty is the standard deviation of the recorded prompt decay constants. The impact of time is observed to affect the prompt decay constant, exhibiting a distribution that resembles a normal distribution.

### III.D.2. Space Resampling

Once again, we will utilize the unique property of SAFFRON of having multiple detectors dispersed into the core. The aim here is to apply a bootstrap resampling by altering the detectors used. In the usual case, we use the 149 detectors in-core. In this section we will choose the detectors randomly, and a detector in question could be chosen multiple times (sampling with replacement). We generated 10 000 samples to build a distribution for the prompt decay constant as shown in Fig. 8.

The distribution from the space bootstrap approach looks like a normal distribution, as shown in Fig. 8. The average prompt decay constant is (166 ± 12) 1/s. The

<figure><img src="image_15.png" /><figcaption>Fig. 8. Bootstrapping the MiMi detectors of SAFFRON to obtain an estimate of the prompt decay constant $α$ with respect to space variation.</figcaption></figure>

uncertainty is the standard deviation of the recordings of $α$. We see that space affects the prompt decay constant leading to a larger variance than in the time bootstrap case.

### III.D.3. Uncertainty Discussion

The bootstrap method is important in accurately estimating the uncertainty of the prompt decay constant, which is often underestimated using nonlinear least squares methods. Bootstrap-derived uncertainties in space provide more reliable values, aligning well with other experiments and avoiding the false impression of physical effects.

Therefore, using bootstrap methods helps discard such misleading possibilities, strengthening our confidence in the pointlike spatial nature of the CROCUS reactor. More specifically, bootstrapping in time helps to identify the uncertainties embedded in the time signal. For noise methods, a steady-state signal is required, but imperfections might arise due to operational irregularities. Bootstrapping in space unravels the possibility of detecting spatial effects beyond point kinetics.[4,32] Based on the space bootstrap data, there is no mean offset of the prompt decay constant, and it has large uncertainty, indicating that the spatial effect seems small, reinforcing the results interpreted for the CPSD, Sec. III.C. The large uncertainty might be explained by the presence of some faulty detectors in the system's array.

## IV. CONCLUSION

In this study, we assessed the feasibility and performance of SAFFRON—a new 3D detector array installed

<figure><img src="image_16.png" /><figcaption>Fig. 7. Bootstrapping the time signal coming from SAFFRON to obtain an estimate of the prompt decay constant $α$ with respect to time variation.</figcaption></figure>

<figure><img src="image_17.png" /></figure>
in the CROCUS zero-power reactor—for neutron noise measurements. Using the Rossi-α method, we measured the prompt decay constant and found it to agree with Serpent 2 predictions within a 1.89% discrepancy at 1σ. Notably, SAFFRON's efficiency in counts per fission was lower than the previously established empirical limit for observing the prompt decay constant. We hypothesize that SAFFRON's 3D distributed geometry contributes to this unexpected “over-efficiency,” an aspect not accounted for in standard point-kinetics equations. Additionally, the CPSD method was explored and showed no spatial effects in the reactor. Bootstrap analysis was conducted in both time and space to accurately quantify the uncertainty around the mean value of the prompt decay constant. Moreover, there was no established method to obtain the uncertainty for such a large array of detectors, and we explored a method to do so for the first time. Future work will explore the phenomenon of this over-efficiency and investigate other 3D neutronics-related experiments.

## Acknowledgments

The authors thank the CROCUS team members for their continuous efforts to maintain the reactor.

## Disclosure Statement

No potential conflict of interest was reported by the author(s).

## ORCID

Vincent Lamirand http://orcid.org/0000-0001-9403-441X

Andreas Pautz http://orcid.org/0000-0002-3242-4305

Oskari Pakari http://orcid.org/0000-0003-3704-8190

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