Patchy Nuclear Chain Reactions

The study addresses how and when stochastic fluctuations in reactor neutron populations disappear, a prerequisite for deterministic power control and safe operation, particularly during startup and shutdown. Prior simulations suggested possible non-Poissonian spatial patterns and clustering at low neutron density, but experimental confirmation was lacking. The authors designed and executed dedicated low-power experiments at the Rensselaer Polytechnic Institute Reactor Critical Facility (RCF) to probe time and space fluctuations without detector saturation, developed a high-fidelity Monte Carlo “numerical twin” of the core, and constructed a stochastic branching random walk model including spontaneous fissions. The goal is to identify conditions under which fluctuations persist, characterize spatial correlations, and assess implications for nuclear safety systems.

Past work has extensively investigated fission-induced correlations and neutron noise, including simulation studies reporting non-Poissonian spatial patterns in decoupled systems and spontaneous clustering. Clustering phenomena are well documented in branching-diffusion processes across life sciences (epidemics, bacterial growth, ecological communities, gene propagation), where local reproduction and global death with diffusion lead to patchiness, especially in low dimensions and at low densities. In reactor physics, such mechanisms suggest clustering at low power due to diffusion, fission (birth), and capture (death). The classical “critical catastrophe” model (Williams) predicts unbounded variance growth at exact criticality if intrinsic sources are neglected, but practical feedbacks are known to limit this in operating reactors. This work builds on and tests these ideas experimentally, incorporating spontaneous fission sources and delayed neutrons into a stochastic framework.

• Experimental site and reactor: Walthousen RCF (zero-power training reactor; 4.81% enriched UO2 fuel in stainless steel cladding; licensed to 100 W; essentially fresh fuel; square lattice, 36-inch active length). Core size enables spatial effects at low power.
• Detectors: Multiple uncompensated ion chambers and BF3 for power; four small 3He tubes installed in a central pincell (in-core) to measure axial flux and correlations at very low power; two ex-core LANL NoMAD 3He array detectors (15 tubes each) positioned laterally to reconstruct axial power distribution and correlations up to ~100 mW without saturation.
• Experimental runs: Low-power operations (sub-mW to several mW up to ~10 mW for in-core, higher for ex-core). “Critical catastrophe” runs placed core near critical without operator intervention for 30–120 min to observe time fluctuations. Time-gate widths for analysis: 1 ms (short) and 1 s (long), among others.
• Numerical twin and simulations: MORET6 Monte Carlo neutron transport with fully analog neutron transport for fluctuations and correlations. Event-by-event fission modeling using the FREYA-based fission library and evaluated data (ENDF/B-VII.1) for cross sections and averages; multiplicity distributions for induced and spontaneous fission of 235U and 238U per literature. Massive parallelism (Intel Xeon E5-2680 cores); typical runs required ~10^5 processor-days to simulate startup and reach stationary populations. Simulations included realistic detector geometry and response to compare directly to measured count rates and correlations.
• Stochastic modeling: Branching random walk model including diffusion, fission (birth), capture (death), leakage, and intrinsic spontaneous fission sources. Average behavior described by a reaction-diffusion equation with reactivity and a source term from spontaneous fission; predicts axial profiles transitioning from source-driven shapes at very low power to cosine-like shapes above ~1 W. Time fluctuations analyzed via variance-to-mean (V/M) ratios, contrasting the classical unbounded growth model without sources to a bounded asymptote when intrinsic sources are included. Extended model includes delayed neutrons and detector time integration to predict power- and gate-dependent noise behavior. Spatial correlations modeled in a simplified homogeneous infinite-medium diffusion framework to derive analytic forms for the two-point correlation function g versus distance and power.
• Data analysis: Comparison of measured and simulated axial flux profiles (2-σ agreement), time series of power, V/M ratios versus power for different gate widths, and spatial correlation functions g versus power and distance. Statistical uncertainties via bootstrap; systematic positioning uncertainties (~1 cm) for detector placement considered. Simulations tuned via control rod position or 235U enrichment adjustments to match asymptotic power observed experimentally.

• Average spatial profiles: Measured and MORET6-simulated axial flux distributions agree within 2-σ for both in-core and ex-core detectors at ~0.66–0.79 mW; inner detectors show a flat central region with sharp boundary decays consistent with source-driven behavior at low power.
• Time fluctuations and “blinking”: Near-critical runs show that after transients the power attains a stationary regime with bounded stochastic noise. At 1 ms time-gate width, the variance-to-mean (V/M) ratio becomes stationary and is approximately independent of reactor power (experiment and simulation), contrary to the classical unbounded “critical catastrophe” prediction without sources. Simulations reveal a blinking behavior: long periods of near-zero power punctuated by bursts triggered by spontaneous fission-induced chains.
• Role of intrinsic sources and delayed neutrons: Including spontaneous fissions yields bounded V/M in time and a power-independent noise at small time gates. For large gate widths (e.g., 1 s), the V/M of detected counts grows approximately as P^2, implying relative fluctuations scaling as P/Δt, potentially recovering a form of “critical catastrophe” at sufficiently large P and integration times.
• Gate-width dependence: Simulated V/M vs P at 1 s gate shows a clear quadratic increase (square-law), while at 1 ms gate V/M saturates with power (consistent with RCF data). This indicates delayed-neutron time scales govern whether power-dependent noise emerges.
• Spatial correlations and clustering: Two-point spatial correlation function g measured with ex-core NoMAD arrays scales inversely with power, g ∝ 1/P. Fits yield exponents near unity: simulation a = 0.96 ± 0.046, experiment a = 1.11 ± 0.044. In-core measurements with four 3He tubes show g(z) decreases approximately linearly with axial distance z; the slope |dg/dz| ∝ 1/P, so an inferred cluster size ~ (dg/dz)^{-1} increases with P up to core dimensions. Snapshot simulations at 1.2 mW show non-Poissonian clustered neutron distributions with average g ≈ 5% at short distances, contrasting with time-averaged smooth profiles.
• Safety implication: For close-to-critical systems and large time gates, fluctuations can grow with power; scaling from the model suggests at P ~ 1 MW and sufficiently large gate widths, fluctuations could approach the same order as the mean signal, potentially impacting protection system stability if gate selection is not appropriate.

The experiments confirm that stochastic fluctuations persist at low power and that spontaneous fission sources critically modify reactor noise behavior. By maintaining a baseline neutron population and introducing effective negative reactivity from the perspective of chain propagation, intrinsic sources prevent extinction and bound the variance growth, thereby averting the classical critical catastrophe at short integration times. However, when detector integration times exceed delayed-neutron time scales, the system has time to amplify stochastic chains, leading to V/M increasing strongly with power and a practical recovery of catastrophic-like noise growth. Spatially, the linear decay of g with distance and its inverse scaling with power are hallmarks of clustering due to local birth correlations and diffusion smoothing; larger power reduces relative fluctuations and increases apparent cluster sizes until limited by core dimensions. These findings validate the branching random walk framework and highlight how time gating, delayed neutrons, and source strength jointly determine observable noise, directly informing reactor startup procedures and protection system signal processing.

The study provides the first experimental evidence, supported by high-fidelity simulations and stochastic theory, of persistent neutron clustering and a blinking power behavior in a low-power research reactor. It demonstrates that intrinsic spontaneous fission sources bound time fluctuations at short gate widths, preventing the classical critical catastrophe, yet large integration times can reintroduce strong power-dependent noise. Spatial correlations scale as 1/P and decay linearly with distance, indicating patchy neutron distributions whose characteristic size grows with power. Practically, appropriate selection of detector integration times is crucial during startup and low-source operations to avoid spurious trips. Future work should quantify how spatial correlations scale with reactor dominance ratio and geometry, explore links to power tilts in large reactors, and extend measurements across different coupling regimes to refine models beyond homogeneous diffusion approximations.

• Modeling: Spatial-correlation theory used a simplified homogeneous infinite-medium diffusion model; heterogeneities and finite-geometry transport effects are not fully captured. Verifying predictions tied to transverse vs axial decoupling is challenging in the heterogeneous RCF core.
• Measurements: Ex-core detector positioning limited low-power spatial-correlation accuracy; in-core measurements mitigated this but with limited detector number and spatial coverage. Large time-gate experimental acquisitions (e.g., 1 s) were impractical, so square-law noise scaling at long gates is supported primarily by simulations.
• Simulations: Achieving stationary behavior near criticality required very long runs; parameter tuning (control rod position or enrichment) was needed to match experimental asymptotic power. Some simulated long-gate results lacked variance error bars due to limited statistics.
• Data/code access: Experimental data require request due to export-controlled codes; analysis relies on specific fission libraries and evaluated data choices, though average results for correlations are less sensitive than reactivity calculations.