[Paper Review] On the Feynman-alpha formula for fast neutrons

Johan Anderson, Pal Lenard+1 more|arXiv (Cornell University)|May 4, 2011

Statistical Mechanics and Entropy1 references3 citations

TL;DR

This paper derives a generalized Feynman-alpha formula for fast and thermal neutrons using a two-energy-group stochastic model based on the Kolmogorov forward (master) equation. It incorporates a compound Poisson source, neutron absorption, down-scattering, fission, and detection, yielding a variance-to-mean formula with two distinct decay exponentials that govern the correlation behavior in pulsed neutron measurements, particularly relevant for nuclear safeguards and subcritical system diagnostics.

ABSTRACT

In this contribution, a stochastic theory for a branching process in a neutron population with two energy levels is investigated. In particular, a variance to mean or Feynman-alpha formula is derived in this generalized scenario using the Kolmogorov forward or master equation theory for the probabilities in a system with a compound Poisson source.

Motivation & Objective

• To develop a stochastic theory for neutron population dynamics with two energy groups (fast and thermal) using the Kolmogorov forward equation.
• To incorporate realistic physical processes: compound Poisson source, absorption, down-scattering, fission, and detection.
• To derive a variance-to-mean (Feynman-alpha) formula applicable to pulsed neutron experiments in fast-reactor-like systems.
• To enable accurate modeling of neutron correlation decay in applications such as differential die-away self-interrogation (DDSI) and accelerator-driven subcritical systems.
• To assess the diagnostic potential of the two-exponential decay structure in correlation functions for fissile material detection.

Proposed method

• Models the neutron population using a continuous-time Markov process with two particle types: fast (N₁) and thermal (N₂) neutrons.
• Formulates the master equation (Kolmogorov forward equation) for the joint probability P(N₁, N₂, Z₁, t), including transitions due to absorption, scattering, fission, detection, and source injection.
• Uses a generating function G(X, Y, Z, t) to transform the master equation into a partial differential equation governing the time evolution of factorial moments.
• Derives first and second-order moment equations via differentiation of the generating function at X=Y=Z=1, incorporating source statistics (r₁, r₂) and fission multiplicity (ν₁, ν₂).
• Applies Laplace transforms to solve the moment equations, yielding a solution with two characteristic decay constants ω₁ and ω₂ from the denominator H(s) = s² + (ω₁+ω₂)s + ω₁ω₂.
• Constructs the final Feynman-alpha formula as σ²(Z)/⟨Z⟩ = 1 + Y₁(1 - (1−e⁻ω₁T)/(ω₁T)) + Y₂(1 - (1−e⁻ω₂T)/(ω₂T)), where Y₁ and Y₂ are complex expressions involving system parameters.

Experimental results

Research questions

• RQ1 How does the inclusion of two energy groups (fast and thermal neutrons) modify the Feynman-alpha formula compared to the single-group case?
• RQ2 What is the functional form of the variance-to-mean ratio in a two-group neutron system with a compound Poisson source and detection?
• RQ3 How do the decay constants ω₁ and ω₂, derived from the system's reaction rates, influence the time-dependent behavior of neutron correlation functions?
• RQ4 To what extent do the two-exponential components in the correlation function remain observable in practice, especially compared to the Rossi-alpha method?
• RQ5 What is the asymptotic behavior of the Feynman-alpha formula at long measurement times T → ∞, and how does it depend on system parameters?

Key findings

• The derived Feynman-alpha formula exhibits two distinct exponential decay components governed by ω₁ and ω₂, which emerge from the characteristic equation H(s) = s² + (ω₁+ω₂)s + ω₁ω₂.
• The asymptotic value of the Feynman-alpha at T → ∞ is determined by the sum Y₀ = Y₁ + Y₂ = q₂(λ_d λ₂ λ_R λ₂f)/(ω₁² ω₂²), which depends on detection and reaction rates.
• Numerical results show that the maximum value of the Feynman-alpha curve is highly sensitive to the ratio of decay constants and the thermalization rate λ_R, with increasing λ_R leading to higher asymptotic values.
• The two-exponential structure is not clearly visible to the naked eye in typical correlation curves, suggesting that detection of fissile material via this method may be less intuitive than with the Rossi-alpha method.
• The model's results agree with those from the backward Kolmogorov approach, validating the consistency of the two-group stochastic formulation.
• The formula enables accurate curve fitting to experimental data, potentially allowing more precise extraction of system parameters like λ_R and λ₂f than in the DDSI method, though diagnostic interpretation remains to be fully established.