Statistical properties of neutron-induced reaction cross sections using random-matrix approach

TL;DR

The paper addresses the statistical properties of neutron-induced nuclear reactions in the unresolved resonance region by employing a GOE-$S$-matrix approach that embeds a GOE Hamiltonian into the scattering matrix and is fully specified by channel transmission coefficients. It demonstrates that elastic decay widths follow a chi-squared distribution with $\nu>1$ while neutron-capture widths approach Gaussian behavior as the number of effective $\gamma$-channels increases, and that fluctuating cross sections are consistent with Hauser-Feshbach plus width fluctuations, matching SLBW and Reich–Moore results in the RRR–URR transition. The work provides a unitarity-preserving, transmission-coefficient-driven framework that naturally connects the RRR described by $R$-matrix theory to the higher-energy URR, enabling realistic cross-section fluctuations without requiring empirical $D$ or $\langle \Gamma \rangle$ inputs. Applied to $^{238}$U, the GOE-$S$-matrix model yields width and cross-section statistics in agreement with established formalisms and reveals the role of resonance interference in shaping fluctuations, offering a practical path to unify statistical descriptions across resonance regimes.

Abstract

We investigate the statistical properties of neutron-induced nuclear reactions on $^{238}$U using the GOE-$S$-matrix model, in which the Gaussian Orthogonal Ensemble (GOE) is embedded into the scattering ($S$) matrix. The GOE-$S$-matrix model does not require any experimental values of the average level spacing $D$ and average decay width $Γ$ with their statistical distributions, but the model is fully characterized by the channel transmission coefficients used in the Hauser-Feshbach theory. We demonstrate that the obtained compound nucleus decay width distribution resembles the $χ$-squared distribution with the degree of freedom greater than unity. This approach enables us to generate fluctuating cross sections while preserving requisite unitarity and accounting for interference between resonances. By comparing the calculated cross section distribution with that from $R$-matrix theory, we demonstrate a smooth transition from the resolved resonance region to the continuum region.

Figures (10)

• Figure 1: Width distribution of the elastic channel at an incident neutron energy of 0.02 MeV for the $s$-wave (top panel) and the ratio to the Porter–Thomas distribution (bottom panel).
• Figure 2: Width distribution of neutron capture channel at an incident neutron energy of 0.02 MeV for the $s$-wave (top panel) and the ratio to the Porter–Thomas distribution (bottom panel). The gray dashed line corresponds to a distribution obtained by summing 10 independent $\chi$-squared variables, namely, the $\chi$-squared distribution with the 10 degree of freedom.
• Figure 3: Total cross sections calculated by the GOE-$S$-matrix model ([a] and [c]) and those from SLBW ([b] and [d]) in the vicinity of incident neutron energies of 0.02 and 0.15 MeV. The right figures are the logarithmic-scale plots of the left figures.
• Figure 4: Calculated average total cross sections as a function of incident neutron energy.
• Figure 5: Calculated average neutron capture cross sections as a function of incident neutron energy.