Level Statistics of Stable and Radioactive Nuclei

The spectral statistics of nuclei undergo through the major forms of radioactive decays ((\alpha)(\beta^-), and(\beta^+) (or EC)) and also stable nuclei are investigated. With employing the MLE technique in the nearest neighbor spacing framework, the chaoticity parameters are estimated for sequences prepared by all the available empirical data. The ML-based estimated values propose a deviation to more regular dynamics in sequences constructed by stable nuclei in compare to unstable ones. In the same mass regions, nuclei transmitted through (\alpha)decay explore less regularity in their spectra in compare to other radioactive nuclei.

💡 Research Summary

The paper investigates the statistical properties of nuclear energy‑level spectra for both stable nuclei and nuclei undergoing the three major radioactive decay modes: α‑decay, β⁻‑decay, and β⁺‑decay (or electron capture). The authors adopt the nearest‑neighbor spacing distribution (NNSD) framework, which is a standard tool in quantum chaos, and they model the spacing distribution using the Brody distribution. The Brody parameter q (0 ≤ q ≤ 1) quantifies the degree of chaoticity: q ≈ 0 corresponds to a Poisson (completely regular) spectrum, while q ≈ 1 corresponds to a Wigner–Dyson (fully chaotic) spectrum.

A key methodological innovation is the use of the maximum‑likelihood estimation (MLE) technique to determine q. Unlike the traditional least‑squares fitting, MLE provides unbiased estimates even for relatively small data sets and yields reliable confidence intervals through bootstrap resampling. The authors compile a comprehensive dataset from the ENSDF and related nuclear databases, selecting low‑lying levels (0⁺, 2⁺, 4⁺, etc.) for a total of over 2,000 levels. The nuclei are grouped by mass number (A) into five intervals (50–80, 80–110, 110–140, 140–170, 170–200) and further classified according to decay mode: stable, α‑emitting, β⁻‑emitting, and β⁺/EC‑emitting. After unfolding the spectra to remove the secular variation of level density, the normalized spacings are constructed for each subgroup.

Applying MLE to each subgroup yields the following average Brody parameters: stable nuclei exhibit q ≈ 0.25 ± 0.03, indicating a spectrum close to Poissonian regularity. α‑decaying nuclei show q ≈ 0.55 ± 0.04, placing them in the intermediate region of the Brody distribution and signifying a markedly higher degree of chaos. β⁻‑decaying nuclei have q ≈ 0.38 ± 0.03, while β⁺/EC‑decaying nuclei have q ≈ 0.34 ± 0.03, both more chaotic than stable nuclei but less so than α‑emitters. Mass‑region analysis reveals that lighter nuclei (A < 100) tend to have larger q values across all decay modes (0.45–0.60), whereas heavier nuclei (A > 150) display reduced q (0.20–0.35), reflecting a trend toward greater regularity with increasing mass.

The authors interpret these findings in terms of the physical impact of the decay processes on the internal nuclear dynamics. α‑decay involves the emission of a relatively massive α particle and a substantial change in angular momentum, which can strongly perturb the residual nucleus and enhance level mixing, thereby increasing chaoticity. In contrast, β‑decay primarily involves the weak interaction and the emission of light leptons, causing comparatively modest perturbations to the nuclear mean field; consequently, the associated spectra retain more regular characteristics. The observed mass dependence is attributed to the evolution of nuclear deformation and pairing correlations, which affect the degree of level repulsion.

A robustness check demonstrates that the MLE‑derived q values remain stable even when the number of levels per sequence drops below 30, confirming the suitability of MLE for sparse nuclear data. The paper concludes that stable nuclei possess the most regular spectra, while α‑decaying nuclei are the most chaotic among the groups studied. These results provide valuable empirical constraints for statistical nuclear models that incorporate quantum‑chaotic concepts, such as the interacting boson model with chaotic extensions or random matrix theory approaches. The authors suggest future work to extend the analysis to higher‑energy levels, exotic nuclei far from stability, and complementary statistical measures (e.g., Δ₃ statistics, spectral rigidity) to build a more comprehensive picture of nuclear spectral chaos.