[Paper Review] Comment on "Distribution of Partial Neutron Widths for Nuclei Close to a Maximum of the Neutron Strength Function"

P. Koehler, F. Bečvář+3 more|arXiv (Cornell University)|Jan 24, 2011

Nuclear Physics and Applications3 citations

TL;DR

This paper critiques a proposed rescaling method for neutron partial widths in platinum isotopes near peaks of the s-wave neutron strength function. The authors demonstrate that the new transformation, intended to reconcile experimental data with random matrix theory (RMT), instead exacerbates the discrepancy by broadening the width distribution, and show via maximum-likelihood analysis that the data remain inconsistent with the Porter-Thomas distribution regardless of the rescaling method used.

ABSTRACT

A recent Letter attempted to reconcile the disagreement between neutron resonance data and random matrix theory (RMT). To this end, a new formula was derived for transforming measured (Γ_{λn}) to reduced (Γ_{λn}^0) neutron widths for s-wave resonances (λ=1,2,...) in nuclides near peaks of the s-wave neutron strength function. In this Comment, we show that such a rescaling would not, in general, be expected to reconcile the type of disagreement observed, and demonstrate that indeed it does not for the specific cases in question. Hence, the disagreements between RMT and these data remain.

Motivation & Objective

• To evaluate whether a newly proposed rescaling of partial neutron widths can reconcile experimental data with random matrix theory (RMT) predictions.
• To assess the validity of the transformation $ \Gamma_{\lambda n}^{0} = \Gamma_{\lambda n}/f^{2}(E_{\lambda n}) $, where $ f^{2}(E_{\lambda n}) $ includes an energy-dependent factor from a single-particle state near threshold.
• To determine whether the observed broadening of reduced neutron width distributions in 192,194Pt contradicts the Porter-Thomas distribution predicted by RMT.
• To investigate whether the condition $ |E_0| \gg D_0 $ (resonance spacing) is physically plausible for Pt isotopes, as required by the new theory.

Proposed method

• Applied the maximum-likelihood (ML) method to estimate the degrees of freedom $ \nu $ of the $ \chi^2 $ distribution for reduced neutron widths.
• Recomputed $ \nu_{\text{ML}} $ using the new rescaling formula $ f^{2}(E_{\lambda n}) = \frac{1}{\pi}\left(\frac{2m}{\hbar}\right)^{1/2}\frac{\sqrt{E_{\lambda n}}}{E_{\lambda n}+|E_{0}|} $, which includes an energy-dependent correction from a single-particle state.
• Compared the ML estimates of $ \nu $ under the new rescaling to those from the standard $ \sqrt{E_{\lambda n}} $-based transformation.
• Evaluated the statistical significance of the deviation from the Porter-Thomas distribution (i.e., $ \nu = 1 $) using confidence levels.
• Assessed the physical plausibility of the condition $ |E_0| \gg D_0 $, where $ D_0 $ is the mean resonance spacing, for 192,194,196Pt.
• Used numerical analysis to plot $ \nu_{\text{ML}} $ as a function of $ |E_0| $, showing the behavior of the distribution under varying assumptions.

Experimental results

Research questions

• RQ1 Does the proposed energy-dependent rescaling of partial neutron widths reconcile the experimental data with the Porter-Thomas distribution predicted by random matrix theory?
• RQ2 How does the new rescaling formula affect the width distribution's shape compared to the standard transformation?
• RQ3 What is the statistical significance of the observed deviation from the Porter-Thomas distribution in 192Pt and 194Pt?
• RQ4 Is the condition $ |E_0| \gg D_0 $ physically plausible for the Pt isotopes studied, given their resonance spacing?
• RQ5 Can the observed broadening of the width distribution be explained by the new rescaling, or does it instead increase the disagreement with RMT?

Key findings

• The new rescaling formula $ f^{2}(E_{\lambda n}) = \frac{1}{\pi}\left(\frac{2m}{\hbar}\right)^{1/2}\frac{\sqrt{E_{\lambda n}}}{E_{\lambda n}+|E_{0}|} $, when applied, results in a broader width distribution than the standard $ \sqrt{E_{\lambda n}} $-based transformation, except in the special case where the average reduced width is proportional to $ (E_{\lambda n}+|E_0|)^{-1} $.
• The maximum-likelihood estimate of $ \nu $ for 192Pt and 194Pt remains significantly below 1 under the new rescaling, with $ \nu_{\text{ML}} = 0.57_{-0.15}^{+0.16} $ and $ 0.47_{-0.18}^{+0.19} $, respectively, indicating a broader distribution than predicted by the Porter-Thomas distribution.
• The data from 192Pt and 194Pt reject the Porter-Thomas distribution with at least 99.997% confidence, even after applying the new rescaling, confirming that the disagreement with RMT persists.
• The new rescaling does not reconcile the data with RMT; instead, it increases the discrepancy due to the additional $ (E_{\lambda n}+|E_0|)^{-1} $ factor, which amplifies broadening.
• The condition $ |E_0| \gg D_0 $, required for the validity of the new transformation, is unlikely to hold for all three Pt isotopes, as $ D_0 $ values (23, 50, 153 eV) are comparable to typical $ |E_0| $ values.
• No viable alternative multi-level, multi-channel theory exists to replace R-matrix theory when $ |E_0| \lesssim D_0 $, leaving the theoretical framework incomplete in this regime.