On fluctuation properties of MACS

TL;DR

This work demonstrates that Maxwellian-averaged cross sections (MACS) can fluctuate significantly around their mean values due to resonance-by-resonance variability in neutron capture, especially when the $s$-wave resonance spacing $D_0$ is in the eV–keV range. The authors simulate thousands of resonance sequences incorporating fluctuations in neutron widths, gamma widths, and resonance positions under different spacing models (Poisson, Wigner, GOE) and compute MACS to quantify the resulting distributions. They derive simple empirical relations for the relative MACS fluctuation $\delta_{MACS}$ as a function of $D_0$ and $kT$ and validate these against real nuclei, also quantifying the contributions from individual angular momenta $\ell$ and from low-energy data gaps. The findings provide practical guidance on when statistical Hauser-Feshbach predictions are reliable and when direct measurements or detailed energy-dependent cross sections are necessary to constrain MACS uncertainties, with implications for stellar reaction-rate calculations and nucleosynthesis modeling.

Abstract

The Maxwellian Average Cross Section (MACS) is usually calculated with help of the statistical codes that do not take into account fluctuations of individual resonance parameters. The actual MACS can substantially deviate from its expectation value. This work focuses on description of various sources and aspects of these fluctuations. Simulated resonance sequences considering all the fluctuations involved in the statistical model are used for this purpose. Contribution of various sources of the fluctuations and neutrons with different orbital momenta are thoroughly investigated and described. Impact of available cross section data at low neutron energies is also checked. Based on this analysis simple empirical formulae for estimating relative fluctuations of MACS using solely $s$-wave resonance spacing $D_0$ and the temperature of the stellar environment are derived. It is shown that real nuclei follow the proposed formula well. The expected MACS fluctuations increase with $D_0$ and can be significant. While for isotopes with $D_0$ in eV range the possible deviation from calculated values are small, for nuclei with $D_0$ in the keV range the full-width half-maximum of the expected distribution is larger than about 25\% and 10\% for astrophysically relevant temperatures of $kT=8$ and 30 keV. At least for these nuclei the predictions from statistical codes must be taken with caution.

Figures (14)

• Figure 1: Distribution of individual simulated MACS values for $^{63}$Ni target nucleus corresponding to $kT=5$ keV. Experimental values from Lederer et al.Lederer13 and Weigand et al.Weigand15 are also shown.
• Figure 2: Dependence of $\langle\Gamma_n\rangle$ on $E_n$ for three $D_0$ values and $S_\ell=1.5\times 10^{-4}$; artificial nucleus was used in simulations. The horizontal band indicate typical range of $\Gamma_\gamma=50-500$ meV. The line style, which is indicated only for $\ell=0$, holds also for $\ell>0$.
• Figure 3: Average ratio $\langle R_k/\Gamma_\gamma \rangle$ (assuming $g=1$) as a function of $\langle \Gamma_n \rangle / \Gamma_\gamma$. Red line indicates the dependence if the fluctuation of $\Gamma_n$ is switched off.
• Figure 4: Relative contribution of individual $\ell$ to the cross section for three different $kT=5, 30,$ and 90 keV. An impact of different $\Gamma_\gamma$ and $S_\ell$ (uncertainty shown only for one combination) is indicated. Total assumed cross section included also $\ell=3$ contribution that is not shown (the sum of all curves then can be smaller than one for large $D_0$). Relative fluctuation of $\Gamma_\gamma$ is $\delta_{\Gamma_\gamma}=25\%$. Spacing fluctuated according to Wigner distribution. Symbols correspond to the same $\Gamma_\gamma$ and $S_\ell$ combination for all $\ell$.
• Figure 5: Effective number of levels $N_{\rm eff}$, see Sec. \ref{['sec:effective']}, for $\ell=0$ as a function of $kT$ for artificial nucleus. If not indicated in the legend, the models correspond to $\Gamma_\gamma=200$ meV, $S_0=1.5\times 10^{-4}$. Similar difference of $N_{\rm eff}$ as indicated for $D_0=10$ eV due to different combinations of $\Gamma_\gamma$ and $S_0$ is observed also for other $D_0$. Lines correspond to Eq. (\ref{['eq:Neff']}). Figures for $\ell=1$ and 2 can be found in Supplemental Material SupplMat.