Understanding the total width of the $3_{1}^{-}$ state in $\mathrm{^{12}C}$

TL;DR

This paper accurately determines the total width of the $3_{1}^{-}$ resonance in $^{12}C$, correcting previous uncertainties and providing a crucial parameter for modeling the triple-$eta$ reaction at high temperatures.

Contribution

It presents a new R-matrix analysis that refines the total width of the $3_{1}^{-}$ resonance, correcting prior estimates and invalidating the previous ENSDF average.

Findings

New total width: 46(2) keV (formal

Observed total width: 38(2) keV

Revised width should be used in future models

Abstract

Recent measurements indicate that the previously established upper limit for the $\gamma$-decay branch of the $3_{1}^{-}$ resonance in $^{12}\textrm{C}$ at $E_{x} = 9.641(5)$ MeV may be incorrect. As a result, the $3_{1}^{-}$ resonance has been suggested as a significant resonance for mediating the triple-$\alpha$ reaction at high temperatures above 2 GK. Accurate estimations of the $3_{1}^{-}$ contribution to the triple-$\alpha$ reaction rate require accurate knowledge of not only the radiative width, but also the total width. In anticipation of future measurements to more accurately determine the $\gamma$-decay branch of the $3_{1}^{-}$ resonance, the objective of this work is to accurately determine the total width of the $3_{1}^{-}$ resonance. An evaluation was performed on all previous results considered in the current ENSDF average of 46(3) keV for the physical total width (FWHM) of the $3_{1}^{-}$ resonance in $^{12}\textrm{C}$. Significant unaccounted-for uncertainties and a misstated result were discovered in these previous results, leading to an invalid ENSDF average. In this work, the new global \textbf{R}-matrix analysis performed on direct-reaction data yields a formal total width of $\Gamma(E_{r}) = 46(2)$ keV and an observed total width of $\Gamma_{\textrm{obs}}(E_{r}) = 38(2)$ keV for the $3_{1}^{-}$ resonance. An observed total width of $\Gamma_{\textrm{obs}}(E_{r}) = 38(2)$ keV is recommended for the $3_{1}^{-}$ resonance in $\mathrm{^{12}C}$. This observed total width should be employed for future evaluations of the observed total radiative width for the $3_{1}^{-}$ resonance and its contribution to the high-temperature triple-$\alpha$ reaction rate.