Understanding the total width of the $3_1^{-}$ state in ${}^{12}\mathrm{C}$

Background: Recent measurements indicate that the previously established upper limit for the $\gamma$-decay branch of the $3_1^{-}$ resonance in ${}^{12}\mathrm{C}$ at $E_x=9.641\left(5\right)$ 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.

Purpose: 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.

Method: 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 ${}^{12}\mathrm{C}$. A new $\mathbf{R}$-matrix analysis for the $3_1^{-}$ resonance was performed with a self-consistent, simultaneous fit of several high-resolution ${}^{12}\mathrm{C}$ excitation-energy spectra populated with direct reactions.

Results: The global analysis performed in this work yields a formal total width of $\mathrm{\Gamma }\left(E_r\right)=46\left(2\right)$ keV and an observed total width of ${\mathrm{\Gamma }}_{\text{obs}}\left(E_r\right)=38\left(2\right)$ keV for the $3_1^{-}$ resonance.

Conclusions: Significant unaccounted-for uncertainties and a misstated result were discovered in the previous results employed in the ENSDF for the physical (or observed) total width of the $3_1^{-}$ resonance. These previously reported widths are fundamentally different quantities, leading to an invalid ENSDF average. An observed total width of ${\mathrm{\Gamma }}_{\text{obs}}\left(E_r\right)=38\left(2\right)$ keV is recommended for the $3_1^{-}$ resonance in ${}^{12}\mathrm{C}$. 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.

Figure 1

Decomposition of the optimal fit for the inclusive excitation-energy spectra. On each spectrum, the $3_1^{-}$ resonance is superimposed on the total contribution from all other resonances and the instrumental background. For each spectrum, the reaction, measurement angle, and beam energy are indicated in the top right of the corresponding panel. On panel (d), the contaminant state from ${}^{61}\mathrm{Cu}$ ($E_x=4.756$ MeV) is indicated; see Ref. [7] for details of the contaminants.

Figure 2

The channel-radius dependence of the $3_1^{-}$ resonance for the intrinsic line shape with (a) constant $\mathrm{\Gamma }\left(E_r\right)=40$ keV, (b) constant ${\mathrm{\Gamma }}_{\text{obs}}\left(E_r\right)=40$ keV, (c) the energy shift, and (d) the FWHM of the intrinsic lineshape.

Figure 3

The channel-radius dependence of the $2_2^{+}$ resonance for (a) the energy shift and (b) the numerically determined FWHM of the intrinsic lineshape.

Figure 4

The differences in the (a) intrinsic and (b) experimentally observed line shapes caused by the considered approximations (relative to the peak maximum). The observed line shapes were produced by convoluting the intrinsic line shapes with the experimental response observed in the ${}^{12}\mathrm{C}{(p,p^{\prime })}^{12}\mathrm{C}$ data analyzed in this work.

Figure 5

Decomposition of the fit for case (4) in Table 4. The $3_1^{-}$ resonance is superimposed on the total contribution of the $2_2^{+}$ resonance (parameterized with a Gaussian line shape) and a polynomial background. The reaction, measurement angle and beam energy are indicated.

Figure 6

For the $3_1^{-}$ resonance: The channel-radius dependence of (a) ${\chi }_{\text{red}}^2$, (b) $\mathrm{\Gamma }\left(E_r\right)$, and (c) ${\mathrm{\Gamma }}_{\text{obs}}\left(E_r\right)$. The range of channel radii which yield significant discontinuities (4.6–5.4 fm) is highlighted in filled gray. (d) and (e), respectively, present the intrinsic and observed line shapes corresponding to the channel radii of 4.5, 5.1, and 6.0 fm (see legend), corresponding to (a)–(c). The fit results at different channel radii correspond to the fit conditions of case (4) (see Table 4), with the exception of $a_c=4.7$ fm in (e), which includes the Landau energy loss of case (5).

Figure 7

Decomposition of the fit for case (4) in Table 4. The $3_1^{-}$ resonance is superimposed on the total contribution of the $2_2^{+}$ resonance (parameterized with a Gaussian line shape) and a polynomial background. The reaction, measurement angle, and beam energy are indicated.

Figure 8

(a) and (b) present artificial spectra which mimic the relevant data in Figs. 1(a) and 2(a) in Ref. [20], respectively. The purpose of this analysis is to estimate the unaccounted-for systematic error in Ref. [20]. See text for details.

Figure 9

The simulated spectrum which mimics the nuclear-emulsion-plate data in Fig. 2 of Ref. [19]. The purpose of this analysis is to estimate the unaccounted-for systematic error in Ref. [19]. See text for details.