$R$-matrix analysis of elastic scattering, phase shifts, and radiative capture reaction cross sections in the $\alpha +\alpha$ system

The unstable nucleus ${}^8\mathrm{Be}$, with its two $\alpha$-cluster configuration, is the doorway to the formation of heavier $\alpha$-cluster nuclei. Most importantly, its the precursor of the production of ${}^{12}\mathrm{C}$ through the Hoyle state, a resonance state of three $\alpha$ clusters, in the helium burning phase of a massive star. The nucleus exhibits a ground state band of rotational states established through $\alpha \text{−}\alpha$ scattering experiments. A subsequent precision particle-$\gamma$ coincidence measurement of the electromagnetic transition between the $4^{+}\to 2^{+}$ excited states also corroborated the evidence for a highly deformed dumb-bell shaped structure of ${}^8\mathrm{Be}$. A simultaneous phenomenological $R$-matrix analysis of the measured capture reaction cross sections along with the elastic excitation function and phase shift data has been performed. The resulting reduced transition strength of $21.96\pm 3.86e^2{\mathrm{fm}}^4$ compares well with the estimated experimental value of $21.0\pm 2.3e^2{\mathrm{fm}}^4$. The $R$-matrix yield of the $B(E2$) value is closer to the prediction of cluster model but about $19\%$ less than the ab initio result.

Figure 1

Excitation functions for $\alpha \text{−}\alpha$ elastic scattering at the center of mass angles of $90.0^{\circ }, 54.8^{\circ }, 40.0^{\circ }$, and $30.6^{\circ }$. The solid curves are the fits obtained from $R$-matrix analysis.

Figure 2

Fits (solid curves) to the phase shift data (black bullets) from Refs. [5, 6, 7] and [18, 19] (see text for the details) of $\alpha \text{−}\alpha$ scattering for the partial waves $l=0$, 2, and 4 in the energy range of 0.45 to 25 MeV. The low energy data points of Russell et al. [33] (orange dotted square) and Jones et al. [34] (magenta triangle) for $s$- and $d$-wave phase shifts, respectively, are plotted on the calculated curves for comparison only. Pink stars on the $g$-wave phase shift curve are the calculated points for $l=4$ phase shift from [13].

Figure 3

$R$-matrix description (solid curve) of $L=4$ phase shift data. Green stars are the calculated points from Ref. [13].

Figure 4

$R$-matrix description (solid curve) of elastic angular distribution data from Ref. [5].

Figure 5

$R$-matrix fit to $4^{+}\to 2^{+}$ capture reaction cross section data of ${}^4\mathrm{He}(\alpha ,\gamma ){}^8\mathrm{Be}$ from Ref. [16]. The calculation is extended to lower energies to include the $2^{+}\to 0^{+}$ capture reaction cross section.