$S$ factor of $^{13}$C($α$,$n$)$^{16}$O at low energies in cluster effective field theory

Shung-Ichi Ando

$S$ factor of $^{13}$C($α$,$n$)$^{16}$O at low energies in cluster effective field theory

Shung-Ichi Ando

Abstract

The $^{13}$C($α$,$n$)$^{16}$O reaction at low energies is studied by constructing an effective field theory. We choose a separation scale at $E=1$~MeV, where $E$ is the initial $α$-$^{13}$C energy in center-of-mass frame, just below the sharp resonant $5/2^+$ state of $^{17}$O, and include two open channels, $α$-$^{13}$C and $n$-$^{16}$O, and resonant $1/2^+$, $5/2^-$, $3/2^+$ states of $^{17}$O in the study. Parameters of the theory are fitted to experimental data, $S$ factor of $^{13}$C($α$,$n$)$^{16}$O at the energies below $E=1$~MeV, including the data sets recently reported by the LUNA and JUNA collaborations, and the $S$ factor of $^{13}$C($α$,$n$)$^{16}$O is extrapolated to the Gamow peak energy $E_G= 0.19$~MeV in the low mass AGB stars. We discuss an uncertainty in the estimate of the $S$ factor and confirm that the main part of the uncertainty emerges from the parameter fit of the near-breakup threshold $1/2^+$ state of $^{17}$O.

$S$ factor of $^{13}$C($α$,$n$)$^{16}$O at low energies in cluster effective field theory

Abstract

The

)

O reaction at low energies is studied by constructing an effective field theory. We choose a separation scale at

~MeV, where

is the initial

C energy in center-of-mass frame, just below the sharp resonant

state of

O, and include two open channels,

C and

O, and resonant

states of

O in the study. Parameters of the theory are fitted to experimental data,

factor of

)

O at the energies below

~MeV, including the data sets recently reported by the LUNA and JUNA collaborations, and the

factor of

)

O is extrapolated to the Gamow peak energy

~MeV in the low mass AGB stars. We discuss an uncertainty in the estimate of the

factor and confirm that the main part of the uncertainty emerges from the parameter fit of the near-breakup threshold

state of

Paper Structure (93 equations, 4 figures, 2 tables)

This paper contains 93 equations, 4 figures, 2 tables.

Figures (4)

Figure 1: Diagrams for dressed $^{17}$O propagators. A thick and thin double-dashed line with and without a filled circle represent a dressed and bare $^{17}$O propagators, respectively. A loop diagram with (without) a shaded ellipse represents the $n$-$^{16}$O ($\alpha$-$^{13}$C) self-energy term. A shaded ellipse represents the Coulomb Green's function.
Figure 2: Feynman diagram of the amplitudes of $^{13}$C($\alpha$,$n$)$^{16}$O reaction. A line (dashed line) in the initial state represents a propagator of $^{13}$C ($\alpha$), and a line (dashed line) in the final state represents a propagator of n ($^{16}$O). A shaded ellipse in the initial state represents the Coulomb wavefunction. See the caption in Fig. \ref{['fig;propagators']} as well.
Figure 3: S factor of $^{13}$C($\alpha$,$n$)$^{16}$O as a function of energy, $E$, of the initial $\alpha$-$^{13}$C state in the center-of-mass frame. A line is plotted by using fitted values of ten parameters to experimental data, presented in the second column in Table \ref{['table;fitted_parameters']}. A band of the plotted line is obtained by 16 to 84 % samples of the MCMC analysis. The experimental data are displayed in the figure as well. A vertical band represents a range of the Gamow peak energy in the low mass AGB stars.
Figure 4: S factor of $^{13}$C($\alpha$,$n$)$^{16}$O as a function of energy, $E$, of the initial $\alpha$-$^{13}$C state in the center-of-mass frame. A line is plotted by using fitted values of seven parameters to experimental data of LUNA and JUNA collaborations, presented in the third column in Table \ref{['table;fitted_parameters']}. See the caption of Fig. \ref{['fig;S-factor10we']} as well.