Subthreshold resonances and resonances in the $R$-matrix method for binary reactions and in the Trojan horse method

In this paper we discuss the $R$-matrix approach to treat the subthreshold resonances for the single-level and one-channel and for the single-level and two-channel cases. In particular, the expression relating the asymptotic normalization coefficient (ANC) with the observable reduced width, when the subthreshold bound state is the only channel or coupled with an open channel, which is a resonance, is formulated. Since the ANC plays a very important role in nuclear astrophysics, these relations significantly enhance the power of the derived equations. We present the relationship between the resonance width and the ANC for the general case and consider two limiting cases: wide and narrow resonances. Different equations for the astrophysical $S$ factors in the $R$-matrix approach are presented. After that we discuss the Trojan horse method (THM) formalism. The developed equations are obtained using the surface-integral formalism and the generalized $R$-matrix approach for the three-body resonant reactions. It is shown how the Trojan horse (TH) double-differential cross section can be expressed in terms of the on-the-energy-shell astrophysical $S$ factor for the binary subreaction. Finally, we demonstrate how the THM can be used to calculate the astrophysical $S$ factor for the neutron generator ${}^{13}\mathrm{C}(\alpha ,n){}^{16}\mathrm{O}$ in low-mass AGB stars. At astrophysically relevant energies this astrophysical $S$ factor is controlled by the threshold level $1/2^{+},E_x=6356$ keV. Here, we reanalyzed recent TH data taking into account more accurately the three-body effects and using both assumptions that the threshold level is a subthreshold bound state or it is a resonance state.

Figure 1

Pole diagram describing the TH reaction mechanism

Figure 2

$S$ factors for the ${}^{13}\mathrm{C}(\alpha ,n){}^{16}\mathrm{O}$ reaction as a function of the $\alpha -{}^{13}\mathrm{C}$ relative kinetic energy proceeding through four resonances: black dotted-dashed line ($5/2^{-},l_i=2,E_x=7.165$ MeV); solid red line ($3/2^{+},l_i=1,E_x=7.216$ MeV); dashed brown line ($5/2^{+},l_i=3,E_x=7.379$ MeV); dotted blue line ($5/2^{-},l_i=2,E_x=7.382$ MeV). All the resonant parameters are taken from [17].

Figure 3

Astrophysical $S$ factor for the ${}^{13}\mathrm{C}(\alpha ,n){}^{16}\mathrm{O}$ reaction as a function of the $\alpha -{}^{13}\mathrm{C}$ relative kinetic energy. Square black boxes, solid green dots, and shaded orange band are data from Refs. [13], [17], and [19], respectively. Red solid lines correspond to our calculations for the fit to the lower and upper limits of the TH data considering $1/2^{+}$ state as $-3$ keV subthreshold resonance with ${\mathrm{\Gamma }}_n=158.1$ keV [17]. The lower and upper limits of the ANC square are 2.89 and 4.7 ${\mathrm{fm}}^{-1}$, respectively. Whereas, the blue dotted-dashed lines correspond to our calculations for the fit to TH data, considering the $1/2^{+}$ state as the 4.7 keV threshold resonance with ${\mathrm{\Gamma }}_n=136$ keV [18] and the corresponding lower and upper values of the observable reduced width are 2.81 and 3.6 ${\mathrm{keV}}^{1/2}$, respectively. For our calculations we have used $R_{\alpha {}^{13}\mathrm{C}}=7.5$ fm, and $R_{n{}^{16}\mathrm{O}}=6.0$ fm. The insert in the figure shows the enlarged low-energy $S$ factor.