Radiative capture reactions via indirect methods

Many radiative capture reactions of astrophysical interest occur at such low energies that their direct measurement is hardly possible. Until now the only indirect method, which was used to determine the astrophysical factor of the astrophysical radiative capture process, was the Coulomb dissociation. In this paper we address another indirect method, which can provide information about resonant radiative capture reactions at astrophysically relevant energies. This method can be considered an extension of the Trojan horse method for resonant radiative capture reactions. The idea of the suggested indirect method is to use the indirect reaction $A(a,s\gamma )F$ to obtain information about the radiative capture reaction $A(x,\gamma )F$, where $a=\left(sx\right)$ and $F=\left(xA\right)$. The main advantage of using the indirect reactions is the absence of the penetrability factor in the channel $x+A$, which suppresses the low-energy cross sections of the $A(x,\gamma )F$ reactions and does not allow one to measure these reactions at astrophysical energies. A general formalism to treat indirect resonant radiative capture reactions is developed when only a few intermediate states contribute and a statistical approach cannot be applied. The indirect method requires coincidence measurements of the triple differential cross section, which is a function of the photon scattering angle, energy, and the scattering angle of the outgoing spectator particle $s$. Angular dependence of the triple differential cross section at fixed scattering angle of the spectator $s$ is the angular $\gamma -s$ correlation function. Using indirect resonant radiative capture reactions, one can obtain information about important astrophysical resonant radiative capture reactions such as $(p,\gamma )$, $(\alpha ,\gamma )$, and $(n,\gamma )$ on stable and unstable isotopes. The indirect technique makes accessible low-lying resonances, which are close to the threshold, and even subthreshold bound states located at negative energies. In this paper, after developing the general formalism, we demonstrate the application of the indirect reaction ${}^{12}\mathrm{C}({}^6\mathrm{Li},d\gamma ){}^{16}\mathrm{O}$ proceeding through $1^{-}$ and $2^{+}$ subthreshold bound states and resonances to obtain the information about the ${}^{12}\mathrm{C}(\alpha ,\gamma ){}^{16}\mathrm{O}$ radiative capture at the astrophysically most effective energy 0.3 MeV, which is impossible using standard direct measurements. Feasibility of the suggested approach is discussed.

Figure 1

Pole diagram describing the indirect radiative capture reaction proceeding through the intermediate excited state $F^{*}$.

Figure 2

Square of the $d-\alpha$ bound state wave function in the momentum space.

Figure 3

Low-energy astrophysical $S_{E1}\left(E_{\alpha -{}^{12}\mathrm{C}}\right)$ and $S_{E2}\left(E_{\alpha -{}^{12}\mathrm{C}}\right)$ factors for $E1$ and $E2$ transitions for the ${}^{12}\mathrm{C}(\alpha ,\gamma ){}^{16}\mathrm{O}$ radiative capture. Black dots are astrophysical factors from [20] and the solid red line is the present paper's fit. (a) $S_{E1}\left(E_{\alpha -{}^{12}\mathrm{C}}\right)$ astrophysical factor; (b) $S_{E2}\left(E_{\alpha -{}^{12}\mathrm{C}}\right)$ astrophysical factor.

Figure 4

Angular distribution of the photons emitted from the reaction ${}^{12}\mathrm{C}({}^6\mathrm{Li},d\gamma ){}^{16}\mathrm{O}$ proceeding through the wings of two subthreshold resonances, $1^{-},E_{\alpha -{}^{12}\mathrm{C}}=-0.045$ MeV and $2^{+},E_{\alpha -{}^{12}\mathrm{C}}=-0.245$ MeV, and the resonances at $E_{\alpha -{}^{12}\mathrm{C}}>0$. The green dashed-dotted line is the angular distribution for the electric dipole transition $E1$, the blue dashed line is the angular distribution generated by the electric quadrupole $E2$ transition, and the red solid line is the total angular distribution resulting from the interference of the $E1$ and $E2$ radiative captures. (a) $E_{\alpha -{}^{12}\mathrm{C}}=0.3$ MeV, constructive interference of the $E1$ transitions through the wing of $1^{-},E_{\alpha -{}^{12}\mathrm{C}}=-0.045$ MeV and the resonance $1^{-},E_R=2.423$ MeV; (b) $E_{\alpha -{}^{12}\mathrm{C}}=0.3$ MeV, destructive interference of the $E1$ transitions through the wing of $1^{-},E_{\alpha -{}^{12}\mathrm{C}}=-0.045$ MeV and the resonance $1^{-},E_R=2.423$ MeV; (c) the same as panel (a) for $E_{\alpha -{}^{12}\mathrm{C}}=0.9$ MeV; (d) the same as panel (b) for $E_{\alpha -{}^{12}\mathrm{C}}=0.9$ MeV.

Figure 5

Angular distribution of the photons emitted from the reaction ${}^{12}\mathrm{C}({}^6\mathrm{Li},d\gamma ){}^{16}\mathrm{O}$ proceeding through the wings of the two subthreshold resonances, $1^{-},E_{\alpha -{}^{12}\mathrm{C}}=-0.045$ MeV and $2^{+},E_{\alpha -{}^{12}\mathrm{C}}=-0.245$ MeV, and the resonances at $E_{\alpha -{}^{12}\mathrm{C}}>0$. Notations of the lines are the same as in Fig. 4. (a) The same as Fig. 4 for $E_{\alpha -{}^{12}\mathrm{C}}=2.1$ MeV; (b) the same as Fig. 4 for $E_{\alpha -{}^{12}\mathrm{C}}=2.1$ MeV; (c) the same as Fig. 4 for $E_{\alpha -{}^{12}\mathrm{C}}=2.28$ MeV; (d) the same as Fig. 4(d) for $E_{\alpha -{}^{12}\mathrm{C}}=2.28$ MeV.

Figure 6

The same as in Fig. 4, but the calculations are done with three modified $R$-matrix parameters generating a lower $S_{E1}\left(0.3\text{MeV}\right)$ astrophysical factor.

Figure 7

The same as in Fig. 5, but calculations are done with three modified $R$-matrix parameters generating a lower $S_{E1}\left(0.3\text{MeV}\right)$ astrophysical factor.