Asymptotic normalization coefficient for ${{}^{12}\mathrm{C}}+p\rightarrow {{}^{13}\mathrm{N}}$ from the ${}^{12}{\mathrm{C}}({}^{10}\mathrm{B},{}^{9}{\mathrm{Be}}){}^{13}\mathrm{N}$ reaction and the ${}^{12}\mathrm{C}(p,\gamma ){}^{13}\mathrm{N}$ astrophysical S factor

Abstract

This work is aimed at clarifying the contribution of the proton direct radiative capture to the ${}^{12}\mathrm{C}(p,\gamma ){}^{13}\mathrm{N}$ reaction by specifying the value of the asymptotic normalization coefficient (ANC) for ${}^{12}\mathrm{C}+p\rightarrow {}^{13}\mathrm{N}_\mathrm{g.s.}$. In order to do this, the differential cross section of the proton transfer in the ${}^{12}\mathrm{C}({}^{10}\mathrm{B},{}^9\mathrm{Be})^{13}\mathrm{N}$ reaction at an energy of 41.3 MeV has been measured and analyzed through the modified distorted wave Born approximation (MDWBA) method taking into account the reaction channel coupling and ${}^{3}{\mathrm{He}}$ cluster transfer contributions. The value of the ANC was derived to be 1.63±0.13 fm$^{-1/2}$, which was used in estimating the astrophysical S(E) factor and the reaction rate of the proton radiative capture by the ${}^{12}{\mathrm{C}}$ nucleus at energies of astrophysical relevance.

Appendices

Appendix A: peripherality conditions and the basic formulas for determining the “experimental” ANC for ${}^{12}{\mathrm{C}}+p\rightarrow {}^{13}\mathrm{N}$

The peripheral character for the ${}^{12}{\mathrm{C}}$(x, y)${}^{13}{\mathrm{N}}$ proton transfer reaction (x is either ${}^{10}{\mathrm{B}}$ or ${}^3{\mathrm{He}}$) is conditioned by fulfillment of the relation [22,23,24, 35]

that is, the left-hand side of Eq. (A1) must not depend on $b_{Ap}$ for each fixed energy $E_i$ and scattering angle $\theta$ belonging to the main peak. Fulfillment of the condition (A1) for the specific peripheral reactions under consideration means that the contribution of the nuclear interior to the calculated $R ({\mathrm{E}}_i,\theta ;b_{Ap}{\mathrm{}})$ and $\sigma ^{{\mathrm{(DWBA)}}}({\mathrm{E}}_i,\theta ;b_{Ap})$ functions [44, 73] must be strongly suppressed. This is due to the fact that the dependence of the calculated R and $\sigma ^{{\mathrm{(DWBA)}}}$ functions on the free parameter $b_{Ap}$, which is mainly determined by the interior nuclear part of the reaction amplitude, is conditioned only by the factors $\varphi _{Ap}(\rho ;b_{Ap})/b_{Ap}$ and $Z_{Ap}^{{\mathrm{1/2}}}\varphi _{Ap}(\rho ;b_{Ap})$ [56], respectively. The letters are included in the integrand functions of the interior part of the radial integrals of the matrix element corresponding to the R and $\sigma ^{{\mathrm{(DWBA)}}}$ functions, respectively. Here, $\varphi _{Ap}(\rho ;b_{Ap})$ is the shell model bound-state wave function for the ${}^{13}\mathrm{N}$ nucleus, and $Z_{Ap}$ is the spectroscopic factor for the ${}^{13}\mathrm{N}$ nucleus in the (${}^{12}{\mathrm{C}}+p$) configuration. Therefore, this interior part of the matrix element becomes really model-dependent due to the uncertain free parameters $b_{Ap}$ and $Z_{Ap}$. As a rule, the uncertainty associated with the interior nuclear parts above can also be increased by ambiguities in the optical potentials, and this results in a large uncertainty in the absolute value of $Z_{Ap}$. It follows from this and from Eqs. (2) and (3) that for the peripheral reactions under consideration, the next condition [58, 59]

must also be fulfilled for each fixed energy $E_i$ and angle $\theta$ of the measurement $\theta$= $\theta _j^{\mathrm{{exp}}}$, j = 1,2,...). The fulfillment of the relations (A1) and (A2) (or their violation within the experimental error limits for $d\sigma ^{{\mathrm{exp}}}/d\Omega$) allows one to determine the interval of scattering angles $\theta$ where the dominance of the external interaction occurs for the $R({\mathrm{E}}_i,\theta ;b_{Ap})$ function calculated at a fixed energy $E_i$. In this case, Eq. (A2) can be applied to extract the “indirectly determined” value ($C_{Ap}^{{\mathrm{exp}}}$)$^{{\mathrm{2}}}$ using the $d\sigma ^{{\mathrm{exp}}}/d\Omega$ measured within the main maximum of the angular distribution instead of $d\sigma /d\Omega$.

Appendix B: basic formulas of the modified R-matrix method

Here, we present only the concepts and the essential formulas for the calculation of the astrophysical S factors in the framework of the modified R-matrix method (see, for example, Ref.  [12, 74] and references therein) specific to the ${}^{12}\mathrm{C}(p,\gamma ){}^{13}\mathrm{N}$ reaction. The channel spin and the orbital angular momentum of the resonant proton capture are equal to 1/2 and 0 (or 1), respectively.

We use the system of units in which $\hbar =c=$ 1.

The astrophysical S factor is determined by the relation

where $\sigma ({\mathrm{E}})$ is the reaction cross section, and E and $\eta$ are the relative kinetic energy and the Coulomb parameter, respectively, of the colliding proton and ${}^{12}{\mathrm{C}}$.

According to [12, 74], within the framework of the modified R-matrix method, the cross section for the reaction ${}^{12}\mathrm{C}(p,\gamma ){}^{13}\mathrm{N}$ populating the ground state of the residual ${}^{13}\mathrm{N}$ nucleus is given by

where I and $l_i$ are the channel spin and the relative orbital momentum of the $p{}^{12}{\mathrm{C}}$-scattering, respectively, $\lambda$ is the multipolarity of the electromagnetic transition, and $k=\sqrt{{\mathrm{2}}\mu E}$, in which $\mu$ is the reduced mass of p and ${}^{12}{\mathrm{C}}$. In Eq. (B2), $M_{Jl_i\lambda }({\mathrm{E}})$ is the amplitude of the electromagnetic ($E\lambda$ and $M\lambda$) transition, which is represented in the form

where $M^{({\mathrm{R}}_{J};\,({\mathrm{E}}\lambda ,M\lambda ))}_{J\,l_i\,\lambda }$ is the proton capture amplitude of the resonance state with the spin J; $M^{{\mathrm{(DC}};\,E\lambda )}_{l_i\,\lambda }$

and $M^{{\mathrm{(DC}};\,M\lambda )}_{l_i\,\lambda }({\mathrm{E }})$ are the direct proton $E\lambda$ and $M\lambda$ capture amplitudes in the ground state of ${}^{13}\mathrm{N}$, respectively. In the single-level approximation, the $M^({\mathrm{R}}_{{j_{{\mathrm{0}}}}};\,({\mathrm{E}}\lambda ,M\lambda ))_{J{_{{}^{10}B}}_{f}}\,J\,I l_i\,\lambda$ amplitude can be represented in the form [74, 75]

Here, $\sigma ^{\mathrm{(c)}}_{l_i}$ and $\delta ^{{\mathrm{(HS)}}}_{l_i}$ are the Coulomb and hard-sphere phase shifts, respectively, for the $p{}^{12}{\mathrm{C}}$-scattering; $\Gamma _{Jl_i}^p({\mathrm{E}})$ and $\Gamma _{J\,\lambda }^{\gamma }({\mathrm{E}})$ are the partial proton and radiative $\gamma$ widths for the resonant decays ${}^{13}{N^*}\rightarrow {}^{12}{\mathrm{C}} +p$ and ${}^{13}\hbox {N}{}^*\rightarrow {}^{13}\mathrm{N}+\gamma$, respectively, and $\Gamma _{J}({\mathrm{E}})$ is the total width. The energy dependencies of the proton and radiative $\gamma$ widths are given by the expressions

and

where $k_{\gamma }$ is the photon momentum, $P_{l_i}$ is the penetrability factor, $S_c$ is the Thomas shift factor [76], and $\gamma _{J\,l_i}^p$ and $\gamma _{J\,\lambda }^{\gamma }$ are the partial reduced proton and radiative $\gamma$-ray widths, respectively. The reduced $\gamma _{J\,\lambda }^{\gamma }$ consists of two parts: the internal [$\gamma _{J\,\lambda }^{\gamma }$(int.)] and external channel [$\gamma _{J\,\lambda }^{\gamma }$(ext.)] parts; i.e., [75]:

$$\begin{aligned} \gamma _{J\,\lambda }^{\gamma }=\gamma _{J\,\lambda }^{\gamma }{\mathrm{(int.)}}+\gamma _{J\,\lambda }^{\gamma }{\mathrm{(ext.)}} \end{aligned}$$

Here, notations “int” and “ext” refer to the distance from the center of the kernel, where the border is the channel radius. The observable partial proton and radiative $\gamma$ widths are given by

It should be noted that in order to set the value of $\Gamma _{J\,\lambda }^{\gamma }$, it will be necessary to adjust $\gamma _{J\,\lambda }^{\gamma }$(int.), while $\gamma _{J\,\lambda }^{\gamma }$(ext.) is a quantity being calculated exactly. The explicit expressions for the direct capture amplitudes for the $E\lambda$ and M1 transitions are presented in [5]. $\gamma _{J\,\lambda }^{\gamma }$(ext.) and the amplitudes of the direct radiative capture are characterized by the asymptotic wave functions of the input channel and bound state. In this case, the radial wave function of the final state is normalized using the ANC of the corresponding channel. In other words, $\gamma _{J\,\lambda }^{\gamma }$(ext.) (but not $\gamma _{J\,\lambda }^{\gamma }$ in total) and the amplitudes of direct radiation captures are directly proportional to ANC, which is defined as

$$\begin{aligned} C_{Ap}=\sqrt{\frac{2}{r_{\mathrm{ch}}}}\left( Z_{Ap}\theta _{Ap}^p\right) ^{\mathrm{1/2}}N_f^{1/2}\left[ W_{-\eta _p;l_p+{\mathrm{1/2}}}(2\kappa _p r_{\mathrm{ch}})\right] ^{-1} \end{aligned}$$

where $N_f$ is the normalization factor and $\theta _{Ap}^p$ is the dimensionless reduced width amplitude from [62]. Nevertheless, we note only that, in the long-wavelength approximation, they contain a radial integral, which has the form

where ${\tilde{\lambda }}=\lambda$ and ($\lambda$-1) for the $E\lambda$ and $M\lambda$ transitions, respectively; $W_{-\eta _p;l_p+{\mathrm{1/2}}}$($\cdot \cdot \cdot$) is the Whittaker function; $\kappa _p=\sqrt{{\mathrm{2}}\mu \varepsilon _p}/\hbar$ in which $\varepsilon _p$ is the binding energy of the ${}^{13}\mathrm{N}$ in the (${}^{12}{\mathrm{C}}+p$) channel, $\eta _p$ is the Coulomb parameter for the bound ${}^{13}\mathrm{N}$(${}^{12}{\mathrm{C}}+p$) nucleus, and $I_{l_i}({\mathrm{kr}})$ and $O_{l_i}({\mathrm{kr}})$ are the incoming and outgoing solutions, respectively, of the radial Schrödinger equation.

As is seen from expression (B7), the strengths of the total direct capture amplitude and the channel radiative $\gamma$ width are determined by the ANC for ${}^{12}{\mathrm{C}}+p\rightarrow {}^{13}\mathrm{N}$. Hence, introduction of information about the “experimental” ANC to the resonance and direct capture amplitudes makes it possible to reduce the uncertainty of the total S(E) astrophysical S factors calculated for the ${}^{12}{\mathrm{C}}(p,\gamma ){}^{13}{\mathrm{N}}$ reaction in the thermonuclear energy region to a minimum.

The Maxwellian-averaged reaction rates $N_A\langle \sigma v\rangle$ are given by [1, 2]

as a function of the temperature T. $N_A$ is the Avogadro number; $k_B$ is the Boltzmann constant; and $v=\sqrt{{\mathrm{2}}E/\mu }$, where $Z_ke$ is the charge of the particle k.