Examination of gamma radiation from $\mathrm{^{10}B(p,\gamma )^{11}C}$ reaction at low energies

Abstract

In this study, we have conducted an assessment of direct capture cross sections, astrophysical S-factors, phase shifts, and their derivatives at low energies within the framework of the potential model. We present the first investigation of the $\mathrm{^{10}B(p,\gamma )^{11}C}$ reaction’s cross sections and astrophysical S-factors using the Woods–Saxon potential. The $\mathrm{^{10}B(p,\gamma )^{11}C}$ reaction plays a key role in the production of carbon-11 in stars, providing insights into stellar evolution and the synthesis of elements in the universe. Our analysis focuses on the astrophysical S-factor for the electric dipole (E1) transition within the energy range of 0–500 keV. The chosen parametrization reproduces the available experimental and theoretical data for cross sections and astrophysical S-factors in this reaction accurately. Furthermore, we have extrapolated the S-factor values to zero energy (S(0)) for the $(5/2)^+$ to $(3/2)^-$, $(5/2)^-$ and $(7/2)^-$ energy levels. The calculated S(0) values are found to be 9.25 keV barn, 9.37 keV barn, and 8.54 keV barn, respectively.

Appendices

Appendices

1.1 Appendix A: Electromagnetic transition

The matrix elements for transition $J_0M_0\longrightarrow JM$ are calculated from the following equation:

$$\begin{aligned} \left\langle J_f M_f\left|\mathcal{\hat{Q}}(E\lambda )\right|J_{i} M_{i}\right\rangle = \ O_{if}(E\lambda )\left\langle J_{i}M_{i}\lambda \mu |J_fM_f\right\rangle ,\nonumber \\ \end{aligned}$$

(15)

where O$_{if}(E\lambda )$ the reduced matrix elements are equal to

$$\begin{aligned} O_{if}(E\lambda )&=\frac{\left( -1\right) ^{j_f+I_{a}+J_{i}+\lambda }}{\sqrt{2J+1}}\ \left\{ \begin{array}{ccc} j_f &{} J_f &{} I_{a}\\ J_{i} &{} j_{i} &{} \lambda \end{array} \right\} \left[ \left( 2J_f\!+\!1\right) \left( 2J_{i}\!+\!1 \right) \right] ^{1/2}\ \nonumber \\&\quad \times \left\langle l_fj_f\left\| \mathcal{\hat{Q}}(E\lambda )\right\| l_{i}j_{i}\right\rangle _{J_f}, \end{aligned}$$

(16)

In this equation:

$$\begin{aligned} \displaystyle \left\langle l_fj_f\left\| \mathcal{\hat{Q}}(E\lambda )\right\| l_{i}j_{i}\right\rangle _{J_f}&=\frac{e_\lambda }{\sqrt{4\pi }}(-1)^{l_i+l+j_i-j}\frac{\hat{\lambda }\hat{j_i}}{\hat{J}} \left\langle j_i\frac{1}{2}\lambda _i\mid j\frac{1}{2}\lambda \right\rangle \nonumber \\&\quad \displaystyle \times \int \limits _{0}^{\infty }dr r^{\lambda } u^J_lj(r)u^{J_i}_{l_ij_i} \end{aligned}$$

(17)

The transition $M_{\lambda }$ is as follows [17]:

$$\begin{aligned}{} & {} \left\langle l_fj_f\left\| \mathcal{\hat{Q}}(M\lambda )\right\| l_{i}j_{i} \right\rangle _{J_f}=\left( -1\right) ^{j_f+I_{a}+J_{i}+1}\ \nonumber \\{} & {} \quad \times \sqrt{\frac{3}{4\pi }}\mu _{N}\ \widehat{J_f}\widehat{J}_{i}\left\{ \begin{array}{ccc} j_f &{} J_f &{} I_{a}\\ J_{i} &{} j_{i} &{} 1 \end{array} \right\} \nonumber \\{} & {} \quad \times \left\{ b\frac{1}{\widehat{l}_{i}}e_{M}\left[ \frac{2\widetilde{j}_{i}}{\widehat{l}_{i}}\mathcal{I}_{ij} +\left( -1\right) ^{l_{i}+1/2-j}\frac{\widehat{j}_{i}}{\sqrt{2}}\delta _{j_{i},\ l_{i}\pm 1/2}\delta _{j,\ l_{i}\mp 1/2}\right] \right. \nonumber \\{} & {} \quad +g_{N}\frac{1}{\widehat{l}_{i}^{2}}\left[ \left( -1\right) ^{l_{i}+1/2-j_{i}}\hat{j}_{i}\delta _{j_f,\ j_{i}}-\mathcal{T}_{ij}\right] \nonumber \\{} & {} \quad \left. +g_{a}\left( -1\right) ^{I_{a}+j_{i}+J+1}\widehat{J}_{i} \widehat{J}\widehat{I}_{a}\widehat{I}_{a}\left\{ \begin{array}{ccc} I_{a} &{} J_f &{} j_{i}\\ J_{i} &{} I_{a} &{} 1 \end{array} \right\} \right\} \nonumber \\{} & {} \quad \times \int _{}^{}dr\ u_{l_0j_0}^{J_0}\left( r\right) \ u_{l_{}j_{}}^{J_{}}\left( r\right) , \end{aligned}$$

(18)

where $\mathcal{I}_{ij}$ and $\mathcal{T}_{ij}$ are defined as follows:

$$\begin{aligned} \mathcal{I}_{ij}= & {} \left( l_{i}\delta _{j_{i},\ l_{i}+1/2}+\left( l_{i}+1\right) \delta _{j_{i},\ l_{i}-1/2}\right) , \nonumber \\ \mathcal{T}_{ij}= & {} \left( -1\right) ^{l_{i}+1/2-j}\frac{\widehat{j}_{i}}{\sqrt{2}}\delta _{j_{i},\ l_{i}\pm 1/2}\delta _{j,\ l_{i}\mp 1/2}. \end{aligned}$$

(19)

The M1 and E2 transitions can are used for [18]

$$\begin{aligned} \Gamma _{E2}[eV]= & {} 8.13 \times 10^-7E^5_\gamma [MeV]B(E2)[e^2fm^4],\nonumber \\ \Gamma _{M1}[eV]= & {} 1.16\times 10^-2E^3_\gamma [MeV]B(M1)[\mu _N^2], \end{aligned}$$

(20)

and

$$\begin{aligned} S(E1)= & {} k_r^3(|{\mathcal{H}}_{if}|^2+|{\mathcal{H}}_{if}|^2),\nonumber \\ S(E2)= & {} \frac{75}{98}k_r^5(|{\mathcal{H}}_{if}\Vert ^2+\frac{3}{2}|{\mathcal{H}}_{if}\Vert ^2). \end{aligned}$$

(21)

In this equation k is photon number and ${\mathcal{H}}_{if}$ (i,f=are a and b nucleous) given by:

$$\begin{aligned} {\mathcal{H}}_{if}=\int _0^\infty r^2dr\bigg [\psi _f(r)r^\lambda \frac{\varphi _l(r)}{kr}\bigg ]e^{r\eta }(2\pi \eta )^{\frac{1}{2}} \end{aligned}$$

(22)

In above equation $\psi _l$ is continum wave function describing the lth partial wave while $\psi _f$ is the radial p-wave bound-state wave function.

1.2 Appendix B: Resonance strength

The resonance strength is showned by

$$\begin{aligned} (\omega _\gamma )_R=F_\Gamma (1+\delta _{ab}) \dfrac{2J_R+1}{(2J_a+1)(2J_b+1)} \end{aligned}$$

(23)

$F_\Gamma$ is defined by $\frac{\Gamma _p\Gamma _\gamma }{\Gamma _{tot}}$. For the case of narrow resonances, $(\omega _\gamma )_R$ can be used for calculation expectation value of product $\sigma v$ $(\langle \sigma v\rangle )$.

$$\begin{aligned} <\sigma v\>=\bigg (\frac{2\pi }{m_{ab}kT}\bigg )^{\frac{3}{2}}\hbar ^2(\omega _\gamma )_Rexp\Bigg (-\frac{E_r}{kT}\Bigg ) \end{aligned}$$

(24)

and for broad resonance we can use:

$$\begin{aligned} <\sigma v\>=C(T)\int ^\infty _0\sigma (E)exp\Bigg (-\frac{E}{k_BT}\Bigg )dE \end{aligned}$$

(25)

where $C(T)=4(\frac{2}{\pi m_{ab}(k_BT)^3})^{\frac{1}{2}}$. The resonance cross section for a narrow resonance is approximated by the Wigner-Bright expression [19,20,21,22].

$$\begin{aligned} \sigma _r(E)=\frac{\pi \hbar ^2}{2\mu E}\dfrac{(2J_R+1)}{(2J_a+1)(2J_b+1)}\dfrac{\Gamma p \Gamma _\gamma }{(E_r-E)^2+(\frac{\Gamma _{tot}}{2})}\nonumber \\ \end{aligned}$$

(26)

The radiative capture cross section for the reaction $a+b\longrightarrow c+\gamma$, where $(\pi \lambda )$ $=E,(M)$=E(electric) or M(magnetic) transition, is defined as follows:

$$\begin{aligned} \sigma ^{d.c}_{(\pi \lambda )}(E)=I_{ac} \bigg (\frac{E_\gamma }{\hbar c}\bigg )^{2\lambda -1}\sigma _\gamma ^{(\lambda )}(E_\gamma ) \end{aligned}$$

(27)

where $I_{ac}$ =$\dfrac{2(2I_c+1)}{(2I_a+1)(2s+1)}$. In the phase shift section, we can use the following formula:

$$\begin{aligned} \delta _R(E)\simeq \frac{\pi }{2}-(E_r-E)\frac{d\delta }{dE}\bigg |_R \end{aligned}$$

(28)

In the case of a(b,$\gamma$)c charged particles(a anb b) S(E) is presented to be a slowly varying function compared to cross section in energy for non resonant nuclear reactions. In this senario, S(E) can be evaluated in a McLaurin series:

$$\begin{aligned} S(E)=S(0)+\dot{S}(E)+\frac{1}{2}\ddot{S}(0) E^2+\cdots \end{aligned}$$

(29)