Determination of the ${}^{36}\mathrm{Mg}(n,\gamma ){}^{37}\mathrm{Mg}$ reaction rate from Coulomb dissociation of ${}^{37}\mathrm{Mg}$

We use the Coulomb dissociation (CD) method to calculate the rate of the ${}^{36}\mathrm{Mg}\left(n,\gamma \right){}^{37}\mathrm{Mg}$ radiative capture reaction. The CD cross sections of the ${}^{37}\mathrm{Mg}$ nucleus on a ${}^{208}\mathrm{Pb}$ target at the beam energy of 244 MeV/nucleon, for which new experimental data have recently become available, were calculated within the framework of a finite-range distorted-wave Born approximation theory that is extended to include the projectile deformation effects. Invoking the principle of detailed balance, these cross sections are used to determine the excitation function and subsequently the rate of the ${}^{36}\mathrm{Mg}(n,\gamma ){}^{37}\mathrm{Mg}$ reaction. We compare these rates to those of the ${}^{36}\mathrm{Mg}(\alpha ,n){}^{39}\mathrm{Si}$ reaction calculated within a Hauser-Feshbach model. We find that for ${\mathrm{T}}_9$ as large as up to 1.0 (in units of ${10}^9$ K) the ${}^{36}\mathrm{Mg}(n,\gamma ){}^{37}\mathrm{Mg}$ reaction is much faster than the ${}^{36}\mathrm{Mg}(\alpha ,n){}^{39}\mathrm{Si}$ one. The inclusion of the effects of ${}^{37}\mathrm{Mg}$ projectile deformation in the breakup calculations enhances the ($n,\gamma$) reaction rate even further. Therefore, it is highly unlikely that the $(n,\gamma )\beta$-decay $r$-process flow will be broken at the ${}^{36}\mathrm{Mg}$ isotope by the $\alpha$ process.

Figure 1

Comparison of the wave functions calculated by solving the Schrödinger equation with potential given by Eq. (7). The $S_n$ value is taken to be 0.35 MeV for the state $2p_{3/2}$. The wave function $\left[ru\right(r\left)\right]$ obtained with the deformations parameter ${\beta }_2=0.0$ (i.e., calculated with spherical potential) is shown by the solid line. $ru\left(r\right)$ obtained with ${\beta }_2=0.2$ but including only the component corresponding to $\ell =1$ and $j=3/2$ is shown by the dashed line while that including components with $\ell =1,3,5$ and all the allowed $j$ values is displayed by the dashed-dotted line. All the wave functions are normalized to unity.

Figure 2

Direct capture (DC) cross sections to the ground state (GS) of ${}^{37}\mathrm{Mg}$, obtained by calculating the Coulomb dissociation of ${}^{37}\mathrm{Mg}$ on a ${}^{208}\mathrm{Pb}$ target at the beam energy of 244 MeV/nucleon for different values of one-neutron separation energies ($S_n$). Results for $S_n$ values of 0.10, 0.35, 0.50, and 0.70 MeV, are shown by dotted, solid, dashed, and dashed-dotted lines, respectively. In these calculations the deformation parameter of the ${}^{37}\mathrm{Mg}$ ground state was taken to be 0 throughout. The spectroscopic factor $C^{2}S$ was unity in each case.

Figure 3

(a) The Coulomb dissociation cross section $d\sigma /d{E}_{\mathrm{c}.\mathrm{m}.}$ calculated for different values of $S_n$ as a function of $E_{\text{c.m.}}$. (b) The kinematical factor $F\left(E_{\text{c.m.}}\right)$ = $[{(E_{\text{c.m}.}+S_n)}^3/E_{\text{c.m}.}]$ as a function of $E_{\text{c.m}.}$ for various values of $S_n$. (c) The product of $F\left(E_{\text{c.m}.}\right)$ and $d\sigma /d{E}_{\text{c.m.}}$ as a function of $E_{\text{c.m}.}$ for various values of $S_n$.

Figure 4

Same as in Fig. 2 obtained by using CD cross sections calculated with different ${\beta }_2$ values for a fixed $S_n$ of 0.35 MeV. Results for ${\beta }_2$ values of 0.0, 0.2, 0.4, and 0.5, are shown by solid, dotted, dashed, and dashed-dotted lines, respectively.

Figure 5

Total wave function $\left[ru\right(r\left)\right]$ for the $\ell =1$ and $j=3/2$ state including components with $\ell =1,3,5$ and all the allowed $j$ values for different ${\beta }_2$ parameters. All the wave functions are normalized to unity.

Figure 6

Capture rates for the ${}^{36}\mathrm{Mg}(n,\gamma ){}^{37}\mathrm{Mg}$ reaction as a function of temperature in units of ${10}^9\mathrm{K}$ (${\mathrm{T}}_9$) for different values of $S_n$ for a fixed ${\beta }_2$ of 0.0.

Figure 7

Same as in Fig. 4 for different values of ${\beta }_2$ for a fixed $S_n$ of 0.35 MeV.

Figure 8

Integrand of the reaction rate expression Eq. (1) as a function of $E_{\mathrm{c}.\mathrm{m}.}$ for the ${}^{36}\mathrm{Mg}(n,\gamma ){}^{37}\mathrm{Mg}$ reaction at different values of $S_n$ for a fixed ${\beta }_2$ of 0.0.

Figure 9

Comparison of the rates of ${}^{36}\mathrm{Mg}(n,\gamma ){}^{37}\mathrm{Mg}$ reaction calculated by the CD method using ${\beta }_2$ parameters of 0 (solid line) and 0.5 (dashed line) (the neutron separation energy $S_n$ was 0.35 MeV in both cases), with those of the ${}^{36}\mathrm{Mg}(\alpha ,n){}^{39}\mathrm{Si}$ (dashed-dotted line) and ${}^{36}\mathrm{Mg}(n,\gamma ){}^{37}\mathrm{Mg}$ (dotted line) reactions obtained from the Hauser Feshbach (HF) cross sections adopted from the Ref. [20] where they were obtained from the code non-smoker(NS). The $x$ axis represents the temperature $T_9$.