Microscopic collective inertial masses for nuclear reaction in the presence of nucleonic effective mass

Collective inertial mass coefficients with respect to translational, relative, and rotational motions are microscopically calculated, along the collective reaction path self-consistently determined, based on the adiabatic self-consistent collective coordinate (ASCC) method. The impact of the time-odd component of the mean-field potential on the inertial masses are investigated. The results are compared with those calculated with the cranking formulas. The inertial masses based on the ASCC method reproduce the exact total nuclear mass for the translational motion as well as the exact reduced masses as the asymptotic values for the relative and rotational motions. In contrast, the cranking formulas fail to do so. This is due to the fact that the (local) Galilean invariance is properly restored in the ASCC method but violated in the cranking formulas. A model Hamiltonian for low-energy nuclear reaction is constructed with the microscopically derived potentials and inertial masses. The astrophysical $S$ factors are calculated, which indicates the importance of microscopic calculation of proper inertial masses.

Figure 1

Translational mass of the $\alpha$ particle in units of the nucleon mass as a function of the parameter $B_3$. The solid (red), dashed (green), and dash-dotted (blue) curves show the results of the ASCC, the nonperturbative cranking formula (23) and $A$ times the average effective mass, respectively. See the text for details.

Figure 2

Inertial mass $M\left(R\right)$ for the reaction $\alpha +\alpha$ as a function of the relative distance $R$. The top panel shows the results of the ASCC method, while the middle panel shows those of the perturbative and nonperturbative cranking formulas $M_{\mathrm{cr}}^{\mathrm{p}}$ (thin lines) and $M_{\mathrm{cr}}^{\mathrm{np}}$ (thick lines). The bottom panel shows the potential energies along the ASCC collective paths. Solid (red) and dashed (blue) lines indicate those of $B_3=0$ and 75 MeV ${\mathrm{fm}}^5$, respectively. In the upper panel, the dots correspond to the positions of the minimum energy in the bottom panel.

Figure 3

Nonperturbative cranking inertial masses calculated at $R=8$ fm for the relative motion between two $\alpha$ particles divided by the reduced mass ${\mu }_{\mathrm{red}}=2m$, as a function of $B_3$. The blue dash-dotted curve indicates ${(m^{*}/m)}_{\mathrm{av}}$ of Eq. (30).

Figure 4

Inertial mass $M\left(R\right)$ for the reaction path $\alpha +{}^{16}\mathrm{O}$ as a function of $R$, in units of the nucleon mass. The top panel shows the results of the ASCC method, the middle panel shows those of the perturbative and nonperturbative cranking formulas $M_{\mathrm{cr}}^{\mathrm{p}}$ (thin lines) and $M_{\mathrm{cr}}^{\mathrm{np}}$ (thick lines), and the bottom panel shows the potential energies along the three ASCC collective paths. Solid (red) and dashed (blue) lines indicate those of $B_3=0$ and 75 MeV ${\mathrm{fm}}^5$. In the bottom panel, in order to show it in the given scale, the potential for $B_3=75$ MeV ${\mathrm{fm}}^5$ (dashed line) is shifted downwards by 60 MeV.

Figure 5

Same as Fig. 4, but for ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$. Solid (red) and dashed (blue) lines indicate results of $B_3=0$ and 30 MeV ${\mathrm{fm}}^5$. In the bottom panel, the potential for $B_3=30$ MeV ${\mathrm{fm}}^5$ (dashed line) is shifted downwards by 45 MeV.

Figure 6

Calculated rotational moments of inertias for $B_3=0$ as a function of relative distance $R$. The upper panel shows the results for the system $\alpha +{}^{16}\mathrm{O}$, the lower panel shows the results for the system ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$. The solid (red), dashed (pink), dotted (blue), and dash-dotted (green) curves indicate the results of ASCC, nonperturbative cranking formula, rigid body approximation, and point-mass approximation, respectively.

Figure 7

Same as Fig. 6, but for $B_3=25$ MeV ${\mathrm{fm}}^5$.

Figure 8

Astrophysical $S$ factor calculated with $B_3=0$ for $\alpha +{}^{16}\mathrm{O}$ (upper panel) and for ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$ (lower panel), as a function of incident energy $E_{\mathrm{c}.\mathrm{m}.}$. (1) The point-particle approximation both for the relative and the rotational degrees of freedom, ${\mu }^{\mathrm{red}}$ and ${\mu }_{\mathrm{red}}{R}^2$, respectively. (2) The ASCC mass $M\left(R\right)$ for the relative motion together with ${\mu }_{\mathrm{red}}{R}^2$ for the rotation. (3) ASCC $M\left(R\right)$ and rigid-body ${\mathcal{J}}_{\mathrm{rig}}\left(R\right)$. (4) ASCC $M\left(R\right)$ and ASCC $\mathcal{J}\left(R\right)$. See text for details.

Figure 9

Same as Fig. 8, but calculated with the inertial masses for $B_3=25$ MeV ${\mathrm{fm}}^5$. The same potentials as those of $B_3=0$ are used. See text for details.