Gauge angle dependence in time-dependent Hartree-Fock-Bogoliubov calculations of ${}^{20}\mathrm{O}+{}^{20}\mathrm{O}$ head-on collisions with the Gogny interaction

A numerical method to solve time-dependent Hartree-Fock-Bogoliubov equations by using a hybrid basis of two-dimensional harmonic oscillator eigenfunctions and a one-dimensional Lagrange mesh with the Gogny effective interaction is applied to head-on collisions of superfluid ${}^{20}\mathrm{O}$ nuclei. Taking the energies around the barrier top, the trajectories, pairing energies, and numbers of transferred nucleons are displayed. Their dependence on the relative gauge angle at the initial time is studied.

Figure 1

Initial condition of the TDHFB equation (4). A HFB ground state is calculated by using basis functions with the number of grid points $N_{\mathrm{grid}}^{\left(0\right)}=N_{\mathrm{grid}}/2$ (a). The HFB ground state in (a) is mapped on the space of the basis functions with the number of grid points $N_{\mathrm{grid}}$ [L or R in (b)].

Figure 2

Densities in the $y\text{−}z$ plane in the collisions ${}^{20}\mathrm{O}+{}^{20}\mathrm{O}$ for the center-of-mass energies $E_{\mathrm{c}.\mathrm{m}.}=9.21\mathrm{MeV}$ (left) and $E_{\mathrm{c}.\mathrm{m}.}=9.61\mathrm{MeV}$ (right). The densities are shown with respect to the time at times $ct=180$, 360, 540, 720, and 900 fm from top to bottom, respectively. $c$ is light speed.

Figure 3

The frozen density potential $V_{\mathrm{FD}}\left(R\right)$ with respect to relative coordinate $R$ of the head-on collision of two ${}^{20}\mathrm{O}$ nuclei. The inset is the magnification of the region around the top of the frozen density potential. In the inset, three energies are described with dashed, dash-dotted, and solid lines for $E_{\mathrm{c}.\mathrm{m}.}=9.21$, 9.41, and 9.61 MeV, respectively.

Figure 4

Trajectories in the phase space of the relative coordinate $R$ and the relative momentum $P_z$. The solid (dashed, dash-dotted) curve corresponds to the initial energy $E_{\mathrm{c}.\mathrm{m}.}=9.21$ (9.41, 9.61) MeV. The two trajectories with $E_{\mathrm{c}.\mathrm{m}.}=9.41$ and 9.61 MeV almost fully overlap in the region $R<$ 8.5 fm.

Figure 5

Pairing energies of the trajectories with the energies $E_{\mathrm{c}.\mathrm{m}.}=9.61\mathrm{MeV}$ (a), 9.41 MeV (b), and 9.21 MeV (c) are plotted with respect to the relative distance $R$.

Figure 6

Numbers of transferred protons (thin curve) and neutrons (thick curve) with respect to the relative distance along the trajectory with energy $E_{\mathrm{c}.\mathrm{m}.}=9.61\mathrm{MeV}$. The broken curve is plotted by multiplying the number of transferred protons by a factor of 1.5.

Figure 7

Numbers of transferred protons (broken curve) and neutrons (solid curve) with respect to the relative distance in the trajectory with energy $E_{\mathrm{c}.\mathrm{m}.}=9.21\mathrm{MeV}$. The dash-dotted curve is for the number of transferred neutrons in Fig. 6 as a reference.

Figure 8

Trajectories of head-on collisions of ${}^{16}\mathrm{O}$ and ${}^{20}\mathrm{O}$ with energy $E_{\mathrm{c}.\mathrm{m}.}=11.41\mathrm{MeV}$ in the phase space of the relative distance $R$ and relative momentum $P_z$. The solid (thin solid, dot-dashed, and dashed) curve is for the case with initial gauge angle $\chi =0$ (45, 90, and 135) degrees. All of the trajectories overlap each other. The curve with $\chi =45$ (90, 135) degrees is shifted upward by 0.2 (0.4, 0.6) $1/\mathrm{fm}$ for readability.

Figure 9

Variations of the pairing energies with respect to the relative distance in the collision processes in Fig. 8. The solid (thin solid, dot-dahsed, and dashed) curve is for the case with initial gauge angle $\chi =0$ (45, 90, and 135) degrees. All of the curves overlap each other. The curve with $\chi =45$ (90, 135) degrees is shifted upward by 0.3 (0.6, 0.9) MeV for readability.

Figure 10

Trajectories of head-on collisions of ${}^{20}\mathrm{O}$ and ${}^{20}\mathrm{O}$ with energy $E_{\mathrm{c}.\mathrm{m}.}=11.41\mathrm{MeV}$ in the phase space of the relative distance $R$ and relative momentum $P_z$. Each of the trajectories is the result of the first 2000 steps of the time integration of the TDHFB equations (4). The solid (thin solid, dot-dashed, and dashed) curve is for the case with initial gauge angle $\chi =0$ (45, 90, and 135) degrees. The trajectories with initial gauge angles $\chi =45$ and 135 degrees overlap each other.

Figure 11

Magnification of the region of relative distance $R\le$ 11 fm in Fig. 10.

Figure 12

Variation of the pairing energies with respect to the relative distance $R$ in the trajectories in Fig. 11.

Figure 13

Trajectories of head-on collisions of ${}^{20}\mathrm{O}$ and ${}^{20}\mathrm{O}$ with energy $E_{\mathrm{c}.\mathrm{m}.}=9.21\mathrm{MeV}$ in the phase space of the relative distance $R$ and relative momentum $P_z$. The solid (dash-dotted) curve is for the case with initial gauge angle $\chi =0$ (90) degrees. The thin solid curve is for both $\chi =45$ and 135 degrees. The trajectories with angles $\chi =45$ and 135 degrees overlap each other.

Figure 14

Magnification of the region of relative distance $9.8\le R\le 11$ fm in Fig. 13. The solid (thin solid, dashed, and dash-dotted) curve is for the case with gauge angle $\chi =0$ (45, 90, and 135) degrees.

Figure 15

The frozen density potentials $V_{\mathrm{FD}}\left(R\right)$ with respect to the relative distance $R$ with the initial gauge angles $\chi =0$ (solid), 45 (thin solid), 90 (dash-dotted), and 135 (dashed) degrees. The thin dashed line stands for the energy $E_{\mathrm{c}.\mathrm{m}.}=9.21$ MeV.

Figure 16

Variation of the number $N_{\mathrm{L}}$ of the neutrons in the left region with respect to the relative distance $R$ for energy $E_{\mathrm{c}.\mathrm{m}.}=11.41$ MeV. The thin solid (dashed) curve is for the case with gauge angle $\chi =45$ (135) degrees. The dot-dashed curve is for the cases with $\chi =0$ and 90 degrees. The two curves of the cases with $\chi =0$ and 90 degrees overlap each other.

Figure 17

The same as in Fig. 16, but for the energy $E_{\mathrm{c}.\mathrm{m}.}=9.21\mathrm{MeV}$.

Figure 18

The same as in Fig. 17, but the variations of the number $N_{\mathrm{L}}$ of the neutrons in the left region are plotted with respect to the elapsed time.