Energy dependence of the nucleus-nucleus potential close to the Coulomb barrier

The nucleus-nucleus interaction potentials in heavy-ion fusion reactions are extracted from the microscopic time-dependent Hartree-Fock theory for the mass symmetric reactions ${}^{16}\mathrm{O}\hspace{0.5em}+\hspace{0.5em}{}^{16}\mathrm{O}$, ${}^{40}\mathrm{Ca}\hspace{0.5em}+\hspace{0.5em}{}^{40}\mathrm{Ca}$, and ${}^{48}\mathrm{Ca}\hspace{0.5em}+\hspace{0.5em}{}^{48}\mathrm{Ca}$ and the mass asymmetric reactions ${}^{16}\mathrm{O}\hspace{0.5em}+\hspace{0.5em}{}^{40,\hspace{0.5em}48}\mathrm{Ca}$, ${}^{40}\mathrm{Ca}\hspace{0.5em}+\hspace{0.5em}{}^{48}\mathrm{Ca}$, ${}^{16}\mathrm{O}\hspace{0.5em}+\hspace{0.5em}{}^{208}\mathrm{Pb}$, and ${}^{40}\mathrm{Ca}\hspace{0.5em}+\hspace{0.5em}{}^{90}\mathrm{Zr}$. When the c.m. energy is much higher than the Coulomb barrier energy, potentials deduced with the microscopic theory identify with the frozen density approximation. As the c.m. energy decreases and approaches the Coulomb barrier, potentials become energy dependent. This dependence indicates dynamical reorganization of internal degrees of freedom and leads to a reduction of the “apparent” barrier felt by the two nuclei during fusion of the order of 2–3% compared to the frozen density case. Several examples illustrate that the potential landscape changes rapidly when the c.m. energy is in the vicinity of the Coulomb barrier energy. The energy dependence is expected to have a significant role on fusion around the Coulomb barrier.

Figure 1

Top: Density profiles $\rho (X,\hspace{0.5em}Y,\hspace{0.5em}0)$ obtained with TDHF for the ${}^{16}\mathrm{O}\hspace{0.5em}+\hspace{0.5em}{}^{208}\mathrm{Pb}$ head-on collision at $E_{\mathrm{c}.\mathrm{m}.}=120$ MeV at three different relative distances. The isodensities (contour lines) are plotted at each 0.025 ${\mathrm{fm}}^{-3}$. The vertical lines indicate the positions of separation planes (see text). Bottom: Total one-dimensional density $\rho (X,\hspace{0.5em}0,\hspace{0.5em}0)$ (dashed line) obtained at the same relative distances. In each case, the two solid curves denote, respectively, ${\rho }_T(X,\hspace{0.5em}0,\hspace{0.5em}0)$ and ${\rho }_P(X,\hspace{0.5em}0,\hspace{0.5em}0)$ (see text). Again, the separation plane is presented by the vertical line.

Figure 2

Comparison of potential energies for the ${}^{16}\mathrm{O}$${+}^{16}$O reaction obtained from the different models. The solid, dashed, and filled circles-dotted line correspond to the DD-TDHF, DC-TDHF [41], and FD potentials, respectively. A zoom on the Coulomb barrier region is also shown in the insert.

Figure 3

Left: Density profiles $\rho (X,\hspace{0.5em}0,\hspace{0.5em}0)$ (dashed) and ${\rho }_{P/T}(t)$ (solid lines) obtained with TDHF for the head-on ${}^{16}\mathrm{O}\hspace{0.5em}+\hspace{0.5em}{}^{16}\mathrm{O}$ collision at $E_{\mathrm{c}.\mathrm{m}.}=34$ MeV at three different times. Each value of relative distance $R$ is indicated in each left panel. Right: Densities $\rho (X,\hspace{0.5em}0,\hspace{0.5em}0)$ (dashed lines) obtained at the same relative distances within the FD approximation. In this case, ${\rho }_{P/T}(X,\hspace{0.5em}0,\hspace{0.5em}0)$ (solid lines) identify with the ground state densities of the ${}^{16}\mathrm{O}$ nucleus.

Figure 4

Potentials extracted from the DD-TDHF (solid lines) at high c.m. energies compared with $V^{\mathrm{FD}}$ (filled circles- and triangles-dotted lines).

Figure 5

Minimal c.m. energy $E_{\mathrm{c}.\mathrm{m}.}^{\mathrm{FD}}$ for which the potentials deduced from the DD-TDHF method identify with the FD case. This quantity is presented as a function of the FD barrier energy.

Figure 6

Potential energy for the ${}^{40}\mathrm{Ca}\hspace{0.5em}+\hspace{0.5em}{}^{40}\mathrm{Ca}$ reaction extracted at different c.m. energies. The FD potential is displayed with the filled circles-dotted line.

Figure 7

Barrier energy $V_B^{\mathrm{DD}}$ for the ${}^{40}\mathrm{Ca}\hspace{0.5em}+\hspace{0.5em}{}^{40}\mathrm{Ca}$ reaction extracted at different c.m. energies. The horizontal dashed line indicates the FD reference.

Figure 8

Density profiles obtained in TDHF for different relative distances $R$ for the ${}^{40}\mathrm{Ca}\hspace{0.5em}+\hspace{0.5em}{}^{40}\mathrm{Ca}$ reaction at three different c.m. energies. These energies correspond to those used in Fig. 7 to obtain $V^{\mathrm{DD}}(R)$.

Figure 9

Difference between the barrier obtained with the FD approximation and the lowest barrier with DD-TDHF as a function of $V_B^{\mathrm{FD}}$. In practice, the lowest barrier is obtained by using c.m. energy at or close to $V_B^{\mathrm{FD}}$.

Figure 10

Potential energy obtained with the DD-TDHF method for the ${}^{16}\mathrm{O}\hspace{0.5em}+\hspace{0.5em}{}^{208}\mathrm{Pb}$ reaction using different c.m. energies. The filled circles correspond to the FD approximation.

Figure 11

Coulomb barrier energy $V_B^{\mathrm{DD}}$ for the ${}^{16}\mathrm{O}\hspace{0.5em}+\hspace{0.5em}{}^{208}\mathrm{Pb}$ reaction as a function of the c.m. energy. The horizontal dashed line indicates the FD reference; the arrow indicates the low-energy fusion TDHF threshold obtained with TDHF calculations in Ref. [36].

Figure 12

Reduced mass deduced from the procedure described in Sec. 3a divided by its initial value ${\mu }_{\mathrm{ini}}$ as a function of relative distance divided by the Coulomb barrier radius for the FD case ($R_B^{\mathrm{FD}}$) for the three mass-asymmetric reactions. The c.m. energies used are reported in each panel.

Figure 13

Comparison of potentials extracted from Eq. (6) (solid line) and from Eq. (6) without the term $\frac{1}{2}\frac{d\mu (R)}{\mathit{dR}}{\stackrel{·}{R}}^2$ (filled triangles-dotted line) as a function of $R/R_B^{\mathrm{FD}}$. The c.m. energies used are the same as in Fig. 12.