Energy dependence of the nucleus-nucleus potential and the friction parameter in fusion reactions

Applying a macroscopic reduction procedure to the improved quantum molecular dynamics (ImQMD) model, the energy dependences of the nucleus-nucleus potential, the friction parameter, and the random force characterizing a one-dimensional Langevin-type description of the heavy-ion fusion process are investigated. Systematic calculations with the ImQMD model show that the fluctuation-dissipation relation found in symmetric head-on fusion reactions at energies just above the Coulomb barrier fades out when the incident energy increases. It turns out that this dynamical change with increasing incident energy is caused by a specific behavior of the friction parameter which directly depends on the microscopic dynamical process, i.e., on how the collective energy of the relative motion is transferred into the intrinsic excitation energy. It is shown microscopically that the energy dissipation in the fusion process is governed by two mechanisms: One is caused by the nucleon exchanges between two fusing nuclei, and the other is due to a rearrangement of nucleons in the intrinsic system. The former mechanism monotonically increases the dissipative energy and shows a weak dependence on the incident energy, while the latter depends on both the relative distance between two fusing nuclei and the incident energy. It is shown that the latter mechanism is responsible for the energy dependence of the fusion potential and explains the fading out of the fluctuation-dissipation relation.

Figure 1

Root mean square of the random force distribution as a function of relative distance $R$. Each inset shows a typical distribution in regions I, II, and III respectively. In the inset corresponding to a typical distribution in region II, the distribution is shifted so that the symmetric Gaussian distribution centers at $\delta F=0$, but the mean value of the random force is still zero. The pink diamond is the root mean square of the random force after eliminating events in the asymmetric tail as shown by the inset in region II.

Figure 2

The strength of random force ${\langle \delta F\left(R\right)\delta F\left(R\right)\rangle }^{\mathrm{tot}}$ [red dots, in (MeV/fm)${}^2$] and the friction coefficient ${\gamma }_0\left(R\right)$ (blue squares, in $0.001c/\mathrm{fm}$) at $E_{\mathrm{c}.\mathrm{m}.}=195$, 205, 215, and 235 MeV, respectively. The grey line shows the potential $U\left(R\right)$. Pink diamonds represent ${\langle \delta F\left(R\right)\delta F\left(R\right)\rangle }^{\mathrm{sym}}$ which is obtained by eliminating events in the asymmetric tail.

Figure 3

(a) Collective kinetic energy $T_{\mathrm{coll}}\left(R\right)=E_{\mathrm{coll}}\left(R\right)-U\left(R\right)$, (b) potential energy $U\left(R\right)$, and (c) intrinsic energy $E_{\mathrm{intr}}\left(R\right)$ as functions of relative distance $R$. The red line represents results at $E_{\mathrm{c}.\mathrm{m}.}=195$ MeV and the blue line shows results at $E_{\mathrm{c}.\mathrm{m}.}=195$ MeV. In (a), the inset shows the kinetic energy from 8 to 10 fm at $E_{\mathrm{c}.\mathrm{m}.}=195$ MeV. In (b), the light grey line shows the potential energy calculated under the frozen approximation.

Figure 4

$F_{\mathrm{intr}}\left(R\right)$, the difference between the total nucleus force $F_{\mathrm{tot}}=-\partial U\left(R\right)/\partial R$ and the collective force $F_{\mathrm{coll}}\left(R\right)$ for the cases of (a) $E_{\mathrm{c}.\mathrm{m}.}=195$ MeV and (b) $E_{\mathrm{c}.\mathrm{m}.}=235$ MeV. “I” and “II” in each subfigure correspond to regions I and II defined in Fig. 1.

Figure 5

Two microscopic sources of dissipation and their comparison with an increase of intrinsic energy. (a) $T_{\mathrm{diss}}\left(R\right)$ is from the nucleon exchange process. (b) $W\left(R\right)$ is due to the rearrangement effects in the intrinsic system. (c) and (d) show the difference between two processes $E_{\mathrm{diss}}=T_{\mathrm{diss}}\left(R\right)-W\left(R\right)$ compared to the intrinsic energy $E_{\mathrm{intr}}\left(R\right)$ at $E_{\mathrm{c}.\mathrm{m}.}=195$ MeV and $E_{\mathrm{c}.\mathrm{m}.}=235$ MeV respectively.

Figure 6

(a) Calculated value of $U_{\mathrm{coll}}\left(R\right)$ as a function $R$. (b) Calculated value of $K_{\mathrm{diss}}\left(R\right)=E_{\mathrm{c}.\mathrm{m}.}-P^2/2\mu -U_{\mathrm{coll}}\left(R\right)$ as a function $R$ defined in Eq. (25).

Figure 7

(a) Nucleon exchanging rate as a function of relative distance. (b) Total number of nucleons which have ever transferred from the left (or right) part to the right (or left) part. The red and blue lines represent two cases with $E_{\mathrm{c}.\mathrm{m}.}=195$ and 235 MeV, respectively.

Figure 8

(a) Density profiles obtained in ImQMD for different relative distances $R$ at $E_{\mathrm{c}.\mathrm{m}.}=195$ MeV and 235 MeV and those with the frozen density. The abscissa axis is the reaction axis $Z$ and the vertical axis is axis $Y$. (b) Nucleon number density along the reaction axis $Z$ with $X=Y=0$ at relative $R=10.5$ fm obtained by the ImQMD model. The red line represents the density at $E_{\mathrm{c}.\mathrm{m}.}=195$ MeV, the blue line at $E_{\mathrm{c}.\mathrm{m}.}=235$ MeV. The dark grey dashed line is the density with frozen density; the light grey dashed-dotted lines are densities for projectile and target.

Figure 9

$R$ dependence of the friction parameter caused by two competitive microscopic processes at (a) $E_{\mathrm{c}.\mathrm{m}.}=195$ MeV and (b) $E_{\mathrm{c}.\mathrm{m}.}=235$ MeV. The blue dashed line shows effects of nucleon exchange. The pink dashed line shows effects of rearrangement in the intrinsic system.

Figure 10

The basic processes of nucleon exchange between two fusing nuclei.