Reexamining an extended-mean-field approach in heavy-ion collisions near the Fermi energy

Static and dynamical aspects of nuclear systems are described through an extended time-dependent mean-field approach. The foundations of the formalism are presented, with highlights on the estimation of average values and their corresponding dispersions. In contrast to semiclassical transport models, the particular interest of this description lies on its intrinsic quantal character. The reliability of this approach is discussed by means of stopping-sensitive observables analysis in heavy-ion collisions in the range of 20 to 120 MeV per nucleon.

Figure 1

Each plot represents $P_{\lambda }(z,k_z)$, which is the contribution to the spatial density of a particular SP state $\lambda$ given by Eq. (11) corresponding to $n_z$ from 0 to 5 and $n_x=n_y=0$, projected onto the $(z,k_z)$ plane. The scale of the $z$-color axis is the same for all panels.

Figure 2

Ground state properties of nuclei addressed in this work (red squares) compared with experimental values (black dots): top panel, binding energy per nucleon, $E_b$; bottom panel, mean-square-radius $\langle R\rangle$.

Figure 3

Time evolution of momentum (top) and spatial widths (bottom) of an isolated coherent state contributing to the ${}^{120}\mathrm{Sn}$ nucleus.

Figure 4

The complete set of coherent-state centroids of the ${}^{129}\mathrm{Xe}+{}^{120}\mathrm{Sn}$ collision at 20 MeV/nucleon and $b=3$ fm projected at selected times onto the $(z,k_z)$ phase-space plane (left column) and the density-profile contours projected onto the reaction plane $(z,x)$ in the configuration space (right column). The $z$ axis is along the incident beam. The color palette is normalized to correspond to the number of system nucleons when integrated over the $(z,k_z)$ plane.

Figure 5

The collision rate integrated over $140\mathrm{fm}/c$ for the ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ reaction at 100 MeV/nucleon and $b=7$ fm as a function of c.m. energy $\sqrt{s}$. The errors are statistical, i.e., they are the square root of the number of accepted collisions.

Figure 6

The isotropy ratio $E_{\mathrm{iso}}$ as given by Eq. (27) for the ${}^{129}\mathrm{Xe}+{}^{120}\mathrm{Sn}$ reaction at 50 MeV/nucleon and $b$=3 fm as a function of time. The red dots show the results of the dissipative evolution [Eq. (7)]. The bluish background is due to the dispersion over 1000 CSSDs according to Eq. (26).

Figure 7

The global cross-section reduction factor $\mathcal{F}$ of Eq. (29) for $\nu =0.85$, 0.6, and 0.4, respectively as a function of $E_{\mathrm{inc}}$. The heavy dots and the background gray zone are $\mathcal{F}$ values and attributed uncertainties extracted from experiment [17]. The open squares stand for the $\mathcal{F}$ values deduced in the LV semiclassical analysis of the observable $R_E$ [25], which used the Zamick parametrization of the nuclear equation of state [32].

Figure 8

Isotropy ratio $R_E$ of Eq. (30) for (a) ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ and (b) ${}^{129}\mathrm{Xe}+{}^{120}\mathrm{Sn}$ reactions as a function of $E_{\mathrm{inc}}$. Lines show simulation results with varied parameter $\nu$ of Eq. (28). The value of impact parameter $b$ is chosen according to [25]. The dimmed background stands for the estimated errors. Symbols represent experimental data [17].

Figure 9

Linear momentum transfer for (a) ${}^{40}\mathrm{Ar}+{}^{63}\mathrm{Cu}$, (b) ${}^{40}\mathrm{Ar}+{}^{107}\mathrm{Ag}$, and (c) ${}^{40}\mathrm{Ar}+{}^{197}\mathrm{Au}$ reactions as a function of $E_{\mathrm{inc}}$. Lines represent simulation results with varied parameter $\nu$ of Eq. (28). The value of impact parameter $b$ corresponds to $b_{\mathrm{eff}}$ for the assumed geometrical $0.08{\sigma }_R$. The dimmed background stands for the estimated errors. Symbols represent experimental data [27].

Figure 10

Same as Fig. 9 but as a function of $E_{\mathrm{avail}}$. Each of the colored zones is due to the simulation with the given value of the parameter $\nu$ or for ${\sigma }_{NN}={\sigma }^{\mathrm{free}}$. The broken line delimiting a given zone corresponds to the simulation result including all three reactions studied.

Figure 11

Linear momentum transfer for the ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ reaction at $b=1.5$ fm as a function of $E_{\mathrm{inc}}$. Lines represent simulation results with varied parameter $\nu$ of Eq. (28). The dimmed background stands for the estimated errors.