Collective aspects deduced from time-dependent microscopic mean-field with pairing: Application to the fission process

Given a set of collective variables, a method is proposed to obtain the associated conjugated collective momenta and masses starting from a microscopic time-dependent mean-field theory. The construction of pairs of conjugated variables is the first step to bridge microscopic and macroscopic approaches. The method is versatile and can be applied to study a large class of nuclear processes. An illustration is given here with the fission of ${}^{258}\mathrm{Fm}$. Using the quadrupole moment and eventually higher-order multipole moments, the associated collective masses are estimated along the microscopic mean-field evolution. When more than one collective variable is considered, it is shown that the off-diagonal matrix elements of the inertia play a crucial role. Using the information on the quadrupole moment and associated momentum, the collective evolution is studied. It is shown that dynamical effects beyond the adiabatic limit are important. Nuclei formed after fission tend to stick together for a longer time leading to a dynamical scission point at a larger distance between nuclei compared to the one anticipated from the adiabatic energy landscape. The effective nucleus-nucleus potential felt by the emitted nuclei is finally extracted.

Figure 1

Potential energy curve of ${}^{258}\mathrm{Fm}$ nucleus as a function of the quadrupole deformation parameter (in barn units). Isosurfaces of the total density drawn at half the maximum value at $Q_2=34$, 80, 194, and 399 b are also shown. The horizontal lines indicate the different starting points that are used in this work as initial conditions for the time-dependent evolution. The different vertical dashed lines corresponds from left to right to $Q_2=34.2\text{b}$, $Q_2=80\text{b}$ (barrier position), $Q_2=160\text{b}$ (spontaneous fission threshold $Q_2^{\mathrm{th}}$), $Q_2=182\text{b}$, $Q_2=194\text{b}$, $Q_2=296\text{b}$, and $Q_2=400\text{b}$. The two thick arrows indicate the spontaneous fission threshold $Q_2^{\mathrm{th}}$ and the adiabatic scission point $Q_2^{\mathrm{sc}}$.

Figure 2

Single-particle energies in ${}^{258}\mathrm{Fm}$ nucleus along the adiabatic PEC. The green solid curves show the neutron and proton Fermi energies. Positive and negative parity states are respectively shown with red filled and blue open triangles. The vertical lines indicate the initial values of the quadrupole moment taken in the present dynamical calculations. The shaded area presenting the region where the system does not spontaneously fission is also shown.

Figure 3

Density profiles obtained for different initial $Q_2$. The densities are shown as a function of time. For initial $Q_2$ below the fission threshold, a quadrupole boost has been imposed initially. From left to right, the initial $Q_2$ values are $Q_2=34.2\text{b}$, $Q_2=80\text{b}$, $Q_2=160\text{b}$, and $Q_2=194\text{b}$. The system is eventually initially boosted leading to nonzero values of $E_2^{\mathrm{ini}}$ directly indicated in the figure. The isodensity curves are drawn from 0.03 to 0.15 ${\mathrm{fm}}^{-3}$ with increments of 0.03 ${\mathrm{fm}}^{-3}$.

Figure 4

Top: Quadrupole mass parameter calculated from TD-EDF paths. In all cases, the initial quadrupole moment is $Q_2^{\mathrm{ini}}=160\text{b}$. Different trajectories correspond to different initial boosts. The corresponding initial collective energies $E_2^{\mathrm{ini}}$ are systematically reported in the figure. The quadrupole mass obtained using Eq. (19) assuming that the system follow the adiabatic PEC is also shown (solid line) for comparison. Bottom: Quadrupole mass obtained with varying initial quadrupole deformation, $Q_2^{\mathrm{ini}}=80$, 114, 160, and 194 b.

Figure 5

Minimum collective energy after a boost that should be initially deposited into the system to induce the fission of ${}^{258}\mathrm{Fm}$ for $Q_2^{\mathrm{ini}}\le Q_2^{\mathrm{th}}$. This energy is plotted as a function of the initial quadrupole deformation considered. For each $Q_2$, the highest (respectively lowest) value of the initial collective energy where the scission is not (respectively is) observed is reported.

Figure 6

Evolution of the total collective kinetic energy $E_{\mathrm{kin}}^{\mathrm{tot}}$ as a function of time. The initial systems correspond respectively to a quadrupole moment (a) $Q_2^{\mathrm{ini}}=171.0\text{b}$ without boost or (b) to $Q_2^{\mathrm{ini}}=79.8\text{b}$ with initial boost. In both cases, the CKE obtained using Eq. (24) and associated to $Q_2$ only ($E_{\mathrm{kin}}^2$), $Q_4$ only ($E_{\mathrm{kin}}^4$), or both ($E_{\mathrm{kin}}^{2+4}$) are also shown.

Figure 7

Evolution of the collective momentum as a function of time for different initial quadrupole deformations (with $Q_2^{\mathrm{ini}}\ge Q_2^{\mathrm{th}}$). The arrow indicates the scission point associated to the adiabatic potential.

Figure 8

(a) Function $F\left(Q_2\right)$ obtained with TD-EDF using Eq. (31) for the three evolutions with $Q_2^{\mathrm{ini}}\le Q_2^{\mathrm{sc}}$ displayed in Fig. 7. For comparison, we also show the forces acting on $Q_2$ that would be induced either by the adiabatic potential (black dashed line) or solely by the Coulomb field (black filled circle). The arrows in the figure indicate the $Q_2$ value where the neck density ${\rho }_{\mathrm{neck}}$ becomes ten times less than the saturation density ${\rho }_{\mathrm{sat}}=0.16{\mathrm{fm}}^{-3}$. The two arrows indicate the adiabatic path (static) and dynamical path (dynamic). In the latter case, the position where ${\rho }_{\mathrm{neck}}/{\rho }_{\mathrm{sat}}=1/10$ is almost independent of $Q_2^{\mathrm{ini}}$. (b) Dynamical potential curve obtained by integrating $F\left(Q_2\right)$ using Eq. (35). Again for comparison, the adiabatic potential and the Coulomb field are also shown. Here, the origin of energy is taken such that $E=0$ for $Q_2\to \infty$.

Figure 9

The relative kinetic energy $T_{\mathrm{rel}}$, the Coulomb energy $V_C$, and the total kinetic energy (TKE) of the fission fragments as a function of the relative distance $R$ for $Q_2^{\mathrm{ini}}=197\text{b}$.