Conservation properties in the time-dependent Hartree Fock theory

We discuss the conservation of angular momentum in nuclear time-dependent Hartree-Fock calculations for a numerical representation of wave functions and potentials on a three-dimensional Cartesian grid. Free rotation of a deformed nucleus performs extremely well even for relatively coarse spatial grids. Heavy ion collisions produce a highly excited compound system associated with substantial nucleon emission. These emitted nucleons reach the bounds of the numerical box which leads to a decrease of angular momentum. We discuss strategies to distinguish the physically justified loss from numerical artifacts.

Figure 1

Simple illustration of the effect of boundary conditions on total angular momentum. The boxes indicate the computational boundaries and the central dot the reference point for the angular momentum. For periodic boundary conditions (left) it is easy to even revert the sign of the particle's angular momentum. For approximately reflecting boundary conditions (right) the situation is not quite as pronounced: for the case ${\overrightarrow{v}}_1$ there is no change by reflection, while for ${\overrightarrow{v}}_2$ the sign also changes.

Figure 2

Relative deviations of the angular momentum expectation value (full curve) and the three principal moments of inertia (smallest: dotted, intermediate: dashed, and largest: dot-dashed curve) from the temporal average during rotation of a ${}^{24}\mathrm{Mg}$ nucleus. The abscissa denotes the rotation angle calculated from the instantaneous tensor of inertia.

Figure 3

Angular momentum as a function of time for collisions of ${}^{16}\mathrm{O}$$+{}^{16}$O at $E_{\mathrm{c}.\mathrm{m}.}=25$ MeV (left) and $125$ MeV (right). The different cases are: “small”: calculation in a grid of $24\times 32\times 32{\mathrm{fm}}^3$, “large”: calculation in a grid doubled in total size in every direction, “restricted”: same as large, but angular momentum is summed up only over the small grid; “absorb”: small grid with a 6 fm absorbing layer around the boundary.