Microscopic description of the fission path with the Gogny interaction

The Gogny D1M potential energy surfaces are used to extract the fission barrier of a large number of nuclei and to study in a systematic way the evolution of the fission barrier height depending on the proton and neutron numbers to deduce possible links with nuclear shell closures and are compared with evaluated nuclear data. The least-action paths are also computed, using Dijkstra's algorithm, and compared to the least-energy paths. Spontaneous fission half-lives are also calculated and compared with experimental data.

Figure 1

Illustration of the shape isomer issue, located at $W_2$, when applying the flooding procedure to the PES of ${}_{100}{}^{260}\mathrm{Fm}_{160}$. Due to this shape isomer, the iterative flooding method leads to an incorrect LEP (magenta squares) with a dead-end from $S_1$ to $W_2$ and a pseudo saddle point $X$. The LEP obtained with the comprehensive BST approach (black squares) has no pseudo saddle point. The $E<E_{S_3}$ area is included in the $E\le E_X$ area because the well $W_3$ is the latter area.

Figure 2

Illustration of the LEP determination by the BST method for ${}_{106}{}^{286}\mathrm{Sg}_{180}$. The search starts with the computation of the highest saddle point of the LEP $S_{00}$ (00 means that it is the first saddle-point search). After that, the LEP computation is reduced to two subproblems. Their resolutions give $S_{11}$ and $S_{12}$; the first index corresponds to the first dichotomy, and the second to the number of the subproblem starting from the ground state. $W_{\mathrm{bxx}}$ and $W_{\mathrm{axx}}$ are wells before and after the saddle point $S_{\mathrm{xx}}$ obtained by the steepest descent from the latter. The ground-state well $W_{\mathrm{GS}}$ is determined by the steepest descent from point (0,0). Inset: BST built during the saddle-point search.

Figure 3

LEP projected on the $Q_{20}$ axis for (a) ${}_{92}{}^{236}\mathrm{U}_{144}$ and (b) ${}_{94}{}^{240}\mathrm{Pu}_{146}$ with different corrections: no corrections, triaxial, and triaxial + ZPE in the ATDHF framework and in the GCM framework for collective corrections.

Figure 4

Comparison of D1M primary and secondary fission barrier height in the ADTHF and GCM frameworks with the empirical values [31] for ${}^{230-232}\mathrm{Th}, {}^{232-238}\mathrm{U}, {}^{238-244}\mathrm{Pu}$, and ${}^{242-248}\mathrm{Cu}$. The rms of the primary fission barrier is equal to 0.52 MeV for the ATDHF framework and 2 MeV for the GCM one. The rms of the secondary fission barrier is equal to 0.45 MeV for the ATDHF framework and 1.94 MeV for the GCM one. For isobaric nuclei, one of the names is displayed below its corresponding symbol.

Figure 5

Color representation of the LEP for (a) U and (b) Fm isotopes (b). Green stars represent the location of the highest barrier along each LEP. The left panels represent the fission barrier height for the whole isotopic chain with respect to the corresponding $0^{+}$ level.

Figure 6

Height of D1M primary fission barriers compared with HFB14 ones [6] for all even-even nuclei with $90\le Z\le 110$ lying between the D1M proton and the D1M neutron driplines.

Figure 7

Color representation in the ($N,Z$) plane of D1M fission barrier properties: (a) barrier height, (b) reduced quadrupole deformation ${\tilde{q}}_{20}$, and (c) reduced octupole deformation ${\tilde{q}}_{30}$.

Figure 8

Evolution of the tree data structure, represented by white arrows, during the LAP computation of ${}^{236}\mathrm{U}$ for an initial excitation energy $E^{*}=E_{0^{+}}$. Black lines represent isoenergies of the PES, and the color scale is the action of the path between the ground state and the various points of the PES. The LAP is shown byh black squares. The scission line is not represented here. The LAP from ground state to ($Q_{20}$[b],$Q_{30}\left[{\mathrm{b}}^{3/2}\right])=(164,0)$ is represented by red plus symbols, while the LAP from ground state to (166,0) is depicted by the magenta $\mathsf{x}$'s.

Figure 9

(a)${}^{236}\mathrm{U}$ LAP calculated for $E^{*}=E_{0^{+}}$ and different inertia, namely, ATD, GCM, and SEMP. (b) Effective inertia along the LAP. (c) Distance $dS$ between two neighboring deformation points of the LAP. We have $S_{\mathrm{ATD}}=63.3\hslash , S_{\mathrm{GCM}}=66.7\hslash$, and $S_{\mathrm{SEMP}}=122.6\hslash$. The black curve in (a) is hidden by the red one.

Figure 10

(a) Energy $E\left(Q_{20}\right)$ (left $y$-axis scale) and mass asymetry $Q_{30}\left(Q_{20}\right)$ (right $y$-axis scale) of the LEP and LAP of ${}^{226}\mathrm{Th}$ calculated with $E^{*}=E_{0^{+}}=2.7$ MeV. (b) ATDHF inertia along the LEP and LAP.

Figure 11

(a) Energy $E\left(Q_{20}\right)$ (left $y$-axis scale) and mass asymetry $Q_{30}\left(Q_{20}\right)$ (right $y$-axis scale) of the LEP and LAP of ${}^{238}\mathrm{U}$ calculated with $E^{*}=E_{0^{+}}=3.3$ MeV. (b) ATDHF inertia along the LEP and LAP.

Figure 12

(a) Energy, (b) mass asymetry $Q_{30}$, and (c) inertia of ${}^{226}\mathrm{Th}$ LEP and LAP calculated with $E^{*}=E_{0^{+}}=2.4$, 5.1, and 5.2 MeV.

Figure 13

LAP's action of ${}^{226}\mathrm{Th}$ depending on the excitation energy, from $E_{0^{+}}$ to the fission barrier height of the LEP $E_{\mathrm{barr},\mathrm{LEP}}$. The arrow at 5 MeV refers to the transition of the LAP between an akin to spontaneous fission LAP to an akin to LEP.

Figure 14

Spontaneous fission half-lives $T_{1/2}^{\mathrm{sf}}$ obtained for ${}^{232-238}\mathrm{U}, {}^{240}\mathrm{Pu}, {}^{248}\mathrm{Cm}, {}^{250-256}\mathrm{Fm}, {}^{252-256}\mathrm{No}, {}^{256-260}\mathrm{Rf}, {}^{258-262}\mathrm{Sg}$, and ${}^{264}\mathrm{Hs}$ as a function of the fissibility parameter $Z^2/A$ with $E^{*}=E_{0^{+}}$ and $E^{*}=E_{0^{+}}+0.5$ MeV. Results were obtained with either the ATDHF or the GCM correction. Theoretical results are compared with experimental data [41].