Fission modes of ${}^{256}\mathrm{Fm}$ and ${}^{258}\mathrm{Fm}$ in a microscopic approach

A static microscopic study of potential-energy surfaces within the Skyrme-Hartree-Fock-plus-BCS model is carried out for the ${}^{256}\mathrm{Fm}$ and ${}^{258}\mathrm{Fm}$ isotopes with the goal of deducing some properties of spontaneous fission. The calculated fission modes are found to be in agreement with the experimentally observed asymmetric-to-symmetric transition in the fragment-mass distributions and with the high- and low-total-kinetic-energy modes experimentally observed in ${}^{258}\mathrm{Fm}$. Most of the results are similar to those obtained in macroscopic-microscopic models as well as in recent Hartree-Fock-Bogolyubov calculations with the Gogny interaction, with a few differences in their interpretations. In particular an alternative explanation is proposed for the low-energy fission mode of ${}^{258}\mathrm{Fm}$.

Figure 1

Potential energy of deformation $E_{\mathrm{def}}$ as a function of the quadrupole moment $Q_{20}$ along the different fission paths obtained for the ${}^{256}\mathrm{Fm}$ isotope. The solid line from 0 to 160 barns (b) represent left-right symmetric solutions including the ground-state (GS) and the superdeformed (SD) minima. The solid and the dotted lines beyond 250 b correspond to symmetric and asymmetric compact fission (CF) paths, respectively. The dashed line is the bottom of the asymmetric elongated fission (EF) valley.

Figure 2

Cuts in the potential-energy surface along the axial octupole moment $Q_{30}$ direction at different fixed elongations $Q_{20}$ for ${}^{256}\mathrm{Fm}$. The solid circles represent calculated points, whereas the solid lines are drawn as eye guides.

Figure 3

Cut of the potential-energy surface along the $A_H$ direction at a fixed elongation $Q_{20}=280$ b for ${}^{256}\mathrm{Fm}$. The solid line represents solutions obtained by increasing $A_H$ from 128 (symmetric solution) and the dashed line is obtained by decreasing $A_H$ from about 141.

Figure 4

Same as described in the legend to Fig. 1 for the ${}^{258}\mathrm{Fm}$ isotope. Solid circles and triangles are calculated points along the symmetric compact fission (sym. CF) and symmetric elongated fission (sym. EF) paths, respectively.

Figure 5

Surface of deformation energy as a function of the center-of-mass distance $D_{\mathrm{c}.\mathrm{m}.}$ and the fragment elongation $Q_{2\mathrm{f}}$ for ${}^{258}\mathrm{Fm}$.

Figure 6

Contour diagram of the deformation energy as a function of the center-of-mass distance $D_{\mathrm{c}.\mathrm{m}.}$ and the fragment elongation $Q_{2\mathrm{f}}$ for ${}^{258}\mathrm{Fm}$. The energy interval between the contour lines (${\mathit{dE}}_{\mathrm{def}}$) is 3 MeV.

Figure 7

Cuts in the potential-energy surface along the $Q_{2\mathrm{f}}$ direction at different fixed center-of-mass distances $D_{\mathrm{c}.\mathrm{m}.}$ for ${}^{258}\mathrm{Fm}$.

Figure 8

Deformation energy along both symmetric fission valleys as a function of $D_{\mathrm{c}.\mathrm{m}.}$ (upper panel) and as a function of $Q_{20}$ (lower panel), obtained with $N_0=20$ and $b=0.42$.

Figure 9

Variation of $D_{\mathrm{c}.\mathrm{m}.}$ along both symmetric fission valleys as a function of $Q_{20}$, obtained with $N_0=20$ and $b=0.42$.

Figure 10

Sequence of nuclear shapes along the symmetric CF valley of ${}^{258}\mathrm{Fm}$ between $D_{\mathrm{c}.\mathrm{m}.}=13.0$ and 15.0 fm. The solid lines correspond to equal nuclear density contours ranging from $\rho (\mathbf{r})=0.01{\mathrm{fm}}^{-3}$ (outermost contour) to $\rho (\mathbf{r})=0.15{\mathrm{fm}}^{-3}$ (innermost contour) with $0.01{\mathrm{fm}}^{-3}$ steps. The thick solid line marks the density contour at half the saturation density, namely $\rho (\mathbf{r})={\rho }_{\mathrm{sat}}/2=0.08{\mathrm{fm}}^{-3}$.

Figure 11

Same as described in the legend to Fig. 10 for nuclear shapes along the symmetric EF valley between $D_{\mathrm{c}.\mathrm{m}.}=12.5$ 22.0 fm.