Structure properties of ${}^{226}\mathrm{Th}$ and ${}^{256,258,260}\mathrm{Fm}$ fission fragments: Mean-field analysis with the Gogny force

The constrained Hartree-Fock-Bogoliubov method is used with the Gogny interaction D1S to calculate potential energy surfaces of fissioning nuclei ${}^{226}\mathrm{Th}$ and ${}^{256,258,260}\mathrm{Fm}$ up to very large deformations. The constraints employed are the mass quadrupole and octupole moments. In this subspace of collective coordinates, many scission configurations are identified ranging from symmetric to highly asymmetric fragmentations. Corresponding fragment properties at scission are derived yielding fragment deformations, deformation energies, energy partitioning, neutron binding energies at scission, neutron multiplicities, charge polarization, and total fragment kinetic energies.

Figure 1

Comparison between symmetric fission of the ${}^{226}\mathrm{Th}$ and ${}^{256}\mathrm{Fm}$ nuclei through total energy, hexadecapole moment, and minimal density along $z$ axis in the neck ${\rho }_{\mathrm{N}}$. $q_{20}^{(\mathrm{ps})}$ and $E^{(\mathrm{ps})}$ represent elongation and HFB energy of the first postscission (ps) point for each fissioning system, respectively.

Figure 2

Symmetric scission configurations of ${}^{226}\mathrm{Th}$ in the ($z,r$) space coordinates before and after scission (upper and lower panels, respectively). The isolines are separated by $0.01$ fm${}^{-3}$. The dashed isoline corresponds to $\rho =0.16$ fm${}^{-3}$.

Figure 3

Same as described in the caption to Fig. 2 for the symmetric scission of ${}^{256}\mathrm{Fm}$.

Figure 4

Potential energies (MeV) as functions of the $q_{20}$ (b) and $q_{30}$ (b${}^{3/2}$) mass moments for ${}^{226}\mathrm{Th}$ (a), ${}^{256}\mathrm{Fm}$ (b), ${}^{258}\mathrm{Fm}$ (c), and ${}^{260}\mathrm{Fm}$ (d). Postscission points are not plotted.

Figure 5

(Upper panel) The scission line for ${}^{226}\mathrm{Th}$ is shown over the ($q_{20},q_{30}$) plane as a continuous curve in red color along which are marked symbols $\mathrm{a},\mathrm{b},\mathrm{c},\dots ,\mathrm{j}$. The black dots, representing single HFB calculations, are shown to illustrate the densifying of the mesh used close to the scission line. (Lower panel) Scission lines for ${}^{256-260}\mathrm{Fm}$.

Figure 6

${}^{226}\mathrm{Th}$. Potential energy along the scission line as a function of fragment mass. The symbols $\mathrm{a},\mathrm{b},\mathrm{c},\dots ,\mathrm{j}$ have the same meaning as in Fig. 5. See text for more details.

Figure 7

Same as described in the caption to Fig. 6 for ${}^{256-260}\mathrm{Fm}$.

Figure 8

Axial mass quadrupole moments $\langle {\hat{Q}}_{20}\rangle$ of the nascent fission fragments for ${}^{226}\mathrm{Th}$ and ${}^{256-260}\mathrm{Fm}$.

Figure 9

${}^{112}\mathrm{Ru}$ and ${}^{150}\mathrm{Ce}$ potential energy curves from constrained HFB calculations restricted to axially symmetric and left-right-symmetric shapes as a function of axial quadrupole deformation [41]. The potential energy of ${}^{150}\mathrm{Ce}$ has been arbitrarily increased by 295 MeV to ease comparison between curves.

Figure 10

Axial mass octupole moments $\langle {\hat{Q}}_{30}\rangle$ of nascent fission fragments for ${}^{226}\mathrm{Th}$ and ${}^{256-260}\mathrm{Fm}$.

Figure 11

Nascent fragment deformation energies for ${}^{226}\mathrm{Th}$ and ${}^{256-260}\mathrm{Fm}$ as functions of (a) fragment mass and (b) fragment charge. The differences between light (L) and heavy (H) fragment energies as functions of light fragment mass are shown in (c).

Figure 12

One-neutron binding energies of nascent fission fragments as functions of fragment mass for ${}^{226}\mathrm{Th}$ and ${}^{256-260}\mathrm{Fm}$.

Figure 13

Differences between one-neutron binding energies of fragments as calculated for scission configurations and for ground states, plotted as functions of fragment mass.

Figure 14

${}^{226}\mathrm{Th}$. Calculated neutron multiplicity as a function of fragment mass. The solid line is to guide the eye.

Figure 15

${}^{256}\mathrm{Fm}$. Neutron multiplicity versus fragment mass. Comparison between predictions (solid symbols) and data [47] (empty symbols).

Figure 16

Neutron multiplicities for ${}^{258}\mathrm{Fm}$ (upper panel) and ${}^{260}\mathrm{Fm}$ (lower panel). Lines are to guide the eye.

Figure 17

Nascent fission fragment proton pairing energy (upper panel) and deviation from unchanged charge distribution (bottom panel) as functions of fragment charge for ${}^{226}\mathrm{Th}$.

Figure 18

Same as described in the caption to Fig. 17 for ${}^{258}\mathrm{Fm}$.

Figure 19

Distances between nascent fragment centers of charge calculated as functions of fragment mass for ${}^{226}\mathrm{Th}$ and ${}^{256-260}\mathrm{Fm}$.

Figure 20

${}^{226}\mathrm{Th}$. TKE values of nascent fission fragments as a function of fragment mass. Comparison between predictions (solid dots) and data [40, 51].

Figure 21

${}^{256-260}\mathrm{Fm}$. TKE values of nascent fission fragments as functions of fragment mass.

Figure 22

${}^{256}\mathrm{Fm}$. TKE values of fission fragments as functions of fragment mass. Results of theoretical calculations (dots) are displayed with pre-neutron-emission data (contour diagram). The solid line represents the experimental average TKE [54].