Collinear versus triangular geometry: A ternary fission study

We study in this work the preference of the arrangements of fragments leading to ternary fission. Earlier experimental investigations on this subject have established the fact that the emission of a third particle happens in a direction perpendicular to the direction of the fission axis. Recently, within the missing mass approach, it has been reported that for fragments of comparable masses, the so-called collinear cluster tripartition (CCT), collinear emission of the fragments occurs. The ternary potential energy surface (PES) of three-body fragmentation of ${}^{252}\mathrm{Cf}$ is studied. The PESs are calculated for the fragment arrangements starting from a collinear configuration to a triangular configuration by varying the angle between the end fragments with respect to the fragment positioned in the middle. Furthermore, the role of the positioning of the three fragments is analyzed. Our results indicate that there is a clear preference for an arrangement in which the lightest fragment is positioned in the middle. Furthermore, for all possible third fragments, collinear geometry is found to be favored over orthogonal geometry.

Figure 1

A schematic geometrical illustration of the evolution of the orientation of three fragments from a collinear configuration towards an oblate, triangular configuration. (a) The lightest fragment is kept in the middle corresponding to the angle $\theta$ = 0, where $\theta$ is defined as the angle between the $x$ axis and the line passing through the centers of the middle fragment and the other two fragments [labeled in panel (c)]. The center-to-center distances, $R_{ij}$, between the fragments are labeled. (b) The center of mass (c.m.) of the three fragments is marked as $\times$. The position vectors ${\mathbf{r}}_{\mathbf{k}}$ from the position of the center of mass to the centers of the fragments are shown. (c) A triangular oblate configuration in which all three fragments are touching. The radius vector $R_x$ of the fragments and the components of the vectors between the fragments $A_1$ and $A_3$ are labeled.

Figure 2

The total ternary fragmentation potential as a function of the orientation angle is presented for three different fragment combinations in three different arrangements for the angular momentum $\ell =0\hslash$ and 40$\hslash$. The intersecting point of the vertical and horizontal dotted lines corresponds to the triangular configuration, and the value of the touching angle is labeled. For details, see text.

Figure 3

Ternary fragmentation potential as a function of the fragment mass number $A_3$ corresponding to the triangular arrangement and three possible collinear arrangements (Cases I, II, and III).

Figure 4

Ternary fragmentation potential (or PES) for the fragment arrangement Case II, as a function of the angle and fragment mass number $A_3$. The solid line in the PES corresponds to the touching angle.