Heavy-element fission barriers

We present calculations of fission properties for heavy elements. The calculations are based on the macroscopic-microscopic finite-range liquid-drop model with a 2002 parameter set. For each nucleus we have calculated the potential energy in three different shape parametrizations: (1) for 5 009 325 different shapes in a five-dimensional deformation space given by the three-quadratic-surface parametrization, (2) for 10 850 different shapes in a three-dimensional deformation space spanned by ${\ensuremath{\epsilon}}_{2}$, ${\ensuremath{\epsilon}}_{4}$, and $\ensuremath{\gamma}$ in the Nilsson perturbed-spheroid parametrization, supplemented by a densely spaced grid in ${\ensuremath{\epsilon}}_{2}$, ${\ensuremath{\epsilon}}_{3}$, ${\ensuremath{\epsilon}}_{4}$, and ${\ensuremath{\epsilon}}_{6}$ for axially symmetric deformations in the neighborhood of the ground state, and (3) an axially symmetric multipole expansion of the shape of the nuclear surface using ${\ensuremath{\beta}}_{2}$, ${\ensuremath{\beta}}_{3}$, ${\ensuremath{\beta}}_{4}$, and ${\ensuremath{\beta}}_{6}$ for intermediate deformations. For a fissioning system, it is always possible to define uniquely one saddle or fission threshold on the optimum trajectory between the ground state and separated fission fragments. We present such calculated barrier heights for 1585 nuclei from $Z=78$ to $Z=125$. Traditionally, actinide barriers have been characterized in terms of a ``double-humped'' structure. Following this custom we present calculated energies of the first peak, second minimum, and second peak in the barrier for 135 actinide nuclei from Th to Es. However, for some of these nuclei which exhibit a more complex barrier structure, there is no unique way to extract a double-humped structure from the calculations. We give examples of such more complex structures, in particular the structure of the outer barrier region near $^{232}\mathrm{Th}$ and the occurrence of multiple fission modes. Because our complete results are too extensive to present in a paper of this type, our aim here is limited: (1) to fully present our model and the methods for determining the structure of the potential-energy surface, (2) to present fission thresholds for a large number of heavy elements, (3) to compare our results with the two-humped barrier structure deduced from experiment for actinide nuclei, and (4) to compare to additional fission-related data and other fission models.