# HEAVY-ELEMENT FISSION BARRIERS

P. MÖLLER and A. J. SIERK
Theoretical Division, Los Alamos National Laboratory, New Mexico 87545, USA

T. ICHIKAWA
RIKEN Nishina Center, Riken, Wako, Saitama, 351-0198, Japan

A. IWAMOTO
Japan Atomic Energy Agency (JAEA),
Tokai-mura, Naka-gun, Ibaraki, 319-1195, Japan

R. BENGTSSON, H. UHRENHOLT, and S. ÅBERG
Department of Mathematical Physics, Lund Institute of Technology,
SE-22100 Lund, Sweden

This paper was published in
Physical Review C 79 064304 (2009)
It has been assigned Los Alamos National Laboratory Preprint No LA-UR-08-4190.

Abstract:

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 3 different shape parameterizations: (1) for 5009325 different shapes in a five-dimensional deformation space given by the three-quadratic-surface parameterization, (2) for 10850 different shapes in a three-dimensional deformation space spanned by $\epsilon$2, $\epsilon$4, and $\gamma$ in the Nilsson perturbed-spheroid parameterization, supplemented by a densely spaced grid in $\epsilon$2, $\epsilon$3, $\epsilon$4, and $\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 $\beta$2, $\beta$3, $\beta$4, and $\beta$6 for intermediate deformations. For a fissioning system it is always possible to define uniquely {\it 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 $232$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 to other fission models.
Many of the 35 Figures are in color, so the paper should be printed on a color printer. In the paper we only show 6 potential-energy contour maps versus ε2 and γ. We have such calculated surfaces for 5254 heavy nuclei; these contour plots can be accessed here. As explained in the paper they are based on a 3-dimensional calculations in the coordinates ε24 and γ; but to plot them in 2D we have "minimized" with respect to ε4. There are issues with this strategy but for small deformations we usually get reasonable contour maps. However, sometimes the dots marking minima and crossed lines marking saddle points can seem offset from the minima and saddles in the 2D contour diagrams. This occurs because the 2D representation of the 3D space does not quite correctly show all aspects of the 3D space. It is in practice impossible to reduce our calculations in a 5D space in the three-quadratic-surface parameterization to 2D contour plots, so we do not show such surfaces here.

The complete manuscript in color as a .ps.gz file is available for download.

The complete manuscript in color as a .pdf file is also available for download. The .ps file may yield better quality when printed, but if you have trouble printing it use the .pdf file, read it into acroread and request "fit to printing area" as one of the options, before hitting the print button.

Table 2 (TABLE 2) , in computer-readable format, is available for download.
The format is (3I5,F10.2)
The variables are: Z, N, A, $B$f

We provide the 35 figures, formatted for printing full-page, as individual .ps.gz files:

Those figures that are in "landscape" mode can be converted to portrait, which may be preferable if they are going to be manipulated electronically or incorporated into a manuscript. To do that look for
(search for "main" in an editor)

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and comment out the last two lines so that they read

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Peter Moller
Created: Sept 24 2007 --> Last modified: Jul 5, 2012