New Calculation of Fission Barriers for Heavy and Superheavy Nuclei

On the basis of the macroscopic-microscopic method we have performed a new calculation of the nuclear potential energy of deformation for heavy and superheavy nuclei. Our primary emphasis has been to develop techniques that permit more accurate extrapolations both to the large deformations encountered in fission and heavy-ion reactions and to new regions of nuclei. With this purpose in mind, we specify the nuclear shape with five degrees of freedom, in terms of smoothly joined portions of three quadratic surfaces of revolution (e.g., two spheroids connected by a hyperboloidal neck). The single-particle potential is obtained by folding a Yukawa function with a uniform sharp-surface generating potential of appropriate shape. The parameters describing the potential are obtained from statistical (Thomas-Fermi) calculations that reproduce correctly the average trends throughout the Periodic Table of a variety of nuclear properties. To solve the Schrödinger equation for the single-particle energies and wave functions, we have used mainly an expansion of the wave function in a set of deformed harmonic-oscillator basis functions, but have also investigated a finite-difference method. Shell and pairing corrections are calculated from the single-particle energies by means of the methods developed by Strutinsky and added to the surface and Coulomb energies of the liquid-drop model to obtain the total potential energy of deformation.

This approach is found to reproduce reasonably well the over-all trends of experimental fission-barrier heights of nuclei ranging from rare earths to actinides. The calculated fission barriers of actinide nuclei contain two peaks separated by a secondary minimum. The over-all variation in the relative heights of the two peaks agrees with experimental results, but the calculated first peak is somewhat low for thorium isotopes and somewhat high for curium isotopes. For the actinides the first saddle point is found to be symmetric in shape and the second saddle point asymmetric. The highest saddle point is asymmetric for ${}_{}{}^{226}{}_{}{}^{}\mathrm{Ra}$ and ${}_{}{}^{210}{}_{}{}^{}\mathrm{Po}$ and symmetric for ${}_{}{}^{188}{}_{}{}^{}\mathrm{Os}$. The fission barriers of superheavy nuclei near ${}_{}{}^{298}{}_{}{}^{}114$ are even higher than previously supposed. With respect to spontaneous fission, the island of superheavy nuclei is a mountain ridge extending from 114 protons to about 124 protons. The descent from the mountain down to the sea of instability is rather gentle for decreasing neutron numbers below 184, but is more rapid on the other three sides.