Hydrodynamic Theory of Spontaneous Fission

The penetration factor for spontaneous fission has been calculated from the liquid-drop model. The transformation of the Gamow integral over the nucleon coordinates into an integral over the deformation parameters $a_n$ has been carried out hydrodynamically, assuming irrotational motion. The transformation requires evaluation of the kinetic energy in terms of $a_n$ and ${\dot{a}}_n$. Series expansions are used for the kinetic energy and for the potential energy of deformation. We have neglected all parameters but $a_2$ and carried the hydrodynamic calculations through terms in ${a_2}^4$. While the potential barrier is subject to several uncertainties, it has nevertheless been possible to estimate the spontaneous fission hindrance factor for the highest $Z$ elements. We find for $Z=100\mathrm{}$ and $\frac{Z^2}{A}\approx 39$ that a 1-Mev increase in barrier height should correspond to a ${10}^{3.7}$-fold increase in the half-life. This result agrees closely with the empirical hindrance factor formula deduced by Swiatecki from a correlation of fluctuations in half-lives with deviations of ground-state masses from the semiempirical mass formula. We have included some details of both the hydrodynamic and the electrostatic calculations.