The shortest path problem

The shortest path problem is solved and applied to the calculations of half lives with respect to spontaneous fission of heavy nuclei. The dynamical programming method of Bellman and Kalaba is used to find the fission path in the d-dimensional space of deformation parameters. Fission half lives of heavy isotopes (100<Z<110) are shown and compared to the experimental data. 1. Introductory remarks A half life of a nucleus A Z X with respect to the spontaneous fission into two nearly equal fragments is inversely proportional to the probability of fission and reads [1] ~ 1 T nP : , (1) where ( ) ( ) 1 P= 1+exp 2S − , (2) is the probability of fission decay, n is a frequency of fission mode equal to 10sec, S is the functional of path x in the d-dimensional space of nuclear deformations {x} and is given in units of Plank constant h=2π. The path x joins two special points of the deformation space: one corresponding to the ground state of the nucleus (a) and the other one to the elongated nuclear shape or "two nuclear fragments" (b). The functional S is given by [ ] ( )