Adiabatic Approximation in Multi-Dimensional Penetration Problem

Abstract

Adiabatic approximation for multi-dimensional penetration problem in which mass tensors are functions of position is discussed. Definition of adiabatic path is shown to be closely related to transformation to a standard coordinate system where mass tensors are diagonal and, at the same time, the potential is flat except for one dimension which corresponds to fissioning degree of freedom. We discuss the general condition of the validity of the adiabatic approximation using this coordinate system and show that the adiabatic path is a good approximation when the potential has a simple geometry and mass tensors are changing smoothly as is realized in our fission calculation using the . potential and mass tensors calculated by macroscopic model. Calculation of the penetrability through a multi-dimensional potential barrier is not only an attractive problem in itself but also is a necessity to understand several quantum phenomena. A typical example is the nuclear fission in which we need several shape parameters to specify the potential energy surface. In Ref. 1), two methods were proposed to determine the fission path in a multi-dimensional potential barrier. One is the method of SchmiCI,2) where they fixed the minimum action path in a semiclassical way. The other is an adiabatic method where the path connecting the ground-state deformation to· the saddle point and beyond it was defined in a coordinate-independent way. The latter is a generalization of the adiabatic path method which is frequently used in actual calculations of fission half-life. A significant result of Ref. 1) is that the adiabatic path deviates little from the minimum action path and two action integrals are nearly the same. In this paper, we will investigate why this happens and show that the adiabatic approximation is useful in fission study. In § 2, we introduce a standard coordinate system where the mass tensors are diagonal and the potential is flat except for one dimension. The transformation to this coordinate system is closely related to the definition of the adiabatic path. In § 3, we investigate the general condition for the validity of the adiabatic approximation using a special coordinate system introduced in § 2. In § 4, the validity of the adiabatic approximation in the case of spontaneous fission calculation is shown. In § 5, we give a short summary.