Calculations of particle decay widths of hyper-deformed nuclei using prolate spheroidal coordinates

Abstract

We address the problem of calculating the particle decay widths from hyper-deformed nuclei. To simplify the calculation we treat the deformed nucleus as a prolate spheroid, and the particle-nucleus interaction as sharp-edged. This enables us to describe the deformed nuclear shape as an equipotential surface in a prolate spheroidal coordinate system. We model the particle-nucleus interaction by means of a spheroidal “square well” potential. Next we make the adiabatic approximation. This enables us to consider the deformed nucleus as being fixed in space. We then solve a system of coupled Schrödinger equations which describe the scattering of the particle from the deformed nucleus. These partial differential equations are separable in our chosen coordinate system. We generate prolate spheroidal wave functions to describe the particle both inside and outside the nuclear volume using a matrix diagonalization technique. The scattering problem is then solved by matching appropriate linear combinations of internal and external wave functions, and their first derivatives, at the nuclear boundary and searching for resonances as the Q-value for particle emission is varied. Using this formalism we calculate the alpha-decay widths of the recently discovered α-chain state in 24Mg and of the proposed 16O+ 16O+ 16O configuration in 48Cr.