Similarity between Shell Model and Deformed Nucleus Wave Functions

Abstract

A body coordinate system $x^{\prime }{y}^{\prime }{z}^{\prime }$ is defined in which the density of nuclear matter is assumed axially symmetric about the $z^{\prime }$ axis, the angular momentum $j_{z^{\prime }}$ of each of the $A$ nucleons in the nucleus about this axis being quantized. The wave function of a nucleon is ${\varphi }_{\kappa }$, with $j_{z^{\prime }}{\varphi }_{\kappa }=\kappa {\varphi }_{\kappa }$. The system $x^{\prime }{y}^{\prime }{z}^{\prime }$ is rotated relative to a system $\mathrm{xyz}$ fixed in space in such a way that the nucleus is in a state of total angular momentum $J$, with $z$ component $M$, while the $z^{\prime }$ component remains $K={\kappa }_1+\cdots +{\kappa }_A$. The nuclear wave function is ${{\Psi }_{\mathrm{MK}}}^J$. The ${\varphi }_{\kappa }$ are so defined that the ${{\Psi }_{\mathrm{MK}}}^J$ are similar to wave functions obtained for shell-theoretical calculations. In order to represent at least 13 low states with $A=18\mathrm{} \mathrm{and} 19$, three ${\varphi }_{\kappa }$ are needed. These ${\varphi }_{\kappa }$ resemble ${\chi }_{\kappa }$, the wave functions of particles in a slightly deformed harmonic-oscillator well. Calculated $\mathrm{ft}$ values and magnetic moments are also similar to those for shell theory.