Collective Modes in Nuclei

Abstract

The Hamiltonian of the "unified model" of Bohr and Mottelson has been derived from many-particle quantum mechanics. A point transformation in the configuration space of the $N$ nucleons of the core introduces 6 collective coordinates describing the size and shape of the nucleus and $3N\ensuremath{-}6$ internal coordinates. The collective coordinates are essentially the components of the quadrupole moment tensor of the nucleon distribution. It is assumed that the wave function $\ensuremath{\Psi}$ of the nucleus for the ground state and the low lying excited states can be approximated by the product ${\ensuremath{\Psi}}_{0}\ensuremath{\Phi}$ where ${\ensuremath{\Psi}}_{0}$ depends only on the internal coordinates of the core and $\ensuremath{\Phi}$ depends on the collective coordinates of the core and the coordinates of the external nucleons. ${\ensuremath{\Psi}}_{0}\ensuremath{\Phi}$ is treated as a trial function in the Schr\"odinger variational principle. For fixed ${\ensuremath{\Psi}}_{0}$ this yields a Schr\"odinger equation for $\ensuremath{\Phi}$. The Hamiltonian in this equation is the Hamiltonian of Bohr and Mottelson plus certain correction terms. ${\ensuremath{\Psi}}_{0}$ influences the values of the constant parameters occurring in this Hamiltonian but not its structure. In the strong-coupling approximation the relationship between the moments of inertia of the core and the nuclear quadrupole moment is essentially the same as in the liquid drop model. This result is also independent of detailed assumptions about ${\ensuremath{\Psi}}_{0}$.