Nuclear Vibrations as a Constrained Hartree-Fock Problem

The constrained Hartree-Fock equations and the adiabatic approximation are combined in a model for describing collective vibrations of deformed nuclei. The $\beta$ vibrations of the Hartree-Fock ground-state determinant are affected by imposing the constraint on the quadropole single-particle operator $Q_{20}$. The geometrical parameter $q$ of the vibrations is thus identified with the expectation value of $Q_{20}$ in a state $\phi \mathrm{}\left(q\right)\mathrm{}$ given by the self-consistency determinant. The energy surface, force constant, and mass parameter are directly related to the intrinsic structure of the nucleus. A Hill-Wheeler representation for the vibrational states is used to extract the one-phonon intrinsic wave function out of the ground-state determinant. The model is tested numerically by applying it to ${\mathrm{Ne}}^{20}$ and identifying the first excited $O^{+}$ with the one-phonon vibrational state. Results of the present adiabatic model are compared explicitly with those of the alternative particle-hole model. The comparison between the two models shows that when adiabaticity conditions exist in the self-consistent field, both models are equally capable of giving a fair description of the vibration of deformed nuclei.