Time-dependent Hartree-Fock equations and rotational states of nuclei

Abstract

The coneection between translational and rotational degeneracies in the solutions of the time independent Hartree-Fock equations and the existence of solutions of the time-dependent equations which represent uniform translational or rotational motions is investigated. With a non-spherical equilibrium shape, the equations give rotating solutions in which the self-consistent rotating potentials are determined by the motion of the particles. With no axial symmetry, and neglecting higher order effects in the angular velocities, there are time-dependent solutions corresponding to the general motion of a classical top. For an equilibrium shape with axial symmetry, the only rotational motions are uniform rotations about an axis perpendicular to the symmetry axis. The equations are studied in the density matrix formulation in which the results can be immediately extended to include the effect of pairing correlations. Solutions representing a uniform translational motion can be explicitly constructed. For the rotational motion, the changes in the density matrix are determined for small angular velocities by a linear equation. This contains a correction to the cranking model equation which expresses the effect of a transformation to a moving frame of reference on the velocity-dependent self-consistent potentials. The connection between vibrational and rotational solutions of the time-dependent Hartree-Fock equations is discussed. The rotational models are related to zero-frequency solutions of the equations for small oscillations, and in the linear space of small variations the infinitesimal rotational and translational changes define an independent subspace. This leads to a separation of the rotational and translational energies in the Hamiltonian of the random phase approximation.