A microscopic quantal self-consistent cranking model for oscillations in spherical nuclei
In this article, we transform the previously-derived microscopic rotational-model Schrodinger equation into a form suitable for describing oscillations-coupled-to-intrinsic motion in spherical nuclei. The resulting equation is decomposed into two coupled cranking-type equations, one for the oscillation and another for the intrinsic motion, using a product wavefunction and a constrained variational method. The energy and cranking parameters in the coupled equations are self-consistently determined as functions of the system parameters by the solutions of the equations themselves. This self-consistency makes the two equations time-reversal invariant, unlike the conventional phenomenological cranking models. The self-consistency and time-reversal invariance accept only real solutions to the equations. For the harmonic oscillator mean-field potential, we explicitly determine these solutions and the corresponding eigenvalues, and derive the set of equations that determine self-consistently the parameters. To explore the relative importance of the various model features and approximations, we perform a preliminary scoping calculation of the excitation energy of the first excited states in the light nuclei using a sum rule to determine the oscillation frequency. The preliminary results indicate that, except in the lightest nuclei, the excitation energies are significantly overpredicted in the light nuclei due to the neglect, among other factors, of the deformation degree of freedom. The model derivation presented here serves as guide for eventually developing a corresponding model for the vibrational-rotational motion in deformed nuclei.