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In this article, we transform the previously-derived microscopicrotational-model Schrodinger equation into a form suitable for describingoscillations-coupled-to-intrinsic motion in spherical nuclei. The resultingequation is decomposed into two coupled cranking-type equations, one for theoscillation and another for the intrinsic motion, using a product wavefunctionand a constrained variational method. The energy and cranking parameters in thecoupled equations are self-consistently determined as functions of the systemparameters by the solutions of the equations themselves. This self-consistencymakes the two equations time-reversal invariant, unlike the conventionalphenomenological cranking models. The self-consistency and time-reversalinvariance accept only real solutions to the equations. For the harmonicoscillator mean-field potential, we explicitly determine these solutions andthe corresponding eigenvalues, and derive the set of equations that determineself-consistently the parameters. To explore the relative importance of thevarious model features and approximations, we perform a preliminary scopingcalculation of the excitation energy of the first excited states in the lightnuclei using a sum rule to determine the oscillation frequency. The preliminaryresults indicate that, except in the lightest nuclei, the excitation energiesare significantly overpredicted in the light nuclei due to the neglect, amongother factors, of the deformation degree of freedom. The model derivationpresented here serves as guide for eventually developing a corresponding modelfor the vibrational-rotational motion in deformed nuclei.

A microscopic quantal self-consistent cranking model for oscillations in spherical nuclei