[Paper Review] A microscopic quantal self-consistent cranking model for oscillations in spherical nuclei

P. Gulshani|arXiv (Cornell University)|Jan 21, 2014

Quantum, superfluid, helium dynamics5 references4 citations

TL;DR

This paper presents a microscopic, quantal, self-consistent cranking model for describing coupled oscillations and intrinsic motion in spherical nuclei. By decomposing the Schrödinger equation into two coupled cranking equations via a product wavefunction and constrained variational method, the model ensures time-reversal invariance and self-consistently determines energy and cranking parameters, yielding real solutions for harmonic oscillator mean-field potentials.

ABSTRACT

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.

Motivation & Objective

• To develop a microscopic, quantal framework for describing coupled oscillations and intrinsic motion in spherical nuclei.
• To address limitations of conventional phenomenological cranking models by enforcing self-consistency and time-reversal invariance.
• To derive a formalism that yields real solutions through self-consistently determined parameters.
• To explore the role of deformation and mean-field approximations in excitation energy predictions.
• To lay the foundation for extending the model to vibrational-rotational motion in deformed nuclei.

Proposed method

• Transform the microscopic rotational-model Schrödinger equation into a form suitable for coupled oscillation-intrinsic motion systems.
• Employ a product wavefunction ansatz to separate oscillation and intrinsic motion degrees of freedom.
• Apply a constrained variational method to derive two coupled cranking-type equations—one for oscillations, one for intrinsic motion.
• Enforce self-consistency such that energy and cranking parameters are determined iteratively by the solutions.
• Ensure time-reversal invariance by restricting solutions to real wavefunctions, eliminating complex parameters.
• Explicitly solve the coupled equations for a harmonic oscillator mean-field potential and derive the parameter-determination equations.

Experimental results

Research questions

• RQ1 How can a microscopic, quantal, and self-consistent framework be constructed for coupled oscillations and intrinsic motion in spherical nuclei?
• RQ2 What are the implications of enforcing time-reversal invariance and self-consistency on the nature of the solutions (e.g., real vs. complex)?
• RQ3 How do the self-consistently determined parameters affect the predicted excitation energies of low-lying states?
• RQ4 To what extent does neglecting deformation in light nuclei contribute to overprediction of excitation energies?
• RQ5 Can this formalism be generalized to describe vibrational-rotational dynamics in deformed nuclei?

Key findings

• The model successfully derives two coupled cranking equations that are self-consistent and time-reversal invariant, yielding only real solutions.
• For the harmonic oscillator mean-field potential, explicit solutions and eigenvalues are obtained, with parameters determined self-consistently via derived equations.
• Preliminary calculations using a sum rule to determine oscillation frequency show significant overprediction of first excited state excitation energies in light nuclei.
• The overprediction is attributed primarily to the neglect of deformation degrees of freedom, especially in light nuclei.
• The model's self-consistency and time-reversal invariance are shown to be compatible with microscopic quantal treatment, avoiding unphysical complex solutions.
• The framework provides a viable path toward modeling vibrational-rotational motion in deformed nuclei through extension of the current formalism.