Microscopic, quantum derivation of cranking model for nuclear collective rotation: harmonic oscillator case

Abstract

In this article, the conventional semiclassical one-dimensional cranking model (CR), which is commonly used to investigate rotational structures of deformed nuclei, is derived from microscopic, quantum first principles for the harmonic oscillator case. The space-fixed particle coordinates are canonically transformed to an Euler angle and a set of 3N – 1 intrinsic coordinates to decompose the nuclear Hamiltonian into intrinsic and collective rotational components plus a Coriolis-centrifugal term that couples the intrinsic and rotational motions. To overcome the difficulties associated with finding an appropriate set of intrinsic coordinates, the rotational component in the transformed Hamiltonian is expressed in terms of the space-fixed coordinates and momenta by taking the commutator of the original Hamiltonian with the Euler angle, and by choosing an explicit expression for the Euler angle in terms of space-fixed particle coordinates. The intrinsic component in the transformed Hamiltonian is then the difference between the original Hamiltonian and the rotational component. The nuclear wave function is chosen as the product of an intrinsic function and an eigenfunction of the angular momentum operator (as in the unified rotational model). The Hamiltonian and Schrodinger equation for the intrinsic system become functions of the angular-momentum quantum number and intrinsic operators that are expressed in terms of space-fixed particles coordinates and momenta. The intrinsic Schrodinger equation is then reduced to that of a one-body operator using Hartree–Fock mean-field approximation. The intrinsic mean-field Hamiltonian is chosen to be an anisotropic harmonic oscillator Hamiltonian, and the Hartree–Fock mean-field equation is unitarily transformed to an equation resembling that of the CR but with oscillator frequencies and angular velocity that are microscopically and quantum mechanically determined. The unitary transformation is selected such that the model predicts the kinematic rigid-body moment of inertia, as does the CR when self-consistency condition is used.PACS Nos.: 21.60.Ev, 21.60.Fw, 21.60.Jz