Variation after projection with a triaxially deformed nuclear mean field

Abstract

We implemented a variation after projection (VAP) algorithm based on a triaxially deformed Hartree-Fock-Bogoliubov vacuum state. This is the first projected mean field study that includes all the quantum numbers (except parity), i.e., spin (J), isospin (T ) and mass number (A). Systematic VAP calculations with JT A-projection have been performed for the even-even sd-shell nuclei with the USDB Hamiltonian. All the VAP ground state energies are within 500 keV above the exact shell model values. Our VAP calculations show that the spin projection has two important effects: (1) the spin projection is crucial in achieving good approximation of the full shell model calculation.(2) the intrinsic shapes of the VAP wavefunctions with spin projection are always triaxial, while the Hartree-Fock-Bogoliubov methods likely provide axial intrinsic shapes. Finally, our analysis suggests that one may not be possible to associate an intrinsic shape to an exact shell model wave function.

Projection of angular momentum via linear algebra

“…Table I shows the low-lying projected Hartree-Fock spectra of 48 Cr computed in the pf shell with GXPF1A [17,18]. We show four calculations, two quadrature calculations with 20 and 40 points, and two linear algebra projection (LAP) calculations, one with a full inversion and one with the 'need-to-know' modification described in section V. The The quadrature calculation with 20 points on each angle, or 8000 evaluations, agrees for small J but breaks down for larger angular momentum; although we don't show it, this breakdown is also signaled by a failure in the sum rule (16).…”

“…Because the inversion becomes sensitive to the choice of angles, further investigation into these improved inversions is suggested. We also leave to future work application to systems beyond shell-model spaces (i.e., coordinate-space mean-field wave functions), to cases with multiple initial states, e.g., generator-coordinator and relative methods, to transitions, and other finite quantum numbers such as isospin and particles number [16].…”

“…We found a tolerance of ǫ ∼ 0.001 worked satisfactorily. One also has to choose a tolerance for satisfying the sum rule (16). This essentially dictates the maximal J used in inversions.…”

“…Projecting out from fully triaxial states, or projecting additional quantum numbers such as isospin or particle number, is so computationally intensive one often has to severely restrict the model space [16]. Given the applications of angular momentum projection and the computational burden, we were motivated to find an alternate approach, not by speeding up the evaluation of the integrands for any set of Euler angles, but rather to reduce the number of mesh points needed.…”

Occupation numbers of spherical orbits in self-consistent beyond-mean-field methods

“…Using these PNAMP states with J = 0, |J ; N,Z; β 2 , and the spherical HFB state computed for 64 Cr as the reference state to define the spherical orbits, the occupancies of these orbits as a function of the axial quadrupole deformation [Eq. (16) combination with Figs. 1(b) and 1(c).…”