Nucleon localization function in rotating nuclei

Abstract

Background: An electron localization function was originally introduced to visualize in positional space bond structures in molecules. It became a useful tool to describe electron configurations in atoms, molecules, and solids. In nuclear physics, a nucleon localization function (NLF) has been used to characterize cluster structures in light nuclei, formation of fragments in fission, and pasta phases appearing in the inner crust of neutron stars. Purpose: We use the NLF to study the nuclear response to fast rotation. Methods: We generalize the NLF to the case of nuclear rotation. The extended expressions involve both timeeven and time-odd local particle and spin densities and currents. Since the current density and density gradient contribute to the NLF primarily at the surface, we propose a simpler spatial measure given by the kinetic-energy density. Illustrative calculations for the superdeformed yrast band of 152 Dy were carried out by using the cranked Skyrme-Hartree-Fock method. We also employed the cranked harmonic-oscillator model to gain insights into spatial patterns revealed by the NLF at high angular momentum. Results: In the case of a deformed rotating nucleus, several NLFs can be introduced, depending on the definition of the spin-quantization axis, direction of the total angular momentum, and self-consistent symmetries of the system. Contributions to the NLF from the current density, spin-current tensor density, and density gradient terms are negligible in the nuclear interior. The oscillating pattern of the simplified NLF can be explained in terms of a constructive interference between kinetic-energy and particle densities. The characteristic nodal pattern seen in the NLF in the direction of major axis of a rotating nucleus comes from single-particle orbits carrying large aligned angular momentum. The variation of the NLF along the minor axis of the nucleus can be traced back to deformation-aligned orbits. Conclusions: The NLF allows a simple interpretation of the shell structure evolution in the rotating nucleus in terms of the angular-momentum alignment of individual nucleons. We expect that the NLF will be very useful for the characterization and visualization of other collective modes in nuclei and time-dependent processes.

Pauli energy contribution to nucleus-nucleus interaction

“…The measure of localization has been originally developed in the context of a mean-field description for electronic systems [53][54][55], and subsequently introduced to nuclear systems [57,65]. We first realize that a fermionic mean-field state is fully characterized by the one-body density-matrix ρ q (rs, r s ).…”

“…The short-range behavior of R qs can be obtained using techniques similar to the local density approximation [57,65]. The leading term in the expansion yields the localization measure…”

“…This measure is the most general form that is appropriate for deformed nuclei and without assuming time-reversal invariance, thus also including the time-odd terms important in applications such as cranking or TDHF. The densities and currents are given in their most unrestricted form [88][89][90] for µaxis denoting the spin-quantization axis by [65]…”

“…The explicit expressions of the local densities and currents are given in Refs. [65,88]. We note that the localization measure includes the spin-density s qµ (r), the time-odd part of the kinetic density T qµ (r), as well as the full spin-orbit tensor J q (r), which is a pseudotensor.…”

“…The same idea was introduced to nuclear physics to bring out the clustering features of the nuclear densities obtained in mean-field calculations [57] and employed to bring out the cluster features of light nuclei [57,58], study of reactions [59], and shell effects in fission [60][61][62][63][64]. In a recent work, the NLF function was generalized to include all of the time-odd and spin dependent terms of the Skyrme energy density functional [65]. As a byproduct of these developments we also have an accurate way to calculate the Pauli energy in a similar way it is employed in DFT calculations [4,5] for electronic systems while including all the nuclear energy density functional contributions.…”