Decomposition of angular momentum projected nuclear wave function

TL;DR

The work develops a general identity that expands the full angular-momentum projection operator into a sum over coupled neutron and proton projections, enabling a decomposition of conventional projected nuclear states into coupled bases. This decomposition yields the coefficients $C(J_,J_ u)$ that quantify the contribution of each $(J_,J_ u)$ channel, and is applicable to arbitrary reference states beyond axial or even-even systems. Using VAPSM in sd-shell nuclei, the authors show that ground states can contain nonzero $(J_,J_ u)$ components due to neutron-proton interactions, and that the distribution of these components aligns closely with full shell-model results. They further demonstrate an improvement by reexpressing the wave function in coupled bases and diagonalizing the Hamiltonian in this expanded space, with larger gains for odd-mass and odd-odd nuclei, pointing to enhanced descriptions of nuclear structure and potential applications to heavy deformed systems and scissors-mode phenomena.

Abstract

Angular momentum projection is a basic technique in constructing nuclear wave functions with good spins. Traditionally, a projected nuclear wave function is expressed in terms of the bases built by performing the angular momentum projection directly on reference states for the whole nuclear system. Alternatively, one can construct nuclear wave function with another kind of projected bases, called as the coupled projected bases, which are generated by first performing the angular momentum projections on the reference states for neutrons and protons, respectively, then coupling the neutron projected states with the proton ones via Clebsch-Gordon coefficients. In the present work, we derive a new identity, which provides a decomposition of the conventional angular momentum projected nuclear wave function in terms of the coupled projected bases. This decomposition offers direct insight into the underlying structure of nuclear states. To show this point, we present the decompositions of variation after projection shell model (VAPSM) wave functions for the ground states in some $sd$ shell nuclei. It is interesting to see that even for the ground states in even-even nuclei, the nucleons are not fully paired. Finally, we demonstrate that the VAPSM wave function can be further improved by adopting the coupled projected bases.

Figures (6)

• Figure 1: Calculated $C(J_\pi,J_\nu)$ values for the ground states in even-even $sd$ shell nuclei by the shell model and by the VAPSM. The USDB interaction is adopted.
• Figure 2: Distribution of the calculated $C(J_\pi,J_\nu)$ values for the yrast $2^+$ state in $^{24}$Mg by the VAPSM (upper panel) and by the shell model (lower panel). The USDB interaction is adopted.
• Figure 3: The same as Fig. \ref{['mg24']} but for the ground $5/2^+$ state in $^{25}$Mg.
• Figure 4: The same as Fig. \ref{['mg24']} but for the ground $5^+$ state in $^{26}$Al.
• Figure 5: Distribution of the calculated $C(J_\pi,J_\nu)$ values for the yrast $0^+$ state in $^{26}$Al by the shell model and the VAPSM. The USDB interaction is adopted.