Robust ab initio predictions for dimensionless ratios of E2 and radius observables. I. Electric quadrupole moments and deformation

TL;DR

This work tackles the slow convergence of ab initio $E2$ observables and charge radii in no-core configuration interaction by exploiting correlations between these two observables to form robust, dimensionless ratios such as $B(E2)/(e^2 r_p^4)$ and $Q/r_p^2$. By focusing on the ratio $Q/r^2$ (and its proton/neutron variants) in the $p$-shell, the authors demonstrate improved convergence and derive microscopic links to quadrupole deformation within an axial rotor framework. The study combines convergence diagnostics, exponential extrapolations, and comparisons with GFMC data across several interactions, illustrating that these ratios provide meaningful deformation information and can predict unmeasured radii or moments. The results lay the groundwork for Part II, which will calibrate $E2$ strengths against charge radii to yield quantitatively reliable ab initio predictions of quadrupole observables across light nuclei.

Abstract

Converged results for E2 observables are notoriously challenging to obtain in ab initio no-core configuration interaction (NCCI) approaches. Matrix elements of the E2 operator are sensitive to the large-distance tails of the nuclear wave function, which converge slowly in an oscillator basis expansion. Similar convergence challenges beset ab initio prediction of the nuclear charge radius. However, we exploit systematic correlations between the calculated E2 and radius observables to yield meaningful predictions for relations among these observables. In particular, we examine ab initio predictions for dimensionless ratios of the form Q/r^2, for nuclei throughout the $p$ shell. Meaningful predictions for electric quadrupole moments may then be made by calibrating to the ground-state charge radius, if experimentally known, or vice versa. Moreover, these dimensionless ratios provide ab initio insight into the nuclear quadrupole deformation.

Figures (11)

• Figure 1: Overview of particle-bound nuclides in the $p$ shell, where those with measured ground-state quadrupole moments stone2016:e2-moments and charge radii angeli2013:charge-radiinpa2017:012 are indicated by the letter "$Q$" or "$R$", respectively. Brackets indicate a particle-unbound but narrow ($\lesssim 1\,{\mathrm{keV}}$) ground-state resonance, shading indicates beta-stable nuclides, and the experimental ground-state angular momentum and parity are given npa2002:005-007npa2004:008-010npa2012:011npa2017:012npa1991:013-015. Nuclei for which the ground-state angular momentum does not support a quadrupole moment ($J\leq1/2$) are crossed out with a diagonal line.
• Figure 2: Calculated ground state observables for $\isotope[9]{Be}$: $E(3/2^-_1)$ (top), $Q(3/2^-_1)$ (upper middle), $r_p(3/2^-_1)$ (lower middle), and the dimensionless ratio $Q/r_p^2$ constructed from the preceding two observables (bottom). Results are shown for the Daejeon16 (left), JISP16 (center), and LENPIC (right) interactions. Calculated values are shown as functions of the basis parameter ${\hbar\omega}$, for successive even values of ${N_\text{max}}$, from ${N_\text{max}}=4$ to $12$ (as labeled). When calibrated to the experimentally deduced value for $r_p$, the ratio provides a prediction for the absolute $Q$ (scale at right). Exponential extrapolations (small circles, dotted lines) are provided, for the Daejeon16 results only (${\hbar\omega}\geq17.5\,{\mathrm{MeV}}$). For comparison, experimental values stone2016:e2-momentsangeli2013:charge-radiinpa2004:008-010 (squares) and GFMC AV18+IL7 predictions pastore2013:qmc-em-alt9 (crosses) are also shown.
• Figure 3: Diagnostics of convergence for ground state observables for $\isotope[9]{Be}$: the relative difference $\Delta_{\mathrm{rel}}$ (left) and ratio of successive differences $\eta$ (right), for $Q(3/2^-_1)$ (top), $r_p(3/2^-_1)$ (middle), and the dimensionless ratio $Q/r_p^2$ (bottom). Calculated values, for the Daejeon16 interaction, are shown as functions of the basis parameter ${\hbar\omega}$, for successive even values of ${N_\text{max}}$, from ${N_\text{max}}=6$ or $8$ (as appropriate, given observables calculated starting with ${N_\text{max}}=4$) to $12$ (as labeled).
• Figure 4: Calculated ratios $Q/r_p^2$, for the ground states of proton-rich (top) and neutron-rich (bottom) nuclides in the $p$ shell, obtained with the Daejeon16, JISP16, and LENPIC interactions (from left to right, within each panel). Calculated values are shown at fixed ${\hbar\omega}=20\,{\mathrm{MeV}}$ and varying ${N_\text{max}}$ (increasing tick size), from ${N_\text{max}}=4$ to the maximum value indicated (at top). For comparison, the experimental ratios stone2016:e2-momentsangeli2013:charge-radii are shown (horizontal line and error band, where the signs of some quadrupole moments are experimentally undetermined), as are the GFMC AV18+IL7 predictions pastore2013:qmc-em-alt9pieper:cited (crosses).
• Figure 5: Calculated ratios $Q/r_p^2$, for the ground states of $N=Z$ nuclides in the $p$ shell, obtained with the Daejeon16, JISP16, and LENPIC interactions (from left to right, within each panel). Calculated values are shown at fixed ${\hbar\omega}=20\,{\mathrm{MeV}}$ and varying ${N_\text{max}}$ (increasing tick size), from ${N_\text{max}}=4$ to the maximum value indicated (at top). For comparison, the experimental ratios stone2016:e2-momentsangeli2013:charge-radii are shown (horizontal line and error band, where the signs of some quadrupole moments are experimentally undetermined), as are the GFMC AV18+IL7 predictions pastore2013:qmc-em-alt9 (crosses).