Robust ab initio predictions for dimensionless ratios of E2 and radius observables. II. Estimation of E2 transition strengths by calibration to the charge radius

TL;DR

This study tackles the slow convergence of ab initio $E2$ transition strengths in NCCI by exploiting robust correlations with the ground-state radius to form dimensionless ratios such as $B(E2)/(eQ)^2$ and $B(E2)/(e^2 r_p^4)$. By calibrating these ratios to known ground-state properties (when available), the authors derive meaningful predictions for $E2$ strengths across selected $p$-shell nuclei, including $^{9}$Be, $^{10}$Be, $^{12}$C, $^{11}$Be, $^{13}$C, and $^{13}$N, using several interactions (Daejeon16, JISP16, LENPIC). They find that $B(E2)$ calibrated to $r_p$ can yield rough but informative estimates, particularly for transitions where the ground state quadrupole moment is unavailable, and that the ratio $B(E2)/(e^2 r_p^4)$ provides a window into intrinsic deformation under an axial-rotation framework, albeit with varying convergence depending on the nucleus and interaction. The work also discusses the limitations imposed by intruder-state mixing and non-axial effects, and demonstrates how deformation parameters $eta_p$ and $eta_n$ can be inferred from these ratios, revealing mirror-symmetric patterns and areas where neutron deformations are less reliably converged.

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 B(E2)/(e^2r^4), for nuclei throughout the p shell. Meaningful predictions for E2 transition strengths may then be made by calibrating to the ground-state charge radius, if experimentally known.

Figures (11)

• Figure 1: Overview of particle-bound nuclides in the $p$ shell, highlighting nuclides for which $E2$ strengths are considered, in relation to the radius, in this work: the nuclide $\isotope[9]{Be}$ (dashed circle), for which the ground-state angular momentum does also support a quadrupole moment, and the remaining nuclides (solid circle), for which it does not. Nuclides 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, while a measured excited-state quadrupole moment stone2016:e2-moments is indicated by "$Q^*$". 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 transition observables for $\isotope[9]{Be}$: $E_x(5/2^-)$, $B(E2;5/2^-_1\rightarrow 3/2^-_1)$, the dimensionless ratio $B(E2)/(eQ)^2$, and the dimensionless ratio $B(E2)/(e^2r_p^4)$ (from top to 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 $Q$ or $r_p$, the ratio provides a prediction for the absolute $B(E2)$ (scale at right). Exponential extrapolations (small circles, dotted lines) are provided, for selected observables, for the Daejeon16 results only (${\hbar\omega}\geq17.5\,{\mathrm{MeV}}$). For comparison, experimental values stone2016:e2-momentsangeli2013:charge-radiinpa2004:008-010 (squares), GFMC AV18+IL7 predictions pastore2013:qmc-em-alt9 (see also Table III of Ref. carlson2015:qmc-nuclear for $E_x$) (crosses), and the rotational $E2$ ratio (asterisk) are also shown. Includes results for $B(E2)/(eQ)^2$ previously shown (for ${N_\text{max}}\leq10$) in Ref. caprio2022:8li-trans.
• Figure 3: Calculated ground state observables for $\isotope[9]{Be}$: (a) $Q(3/2^-_1)$, (b) $r_p(3/2^-_1)$, and (c) the dimensionless ratio $Q/r_p^2$. Results are shown for the Daejeon16 interaction. 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 (${\hbar\omega}\geq17.5\,{\mathrm{MeV}}$). For comparison, experimental values stone2016:e2-momentsangeli2013:charge-radii (squares) and GFMC AV18+IL7 predictions pastore2013:qmc-em-alt9 (crosses) are also shown. Results reproduced from Fig. \ref{['part1:fig:q-norm-rp-scan-9be']} of Part I.
• Figure 4: Diagnostics of convergence for transition observables for $\isotope[9]{Be}$: the relative difference $\Delta_{\mathrm{rel}}$ (left) and ratio of successive differences $\eta$ (right), for $B(E2;5/2^-_1\rightarrow 3/2^-_1)$ (top) and the dimensionless ratio $B(E2)/(e^2r_p^4)$ (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 from ${N_\text{max}}=6$ or $8$ (as appropriate, given observables calculated starting with ${N_\text{max}}=4$) to $12$ (as labeled).
• Figure 5: Calculated ratios $B(E2)/(e^2r_p^4)$, for transitions involving the ground states of nuclides in the $p$ shell, considering cases in which the ground-state angular momentum does not support a quadrupole moment, namely, $\isotope[10]{Be}$, $\isotope[10]{C}$, $\isotope[11]{Be}$, $\isotope[12]{C}$, $\isotope[13]{C}$, and $\isotope[13]{N}$ (left to right). Results are obtained with the Daejeon16, JISP16, and LENPIC interactions (from left to right, within each panel). Exponential extrapolations for $B(E2)/(e^2r_p^4)$ (small circles, dotted lines) are provided, for the Daejeon16 results only (plotted with ${N_\text{max}}$ increasing from left to right). For comparison, the experimental ratios angeli2013:charge-radiipritychenko2016:e2-systematicsnpa1991:013-015 are shown (horizontal line and error band), as are the GFMC AV18+IL7 predictions mccutchan2012:10c-dsamcarlson2015:qmc-nuclear for $A=10$ (crosses). For $\isotope[12]{C}$, the ratio deduced from the more recent $B(E2)$ measurement of D'Alessio et al.dalessio2020:12c-escatt-be2 is also included (horizontal line and error band, shown as narrow open box).