Ab initio estimation of $E2$ strengths in ${}^8\mathrm{Li}$ and its neighbors by normalization to the measured quadrupole moment

For electric quadrupole ($E2$) observables, which depend on the large-distance tails of the nuclear wave function, ab initio no-core configuration interaction calculations converge slowly, making meaningful predictions challenging to obtain. Nonetheless, the calculated values for different $E2$ matrix elements, particularly those involving levels with closely related structure (e.g., within the same rotational band) are found to be robustly proportional. This observation suggests that a known value for one observable may be used to determine the overall scale of $E2$ strengths, and thereby provide predictions for others. In particular, we demonstrate that meaningful predictions for $E2$ transitions may be obtained by calibration to the ground-state quadrupole moment. We test this approach for well-measured low-lying $E2$ transitions in ${}^7\mathrm{Li}$ and ${}^9\mathrm{Be}$, then provide predictions for transitions in ${}^8\mathrm{Li}$ and ${}^9\mathrm{Li}$. In particular, we address the $2^{+}\to 1^{+}$ transition in ${}^8\mathrm{Li}$, for which the reported measured strength exceeds ab initio Green's function Monte Carlo predictions by over an order of magnitude.

Figure 1

Nuclides with measured ground-state quadrupole moments [19] (indicated with the letter “$Q$”) in the $p$ shell. Particle-bound nuclides are designated by name, while brackets indicate a particle-unbound but narrow ($\lesssim 1\mathrm{keV}$) ground-state resonance, and shading indicates stable nuclides. The ground-state angular momentum and parity are given [20, 21, 22, 23, 24] (upper right), while slashes serve to exclude those nuclei (with $J\le 1/2$) for which the ground-state angular momentum does not support a quadrupole moment. The nuclide ${}^8\mathrm{Li}$ and its neighbors considered in this work are highlighted (dashed circles). Figure adapted from Ref. [18].

Figure 2

Convergence of ab initio NCCI calculated observables for ${}^7\mathrm{Li}$: (top) the $3/2^{-}\to 1/2^{-} E2$ strength, (middle) the electric quadrupole moment of the $3/2^{-}$ ground state, and (bottom) the dimensionless ratio $B\left(E2\right)/{\left(eQ\right)}^2$ constructed from the preceding two observables. Results are shown for the (left) Daejeon16, (center) JISP16, and (right) LENPIC interactions. When calibrated to the experimental quadrupole moment, the ratio provides a prediction for the absolute $B\left(E2\right)$ (scale at right). Calculated values are shown as functions of the basis parameter $\hslash \omega$, for successive even values of $N_{\text{max}}$ (increasing symbol size and longer dashing), from $N_{\text{max}}=4$ (short dashed curves) to 16 (solid curves). For comparison, experimental values [19, 20] (squares), $\mathrm{GFMC}\mathrm{AV}18+\mathrm{IL}7$ predictions [28] (crosses), and the rotational ratio (asterisk) are also shown.

Figure 3

Calculated excitation energies of low-lying states in (a) ${}^7\mathrm{Li}$ and (b) ${}^9\mathrm{Be}$, with angular momentum and parity as indicated at bottom, as obtained with the Daejeon16 interaction. Calculated values are shown at fixed $\hslash \omega =20\mathrm{MeV}$ and varying $N_{\text{max}}$ (increasing symbol size), from $N_{\text{max}}=4$ to the maximum value indicated (at top). Experimental energies [20, 21] are shown (horizontal line and error band) where available, as are the $\mathrm{GFMC}\mathrm{AV}18+\mathrm{IL}7$ predictions [28] (crosses) (see Table III of Ref. [37]).

Figure 4

Calculated ratios of the form $B\left(E2\right)/{\left(eQ\right)}^2$, for excitation to low-lying states in ${}^7\mathrm{Li}$ and ${}^9\mathrm{Be}$, obtained with the Daejeon16, JISP16, and LENPIC interactions (from left to right, for each transition). Calculated values are shown at fixed $\hslash \omega =20\mathrm{MeV}$ and varying $N_{\text{max}}$ (increasing symbol size), from $N_{\text{max}}=4$ to the maximum value indicated (at top). When calibrated to the experimental quadrupole moment [19], this ratio provides an estimate for the absolute $B\left(E2\right)$ (scale at right). Experimental results [19, 20, 21] are shown (horizontal line and error band) where available, as are the $\mathrm{GFMC}\mathrm{AV}18+\mathrm{IL}7$ predictions [28] (crosses) and rotational ratios (asterisks).

Figure 5

Convergence of the ab initio NCCI calculated dimensionless ratio $B\left(E2\right)/{\left(eQ\right)}^2$, for ${}^9\mathrm{Be}$, constructed from the $3/2^{-}\to 5/2^{-} E2$ strength and the electric quadrupole moment of the $3/2^{-}$ ground state. Results are shown for the Daejeon16 interaction. When calibrated to the experimental quadrupole moment, the ratio provides a prediction for the absolute $B\left(E2\right)$ (scale at right). Calculated values are shown as functions of the basis parameter $\hslash \omega$, for successive even values of $N_{\text{max}}$ (increasing symbol size and longer dashing), from $N_{\text{max}}=4$ (short dashed curves) to 10 (solid curves). For comparison, the experimental ratio [19, 21] (square), $\mathrm{GFMC}\mathrm{AV}18+\mathrm{IL}7$ prediction [28] (cross), and rotational ratio (asterisk) are also shown.

Figure 6

Convergence of the ab initio NCCI calculated dimensionless ratio $B\left(E2\right)/{\left(eQ\right)}^2$, for ${}^8\mathrm{Li}$, constructed from the $2^{+}\to 1^{+} E2$ strength and the electric quadrupole moment of the $2^{+}$ ground state. Results are shown for the (a) Daejeon16, (b) JISP16, and (c) LENPIC interactions. When calibrated to the experimental quadrupole moment, the ratio provides a prediction for the absolute $B\left(E2\right)$ (scale at right). Calculated values are shown as functions of the basis parameter $\hslash \omega$, for successive even values of $N_{\text{max}}$ (increasing symbol size and longer dashing), from $N_{\text{max}}=4$ (short dashed curves) to 14 (solid curves). For comparison, the $\mathrm{GFMC}\mathrm{AV}18+\mathrm{IL}7$ prediction [28] (cross) is shown, while the experimental ratio [19, 21], corresponding to the reported $E2$ strength of $55\left(15\right)e^2{\mathrm{fm}}^4$ [26], lies off scale.

Figure 7

Calculated excitation energies of low-lying states in (a) ${}^8\mathrm{Li}$ and (b) ${}^9\mathrm{Li}$, with angular momentum and parity as indicated at bottom, as obtained with the Daejeon16 interaction. Calculated values are shown at fixed $\hslash \omega =20\mathrm{MeV}$ and varying $N_{\text{max}}$ (increasing symbol size), from $N_{\text{max}}=4$ to the maximum value indicated (at top). Experimental energies [21] are shown (horizontal line and error band) where available, as are the $\mathrm{GFMC}\mathrm{AV}18+\mathrm{IL}7$ predictions [28] (crosses) (see Table III of Ref. [37]).

Figure 8

Calculated ratios of the form $B\left(E2\right)/{\left(eQ\right)}^2$, for excitation to low-lying states in ${}^8\mathrm{Li}$ and ${}^9\mathrm{Li}$, obtained with the Daejeon16, JISP16, and LENPIC interactions (from left to right, for each transition). Calculated values are shown at fixed $\hslash \omega =20\mathrm{MeV}$ and varying $N_{\text{max}}$ (increasing symbol size), from $N_{\text{max}}=4$ to the maximum value indicated (at top). When calibrated to the experimental quadrupole moment [19], this ratio provides an estimate for the absolute $B\left(E2\right)$ (scale at right). The experimental result [19, 21] for the ${}^8\mathrm{Li} 2^{+}\to 1^{+}$ transition strength lies off scale. The $\mathrm{GFMC}\mathrm{AV}18+\mathrm{IL}7$ predictions [28] (crosses) and rotational ratios (asterisks) are shown.