Precision measurement of the $E2$ transition strength to the $2_1^{+}$ state of ${}^{12}\mathrm{C}$

The form factor of the electromagnetic excitation of ${}^{12}\mathrm{C}$ to its $2_1^{+}$ state was measured at extremely low momentum transfers in an electron-scattering experiment at the Superconducting Darmstadt Electron Linear Accelerator (S-DALINAC). A combined analysis with the world form-factor data results in a reduced transition strength $B(E2;2_1^{+}\to 0_1^{+})=7.63\left(19\right)e^2{\mathrm{fm}}^4$ with an accuracy improved to 2.5%. In-medium no-core shell-model results with interactions derived from chiral effective field theory are able to reproduce the result. A quadrupole moment $Q\left(2_1^{+}\right)=5.97\left(30\right)e{\mathrm{fm}}^2$ can be extracted from the strict correlation with the $B\left(E2\right)$ strength emerging in the calculations.

Figure 1

Elastic electron-scattering spectrum taken at a beam energy of 42.5 MeV and a scattering angle of ${69}^{\circ }$. The inset shows the excitation of the $2_1^{+}$ state. The light gray (red) lines display a fit using Eq. (1) and the dark gray (blue) lines a linear background.

Figure 2

Area/charge ratios for slices of the elastic line measurements at ${69}^{\circ }$ (circles) and uncertainty-weighted average band. The inset shows corresponding values for the excitation of the $2_1^{+}$ state.

Figure 3

Experimental form factor of the transition to the $2_1^{+}$ state in ${}^{12}\mathrm{C}$ after the DWBA corrections described in the text. Data from the present work are shown as (blue) squares and previous measurements [35, 36, 37, 38] as (green) circles. Most of the error bars are smaller than the displayed data points. The (red) band shows a fit of Eq. (3) with a $1\sigma$ uncertainty. The arrow indicates the photon point. The inset shows the scattering of the residuals around the fit.

Figure 4

Excitation spectrum (a), $B\left(E2\right)$ transition strength (b), and quadrupole moment (c) for excited states of ${}^{12}\mathrm{C}$ obtained in the IM-NCSM for different reference-space truncations $N_{\max }^{\mathrm{ref}}$ (panels left to right) as function of $N_{\text{max}}$. All calculations are performed with the chiral two- plus three-body interaction at ${\mathrm{N}}^3\mathrm{LO}$ with cutoff $\mathrm{\Lambda }=500\mathrm{MeV}/c$. The error bars indicate the many-body uncertainties (see text).

Figure 5

Excitation spectrum (a), $B\left(E2\right)$ transition strength (b), and quadrupole moment (c) for excited states of ${}^{12}\mathrm{C}$ obtained in the IM-NCSM for different reference-space truncations $N_{\max }^{\mathrm{ref}}$ (panels left to right) with interactions from NLO to ${\mathrm{N}}^3\mathrm{LO}$ with cutoff $\mathrm{\Lambda }=500\mathrm{MeV}/c$. The error bars represent many-body uncertainties and the shaded bars indicate the interaction uncertainties (see text).

Figure 6

Correlation of the quadrupole observables $B(E2,2_1^{+}\to 0_1^{+})$ and $Q\left(2_1^{+}\right)$ for ${}^{12}\mathrm{C}$ obtained with ${\mathrm{N}}^2\mathrm{LO}$ (open symbols) and ${\mathrm{N}}^3\mathrm{LO}$ (full symbols) interactions for three different cutoffs $450\mathrm{MeV}/c$ (squares), $500\mathrm{MeV}/c$ (circles), and $550\mathrm{MeV}/c$ (diamonds). All IM-NCSM calculations are performed with $N_{\text{max}}=N_{\text{max}}^{\text{ref}}=4$. The error bars indicate the combined many-body and interaction uncertainties. The lines show the prediction of a simple rigid rotor (dashed) and a fitted (solid) rotor model, see text. The horizontal and vertical shaded bands indicate the experimental $B\left(E2\right)$ value and the $Q(2_1^{+}$) value derived from the intersection with the model correlation. The light gray and dark gray (red) areas indicate the experimental limits from literature values [25, 40] and from the present work, respectively.