The nuclear charge radius of 13C

Abstract

The size is a key property of a nucleus. Accurate nuclear radii are extracted from elastic electron scattering, laser spectroscopy, and muonic atom spectroscopy. The results are not always compatible, as the proton-radius puzzle has shown most dramatically. Beyond helium, precision data from muonic and electronic sources are scarce in the light-mass region. The stable isotopes of carbon are an exception. We present a laser spectroscopic measurement of the root-mean-square (rms) charge radius of ¹³C and compare this with ab initio nuclear structure calculations. Measuring all hyperfine components of the 2 ³S $\to$ 2 ³P fine-structure triplet in ¹³C⁴⁺ ions referenced to a frequency comb allows us to determine its center-of-gravity with accuracy better than 2 MHz although second-order hyperfine-structure effects shift individual lines by several GHz. We improved the uncertainty of R_c(¹³C) determined with electrons by a factor of 6 and found a 3σ discrepancy with the muonic atom result of similar accuracy.

Fig. 1. Experimental setup.

Sketch of the measurement principle including the electron-beam ion source (EBIS), the electrostatic switchyard, the beam alignment irises, the laser system, and the fluorescence detection region (FDR). The laser system consists of two Millennia pump lasers (Nd:YVO₄) that drive two Ti:Sapphire lasers, each followed by two frequency doublers, one operated with lithium-triborate (LBO) and second one with barium betaborate (BBO). The lasers are locked to a wavemeter and a frequency comb. The  227-nm light is then transported through air to the COALA beamline. The two charge-breeding processes, electron-impact ionization and electron capture, are shown in the inset above the EBIS potential.

Fig. 2. Atomic spectroscopy data of ^12,13C⁴⁺.

a Level scheme of ^12,13C⁴⁺ and the electric dipole transitions that are addressed by the laser. b Hyperfine-structure (HFS) spectrum of the 1s2s ³S₁ $\to$ 1s2p ³P_0,1,2 transitions in ¹³C⁴⁺ simulated with the experimentally determined frequencies and linewidths. The x-axis represents the laser frequency in the rest-frame of the ion ν relative to the center-of-gravity frequency of the respective fine-structure transition. The peak heights were set to the theoretical transition strengths used in Eq. (7). The two insets show measured spectra of the marked transitions. Next to the resonances, the contributing quantum numbers $F\to F^{\prime }$ of the lower and upper states are shown, respectively.

Fig. 3. Absolute and differential nuclear charge radii of ^12,13C.

Experimentally determined absolute $R_{\mathrm{c}}^{12}$, $R_{\mathrm{c}}^{13}$ and nuclear charge radius difference $\delta R_{\mathrm{c}}^{12,13}=R_{\mathrm{c}}^{13}-R_{\mathrm{c}}^{12}$ of ^12,13C determined with elastic electron scattering (blue), muonic atom spectroscopy (μ-atoms, purple) and collinear laser spectroscopy (CLS, black). Results from CLS and e⁻-scattering were combined to obtain an improved $R_{\mathrm{c}}^{13}$ (black & blue) purely from electronic measurements. The differential rms nuclear charge radii from e⁻-scattering and μ-atoms are differences of absolute radii while the CLS result is determined directly from the isotope shift and ab initio atomic structure calculations using Eq. (2). $\delta R_{\mathrm{c}}^{12,13}$ is also compared to ab initio valence-space in-medium similarity renormalization group (VS-IMSRG, red) and in-medium no-core shell-model (IM-NSCM, orange) calculations. The lower-order IM-NSCM results are plotted with open symbols. Results from nuclear-lattice effective field theory (NLEFT, brown) were published by Elhatisari et al.. The numerical values of this plot are listed in Table 2.

Fig. 4. Theoretical absolute nuclear charge radii of ^12,13C.

The radii $R_{\mathrm{c}}^{12,13}$ have been determined from ab initio valence-space in-medium similarity renormalization group (VS-IMSRG, red) and in-medium no-core shell model (IM-NSCM, orange) calculations. The lower-order IM-NSCM results are plotted with open symbols. Results from nuclear-lattice effective field theory (NLEFT, brown) were published by Elhatisari et al.. The numerical values of this plot are listed in Table 2.