Lamb shifts and hyperfine structure in ${}_{}{}^6{}_{}{}^{}\mathrm{Li}_{}^{+}{}_{}{}^{}$ and ${}_{}{}^7{}_{}{}^{}\mathrm{Li}_{}^{+}{}_{}{}^{}$: Theory and experiment

High-precision laser-resonance measurements accurate to ±0.5 MHz or better are reported for transitions among the 1s2s ${}_{}{}^3{}_{}{}^{}$${\mathit{S}}_{1–}$1s2p ${}_{}{}^3{}_{}{}^{}$${\mathit{P}}_{\mathit{J}}$ hyperfine manifolds for each of J=0, 1, and 2 in both ${}_{}{}^6{}_{}{}^{}\mathrm{Li}_{}^{+}{}_{}{}^{}$ and ${}_{}{}^7{}_{}{}^{}\mathrm{Li}_{}^{+}{}_{}{}^{}$. A detailed analysis of hyperfine structure is performed for both the S and P states, using newly calculated values for the magnetic dipole and electric quadrupole coupling constants, and the hyperfine shifts subtracted from the measurements. The resulting transition frequencies are then analyzed on three different levels. First, the isotope shifts in the fine-structure splittings are calculated from the relativistic reduced mass and recoil terms in the Breit interaction, and compared with experiment at the ±0.5-MHz level of accuracy. This comparison is particularly significant because J-independent theoretical uncertainties reduce through cancellation to the ±0.01-MHz level. Second, the isotope shifts in the full transition frequencies are used to deduce the difference in rms nuclear radii.

The result is ${\mathit{R}}_{\mathrm{rms}}$${(}^6$Li)-${\mathit{R}}_{\mathrm{rms}}$${(}^7$Li)=0.15±0.01 fm, in agreement with nuclear scattering data, but with substantially improved accuracy. Third, high-precision calculations of the low-order non-QED contributions to the transition frequencies are subtracted from the measurements to obtain the residual QED shifts. The isotope-averaged and spin-averaged effective shift for ${}_{}{}^7{}_{}{}^{}\mathrm{Li}_{}^{+}{}_{}{}^{}$ is 37 429.40±0.39 MHz, with an additional uncertainty of ±1.5 MHz due to finite nuclear size corrections. The accuracy of 11 parts per million is the best two-electron Lamb shift measurement in the literature, and is comparable to the accuracies achieved in hydrogen. Theoretical contributions to the two-electron Lamb shift are discussed, including terms of order (αZ${)}^4$ recently obtained by Chen, Cheng, and Johnson [Phys. Rev. A 47, 3692 (1993)], and the results used to extract a QED shift for the 2 ${}_{}{}^3{}_{}{}^{}$${\mathit{S}}_1$ state. The result of 30 254±12 MHz is shown to be in good accord with theory (30 250±30 MHz) when two-electron corrections to the Bethe logarithm are taken into account by a 1/Z expansion method.