Frequency-comb referenced collinear laser spectroscopy of Be+ for nuclear structure investigations and many-body QED tests

Transition frequencies of the \( 2{s}^2{\mathrm{S}}_{1/2}\to 2{p}^2{\mathrm{P}}_{1/2,3/2} \) transitions in Be\( {}^{+} \) were measured in stable and short-lived isotopes at ISOLDE (CERN) using collinear laser spectroscopy and frequency-comb-referenced dye lasers. Quasi-simultaneous measurements in copropagating and counterpropagating geometry were performed to become independent from acceleration voltage determinations for Doppler-shift corrections of the fast ion beam. Isotope shifts and fine-structure splittings were obtained from the transition frequencies measured with a frequency comb with accuracies better than 1 MHz and led to a precise determination of the nuclear charge radii of \( {}^{7,10-12} \)Be relative to the stable isotope 9Be. Moreover, an accurate determination of the 2p fine-structure splitting allowed a test of high-precision bound-state QED calculations in the three-electron system. Here, we describe the laser spectroscopic method in detail, including several tests that were carried out to determine or estimate systematic uncertainties. Final values from two experimental runs at ISOLDE are presented, and the results are discussed.