Laser spectroscopy measurement of the 2s-hyperfine splitting in lithium-like bismuth

We have recently reported on the first direct measurement of the $2s$ hyperfine transition in lithium-like bismuth (209Bi80+) at the GSI Helmholtz Centre for Heavy Ion Research in Darmstadt, Germany. Combined with a new measurement of the $1s$ hyperfine splitting (HFS) in hydrogen-like (209Bi82+) the so-called specific difference ${\rm{\Delta }}^{\prime} E=-61.37(36)$ meV could be determined and was found to be in good agreement with its prediction from strong-field bound-state quantum electrodynamics. Here we report on additional investigations performed to estimate systematic uncertainties of these results and on details of the experimental setup. We show that the dominating uncertainty arises from insufficient knowledge of the ion beam velocity which is determined by the electron-cooler voltage. Two routes to obtain a cooler-voltage calibration are discussed and it is shown that agreement can be reached either between the experimental ${\rm{\Delta }}^{\prime} E$ and the theoretical result, or between the two measurements of the HFS in hydrogen-like bismuth, but not both at the same time