Laser spectroscopy measurement of the 2s-hyperfinesplitting 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 Centrefor Heavy Ion Research (GmbH) in Darmstadt, Germany. Combined with a newmeasurement of the 1s hyperfine splitting in hydrogen-like (209Bi82+) the so-called specific difference Δ’E = -61.37(36) meV could be determined and was found to bein good agreement with its prediction from strong-field bound-state QED. Here wereport 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 of the experimental Δ’E with the theoretical result, or between the two measurements of the hyperfine splitting in hydrogen-like bismuth, but not both at the same time.