Devices and methods for the acquisition of mass spectra with very high mass resolution in ion cyclotron resonance mass spectrometers include cylindrical ICR measuring cells with special electrode geometries to generate harmonic trapping potentials for orbiting ions. The sheath of the cylindrical cell is divided by longitudinal gaps into a multitude of sheath electrodes, which either have to carry layers with resistance profiles able to generate parabolic voltage profiles along the sheath electrodes, or which form sheath electrodes of varying width by parabolic gaps. Orbiting ions of a given mass m/z oscillate harmonically in an axial direction with the same frequency, independent of the radius of their orbit and their oscillation amplitude. Ideally, the cylinders are closed by endcaps with rotationally hyperbolic form, divided into partial electrodes. The ions are excited by dipolar excitation fields. The orbiting ion clouds are kept together for much longer periods than was possible hitherto.