The true cyclotron frequency for particles and ions in a Penning trap

The true cyclotron frequency of a particle or ion, needed for mass spectrometry and other accurate measurements in a Penning trap, cannot be measured directly. It is not one of the oscillation frequencies of the trapped particle, and the three oscillation frequencies that can be measured vary with the misalignment and the harmonic distortion of the trap potential. Two methods to determine the cyclotron frequency are discussed. First, when all three eigenfrequencies of a trapped particle can be measured, the true cyclotron frequency is given by the prescription of the Brown-Gabrielse invariance theorem. This prescription makes possible a surprising number of the most accurate measurements in particle, nuclear and atomic physics because it accounts exactly for the lowest order electrostatic imperfections and magnetic misalignments. Second, when less accuracy is required, as when the masses of unstable nuclei are measured, a single sideband frequency is often measured instead-the frequency of a driving force that optimally couples two of the motions of the ion in the trap. A missing theoretical justification for this alternate method is provided using an expansion of the same invariance theorem. A remarkable suppression of systematic measurement errors is predicted, showing why these are not larger than reported measurement uncertainties, despite the contrary indication of simple estimates.