First-order perturbative calculation of the frequency-shifts caused by static cylindrically-symmetric electric and magnetic imperfections of a Penning trap

The ideal Penning trap consists of a uniform magnetic field and an electrostatic quadrupole potential. Cylindrically-symmetric deviations thereof are parametrized by the coefficients Bn and Cn, respectively. Relativistic mass-increase aside, the three characteristic eigenfrequencies of a charged particle stored in an ideal Penning trap are independent of the three motional amplitudes. This three-fold harmonicity is a highly-coveted virtue for precision experiments that rely on the measurement of at least one eigenfrequency in order to determine fundamental properties of the stored particle, such as its mass. However, higher-order contributions to the ideal fields result in amplitude-dependent frequency-shifts. In turn, these frequency-shifts need to be understood for estimating systematic experimental errors, and eventually for correcting them by means of calibrating the imperfections. The problem of calculating the frequency-shifts caused by small imperfections of a near-ideal trap yields nicely to perturbation theory, producing analytic formulas that are easy to evaluate for the relevant parameters of an experiment. In particular, the frequency-shifts can be understood on physical rather than purely mathematical grounds by considering which terms actually drive them. Based on identifying these terms, we derive general formulas for the first-order frequency-shifts caused by any perturbation parameter Bn or Cn.