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Abstract

The ideal Penning trap consists of a homogeneous magnetic field and an electrostatic quadrupole potential. In this configuration, the three characteristic eigenfrequencies of a trapped particle do not depend on its motional amplitudes from a classical point of view. However, this three-fold harmonicity of the eigenmotions is compromised by higher-order terms in the magnetic field and electric potential, and ultimately by special relativity. Understanding the systematic effect of these deviations on the motional frequencies is crucial for accurate measurements. This thesis calculates numerous frequency-shifts in the framework of classical perturbation theory working with equations of motion for the particle’s trajectory. Starting from a general parametrization of cylindrically-symmetric electric and magnetic imperfections in cylindrical coordinates, it is shown how to calculate the corresponding first-order frequency-shift consistently. Relativistic frequency-shifts are handled perturbatively in the relativistic equations of motion rather than via a quantum-mechanical operator formalism. Other frequency-shifts considered include the effect of a slightly elliptic quadrupole potential, the interaction of an ion with its image charges induced in the trap electrodes, and a small modulation of the quadrupole potential. The frequency-shifts derived are translated into shifts under the operation mode of locked axial-frequency used by the THe-Trap experiment.