# Relativistic shifts of eigenfrequencies in an ideal Penning trap

## TL;DR

This paper calculates how special relativity causes frequency shifts in a charged particle's motion within an ideal Penning trap, using perturbation theory and elliptic functions to describe the relativistic corrections.

## Contribution

It provides a first-order perturbative analysis of relativistic frequency shifts in a Penning trap, including amplitude-dependent shifts and modular angle contributions, extending previous models.

## Key findings

- Relativistic corrections to oscillation amplitudes and eigenfrequencies are derived.
- The orbit deformation is described by Jacobian elliptic functions, revealing amplitude dependence.
- Results agree with existing models in the low-energy limit.

## Abstract

First-order perturbative calculation of the frequency-shifts caused by special relativity is performed for a charged particle confined in a Penning trap. The perturbed motion is approximated by the Jacobian elliptic functions which describe the periodic orbit repeating itself sinuously with a period that exceeds $2\pi$. We find relativistic corrections to amplitudes of oscillating modes as well as shifts of eigenfrequencies which depend on amplitudes. Besides we find the relativistic contributions to modular angles. In the low-energy limit the deformed orbit simplifies to the well-known combination of axial oscillation and in-plane motion consisting of two circular modes. We compare the results with the model of relativistic frequency-shifts developed by J. Ketter {\em et al.}