Extension of Busch's Theorem to Particle Beams

TL;DR

This paper extends Busch's theorem from single particles to particle beams with complex geometries, providing a conserved quantity useful for modeling beam dynamics and emittance re-partitioning in accelerators.

Contribution

It introduces a generalized theorem for particle beams without cylindrical symmetry, linking multiple beam parameters and validated through simulations and experiments.

Findings

Validated the extended theorem with simulations and analytical calculations.

Applied the theorem to model emittance re-partitioning in accelerator experiments.

Demonstrated the theorem's utility in fast beam dynamics modeling.

Abstract

In 1926, H. Busch formulated a theorem for one single charged particle moving along a region with a longitudinal magnetic field [H. Busch, Berechnung der Bahn von Kathodenstrahlen in axial symmetrischen electromagnetischen Felde, Z. Phys. 81 (5) p. 974, (1926)]. The theorem relates particle angular momentum to the amount of field lines being enclosed by the particle cyclotron motion. This paper extends the theorem to many particles forming a beam without cylindrical symmetry. A quantity being preserved is derived, which represents the sum of difference of eigen-emittances, magnetic flux through the beam area, and beam rms-vorticity multiplied by the magnetic flux. Tracking simulations and analytical calculations using the generalized Courant-Snyder formalism confirm the validity of the extended theorem. The new theorem has been applied for fast modelling of experiments with electron and ion beams on transverse emittance re-partitioning conducted at FERMILAB and at GSI.