Radial distribution of charged particles in a magnetic field

The radial spread of charged particles emitted from a point source in a magnetic field is a potential source of systematic error for any experiment where magnetic fields guide charged particles to detectors with finite size. Assuming uniform probability as a function of the phase along the particle's helical trajectory, an analytic solution for the radial probability distribution function follows which applies to experiments in which particles are generated throughout a volume that spans a sufficient length along the axis of a homogeneous magnetic field. This approach leads to the same result as a different derivation given by Dubbers et al., Nucl. Instrum. Methods Phys. Res., Sect. A 763, 112-119 (2014). But the constant phase approximation does not strictly apply to finite source volumes or fixed positions, which lead to local maxima in the radial distribution of emitted particles at the plane of the detector. A simple method is given to calculate such distributions, then the effect is demonstrated with data from a (207)Bi electron-conversion source in the superconducting solenoid magnet spectrometer of the Ultracold Neutron facility at the Los Alamos Neutron Science Center. Implications for neutron beta decay spectroscopy are discussed.