Charged particle trajectories in an ideal paracentric hemispherical deflection analyser

摘要

We present an exact analytic solution for the trajectory of a charged particle moving in the ideal potential V(r) = - k/r + c inside a hemispherical deflector analyser (HDA). Our treatment extends the known solutions to also include paracentric entry for which R_0 ≠ R ≡ 1/2 (R_1 + R_2) and V(R_0) is not necessarily zero, where R_0 is the centre of the HDA entry aperture. We also account for particle refraction at the potential boundary that cannot be neglected when V(R_0) ≠ 0. A general 3-D vector treatment for calculating trajectories in a fixed frame is also described based on the conservation of the angular momentum and eccentricity vectors. These results find applications in modern hemispherical spectrographs incorporating large diameter position sensitive detectors (PSD) as for example in ESCA.