The hemispherical deflector analyser revisited. I. Motion in the ideal 1/ r potential, generalized entry conditions, Kepler orbits and spectrometer basic equation

Abstract

We re-examine the orbits of non-relativistic charged particles in a hemispherical deflector analyser (HDA) assuming an ideal 1/ r potential. The particles start their trajectory within the HDA at the arbitrary entry radius r 0, within a circular entry aperture centered at R 0 at an arbitrary potential V 0= V( R 0). We present a vector treatment of the trajectories deriving many useful relations expressed as a function of the launching angle α. Refraction at the potential boundary at the entry of the HDA (modelled by an idealized step potential) is also considered and found to be important when V 0≠ V p, where V p is the plate voltage used for preretardation. We derive the analyser’s generalized basic equation for deflection through 180° for which the principal reference ray is an ellipse rather than a circle as in the conventional HDA treatment. Both the conventional HDA, for which R 0= R ̄ and V 0= V p, as well as the paracentric HDA for which R 0≠ R ̄ and V 0≠ V p, where R̄ is the mean radius, are thus described as special cases of the same trajectory equation. Our results are expected to be of interest to all fields of electron spectroscopy, but particularly to those utilizing modern spherical sector analysers with sizeable interradial separation for accommodating large area position-sensitive detectors. This investigation is part of a concerted effort to investigate the refocusing properties of the paracentric HDA recently reported by Benis and Zouros [Nucl. Instr. & Meth. A 440 (2000) 462].