The parallel plate condenser as a relativistic particle analyzer

A fully analytical study of the properties of the parallel plate electrostatic spectrograph in the relativistic region is presented. The formalism used allows for an exact tridimensional treatment whereby the coupling between the polar angles θ, φ appears through the quantity α = −dφ/dθ only while the explicit expression of α(θ, φ) remains unspecified. This freedom in the choice of the relationship φ = f (θ) may be used to improve on the properties of the instrument. The main consequence of the relativistic effect is to alter the shape of the focal surface in a way which reduces the versatility of the instrument when changes in energy range are considered. This stems from the dependence of θ, φ on β = v/c introduced by the second order focusing condition. It is shown that versatility is restored to a considerable extent for suitable choices of φ = f(θ). The relative merits of various choices are discussed. A numerical calculation of the basic parameters of a spectrograph suited for astrophysical research is given stressing the possibility of flying such an instrument aboard a Space Shuttle within the Post Apollo program.