Hemispherical deflection analyzer.
For concentric spheres of radii R1 and R2 and potentials V1 and V2 respectively, the potential V and field F at radius R between the spheres are
V = b/R + c, F = b/R^2,
b = R1*R2*(V1 - V2)/(R2-R1), c = (V2*R2 - V1*R1)/(R2 - R1).
At the mean radius R0 the potential is
V0 = (R1*V1 + R2*V2)/(R1 + R2).
An electron of kinetic energy E0 = e*V0 (where e is taken to be positive) that starts at R = R0 will follow a circular path if
V = V0*(2*R0/R - 1).
From here on we use the reduced parameters
r = R/R0, e = E/E0,
where E is the kinetic energy of the electron.
If the initial value of r is ri, the initial angle of the ray to the median ray is a and the initial angular position theta about the centre of the spheres is taken to be ti = 0, then the key equation that relates r and t is
ri/r = (1 - cos(t))/(ri*ei*cos(a)^2) + cos(t) - tan(a)*sin(t).
(see F Hadjarab and J L Erskine, J Electron Spect 36, 227 (1985), P Louette et al, J Electron Spect 52, 867 (1990), F H Read and S C Page, Nucl. Inst. and Methods A363, 249-253, 1995).
For the particular case ri = 1, t = pi,
rf = (e*d)/(2 - e*d),
where d = cos(a)^2. This relationship can be rewritten as
e = (2*rf)/((1 + rf)*d),
which for small values of (rf - 1) and a can be expanded as
e = 1 + (rf - 1)/2 + a^2 - (rf - 1)^2/4 + ...
Examples of the use of some of the above formulas can be found in the test files test2d09 and test3d09.