Cylindrical deflection analyzer.

For coaxial cylinders of radii R1 and R2 and potentials V1 and V2 respectively, the potential V and field F at radius R between the cylinders are

V = b*ln(R) + c, F = -b/R,

where

b = (V1 - V2)/(ln(R1)-ln(R2)),

c = (V2*ln(R1) - V1*ln(R2))/(ln(R1) - ln(R2)).

An electron that starts at R=R0, at potential V0 and with kinetic energy E0 = e*V0 (where e is taken to be positive) will follow a circular path of radius R0 if

V0 = -b/2.

Then the expression for the potential V becomes

V = V0 - 2*V0*ln(R/R0),

and the cylinders have the potentials

V1 = V0 - 2*V0*ln(R1/R0), V2 = V0 - 2*V0*ln(R2/R0).

The second order focus is at the azimuthal angle pi/sqrt(2) (=127.28 degrees).