ELECTROSTATIC ELECTRON-OPTICS 



25 



In complete lens systems, where the symbolic calculations are 

 complicated, it is frequently simpler to introduce specific numerical 

 values and carry the successive steps of the calculation through in a 

 numerical manner. By doing this for a few suitably chosen numerical 

 values one can obtain the particular information that is desired. 



Appendix II — Concentric Tubes 

 Two concentric tubes at different potentials form an electron lens 

 that is well adapted to practical tube construction. When the two 

 tubes are of the same diameter, the approximate constants of the 

 lens may be determined as follows. ^^ 



Fig. 16 — Concentric tubes. 



In this type of lens, the electric intensity is symmetrical with 

 respect to an imaginary plane drawn between the two tubes — as 

 illustrated in Fig. 16 — and the plane is therefore an equipotential 

 surface. Its potential Vq is the mean potential of the two tubes. 

 This plane is regarded as a division plane separating the lens into two 

 component electric fields. 



We first consider the component to the right of the plane. The 

 solution for the potential inside of the tube may be obtained in the 

 form of a Bessel Function series, and it follows from this series that 

 the potential on the axis is 



2 



V ^ V2 — (V2 — Vo)H 



fxJiifJ.) 



exp. 



^JLZ 

 R 



(1) 



where R is the radius of the tubes, and lu. takes on discrete values 

 equal to the successive roots of 



0. 



(2) 



We find that an approximation to the exponential series is given by 



E 



■;:' fiJi(n) 



exp. 



-^1=1 

 R ' 



tanh co2. 



(3) 



^^ We assume that the separation between their ends is negligibly small compared 

 to their diameter. 



