ELECTROSTATIC ELECTRON-OPTICS 13 



The apertured plate between two uniform fields is the only lens 

 that permits such a simple calculation of focal distances. In all 

 other lens structures the potential varies appreciably throughout the 

 lens and the integration for the focal term is complicated. The 

 actual numerical calculations have been carried out for only a few of 

 these cases. 



The second type of lens deserving special consideration is a lens 

 bounded on both sides by field-free space. For such boundaries the 

 first term in the last member of equation 34 vanishes, and \IF is 

 determined by the integral term alone. This integral in inherently 

 positive, and a lens bounded on both sides by field-free space is thus 

 always a convergent lens. The two concentric tubes of Fig. 7 give 



Z7777//////^///M :V(///////////7777A 



^ /////////)/7 7f J7777ZZ7///////// . 

 Fig. 7 — Concentric tubes — lines of force and electron paths. 



a lens of this type, the electric field in each tube dropping to zero at 

 a short distance from its end. It is true that there is a divergent 

 field of the same intensity as the convergent field ; but an electron is 

 at a higher potential in the divergent field and traveling faster, so it 

 receives a smaller radial deflection in that field and the lens is con- 

 vergent. It is interesting to note that the lens is still positive even 

 when the potentials on the electrodes are reversed; in other words, 

 a lens of this type is positive irrespective of the direction of the electric 

 field.is 



An Approximation j or Certain Thick Lenses 



In certain electron lenses there is a short region of strong lens 

 action accompanied by more extended regions of weaker action ; the 

 large values of the derivative v" are confined to a short distance along 

 the axis, but the derivative does have appreciable values over a more 

 extended region. A lens of this type can be treated in the following 

 approximate manner, provided that there is but one maximum of 

 I v" I in the lens. 



For this purpose, the differential equation 31 is rewritten in the form 



cf(^)=y(l+|.)~'^/, dt = dzl^v (42) 



" The principal focal distances of concentric tubes are calculated in Appendix 2. 



