ELECTROSTATIC ELECTRON-OPTICS 9 



For such rays the higher terms in r may be neglected in the general 

 equations and we obtain the following useful relations 



2v 

 d 



A = r/r = — 

 z = V2i;, 



T = d|^|2i = - ^ ' 



dt — dz/z = dzl^2v. 



(22) 

 (23) 

 (24) 



(25) 



A Uniform Electric Field 

 A uniform electric field parallel to the axis is not usually regarded 

 as an electron lens,^^ but it does shift the focal point of a beam of 

 electrons passing through it. In a uniform field, v" is zero and the 

 differential equation for A may be written in the form 



dA 



^2 



dz 



\2v 



(26) 



An integration of this equation from any point Si to any other point Z2, 

 in the uniform field-, gives 



J 1_ 2(22 - Si) 



^2 Ai V2i;2 + A'2yi' 



(27) 



where Ai and A 2 are the values of A at Zi and Z2. On substituting 

 — -^^Id for A in this equation, it may be transformed to 



1 + 



^1 - 1 + 



-')^2 = 2(s2-zi), (28) 



V2 



which is the equation relating the conjugate focal distances at any 

 two planes — located at Zi and Z2 — in the uniform field. 



The shift in the focal point of an electron beam as it passes through 

 a uniform field is illustrated in Fig. 4. 



-^ 



^1^ 



-P2 



-^=0— 



-d2- 



Fig. 4 — Focal distances in a uniform field. 



1^ Electron rays parallel to the axis are not bent by the field, and it does not 

 magnify an electron image. 



