ELECTROSTATIC ELECTRON-OPTICS 3 



The theory of electron-optics is thus well established and any 

 further attempts at the subject must lead to substantially the same 

 results. There is, however, a need for a precise development of the 

 theory in a simpler manner. With this need in mind, the present 

 article approaches the subject in a manner that appeals to the reader 

 who is more familiar with electrical theory than he is with the concepts 

 of geometrical optics, and this approach leads clearly to the various 

 approximations that are needed in the development of the theory. 

 With the aid of two velocity functions, the partial differential equations 

 of electron motion are briefly and exactly reduced to a series of ordinary 

 differential equations ; the theory is then developed in terms of their 

 approximate solutions. 



Attention is confined to systems in which the electric fields are 

 symmetrical about a central axis. In such systems any field having 

 a radial component of electric intensity changes the radial velocity of 

 an electron passing through it, and thus behaves — to some extent at 

 least — as an electron lens. A uniform field parallel to the axis and 

 field-free space are the only regions in which there is no lens action. 

 Typical electron lenses are shown in the figures on the second page. 

 As illustrated by these examples, a practical electron lens is character- 

 ized by a short region in which there is an abrupt change in the electric 

 intensity parallel to the axis. Lines of force are continuous, and the 

 field parallel to the axis can change only by lines of force coming 

 into it, or going out from it, in a radial direction. In the region of 

 the abrupt change, there are consequently strong radial fields which 

 can deflect an electron in a radial direction. The region changes the 

 focus of an electron beam passing through it, and its action is analogous 

 to that of an optical lens. 



Section I — The General Equations 



In the present paper it is assumed that the initial electron source 

 has perfect symmetry of form about the central axis, and that the 

 electrons have no appreciable velocities of emission from the source. 

 An electron thus has no angular velocity about the axis, and its motion 

 may be described in terms of a coordinate z taken along the axis and 

 a radial coordinate r measured from the axis. 



If an electron's velocity vector is projected at any point along its 

 path, it intersects the axis at some point p, as illustrated in Fig. 2, 

 and the electron may be regarded as instantaneously moving either 

 away from, or else toward, this point of intersection. The distance d 

 along the axis from the electron to the point of intersection is called 



