ELECTROSTATIC ELECTRON-OPTICS 5 



Before passing on to such a derivation, it is well to introduce 

 another quantity, which is analogous to focal distance and very useful 

 in making approximations. Suppose that, from any point along its 

 path, an electron were to continue on with its instantaneous velocity 

 in a straight line. Its velocity along the axis would continue to have 

 the instantaneous value i, and the electron would travel over the 

 distance d and arrive at the focal point in a period of time T given by 



T = djz (2) 



or from equation 1 



T = - r/r. (3) 



This period of time is analogous to focal distance, and we therefore 

 call it focal time. The values of T at the two sides of an electron 

 lens, for any electron path, are in a corresponding manner called 

 conjugate focal times of the lens. 



To obtain an equation relating the conjugate focal distances of a 

 lens, we must consider the path of an electron through the lens. 

 The path is determined by the initial velocity and coordinates of the 

 electron as it enters the lens and by its acceleration in the electric 

 field of the lens. By defining electrical units in the proper manner 

 the ratio e/m is eliminated from the equations of acceleration and they 

 assume the simple form 



where $ is the potential at points in space. ^ 



The first solution of these equations gives the well known energy 

 relation 



r^ -\- z^ = 2$, (6) 



where the electron source is taken as zero potential. 



With the exception of special cases, the equations are not further 

 soluble in the usual sense, and one resorts to solution in series. 



As they stand, the two equations for acceleration are inconvenient; 

 they involve partial derivatives of potential with respect to space and 

 ordinary derivatives of velocity with respect to time, and the latter 

 cannot be transformed to partial derivatives with respect to space, 

 for the simple reason that the velocity of an electron does not exist 



^ The final equations of electron-optics involve the potentials only in the form of 

 ratios which are independent of the electrical units. 



