392 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



We have already investigated the region of the anode hole in some 

 detail in Section 3 and have found it worth while to modify the ideal 

 Davisson expression for focal length of an equivalent anode lens. In 

 particular, let us define a quantity F by 



F = focal length = Fd/T (19) 



where Fd is the Davisson focal length. Thus T represents a corrective 

 factor to be applied to Fd to give a more accurate value for the focal 

 length. In so far as any thin lens is capable of describing the effects of 

 diverging fields in the anode region, we may then use the appropriate 

 optical formulas to transfer our knowledge of the electron trajectories 

 (calculated in the anode region as outlined above) to the start of the drift 

 region. In particular, 



-f (20) 



where {dr/dz)i and {dr/dz)^ are the slopes of the path just before and 

 just after the lens, and r is the distance from the axis to the point where 

 the ideal path crosses the lens plane. 



B. The Drift Region 



Although Te/a- was found to be large at the anode plane for most guns 

 of interest, this ratio often shrinks to 1 or less at an axial distance of 

 only a few beam diameters from the lens. Therefore, the assumption that 

 electron trajectories may be found by using the space charge forces 

 which would exist in the absence of thermal velocities of emission (i.e., 

 forces consistant with the universal beam spread curve) may lead to very 

 appreciable error. For example, if ecjual normal (Gaussian) distributions 

 of points about a central point are superposed so that the central points 

 are equally dense throughout a circle of radius Te , and if the standard de- 

 viation for each of the normal distributions is cr = r^ , the relative density 

 of points in the center of the circle is only about 39 per cent of what it 

 would be Avith a < (re/5). 



In order to minimize errors of this type we have modified the Hines- 

 Cutler treatment of the drift space in two ways: (1) The forces influenc- 

 ing the trajectories of the non- thermal electrons are calculated from a 

 progressive estimation of the actual space charge configuration as modi- 

 fied by the presence of thermal electrons. (2) Some account is taken of 

 the fact that, as the space charge density in the beam becomes less uni- 

 form as a function of radius, the spread of electrons near the center of 

 the beam increases more rapidly than does the corresponding spread 



