BEAM FORMATION WITH ELECTRON GUNS 379 



vestigation of the anode lens effect was called for; and second, there was 

 a need to extend thermal velocity calculations to include cases where 

 the percentage increase in beam size due to thermal electrons was as 

 large as 100 per cent or 200 per cent. Some suggestions toward meeting 

 this second need have been included in a paper by M. E. Hines.* They 

 have been applied to two-dimensional beams by R. L. Schrag.^ The 

 particular assumptions and methods of the present paper as applied to 

 the two needs cited above are somewhat different from those of Refer- 

 ences 8 and 9, and are fully treated in the sections which follow. 



3. TREATMENT OF THE ANODE LENS PROBLEM 



Using thermal velocity calculations of the type made in Reference 6, 

 it can easily be shown that at the anode plane of a typical moderate 

 perveance Pierce type electron gun, the average spread in radial posi- 

 tion of those electrons which originate from the same point of the cathode 

 is several times smaller than the beam diameter. For guns of this type, 

 then, we may look for the effect of the anode aperture on an electron 

 beam for the idealized case in which thermal velocities are absent and 

 confidently apply the correction to the anode lens formula so obtained 

 to the case of a real beam. 



Several authors have been concerned with the diverging effect of a 

 hole in an accelerating electrode where the field drops to zero in the 

 space beyond, ^° but these treatments do not include space charge effects 

 except as given by the Davisson formula for the focal length, Fd , of 

 the lens: 



F. = -^ (1) 



where V would be the magnitude of the electric field at the aperture if 

 it were gridded, and V would be the voltage there. 



In attempting to describe the effect of the anode hole with more ac- 

 curacy than (1) affords, we have combined analytical methods with 

 electrolytic tank measurements in two i-ather different ways. The first 

 method to be given is more rigorous than the second, hut a modification 

 of the second method is much easier to use and gives essentially the 

 same result. 



A. Siipcrposition Approach to the Anode Lens Problem 



Special techniques are required for finding electron trajectories in a 

 space charge limited Pierce gun having a non-gridded anode. M. E. 



