I 



388 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



Electrolytic tank measurements are normally made in the cathode- 

 anode region to give the potential variation along the outside edge of 

 the electron beam (for comparison with the Langmuir potential) ; hence, 

 by tracing out a suitable equipotential line, the shape of the false cathode 

 can easily be obtained. With the false cathode in place and at the proper 

 potential, the approximate value for Vx is then obtained by a direct tank 

 measurement of the potential at an axial point whose distance from the 

 true cathode center is (fc — fa) as outlined above. Although finite elec- 

 tron emission velocities typically do not much influence the trajectory 

 of an electron at the anode, they do nevertheless significantly alter the 

 beam in the region beyond. It is in this affected region where experi- 

 mental data can be conveniently taken. We must, therefore, postpone a 

 comparison of lens theory with experiment until the effect of thermal 

 velocities has been treated. At that time theoretical predictions com- 

 bining the effects of both thermal velocities and the anode lens can be 

 made and compared with experiment. Such a comparison is made in 

 Section 6. 



4. TREATMENT OF BEAM SPREADING, INCLUDING THE EFFECT OF THERMAL 

 ELECTRONS 



Jn Section 2 the desirability of having an approach to the thermal 

 spreading of a beam which would be applicable under a wide variety of 

 conditions was stressed. In particular, there was a need to extend ther- 

 mal velocity calculations to include the effects of thermal velocities even 

 when electrons with high average transverse velocities perturb the beam 

 size by as much as 100 or 200 per cent. Furthermore, a realistic mathe- 

 matical description which would allow electrons to cross the axis seemed 

 essential. The method described below is intended adequately to answer 

 these requirements. 



A. The Gun Region 



The Hines-Cutler method of including the effect of thermal velocities 

 on beam size and shape leads one to conclude that, for usual anode 

 voltages and gun perveance, the beam density profile in the plane of 

 the anode hole is not appreciably altered by thermal velocities of emis- 

 sion. (This statement will be verified and put on a more quantitative 

 basis below.) Under these conditions, the beam at the anode is ade- 

 quately described by the Hines-Cutler treatment. We will therefore find 

 it convenient to adopt their notation where possible, and it will be 

 worthwhile to review their approach to the thermal problem. 



