378 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



necessary to produce electron beams under circumstances where both 

 effects are important and so must be dealt with simultaneously and more 

 precisely than has until now been possible. It is the purpose of this paper 

 to provide a simple design procedure for typical Pierce guns which in- 

 cludes both effects. Satisfactory agreement has been obtained between 

 measured l^eam contours and those predicted for several guns having 

 per\'eances (i.e., ratios of beam current to the ^^ power of the anode 

 voltage) from 0.07 X 10-« to 0.7 X 10"^ amp (volt)-3/2. 



2. PRESENT STATUS OF GUN DESIGN — LIMITATIONS 



Gun design techniques of the type originally suggested by J. R. Pierce 

 were enlarged in papers by SamueP and by Field^ in 1945 and 1946. 

 Samuel's work did not consider the effect of thermal velocities on beam 

 shape and, although Field pointed out the importance of thermal veloci- 

 ties in limiting the theoretically attainable current density, no method 

 for predicting beam size and shape by including thermal effects was 

 suggested. The problem of the divergent effect of the anode lens was 

 treated in terms of the Davisson"* electrostatic lens formula, and no 

 corrections were applied.* 



More recently. Cutler and Hines^ and also Cutler and Saloom^ have 

 presented theoretical and experimental work which shows the pro- 

 nounced effects of the thermal velocity distribution on the size and shape 

 of beams produced by Pierce guns. Cutler and Saloom also point to the 

 critical role of the beam-forming electrode in minimizing beam distor- 

 tion due to improper fields in the region where the cathode and the 

 beam-forming electrode would ideally meet. With regard to the anode 

 lens effect, these authors also show experimental data which strongly 

 suggest a more divergent lens than given by the Davisson formula. The 

 Hines and Cutler thermal velocity calculations have been used"' "^ to 

 predict departures in current density from that which should prevail in 

 ideal beams where thermal electrons are absent. Their theory is limited, 

 however, by the assumption that the beam-spreading caused by thermal 

 velocities is small compared to the nominal beam size. 



In reviewing the various successes of the above mentioned papers in 

 affording valuable tools for electron beam design, it appeared to the 

 present authors that significant improvement could be made, in two 

 respects, by extensions of existing theories. First, a more thorough in- 



* It is in fact erroneously statoci in Reference 5 that the lens action of an actual 

 structure must be somewhat weaker than i)re(licted by the Davisson formula so 

 that the beam on leaving the anode hole is more convergent than would be calcu- 

 lated by llie Davisson method. This cjuestion is discussed further in Section 3. 



