414 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



beam with rectilinear flow, by averaging El oxer the Ijeam area (with 

 E, for the empty hehx) : 



{Ks/Krf" - [Il(yb) - Il(yb)]%^, (87) 



Fletcher has improved upon this calculation by replacing the solid beam 

 with a thin hollow beam of different radius and dc current. This has the 

 same electronic admittance Ye and deri\'ative dYe/dy when R = 1. 



The impedance reduction factors for the three types of beams have 

 been plotted in Fig. 4, using a typical value of h/a. For the same h , 

 Vo , and h/a, the gain parameters C are found to be greatest for the 

 hollow beam, and least for the solid rectilinear beam. 



The high gain of the hollow beam is due to its concentration in the 

 region of greatest field strength. The greater gain of the beam \vith 

 Brillouin flow, relative to that of a similar beam with confined flow, is 

 probably due to transverse electron motion, in two ways: 



(1) causing electrons to interact with the transverse as well as longi- 

 tudinal fields; and 



O 4 



7a 



Fig. 3 — The factor {Kb/KtY^^, or Fi , by which the gain parameter Ct for a 

 thin beam should be multiplied to give Cb , the gain parameter for a cylindrical 

 beam with Brillouin flow, of the same current and voltage. Computation of Cg using ! 

 this factor is described in text following equation (85). 



