BEAM FORMATION WITH ELECTRON GUNS 377 



V(f, /■), Vc.(f, potential distributions used in the anode lens study 

 r), etc. 



V' voltage gradient 



z distance along the beam from the anode lens 



2n,in distance to the point where rgs is a minimum 



( — a) Langmuir potential parameter for spherical cathode- 



anode gun geometry 



7 slope of an electron's path after coming into a space 



charge free region just beyond the anode lens 



r the factor which divides Fd to give the modified anode 



focal length 



5 dimensionless radius parameter 

 €o dielectric constant of free space 

 f dimensionless voltage parameter 



6 slope of an electron's path in the gun region 

 r} charge to mass ratio for the electron 



fx normalized radial position in a beam 



a the radial position of an electron which left the cathode 



center with "normal" transverse velocity 

 (T+' slope of o--electron on drift side of anode lens 



a J slope of (T-electron on gun side of anode lens 



^ electric flux 



1. INTRODUCTION 



During the past few years there have been several additions to the 

 family of microwave tubes rec}uiring long electron beams of small diame- 

 ter and high current density. Due to the limited electron current which 

 can be "drawn from unit area of a cathode surface with some assurance 

 of long cathode operating life, high density electron beams have been 

 produced largely through the use of convergent electron guns which 

 increase markedly the current density in the beam over that at the 

 cathode surface. 



An elegant approach to the design of convergent electron guns was 

 provided by J. R. Pierce^ in 1940. Electron guns designed by this method 

 are known as Pierce guns and have found extensive use in the produc- 

 tion of long, high density beams for microwave tubes. 



]\Iore recent studies, reviewed in Section 2, have led to a better under- 

 standing of the influence on the electron beam of (a) the finite velocities 

 with which electrons are emitted from the cathode surface, and (b) the 

 defocusing electric fields associated with the transition from the ac- 

 celerating region of the gun to the drift region beyond. Although these 

 two effects have heretofore been treated separately, it is in many cases 



