BEAM FORMATION WITH ELECTRON GUNS 389 



It is assumed that electrons are emitted from the cathode of a therm- 

 ionic gun with a IMaxwelhan distribution of transverse velocities 



ZTTfC 1 



where Jc is the cathode current density in the z direction, T is the cath- 

 jode temperature, and v^: and Vy are transverse velocities. The number 

 iof electrons emitted per second with radially directed voltages between 



V and V + dV is then 



-(.Ve/kT) 



(S) 



^J. = /.e— -^^^(^^j (12) 



Now in the accelerating region of an ideal Pierce gun (and more generally 

 I in any beam exhibiting laminar flow and having constant current density 

 ()\'er its cross section) the electric field component perpendicular to the 

 axis of symmetry must vary linearly with radius. Conseciuently Hines 

 and Cutler measure radial position in the electron beam as a fraction, 

 ^, of the outer beam radius (re) at the same longitudinal position, 



r = fire (13) 



The laminar flow assumption for constant current densities and small 

 beam angles implies a radius of curvature for laminar electrons which 

 so varies linearly with radius at any given cross section so that 



a 



Substituting for r from (13), (14) becomes 



rfV , /2 dre\ dfj. 



d^^VcTt)dt=^ ^^^^ 



where Ve and dr /dt can be easily obtained from the ideal Langmuir 

 solution. Since the eciuation is linear in /x, we are assured that the radial 

 position of a non-ideal electron that is emitted with finite transverse 

 velocity from the cathode center (where ^ = 0) will, at any axial point, 

 be proportional to dii/dt at the cathode. 



Let us now define a quantity "o-" such that n = a/re is the solution 

 to (15) with the boundary conditions /Xr = and 



_ 1 

 where the subscript c denotes evaluation at the cathode surface, k is 



