394 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



The non-thermal paths remain essentially laminar, and with r^ de- 

 noting the radial coordinate of the outermost non-thermal electron, we 

 will make little error in assuming that the current density of non-ther- 

 mal electrons is constant for r < Ve . Consequently, if equal numbers of 

 thermal electrons are assumed to be normally distributed about the cor- 

 responding non-thermal paths, the longitudinal current density as a 

 function of radius can be found in a straightforward way by using (18). 

 The result is 



J ^ ^_(..,,..) n" R ^-(«^/2.^)^^ frR\ ^ /R\ ^23) 



Jd Jo a \a^/ \(t/ 



where /o is the zero order modified Bessel function and the total current 

 is Id = TTVe Jd ' Equation (23) was integrated to give a plot of Jr/Jo 

 versus r/a, with re/a as a parameter and is given as Fig. 6 in Reference 

 6. It is reproduced here as Fig. 6. Since the only forces acting on elec- 

 trons in the drift region are due to space charge, we may write the equa- 

 tion of motion as 



where Er is the radial electrical field acting on an electron with radial 

 coordinate r. Since the beam is long and narrow, all electric lines of force 

 may be considered to leave the beam radially so that Er is simpl}^ ob- 

 tained from Gauss' law. Equation (24) therefore becomes 



-— = --^— / 2irp dr = -— ! — / ■ Iirr dr 



dt^ zireor Jo Zireor Jo \/2t]V a. 



(25) 

 2irenr Jo 



27reor 



From (23) we note that the fraction of the total current within any 

 radius depends only on fe/o- and j'/ct: 



:il 



dr 



^ / J0')2irr ar / xo ,/o 



r = - = H-) f 



'- r.J(r)2.rdr ^''''° (2«) ' 



Jo 



■•r I a 



C 



'^dV^]^Fr-j- 



\(X a t 



\ 



