APPENDIX VI 453 



where 9 = yj) and ye is the value of 7 at « = 0; i.e. for 7e ]:^ )3o 



^^^0 ^/3e. (19) 



7e = /3e + 





In the vicinity of /; = 0, n varies very rapidly with 7, and hence matching 



— ^ ) is practically the same as matching -j-^ . With this approximation 



eqs. (16) and (17) can be differentiated with respect to n and set equal at 



0.6 



7eb 

 Fig. A6.3 — Parameters of the hollow electron beam which is matched to the solid 

 electron beam of radius b and current /o at 7 = 7« c^ /3« , where n = Q. sb is the radius 

 and th is the current of the equivalent hollow beam. 



« = to yield the second relation 

 = d'll(d)ll{sd) 



1 ^2.2,.^.2,..^ Ko(se) , K,{er'' 



t 



L h{sd) "^ h{d) J 



(20) 



Equations (18) and (20) can then be solved to give the implicit equation 

 for 5 as 



K,{sd) 



KM , _A_ 



"T r>r2/ 



lo(sd) hid) ' 211(d) 



and the simpler equation for / 



e' nisd) 



(21) 



(22) 



