412 



THE BELL SYSTEM TECHNICAL JOURNAL, IVtARCH 1954 



this product, for the slope of the curve 5o versus yb increases with 76. 

 Putting F = 0, therefore, leads at most to a very shght error in jj, and 



dYs\ 



dy /7=7o 



Outside of the region (70 , 7p) , 5o grows large rapidly, and the expres- 

 sion for Yb is hardly affected at all by this assumption. 



Physically, the negligible role played by F in the admittance equation 

 means that 8 O!:^ 80 , i.e., the TM helix admittance is not appreciably per- 

 turbed by the TE fields excited by Go . 



With F = 0, (67) may be re-written: 



7^ /o(7&) V 



Yn = 



2 h{yh) 



jcoe r /i(76) - 8o'Ki{yb) _ /i(7b) 

 7 ih{yh) + 8o-Koiyh) h{yh)_ 



(72) 

 (73) 



Here Yh is the helix admittance seen by a thin cyUndrical hollow beam, 

 with rectilinear electron flow. As in the case of Yb , it may be replaced 

 in the vicinity of (70 , 7p), by the approximation: 



Yh 



(Tp - To) 



dY, 



To 



^T /7=7o T - T; 



(74) 



Fletcher has evaluated the normal-mode parameters for Yh as fol- 

 lows: 



■p2 2 



To = -To 



Qh 



1 + 



To. 



-1/2 



1 



O 2 2 



2 Tp - To 



T^o 



= -jTrbyo 



1 + 



To. 



,2-|3/2 



BYh 



dy 



(75) 

 (76) 



(77) 



We have used the subscript H to refer the parameters to the hollow beam, 

 and will use the subscript B to refer to the solid-cylindrical Brillouin 

 beam. 



As Yb and Yh have the same zero and pole, they have the same natural 

 propagation constant To , and the same space-charge parameter Q: 



Qb = Qh (78) 



This quantity can be found plotted in Fig. 1 of Reference 4, or in Fig. 

 A6.1 of Reference 1. 



