REFLEX OSCILLATORS 



631 



Here I' is the voltage between a and b. Hence, the modulation coefficient 

 jS is given for small signals by 



/3 = (1/F) 



•'a 



dx 



V = V{b) - V{a) 

 7 = w/z'o 

 Wo = V27/F0 



(b9) 



(MO) 

 (Ml) 

 (M2) 



Thus Uii is the electron velocity. 



It is sometimes convenient to integrate by parts, giving the mathematical 

 expression for /3 a different form 



^ = (1/F) 



V'{x)e^ 



h 



as F'(.v) is zero at a and h 



13 = (1/F) 



- ~ Vixy"" d.y 



31 -la 



- [ F"(.r)e^'^" rfx 



T ''a 



(bl3) 



An interesting and important case is that of a uniform held between 

 grids. Let the tirst grid be at .v = and the second at .v = d. There is an 

 abrupt transition to a gradient V/d at .v = 0, and another abrupt transition 

 to zero gradient at x = d. Thus, the integral (bl3) is reduced to these two 

 contributions, and we obtain 



/3 =: (1/F) 



F 



yd 



1 - e 



jyd 



(bl4) 



This is easily seen to be 



/3 = sin (7 d/2)/(y d/2) ' (bl4) 



This function, the modulation coefficient for fine parallel grids, is plotted in 

 Fig. 119. 



Sometimes apertures, as, circular apertures, or long narrow slits are used 

 without grids. There are important relations between the modulation 

 coefficient for a path on the axis and one parallel to the axis for such systems."^ 



In a two-dimensional gap system with axial symmetry, if the modulation 

 coefficient for a path along the axis is /?o , the modulation coefficient for a 

 path y away from the axis is 



^y = ^0 cosh yy 



(bl5) 



-■' These relations first came to the attention of the writers through unpublished work of 

 D. P. R. Petrie. C, Strachey and P. J. Wallis of .Standard Telephones and Cables. 



