REFLEX OSCILLATORS 659 



The function F{V r/Vo) expresses the effect on r of different penetrations of 

 the electron into the drift field and the factor V^" tell us that if V h and 

 Fo are changed in the same ratio, the drift time changes as one over the 

 square root of either voltage. 

 We can differentiate, obtaining 



dr/dW = VrF'iVn/V,) (f21) 



dr/dV, - ~Vr{{Vn/V,)F'{Vr/V,) + {\/2)F{Vr/V,)) (f22) 



If the electron gains an energy ^V fh crossing the gap, the effect on r 

 is the same as if Fo were increased by /3F and V r were changed by an 

 amount — jSF, because in an acceleration of an electron in crossing the gap 

 the electron gains energy with respect to both the resonator (where the 

 energy is specified by Fo for an unaccelerated electron) and with respect to 

 the repeller {—Vr for an unaccelerated electron). We may thus write 



dr/di^V) = dr/dVn - dr/dVR 



= -Fr[(l + Vr/Vo)F'{Vr/Vo) + hF(VR/Vo)] (f23) 



This expression (f23) is for the same quantity as (f7). Fo of (f23) is 

 F(0 of (f7). Expressions (f21), (f22) and (f23) compare the effects on 

 drift time of changing the repeller voltage alone, as in electronic tuning, the 

 resonator voltage alone, and of accelerating the electrons in crossing the 

 gap. As making the repeller more negative always decreases the drift 

 time, we see that the two terms of (f22) subtract, and usually | dr/dVo \ 

 will be less than | dr/dYR \ . In fact, for a linear variation potential in 

 the drift space and for Fo = Vr , dr/dVo = 0. Weak fields at the zero 

 potential point make the absolute value of F'(Vr/Vo) larger and hence 

 tend to make both ] dr/dVR | and | dr/di^V) \ larger. However, these 

 quantities are not changed in quite the same way. 



The reader should be warned that (fl4) and (f20)-(f23) apply only for 

 fields not affected by the space charge of the electron beam. For instance, 

 suppose we had a gap with a fiat grid and a parallel plane repeller a long 

 way off at zero potential. If edge effects and thermal velocities were 

 neglected, we would have a Child's law discharge. The potential would be 

 zero beyond a certain distance from the repeller, and we would have 



V(z) = Az^ 



According to (fl4), F should be infinite. There is no reason to expect 

 infinite drift action, however, for the drift field, which is affected by the 

 fluctuating electron density in the beam, is a function of time, and (fl4) 

 does not apply. 



