630 BELL SYSTEM TECHNICAL JOURNAL 



to be small compared with a wa\elength and to have little a-c magnetic 

 field in it, so thai we can pretty accurately represent the field in this re- 

 stricted region as the gradient of a potential. Along the path the potential 

 is taken as the real part of 



V(x)e^''' (61) 



For small gap voltages, to first order, the time that an electron reaches a 

 given position may be taken as unaffected by the signal, so that 



/ = .r/w„ + /„ (62) 



Then the gradient along the path is 



-,- = Real 7'(.v)e>(-'«o+'o«o)_ .^^ 



dx 



The change in momentum in passing through the field may be obtained by 

 integrating the force on an electron times the time through the field. Let 

 points a and b be in the field-free region to the left and right of the gap. 

 Then we have 



r'' 



A(x) = Real ^ |/'(.v)^/-/"o+-'o^ ^^. ^^^-^ 



A{x) = Real''— [ V'(x)e'''' dx (b5) 



y = co/tiQ . (b6) 



The integral will be a complex quantity. The exponential factor involving 

 the starting time In will rotate this. A(.v) will have a maximum value when 

 the rotation causes the vector to lie along the real axis, and thi^ maximum 

 value is thus the absolute value of the integral. Hence 



A(x)„,ax =^ - 



I V'(x)e'' 



''a 



dx 



(b7) 



For a-c voltages small compared with the voltage specifying the speed 

 Uo , the energy change is proportional to the momentum change. For an 

 electron transported instantly from one side of the gap to the other, the 

 momentum change can be obtained by setting y = 0. 





dx 



(b8) 





