186 BELL SYSTEM TECHNICAL JOURNAL 



lose energy gained from the DC field to the RF field at successive gaps. 

 Electrons which gain energy from the RF field are driven back into the 

 cathode after only one orbital loop and are removed from further motion 

 detrimental to the oscillation. This process of selection and rejection of 

 electrons forms the groups of bunches, shown in Fig. 2(c), which sweep past 

 the anode slots in phase to be retarded by the RF field component. The 

 criterion that the electron drift velocity shall be such as to keep these 

 bunches in proper phase is analogous to the condition that the drift angle 

 in a velocity variation oscillator [Fig. 2(b)] be such as to cause the bunches 

 to cross the gap of the second or "catcher" cavity in phase to lose energy 

 to the RF field across the gap. 



The condition placed upon the mean angular velocity of the electrons may 

 be discussed more readily by reference to Figs. 9, 10, and 11. Consider 

 first, however, only Fig. 9 for the standing potential wave of the n = ^ mode, 

 and focus attention on an electron which crosses the gap between anode 

 segments 1 and 2 at the instant t when the RF field is maximum retarding, 

 that is, the potential on segment 1 is maximum and on segment 2 minimum. 

 It is clear that this electron can cross the next gap in the same phase if the 

 time required to reach it is (|^| + |) T, in which p is any integer and T is 

 the period of RF oscillation. In Fig. 9, four lines are drawn representing 

 the mean paths of electrons moving with such velocities as to make p = 

 0, 1, 2, and 3. Each line crosses a gap when the RF field is maximum 

 retarding, that is, when the potential has the maximum -negative slope at 

 the center of the gap. As will be seen later, a more convenient parameter, 

 to be called k, is that whose absolute magnitude, |^|, specifies the number 

 of RF cycles required for the electron to move once around the interaction 



space. — is then the number of cycles between crossings of successive 



anode gaps, which for the tt mode of Fig. 9 must take on the values: 



1^1 ,, 1 



^=1^1 + 2' P = 0,±1,±2,--., 



or the values given by the more general expression, applicable to any mode: 



\k\ n 



ft 



In this expression, — is the phase difference between adjacent resonators, 

 expressed as a fraction of a cycle, k may thus assume the values given by 



k = n + pN, ] 



(10) 

 ^ = 0, ±1, ±2, .-.. 



