Sec. 23.2] 



THE SYNCHROTRON 



497 



b. Phase Oscillations. Electrons that were not started in synchronous 

 orbits or that arrive at the gap at a time different than for the equilibrium 

 phase are retarded or advanced on successive transits and thus continue to 

 execute stable oscillations in phase as well as in radius and energy about the 

 synchronous values. The equation of motion of the phase <p is given [1,3,4] 

 by the relations 



d 



dt 



vw 



+ 



r 8 L' s c 2 <p . eV . eV . 



—— -=- h ir- sin tp = -7T- sin ip s 



2ir(l — n)Avj(t) s Zir It 



where 



K = l + 



n 



1 



n = 



(1 — n) v\ 

 <Z(ln H) 

 d(\n r) 



1 — n 



L' = 



d_Ls 

 dr 



v g f**' c = velocity of light 



For oscillations of small amplitude, the energy also oscillates about the 

 equilibrium value by the amount 



eVEg cos 



AE= - 



(eVE 8 cos <p\ H , v 



where <p m = maximum phase amplitude 



The corresponding change in radius is given by 



Ar = 



AE 



vl (1 — n) E s 



In general, the amplitudes <p m — <p„ AE and Ar vary as 



*.-*.~[(l -n)VE,]-* 

 AE~E S 



V 



o— ) S 



>4 



Ar r^ r s 



■a 



1(1 -n)*El\ 



The frequency of phase oscillation is small compared to the orbital fre- 

 quency of the electrons. From the equation for the phase, its frequency is 

 found [3] to be 



(eVK cos <p,\* l\ l 

 U °\ 2.E a ) L 16 



fi = 



(<Pm — <PY~ 



These oscillations were shown to be stable for adiabatic changes in the 

 magnetic field [3]. 



c. Orbital Oscillations. Both vertical and radial oscillations in the orbits 

 may occur witn frequencies comparable to the rotational frequency of the 

 electron. Calculations of these frequencies [5] gives values of 



