Sec. 22.4] THE BETATRON 491 



magnetic field at the orbit increases and, hence, spiral outward until they 

 strike the tungsten target mounted on the back of the injector. 



A second and more practicable method of beam extraction alters the flux 

 condition by the use of auxiliary coils at the pole pieces [3]. At the end of 

 the first quarter cycle of the magnetic field these coils are energized by the 

 discharge from a condenser bank and, depending on the polarity, the field 

 at the equilibrium orbit is increased or decreased while the linkage flux is 

 little altered. The electrons correspondingly spiral inward or outward to 

 reach the target. 



22.3. Orbital Oscillations. The existence of an equilibrium orbit when 

 the flux condition is maintained suggests that oscillations about this orbit 

 may also be stable. This is a property of special importance because not all 

 electrons can be started or maintained in precisely the equilibrium orbit. 

 If the magnetic field is radially symmetrical, the principal oscillations are 

 vertical and radial. As shown by Kerst and Serber [4], such oscillations will 

 have frequencies given by 



d(ln r) 



where to = angular velocity of electrons in equilibrium orbit 



Excitation of these modes of vibration accompanies injection as a result of 

 the angular spread in the injected electrons and the radial displacement of the 

 injector from the equilibrium orbit. Electrons leaving the injector must 

 spiral inward to approach the stable orbit whose radius satisfies the flux 

 condition; but, depending upon the initial angle of injection, some electrons 

 may overshoot and others undershoot the equilibrium orbit and subsequently 

 oscillate with simple harmonic motion about the correct radius and the 

 median plane with a frequency usually less than that of rotation. Further 

 excitation of these oscillations may occur during acceleration both from the 

 effect of space charge spreading and from scattering by the residual gas in the 

 vacuum chamber. 



Two damping factors appear once oscillations are excited. As the mag- 

 netic field increases, the amplitude of oscillation is diminished in proportion 

 to H~M. At high energies further damping is introduced by radiation loss. 

 For a radially symmetrical field, no forcing factors are present to maintain 

 the amplitude of oscillations following the initial exciting pulse resulting from 

 overshooting or scattering. 



22.4. Focusing. Although the equilibrium radius is fixed by the flux 

 condition, it is necessary to provide vertical magnetic focusing to keep the 

 beam in or near the median plane during the long total path length which, in 



