CHAPTER 22 

 THE BETATRON 



22.1. Description. The betatron was first successfully developed by 

 Kerst [1,2] for the acceleration of electrons by electromagnetic induction. In 

 principle, it is similar in operation to the common shell-type transformer, but 

 instead of employing a secondary coil, electrons are injected into an evacuated 

 "doughnut" to move freely in a circular path to which they are constrained 

 by a magnetic field. A large part of the magnetic flux produced in the 

 magnet core is made to loop through the circular orbits by an arrangement of 

 narrow air gaps and iron cores. Thus, as the flux is increased or decreased, a 

 tangentially induced electric field is experienced by an electron in an orbit 

 of radius r identical to the field induced in a secondary winding. 



E = — — 



2-ktc dt 



where c = velocity of light 



$ = magnetic flux through the electronic orbit 

 The magnetic field H at the orbit must also increase at the same rate as the 

 flux. If both the electron momentum and the magnetic field increase in pro- 

 portion, the radius of the orbit will remain constant during the interval the 

 electron is accelerated. The condition necessary for the maintenance of an 

 equilibrium orbit of fixed radius r is given by the betatron flux condition 



where <£ = initial flux when H = 

 <£ = flux at any instant t 



H = magnetic field at equilibrium orbit at same instant 

 The flux through the orbit must be proportional to twice the field strength 

 at the orbit; this condition is easily realized in practice since both the linkage 

 flux and the magnetic field at the orbit are produced in the same gap by a 

 single magnet. 



Acceleration of electrons from the initial injection energy to the maximum 

 energy occurs in one quarter of the magnetic field cycle, either when the 

 field is increasing or decreasing; in either case the direction of rotation is 

 reversed. The low-frequency alternating magnetic field is produced by coils 

 wound on the upper and lower pole pieces and, in the most efficient form, 

 coupled to a bank of condensers of appropriate capacity to form either a 



489 



