498 ISOTOPIC TRACERS AND NUCLEAR RADIATIONS [Chap. 23 



fivert = n^U 



firad = (1 - n)U<a 

 d{\n H ) 

 n ~ d (In r) 



which are damped in part by the increasing magnetic field and in part by- 

 radiation loss at high energies. 



23.3. Energy Loss by Radiation. Electrons moving in circular orbits 

 with velocities very close to the velocity of light lose energy by radiation 

 arising from radial acceleration. The energy loss per turn due to incoherent 

 radiation is given by 



L = 



3r 



(A)' 



\mc 2 / 



Additional radiation resulting from orbital and phase accelerations has 

 the effect of damping such oscillations. For phase oscillation the decrease 

 in phase amplitude per turn due to radiation damping is 



AO m - <p g ) = ^ (3 - 4») -gr 



23.4. Synchrotron Operation. Shortly after the magnetic field cycle 

 begins, electrons are injected into the accelerating region with an energy of 

 approximately 60 kev. By betatron action (radio-frequency field off) they 

 are then accelerated by the constant torque of the induced field to an energy 

 of the order of 2 mev and a velocity of approximately 0.99 c. By this 

 means the orbits are rapidly brought clear of the injector and to a radius 

 that changes very little on subsequent acceleration. The transition from 

 betatron to synchrotron operation takes place when Ep = E s or r$ = r s . 

 During the period of transition the electrons increase in angular velocity 

 until their period matches that of the radio-frequency field which, during 

 this time, is slowly increasing in amplitude to its peak value. Those electrons 

 lying within the proper phase angles are then "locked" into synchronism 

 and gradually bunched. Betatron action effectively ceases with saturation 

 of the flux bars, and the electrons thereafter rapidly derive energy from the 

 radio-frequency field. 



Despite the bunching effect an energy spread persists because of the 

 finite duration of the injection time. Assuming that the injection interval 

 corresponds to the time required for an electron to increase from radius r to 

 f/j, the energy spread can be calculated [3] from the expression 



8E = (1 - n)(2mc 2 T) r *^* 



where T = kinetic energy of electrons at injection 

 r = injector radius 

 r$ = betatron equilibrium radius 



