54 ISOTOPIC TRACERS AND NUCLEAR RADIATIONS [Chap. 3 



For Ey> 137m c 2_1/ * (complete screening) 



where Z = atomic number of absorber 

 e = electronic charge 

 m = electronic rest mass 

 E = energy of electron in units of m c 2 

 c = velocity of light 

 In energy ranges other than those indicated above the complete expression for 

 4>„ should be used and integrated numerically. 



Neglecting the logarithmic terms in the stopping formula for ionization 

 and in the formula above for radiative collisions, a simple but approximate 

 formula is found for the ratio of the rates of energy loss by these two processes 

 as a function of energy and atomic number. 



(dE/dx)i on EZ EZ 



(dE/dx) I&d ~ 1,600 mc 2 == 800 



where E = energy, mev 



A second form of radiative energy loss, known as Cerenkov radiation [20], 

 occurs when high-speed electrons traverse dielectric media. Radiation is 

 emitted in the frequency range for which the phase velocity in the medium is 

 smaller than the velocity of the electron. The theory of the process, devel- 

 oped by Frank and Tomm [21], leads to expressions for the rate of emission 

 given also by Fermi [28] in the form 

 For v < c<T Yi , 



For v > ce~ y2 , 



-(f) =^( 1 -^ + log^ 1 ) 



\dx /cer tnv \ e — 1 e — 1/ 



where e = dielectric constant 



n = number of electrons per cc 

 (8 = v/c 



In calculations of the total rate of energy loss from both ionization and 

 radiation, the contribution from the Cerenkov effect given in the formula 

 above should not be added since they are contained implicitly in Fermi's 

 complete stopping formulas which should be used in the velocity range where 

 the Cerenkov effect appears (see Sec. 4.4). 



3.5. Specific Ionization. The total ionization produced by a beta particle 

 is the sum of the primary ions produced directly by the particle plus the 



