84 ISOTOPIC TRACERS AND NUCLEAR RADIATIONS [Chap. 4 



The model used by Fermi as the basis for calculating the reduction in 

 energy loss clue to polarization is that in which electrons are regarded as 

 classical oscillators and the influence of damping and conduction electrons 

 is negligible. Further, it is assumed that each absorbing substance can be 

 characterized by a single dispersion frequency v , which can be obtained from 

 spectroscopic and x-ray data. With these assumptions the dielectric 

 constant of the absorbing medium may be expressed in the simple form 



. Airne 2 

 e = 1 H »- 



When Fermi's corrections are combined with the simple stopping formula, 

 the expressions applicable to particles with very high energies are [15] 



For v < ct 



-i.. 



dE = 2tt NZe'z 1 

 dx mv 2 



For v > ce~ Vl , 



dE 2ivNZe A z 



dx mv- 



log^jf- - log (1 - 2 ) + (1 - p) - loge 



lo g — n h 1 — log (e — lj 



1 



where / = average excitation potential 



e = dielectric constant 

 The factor W is the maximum energy that can be transferred to an electron. 

 According to Bhabha [13], W has the form 



2m(E 2 - M 2 c A ) 



W = 



m 2 c 2 + M 2 c 2 + 2mE 



where m = mass of electron 

 M = mass of particle 

 E = energy of particle 

 c = velocity of light 

 The correction contained in the first formula is always negligible either when 

 v <K ce~W or when, the particle passes through media of low density, i.e., 

 with c~l. Under these conditions it reduces to the simple stopping 

 formula. 



The second expression diverges rapidly from the simple stopping formula 

 for increasing velocity, and as v — > c it approaches asymptotically the 

 constant value given by the expression [15] 



dE __ 2xiV^ 4 2 2 

 dx mc 2 



where h = Planck's constant 



/, m 2 c 2 W , A 

 V° g ZAW 2 +V 



