78 ISOTOPIC TRACERS AND NUCLEAR RADIATIONS [Chap. 4 



N = number of atoms (not molecules) per cc 



2 = charge number of particle 

 Z = atomic number of absorbing atoms 



v = velocity of particle 



= v/c 



1 = average excitation potential of atom 



The range in energy over which the energy loss or stopping formula remains 

 valid is determined (1) by the upper and lower limits inherent in its derivation 

 and (2) by the appearance of effects at both low and very high energies which 

 are not taken into account by the formula above. 



The lower limit arises from the assumption that the particle's velocity 

 is greater than the highest orbital electron velocity in an absorbing atom, 

 i.e., E ^>> MEk/wi, where E K is the K electron ionization potential and M and 

 m are the masses of the particle and electron, respectively. This condition is 

 necessary to ensure the constancy of the particle's charge ze. When the 

 velocities of the incident particle and the electron are comparable, the prob- 

 ability of electron capture is no longer negligible and the charge of the particle 

 becomes indeterminate since it then fluctuates with successive collisions. 

 The probability of electron capture and loss as the particle is slowed down 

 and brought to rest has not been completely formulated as yet, and only 

 empirical corrections for its effect on the stopping formula are possible. 

 This uncertainty in the charge leads to considerable error in attempting to 

 calculate the rate of energy loss for protons with energies less than ~ 0.2 mev 

 and alpha particles with less than ~ 1.0 mev. The same difficulty makes the 

 stopping formula generally inapplicable to very heavy and multiply charged 

 particles such as fission fragments since the charge varies more rapidly and in 

 some unknown way with the velocity even at very high energies (> 100 mev). 



The highest energies for which the simple stopping formula above can be 

 applied are limited by the appearance at very great energies of the Fermi 

 effect described in the next section. 



The average excitation potential I used in the stopping formula is usually 

 considered to be a function only of the atomic number of the absorbing 

 atoms and, in particular, to be independent of the kind of incident particle 

 and its energy. As yet no completely satisfactory formula for its calculation 

 has been provided. Its value can be determined empirically, however, with 

 sufficient precision from accurately known range-energy data for protons or 

 alpha particles in the absorber for which I is to be found. This is often done 

 by adjusting the calculated range-energy curve to pass through the experi- 

 mentally determined points by allowing / to be an arbitrary parameter 

 [6,8,9,10]. The simplest and perhaps best theoretical value appears to be 

 given by Bloch's conclusion [11] that / is directly proportional to the atomic 

 number and by using Wilson's [30] empirical constant of proportionality, or 



