80 ISOTOPIC TRACERS AND NUCLEAR RADIATIONS [Chap. 4 



energy loss and range data for various kinds of particles when detailed 

 calculations of the stopping number have been made for a wide range in 

 velocity for any one particle. 



4.3. Stopping-formula Corrections. When applied to protons, deuterons, 

 and alpha particles, the stopping formula given in the last section is reliable 

 over a wide range of energy without further modification, but at low energies 

 (< 10 mev) and very high energies (> 1,000 mev) corrections for more 

 complicated effects must be inserted. 



The largest error at very low energies is introduced by electron capture and 

 loss, making the particles' time-averaged charge a function of velocity, The 

 actual energy loss therefore is less than computed from the formula and for 

 obvious physical reasons does not become infinite when v — > as indicated 

 by the stopping formula. As yet, however, no general theoretical formula- 

 tion of a correction for this effect has been given. The error becomes 

 appreciable for alpha particles below 1.0 mev and for protons below approxi- 

 mately 0.2 mev. 



A second correction, formulated by Bethe [6], for low-energy particles 

 arises from the reduced contribution of the K electrons, and in principle, also 

 the L and M electrons, to the stopping power of medium and heavy atoms due 

 to the screening effect of the outer orbital electrons. The correction for the 

 reduced stopping power of the K electrons is contained in modifications of the 

 stopping number B which, as given by Bethe [6], have the forms 



For E < — Z%J H , 



m 



B= (Z- 1.81) log ~+£* 



M 



For E > — Z\ {i I H , 



m 



where E = energy of particle 

 M = mass of particle 

 m = electronic mass 



I h = ionization potential of hydrogen atom, 13.60 ev 

 /' = average excitation potential of electrons outside K shell 

 / = average excitation potential of entire atom 

 Zeff = effective nuclear charge in K shell 

 The numbers B K and C K are plotted in Fig. 15 as functions of r\ where 



mE 



71 ~ MT^zJ (f 



