Sec. 3.4] BETA PARTICLES 53 



field of the electron, and with it the rate of energy loss. Fermi's stopping 

 formulas should be used when v ~ ce -I/ -, where e is the dielectric constant of 

 the absorber (see Sec. 4.4). 



Although these formulas may account adequately for the average energy 

 loss by ionization, in practice they are of little use in calculating the actual 

 absorption and range of beta particles since other factors also strongly 

 influence absorption. (1) Unlike heavy charged particles, an electron may 

 lose a large fraction of its energy in a single collision, and hence straggling 

 occurs over a large portion of the range. (2) Beta particles are repeatedly 

 scattered, often into large angles, making it impossible to correlate their 

 actual path length with linear absorber thickness and also enhancing the 

 straggling in any given direction. (3) Radiative loss, which is not taken into 

 account in the stopping formula, increases with energy and becomes, at high 

 energies, the more important factor in energy loss. 



3.4. Radiative Collision Losses. At energies greater than several million 

 electron volts radiative collisions account for an appreciable fraction of the 

 energy loss. For this and higher energies, ionization losses increase very 

 slowly, approximately as the log E, whereas radiative loss increases directly in 

 proportion to E. At energies higher than 10 to 100 mev, depending on the 

 atomic number of the absorber, radiation is the principal process for energy 

 loss. Radiation emitted directly from beta particles traversing an absorber 

 is known as Bremsstrahlung and is the source of the continuous or white x-ray 

 spectrum. 



The average energy lost per centimeter path length by radiation is given 

 by an integral of the form 



\dx /rad J0 



hv$ v dv 



where N = number of atoms per cc 



v = frequency of radiation 

 The function <£„ has been derived by Bethe and Heitler [3], but in its general 

 form, which includes the effect of screening by the outer orbital electrons, the 

 function can be integrated only numerically. In two special cases of physical 

 interest, however, the function reduces to integrable expressions and leads to 

 the following formulas for calculating the average rate of energy loss by 

 radiation: 

 For m c 2 <K E <<C l37m c 2 Z~^ (no screening), 



_(^_E\ _NZ*(e* Y E ( ilo 2E 4\ 



\dx / rad 137 \m c 2 ) \ 3 m c 2 3/ 



