38 ISOTOPIC TRACERS AND NUCLEAR RADIATIONS [Chap. 2 



A quantum mechanical treatment of scattering that is valid for all gamma- 

 ray energies has been carried out by Klein and Nishina [2] on the basis of 

 Dirac's relativistic theory of the electron [7]. The total electronic cross 

 section is given as the sum of two scattering terms. The first, denoted by 

 s a e , accounts for the reduction in intensity due to loss of gamma photons 

 scattered out of the beam. The second, denoted by a o e , is the scattering- 

 absorption cross section arising from the loss in energy suffered by scattered 

 photons. The total electronic scattering cross section is given in the form [2] 



2tt^ 



2(1 -f a) 2 1 + a . .. _ . 1 + 3a 1 . . 



log (1 + 2a) - ,., , x 2 + w- log (1 + 2a) 



_a 2 (l + 2a) a 3 &v ' (1 + 2a) 2 ' 2a 



hv E 



where a = — ^ = jr^ = gamma-ray energy in units of m c 2 where E y is 



energy, mev 

 e = electronic charge 

 m = electronic rest mass 

 For very low and very high energies reduced expressions may be used. For 

 low energies a < 1, a e can be expanded as [3] 



Sire* 



3m 2 c 4 



CFe = 



(1 - 2a + 5.2a 2 - 13.3a 3 + 32.7a 4 +••• ) 



For high energies a ^>> 1, s a e reduces to 



2ire 

 m 2 c 



1 (-T + y- ^g 2a - ~ 2 log 2a) 



4 \4a la a- / 



The scattering-absorption coefficient a c e is given by the equation [2] 



2«H 



aO e o a 



2(1 + a) 2 1 +3a (1 + a)(l + 2a - 2a 2 ) 



a 2 (l + 2a) 2 (1 + 2a) 2 + a 2 ( 1+ 2a) 2 



4a 2 ( \ + cl 1 , 1 \ . f s, , 



~ 3(1 + 2a) 3 " \~a^ ~ Ta + W) bg (1 + 2a) 



As for <x e , similar reduced expressions may be used for low and high gamma 

 energies. For low energies a < 1, and 



87T6 4 



iO"e — 



3m 2 c 



2,-4 



(a - 4.2a 2 4- 14.7a 3 - 46.17a 4 + • • • ) 



For high energies a >>> 1, 



« 4 /l, ry 1 \ 



a<Te = —2-1 I log 2a — — I 



m^c* \ a 6a/ 



The total electronic scattering cross section as indicated before is the sum 



