62 ISOTOPIC TRACERS AND NUCLEAR RADIATIONS [Chap. 3 



large deflections in a single collision may occur frequently, particularly for 

 beta particles of low energies scattered by heavy elements. For this reason, 

 the effects of elastic scattering should never be underestimated in the evalua- 

 tion of measurements on beta particles. 



Single elastic collisions of slow electrons with nuclei can be calculated from 

 Rutherford's formula [6] since elastic collisions involve only interaction of the 

 coulomb fields of the electron and nucleus. For collisions of high-speed beta 

 particles where relativistic effects are important, a formula has been given by 

 Mott [7] for the intensity (or number) of beta particles scattered into the 

 solid angle doi at an angle 9 from the initial direction. 



l7 /« 2 ZV/i o^\ 1 P 2 i o 7 cos 2 (0/2)' 



n(9) = n N I ^ s ) (1-0) . ia/1 ~ ■ , an + T(3aZ . )' ' 



\2m v 2 / |_ sin 4 9/2 sin 2 6/2 sm 6 (8/2) 



which for small angles and all values of Z reduces to 



n = n N ( ^-, ) (1 - /3 2 ) cosec 4 Q - 

 \2m v 2 / 2 



and for large angles and small Z is 



» ■ n ° N {&} (1 - ^ - "* si,,: !) cosec< \ 



where a = 2ire-/hc = \\zi 



n = number of particles per cm 2 scattered into solid angle dco at 

 angle 9 



n = initial number of particles per cm 2 in beam 



N = number of atoms per cc 



Z — atomic number of scatterer 

 m = electronic mass 

 In all cases the scattering coefficient is proportional to Z-(\ — /3 2 )/V. These 

 formulas apply only to very thin foils in which the probability of more than 

 one collision is small. A criterion for determining if single scattering can be 

 expected in an experimental arrangement has been given by Wentzel [8] 

 which provides that the minimum observed angle of deflection 9 and the foil 

 thickness / must be chosen so that 9 > 4<p where 



cot 



<p __ m v 2 J 2 

 2 '' ~1?Z \wN~t 



. rNt 



where / = foil thickness, cm 



This criterion permits several deflections through angles of the order of <p 

 but determines the magnitudes of 9 m \n and / such that the contributions due to 

 multiple collisions are negligible at an angle > 9 where the scattered beam is 

 detected. For aluminum the maximum thickness for single scattering is 



