Scattering of cc and (3 Particles by Matter. 679 



Neglecting the second term, the average deflexion per 



atom is — . We are now in a position to consider the 



relative effects on the distribution of particles due to single 

 and to compound scattering. Following J. J . Thomson's 

 argument, the average deflexion 6 t after passing through a 

 thickness t of matter is proportional to the square root of the 

 number of encounters and is given by 



Q t — — - slir\X" . n . t = -g~ s/iriit, 



where n as before is equal to the number of atoms per unit 

 volume. 



The probability p x for compound scattering that the 

 deflexion of the particle is greater than <£ is equal to e~$ < 9t • 



0,7T 3 



Consequently $ 2 = -rr- b 2 nt log p v 



Next suppose that single scattering alone is operative. We 

 have seen (§ 3) that the probability p 2 of a deflexion greater 

 than </> is given by 



p 2 =?-b 2 . n. £cot 2 c/>/2. 



By comparing these two equations 



^logp^-'lSl^cot 2 ^, 



$ is sufficiently small that 



tan 0/2 = #2, 



p 2 logp 1 =—'72. 



If we suppose p 2 = '5, then p l — '24r. 



If P2 = % pi = *0004. 



It is evident from this comparison, that the probabilitv for 

 any given deflexion is always greater for single than for 

 compound scattering. The difference is especially marked 

 when only a small fraction of the particles are scattered 

 through any given angle. It follows from this result that 

 the distribution of particles due to encounters with the atoms 

 is for small thicknesses mainly governed by single scattering. 

 No doubt compound scattering produces some effect in 

 equalizing the distribution oF the scattered particles ; but its 

 effect becomes relatively smaller, the smaller the Fraction 

 of the particles scattered through a given angle. 



