PRINCIPLES OF RADIOLOGICAL PHYSICS 



89 



4-1 b. Deflection of the Track. We consider here briefly the deflections experi- 

 enced by a heavy particle along its track, which we have disregarded so far. The 

 rare occurrence of rather large sharp deflections is governed by Rutherford's 

 formula, Eq. (13). Very small deflections take place steadily in the course of 

 every collision, elastic or inelastic, experienced by a particle. These successive 

 deflections are directed in various ways and frequently tend to cancel each other. 



2.4 

 2.3 



CD 2.2 



z 

 o 

 2 1 



2.0 



1.9 



1.8 



1.7 



o 



ir 

 a. 



oc 

 o 



1.20 



1.15 



1.10 



ifi 



1. 05 E 

 < 



Q. 



1.00 V 



0.95 £ 



0.90 



0.85 



3 4 5 6 8 10 15 20 30 60 100 200 



MEAN RANGE iR), cm 



Fig. 1-54. Straggling of protons and a. particles. The difference between the extra- 

 polated range fextr and the mean range R, in per cent of the latter, is given as a func- 

 tion of the mean range. {Livingston and Bethe, 1937.) 



On the whole, however, the direction of the track eventually departs more and 

 more from the initial direction, in the same way as a particle in Brownian motion 

 eventually gets away from its initial position. The distribution of particles 

 about their initial direction can be described approximately as a gaussian dis- 

 tribution with an additional extended tail which arises primarily from occasional 

 large-angle scattering (see Hanson et al., 1951). The gaussian portion has a 

 width corresponding to a mean square deflection of 



<,.> = ^rN(Z'J^ Z).'e< ^^ 



p2p2 



l.2)t 



(30) 



where B is the solution of the equation 



B -InB = \n 



NiZ^ + Z)z^ 



t 



h^ 



5.27Z?^[1 + 3.33(Zc/137y)2] mh>^ 



(30') 



ze, p, and v are the charge, momentum, and velocity of the incident particle, Z is 

 the atomic number of the material, A^ the number of atoms per unit volume of 

 the material, t the thickness of material traversed, ni the mass of the atomic elec- 

 trons, h the Planck constant, and c the velocity of light. This formula is written 

 so that it holds for relativistic speeds of the incident particles and for incident 

 electrons as well as for heavy particles. [Notice the resemblance of the first 

 factor of Eq. (30) to the Rutherford equation, Eq. (13) ; the addition of Z to Z^ 

 in the numerator is a rough attempt to take into account the scattering by the 

 atomic electrons.] Occasionally minute deflections may not tend to average out 

 but pile up to simulate a steady curvature of a track. 



