PRINCIPLES OF RADIOLOGICAL PHYSICS 



79 



and is independent of the distance x already traveled. The coefficient v equals 

 the expected number of collisions per unit distance, i.e., the average number of 

 collisions 91 calculated according to the formulas of Sect. 2-4c for a layer of thick- 

 ness t = I. The reciprocal of v represents the average distance I of two successive 

 collisions. Therefore, the number A' decreases along dx by 



-dN = Ndf = Nv dx = N dx/l 



(26) 



This equation requires that the number of particles A' which have traveled a 

 distance k without any collision vary according to the law 



N/No =f=e- 



vx -= Q—x/l 



(27; 



where e represents the number 2.718 . . . and A'o indicates the initial number of 

 particles. The ratio / = N/Nq is the fraction of all particles which travel a 

 distance x without collision, i.e., the probability for any particle to travel that 

 distance without collision. Law (27) may be apphed to the distance traveled 

 without any collision whatsoever or to the distance without any collision of any 

 given type ; the appropriate value of the expected number of collisions v must be 

 introduced in each case. 



en 



S 0.05 



0.02 



Fig. 1-46. Illustration of the exponential law of probability for the distance of succes- 

 sive collisions. The stepwise plot in (a) illustrates the argument presented in the 

 text in the derivation of Eq. (27). The semilogarithmic plot (6) corresponds to the 

 form [Eq. (27')] of the exponential law. 



Figure 1-46 also shows that the semilogarithmic plot of N vs. x follows 

 a straight line with the slope v which is described by the equation 



hiN = In No — vx 



(270 



