59 



the only mechanism I can think of, but we still have the radical diffusion picture 

 to build up. Regardless of the details of this mechanism, it is of some interest 

 to see whether it gives anything reasonable or not. 



I shall make this development as brief as possible. The radicals are all 

 confined within a spherical region of radius r, where r increases with time ac- 

 cording to r z = \ u R t in which X is a distance of the order of a radical jump dis- 

 tance; u R is a velocity, and t is the time. We shall not define precisely what is 

 meant by r, but it is clearly something like the square root of the mean square 

 radius for the diffusing radicals. 



The random motion of the radicals is actually a sort of confinement. If we 

 say that N radicals are within the volume v = 4 tt r^, the rate at which they dis- 

 appear by mutual reaction is 3 



- dN_ _ (N) 2 

 dt K v 



where k is the reaction rate constant. Substitution for v in terms of r, and r in 

 terms of t yields: 



- dN_ 3k N 2 



dt " 4tt (\u R t) 3 ' 2 



This equation integrates to: 



N _ 1 



N o 



2k N t r 



R o 



where the initial conditions v D = 4 it rX* - 4 tt \ u t 3 have been inserted. 



"T T 



At the time t = CD , the fraction of radicals 



Nco 1 



N o j + 2k N otp 



v o 



remain uncombined. The fraction 1 - Nco have combined with each other. 



No 



Thus a certain fraction of the radicals will combine and a certain fraction will 

 get away. This is exactly the kind of separation into two classes we wanted in 

 order to explain the F and R reactions. There is a distribution between the two 

 processes; it depends on the magnitude of the parameter y - 2kN t Q 



I think probably the most efficient procedure is just to summarize our calcu- 

 lations. For details you can refer to the paper by Samuel and Magee (7). We had 

 to put in a distribution function for the number of radicals formed in primary 

 events and definite values for the r 's. We don't have good information as to what 

 the actual distribution is of N Q for the actual events of ionizing particles, but the 

 calculation does not turn out to be too sensitive to that. If you adjust parameters 

 to obtain the observed value, say, for what fraction of radicals goes the F re- 

 action and what fraction goes the R reaction for gamma rays, you find that the 



