80 RADIATION BIOLOGY 



The same type of consideration serves to find the probabihty Pn(x) 

 that any one particle undergoes a given number n of colUsions along a 

 track distance x, 



Pn{x) = exp(-vx){vxy/nl (28) 



where n! indicates the "factorial of n," i.e., the product 1 • 2 • 3 • • • • n. 

 This is the well-known Poisson distribution law. The expected number 

 of collisions along a track length x equals vx; the actual number is n. 

 The calculation of the probability P„ for various values of n shows that the 

 most probable value of n equals the next integral number below x. An 

 estimate of the likely departures of n from x is obtained by calculating the 

 variance, or mean square deviation, of n. This quantity turns out to equal 

 vx itself, meaning that departures of the order of the root mean square 

 deviation, ■\/vx, are frequent. 



3-6b. Distribution of Activations Produced by Secondary Electrons. As 

 we have seen in Sects. 2-4c and 3-ld, the great majority of the secondary 

 electrons have a rather low energy, even though their aggregate energy 

 amounts to about two-thirds of the energy lost by a fast particle. Elec- 

 trons whose energy amounts to no more than 100 or 200 ev can transfer 

 energy only to the external electrons of atoms, and this only when passing 

 right through or very close to an atom. On the other hand, every passage 

 in the proximity of an atom has a fair chance of leading to a collision with 

 energy transfer because these electrons are slow. Furthermore, low- 

 energy electrons experience frequent, repeated large-angle deflections. 



Therefore low-energy secondaries dissipate most of their energy within 

 a short distance from their point of origin. This distance is of the order 

 of 10 A in solid or liquid materials and about 1000 times as large in gases 

 at atmospheric pressure. This energy is dealt out in the form of activa- 

 tions (excitations or ionizations) at points irregularly scattered in the 

 proximity of the atoms from which each electron was ejected. These 

 activations are said to form a "cluster." 



Not all the activations in a cluster are produced directly by the secondary 

 electron which originates the cluster. Some are produced by other electrons 

 which are "secondary to the secondary," i.e., which are ejected with sufficient 

 energy as a result of ionizing collisions within the same cluster. 



Little detailed information is available regarding the statistical distribution of 

 the location of activations within clusters. The distance traveled by a secondary 

 electron depends on its initial energy, is roughly inversely proportional to the 

 density of the material (see Sect. 4-2a), but depends little on the chemical structure 

 of the material. The clusters include the negative ions which arise by capture of 

 very slow electrons after protracted migration (see Sect. 3-ld). In the course of 

 the migration the electrons diffuse far away from their point of origin, probably 

 up to more than 100 A in water and 1000 times as far in air (even though air 

 contains much oxygen). Therefore, negative ions lie, on the whole, much farther 

 from the center of a cluster than the positive ions or the simple excitations. 



