88 RADIATION BIOLOGY 



it is difficult and of no great interest to determine where each particle 

 actually stops after having become " slow." Figure 1-53 gives data on the 

 effective range of protons and on the rate of energy dissipation and 

 mean residual energy. 



The following rules, based on the discussion in Sect. 2-4b, serve as a guide to 

 estimate the ratio of the stopping powers of different materials: 



1. The stopping power is approximately proportional to the number of elec- 

 trons per unit volume of material. 



2. The number of electrons per unit volume equals the product of Avogadro's 

 number N = 6.0 X 10^^, the density p, and the atomic number Z divided l)y the 

 atomic weight A of the material, 



No. of electrons per unit volume = NpZ/A (29) 



3. The ratio of atomic number to atomic weight equals 1 for hydrogen, } 2 for 

 the other light elements, and a little less than >^ for heavier elements (approxi- 

 mately 0.4 for lead). If a standard value of Z/A, e.g., yi, is adopted, the number 

 of electrons appears to be simply proportional to the density of the material. 



4. The internal electrons of heavy elements contribute little to the energy dis- 

 sipation by heavy particles (see Fig. 1-36) and should be discounted accordingly. 



5. When the material under consideration contains different chemical elements, 

 the number of electrons per unit volume corresponding to the different elements 

 must be added. We must therefore consider separately the partial densities 

 Pi, p2, ... of the component elements and the corresponding values Zi/Ai, 

 Z2/A2, . . . of the ratio Z/A. The number of electrons is then 



Nip,Z,/Ar + P2ZJA2 + ■ • •) (29') 



If a standard value of Z/A is adopted, as in item 3 above, then the number of 

 electrons is again proportional to the total density: 



N{p, +P-2+ ■ ■ ■)Z/A = NpZ/A 



6. The stopping power of a material for different heavy particles of equal 

 velocity differs only in proportion to the square of their charges. 



A discussion of various influences on the penetration of particles has been pub- 

 lished recently by Platzman (1952). 



4-1 a. Straggling. The actual energy dissipation along the tracks of individual 

 particles may depart a little from its expected value, owing to statistical fluctua- 

 tions of the distance of collisions and of the amount of the energy lost in individual 

 collisions. This leads to small random variations of the total track length; for 

 example, particles which experience a particularly large energy loss stop short of 

 the effective range. This variability of the actual range is called "straggling." 

 The tail end of the Bragg curve in Fig. 1-52 is rounded off because of straggling. 



The amount of straggling can be estimated from the data on the statistics of 

 collisions involving various amounts of energy loss. Straggling is rather small 

 for heavy particles of moderate energy, but it increases with the energy of the 

 particles. Figure 1-54 gives illustrative data on straggling. 



