LESIONS WITH ACCELERATED PARTICLES 331 



a heavy particle, e.g., oxygen has 4 times less straggling than a proton. For 

 a given range, experimentally minimum range variation is achieved by 

 making A '•* (initial energy spread) as Small as possiblc. Thus ideally, one would 

 wish to accelerate particles with the proper energy for each range used in 

 order to have optimum ionization properties. 



As a parallel beam of particles enters an absorber, due to multiple elastic 

 scattering there will be a radial spread in the beam. Having crossed a dis- 

 tance X in an absorber, the mean square radial distance r~ from theoretical 

 straight line trajectory is 



7 rr 1/3 ^ Z^ 

 where 6" is the mean square of the angular spread (in radians) 



where T is the kinetic energy, Z the atomic number of the absorber, p is its 

 density, z is the atomic number of the fast charged particle, and a c a con- 

 stant that depends on the allowable elastic scattering angles 6. For the same 

 velocity a lighter particle has a longer range and vmdergoes more scattering 

 than a heavier one. Also, the root mean square angle of scattering is propor- 

 tional to the rate of energy loss; thus at high energy, where the rate of energy 

 loss is low, there is less scattering than at low energy; the radial spread of the 

 beam will be less if an absorber of high atomic number is vised. After a beam 

 has emerged from an absorber, to the air, the spread of the scattered beam is 

 much more noticeable than in a solid absorber, since in air there are greatly 

 reduced numbers of collisions. The energy transfer to the absorbing medium 

 along the path of an ionizing particle has been calculated by Bohr (see Evans, 

 1955). Over a wide range of energies the rate of energy loss is almost in- 

 versely proportional to the kinetic energy T of the particles {T'^'') ; thus 

 low energy particles ionize much more heavily than high energy ones. The 

 relationship breaks down near the end of the range where the particles pick 

 up electrons as they stop, and also at very high energies, near 1 Bev per nu- 

 cleon, where relativistic effects appear. Comparing protons to heavier acceler- 

 ated particles of charge z, we find that at the same velocity the heavier particles 

 transfer more energy to the medium, in proportion to z^. The rate of energy 

 loss of various individual particles is plotted in Fig. 2 as a function of their 

 kinetic energy per nucleon. Unlike x-rays, which show great dependence of 

 energy transfer on the atomic number of the absorber, heavy accelerated parti- 

 cles interact mostly with atomic electrons; thus the stopping per electron is 

 nearly the same for all elements. For accurate dose calculations it is necessary 

 to know the stopping power of the tissue, relative to nitrogen or other gas, 

 where the ionization is measured. While more experimental work needs to 

 be done in this field, some stopping powers are accurately known (Tobias 

 et al, 1952). 



