146 PENETRATION PHENOMENA IN LIQUID WATER 



in fact varies rather slowly with the velocity. By a fast particle is meant 

 in this paper one with velocity great compared to vq, the orbital velocity 

 of an electron in the normal Bohr orbit of the hydrogen atom (vq is a 

 fundamental atomic constant compounded of universal constants: vq = 

 e^/h. = 2.188 X 10^ cm per sec = c/137, where the symbols e, h = 27rh, 

 and c have their customary meanings). At the velocity vq an electron 

 has energy 13.6 ev, a proton 25 kev, a deuteron 50 kev, an alpha particle 

 99 kev, and a fission fragment (of mass 120) 3 Mev. For fast particles 

 the stopping power arises almost entirely from individual acts of energy 

 transfer from the particle to electrons of atoms close to the path of the 

 particle; these transfers range in magnitude from small ones, of roughly 

 5-15 ev, which excite the atom, to greater ones, of energy ranging from 

 the ionization potential P upward, which result in a free electron and a 

 positive ion — and so on, with decreasing probability, to very great energy 

 transfers which ionize the atom and produce a very fast secondary elec- 

 tron. The mean energy transfer is always of the order of magnitude of 

 20 ev. 



Theoretical treatment of the stopping power for fast particles has 

 been developed in satisfactory detail, thus far, only for the case of a 

 medium composed of isolated atoms of low atomic number. (In this 

 respect it is fortunate that only light atoms are important for radio- 

 biology.) Indeed, a variety of different modes of treatment is avail- 

 able. A particularly illuminating approach is the so-called "method of 

 impact parameters," which will now be sketched very briefly for later 

 reference. (An alternative model will be demonstrated in Section IV.) 

 We restrict the discussion to fast particles having small positive charge 

 and mass of atomic magnitude, of which the only ones now experimen- 

 tally accessible are alpha particles and ions of hydrogen (H, D, or T). 

 The other practically important cases, namely fast electrons and fast 

 heavij ions ("recoils," fission fragments), will be mentioned later. 



For these heavy particles the momentum is extremely great compared 

 to the momentum change in any collision in which momentum is trans- 

 ferred to an atomic electron of the medium. Therefore the motion of 

 the heavy particle is only insignificantly affected by any one momentum 

 (energy) transfer, and the excitation or ionization act can be treated as 

 caused by a uniformly moving Coulomb center of force. If we consider 

 two concentric cylinders with radii p and p + dp and axes on the path 

 of the particle, we easily find from the laws of classical dynamics the 

 energy transfer to electrons initially lying between the cylinders: 



^TTZ^e^ p dp 

 — dEry = :r-n— —dx 



■'p 



mv^ p^ + R' 



