148 PENETRATION PHENOMENA IN LIQUID WATER 



The considerations above were oversimplified in one important re- 

 spect: it was assumed that there is a unique binding frequency w/27r. 

 The quantum theory of dispersion shows, however, that even for the 

 hydrogen atom, with its single electron, it is necessary to treat the atom 

 as an assembly of an infinite number of different "virtual" oscillating 

 electrons; each type has effective number (or oscillator strength) /„ and 

 frequency 27rajn = ejh. Here, X) //i = 1 • the total effective number of 



n 



such oscillators corresponds to just a single electron. Hence, for the 



stopping power of a gas containing N hydrogen atoms per unit volume, 



we have 



dE ^^ / 2mv\ 2mv^ 



-_^ / 2wr\ 



X;/n( 2/ciVln j = 2KN\n 



In this last expression I is defined by 



In/ = X)/nlnen (3) 



n 



The sum should be understood as embracing both the summation over 

 discrete and the integration over ionization states. The quantity / is 

 called the mean excitation energy or mean excitation potential. It is in 

 effect a geometrical mean of all possible excitation and ionization ener- 

 gies of the atomic system, each weighted by the corresponding oscillator 

 strength. For hydrogen atoms an exact calculation can be carried 

 through and leads to the value I = 15.00 ev, some 10 per cent greater 

 than the ionization potential (13.60 ev). 



For atoms with more than one electron the same treatment is appli- 

 cable, although it must cope with the vastly more complex dispersion 

 model for the electronic frequencies. There is one important restriction, 

 however: in order for the final formula given below to be valid, it is 

 necessary for the incident particle to be "fast" not only with respect to 

 ^0, but also with respect to all orbital electron velocities in an atom of 

 the medium — or, in effect, to the greatest of these, namely that of a K 

 electron, approximately Zvq. This is a most severe restriction. For 

 oxygen, as an example, Z = 8, and at a velocity of S^o a proton has 

 energy 1.6 Mev, a deuteron 3.2 Mev, an alpha particle 6.4 Mev. If, how- 

 ever, the particle is fast compared to Zvq, the stopping power is given 

 by the celebrated formula of Bethe: 



dE 2mv^ 



= 2kNZ In (4) 



dx I 



where N is again the number of atoms of the stopping medium per unit 

 volume, Z the number of electrons per atom (atomic number), and I 



