STOPPING POWER OF A SOLID OR LIQUID 159 



are essentially "free," and naive application of the method of impact 

 parameters, and Eq. 1 with Pmax = 2t;/w, would lead to a paradox for 

 these electrons, for which co '^ 0. Resolution of this paradox, and for- 

 mulation of the theory for the contribution to the stopping power by 

 conduction electrons, have been given by Kramers (23) [cf. also A. 

 Bohr (24)]. In essence, it is found that the effective "radius of action" 

 of the field of the moving charged particle is, after the polarization of 

 the medium by the passage of the particle has been taken into account, 

 not simply p^ax = 2y/c<j, but rather p^ax = 2<;(aj^ + 4xe^n/m)~'^ 

 (Here n is the number of electrons per unit volume ; thus Pmax depends on 

 the density.) For all media except metallic conductors this quantity is 

 effectively the same as Iv/oi, since 47re^n/TO is always smaller than J^ 

 and may be very much smaller, and the state of aggregation is con- 

 sequently without influence. For metals the dynamic screening (by 

 conduction electrons) does not exist, for v/o) -^ oo , and it is p^ax = 

 2r(47re^w/m)~'^ that must be used in Eq. 1. This leads at once to a 

 stopping power for valence electrons different in the gaseous and metallic 

 states. 



Beryllium is the only substance in which this effect has, thus far, been 

 demonstrated. Beryllium vapor cannot be studied, but A. Bohr (24) has 

 computed its stopping power for 1-Mev protons and also that of metallic 

 Be, the latter by adding to the stopping power of the K shell the stopping 

 power of the conduction electrons calculated by the theory of Kramers. 

 The result, which is smaller by 7.4 per cent than the theoretical value 

 for the vapor, agrees very well with accurate measurements on Be foils 

 by Madsen and Venkateswarlu (25), who found a difference of 9.1 ± 

 1.8 per cent. 



As always, the influence of inner electrons, which are essentially un- 

 affected by changes in the state of aggregation of the atoms, is such as 

 to suppress the influence of valence electrons, which are so affected, and 

 with presently achievable experimental accuracy the polarization effect 

 is detectable only for metals of low atomic weight. Metallic lithium 

 should exhibit an effect, but one less pronounced than that in Be because 

 it has only one conduction electron. Recent data do provide a value of / 

 for lithium, but a detailed analysis has not yet been given. 



In the case of non-metallic solids and liquids, virtually nothing is 

 known, either from experiment or theory, about possible effects of state 

 of aggregation on the stopping power. Such effects again involve only 



years, has been treated theoretically by Fermi, Halpern and Hall, Wick, and others. 

 It is negligible (that is, of the order of magnitude 0.1 per cent, at most) in practice 

 for fast (but non-relativistic) particles — particles having velocity v in the range 

 vq<^v <^ c — except in the case of metals, discussed above. 



