156 PENETRATION PHENOMENA IN LIQUID WATER 



of the particle. Both increase, relatively, as the energy declines. The 

 contribution of rotational excitation is, of course, much the smaller of 

 the two. Although for fast particles they are beyond the discrimination 

 of present stopping-power data, it is quite possible that excitation of 

 vibrations may play a detectable role for particles having velocity of the 

 order of magnitude? of, or only a few times greater than, t^o- 



We shall now examine a specific and most important example, molec- 

 ular hydrogen. The difference in stopping power between the molecule 

 and the extreme of isolated atoms has already been emphasized, for 

 /isolated = 15.00 ev, whcrcas /bound ~ 18 ev, the former being an exact 

 value from theory and the latter a very approximate one from experi- 

 ment. A theoretical treatment of the stopping power of molecular hy- 

 drogen, or, in effect, a theoretical calculation of /bound, has not yet been 

 achieved. It would be a valuable contribution and appears to lie 

 within the realm of possibility. Values of 8„ for the molecule are, of 

 course, known with great accuracy from analysis of the molecular 

 spectrum. They show a crude, but necessarily far from close, similarity 

 to the corresponding values for the hydrogen atom. The disparity, 

 which arises chiefly from the difference in binding energies for the dif- 

 ferent states, is enhanced by the fact that transitions induced by the 

 passing particle may not appreciably alter the internuclear separation — 

 this being the demand of the Franck-Condon principle — and hence, be- 

 cause all excited states have considerably greater equilibrium separa- 

 tions than the ground state, always leave the excited molecule in a high 

 vibrational level. This coupled vibrational excitation is about 3^ - 1 ev 

 for most excited (attractive) states. For all ionization processes it is 

 slightly greater than 1 ev. For "repulsive" excited states, of course, a 

 great additional amount of energy is transferred as momentary potential 

 energy of the molecule. 



Although a detailed calculation of /bound for molecular hydrogen will 

 not be attempted here, a rough estimate may readily be given. We write 

 Eq. 3, appropriately generalized to the many-electron case, in the form 



■n^^iE/.lni" (7) 



where P is the ionization potential. In the single instance in which this 

 expression can be evaluated exactly, namely atomic hydrogen, it is 

 found that I/P = 1.102, and it has been common practice to assume 

 that this ratio has the same value for certain other related atoms and 

 molecules; thus, for molecular hydrogen (for which the ionization po- 

 tential is 15.43 ev), it is assumed that / = I.IOP = 17 ev. This pre- 



