STOPPING-POWER THEORY FOR MON ATOMIC GAS 147 



where ze is the charge of the particle, — e that of an electron, ni the elec- 

 tronic mass, V the velocity of the particle, and n the number of electrons 

 per unit volume; R is a parameter called the collision radius which 

 measures the "size" of the Coulomb field and is equal to ze^/mv^. To 

 find the stopping power we simply integrate over all permitted values of 



The quantity /c, defined as 2'Kz'^e^/mv^, is called the stopping parameter. 

 The factor 2/cn appears in all formulae for stopping power, for all media. 

 It determines the order of magnitude of the stopping power; all details, 

 however, are contained in the balance of the expression, in which our 

 interest will therefore be centered exclusively. Many of the intricacies 

 of the problem lie in the determination of p^ax and Pmin- 



Although superficial examination might suggest that Pmin = 0, ap- 

 plication of the laws of quantum mechanics shows that, if one uses the 

 above, essentially classical, formulation, one must set p^i^ = h/mv. 

 (Thus Pyain is just the wave length Xg which an atomic electron has in the 

 coordinate system in which the particle is at rest, and the electron there- 

 fore moves with velocity v; the position of the electron is "uncertain" 

 by Xe, which suggests that Pmin cannot be smaller than Xg, but the fact 

 that Prain = ^e actually requires a careful justification which will not 

 be given here.) Since Xe = h/mv = R{v/zvq), it follows that for fast 

 particles Pnun » ^, and 



_^ = 2«ln^ (1) 



ax Pmin 



It is obviously absurd to set p^ax = °° , for this would predict infinite 

 stopping power. Bohr pointed out in 1913 that an upper limit to p is 

 established by a kind of "dynamic" screening arising from the binding of 

 the electrons in atoms of the medium: a bound electron located at a 

 very large distance p from the path of the particle is perturbed adiabat- 

 ically and no energy is transferred to it. If we consider first that an 

 atom contains one electron bound with frequency aj/27r (binding energy 

 = e = hco), the condition for adiabatic perturbation is "duration of 

 collision" ^p/v » l/oj, so that p^ax ~ v/oi. This value of p^ax is cor- 

 rect in quantum theory as well as in classical theory except for a nu- 

 merical factor, which detailed calculation determines as 2, so that p,nax = 

 2v/w. Hence, for fast particles, 



dE 2mv^ 



= 2Kn In (2) 



dx 8 



