Sec. 4.4] PROTONS, DEUTERONS, AND ALPHA PARTICLES 83 



K or L shell and n is the number of K or L electrons. This correction is 

 similar in its effect to the procedure followed by Duncanson [9] in treating 

 both / and Z as empirical parameters in adjusting the simple stopping formula 

 to the observed data. For very high energies, however, the correction is not 

 important and the actual value of Z can be used. 



4.4. Stopping Formula for High-energy Particles. An important correc- 

 tion to the simple stopping formula was introduced by Fermi [15] for particles 

 with very great energy traversing condensed materials. The effect that 

 necessitates the correction arises from changes induced in the field of the 

 passing particle due to local polarization of atoms in media that possess 

 dielectric properties, or more specifically, media with a dielectric constant 

 differing from unity. The rate of energy loss under these conditions is found 

 to be smaller than that computed from the simple relativistic stopping 

 formula which at very high energies increases as the logarithm of energy. 

 The older stopping theories considered the field of the particle to be inde- 

 pendent of the properties of the absorbing media. Thus, as a result of the 

 Lorentz contraction of the particle's field, the collision radius, and conse- 

 quently also the rate of energy loss, increases without limit as v — ■» c. This 

 concept is no longer valid when polarization of the medium is taken into 

 account. When the field of the particle is analyzed into its Fourier com- 

 ponents, it is apparent that when v > «r ! - each component of the field is 

 propagated with a different velocity in the medium, or is subject to dispersion. 

 Those frequency components with velocities less than v form wave fronts or 

 bow waves at various angles with the direction of the particle. As v — > c the 

 field is greatly altered as a result of dispersion but, most important, it 

 approaches a limiting form in which all frequency components assume fixed 

 phase relations. The distance at which atoms can be excited by the limiting 

 form of the field remains finite and, correspondingly, the rate of energy 

 loss approaches asymptotically a finite limiting value. 



In accordance with this description the reduction in rate of energy loss 

 from the value indicated by the simple stopping formula becomes most 

 important when the velocity of the particle exceeds ce~ ] '-. It is apparent that 

 for electrons the effect of polarization should become appreciable for energies 

 greater than several mev. The correction is of little use in this case, however, 

 because energy loss by radiative collision also becomes significant and a 

 stopping formula based only on ionization energy loss is no longer justified. 

 This applies, fortunately, only to electrons, and the stopping formula can be 

 validly used, so far as is known, for heavier charged particles for all prac- 

 ticable energies. The Fermi corrections, therefore, must be adapted in the 

 appropriate energy range. In general, the corrections become appreciable 

 for protons and alpha particles with E <~ 10 4 mev and for mesons with 

 E ~ 10 3 mev. 



