Sec. 4.2] PROTONS, DEUTERONS, AND ALPHA PARTICLES 77 



the continual transfer of small fractions of the particle's kinetic energy to the 

 excitation and ionization of atoms lying sufficiently near the path of the 

 passing particle to be affected by its field. The greatest distance at which 

 the field of the particle is effective in exciting atoms is of the order of 

 2v/v(\ — /3 2 ) 1 -, where v is the particle's velocity and v is the lowest vibra- 

 tional frequency of the electrons. This corresponds to the radial distance 

 beyond which the force on an electron due to the passing particle changes 

 slowly compared to the electronic period in the atom. The interaction with 

 distant atoms therefore may be treated by an adiabatic approximation which 

 demonstrates that an electron, although temporarily perturbed, is left in its 

 original state in an atom and does not absorb energy from the particle. A 

 minimum radial distance from the path at which an atom can be excited is 

 limited by the De Broglie wavelength X = //(l — /3' 2 ) 1/2 /2ir mv, since an 

 approach closer than this has no significance. The relativistic factor 

 (1 — /3 2 )^- is included to take into account the Lorentz contraction of the 

 field at high velocities. In principle, the rate of energy loss is found by calcu- 

 lating the amount of energy transferred to electrons of all those atoms of the 

 absorber lying within the cylindrical volume surrounding the path of the 

 particle and defined by the maximum and minimum radii indicated above. 

 This calculation has been subject to numerous theoretical studies which, 

 though differing somewhat in detail, have led to formulas that are now highly 

 satisfactory for computing the rate of energy loss and the range of heavy 

 charged particles. The first derivation of a stopping formula was given by 

 Bohr [24] and was subsequently developed mainly by Bloch [25] Bethe 

 [4,5,6], Miller [7], Fermi [15], and Halpern and Hall [26]. 



Assuming that the stopping of charged particles results wholly from 

 excitation and ionization, calculations of the rate of energy loss —dE/dx in 

 simple substances compare remarkably well with values derived from accu- 

 rately measured ranges of alpha particles emitted by the natural radioactive 

 elements. The most commonly used and accepted expression at the present 

 time probably is that derived by Bethe [4,5,6]. It is applicable to mesons, 

 protons, deuterons, and alpha particles over a wide range of energy. In its 

 simple form, including the relativity correction but excluding certain correc- 

 tions discussed below, it is given as 



dE lireWN 



dx mv- 



B 



B = Z 



12] 



log ^j- - log (1 - /3 2 ) - 0« 



where B = a dimensionless quantity called the stopping number 

 e = electronic charge 

 m = electronic mass 



