488 BELL SYSTEM TECIIXICIL JOURNAL 



mass m) were free and stationary, an alpha-particle of mass M moving 

 with speed f/ along a line passing at distance p from the initial position 

 of any one of them would communicate to it an amount of energy: 



where 



m UM 



a = — ::rTJTn — • ^^^ 



Imagine Q alpha-particles passing through this collection of NZ 

 electrons; the number of encounters for which this energy-value lies 

 between two values W and W-\-dW is equal to 2Tp(dp/dW)dW. 

 Multiplying this by W and integrating over all values from 1^=0 

 (corresponding to /)==>o) to W=8e-/mU^a^ (corresponding to ^ = 0), 

 we arrive at a value for the total amount of energy communicated 

 by the alpha-particles to the electrons, which \-alue is infinite. This 

 absurd conclusion rests on the absurd assumption that the electrons 

 are free, which, of course, is not made. Generally it is assumed that 

 whenever p exceeds a certain value, selected for one reason or an- 

 other, equation (8) loses its validity and W is zero; for instance, that 

 whenever p is so great that the value computed by (8) for W is smaller 

 than the least energy sufficing to remove the electron altogether from 

 the atom or to put the atom into a Stationary State, then there is 

 no transfer of energy whatever; but, whenever p is so small that W 

 as computed by (8) exceeds the extraction-energy for the electron in 

 question, then the electron is extracted and carries off, as kinetic 

 energy, the difference between W and its extraction-energy. 



Definite assumptions must be made about the extraction-energies 

 of the various classes of electrons in the atom, the number of electrons 

 in each class, and the Stationary States of the atom ; this being done, 

 formulae are derived for the primary ionization, the scondary ioniza- 

 tion, and the rate at which the alpha-particle (or fast electron) loses 

 energy.-^ Apart from these results of elaborate and careful analysis 

 which lead as I have said to order-of-magnitude agreements (in some 

 cases the agreements approach quantitative value) it may be pointed 

 out that the equation (8) leads, when U is so great that a becomes 

 small relatively to p, to the conclusion that as a fast-flying particle 

 proceeds through matter the fourth power of its speed falls off linearly 

 with increase of distance traversed, which is in agreement with much 



23 See R. H. Fowler, Proc. Canib. Phil. Soc. 21, pp. 521 540 (1923), and G. H. 

 Henderson, Phil. Mag. 44, pp. 680-(1922) for discussion and prior literature as well 

 for their own work. 



