Velocity of Swiftly Moving Electrified Particles. 583 



Consider a collision between an electrified particle moving 

 with a velocity V and an electron initially at rest. Let M, 

 E, ?n, and e be the mass and the electric charge of the 

 particle and the electron respectively t and let the length of 

 the perpendicular from the electron to the path of the particle 

 before the collision be p. If the electron is free, the kinetic 

 energy Q given to the electron during the collision can 

 simply be shown to be 



H ~ mV 2 p 2 + a 2 ' W 



where <?E(M+??i) 



a= « i^ 



(2) 



Consider next an « or j3 particle penetrating through a 

 sheet of some substance of thickness A#, and let the number 

 of atoms in unit volume be N, each atom containing n 

 electrons. The mean value of the number of collisions in 

 which p has a value between p and p + dp is given by 



dA = 2ir'Nn&xpdp (3) 



If we now could neglect the effect of the interatomic 

 forces on the electrons, the average value of the loss ofc' 

 kinetic energy of the swiftly moving particle in penetrating- 

 through the sheet of matter would consequently be 



WE 2 N.„. 

 Ai = :„V2 lTJrr^ • • • • W 



m 



V 



y/A,>' C pdp 



where the integration is to be performed over all the values 

 for p, from jo = to p = yo. The value of this integral, 

 however, is infinite. Vse therefore see that in order to 

 obtain agreement with experiments it is necessary to take 

 the effect of the interatomic forces into consideration. 



Let us assume, as in the electron theory of dispersion, 

 that the electrons normally are kept in positions of stable 

 equilibrium and, if slightly displaced, they will execute 

 vibrations around these positions with a frequency v charac- 

 teristic for the different electrons. In estimating the effect 

 of the interatomic forces it is convenient to introduce the 

 conception of the " time of collision," i. e. a time interval of 

 the same order of magnitude as that which the a or /3 par- 

 ticle will take in travelling through a distance of length j>. 

 If this time interval is very short compared with the time of 

 vibration of the- electron, the interatomic forces will not have 

 time to act before the a or ft particle has escaped again from 



