730 BELL SYSTEM TECHNICAL JOURNAL 



for the number of particles leaping from C into any other cell C of 

 the layer, and an expression for the number leaping reversely. The 

 difference or lack of balance between these numbers, integrated over 

 all the cells C , will be the required quantity (b — a). 



We begin by inquiring how many particles make such impacts that 

 their paths (in the coordinate-space, of course — not the velocity-space) 

 are deflected through angles between say 6 and 9 -j- dd. To be 

 deflected through an angle 6, an electron must strike an atom at a 

 point where its surface is so oriented, that the normal (which is the 

 line of centres at the instant of collision) is inclined to the line of 

 approach of the electron at the angle ^ = hi'"' ~ &)• Denote by R 

 the radius of the atom, and suppose that the radius of the electron 

 is negligibly small. ^^ Think of all the f-d^drjd^ electrons which at 

 some particular moment of time are in unit volume of the metal, and 

 belong to the compartment C of the velocity-space. Imagine each of 

 these to be the centre, in the coordinate-space, of a pair of circles 

 lying in the plane perpendicular to its path, and having radii R sin \l/ 

 and R{s[n x}/ -\- d sin \f/) =i?(sin xj/ -f cos ^l/drp). As time goes on, let 

 these circles travel in the direction normal to their plane with the 

 speed V. During unit time each pair of circles traces out a pair of 

 cylinders of length v, containing between them a cylindrical sheath of 

 volume v-IttR^ sin ^ cos ^pd^p. Multiplying this by the number 

 f-d^drjd^ of the electrons, we get the total volume included in all of 

 these sheaths. Multiplying this by the number N of atoms in unit 

 volume, we get the number of atoms located with their centres in these 

 sheaths — which is the number of atoms so placed that in unit time, 

 electrons of the stated cell impinge on them at angles between \p 

 and \J/ + d^ — which is the number of impacts per unit time in which 

 electrons are deflected through angles between 6 and d + dd, which 

 accordingly is this: 



N-fd^drjdt -lirvR^ sin V' cos xl^d^P = fd^drjd^ • NirRh • I sin Odd. (95) 



It will be convenient to express this as a fraction of the total number 

 of impacts — call it ZdT — experienced per unit time by all the electrons 

 in question, which by integrating (95) is found to be: 



Zdr = TrNRh-f-dr, (96) 



substituting which into (95) we get: 



Z-lsm^l/cosi^dyp = Z-^ sin edd. (97) 



1^ The formulae remain valid even if the diameter of the electron is supposed 

 not negligibly small, provided that we interpret R as the sum of the radii of atom 

 and electron; but the generalization is not, so far as I know, of any practical value. 



