Freedom of Electrons in Metals. 315 



cerned with the duration of the free paths *, the mean free 

 path of the electrons must not much exceed 10~ 8 cm., so that 

 their excursions must in fact be largely confined to the spaces 

 between each molecule and the next. If their free paths 

 were larger than this, complete conduction could not become 

 established in so short a time. 



If a free path longer than this were demanded, then the 

 optical incipient conduction would have to be ascribed mainly 

 to the gradual deflexion of the path by the electric field with- 

 out introducing limitation due to interruption of the path by 

 collision with the molecules ; and the theory of propagation 

 which has been developed above would be the one applicable 

 to ultra-red radiation. The results of Ha gen and Rubens 

 seem emphatically to preclude that type of hypothesis. 



If then metallic conduction is due to free electrons, their 

 freedom is spacially very much restricted, almost in fact 

 within molecular limits. 



While if it is inferred from the approximation of p? to a 

 real negative value, that the incipient conduction, which is 

 the main agent in the optical phenomena of the nobler metals, 

 must be largely due in the above manner to deflexion of paths 

 of electrons, effectively free for times exceeding the period of 

 the vibration, it would appear that the number of them that 

 are concerned is roughly of about the same order as the 

 number of the metallic molecules. 



The value of the mean velocity of the free electrons, above 

 employed, about 0.10°, is involved in Drude's form of the 

 theory (Ann. der Phys. i. 1900, p. 577), in order to get the 

 right (universal) ratio of electric to thermal conduction, on 

 the assumption that both of them are effected through the 

 agency of the free electrons. If, however, their velocity 

 were of the order 3 . 10°, like the electrons from radio-active 

 substances, then the free path might be as much as 5.10 -4 

 cm. without vitiatino- the conditions laid down. The work of 



* Because the mean additional velocity imposed upon the electron by 

 the electric field is proportional to the duration of the free path. 



This view of conduction, as stated by Drude, seems to require tacitly 

 that the average velocity is in most instances restored by collision at the 

 end of each free path. Moreover the conductivit}^ varies as the tem- 

 perature for pure metsls : thus N X free path X velocity must be constant. 

 These conditions are difficult to interpret, unless each molecule may be 

 taken to emit electrons at a constant rate and absorb a definite proportion 

 of those that encounter it, the same at all temperatures, thus establishing 

 an equilibrium ir Avhich N' varies inversely as the velocity. Such 

 increase of N' at low temperatures would make fx 2 , and also the absorbing 

 power, for those shorter wave-lengths for which -p 2 is large, tend to vary 

 inversely as the square root of the temperature. 



