Resistance of Thin Metallic Films. 47 & 



where 



F(a)=e ffla -(l + a 2 ) + 2aV 2 I, ........ (6) 



and 



1 7 _ 'a 



/» = 



6 + 12 € 6 e 



-+(f-£) 



l-(l+a*)e~" aS + 2a 8 I 



If for the moment we neglect the term (JieY) 2 f(a) in 

 order to obtain an approximate value of Y which will be 

 sufficiently accurate to use in the small term we find 



x-^=x I= 3V 



/ l 4Xh\a) ' 



so that v _ <L\lF(a) x 



4\F(a)+3J 



Substituting this value in the small term (JieY) 2 f(a) of 

 equation (5), we obtain, after a little reduction, from equa- 

 tion (5) itself 



x,_ rrr ^ 1r i + -MEW,,. 



Thus since 



we have 



/ 2 ne 2 \i'r 

 /y«A.r X , r (fo/X)'/(q) -, 



a* w ' i + 4\F(«)j 



.... (7) 



a involves Y, and so X (see Appendix, Problem 1) in the 

 form of a small term of the second order of small quantities. 

 It will be noted that the equation indicates a departure from 

 Ohm's law. For a field of 1 volt per cm. 7t?X = 14, and for 

 values of I such as we are dealing with in these films, i. e. 

 values probably of the order 10 ~ 5 cm., the departure from 

 Ohm's law would be quite inappreciable. The formula (7) 

 is, however, not without interest, and we shall return to it 

 later. For the present we shall assume terms of the orders 



/Ye \ 2 

 (JielX) 2 and ( — -i to be negligible, so that we are left with 

 \murj ° 



._ /T nfkv ( X ) 



'-V^ aB }j*F(.)+lf 



in which a 2 may now be taken as lunir. 



