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BELL SYSTEM TECHNICAL JOURNAL 



The above treatment could just as well be applied to the case in which 

 the diffusing quantity is heat instead of ions. 



(«) Resistance of a Localized Contact Disturbed by a Dif using Surface 

 Layer. Here we consider the case of two conductors covered with diffusing 

 surface layers. It is supposed that the conduction from one conductor to 

 the other is distributed Gaussianly with a width A^'^. Finally, it is supposed 

 that the conductivity through the above area varies with the surface con- 

 centration of the surface layer in that region. This situation is well repre- 

 sented by the above general model by taking v = 2, Ai = A2 = A, and 

 bi = bi = b. 



The self-covariance is readily calculated with the result 



RMRAI +«)-fr 4,(^ + On) 



(5.8) 



The corresponding spectral density is 



ZTT 5 Jo 



COS uoi du 

 A + Du 



27r2 s"D 



-COS {coA/D)Ci{coA/D) 



+ sin (coA/£>) (1 - Si{aiA/D) 



(5.8a) 



where Ci(x) and Si{x) are the cosine and sine integrals-^ respectively. When 

 to « D/A 



Sic)^ -l-,j^\og{8coA/D), 



ZTT" 5 U 



8 = 0.5772, 



(5.8b) 



and when w >>> D/A 



5(co) c^ 



1 xb D 1 



2^2 5"A2 0)2 



(5.8c) 



Thus we see that this case does not lead to the experimental form of the 

 spectral density. It must be remarked that here S{co) is very sensitive to 

 the form of the self-covariance for small u. 



b. Contact between Relatively Large Areas of Rough Surfaces Covered with 

 Diffusing Surface Layers. We consider this case in detail since it leads to 

 results in agreement with experiment. Furthermore, the more detailed 

 consideration of this case will illustrate more fully the use of the general 

 mathematical model, which may be of use in studying other diffusional 

 mechanisms should they be postulated at some future time. This mechanism 

 does not fall into the class just considered. 



^ See Jahnke and Emde, "Tables of Functions," p. 3, Dover (1943). 



