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BELL SYSTE}f TECHNICAL JOURNAL 



of hi(r) and Ci(r, /) over Ao vanish. Inserting (5.10) and (5.11) into (5 9) and 

 expanding, we get 



l/R - Ri{l)/U' + . . . = A,F{fi"\ c) 



+ 



'ji)^fjM.ir,6Jr + i{gy}jirUr 



dhd 



+ 



(5.12) 



where the super zero on the derivatives indicates that they are evaluated at 

 h = ^*'^ and c = c. In accordance with previous approximations in this 

 memorandum we neglect-^ terms of the order of Ci and Rl . We also neglect 

 terms of the order of hi . After taking the time average of (5.12) and sub- 

 tracting the result from (5.12) we get 



RiiO = f f{r)ci{r, i) dr, 



•I A 2 



fix) = arf-lhir), 

 \dhdcj 



(5.13) 



Thus we now have a special case of our general mathematical model for the 

 number of dimensions v = 2, provided that we assume that the total con- 

 centration c on both of the rough surfaces fluctuates in the same manner as 

 the concentration of a single adsorbed layer confined to a plane rectangular 

 surface. The spectral density S(co) of Ri{l) is then given by Eq. (3.17) which 

 we repeat here 



5(a;) = - 



2 xAo.D 





(5.14) 



where /j is a two-dimensional vector whose components take the values 

 ki = l-n-tii/Li , Hi = 0, ± 1, ± 2, • • • , and where /^ are the Fourier space- 

 amplitudes of /(r) given by 



/* = ^r f /(r)e-'*'- 



JA2 



dr. 



It may be appropriate at this point to consider the quantity s'' in detail 

 for this particular case. If the energy e per unit area is independent of c, 



d s 



we have s" = — — „ evaluated at c = c where s is here the entropy of the 



absorbate per unit area. For the sake of illustration let us consider a single 

 layer of absorbate in which the molecules are non-interacting. If c, the sur- 



^■' Wc neglect these terms not because they are small compared with Ci or hiCi but, 

 because they are non-fluctuating (in time), are hence to be compared with l/R. 



