B-3 



P(£j> " Pq^Ij) +y^ g.. P^ (r..) E(r., r.,) (B-6) 



These represent the fundamental equations of multiple scattering. The direct 

 method would then consist of solving the set of simultaneous linear equations (B-6) 

 for the pj(r.) and substituting these in (B-5), thus giving p(r) as a function of the 

 positions and scattering parameters of the scatterers. Then taking the configura- 

 tional average of this quantity would give the desired results. Unfortunately, it is 

 not possible to carry this procedure through because of the complexity of the 

 integration. The alternative method used by Foldy involves finding equations 

 satisfied by <p(r) > and then solving these equations for the desired averaged 

 quantities. 



Taking the configurational average of both sides of(B-5), we have 



<(r)> = <p(r)> =2_^ J I g. <p-'(r.)> E(r, r.) i^— i- ds.dr. (B-7) 



J V V 



p (r)+ fG(r.)<p-'(r.)> .E(r, r.)dr 

 o - J -J -J J ] ~] 



where 



G(r) = j g(s, a))n(r,s) ds (B-8) 



V 



The quantity <pl (r.) >^ represents the external field acting on the j-th scatterer 

 averaged over all possible configurations of all the other scatterers. The only 

 rigorous way of evaluating it seems to be to solve the set of equations (B-6); sub- 

 stituting these in (B-7) and carrying out the necessary integrations would then give 

 <p(r) >. As stated before, this does not appear to be feasible. Thus, we "resort to 

 approximating <pj(£j-"i by the average field which would exist at r. if the j-th 

 scatterer were not present. This last quantity differs. from <p(r.)> only by a term 

 of order 1/N. Thus, if N is large, we may substitute <p(r|)> for <pJ(£j)>j in (B-7), 

 obtaining the integral equation 



< p(r)^ «Po(.r) + [ G(r') <p(r') > E(r, r')dr' (B-9) 



;arthur Sl.HittlcJnf. 



S-7001-0307 



