Effect of Dense Packing

So far we have discussed only independent scattering from particles, that is, the particles are so far apart from each other that the effect of interference among the waves scattered by different particles can be ignored. We now consider what happens to the observed intensity as the concentration of particles in the system is increased and the interference effect becomes no longer negligible. 

Let us first consider the simplest case in which the system contains N spherical particles of radius R and uniform scattering length density po. The amplitude of the scattered radiation is then given by 

Here the first term represents the independent scattering and the second term represents the contribution by the interference effect. 

To be able to calculate the second term, we obviously need information on the statistics of interparticle distances. We introduce the pair distribution function g(r) by saying that (n)g(r) dr is the probability of finding another particle in the volume element dr a distance r from a given particle, and (n) is the average number density of the particles in the system (cf. Section 4.1.1). In terms of g(r), Equation (5.48) can be rewritten as, 

Eliminating the null scattering that shows up only at q = 0 (cf. Section 1.6), (5.49) is further rewritten as 

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