Does scattering cross-section of a large number of scatterers tend to sum of their cross-sections? Suppose we know differential cross-section for some type of scatterer particle. Now, consider a large number of such scatterers distributed randomly in some volume. If there's much space between these scatterers, I suppose the cross-sections of individual scatterers can be simply summed, because multiple scattering would be negligible(?).
But in general, it seems, multiple scatterings should distort differential cross-section of individual scatterers, so that simple sum won't work to get the cross-section for the collection. Is this correct? If yes, how can one calculate the differential cross-section of a large collection of randomly-distributed scatterers, given the cross-section for one scatterer?
 A: As long as the target is "thin" (meaning the total odds of being scattered are small), then you are safe simply adding cross-sections together because the odds of being scattered more than once are negligible.
Once the total chance of a incident particle being scattered gets to be significant, then simply summing the cross-sections of scattering centers in the beam will over count—because a significant fraction of incident particles will be scattered by more than one center, but it is still only one particle.
As with other concerns the definition of "negligible" is driven by your precision goals for the measurement and by comparison to other uncertainties. To keep the frequency of individual incident particles experiencing multiple hard scattering events small enough to ignore when you want a 1% final precision (the goal in most of my dissertation work) you'll need a target that scatters less than $\approx 10\%$ of incident beam particles (we used targets up to 6% radiation length).
A: I think you should write the scatterer, consisting of many centers, exactly in the Schroedinger (or a wave) equation and make some approximate treatment. If the wavelength of the incident "projectiles" (waves too) is much larger than the charateristic distance between scattering centers, you will not be able to add cross sections because you will have some sort of a "coherent scattering" from a compound target.
