Scatterers per unit volume

Figure 10.1 Relationships between Iq, Ij, and The light scattered per unit volume i is also shown.

For non-interacting, incompressible polymer systems the dynamic structure factors of Eq. (3) may be significantly simplified. The sums, which in Eq. (3) have to be carried out over all atoms or in the small Q limit over all monomers and solvent molecules in the sample, may be restricted to only one average chain yielding so-called form factors. With the exception of semi-dilute solutions in the following, we will always use this restriction. Thus, S(Q, t) and Sinc(Q, t) will be understood as dynamic structure factors of single chains. Under these circumstances the normalized, so-called macroscopic coherent cross section (scattering per unit volume) follows as... 

We can determine experimentally the scattering per unit volume and multiplying it by v the number of particles per cc can be found,... 

Conventionally, the excess light scattered per unit volume at angle 6 from the transmitted beam, e, is reported as a Rayleigh factor (or ratio). 

Figure 5. Light scattering per unit volume of aerosol material (G) as a function of particle diameter (dp), integrated over the wavelengths 360-680 nm for a refractive index of 1.5. The curve is independent of the particle size distribution. From Friedlander (2000) by Oxford University Press, Inc. Used by permission.

Using the number of the scatterers per unit volume Ns (particles/m3) and assuming independent scattering from each scatterer, the spectral scattering coefficient for uniformly distributed monosize scatterers is defined as... 

Assuming there are A, scatterers per unit volume, each with a radius r it follows that the intensity scattered per unit volume in any given direction is simply proportional to A, times the intensity scattered by one particle. In... 

Consider a dilute solution of macromolecules differing only in molecular weight. Assume first that a M does not depend on the molecular weight of the molecule. Furthermore let the mass concentration, molecular weights, and structure factors of species i be, respectively, c/, Mu and S<(q). Then since each species is independent, we merely sum the scattering from each to obtain the total scattering. Thus from Eq. (8.4.4) for the scattering per unit volume... 

Suppose a beam of cross-sectional area A falls on a sheet sample of thickness t in the symmetrical reflection geometry, as in Figure 2.24a. We assume, for the sake of simplicity, that the rays in the beam are all parallel to each other. Now consider a layer of thickness dx inside the sample, at depth x below the flat surface, where the irradiated volume is equal to dx A/ sin 0. Before reaching this depth the beam has traveled distance Z within the sample, where l is equal to x/sin 0, and has suffered attenuation by a factor exp(—/x/), p being the linear absorption coefficient. The scattered beam must travel the same distance within the sample again on its way out. If i 20) is the intensity of scattering per unit volume of the sample, then the contribution dl(20) to the total scattering intensity by the layer dx at depth x is... 

The quantity Rg is often referred to as the Rayleigh ratio and is equal to (igr Ho), where /g is the intensity of the incident beam, ig is the quantity of light scattered per unit volume by one center at an angle 0 to the incident beam, and r is the distance... 

P24.19 The change in intensity of the beam, d/, is proportional to the number of scatterers per unit volume,, the intensity of the beam, /, and the path length dl. The constant of proportionality is defined as the collision cross-section a. Therefore,... 

The contributions to q(E) so far considered are due to scattering of neutrons of energies E > E but less than So. As previously mentioned, the flux is not defined at S = So, and the contributions to q E) from the first collisions must be taken into account by a separate computation. For the pure scattering medium considered here, there are exactly go first scatterings per unit volume per unit time. Of this total number, a fraction (S — aSo)/(l — a)So yield neutrons of energy total number of neutrons from first collisions which can contribute to q E) is therefore... 

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