Derivation of Guinier Law

We start from Equation (1.73) giving the amplitude A(q) of scattering from a particle with its scattering length density distribution given by p(r). Focusing our attention on the cases where q is very small, we expand the exponential as a power series  

The first term is equal to p v, where p0 is the average scattering length density and v is the volume of the particle. Since the final expression for the intensity does not depend on the choice of the origin, we take the origin of r to be at the center of mass of the particle. Then, in view of (5.3), the second term in (5.35) is seen to vanish. In the third term we make the substitution 

The integral can then be expressed in terms of the various second moments defined, for example, by 

The intensity is obtained by taking the absolute square of (5.35), and in doing so we remember the fact that for small q the quantity represented by (5.38) is much smaller than pqv. In the presence of a large number of identical particles oriented in random directions, the average intensity per particle becomes 

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