In this form, \(k_{0}\) takes a value of \(0.043\) when \(D_{f}\) is 2. [ For comparison to other studies, an approximate relationship with the \(R_{max}\) and \(R_{g}\) is useful to investigate. \citet{Lattuada_2003} found that the ratio of the minimum radius of a sphere encompassing the aggregate, \(R_s\), to the gyration radius to be increasing with \(N\), and almost approaching to an asymptotic value of 1.6-1.7. Note that \(R_{s} = \frac{1}{2}R_{max} + r_m\)].
In order to calculate the scattering field by a fractal aggregate, the commonly used method is to define a monomer pair-correlation function (which represents in average how many monomers are located at a distance, r, as seen by individual monomers.) Then the static structure factor can be calculated as the Fourier transform of the pair-correlation function. The structure factor is proportional to intensity or the square of the amplitude of the scattering field measured at the detector \citep{Sorensen_2001,Filippov_2000,Lattuada_2003} .
Hence the intensity of the scattered field in reciprocal space is written as (reference for both double sum and square form)
The below block is for the thesis. It will show in detail how the S(q) is the fourier transform of g(r). It shows that g(r) is the self convoluted density auto correlation function. and from convolution theorem the s(q) becomes the fourier transform of g(r)