Mass Fractal model

Parameter Description Units Default value
scale Source intensity None 1
background Source background cm-1 0.001
radius Particle radius 10
fractal_dim_mass Mass fractal dimension None 1.9
cutoff_length Cut-off length 100

The returned value is scaled to units of cm-1 sr-1, absolute scale.

Calculates the scattering from fractal-like aggregates based on the Mildner reference.


The scattering intensity \(I(q)\) is calculated as

\[I(q) = scale \times P(q)S(q) + background\]
\[P(q) = F(qR)^2\]
\[F(x) = \frac{3\left[sin(x)-xcos(x)\right]}{x^3}\]
\[S(q) = \frac{\Gamma(D_m-1)\zeta^{D_m-1}}{\left[1+(q\zeta)^2 \right]^{(D_m-1)/2}} \frac{sin\left[(D_m - 1) tan^{-1}(q\zeta) \right]}{q}\]
\[scale = scale\_factor \times NV^2(\rho_{particle} - \rho_{solvent})^2\]
\[V = \frac{4}{3}\pi R^3\]

where \(R\) is the radius of the building block, \(D_m\) is the mass fractal dimension, | zeta| is the cut-off length, \(\rho_{solvent}\) is the scattering length density of the solvent, and \(\rho_{particle}\) is the scattering length density of particles.


The mass fractal dimension ( \(D_m\) ) is only valid if \(0 < mass\_dim < 6\). It is also only valid over a limited \(q\) range (see the reference for details).


Fig. 94 1D plot corresponding to the default parameters of the model.


D Mildner and P Hall, J. Phys. D: Appl. Phys., 19 (1986) 1535-1545 Equation(9)