mass_fractal

Mass Fractal model

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

Definition

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

\[I(q) = scale \times P(q)S(q) + background\]

\[P(q) = F(qR)^2\]

\[F(x) = \frac{3\left[sin(x)-xcos(x)\right]}{x^3}\]

\[S(q) = \frac{\Gamma(D_m-1)\zeta^{D_m-1}}{\left[1+(q\zeta)^2 \right]^{(D_m-1)/2}} \frac{sin\left[(D_m - 1) tan^{-1}(q\zeta) \right]}{q}\]

\[scale = scale\_factor \times NV^2(\rho_\text{particle} - \rho_\text{solvent})^2\]

\[V = \frac{4}{3}\pi R^3\]

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

Note

The mass fractal dimension ( \(D_m\) ) is only valid if \(1 < mass\_dim < 6\). It is also only valid over a limited \(q\) range (see the reference for details).

Fig. 107 1D plot corresponding to the default parameters of the model.

References

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