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## mass_fractal

Mass Fractal model

Parameter | Description | Units | Default value
--- | --- | --- | ---
scale | Scale factor or Volume fraction | 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. 113 Figure 1: 1D plot corresponding to the default parameters of the model.

Source

`mass_fractal.py` \(\ \star\ \) `mass_fractal.c` \(\ \star\ \) `sas_gamma.c` \(\ \star\ \) `sas_3j1x_x.c`

References

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

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