Mass Fractal

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).

References

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

Authorship and Verification

**Author:**
**Last Modified by:**
**Last Reviewed by:**