fractal

Calculates the scattering from fractal-like aggregates of spheres following theTexiera reference.

Parameter Description Units Default value
scale Scale factor or Volume fraction None 1
background Source background cm-1 0.001
volfraction volume fraction of blocks None 0.05
radius radius of particles Å 5
fractal_dim fractal dimension None 2
cor_length cluster correlation length Å 100
sld_block scattering length density of particles 10-6Å-2 2
sld_solvent scattering length density of solvent 10-6Å-2 6.4

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

Definition This model calculates the scattering from fractal-like aggregates of spherical building blocks according the following equation:

\[I(q) = \phi\ V_\text{block} (\rho_\text{block} - \rho_\text{solvent})^2 P(q)S(q) + \text{background}\]

where \(\phi\) is The volume fraction of the spherical “building block” particles of radius \(R_0\), \(V_{block}\) is the volume of a single building block, \(\rho_{solvent}\) is the scattering length density of the solvent, and \(\rho_{block}\) is the scattering length density of the building blocks, and P(q), S(q) are the scattering from randomly distributed spherical particles (the building blocks) and the interference from such building blocks organized in a fractal-like clusters. P(q) and S(q) are calculated as:

\[\begin{split}P(q)&= F(qR_0)^2 \\ F(q)&= \frac{3 (\sin x - x \cos x)}{x^3} \\ V_\text{particle} &= \frac{4}{3}\ \pi R_0 \\ S(q) &= 1 + \frac{D_f\ \Gamma\!(D_f-1)}{[1+1/(q \xi)^2\ ]^{(D_f -1)/2}} \frac{\sin[(D_f-1) \tan^{-1}(q \xi) ]}{(q R_0)^{D_f}}\end{split}\]

where \(\xi\) is the correlation length representing the cluster size and \(D_f\) is the fractal dimension, representing the self similarity of the structure. Note that S(q) here goes negative if \(D_f\) is too large, and the Gamma function diverges at \(D_f=0\) and \(D_f=1\).

Polydispersity on the radius is provided for.

For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the q vector is defined as

\[q = \sqrt{q_x^2 + q_y^2}\]
../../_images/fractal_autogenfig.png

Fig. 98 1D plot corresponding to the default parameters of the model.

Source

fractal.py \(\ \star\ \) fractal.c \(\ \star\ \) fractal_sq.c \(\ \star\ \) sas_gamma.c \(\ \star\ \) sas_3j1x_x.c

References

  1. J Teixeira, J. Appl. Cryst., 21 (1988) 781-785

Authorship and Verification

  • Author: NIST IGOR/DANSE Date: pre 2010
  • Converted to sasmodels by: Paul Butler Date: March 19, 2016
  • Last Modified by: Paul Butler Date: March 12, 2017
  • Last Reviewed by: Paul Butler Date: March 12, 2017