1 | r""" |
---|
2 | Definition |
---|
3 | ---------- |
---|
4 | This model calculates the scattering from fractal-like aggregates of spherical |
---|
5 | building blocks according the following equation: |
---|
6 | |
---|
7 | .. math:: |
---|
8 | |
---|
9 | I(q) &=& \phi\ V_{block} (\rho_{block} - \rho_{solvent})^2 P(q)S(q) |
---|
10 | + background |
---|
11 | |
---|
12 | where $\phi$ is The volume fraction of the spherical "building block" particles |
---|
13 | of radius $R_0$, $V_{block}$ is the volume of a single building block, |
---|
14 | $\rho_{solvent}$ is the scattering length density of the solvent, and |
---|
15 | $\rho_{block}$ is the scattering length density of the building blocks, and |
---|
16 | P(q), S(q) are the scattering from randomly distributed spherical particles |
---|
17 | (the building blocks) and the interference from such building blocks organized |
---|
18 | in a fractal-like clusters. P(q) and S(q) are calculated as: |
---|
19 | |
---|
20 | .. math:: |
---|
21 | |
---|
22 | \begin{eqnarray} |
---|
23 | P(q)&=& F(qR_0)^2 \\ |
---|
24 | F(q)&=& \frac{3 (sinx - x cosx)}{x^3} \\ |
---|
25 | V_{particle} &=& \frac{4}{3}\ \pi R_0 \\ |
---|
26 | S(q) &=& 1 + \frac{D_f\ \Gamma\!(D_f-1)}{[1+1/(q \xi)^2\ ]^{(D_f -1)/2}} |
---|
27 | \frac{sin[(D_f-1) \tan^{-1}(q \xi) ]}{(q R_0)^{D_f}} |
---|
28 | \end{eqnarray} |
---|
29 | |
---|
30 | where $\xi$ is the correlation length representing the cluster size and $D_f$ |
---|
31 | is the fractal dimension, representing the self similarity of the structure. |
---|
32 | |
---|
33 | **Polydispersity on the radius is provided for.** |
---|
34 | |
---|
35 | For 2D data: The 2D scattering intensity is calculated in the same way as |
---|
36 | 1D, where the *q* vector is defined as |
---|
37 | |
---|
38 | .. math:: |
---|
39 | |
---|
40 | q = \sqrt{q_x^2 + q_y^2} |
---|
41 | |
---|
42 | |
---|
43 | References |
---|
44 | ---------- |
---|
45 | |
---|
46 | J Teixeira, *J. Appl. Cryst.*, 21 (1988) 781-785 |
---|
47 | |
---|
48 | **Author:** NIST IGOR/DANSE **on:** pre 2010 |
---|
49 | |
---|
50 | **Last Modified by:** Paul Butler **on:** March 20, 2016 |
---|
51 | |
---|
52 | **Last Reviewed by:** Paul Butler **on:** March 20, 2016 |
---|
53 | |
---|
54 | """ |
---|
55 | |
---|
56 | from numpy import inf |
---|
57 | |
---|
58 | name = "fractal" |
---|
59 | title = "Calculates the scattering from fractal-like aggregates of spheres \ |
---|
60 | following theTexiera reference." |
---|
61 | description = """ |
---|
62 | The scattering intensity is given by |
---|
63 | I(q) = scale * V * delta^(2) * P(q) * S(q) + background, where |
---|
64 | p(q)= F(q*radius)^(2) |
---|
65 | F(x) = 3*[sin(x)-x cos(x)]/x**3 |
---|
66 | delta = sld_block -sld_solv |
---|
67 | scale = scale * volfraction |
---|
68 | radius = Block radius |
---|
69 | sld_block = SDL block |
---|
70 | sld_solv = SDL solvent |
---|
71 | background = background |
---|
72 | and S(q) is the interference term between building blocks given |
---|
73 | in the full documentation and depending on the parameters |
---|
74 | fractal_dim = Fractal dimension |
---|
75 | cor_length = Correlation Length """ |
---|
76 | |
---|
77 | category = "shape-independent" |
---|
78 | |
---|
79 | # pylint: disable=bad-whitespace, line-too-long |
---|
80 | # ["name", "units", default, [lower, upper], "type","description"], |
---|
81 | parameters = [["volfraction", "", 0.05, [0.0, 1], "", |
---|
82 | "volume fraction of blocks"], |
---|
83 | ["radius", "Ang", 5.0, [0.0, inf], "", |
---|
84 | "radius of particles"], |
---|
85 | ["fractal_dim", "", 2.0, [0.0, 6.0], "", |
---|
86 | "fractal dimension"], |
---|
87 | ["cor_length", "Ang", 100.0, [0.0, inf], "", |
---|
88 | "cluster correlation length"], |
---|
89 | ["sld_block", "1e-6/Ang^2", 2.0, [-inf, inf], "", |
---|
90 | "scattering length density of particles"], |
---|
91 | ["sld_solvent", "1e-6/Ang^2", 6.4, [-inf, inf], "", |
---|
92 | "scattering length density of solvent"], |
---|
93 | ] |
---|
94 | # pylint: enable=bad-whitespace, line-too-long |
---|
95 | |
---|
96 | source = ["lib/sph_j1c.c", "lib/sas_gamma.c", "fractal.c"] |
---|
97 | |
---|
98 | demo = dict(volfraction=0.05, |
---|
99 | radius=5.0, |
---|
100 | fractal_dim=2.0, |
---|
101 | cor_length=100.0, |
---|
102 | sld_block=2.0, |
---|
103 | sld_solvent=6.4) |
---|
104 | |
---|
105 | # NOTE: test results taken from values returned by SasView 3.1.2 |
---|
106 | tests = [ |
---|
107 | [{}, 0.0005, 40.4980069872], |
---|
108 | [{}, 0.234734468938, 0.0947143166058], |
---|
109 | [{}, 0.5, 0.0176878183458], |
---|
110 | ] |
---|