[12dbc90] | 1 | r""" |
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| 2 | Definition |
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| 3 | ---------- |
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| 4 | This model calculates the scattering from fractal-like aggregates of spherical |
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| 5 | building blocks according the following equation: |
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| 6 | |
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| 7 | .. math:: |
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| 8 | |
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[785cbec] | 9 | I(q) = \phi\ V_\text{block} (\rho_\text{block} |
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| 10 | - \rho_\text{solvent})^2 P(q)S(q) + \text{background} |
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[12dbc90] | 11 | |
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| 12 | where $\phi$ is The volume fraction of the spherical "building block" particles |
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| 13 | of radius $R_0$, $V_{block}$ is the volume of a single building block, |
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| 14 | $\rho_{solvent}$ is the scattering length density of the solvent, and |
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| 15 | $\rho_{block}$ is the scattering length density of the building blocks, and |
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| 16 | P(q), S(q) are the scattering from randomly distributed spherical particles |
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| 17 | (the building blocks) and the interference from such building blocks organized |
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| 18 | in a fractal-like clusters. P(q) and S(q) are calculated as: |
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| 19 | |
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| 20 | .. math:: |
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| 21 | |
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[df89d77] | 22 | P(q)&= F(qR_0)^2 \\ |
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| 23 | F(q)&= \frac{3 (\sin x - x \cos x)}{x^3} \\ |
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| 24 | V_\text{particle} &= \frac{4}{3}\ \pi R_0 \\ |
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[785cbec] | 25 | S(q) &= 1 + \frac{D_f\ \Gamma\!(D_f-1)}{[1+1/(q \xi)^2\ ]^{(D_f -1)/2}} |
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| 26 | \frac{\sin[(D_f-1) \tan^{-1}(q \xi) ]}{(q R_0)^{D_f}} |
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[12dbc90] | 27 | |
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| 28 | where $\xi$ is the correlation length representing the cluster size and $D_f$ |
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[404ebbd] | 29 | is the fractal dimension, representing the self similarity of the structure. |
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| 30 | Note that S(q) here goes negative if $D_f$ is too large, and the Gamma function |
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[df89d77] | 31 | diverges at $D_f=0$ and $D_f=1$. |
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[12dbc90] | 32 | |
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| 33 | **Polydispersity on the radius is provided for.** |
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| 34 | |
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| 35 | For 2D data: The 2D scattering intensity is calculated in the same way as |
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| 36 | 1D, where the *q* vector is defined as |
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| 37 | |
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| 38 | .. math:: |
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| 39 | |
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| 40 | q = \sqrt{q_x^2 + q_y^2} |
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| 41 | |
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| 42 | |
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| 43 | References |
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| 44 | ---------- |
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| 45 | |
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[4788822] | 46 | .. [#] J Teixeira, *J. Appl. Cryst.*, 21 (1988) 781-785 |
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[12dbc90] | 47 | |
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[4788822] | 48 | Authorship and Verification |
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| 49 | ---------------------------- |
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[12dbc90] | 50 | |
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[4788822] | 51 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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| 52 | * **Converted to sasmodels by:** Paul Butler **Date:** March 19, 2016 |
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| 53 | * **Last Modified by:** Paul Butler **Date:** March 12, 2017 |
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| 54 | * **Last Reviewed by:** Paul Butler **Date:** March 12, 2017 |
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[12dbc90] | 55 | """ |
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[404ebbd] | 56 | from __future__ import division |
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[12dbc90] | 57 | |
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[2d81cfe] | 58 | import numpy as np |
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[12dbc90] | 59 | from numpy import inf |
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| 60 | |
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| 61 | name = "fractal" |
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| 62 | title = "Calculates the scattering from fractal-like aggregates of spheres \ |
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| 63 | following theTexiera reference." |
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| 64 | description = """ |
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| 65 | The scattering intensity is given by |
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| 66 | I(q) = scale * V * delta^(2) * P(q) * S(q) + background, where |
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| 67 | p(q)= F(q*radius)^(2) |
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| 68 | F(x) = 3*[sin(x)-x cos(x)]/x**3 |
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| 69 | delta = sld_block -sld_solv |
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| 70 | scale = scale * volfraction |
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| 71 | radius = Block radius |
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| 72 | sld_block = SDL block |
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| 73 | sld_solv = SDL solvent |
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| 74 | background = background |
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| 75 | and S(q) is the interference term between building blocks given |
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| 76 | in the full documentation and depending on the parameters |
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| 77 | fractal_dim = Fractal dimension |
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| 78 | cor_length = Correlation Length """ |
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| 79 | |
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| 80 | category = "shape-independent" |
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| 81 | |
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| 82 | # pylint: disable=bad-whitespace, line-too-long |
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| 83 | # ["name", "units", default, [lower, upper], "type","description"], |
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| 84 | parameters = [["volfraction", "", 0.05, [0.0, 1], "", |
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| 85 | "volume fraction of blocks"], |
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[4788822] | 86 | ["radius", "Ang", 5.0, [0.0, inf], "volume", |
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[12dbc90] | 87 | "radius of particles"], |
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[217590b] | 88 | ["fractal_dim", "", 2.0, [0.0, 6.0], "", |
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[12dbc90] | 89 | "fractal dimension"], |
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| 90 | ["cor_length", "Ang", 100.0, [0.0, inf], "", |
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| 91 | "cluster correlation length"], |
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[42356c8] | 92 | ["sld_block", "1e-6/Ang^2", 2.0, [-inf, inf], "sld", |
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[12dbc90] | 93 | "scattering length density of particles"], |
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[42356c8] | 94 | ["sld_solvent", "1e-6/Ang^2", 6.4, [-inf, inf], "sld", |
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[12dbc90] | 95 | "scattering length density of solvent"], |
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| 96 | ] |
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| 97 | # pylint: enable=bad-whitespace, line-too-long |
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| 98 | |
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[925ad6e] | 99 | source = ["lib/sas_3j1x_x.c", "lib/sas_gamma.c", "lib/fractal_sq.c", "fractal.c"] |
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[12dbc90] | 100 | |
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[404ebbd] | 101 | def random(): |
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| 102 | radius = 10**np.random.uniform(0.7, 4) |
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| 103 | #radius = 5 |
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| 104 | cor_length = 10**np.random.uniform(0.7, 2)*radius |
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| 105 | #cor_length = 20*radius |
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| 106 | volfraction = 10**np.random.uniform(-3, -1) |
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| 107 | #volfraction = 0.05 |
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| 108 | fractal_dim = 2*np.random.beta(3, 4) + 1 |
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| 109 | #fractal_dim = 2 |
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| 110 | pars = dict( |
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| 111 | #background=0, sld_block=1, sld_solvent=0, |
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| 112 | volfraction=volfraction, |
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| 113 | radius=radius, |
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| 114 | cor_length=cor_length, |
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| 115 | fractal_dim=fractal_dim, |
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| 116 | ) |
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| 117 | return pars |
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| 118 | |
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[dff1088] | 119 | demo = dict(volfraction=0.05, |
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[12dbc90] | 120 | radius=5.0, |
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| 121 | fractal_dim=2.0, |
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| 122 | cor_length=100.0, |
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| 123 | sld_block=2.0, |
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| 124 | sld_solvent=6.4) |
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| 125 | |
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| 126 | # NOTE: test results taken from values returned by SasView 3.1.2 |
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| 127 | tests = [ |
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[dff1088] | 128 | [{}, 0.0005, 40.4980069872], |
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| 129 | [{}, 0.234734468938, 0.0947143166058], |
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| 130 | [{}, 0.5, 0.0176878183458], |
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[12dbc90] | 131 | ] |
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