1 | r""" |
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2 | Definition |
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3 | ---------- |
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4 | Calculates the scattering from a fractal structure with a primary building |
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5 | block of core-shell spheres, as opposed to just homogeneous spheres in |
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6 | the fractal model. It is an extension of the well known Teixeira\ [#teixeira]_ |
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7 | fractal model replacing the $P(q)$ of a solid sphere with that of a core-shell |
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8 | sphere. This model could find use for aggregates of coated particles, or |
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9 | aggregates of vesicles for example. |
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10 | |
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11 | .. math:: |
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12 | |
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13 | I(q) = P(q)S(q) + \text{background} |
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14 | |
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15 | Where $P(q)$ is the core-shell form factor and $S(q)$ is the |
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16 | Teixeira\ [#teixeira]_ fractal structure factor both of which are given again |
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17 | below: |
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18 | |
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19 | .. math:: |
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20 | |
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21 | P(q) &= \frac{\phi}{V_s}\left[3V_c(\rho_c-\rho_s) |
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22 | \frac{\sin(qr_c)-qr_c\cos(qr_c)}{(qr_c)^3}+ |
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23 | 3V_s(\rho_s-\rho_{solv}) |
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24 | \frac{\sin(qr_s)-qr_s\cos(qr_s)}{(qr_s)^3}\right]^2 \\ |
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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)]}{(qr_s)^{D_f}} |
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27 | |
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28 | where $\phi$ is the volume fraction of particles, $V_s$ is the volume of the |
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29 | whole particle, $V_c$ is the volume of the core, $\rho_c$, $\rho_s$, and |
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30 | $\rho_{solv}$ are the scattering length densities of the core, shell, and |
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31 | solvent respectively, $r_c$ and $r_s$ are the radius of the core and the radius |
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32 | of the whole particle respectively, $D_f$ is the fractal dimension, and $\xi$ the |
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33 | correlation length. |
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34 | |
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35 | Polydispersity of radius and thickness are also provided for. |
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36 | |
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37 | This model does not allow for anisotropy and thus the 2D scattering intensity |
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38 | is calculated in the same way as 1D, where the $q$ vector is defined as |
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39 | |
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40 | .. math:: |
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41 | |
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42 | q = \sqrt{q_x^2 + q_y^2} |
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43 | |
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44 | Our model is derived from the form factor calculations implemented in IGOR |
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45 | macros by the NIST Center for Neutron Research\ [#Kline]_ |
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46 | |
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47 | References |
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48 | ---------- |
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49 | |
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50 | .. [#teixeira] J Teixeira, *J. Appl. Cryst.*, 21 (1988) 781-785 |
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51 | .. [#Kline] S R Kline, *J Appl. Cryst.*, 39 (2006) 895 |
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52 | |
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53 | Source |
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54 | ------ |
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55 | |
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56 | `fractal_core_shell.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/fractal_core_shell.py>`_ |
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57 | |
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58 | `fractal_core_shell.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/fractal_core_shell.c>`_ |
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59 | |
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60 | Authorship and Verification |
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61 | ---------------------------- |
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62 | |
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63 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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64 | * **Last Modified by:** Paul Butler and Paul Kienzle **Date:** November 27, 2016 |
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65 | * **Last Reviewed by:** Paul Butler and Paul Kienzle **Date:** November 27, 2016 |
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66 | * **Source added by :** Steve King **Date:** March 25, 2019 |
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67 | """ |
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68 | |
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69 | import numpy as np |
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70 | from numpy import inf |
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71 | |
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72 | name = "fractal_core_shell" |
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73 | title = "Scattering from a fractal structure formed from core shell spheres" |
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74 | description = """\ |
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75 | Model for fractal aggregates of core-shell primary particles. It is based on |
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76 | the Teixeira model for the S(q) of a fractal * P(q) for a core-shell sphere |
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77 | |
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78 | radius = the radius of the core |
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79 | thickness = thickness of the shell |
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80 | thick_layer = thickness of a layer |
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81 | sld_core = the SLD of the core |
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82 | sld_shell = the SLD of the shell |
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83 | sld_solvent = the SLD of the solvent |
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84 | volfraction = volume fraction of core-shell particles |
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85 | fractal_dim = fractal dimension |
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86 | cor_length = correlation length of the fractal like aggretates |
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87 | """ |
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88 | category = "shape-independent" |
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89 | |
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90 | # pylint: disable=bad-whitespace, line-too-long |
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91 | # ["name", "units", default, [lower, upper], "type","description"], |
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92 | parameters = [ |
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93 | ["radius", "Ang", 60.0, [0.0, inf], "volume", "Sphere core radius"], |
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94 | ["thickness", "Ang", 10.0, [0.0, inf], "volume", "Sphere shell thickness"], |
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95 | ["sld_core", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", "Sphere core scattering length density"], |
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96 | ["sld_shell", "1e-6/Ang^2", 2.0, [-inf, inf], "sld", "Sphere shell scattering length density"], |
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97 | ["sld_solvent", "1e-6/Ang^2", 3.0, [-inf, inf], "sld", "Solvent scattering length density"], |
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98 | ["volfraction", "", 0.05, [0.0, inf], "", "Volume fraction of building block spheres"], |
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99 | ["fractal_dim", "", 2.0, [0.0, 6.0], "", "Fractal dimension"], |
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100 | ["cor_length", "Ang", 100.0, [0.0, inf], "", "Correlation length of fractal-like aggregates"], |
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101 | ] |
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102 | # pylint: enable=bad-whitespace, line-too-long |
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103 | |
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104 | source = ["lib/sas_3j1x_x.c", "lib/sas_gamma.c", "lib/core_shell.c", |
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105 | "lib/fractal_sq.c", "fractal_core_shell.c"] |
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106 | |
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107 | def random(): |
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108 | """Return a random parameter set for the model.""" |
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109 | outer_radius = 10**np.random.uniform(0.7, 4) |
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110 | # Use a distribution with a preference for thin shell or thin core |
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111 | # Avoid core,shell radii < 1 |
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112 | thickness = np.random.beta(0.5, 0.5)*(outer_radius-2) + 1 |
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113 | radius = outer_radius - thickness |
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114 | cor_length = 10**np.random.uniform(0.7, 2)*outer_radius |
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115 | volfraction = 10**np.random.uniform(-3, -1) |
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116 | fractal_dim = 2*np.random.beta(3, 4) + 1 |
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117 | pars = dict( |
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118 | #background=0, sld_block=1, sld_solvent=0, |
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119 | volfraction=volfraction, |
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120 | radius=radius, |
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121 | cor_length=cor_length, |
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122 | fractal_dim=fractal_dim, |
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123 | ) |
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124 | return pars |
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125 | |
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126 | demo = dict(scale=0.05, |
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127 | background=0, |
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128 | radius=20, |
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129 | thickness=5, |
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130 | sld_core=3.5, |
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131 | sld_shell=1.0, |
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132 | sld_solvent=6.35, |
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133 | volfraction=0.05, |
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134 | fractal_dim=2.0, |
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135 | cor_length=100.0) |
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136 | |
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137 | # TODO: why is there an ER function here? |
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138 | def ER(radius, thickness): |
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139 | """ |
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140 | Equivalent radius |
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141 | @param radius: core radius |
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142 | @param thickness: shell thickness |
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143 | """ |
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144 | return radius + thickness |
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145 | |
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146 | #tests = [[{'radius': 20.0, 'thickness': 10.0}, 'ER', 30.0], |
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147 | tests = [ |
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148 | # At some point the SasView 3.x test result was deemed incorrect. The |
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149 | # following tests were verified against NIST IGOR macros ver 7.850. |
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150 | # NOTE: NIST macros do only provide for a polydispers core (no option |
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151 | # for a poly shell or for a monodisperse core. The results seemed |
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152 | # extremely sensitive to the core PD, varying non monotonically all |
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153 | # the way to a PD of 1e-6. From 1e-6 to 1e-9 no changes in the |
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154 | # results were observed and the values below were taken using PD=1e-9. |
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155 | # Non-monotonically = I(0.001)=188 to 140 to 177 back to 160 etc. |
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156 | [{'radius': 20.0, |
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157 | 'thickness': 5.0, |
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158 | 'sld_core': 3.5, |
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159 | 'sld_shell': 1.0, |
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160 | 'sld_solvent': 6.35, |
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161 | 'volfraction': 0.05, |
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162 | 'background': 0.0}, |
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163 | [0.001, 0.00291, 0.0107944, 0.029923, 0.100726, 0.476304], |
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164 | [177.146, 165.151, 84.1596, 20.1466, 1.40906, 0.00622666] |
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165 | ] |
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166 | ] |
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