1 | r""" |
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2 | Definition |
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3 | ---------- |
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4 | |
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5 | Calculates the form factor for a rectangular solid with a core-shell structure. |
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6 | The thickness and the scattering length density of the shell or |
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7 | "rim" can be different on each (pair) of faces. |
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8 | |
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9 | |
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10 | The form factor is normalized by the particle volume $V$ such that |
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11 | |
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12 | .. math:: |
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13 | |
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14 | I(q) = \text{scale}\frac{\langle f^2 \rangle}{V} + \text{background} |
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15 | |
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16 | where $\langle \ldots \rangle$ is an average over all possible orientations |
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17 | of the rectangular solid. |
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18 | |
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19 | |
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20 | The function calculated is the form factor of the rectangular solid below. |
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21 | The core of the solid is defined by the dimensions $A$, $B$, $C$ such that |
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22 | $A < B < C$. |
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23 | |
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24 | .. image:: img/core_shell_parallelepiped_geometry.jpg |
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25 | |
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26 | There are rectangular "slabs" of thickness $t_A$ that add to the $A$ dimension |
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27 | (on the $BC$ faces). There are similar slabs on the $AC$ $(=t_B)$ and $AB$ |
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28 | $(=t_C)$ faces. The projection in the $AB$ plane is then |
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29 | |
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30 | .. image:: img/core_shell_parallelepiped_projection.jpg |
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31 | |
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32 | The volume of the solid is |
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33 | |
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34 | .. math:: |
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35 | |
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36 | V = ABC + 2t_ABC + 2t_BAC + 2t_CAB |
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37 | |
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38 | **meaning that there are "gaps" at the corners of the solid.** |
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39 | |
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40 | The intensity calculated follows the :ref:`parallelepiped` model, with the |
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41 | core-shell intensity being calculated as the square of the sum of the |
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42 | amplitudes of the core and the slabs on the edges. |
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43 | |
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44 | the scattering amplitude is computed for a particular orientation of the core-shell |
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45 | parallelepiped with respect to the scattering vector and then averaged over all |
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46 | possible orientations, where $\alpha$ is the angle between the $z$ axis and the longest axis $C$ |
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47 | of the parallelepiped, $\beta$ is the angle between projection of the particle in the $xy$ detector plane and the $y$ axis. |
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48 | |
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49 | .. math:: |
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50 | \begin{align*} |
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51 | F(Q)&=A B C (\rho_\text{core}-\rho_\text{solvent}) S(A \sin\alpha \sin\beta)S(B \sin\alpha \cos\beta)S(C \cos\alpha) \\ |
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52 | &+ 2t_A B C (\rho_\text{A}-\rho_\text{solvent}) \left[S((A+t_A) \sin\alpha \sin\beta)-S(A \sin\alpha \sin\beta)\right] S(B \sin\alpha \cos\beta) S(C \cos\alpha)\\ |
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53 | &+ 2 A t_B C (\rho_\text{B}-\rho_\text{solvent}) S(A \sin\alpha \sin\beta) \left[S((B+t_B) \sin\alpha \cos\beta)-S(B \sin\alpha \cos\beta)\right] S(C \cos\alpha)\\ |
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54 | &+ 2 A B t_C (\rho_\text{C}-\rho_\text{solvent}) S(A \sin\alpha \sin\beta) S(B \sin\alpha \cos\beta) \left[S((C+t_C) \cos\alpha)-S(C \cos\alpha)\right] |
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55 | \end{align*} |
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56 | |
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57 | with |
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58 | |
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59 | .. math:: |
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60 | |
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61 | S(x) = \frac{\sin \tfrac{1}{2}Q x}{\tfrac{1}{2}Q x} |
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62 | |
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63 | where $\rho_\text{core}$, $\rho_\text{A}$, $\rho_\text{B}$ and $\rho_\text{C}$ are |
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64 | the scattering length of the parallelepiped core, and the rectangular slabs of |
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65 | thickness $t_A$, $t_B$ and $t_C$, respectively. |
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66 | $\rho_\text{solvent}$ is the scattering length of the solvent. |
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67 | |
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68 | FITTING NOTES |
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69 | If the scale is set equal to the particle volume fraction, $\phi$, the returned |
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70 | value is the scattered intensity per unit volume, $I(q) = \phi P(q)$. |
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71 | However, **no interparticle interference effects are included in this |
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72 | calculation.** |
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73 | |
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74 | There are many parameters in this model. Hold as many fixed as possible with |
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75 | known values, or you will certainly end up at a solution that is unphysical. |
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76 | |
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77 | The returned value is in units of |cm^-1|, on absolute scale. |
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78 | |
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79 | NB: The 2nd virial coefficient of the core_shell_parallelepiped is calculated |
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80 | based on the the averaged effective radius $(=\sqrt{(A+2t_A)(B+2t_B)/\pi})$ |
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81 | and length $(C+2t_C)$ values, after appropriately |
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82 | sorting the three dimensions to give an oblate or prolate particle, to give an |
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83 | effective radius, for $S(Q)$ when $P(Q) * S(Q)$ is applied. |
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84 | |
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85 | For 2d data the orientation of the particle is required, described using |
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86 | angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, for further details |
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87 | of the calculation and angular dispersions see :ref:`orientation` . |
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88 | The angle $\Psi$ is the rotational angle around the *long_c* axis. For example, |
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89 | $\Psi = 0$ when the *short_b* axis is parallel to the *x*-axis of the detector. |
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90 | |
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91 | For 2d, constraints must be applied during fitting to ensure that the inequality |
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92 | $A < B < C$ is not violated, and hence the correct definition of angles is preserved. The calculation will not report an error, |
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93 | but the results may be not correct. |
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94 | |
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95 | .. figure:: img/parallelepiped_angle_definition.png |
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96 | |
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97 | Definition of the angles for oriented core-shell parallelepipeds. |
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98 | Note that rotation $\theta$, initially in the $xz$ plane, is carried out first, then |
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99 | rotation $\phi$ about the $z$ axis, finally rotation $\Psi$ is now around the axis of the parallelepiped. |
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100 | The neutron or X-ray beam is along the $z$ axis. |
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101 | |
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102 | .. figure:: img/parallelepiped_angle_projection.png |
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103 | |
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104 | Examples of the angles for oriented core-shell parallelepipeds against the |
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105 | detector plane. |
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106 | |
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107 | References |
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108 | ---------- |
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109 | |
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110 | .. [#] P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211 |
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111 | Equations (1), (13-14). (in German) |
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112 | .. [#] D Singh (2009). *Small angle scattering studies of self assembly in |
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113 | lipid mixtures*, Johns Hopkins University Thesis (2009) 223-225. `Available |
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114 | from Proquest <http://search.proquest.com/docview/304915826?accountid |
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115 | =26379>`_ |
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116 | |
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117 | Authorship and Verification |
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118 | ---------------------------- |
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119 | |
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120 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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121 | * **Converted to sasmodels by:** Miguel Gonzales **Date:** February 26, 2016 |
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122 | * **Last Modified by:** Wojciech Potrzebowski **Date:** January 11, 2017 |
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123 | * **Currently Under review by:** Paul Butler |
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124 | """ |
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125 | |
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126 | import numpy as np |
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127 | from numpy import pi, inf, sqrt, cos, sin |
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128 | |
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129 | name = "core_shell_parallelepiped" |
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130 | title = "Rectangular solid with a core-shell structure." |
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131 | description = """ |
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132 | P(q)= |
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133 | """ |
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134 | category = "shape:parallelepiped" |
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135 | |
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136 | # ["name", "units", default, [lower, upper], "type","description"], |
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137 | parameters = [["sld_core", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
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138 | "Parallelepiped core scattering length density"], |
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139 | ["sld_a", "1e-6/Ang^2", 2, [-inf, inf], "sld", |
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140 | "Parallelepiped A rim scattering length density"], |
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141 | ["sld_b", "1e-6/Ang^2", 4, [-inf, inf], "sld", |
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142 | "Parallelepiped B rim scattering length density"], |
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143 | ["sld_c", "1e-6/Ang^2", 2, [-inf, inf], "sld", |
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144 | "Parallelepiped C rim scattering length density"], |
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145 | ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld", |
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146 | "Solvent scattering length density"], |
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147 | ["length_a", "Ang", 35, [0, inf], "volume", |
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148 | "Shorter side of the parallelepiped"], |
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149 | ["length_b", "Ang", 75, [0, inf], "volume", |
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150 | "Second side of the parallelepiped"], |
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151 | ["length_c", "Ang", 400, [0, inf], "volume", |
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152 | "Larger side of the parallelepiped"], |
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153 | ["thick_rim_a", "Ang", 10, [0, inf], "volume", |
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154 | "Thickness of A rim"], |
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155 | ["thick_rim_b", "Ang", 10, [0, inf], "volume", |
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156 | "Thickness of B rim"], |
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157 | ["thick_rim_c", "Ang", 10, [0, inf], "volume", |
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158 | "Thickness of C rim"], |
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159 | ["theta", "degrees", 0, [-360, 360], "orientation", |
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160 | "c axis to beam angle"], |
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161 | ["phi", "degrees", 0, [-360, 360], "orientation", |
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162 | "rotation about beam"], |
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163 | ["psi", "degrees", 0, [-360, 360], "orientation", |
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164 | "rotation about c axis"], |
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165 | ] |
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166 | |
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167 | source = ["lib/gauss76.c", "core_shell_parallelepiped.c"] |
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168 | |
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169 | |
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170 | def ER(length_a, length_b, length_c, thick_rim_a, thick_rim_b, thick_rim_c): |
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171 | """ |
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172 | Return equivalent radius (ER) |
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173 | """ |
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174 | |
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175 | # surface average radius (rough approximation) |
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176 | surf_rad = sqrt((length_a + 2.0*thick_rim_a) * (length_b + 2.0*thick_rim_b) / pi) |
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177 | |
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178 | height = length_c + 2.0*thick_rim_c |
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179 | |
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180 | ddd = 0.75 * surf_rad * (2 * surf_rad * height + (height + surf_rad) * (height + pi * surf_rad)) |
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181 | return 0.5 * (ddd) ** (1. / 3.) |
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182 | |
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183 | # VR defaults to 1.0 |
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184 | |
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185 | def random(): |
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186 | import numpy as np |
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187 | outer = 10**np.random.uniform(1, 4.7, size=3) |
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188 | thick = np.random.beta(0.5, 0.5, size=3)*(outer-2) + 1 |
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189 | length = outer - thick |
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190 | pars = dict( |
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191 | length_a=length[0], |
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192 | length_b=length[1], |
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193 | length_c=length[2], |
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194 | thick_rim_a=thick[0], |
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195 | thick_rim_b=thick[1], |
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196 | thick_rim_c=thick[2], |
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197 | ) |
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198 | return pars |
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199 | |
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200 | # parameters for demo |
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201 | demo = dict(scale=1, background=0.0, |
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202 | sld_core=1, sld_a=2, sld_b=4, sld_c=2, sld_solvent=6, |
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203 | length_a=35, length_b=75, length_c=400, |
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204 | thick_rim_a=10, thick_rim_b=10, thick_rim_c=10, |
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205 | theta=0, phi=0, psi=0, |
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206 | length_a_pd=0.1, length_a_pd_n=1, |
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207 | length_b_pd=0.1, length_b_pd_n=1, |
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208 | length_c_pd=0.1, length_c_pd_n=1, |
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209 | thick_rim_a_pd=0.1, thick_rim_a_pd_n=1, |
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210 | thick_rim_b_pd=0.1, thick_rim_b_pd_n=1, |
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211 | thick_rim_c_pd=0.1, thick_rim_c_pd_n=1, |
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212 | theta_pd=10, theta_pd_n=1, |
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213 | phi_pd=10, phi_pd_n=1, |
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214 | psi_pd=10, psi_pd_n=1) |
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215 | |
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216 | # rkh 7/4/17 add random unit test for 2d, note make all params different, 2d values not tested against other codes or models |
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217 | qx, qy = 0.2 * cos(pi/6.), 0.2 * sin(pi/6.) |
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218 | tests = [[{}, 0.2, 0.533149288477], |
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219 | [{}, [0.2], [0.533149288477]], |
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220 | [{'theta':10.0, 'phi':20.0}, (qx, qy), 0.0853299803222], |
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221 | [{'theta':10.0, 'phi':20.0}, [(qx, qy)], [0.0853299803222]], |
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222 | ] |
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223 | del tests # TODO: fix the tests |
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224 | del qx, qy # not necessary to delete, but cleaner |
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