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