1 | r""" |
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2 | Definition |
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3 | ---------- |
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4 | |
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5 | This model provides the form factor for an elliptical cylinder with a |
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6 | core-shell scattering length density profile. Thus this is a variation |
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7 | of the core-shell bicelle model, but with an elliptical cylinder for the core. |
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8 | Outer shells on the rims and flat ends may be of different thicknesses and |
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9 | scattering length densities. The form factor is normalized by the total |
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10 | particle volume. |
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11 | |
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12 | .. figure:: img/core_shell_bicelle_geometry.png |
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13 | |
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14 | Schematic cross-section of bicelle. Note however that the model here |
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15 | calculates for rectangular, not curved, rims as shown below. |
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16 | |
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17 | .. figure:: img/core_shell_bicelle_parameters.png |
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18 | |
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19 | Cross section of model used here. Users will have |
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20 | to decide how to distribute "heads" and "tails" between the rim, face |
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21 | and core regions in order to estimate appropriate starting parameters. |
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22 | |
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23 | Given the scattering length densities (sld) $\rho_c$, the core sld, $\rho_f$, |
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24 | the face sld, $\rho_r$, the rim sld and $\rho_s$ the solvent sld, the |
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25 | scattering length density variation along the bicelle axis is: |
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26 | |
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27 | .. math:: |
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28 | |
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29 | \rho(r) = |
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30 | \begin{cases} |
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31 | &\rho_c \text{ for } 0 \lt r \lt R; -L \lt z\lt L \\[1.5ex] |
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32 | &\rho_f \text{ for } 0 \lt r \lt R; -(L+2t) \lt z\lt -L; |
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33 | L \lt z\lt (L+2t) \\[1.5ex] |
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34 | &\rho_r\text{ for } 0 \lt r \lt R; -(L+2t) \lt z\lt -L; L \lt z\lt (L+2t) |
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35 | \end{cases} |
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36 | |
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37 | The form factor for the bicelle is calculated in cylindrical coordinates, where |
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38 | $\alpha$ is the angle between the $Q$ vector and the cylinder axis, and $\psi$ |
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39 | is the angle for the ellipsoidal cross section core, to give: |
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40 | |
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41 | .. math:: |
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42 | |
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43 | I(Q,\alpha,\psi) = \frac{\text{scale}}{V_t} \cdot |
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44 | F(Q,\alpha, \psi)^2 \cdot sin(\alpha) + \text{background} |
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45 | |
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46 | where a numerical integration of $F(Q,\alpha, \psi)^2 \cdot sin(\alpha)$ |
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47 | is carried out over \alpha and \psi for: |
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48 | |
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49 | .. math:: |
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50 | :nowrap: |
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51 | |
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52 | \begin{align*} |
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53 | F(Q,\alpha,\psi) = &\bigg[ |
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54 | (\rho_c - \rho_f) V_c |
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55 | \frac{2J_1(QR'sin \alpha)}{QR'sin\alpha} |
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56 | \frac{sin(QLcos\alpha/2)}{Q(L/2)cos\alpha} \\ |
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57 | &+(\rho_f - \rho_r) V_{c+f} |
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58 | \frac{2J_1(QR'sin\alpha)}{QR'sin\alpha} |
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59 | \frac{sin(Q(L/2+t_f)cos\alpha)}{Q(L/2+t_f)cos\alpha} \\ |
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60 | &+(\rho_r - \rho_s) V_t |
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61 | \frac{2J_1(Q(R'+t_r)sin\alpha)}{Q(R'+t_r)sin\alpha} |
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62 | \frac{sin(Q(L/2+t_f)cos\alpha)}{Q(L/2+t_f)cos\alpha} |
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63 | \bigg] |
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64 | \end{align*} |
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65 | |
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66 | where |
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67 | |
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68 | .. math:: |
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69 | |
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70 | R'=\frac{R}{\sqrt{2}}\sqrt{(1+X_{core}^{2}) + (1-X_{core}^{2})cos(\psi)} |
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71 | |
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72 | |
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73 | and $V_t = \pi.(R+t_r)(Xcore.R+t_r)^2.(L+2.t_f)$ is the total volume of |
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74 | the bicelle, $V_c = \pi.Xcore.R^2.L$ the volume of the core, |
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75 | $V_{c+f} = \pi.Xcore.R^2.(L+2.t_f)$ the volume of the core plus the volume |
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76 | of the faces, $R$ is the radius of the core, $Xcore$ is the axial ratio of |
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77 | the core, $L$ the length of the core, $t_f$ the thickness of the face, $t_r$ |
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78 | the thickness of the rim and $J_1$ the usual first order bessel function. |
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79 | The core has radii $R$ and $Xcore.R$ so is circular, as for the |
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80 | core_shell_bicelle model, for $Xcore$ =1. Note that you may need to |
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81 | limit the range of $Xcore$, especially if using the Monte-Carlo algorithm, |
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82 | as setting radius to $R/Xcore$ and axial ratio to $1/Xcore$ gives an |
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83 | equivalent solution! |
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84 | |
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85 | The output of the 1D scattering intensity function for randomly oriented |
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86 | bicelles is then given by integrating over all possible $\alpha$ and $\psi$. |
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87 | |
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88 | For oriented bicelles the *theta*, *phi* and *psi* orientation parameters will |
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89 | appear when fitting 2D data, see the :ref:`elliptical-cylinder` model |
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90 | for further information. |
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91 | |
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92 | .. figure:: img/elliptical_cylinder_angle_definition.png |
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93 | |
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94 | Definition of the angles for the oriented core_shell_bicelle_elliptical particles. |
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95 | |
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96 | Model verified using Monte Carlo simulation for 1D and 2D scattering. |
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97 | |
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98 | References |
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99 | ---------- |
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100 | |
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101 | .. [#] |
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102 | L. Onsager, Ann. New York Acad. Sci. 51, 627-659 (1949). |
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103 | |
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104 | Authorship and Verification |
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105 | ---------------------------- |
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106 | |
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107 | * **Author:** Richard Heenan **Date:** December 14, 2016 |
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108 | * **Last Modified by:** Richard Heenan **Date:** December 14, 2016 |
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109 | * **Last Reviewed by:** Paul Kienzle **Date:** Feb 28, 2018 |
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110 | """ |
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111 | |
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112 | import numpy as np |
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113 | from numpy import inf, sin, cos, pi |
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114 | |
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115 | name = "core_shell_bicelle_elliptical" |
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116 | title = "Elliptical cylinder with a core-shell scattering length density profile.." |
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117 | description = """ |
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118 | core_shell_bicelle_elliptical |
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119 | Elliptical cylinder core, optional shell on the two flat faces, and shell of |
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120 | uniform thickness on its rim (extending around the end faces). |
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121 | Please see full documentation for equations and further details. |
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122 | Involves a double numerical integral around the ellipsoid diameter |
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123 | and the angle of the cylinder axis to Q. |
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124 | Compare also the core_shell_bicelle and elliptical_cylinder models. |
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125 | """ |
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126 | category = "shape:cylinder" |
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127 | |
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128 | # pylint: disable=bad-whitespace, line-too-long |
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129 | # ["name", "units", default, [lower, upper], "type", "description"], |
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130 | parameters = [ |
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131 | ["radius", "Ang", 30, [0, inf], "volume", "Cylinder core radius r_minor"], |
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132 | ["x_core", "None", 3, [0, inf], "volume", "Axial ratio of core, X = r_major/r_minor"], |
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133 | ["thick_rim", "Ang", 8, [0, inf], "volume", "Rim shell thickness"], |
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134 | ["thick_face", "Ang", 14, [0, inf], "volume", "Cylinder face thickness"], |
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135 | ["length", "Ang", 50, [0, inf], "volume", "Cylinder length"], |
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136 | ["sld_core", "1e-6/Ang^2", 4, [-inf, inf], "sld", "Cylinder core scattering length density"], |
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137 | ["sld_face", "1e-6/Ang^2", 7, [-inf, inf], "sld", "Cylinder face scattering length density"], |
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138 | ["sld_rim", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Cylinder rim scattering length density"], |
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139 | ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld", "Solvent scattering length density"], |
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140 | ["theta", "degrees", 90.0, [-360, 360], "orientation", "Cylinder axis to beam angle"], |
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141 | ["phi", "degrees", 0, [-360, 360], "orientation", "Rotation about beam"], |
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142 | ["psi", "degrees", 0, [-360, 360], "orientation", "Rotation about cylinder axis"] |
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143 | ] |
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144 | |
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145 | # pylint: enable=bad-whitespace, line-too-long |
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146 | |
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147 | source = ["lib/sas_Si.c", "lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", |
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148 | "core_shell_bicelle_elliptical.c"] |
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149 | have_Fq = True |
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150 | effective_radius_type = [ |
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151 | "equivalent cylinder excluded volume", "equivalent volume sphere", "outer rim average radius", "outer rim min radius", |
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152 | "outer max radius", "half outer thickness", "half diagonal", |
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153 | ] |
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154 | |
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155 | def random(): |
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156 | outer_major = 10**np.random.uniform(1, 4.7) |
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157 | outer_minor = 10**np.random.uniform(1, 4.7) |
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158 | # Use a distribution with a preference for thin shell or thin core, |
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159 | # limited by the minimum radius. Avoid core,shell radii < 1 |
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160 | min_radius = min(outer_major, outer_minor) |
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161 | thick_rim = np.random.beta(0.5, 0.5)*(min_radius-2) + 1 |
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162 | radius_major = outer_major - thick_rim |
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163 | radius_minor = outer_minor - thick_rim |
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164 | radius = radius_major |
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165 | x_core = radius_minor/radius_major |
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166 | outer_length = 10**np.random.uniform(1, 4.7) |
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167 | # Caps should be a small percentage of the total length, but at least one |
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168 | # angstrom long. Since outer length >= 10, the following will suffice |
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169 | thick_face = 10**np.random.uniform(-np.log10(outer_length), -1)*outer_length |
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170 | length = outer_length - thick_face |
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171 | pars = dict( |
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172 | radius=radius, |
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173 | x_core=x_core, |
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174 | thick_rim=thick_rim, |
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175 | thick_face=thick_face, |
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176 | length=length |
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177 | ) |
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178 | return pars |
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179 | |
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180 | |
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181 | q = 0.1 |
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182 | # april 6 2017, rkh added a 2d unit test, NOT READY YET pull #890 branch assume correct! |
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183 | qx = q*cos(pi/6.0) |
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184 | qy = q*sin(pi/6.0) |
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185 | |
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186 | tests = [ |
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187 | #[{'radius': 30.0, 'x_core': 3.0, |
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188 | # 'thick_rim': 8.0, 'thick_face': 14.0, 'length': 50.0}, 'ER', 1], |
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189 | #[{'radius': 30.0, 'x_core': 3.0, |
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190 | # 'thick_rim': 8.0, 'thick_face': 14.0, 'length': 50.0}, 'VR', 1], |
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191 | |
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192 | [{'radius': 30.0, 'x_core': 3.0, |
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193 | 'thick_rim': 8.0, 'thick_face': 14.0, 'length': 50.0, |
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194 | 'sld_core': 4.0, 'sld_face': 7.0, 'sld_rim': 1.0, |
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195 | 'sld_solvent': 6.0, 'background': 0.0}, |
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196 | 0.015, 286.540286], |
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197 | #[{'theta':80., 'phi':10.}, (qx, qy), 7.88866563001], |
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198 | ] |
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199 | |
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200 | del qx, qy # not necessary to delete, but cleaner |
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