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 | In this version the "rim" or "belt" does NOT extend the full length of |
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9 | the particle, but has the same length as the core. Outer shells on the |
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10 | rims and flat ends may be of different thicknesses and scattering length |
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11 | densities. The form factor is normalized by the total particle volume. |
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12 | This version includes an approximate "interfacial roughness". |
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13 | |
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14 | |
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15 | .. figure:: img/core_shell_bicelle_belt_geometry.png |
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16 | |
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17 | Schematic cross-section of bicelle with belt. Note however that the model |
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18 | here calculates for rectangular, not curved, rims as shown below. |
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19 | |
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20 | .. figure:: img/core_shell_bicelle_belt_parameters.png |
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21 | |
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22 | Cross section of model used here. Users will have |
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23 | to decide how to distribute "heads" and "tails" between the rim, face |
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24 | and core regions in order to estimate appropriate starting parameters. |
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25 | |
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26 | Given the scattering length densities (sld) $\rho_c$, the core sld, $\rho_f$, |
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27 | the face sld, $\rho_r$, the rim sld and $\rho_s$ the solvent sld, the |
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28 | scattering length density variation along the bicelle axis is: |
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29 | |
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30 | .. math:: |
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31 | |
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32 | \rho(r) = |
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33 | \begin{cases} |
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34 | &\rho_c \text{ for } 0 \lt r \lt R; -L/2 \lt z\lt L/2 \\[1.5ex] |
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35 | &\rho_f \text{ for } 0 \lt r \lt R; -(L/2 +t_\text{face}) \lt z\lt -L/2; |
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36 | L/2 \lt z\lt (L/2+t_\text{face}) \\[1.5ex] |
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37 | &\rho_r\text{ for } R \lt r \lt R+t_\text{rim}; -L/2 \lt z\lt L/2 |
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38 | \end{cases} |
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39 | |
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40 | The form factor for the bicelle is calculated in cylindrical coordinates, where |
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41 | $\alpha$ is the angle between the $Q$ vector and the cylinder axis, and $\psi$ |
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42 | is the angle for the ellipsoidal cross section core, to give: |
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43 | |
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44 | .. math:: |
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45 | |
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46 | I(Q,\alpha,\psi) = \frac{\text{scale}}{V_t} |
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47 | \cdot F(Q,\alpha, \psi)^2 \cdot \sin(\alpha) |
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48 | \cdot\exp\left\{ -\frac{1}{2}Q^2\sigma^2 \right\} + \text{background} |
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49 | |
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50 | where a numerical integration of $F(Q,\alpha, \psi)^2\sin(\alpha)$ is |
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51 | carried out over $\alpha$ and $\psi$ for: |
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52 | |
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53 | .. math:: |
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54 | |
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55 | F(Q,\alpha,\psi) = &\bigg[ |
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56 | (\rho_c -\rho_r - \rho_f + \rho_s) V_c |
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57 | \frac{2J_1(QR'\sin \alpha)}{QR'\sin\alpha} |
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58 | \frac{\sin(QL\cos\alpha/2)}{Q(L/2)\cos\alpha} \\ |
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59 | &+(\rho_f - \rho_s) V_{c+f} |
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60 | \frac{2J_1(QR'\sin\alpha)}{QR'\sin\alpha} |
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61 | \frac{\sin(Q(L/2+t_f)\cos\alpha)}{Q(L/2+t_f)\cos\alpha} \\ |
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62 | &+(\rho_r - \rho_s) V_{c+r} |
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63 | \frac{2J_1(Q(R'+t_r)\sin\alpha)}{Q(R'+t_r)\sin\alpha} |
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64 | \frac{\sin(Q(L/2)\cos\alpha)}{Q(L/2)\cos\alpha} |
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65 | \bigg] |
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66 | |
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67 | where |
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68 | |
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69 | .. math:: |
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70 | |
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71 | R' = \frac{R}{\sqrt{2}} |
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72 | \sqrt{(1+X_\text{core}^{2}) + (1-X_\text{core}^{2})\cos(\psi)} |
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73 | |
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74 | |
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75 | and $V_t = \pi (R+t_r)(X_\text{core} R+t_r) L + 2 \pi X_\text{core} R^2 t_f$ is |
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76 | the total volume of the bicelle, $V_c = \pi X_\text{core} R^2 L$ the volume of |
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77 | the core, $V_{c+f} = \pi X_\text{core} R^2 (L+2 t_f)$ the volume of the core |
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78 | plus the volume of the faces, $V_{c+r} = \pi (R+t_r)(X_\text{core} R+t_r) L$ |
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79 | the volume of the core plus the rim, $R$ is the radius of the core, |
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80 | $X_\text{core}$ is the axial ratio of the core, $L$ the length of the core, |
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81 | $t_f$ the thickness of the face, $t_r$ the thickness of the rim and $J_1$ the |
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82 | usual first order bessel function. The core has radii $R$ and $X_\text{core} R$ |
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83 | so is circular, as for the core_shell_bicelle model, for $X_\text{core}=1$. |
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84 | Note that you may need to limit the range of $X_\text{core}$, especially if |
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85 | using the Monte-Carlo algorithm, as setting radius to $R/X_\text{core}$ and |
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86 | axial ratio to $1/X_\text{core}$ gives an equivalent solution! |
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87 | |
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88 | An approximation for the effects of "Gaussian interfacial roughness" $\sigma$ |
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89 | is included, by multiplying $I(Q)$ by |
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90 | $\exp\left \{ -\frac{1}{2}Q^2\sigma^2 \right \}$. This applies, in some way, to |
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91 | all interfaces in the model not just the external ones. (Note that for a one |
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92 | dimensional system convolution of the scattering length density profile with |
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93 | a Gaussian of standard deviation $\sigma$ does exactly this multiplication.) |
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94 | Leave $\sigma$ set to zero for the usual sharp interfaces. |
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95 | |
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96 | The output of the 1D scattering intensity function for randomly oriented |
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97 | bicelles is then given by integrating over all possible $\alpha$ and $\psi$. |
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98 | |
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99 | For oriented bicelles the *theta*, *phi* and *psi* orientation parameters |
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100 | will appear when fitting 2D data, for further details of the calculation |
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101 | and angular dispersions see :ref:`orientation` . |
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102 | |
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103 | .. figure:: img/elliptical_cylinder_angle_definition.png |
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104 | |
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105 | Definition of the angles for the oriented core_shell_bicelle_elliptical |
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106 | particles. |
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107 | |
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108 | |
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109 | |
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110 | References |
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111 | ---------- |
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112 | |
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113 | .. [#] |
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114 | |
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115 | Authorship and Verification |
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116 | ---------------------------- |
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117 | |
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118 | * **Author:** Richard Heenan **Date:** October 5, 2017 |
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119 | * **Last Modified by:** Richard Heenan new 2d orientation **Date:** October 5, 2017 |
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120 | * **Last Reviewed by:** Richard Heenan 2d calc seems agree with 1d **Date:** Nov 2, 2017 |
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121 | """ |
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122 | |
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123 | from numpy import inf, sin, cos, pi |
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124 | |
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125 | name = "core_shell_bicelle_elliptical_belt_rough" |
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126 | title = "Elliptical cylinder with a core-shell scattering length density profile.." |
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127 | description = """ |
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128 | core_shell_bicelle_elliptical_belt_rough |
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129 | Elliptical cylinder core, optional shell on the two flat faces, and "belt" shell of |
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130 | uniform thickness on its rim (in this case NOT extending around the end faces). |
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131 | with approximate interfacial roughness. |
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132 | Please see full documentation for equations and further details. |
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133 | Involves a double numerical integral around the ellipsoid diameter |
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134 | and the angle of the cylinder axis to Q. |
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135 | Compare also the core_shell_bicelle and elliptical_cylinder models. |
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136 | """ |
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137 | category = "shape:cylinder" |
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138 | |
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139 | # pylint: disable=bad-whitespace, line-too-long |
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140 | # ["name", "units", default, [lower, upper], "type", "description"], |
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141 | parameters = [ |
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142 | ["radius", "Ang", 30, [0, inf], "volume", "Cylinder core radius"], |
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143 | ["x_core", "None", 3, [0, inf], "volume", "axial ratio of core, X = r_polar/r_equatorial"], |
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144 | ["thick_rim", "Ang", 8, [0, inf], "volume", "Rim or belt shell thickness"], |
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145 | ["thick_face", "Ang", 14, [0, inf], "volume", "Cylinder face thickness"], |
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146 | ["length", "Ang", 50, [0, inf], "volume", "Cylinder length"], |
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147 | ["sld_core", "1e-6/Ang^2", 4, [-inf, inf], "sld", "Cylinder core scattering length density"], |
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148 | ["sld_face", "1e-6/Ang^2", 7, [-inf, inf], "sld", "Cylinder face scattering length density"], |
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149 | ["sld_rim", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Cylinder rim scattering length density"], |
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150 | ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld", "Solvent scattering length density"], |
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151 | ["theta", "degrees", 90.0, [-360, 360], "orientation", "cylinder axis to beam angle"], |
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152 | ["phi", "degrees", 0, [-360, 360], "orientation", "rotation about beam"], |
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153 | ["psi", "degrees", 0, [-360, 360], "orientation", "rotation about cylinder axis"], |
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154 | ["sigma", "Ang", 0, [0, inf], "", "interfacial roughness"] |
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155 | ] |
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156 | |
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157 | # pylint: enable=bad-whitespace, line-too-long |
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158 | |
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159 | source = ["lib/sas_Si.c", "lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", |
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160 | "core_shell_bicelle_elliptical_belt_rough.c"] |
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161 | |
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162 | demo = dict(scale=1, background=0, |
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163 | radius=30.0, |
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164 | x_core=3.0, |
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165 | thick_rim=8.0, |
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166 | thick_face=14.0, |
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167 | length=50.0, |
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168 | sld_core=4.0, |
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169 | sld_face=7.0, |
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170 | sld_rim=1.0, |
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171 | sld_solvent=6.0, |
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172 | theta=90, |
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173 | phi=0, |
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174 | psi=0, |
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175 | sigma=0) |
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176 | |
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177 | q = 0.1 |
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178 | # april 6 2017, rkh added a 2d unit test, NOT READY YET pull #890 branch assume correct! |
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179 | qx = q*cos(pi/6.0) |
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180 | qy = q*sin(pi/6.0) |
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181 | |
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182 | tests = [ |
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183 | [{'radius': 30.0, 'x_core': 3.0, 'thick_rim':8.0, 'thick_face':14.0, 'length':50.0}, 'ER', 1], |
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184 | [{'radius': 30.0, 'x_core': 3.0, 'thick_rim':8.0, 'thick_face':14.0, 'length':50.0}, 'VR', 1], |
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185 | |
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186 | [{'radius': 30.0, 'x_core': 3.0, 'thick_rim':8.0, 'thick_face':14.0, 'length':50.0, |
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187 | 'sld_core':4.0, 'sld_face':7.0, 'sld_rim':1.0, 'sld_solvent':6.0, 'background':0.0}, |
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188 | 0.015, 189.328], |
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189 | #[{'theta':80., 'phi':10.}, (qx, qy), 7.88866563001 ], |
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190 | ] |
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191 | |
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192 | del qx, qy # not necessary to delete, but cleaner |
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