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 a circular 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 a core-shell cylinder or disc where the shell on the walls and ends |
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8 | may be of different thicknesses and scattering length densities. The form |
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9 | factor is normalized by the particle volume. |
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10 | |
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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 cylindrical symmetry 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 cylinder 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, to give: |
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39 | |
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40 | .. math:: |
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41 | |
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42 | I(Q,\alpha) = \frac{\text{scale}}{V_t} \cdot |
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43 | F(Q,\alpha)^2 \cdot sin(\alpha) + \text{background} |
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44 | |
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45 | where |
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46 | |
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47 | .. math:: |
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48 | :nowrap: |
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49 | |
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50 | \begin{align*} |
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51 | F(Q,\alpha) = &\bigg[ |
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52 | (\rho_c - \rho_f) V_c |
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53 | \frac{2J_1(QRsin \alpha)}{QRsin\alpha} |
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54 | \frac{sin(QLcos\alpha/2)}{Q(L/2)cos\alpha} \\ |
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55 | &+(\rho_f - \rho_r) V_{c+f} |
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56 | \frac{2J_1(QRsin\alpha)}{QRsin\alpha} |
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57 | \frac{sin(Q(L/2+t_f)cos\alpha)}{Q(L/2+t_f)cos\alpha} \\ |
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58 | &+(\rho_r - \rho_s) V_t |
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59 | \frac{2J_1(Q(R+t_r)sin\alpha)}{Q(R+t_r)sin\alpha} |
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60 | \frac{sin(Q(L/2+t_f)cos\alpha)}{Q(L/2+t_f)cos\alpha} |
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61 | \bigg] |
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62 | \end{align*} |
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63 | |
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64 | where $V_t$ is the total volume of the bicelle, $V_c$ the volume of the core, |
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65 | $V_{c+f}$ the volume of the core plus the volume of the faces, $R$ is the radius |
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66 | of the core, $L$ the length of the core, $t_f$ the thickness of the face, $t_r$ |
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67 | the thickness of the rim and $J_1$ the usual first order bessel function. |
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68 | |
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69 | The output of the 1D scattering intensity function for randomly oriented |
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70 | cylinders is then given by integrating over all possible $\theta$ and $\phi$. |
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71 | |
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72 | For oriented bicelles the *theta*, and *phi* orientation parameters will appear |
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73 | when fitting 2D data, see the :ref:`cylinder` model for further information. |
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74 | Our implementation of the scattering kernel and the 1D scattering intensity |
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75 | use the c-library from NIST. |
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76 | |
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77 | .. figure:: img/cylinder_angle_definition.png |
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78 | |
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79 | Definition of the angles for the oriented core shell bicelle model, |
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80 | note that the cylinder axis of the bicelle starts along the beam direction |
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81 | when $\theta = \phi = 0$. |
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82 | |
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83 | |
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84 | References |
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85 | ---------- |
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86 | |
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87 | .. [#] D Singh (2009). *Small angle scattering studies of self assembly in |
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88 | lipid mixtures*, John's Hopkins University Thesis (2009) 223-225. `Available |
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89 | from Proquest <http://search.proquest.com/docview/304915826?accountid |
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90 | =26379>`_ |
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91 | |
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92 | L. Onsager, Ann. New York Acad. Sci. 51, 627-659 (1949). |
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93 | |
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94 | Authorship and Verification |
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95 | ---------------------------- |
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96 | |
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97 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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98 | * **Last Modified by:** Paul Butler **Date:** September 30, 2016 |
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99 | * **Last Reviewed by:** Richard Heenan **Date:** January 4, 2017 |
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100 | """ |
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101 | |
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102 | import numpy as np |
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103 | from numpy import inf, sin, cos, pi |
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104 | |
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105 | name = "core_shell_bicelle" |
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106 | title = "Circular cylinder with a core-shell scattering length density profile.." |
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107 | description = """ |
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108 | P(q,alpha)= (scale/Vs)*f(q)^(2) + bkg, where: |
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109 | f(q)= Vt(sld_rim - sld_solvent)* sin[qLt.cos(alpha)/2] |
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110 | /[qLt.cos(alpha)/2]*J1(qRout.sin(alpha)) |
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111 | /[qRout.sin(alpha)]+ |
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112 | (sld_core-sld_face)*Vc*sin[qLcos(alpha)/2][[qL |
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113 | *cos(alpha)/2]*J1(qRc.sin(alpha)) |
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114 | /qRc.sin(alpha)]+ |
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115 | (sld_face-sld_rim)*(Vc+Vf)*sin[q(L+2.thick_face). |
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116 | cos(alpha)/2][[q(L+2.thick_face)*cos(alpha)/2]* |
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117 | J1(qRc.sin(alpha))/qRc.sin(alpha)] |
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118 | |
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119 | alpha:is the angle between the axis of |
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120 | the cylinder and the q-vector |
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121 | Vt = pi.(Rc + thick_rim)^2.Lt : total volume |
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122 | Vc = pi.Rc^2.L :the volume of the core |
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123 | Vf = 2.pi.Rc^2.thick_face |
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124 | Rc = radius: is the core radius |
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125 | L: the length of the core |
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126 | Lt = L + 2.thick_face: total length |
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127 | Rout = radius + thick_rim |
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128 | sld_core, sld_rim, sld_face:scattering length |
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129 | densities within the particle |
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130 | sld_solvent: the scattering length density |
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131 | of the solvent |
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132 | bkg: the background |
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133 | J1: the first order Bessel function |
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134 | theta: axis_theta of the cylinder |
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135 | phi: the axis_phi of the cylinder... |
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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", 80, [0, inf], "volume", "Cylinder core radius"], |
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143 | ["thick_rim", "Ang", 10, [0, inf], "volume", "Rim shell thickness"], |
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144 | ["thick_face", "Ang", 10, [0, inf], "volume", "Cylinder face thickness"], |
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145 | ["length", "Ang", 50, [0, inf], "volume", "Cylinder length"], |
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146 | ["sld_core", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Cylinder core scattering length density"], |
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147 | ["sld_face", "1e-6/Ang^2", 4, [-inf, inf], "sld", "Cylinder face scattering length density"], |
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148 | ["sld_rim", "1e-6/Ang^2", 4, [-inf, inf], "sld", "Cylinder rim scattering length density"], |
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149 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Solvent scattering length density"], |
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150 | ["theta", "degrees", 90, [-360, 360], "orientation", "cylinder axis to beam angle"], |
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151 | ["phi", "degrees", 0, [-360, 360], "orientation", "rotation about beam"] |
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152 | ] |
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153 | |
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154 | # pylint: enable=bad-whitespace, line-too-long |
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155 | |
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156 | source = ["lib/sas_Si.c", "lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", |
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157 | "core_shell_bicelle.c"] |
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158 | have_Fq = True |
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159 | effective_radius_type = [ |
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160 | "excluded volume","equivalent volume sphere", "outer rim radius", |
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161 | "half outer thickness", "half diagonal", |
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162 | ] |
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163 | |
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164 | def random(): |
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165 | pars = dict( |
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166 | radius=10**np.random.uniform(1.3, 3), |
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167 | length=10**np.random.uniform(1.3, 4), |
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168 | thick_rim=10**np.random.uniform(0, 1.7), |
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169 | thick_face=10**np.random.uniform(0, 1.7), |
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170 | ) |
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171 | return pars |
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172 | |
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173 | demo = dict(scale=1, background=0, |
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174 | radius=20.0, |
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175 | thick_rim=10.0, |
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176 | thick_face=10.0, |
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177 | length=400.0, |
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178 | sld_core=1.0, |
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179 | sld_face=4.0, |
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180 | sld_rim=4.0, |
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181 | sld_solvent=1.0, |
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182 | theta=90, |
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183 | phi=0) |
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184 | q = 0.1 |
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185 | # april 6 2017, rkh add unit tests, NOT compared with any other calc method, assume correct! |
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186 | qx = q*cos(pi/6.0) |
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187 | qy = q*sin(pi/6.0) |
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188 | tests = [ |
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189 | [{}, 0.05, 7.4883545957], |
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190 | [{'theta':80., 'phi':10.}, (qx, qy), 2.81048892474] |
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191 | ] |
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192 | del qx, qy # not necessary to delete, but cleaner |
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