1 | # rectangular_prism model |
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2 | # Note: model title and parameter table are inserted automatically |
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3 | r""" |
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4 | |
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5 | This model provides the form factor, $P(q)$, for a hollow rectangular |
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6 | parallelepiped with a wall of thickness $\Delta$. |
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7 | |
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8 | |
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9 | Definition |
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10 | ---------- |
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11 | |
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12 | The 1D scattering intensity for this model is calculated by forming |
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13 | the difference of the amplitudes of two massive parallelepipeds |
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14 | differing in their outermost dimensions in each direction by the |
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15 | same length increment $2\Delta$ (Nayuk, 2012). |
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16 | |
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17 | As in the case of the massive parallelepiped model (:ref:`rectangular-prism`), |
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18 | the scattering amplitude is computed for a particular orientation of the |
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19 | parallelepiped with respect to the scattering vector and then averaged over all |
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20 | possible orientations, giving |
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21 | |
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22 | .. math:: |
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23 | P(q) = \frac{1}{V^2} \frac{2}{\pi} \times \, \int_0^{\frac{\pi}{2}} \, |
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24 | \int_0^{\frac{\pi}{2}} A_{P\Delta}^2(q) \, \sin\theta \, d\theta \, d\phi |
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25 | |
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26 | where $\theta$ is the angle between the $z$ axis and the longest axis |
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27 | of the parallelepiped, $\phi$ is the angle between the scattering vector |
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28 | (lying in the $xy$ plane) and the $y$ axis, and |
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29 | |
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30 | .. math:: |
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31 | :nowrap: |
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32 | |
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33 | \begin{align*} |
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34 | A_{P\Delta}(q) & = A B C |
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35 | \left[\frac{\sin \bigl( q \frac{C}{2} \cos\theta \bigr)} |
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36 | {\left( q \frac{C}{2} \cos\theta \right)} \right] |
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37 | \left[\frac{\sin \bigl( q \frac{A}{2} \sin\theta \sin\phi \bigr)} |
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38 | {\left( q \frac{A}{2} \sin\theta \sin\phi \right)}\right] |
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39 | \left[\frac{\sin \bigl( q \frac{B}{2} \sin\theta \cos\phi \bigr)} |
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40 | {\left( q \frac{B}{2} \sin\theta \cos\phi \right)}\right] \\ |
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41 | & - 8 |
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42 | \left(\frac{A}{2}-\Delta\right) \left(\frac{B}{2}-\Delta\right) \left(\frac{C}{2}-\Delta\right) |
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43 | \left[ \frac{\sin \bigl[ q \bigl(\frac{C}{2}-\Delta\bigr) \cos\theta \bigr]} |
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44 | {q \bigl(\frac{C}{2}-\Delta\bigr) \cos\theta} \right] |
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45 | \left[ \frac{\sin \bigl[ q \bigl(\frac{A}{2}-\Delta\bigr) \sin\theta \sin\phi \bigr]} |
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46 | {q \bigl(\frac{A}{2}-\Delta\bigr) \sin\theta \sin\phi} \right] |
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47 | \left[ \frac{\sin \bigl[ q \bigl(\frac{B}{2}-\Delta\bigr) \sin\theta \cos\phi \bigr]} |
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48 | {q \bigl(\frac{B}{2}-\Delta\bigr) \sin\theta \cos\phi} \right] |
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49 | \end{align*} |
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50 | |
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51 | where $A$, $B$ and $C$ are the external sides of the parallelepiped fulfilling |
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52 | $A \le B \le C$, and the volume $V$ of the parallelepiped is |
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53 | |
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54 | .. math:: |
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55 | V = A B C \, - \, (A - 2\Delta) (B - 2\Delta) (C - 2\Delta) |
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56 | |
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57 | The 1D scattering intensity is then calculated as |
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58 | |
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59 | .. math:: |
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60 | I(q) = \text{scale} \times V \times (\rho_\text{p} - |
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61 | \rho_\text{solvent})^2 \times P(q) + \text{background} |
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62 | |
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63 | where $\rho_\text{p}$ is the scattering length of the parallelepiped, |
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64 | $\rho_\text{solvent}$ is the scattering length of the solvent, |
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65 | and (if the data are in absolute units) *scale* represents the volume fraction |
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66 | (which is unitless). |
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67 | |
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68 | For 2d data the orientation of the particle is required, described using |
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69 | angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, for further details |
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70 | of the calculation and angular dispersions see :ref:`orientation` . |
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71 | The angle $\Psi$ is the rotational angle around the long *C* axis. For example, |
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72 | $\Psi = 0$ when the *B* axis is parallel to the *x*-axis of the detector. |
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73 | |
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74 | For 2d, constraints must be applied during fitting to ensure that the inequality |
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75 | $A < B < C$ is not violated, and hence the correct definition of angles is preserved. The calculation will not report an error, |
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76 | but the results may be not correct. |
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77 | |
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78 | .. figure:: img/parallelepiped_angle_definition.png |
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79 | |
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80 | Definition of the angles for oriented hollow rectangular prism. |
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81 | Note that rotation $\theta$, initially in the $xz$ plane, is carried out first, then |
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82 | rotation $\phi$ about the $z$ axis, finally rotation $\Psi$ is now around the axis of the prism. |
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83 | The neutron or X-ray beam is along the $z$ axis. |
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84 | |
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85 | .. figure:: img/parallelepiped_angle_projection.png |
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86 | |
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87 | Examples of the angles for oriented hollow rectangular prisms against the |
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88 | detector plane. |
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89 | |
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90 | |
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91 | Validation |
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92 | ---------- |
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93 | |
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94 | Validation of the code was conducted by qualitatively comparing the output |
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95 | of the 1D model to the curves shown in (Nayuk, 2012). |
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96 | |
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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 | R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854 |
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102 | |
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103 | """ |
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104 | |
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105 | from numpy import pi, inf, sqrt |
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106 | |
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107 | name = "hollow_rectangular_prism" |
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108 | title = "Hollow rectangular parallelepiped with uniform scattering length density." |
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109 | description = """ |
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110 | I(q)= scale*V*(sld - sld_solvent)^2*P(q,theta,phi)+background |
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111 | P(q,theta,phi) = (2/pi/V^2) * double integral from 0 to pi/2 of ... |
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112 | (AP1-AP2)^2(q)*sin(theta)*dtheta*dphi |
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113 | AP1 = S(q*C*cos(theta)/2) * S(q*A*sin(theta)*sin(phi)/2) * S(q*B*sin(theta)*cos(phi)/2) |
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114 | AP2 = S(q*C'*cos(theta)) * S(q*A'*sin(theta)*sin(phi)) * S(q*B'*sin(theta)*cos(phi)) |
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115 | C' = (C/2-thickness) |
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116 | B' = (B/2-thickness) |
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117 | A' = (A/2-thickness) |
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118 | S(x) = sin(x)/x |
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119 | """ |
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120 | category = "shape:parallelepiped" |
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121 | |
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122 | # ["name", "units", default, [lower, upper], "type","description"], |
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123 | parameters = [["sld", "1e-6/Ang^2", 6.3, [-inf, inf], "sld", |
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124 | "Parallelepiped scattering length density"], |
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125 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
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126 | "Solvent scattering length density"], |
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127 | ["length_a", "Ang", 35, [0, inf], "volume", |
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128 | "Shorter side of the parallelepiped"], |
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129 | ["b2a_ratio", "Ang", 1, [0, inf], "volume", |
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130 | "Ratio sides b/a"], |
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131 | ["c2a_ratio", "Ang", 1, [0, inf], "volume", |
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132 | "Ratio sides c/a"], |
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133 | ["thickness", "Ang", 1, [0, inf], "volume", |
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134 | "Thickness of parallelepiped"], |
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135 | ["theta", "degrees", 0, [-360, 360], "orientation", |
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136 | "c axis to beam angle"], |
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137 | ["phi", "degrees", 0, [-360, 360], "orientation", |
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138 | "rotation about beam"], |
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139 | ["psi", "degrees", 0, [-360, 360], "orientation", |
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140 | "rotation about c axis"], |
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141 | ] |
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142 | |
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143 | source = ["lib/gauss76.c", "hollow_rectangular_prism.c"] |
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144 | |
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145 | def ER(length_a, b2a_ratio, c2a_ratio, thickness): |
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146 | """ |
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147 | Return equivalent radius (ER) |
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148 | thickness parameter not used |
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149 | """ |
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150 | b_side = length_a * b2a_ratio |
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151 | c_side = length_a * c2a_ratio |
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152 | |
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153 | # surface average radius (rough approximation) |
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154 | surf_rad = sqrt(length_a * b_side / pi) |
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155 | |
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156 | ddd = 0.75 * surf_rad * (2 * surf_rad * c_side + (c_side + surf_rad) * (c_side + pi * surf_rad)) |
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157 | return 0.5 * (ddd) ** (1. / 3.) |
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158 | |
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159 | def VR(length_a, b2a_ratio, c2a_ratio, thickness): |
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160 | """ |
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161 | Return shell volume and total volume |
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162 | """ |
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163 | b_side = length_a * b2a_ratio |
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164 | c_side = length_a * c2a_ratio |
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165 | a_core = length_a - 2.0*thickness |
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166 | b_core = b_side - 2.0*thickness |
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167 | c_core = c_side - 2.0*thickness |
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168 | vol_core = a_core * b_core * c_core |
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169 | vol_total = length_a * b_side * c_side |
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170 | vol_shell = vol_total - vol_core |
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171 | return vol_total, vol_shell |
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172 | |
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173 | |
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174 | def random(): |
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175 | import numpy as np |
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176 | a, b, c = 10**np.random.uniform(1, 4.7, size=3) |
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177 | # Thickness is limited to 1/2 the smallest dimension |
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178 | # Use a distribution with a preference for thin shell or thin core |
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179 | # Avoid core,shell radii < 1 |
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180 | min_dim = 0.5*min(a, b, c) |
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181 | thickness = np.random.beta(0.5, 0.5)*(min_dim-2) + 1 |
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182 | #print(a, b, c, thickness, thickness/min_dim) |
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183 | pars = dict( |
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184 | length_a=a, |
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185 | b2a_ratio=b/a, |
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186 | c2a_ratio=c/a, |
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187 | thickness=thickness, |
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188 | ) |
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189 | return pars |
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190 | |
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191 | |
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192 | # parameters for demo |
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193 | demo = dict(scale=1, background=0, |
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194 | sld=6.3, sld_solvent=1.0, |
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195 | length_a=35, b2a_ratio=1, c2a_ratio=1, thickness=1, |
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196 | length_a_pd=0.1, length_a_pd_n=10, |
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197 | b2a_ratio_pd=0.1, b2a_ratio_pd_n=1, |
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198 | c2a_ratio_pd=0.1, c2a_ratio_pd_n=1) |
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199 | |
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200 | tests = [[{}, 0.2, 0.76687283098], |
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201 | [{}, [0.2], [0.76687283098]], |
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202 | ] |
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