1 | # parallelepiped 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 | The form factor is normalized by the particle volume. |
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5 | |
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6 | Definition |
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7 | ---------- |
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8 | |
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9 | | This model calculates the scattering from a rectangular parallelepiped (:numref:`parallelepiped-image`). |
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10 | | If you need to apply polydispersity, see also :ref:`rectangular-prism`. |
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11 | |
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12 | .. _parallelepiped-image: |
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13 | |
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14 | .. figure:: img/parallelepiped_geometry.jpg |
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15 | |
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16 | Parallelepiped with the corresponding definition of sides. |
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17 | |
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18 | .. note:: |
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19 | |
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20 | The edge of the solid must satisfy the condition that $A < B < C$. |
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21 | This requirement is not enforced in the model, so it is up to the |
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22 | user to check this during the analysis. |
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23 | |
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24 | The 1D scattering intensity $I(q)$ is calculated as: |
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25 | |
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26 | .. Comment by Miguel Gonzalez: |
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27 | I am modifying the original text because I find the notation a little bit |
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28 | confusing. I think that in most textbooks/papers, the notation P(Q) is |
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29 | used for the form factor (adim, P(Q=0)=1), although F(q) seems also to |
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30 | be used. But here (as for many other models), P(q) is used to represent |
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31 | the scattering intensity (in cm-1 normally). It would be good to agree on |
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32 | a common notation. |
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33 | |
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34 | .. math:: |
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35 | |
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36 | I(q) = \frac{\text{scale}}{V} (\Delta\rho \cdot V)^2 \left< P(q, \alpha) \right> |
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37 | + \text{background} |
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38 | |
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39 | where the volume $V = A B C$, the contrast is defined as |
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40 | :math:`\Delta\rho = \rho_{\textstyle p} - \rho_{\textstyle solvent}`, |
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41 | $P(q, \alpha)$ is the form factor corresponding to a parallelepiped oriented |
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42 | at an angle $\alpha$ (angle between the long axis C and :math:`\vec q`), |
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43 | and the averaging $\left<\ldots\right>$ is applied over all orientations. |
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44 | |
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45 | Assuming $a = A/B < 1$, $b = B /B = 1$, and $c = C/B > 1$, the |
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46 | form factor is given by (Mittelbach and Porod, 1961) |
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47 | |
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48 | .. math:: |
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49 | |
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50 | P(q, \alpha) = \int_0^1 \phi_Q\left(\mu \sqrt{1-\sigma^2},a\right) |
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51 | \left[S(\mu c \sigma/2)\right]^2 d\sigma |
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52 | |
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53 | with |
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54 | |
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55 | .. math:: |
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56 | |
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57 | \phi_Q(\mu,a) = \int_0^1 |
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58 | \left\{S\left[\frac{\mu}{2}\cos\left(\frac{\pi}{2}u\right)\right] |
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59 | S\left[\frac{\mu a}{2}\sin\left(\frac{\pi}{2}u\right)\right] |
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60 | \right\}^2 du |
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61 | |
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62 | S(x) = \frac{\sin x}{x} |
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63 | |
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64 | \mu = qB |
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65 | |
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66 | |
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67 | The scattering intensity per unit volume is returned in units of |cm^-1|. |
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68 | |
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69 | NB: The 2nd virial coefficient of the parallelepiped is calculated based on |
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70 | the averaged effective radius $(=\sqrt{A B / \pi})$ and |
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71 | length $(= C)$ values, and used as the effective radius for |
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72 | $S(q)$ when $P(q) \cdot S(q)$ is applied. |
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73 | |
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74 | To provide easy access to the orientation of the parallelepiped, we define |
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75 | three angles $\theta$, $\phi$ and $\Psi$. The definition of $\theta$ and |
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76 | $\phi$ is the same as for the cylinder model (see also figures below). |
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77 | |
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78 | .. Comment by Miguel Gonzalez: |
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79 | The following text has been commented because I think there are two |
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80 | mistakes. Psi is the rotational angle around C (but I cannot understand |
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81 | what it means against the q plane) and psi=0 corresponds to a||x and b||y. |
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82 | |
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83 | The angle $\Psi$ is the rotational angle around the $C$ axis against |
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84 | the $q$ plane. For example, $\Psi = 0$ when the $B$ axis is parallel |
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85 | to the $x$-axis of the detector. |
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86 | |
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87 | The angle $\Psi$ is the rotational angle around the $C$ axis. |
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88 | For $\theta = 0$ and $\phi = 0$, $\Psi = 0$ corresponds to the $B$ axis |
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89 | oriented parallel to the y-axis of the detector with $A$ along the z-axis. |
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90 | For other $\theta$, $\phi$ values, the parallelepiped has to be first rotated |
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91 | $\theta$ degrees around $z$ and $\phi$ degrees around $y$, |
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92 | before doing a final rotation of $\Psi$ degrees around the resulting $C$ to |
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93 | obtain the final orientation of the parallelepiped. |
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94 | For example, for $\theta = 0$ and $\phi = 90$, we have that $\Psi = 0$ |
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95 | corresponds to $A$ along $x$ and $B$ along $y$, |
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96 | while for $\theta = 90$ and $\phi = 0$, $\Psi = 0$ corresponds to |
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97 | $A$ along $z$ and $B$ along $x$. |
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98 | |
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99 | .. _parallelepiped-orientation: |
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100 | |
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101 | .. figure:: img/parallelepiped_angle_definition.jpg |
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102 | |
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103 | Definition of the angles for oriented parallelepipeds. |
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104 | |
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105 | .. figure:: img/parallelepiped_angle_projection.jpg |
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106 | |
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107 | Examples of the angles for oriented parallelepipeds against the detector plane. |
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108 | |
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109 | For a given orientation of the parallelepiped, the 2D form factor is calculated as |
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110 | |
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111 | .. math:: |
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112 | |
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113 | P(q_x, q_y) = \left[\frac{sin(qA\cos\alpha/2)}{(qA\cos\alpha/2)}\right]^2 |
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114 | \left[\frac{sin(qB\cos\beta/2)}{(qB\cos\beta/2)}\right]^2 |
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115 | \left[\frac{sin(qC\cos\gamma/2)}{(qC\cos\gamma/2)}\right]^2 |
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116 | |
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117 | with |
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118 | |
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119 | .. math:: |
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120 | |
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121 | \cos\alpha = \hat A \cdot \hat q, \; |
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122 | \cos\beta = \hat B \cdot \hat q, \; |
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123 | \cos\gamma = \hat C \cdot \hat q |
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124 | |
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125 | and the scattering intensity as: |
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126 | |
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127 | .. math:: |
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128 | |
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129 | I(q_x, q_y) = \frac{\text{scale}}{V} V^2 \Delta\rho^2 P(q_x, q_y) + \text{background} |
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130 | |
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131 | .. Comment by Miguel Gonzalez: |
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132 | This reflects the logic of the code, as in parallelepiped.c the call |
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133 | to _pkernel returns P(q_x, q_y) and then this is multiplied by V^2 * (delta rho)^2. |
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134 | And finally outside parallelepiped.c it will be multiplied by scale, normalized by |
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135 | V and the background added. But mathematically it makes more sense to write |
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136 | I(q_x, q_y) = \text{scale} V \Delta\rho^2 P(q_x, q_y) + \text{background}, |
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137 | with scale being the volume fraction. |
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138 | |
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139 | |
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140 | Validation |
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141 | ---------- |
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142 | |
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143 | Validation of the code was done by comparing the output of the 1D calculation |
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144 | to the angular average of the output of a 2D calculation over all possible |
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145 | angles. |
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146 | |
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147 | This model is based on form factor calculations implemented in a c-library |
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148 | provided by the NIST Center for Neutron Research (Kline, 2006). |
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149 | |
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150 | References |
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151 | ---------- |
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152 | |
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153 | P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211 |
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154 | |
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155 | R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854 |
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156 | """ |
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157 | |
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158 | import numpy as np |
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159 | from numpy import pi, inf, sqrt |
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160 | |
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161 | name = "parallelepiped" |
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162 | title = "Rectangular parallelepiped with uniform scattering length density." |
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163 | description = """ |
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164 | I(q)= scale*V*(sld - sld_solvent)^2*P(q,alpha)+background |
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165 | P(q,alpha) = integral from 0 to 1 of ... |
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166 | phi(mu*sqrt(1-sigma^2),a) * S(mu*c*sigma/2)^2 * dsigma |
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167 | with |
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168 | phi(mu,a) = integral from 0 to 1 of .. |
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169 | (S((mu/2)*cos(pi*u/2))*S((mu*a/2)*sin(pi*u/2)))^2 * du |
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170 | S(x) = sin(x)/x |
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171 | mu = q*B |
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172 | V: Volume of the rectangular parallelepiped |
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173 | alpha: angle between the long axis of the |
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174 | parallelepiped and the q-vector for 1D |
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175 | """ |
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176 | category = "shape:parallelepiped" |
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177 | |
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178 | # ["name", "units", default, [lower, upper], "type","description"], |
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179 | parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "", |
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180 | "Parallelepiped scattering length density"], |
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181 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "", |
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182 | "Solvent scattering length density"], |
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183 | ["a_side", "Ang", 35, [0, inf], "volume", |
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184 | "Shorter side of the parallelepiped"], |
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185 | ["b_side", "Ang", 75, [0, inf], "volume", |
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186 | "Second side of the parallelepiped"], |
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187 | ["c_side", "Ang", 400, [0, inf], "volume", |
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188 | "Larger side of the parallelepiped"], |
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189 | ["theta", "degrees", 60, [-inf, inf], "orientation", |
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190 | "In plane angle"], |
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191 | ["phi", "degrees", 60, [-inf, inf], "orientation", |
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192 | "Out of plane angle"], |
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193 | ["psi", "degrees", 60, [-inf, inf], "orientation", |
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194 | "Rotation angle around its own c axis against q plane"], |
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195 | ] |
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196 | |
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197 | source = ["lib/gauss76.c", "parallelepiped.c"] |
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198 | |
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199 | def ER(a_side, b_side, c_side): |
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200 | """ |
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201 | Return effective radius (ER) for P(q)*S(q) |
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202 | """ |
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203 | |
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204 | # surface average radius (rough approximation) |
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205 | surf_rad = sqrt(a_side * b_side / pi) |
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206 | |
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207 | ddd = 0.75 * surf_rad * (2 * surf_rad * c_side + (c_side + surf_rad) * (c_side + pi * surf_rad)) |
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208 | return 0.5 * (ddd) ** (1. / 3.) |
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209 | |
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210 | # VR defaults to 1.0 |
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211 | |
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212 | # parameters for demo |
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213 | demo = dict(scale=1, background=0, |
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214 | sld=6.3e-6, sld_solvent=1.0e-6, |
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215 | a_side=35, b_side=75, c_side=400, |
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216 | theta=45, phi=30, psi=15, |
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217 | a_side_pd=0.1, a_side_pd_n=10, |
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218 | b_side_pd=0.1, b_side_pd_n=1, |
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219 | c_side_pd=0.1, c_side_pd_n=1, |
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220 | theta_pd=10, theta_pd_n=1, |
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221 | phi_pd=10, phi_pd_n=1, |
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222 | psi_pd=10, psi_pd_n=10) |
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223 | |
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224 | qx, qy = 0.2 * np.cos(2.5), 0.2 * np.sin(2.5) |
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225 | tests = [[{}, 0.2, 0.17758004974], |
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226 | [{}, [0.2], [0.17758004974]], |
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227 | [{'theta':10.0, 'phi':10.0}, (qx, qy), 0.00560296014], |
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228 | [{'theta':10.0, 'phi':10.0}, [(qx, qy)], [0.00560296014]], |
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229 | ] |
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230 | del qx, qy # not necessary to delete, but cleaner |
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