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 | Definition |
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5 | ---------- |
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6 | |
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7 | This model calculates the scattering from a rectangular solid |
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8 | (:numref:`parallelepiped-image`). |
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9 | If you need to apply polydispersity, see also :ref:`rectangular-prism`. For |
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10 | information about polarised and magnetic scattering, see |
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11 | the :ref:`magnetism` documentation. |
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12 | |
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13 | .. _parallelepiped-image: |
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14 | |
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15 | |
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16 | .. figure:: img/parallelepiped_geometry.jpg |
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17 | |
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18 | Parallelepiped with the corresponding definition of sides. |
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19 | |
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20 | The three dimensions of the parallelepiped (strictly here a cuboid) may be |
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21 | given in *any* size order as long as the particles are randomly oriented (i.e. |
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22 | take on all possible orientations see notes on 2D below). To avoid multiple fit |
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23 | solutions, especially with Monte-Carlo fit methods, it may be advisable to |
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24 | restrict their ranges. There may be a number of closely similar "best fits", so |
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25 | some trial and error, or fixing of some dimensions at expected values, may |
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26 | help. |
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27 | |
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28 | The form factor is normalized by the particle volume and the 1D scattering |
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29 | intensity $I(q)$ is then calculated as: |
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30 | |
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31 | .. Comment by Miguel Gonzalez: |
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32 | I am modifying the original text because I find the notation a little bit |
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33 | confusing. I think that in most textbooks/papers, the notation P(Q) is |
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34 | used for the form factor (adim, P(Q=0)=1), although F(q) seems also to |
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35 | be used. But here (as for many other models), P(q) is used to represent |
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36 | the scattering intensity (in cm-1 normally). It would be good to agree on |
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37 | a common notation. |
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38 | |
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39 | .. math:: |
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40 | |
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41 | I(q) = \frac{\text{scale}}{V} (\Delta\rho \cdot V)^2 |
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42 | \left< P(q, \alpha, \beta) \right> + \text{background} |
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43 | |
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44 | where the volume $V = A B C$, the contrast is defined as |
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45 | $\Delta\rho = \rho_\text{p} - \rho_\text{solvent}$, $P(q, \alpha, \beta)$ |
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46 | is the form factor corresponding to a parallelepiped oriented |
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47 | at an angle $\alpha$ (angle between the long axis C and $\vec q$), and $\beta$ |
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48 | (the angle between the projection of the particle in the $xy$ detector plane |
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49 | and the $y$ axis) and the averaging $\left<\ldots\right>$ is applied over all |
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50 | orientations. |
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51 | |
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52 | Assuming $a = A/B < 1$, $b = B /B = 1$, and $c = C/B > 1$, the |
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53 | form factor is given by (Mittelbach and Porod, 1961 [#Mittelbach]_) |
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54 | |
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55 | .. math:: |
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56 | |
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57 | P(q, \alpha) = \int_0^1 \phi_Q\left(\mu \sqrt{1-\sigma^2},a\right) |
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58 | \left[S(\mu c \sigma/2)\right]^2 d\sigma |
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59 | |
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60 | with |
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61 | |
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62 | .. math:: |
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63 | |
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64 | \phi_Q(\mu,a) &= \int_0^1 |
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65 | \left\{S\left[\frac{\mu}{2}\cos\left(\frac{\pi}{2}u\right)\right] |
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66 | S\left[\frac{\mu a}{2}\sin\left(\frac{\pi}{2}u\right)\right] |
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67 | \right\}^2 du \\ |
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68 | S(x) &= \frac{\sin x}{x} \\ |
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69 | \mu &= qB |
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70 | |
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71 | where substitution of $\sigma = cos\alpha$ and $\beta = \pi/2 \ u$ have been |
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72 | applied. |
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73 | |
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74 | For **oriented** particles, the 2D scattering intensity, $I(q_x, q_y)$, is |
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75 | given as: |
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76 | |
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77 | .. math:: |
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78 | |
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79 | I(q_x, q_y) = \frac{\text{scale}}{V} (\Delta\rho \cdot V)^2 P(q_x, q_y) |
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80 | + \text{background} |
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81 | |
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82 | .. Comment by Miguel Gonzalez: |
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83 | This reflects the logic of the code, as in parallelepiped.c the call |
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84 | to _pkernel returns $P(q_x, q_y)$ and then this is multiplied by |
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85 | $V^2 * (\Delta \rho)^2$. And finally outside parallelepiped.c it will be |
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86 | multiplied by scale, normalized by $V$ and the background added. But |
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87 | mathematically it makes more sense to write |
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88 | $I(q_x, q_y) = \text{scale} V \Delta\rho^2 P(q_x, q_y) + \text{background}$, |
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89 | with scale being the volume fraction. |
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90 | |
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91 | Where $P(q_x, q_y)$ for a given orientation of the form factor is calculated as |
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92 | |
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93 | .. math:: |
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94 | |
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95 | P(q_x, q_y) = \left[\frac{\sin(\tfrac{1}{2}qA\cos\alpha)}{(\tfrac{1} |
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96 | {2}qA\cos\alpha)}\right]^2 |
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97 | \left[\frac{\sin(\tfrac{1}{2}qB\cos\beta)}{(\tfrac{1} |
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98 | {2}qB\cos\beta)}\right]^2 |
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99 | \left[\frac{\sin(\tfrac{1}{2}qC\cos\gamma)}{(\tfrac{1} |
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100 | {2}qC\cos\gamma)}\right]^2 |
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101 | |
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102 | with |
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103 | |
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104 | .. math:: |
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105 | |
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106 | \cos\alpha &= \hat A \cdot \hat q, \\ |
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107 | \cos\beta &= \hat B \cdot \hat q, \\ |
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108 | \cos\gamma &= \hat C \cdot \hat q |
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109 | |
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110 | |
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111 | FITTING NOTES |
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112 | ~~~~~~~~~~~~~ |
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113 | |
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114 | #. The 2nd virial coefficient of the parallelepiped is calculated based on |
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115 | the averaged effective radius, after appropriately sorting the three |
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116 | dimensions, to give an oblate or prolate particle, $(=\sqrt{AB/\pi})$ and |
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117 | length $(= C)$ values, and used as the effective radius for |
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118 | $S(q)$ when $P(q) \cdot S(q)$ is applied. |
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119 | |
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120 | #. For 2d data the orientation of the particle is required, described using |
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121 | angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, where $\theta$ |
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122 | and $\phi$ define the orientation of the director in the laboratry reference |
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123 | frame of the beam direction ($z$) and detector plane ($x-y$ plane), while |
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124 | the angle $\Psi$ is effectively the rotational angle around the particle |
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125 | $C$ axis. For $\theta = 0$ and $\phi = 0$, $\Psi = 0$ corresponds to the |
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126 | $B$ axis oriented parallel to the y-axis of the detector with $A$ along |
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127 | the x-axis. For other $\theta$, $\phi$ values, the order of rotations |
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128 | matters. In particular, the parallelepiped must first be rotated $\theta$ |
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129 | degrees in the $x-z$ plane before rotating $\phi$ degrees around the $z$ |
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130 | axis (in the $x-y$ plane). Applying orientational distribution to the |
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131 | particle orientation (i.e `jitter` to one or more of these angles) can get |
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132 | more confusing as `jitter` is defined **NOT** with respect to the laboratory |
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133 | frame but the particle reference frame. It is thus highly recmmended to |
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134 | read :ref:`orientation` for further details of the calculation and angular |
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135 | dispersions. |
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136 | |
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137 | .. note:: For 2d, constraints must be applied during fitting to ensure that the |
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138 | order of sides chosen is not altered, and hence that the correct definition |
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139 | of angles is preserved. For the default choice shown here, that means |
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140 | ensuring that the inequality $A < B < C$ is not violated, The calculation |
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141 | will not report an error, but the results may be not correct. |
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142 | |
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143 | .. _parallelepiped-orientation: |
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144 | |
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145 | .. figure:: img/parallelepiped_angle_definition.png |
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146 | |
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147 | Definition of the angles for oriented parallelepiped, shown with $A<B<C$. |
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148 | |
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149 | .. figure:: img/parallelepiped_angle_projection.png |
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150 | |
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151 | Examples of the angles for an oriented parallelepiped against the |
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152 | detector plane. |
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153 | |
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154 | .. Comment by Paul Butler |
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155 | I am commenting this section out as we are trying to minimize the amount of |
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156 | oritentational detail here and encourage the user to go to the full |
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157 | orientation documentation so that changes can be made in just one place. |
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158 | below is the commented paragrah: |
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159 | On introducing "Orientational Distribution" in the angles, "distribution of |
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160 | theta" and "distribution of phi" parameters will appear. These are actually |
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161 | rotations about axes $\delta_1$ and $\delta_2$ of the parallelepiped, |
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162 | perpendicular to the $a$ x $c$ and $b$ x $c$ faces. (When $\theta = \phi = 0$ |
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163 | these are parallel to the $Y$ and $X$ axes of the instrument.) The third |
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164 | orientation distribution, in $\psi$, is about the $c$ axis of the particle, |
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165 | perpendicular to the $a$ x $b$ face. Some experimentation may be required to |
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166 | understand the 2d patterns fully as discussed in :ref:`orientation` . |
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167 | |
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168 | |
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169 | Validation |
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170 | ---------- |
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171 | |
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172 | Validation of the code was done by comparing the output of the 1D calculation |
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173 | to the angular average of the output of a 2D calculation over all possible |
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174 | angles. |
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175 | |
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176 | References |
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177 | ---------- |
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178 | |
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179 | .. [#Mittelbach] P Mittelbach and G Porod, *Acta Physica Austriaca*, |
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180 | 14 (1961) 185-211 |
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181 | .. [#] R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854 |
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182 | |
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183 | Authorship and Verification |
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184 | ---------------------------- |
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185 | |
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186 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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187 | * **Last Modified by:** Paul Kienzle **Date:** April 05, 2017 |
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188 | * **Last Reviewed by:** Miguel Gonzales and Paul Butler **Date:** May 24, |
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189 | 2018 - documentation updated |
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190 | """ |
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191 | |
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192 | import numpy as np |
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193 | from numpy import pi, inf, sqrt, sin, cos |
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194 | |
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195 | name = "parallelepiped" |
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196 | title = "Rectangular parallelepiped with uniform scattering length density." |
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197 | description = """ |
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198 | I(q)= scale*V*(sld - sld_solvent)^2*P(q,alpha)+background |
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199 | P(q,alpha) = integral from 0 to 1 of ... |
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200 | phi(mu*sqrt(1-sigma^2),a) * S(mu*c*sigma/2)^2 * dsigma |
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201 | with |
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202 | phi(mu,a) = integral from 0 to 1 of .. |
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203 | (S((mu/2)*cos(pi*u/2))*S((mu*a/2)*sin(pi*u/2)))^2 * du |
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204 | S(x) = sin(x)/x |
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205 | mu = q*B |
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206 | V: Volume of the rectangular parallelepiped |
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207 | alpha: angle between the long axis of the |
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208 | parallelepiped and the q-vector for 1D |
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209 | """ |
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210 | category = "shape:parallelepiped" |
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211 | |
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212 | # ["name", "units", default, [lower, upper], "type","description"], |
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213 | parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "sld", |
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214 | "Parallelepiped scattering length density"], |
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215 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
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216 | "Solvent scattering length density"], |
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217 | ["length_a", "Ang", 35, [0, inf], "volume", |
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218 | "Shorter side of the parallelepiped"], |
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219 | ["length_b", "Ang", 75, [0, inf], "volume", |
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220 | "Second side of the parallelepiped"], |
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221 | ["length_c", "Ang", 400, [0, inf], "volume", |
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222 | "Larger side of the parallelepiped"], |
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223 | ["theta", "degrees", 60, [-360, 360], "orientation", |
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224 | "c axis to beam angle"], |
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225 | ["phi", "degrees", 60, [-360, 360], "orientation", |
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226 | "rotation about beam"], |
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227 | ["psi", "degrees", 60, [-360, 360], "orientation", |
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228 | "rotation about c axis"], |
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229 | ] |
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230 | |
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231 | source = ["lib/gauss76.c", "parallelepiped.c"] |
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232 | |
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233 | def ER(length_a, length_b, length_c): |
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234 | """ |
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235 | Return effective radius (ER) for P(q)*S(q) |
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236 | """ |
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237 | # now that axes can be in any size order, need to sort a,b,c |
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238 | # where a~b and c is either much smaller or much larger |
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239 | abc = np.vstack((length_a, length_b, length_c)) |
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240 | abc = np.sort(abc, axis=0) |
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241 | selector = (abc[1] - abc[0]) > (abc[2] - abc[1]) |
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242 | length = np.where(selector, abc[0], abc[2]) |
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243 | # surface average radius (rough approximation) |
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244 | radius = sqrt(np.where(~selector, abc[0]*abc[1], abc[1]*abc[2]) / pi) |
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245 | |
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246 | ddd = 0.75 * radius * (2*radius*length + (length + radius)*(length + pi*radius)) |
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247 | return 0.5 * (ddd) ** (1. / 3.) |
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248 | |
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249 | # VR defaults to 1.0 |
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250 | |
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251 | |
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252 | def random(): |
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253 | length = 10**np.random.uniform(1, 4.7, size=3) |
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254 | pars = dict( |
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255 | length_a=length[0], |
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256 | length_b=length[1], |
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257 | length_c=length[2], |
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258 | ) |
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259 | return pars |
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260 | |
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261 | |
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262 | # parameters for demo |
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263 | demo = dict(scale=1, background=0, |
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264 | sld=6.3, sld_solvent=1.0, |
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265 | length_a=35, length_b=75, length_c=400, |
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266 | theta=45, phi=30, psi=15, |
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267 | length_a_pd=0.1, length_a_pd_n=10, |
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268 | length_b_pd=0.1, length_b_pd_n=1, |
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269 | length_c_pd=0.1, length_c_pd_n=1, |
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270 | theta_pd=10, theta_pd_n=1, |
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271 | phi_pd=10, phi_pd_n=1, |
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272 | psi_pd=10, psi_pd_n=10) |
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273 | # rkh 7/4/17 add random unit test for 2d, note make all params different, |
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274 | # 2d values not tested against other codes or models |
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275 | qx, qy = 0.2 * cos(pi/6.), 0.2 * sin(pi/6.) |
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276 | tests = [[{}, 0.2, 0.17758004974], |
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277 | [{}, [0.2], [0.17758004974]], |
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278 | [{'theta':10.0, 'phi':20.0}, (qx, qy), 0.0089517140475], |
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279 | [{'theta':10.0, 'phi':20.0}, [(qx, qy)], [0.0089517140475]], |
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280 | ] |
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281 | del qx, qy # not necessary to delete, but cleaner |
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