1 | r""" |
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2 | Definition |
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3 | ---------- |
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4 | |
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5 | Calculates the form factor for a rectangular solid with a core-shell structure. |
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6 | The thickness and the scattering length density of the shell or "rim" can be |
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7 | different on each (pair) of faces. The three dimensions of the core of the |
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8 | parallelepiped (strictly here a cuboid) may be given in *any* size order as |
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9 | long as the particles are randomly oriented (i.e. take on all possible |
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10 | orientations see notes on 2D below). To avoid multiple fit solutions, |
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11 | especially with Monte-Carlo fit methods, it may be advisable to restrict their |
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12 | ranges. There may be a number of closely similar "best fits", so some trial and |
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13 | error, or fixing of some dimensions at expected values, may help. |
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14 | |
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15 | The form factor is normalized by the particle volume $V$ such that |
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16 | |
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17 | .. math:: |
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18 | |
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19 | I(q) = \frac{\text{scale}}{V} \langle P(q,\alpha,\beta) \rangle |
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20 | + \text{background} |
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21 | |
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22 | where $\langle \ldots \rangle$ is an average over all possible orientations |
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23 | of the rectangular solid, and the usual $\Delta \rho^2 \ V^2$ term cannot be |
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24 | pulled out of the form factor term due to the multiple slds in the model. |
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25 | |
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26 | The core of the solid is defined by the dimensions $A$, $B$, $C$ here shown |
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27 | such that $A < B < C$. |
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28 | |
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29 | .. figure:: img/parallelepiped_geometry.jpg |
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30 | |
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31 | Core of the core shell parallelepiped with the corresponding definition |
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32 | of sides. |
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33 | |
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34 | |
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35 | There are rectangular "slabs" of thickness $t_A$ that add to the $A$ dimension |
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36 | (on the $BC$ faces). There are similar slabs on the $AC$ $(=t_B)$ and $AB$ |
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37 | $(=t_C)$ faces. The projection in the $AB$ plane is |
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38 | |
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39 | .. figure:: img/core_shell_parallelepiped_projection.jpg |
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40 | |
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41 | AB cut through the core-shell parallelipiped showing the cross secion of |
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42 | four of the six shell slabs. As can be seen, this model leaves **"gaps"** |
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43 | at the corners of the solid. |
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44 | |
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45 | |
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46 | The total volume of the solid is thus given as |
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47 | |
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48 | .. math:: |
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49 | |
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50 | V = ABC + 2t_ABC + 2t_BAC + 2t_CAB |
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51 | |
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52 | The intensity calculated follows the :ref:`parallelepiped` model, with the |
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53 | core-shell intensity being calculated as the square of the sum of the |
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54 | amplitudes of the core and the slabs on the edges. The scattering amplitude is |
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55 | computed for a particular orientation of the core-shell parallelepiped with |
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56 | respect to the scattering vector and then averaged over all possible |
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57 | orientations, where $\alpha$ is the angle between the $z$ axis and the $C$ axis |
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58 | of the parallelepiped, and $\beta$ is the angle between the projection of the |
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59 | particle in the $xy$ detector plane and the $y$ axis. |
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60 | |
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61 | .. math:: |
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62 | |
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63 | P(q)=\frac {\int_{0}^{\pi/2}\int_{0}^{\pi/2}F^2(q,\alpha,\beta) \ sin\alpha |
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64 | \ d\alpha \ d\beta} {\int_{0}^{\pi/2} \ sin\alpha \ d\alpha \ d\beta} |
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65 | |
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66 | and |
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67 | |
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68 | .. math:: |
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69 | |
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70 | F(q,\alpha,\beta) |
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71 | &= (\rho_\text{core}-\rho_\text{solvent}) |
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72 | S(Q_A, A) S(Q_B, B) S(Q_C, C) \\ |
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73 | &+ (\rho_\text{A}-\rho_\text{solvent}) |
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74 | \left[S(Q_A, A+2t_A) - S(Q_A, A)\right] S(Q_B, B) S(Q_C, C) \\ |
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75 | &+ (\rho_\text{B}-\rho_\text{solvent}) |
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76 | S(Q_A, A) \left[S(Q_B, B+2t_B) - S(Q_B, B)\right] S(Q_C, C) \\ |
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77 | &+ (\rho_\text{C}-\rho_\text{solvent}) |
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78 | S(Q_A, A) S(Q_B, B) \left[S(Q_C, C+2t_C) - S(Q_C, C)\right] |
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79 | |
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80 | with |
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81 | |
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82 | .. math:: |
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83 | |
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84 | S(Q_X, L) = L \frac{\sin (\tfrac{1}{2} Q_X L)}{\tfrac{1}{2} Q_X L} |
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85 | |
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86 | and |
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87 | |
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88 | .. math:: |
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89 | |
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90 | Q_A &= q \sin\alpha \sin\beta \\ |
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91 | Q_B &= q \sin\alpha \cos\beta \\ |
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92 | Q_C &= q \cos\alpha |
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93 | |
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94 | |
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95 | where $\rho_\text{core}$, $\rho_\text{A}$, $\rho_\text{B}$ and $\rho_\text{C}$ |
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96 | are the scattering lengths of the parallelepiped core, and the rectangular |
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97 | slabs of thickness $t_A$, $t_B$ and $t_C$, respectively. $\rho_\text{solvent}$ |
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98 | is the scattering length of the solvent. |
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99 | |
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100 | .. note:: |
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101 | |
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102 | the code actually implements two substitutions: $d(cos\alpha)$ is |
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103 | substituted for -$sin\alpha \ d\alpha$ (note that in the |
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104 | :ref:`parallelepiped` code this is explicitly implemented with |
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105 | $\sigma = cos\alpha$), and $\beta$ is set to $\beta = u \pi/2$ so that |
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106 | $du = \pi/2 \ d\beta$. Thus both integrals go from 0 to 1 rather than 0 |
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107 | to $\pi/2$. |
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108 | |
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109 | FITTING NOTES |
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110 | ~~~~~~~~~~~~~ |
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111 | |
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112 | #. There are many parameters in this model. Hold as many fixed as possible with |
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113 | known values, or you will certainly end up at a solution that is unphysical. |
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114 | |
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115 | #. The 2nd virial coefficient of the core_shell_parallelepiped is calculated |
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116 | based on the the averaged effective radius $(=\sqrt{(A+2t_A)(B+2t_B)/\pi})$ |
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117 | and length $(C+2t_C)$ values, after appropriately sorting the three |
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118 | dimensions to give an oblate or prolate particle, to give an effective radius |
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119 | for $S(q)$ when $P(q) * S(q)$ is applied. |
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120 | |
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121 | #. For 2d data the orientation of the particle is required, described using |
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122 | angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, where $\theta$ |
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123 | and $\phi$ define the orientation of the director in the laboratry reference |
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124 | frame of the beam direction ($z$) and detector plane ($x-y$ plane), while |
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125 | the angle $\Psi$ is effectively the rotational angle around the particle |
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126 | $C$ axis. For $\theta = 0$ and $\phi = 0$, $\Psi = 0$ corresponds to the |
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127 | $B$ axis oriented parallel to the y-axis of the detector with $A$ along |
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128 | the x-axis. For other $\theta$, $\phi$ values, the order of rotations |
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129 | matters. In particular, the parallelepiped must first be rotated $\theta$ |
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130 | degrees in the $x-z$ plane before rotating $\phi$ degrees around the $z$ |
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131 | axis (in the $x-y$ plane). Applying orientational distribution to the |
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132 | particle orientation (i.e `jitter` to one or more of these angles) can get |
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133 | more confusing as `jitter` is defined **NOT** with respect to the laboratory |
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134 | frame but the particle reference frame. It is thus highly recmmended to |
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135 | read :ref:`orientation` for further details of the calculation and angular |
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136 | dispersions. |
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137 | |
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138 | .. note:: For 2d, constraints must be applied during fitting to ensure that the |
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139 | order of sides chosen is not altered, and hence that the correct definition |
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140 | of angles is preserved. For the default choice shown here, that means |
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141 | ensuring that the inequality $A < B < C$ is not violated, The calculation |
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142 | will not report an error, but the results may be not correct. |
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143 | |
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144 | .. figure:: img/parallelepiped_angle_definition.png |
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145 | |
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146 | Definition of the angles for oriented core-shell parallelepipeds. |
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147 | Note that rotation $\theta$, initially in the $x-z$ plane, is carried |
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148 | out first, then rotation $\phi$ about the $z$ axis, finally rotation |
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149 | $\Psi$ is now around the $C$ axis of the particle. The neutron or X-ray |
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150 | beam is along the $z$ axis and the detecotr defines the $x-y$ plane. |
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151 | |
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152 | .. figure:: img/parallelepiped_angle_projection.png |
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153 | |
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154 | Examples of the angles for oriented core-shell parallelepipeds against the |
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155 | detector plane. |
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156 | |
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157 | |
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158 | Validation |
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159 | ---------- |
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160 | |
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161 | Cross-checked against hollow rectangular prism and rectangular prism for equal |
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162 | thickness overlapping sides, and by Monte Carlo sampling of points within the |
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163 | shape for non-uniform, non-overlapping sides. |
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164 | |
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165 | |
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166 | References |
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167 | ---------- |
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168 | |
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169 | .. [#] P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211 |
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170 | Equations (1), (13-14). (in German) |
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171 | .. [#] D Singh (2009). *Small angle scattering studies of self assembly in |
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172 | lipid mixtures*, Johns Hopkins University Thesis (2009) 223-225. `Available |
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173 | from Proquest <http://search.proquest.com/docview/304915826?accountid |
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174 | =26379>`_ |
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175 | L. Onsager, Ann. New York Acad. Sci. 51, 627-659 (1949). |
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176 | |
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177 | Authorship and Verification |
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178 | ---------------------------- |
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179 | |
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180 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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181 | * **Converted to sasmodels by:** Miguel Gonzalez **Date:** February 26, 2016 |
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182 | * **Last Modified by:** Paul Kienzle **Date:** October 17, 2017 |
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183 | * **Last Reviewed by:** Paul Butler **Date:** May 24, 2018 - documentation |
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184 | updated |
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185 | """ |
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186 | |
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187 | import numpy as np |
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188 | from numpy import pi, inf, sqrt, cos, sin |
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189 | |
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190 | name = "core_shell_parallelepiped" |
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191 | title = "Rectangular solid with a core-shell structure." |
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192 | description = """ |
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193 | P(q)= |
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194 | """ |
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195 | category = "shape:parallelepiped" |
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196 | |
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197 | # ["name", "units", default, [lower, upper], "type","description"], |
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198 | parameters = [["sld_core", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
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199 | "Parallelepiped core scattering length density"], |
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200 | ["sld_a", "1e-6/Ang^2", 2, [-inf, inf], "sld", |
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201 | "Parallelepiped A rim scattering length density"], |
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202 | ["sld_b", "1e-6/Ang^2", 4, [-inf, inf], "sld", |
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203 | "Parallelepiped B rim scattering length density"], |
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204 | ["sld_c", "1e-6/Ang^2", 2, [-inf, inf], "sld", |
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205 | "Parallelepiped C rim scattering length density"], |
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206 | ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld", |
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207 | "Solvent scattering length density"], |
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208 | ["length_a", "Ang", 35, [0, inf], "volume", |
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209 | "Shorter side of the parallelepiped"], |
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210 | ["length_b", "Ang", 75, [0, inf], "volume", |
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211 | "Second side of the parallelepiped"], |
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212 | ["length_c", "Ang", 400, [0, inf], "volume", |
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213 | "Larger side of the parallelepiped"], |
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214 | ["thick_rim_a", "Ang", 10, [0, inf], "volume", |
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215 | "Thickness of A rim"], |
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216 | ["thick_rim_b", "Ang", 10, [0, inf], "volume", |
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217 | "Thickness of B rim"], |
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218 | ["thick_rim_c", "Ang", 10, [0, inf], "volume", |
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219 | "Thickness of C rim"], |
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220 | ["theta", "degrees", 0, [-360, 360], "orientation", |
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221 | "c axis to beam angle"], |
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222 | ["phi", "degrees", 0, [-360, 360], "orientation", |
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223 | "rotation about beam"], |
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224 | ["psi", "degrees", 0, [-360, 360], "orientation", |
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225 | "rotation about c axis"], |
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226 | ] |
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227 | |
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228 | source = ["lib/gauss76.c", "core_shell_parallelepiped.c"] |
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229 | have_Fq = True |
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230 | effective_radius_type = [ |
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231 | "equivalent cylinder excluded volume", |
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232 | "equivalent volume sphere", |
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233 | "half outer length_a", "half outer length_b", "half outer length_c", |
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234 | "equivalent circular cross-section", |
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235 | "half outer ab diagonal", "half outer diagonal", |
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236 | ] |
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237 | |
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238 | def random(): |
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239 | outer = 10**np.random.uniform(1, 4.7, size=3) |
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240 | thick = np.random.beta(0.5, 0.5, size=3)*(outer-2) + 1 |
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241 | length = outer - thick |
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242 | pars = dict( |
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243 | length_a=length[0], |
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244 | length_b=length[1], |
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245 | length_c=length[2], |
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246 | thick_rim_a=thick[0], |
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247 | thick_rim_b=thick[1], |
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248 | thick_rim_c=thick[2], |
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249 | ) |
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250 | return pars |
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251 | |
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252 | # parameters for demo |
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253 | demo = dict(scale=1, background=0.0, |
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254 | sld_core=1, sld_a=2, sld_b=4, sld_c=2, sld_solvent=6, |
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255 | length_a=35, length_b=75, length_c=400, |
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256 | thick_rim_a=10, thick_rim_b=10, thick_rim_c=10, |
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257 | theta=0, phi=0, psi=0, |
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258 | length_a_pd=0.1, length_a_pd_n=1, |
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259 | length_b_pd=0.1, length_b_pd_n=1, |
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260 | length_c_pd=0.1, length_c_pd_n=1, |
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261 | thick_rim_a_pd=0.1, thick_rim_a_pd_n=1, |
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262 | thick_rim_b_pd=0.1, thick_rim_b_pd_n=1, |
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263 | thick_rim_c_pd=0.1, thick_rim_c_pd_n=1, |
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264 | theta_pd=10, theta_pd_n=1, |
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265 | phi_pd=10, phi_pd_n=1, |
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266 | psi_pd=10, psi_pd_n=1) |
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267 | |
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268 | # rkh 7/4/17 add random unit test for 2d, note make all params different, |
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269 | # 2d values not tested against other codes or models |
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270 | if 0: # pak: model rewrite; need to update tests |
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271 | qx, qy = 0.2 * cos(pi/6.), 0.2 * sin(pi/6.) |
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272 | tests = [[{}, 0.2, 0.533149288477], |
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273 | [{}, [0.2], [0.533149288477]], |
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274 | [{'theta':10.0, 'phi':20.0}, (qx, qy), 0.0853299803222], |
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275 | [{'theta':10.0, 'phi':20.0}, [(qx, qy)], [0.0853299803222]], |
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276 | ] |
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277 | del qx, qy # not necessary to delete, but cleaner |
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