1 | # pylint: disable=line-too-long |
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2 | r""" |
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3 | |
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4 | .. figure:: img/elliptical_cylinder_geometry.png |
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5 | |
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6 | Elliptical cylinder geometry $a = r_\text{minor}$ |
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7 | and $\nu = r_\text{major} / r_\text{minor}$ is the *axis_ratio*. |
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8 | |
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9 | The function calculated is |
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10 | |
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11 | .. math:: |
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12 | |
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13 | I(\vec q)=\frac{1}{V_\text{cyl}}\int{d\psi}\int{d\phi}\int{ |
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14 | p(\theta,\phi,\psi)F^2(\vec q,\alpha,\psi)\sin(\alpha)d\alpha} |
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15 | |
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16 | with the functions |
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17 | |
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18 | .. math:: |
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19 | |
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20 | F(q,\alpha,\psi) = 2\frac{J_1(a)\sin(b)}{ab} |
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21 | |
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22 | where |
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23 | |
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24 | .. math:: |
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25 | |
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26 | a = qr'\sin(\alpha) |
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27 | |
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28 | b = q\frac{L}{2}\cos(\alpha) |
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29 | |
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30 | r'=\frac{r_{minor}}{\sqrt{2}}\sqrt{(1+\nu^{2}) + (1-\nu^{2})cos(\psi)} |
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31 | |
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32 | |
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33 | and the angle $\psi$ is defined as the orientation of the major axis of the |
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34 | ellipse with respect to the vector $\vec q$. The angle $\alpha$ is the angle |
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35 | between the axis of the cylinder and $\vec q$. |
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36 | |
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37 | |
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38 | For 1D scattering, with no preferred orientation, the form factor is averaged over all possible orientations and normalized |
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39 | by the particle volume |
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40 | |
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41 | .. math:: |
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42 | |
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43 | P(q) = \text{scale} <F^2> / V |
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44 | |
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45 | For 2d data the orientation of the particle is required, described using a different set |
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46 | of angles as in the diagrams below, for further details of the calculation and angular |
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47 | dispersions see :ref:`orientation` . |
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48 | |
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49 | |
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50 | .. figure:: img/elliptical_cylinder_angle_definition.png |
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51 | |
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52 | Note that the angles here are not the same as in the equations for the scattering function. |
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53 | Rotation $\theta$, initially in the $xz$ plane, is carried out first, then |
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54 | rotation $\phi$ about the $z$ axis, finally rotation $\Psi$ is now around the axis of the cylinder. |
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55 | The neutron or X-ray beam is along the $z$ axis. |
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56 | |
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57 | .. figure:: img/elliptical_cylinder_angle_projection.png |
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58 | |
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59 | Examples of the angles for oriented elliptical cylinders against the |
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60 | detector plane, with $\Psi$ = 0. |
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61 | |
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62 | The $\theta$ and $\phi$ parameters to orient the cylinder only appear in the model when fitting 2d data. |
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63 | |
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64 | |
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65 | NB: The 2nd virial coefficient of the cylinder is calculated based on the |
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66 | averaged radius $(=\sqrt{r_\text{minor}^2 * \text{axis ratio}})$ and length |
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67 | values, and used as the effective radius for $S(Q)$ when $P(Q)*S(Q)$ is applied. |
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68 | |
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69 | |
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70 | Validation |
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71 | ---------- |
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72 | |
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73 | Validation of our code was done by comparing the output of the 1D calculation |
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74 | to the angular average of the output of the 2D calculation over all possible |
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75 | angles. |
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76 | |
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77 | In the 2D average, more binning in the angle $\phi$ is necessary to get the |
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78 | proper result. The following figure shows the results of the averaging by |
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79 | varying the number of angular bins. |
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80 | |
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81 | .. figure:: img/elliptical_cylinder_averaging.png |
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82 | |
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83 | The intensities averaged from 2D over different numbers of bins and angles. |
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84 | |
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85 | References |
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86 | ---------- |
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87 | |
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88 | L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and |
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89 | Neutron Scattering*, Plenum, New York, (1987) [see table 3.4] |
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90 | L. Onsager, Ann. New York Acad. Sci. 51, 627-659 (1949). |
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91 | |
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92 | Authorship and Verification |
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93 | ---------------------------- |
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94 | |
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95 | * **Author:** |
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96 | * **Last Modified by:** |
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97 | * **Last Reviewed by:** Richard Heenan - corrected equation in docs **Date:** December 21, 2016 |
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98 | """ |
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99 | |
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100 | import numpy as np |
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101 | from numpy import pi, inf, sqrt, sin, cos |
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102 | |
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103 | name = "elliptical_cylinder" |
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104 | title = "Form factor for an elliptical cylinder." |
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105 | description = """ |
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106 | Form factor for an elliptical cylinder. |
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107 | See L A Feigin and D I Svergun, Structure Analysis by Small-Angle X-Ray and Neutron Scattering, Plenum, New York, (1987). |
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108 | """ |
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109 | category = "shape:cylinder" |
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110 | |
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111 | # pylint: disable=bad-whitespace, line-too-long |
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112 | # ["name", "units", default, [lower, upper], "type","description"], |
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113 | parameters = [["radius_minor", "Ang", 20.0, [0, inf], "volume", "Ellipse minor radius"], |
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114 | ["axis_ratio", "", 1.5, [1, inf], "volume", "Ratio of major radius over minor radius"], |
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115 | ["length", "Ang", 400.0, [1, inf], "volume", "Length of the cylinder"], |
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116 | ["sld", "1e-6/Ang^2", 4.0, [-inf, inf], "sld", "Cylinder scattering length density"], |
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117 | ["sld_solvent", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", "Solvent scattering length density"], |
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118 | ["theta", "degrees", 90.0, [-360, 360], "orientation", "cylinder axis to beam angle"], |
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119 | ["phi", "degrees", 0, [-360, 360], "orientation", "rotation about beam"], |
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120 | ["psi", "degrees", 0, [-360, 360], "orientation", "rotation about cylinder axis"]] |
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121 | |
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122 | # pylint: enable=bad-whitespace, line-too-long |
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123 | |
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124 | source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "elliptical_cylinder.c"] |
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125 | have_Fq = True |
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126 | effective_radius_type = [ |
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127 | "equivalent cylinder excluded volume", "equivalent volume sphere", "average radius", "min radius", "max radius", |
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128 | "equivalent circular cross-section", |
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129 | "half length", "half min dimension", "half max dimension", "half diagonal", |
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130 | ] |
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131 | |
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132 | demo = dict(scale=1, background=0, radius_minor=100, axis_ratio=1.5, length=400.0, |
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133 | sld=4.0, sld_solvent=1.0, theta=10.0, phi=20, psi=30, |
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134 | theta_pd=10, phi_pd=2, psi_pd=3) |
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135 | |
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136 | def random(): |
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137 | # V = pi * radius_major * radius_minor * length; |
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138 | volume = 10**np.random.uniform(3, 9) |
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139 | length = 10**np.random.uniform(1, 3) |
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140 | axis_ratio = 10**np.random.uniform(0, 2) |
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141 | radius_minor = np.sqrt(volume/length/axis_ratio) |
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142 | volfrac = 10**np.random.uniform(-4, -2) |
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143 | pars = dict( |
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144 | #background=0, sld=0, sld_solvent=1, |
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145 | scale=1e9*volfrac/volume, |
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146 | length=length, |
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147 | radius_minor=radius_minor, |
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148 | axis_ratio=axis_ratio, |
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149 | ) |
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150 | return pars |
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151 | |
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152 | q = 0.1 |
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153 | # april 6 2017, rkh added a 2d unit test, NOT READY YET pull #890 branch assume correct! |
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154 | qx = q*cos(pi/6.0) |
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155 | qy = q*sin(pi/6.0) |
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156 | |
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157 | tests = [ |
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158 | # [{'radius_minor': 20.0, 'axis_ratio': 1.5, 'length':400.0}, 'ER', 79.89245454155024], |
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159 | # [{'radius_minor': 20.0, 'axis_ratio': 1.2, 'length':300.0}, 'VR', 1], |
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160 | |
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161 | # The SasView test result was 0.00169, with a background of 0.001 |
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162 | [{'radius_minor': 20.0, 'axis_ratio': 1.5, 'sld': 4.0, 'length':400.0, |
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163 | 'sld_solvent':1.0, 'background':0.0}, |
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164 | 0.001, 675.504402], |
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165 | #[{'theta':80., 'phi':10.}, (qx, qy), 7.88866563001 ], |
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166 | ] |
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