1 | # cylinder 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 | For information about polarised and magnetic scattering, see |
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6 | the :ref:`magnetism` documentation. |
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7 | |
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8 | Definition |
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9 | ---------- |
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10 | |
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11 | The output of the 2D scattering intensity function for oriented cylinders is |
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12 | given by (Guinier, 1955) |
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13 | |
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14 | .. math:: |
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15 | |
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16 | P(q,\alpha) = \frac{\text{scale}}{V} F^2(q,\alpha).sin(\alpha) + \text{background} |
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17 | |
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18 | where |
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19 | |
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20 | .. math:: |
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21 | |
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22 | F(q,\alpha) = 2 (\Delta \rho) V |
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23 | \frac{\sin \left(\tfrac12 qL\cos\alpha \right)} |
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24 | {\tfrac12 qL \cos \alpha} |
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25 | \frac{J_1 \left(q R \sin \alpha\right)}{q R \sin \alpha} |
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26 | |
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27 | and $\alpha$ is the angle between the axis of the cylinder and $\vec q$, $V =\pi R^2L$ |
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28 | is the volume of the cylinder, $L$ is the length of the cylinder, $R$ is the |
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29 | radius of the cylinder, and $\Delta\rho$ (contrast) is the scattering length |
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30 | density difference between the scatterer and the solvent. $J_1$ is the |
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31 | first order Bessel function. |
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32 | |
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33 | For randomly oriented particles: |
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34 | |
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35 | .. math:: |
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36 | |
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37 | F^2(q)=\int_{0}^{\pi/2}{F^2(q,\alpha)\sin(\alpha)d\alpha}=\int_{0}^{1}{F^2(q,u)du} |
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38 | |
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39 | |
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40 | Numerical integration is simplified by a change of variable to $u = cos(\alpha)$ with |
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41 | $sin(\alpha)=\sqrt{1-u^2}$. |
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42 | |
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43 | The output of the 1D scattering intensity function for randomly oriented |
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44 | cylinders is thus given by |
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45 | |
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46 | .. math:: |
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47 | |
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48 | P(q) = \frac{\text{scale}}{V} |
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49 | \int_0^{\pi/2} F^2(q,\alpha) \sin \alpha\ d\alpha + \text{background} |
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50 | |
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51 | |
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52 | NB: The 2nd virial coefficient of the cylinder is calculated based on the |
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53 | radius and length values, and used as the effective radius for $S(q)$ |
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54 | when $P(q) \cdot S(q)$ is applied. |
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55 | |
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56 | For oriented cylinders, we define the direction of the |
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57 | axis of the cylinder using two angles $\theta$ (note this is not the |
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58 | same as the scattering angle used in q) and $\phi$. Those angles |
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59 | are defined in :numref:`cylinder-angle-definition` . |
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60 | |
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61 | .. _cylinder-angle-definition: |
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62 | |
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63 | .. figure:: img/cylinder_angle_definition.png |
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64 | |
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65 | Definition of the $\theta$ and $\phi$ orientation angles for a cylinder relative |
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66 | to the beam line coordinates, plus an indication of their orientation distributions |
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67 | which are described as rotations about each of the perpendicular axes $\delta_1$ and $\delta_2$ |
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68 | in the frame of the cylinder itself, which when $\theta = \phi = 0$ are parallel to the $Y$ and $X$ axes. |
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69 | |
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70 | .. figure:: img/cylinder_angle_projection.png |
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71 | |
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72 | Examples for oriented cylinders. |
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73 | |
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74 | The $\theta$ and $\phi$ parameters to orient the cylinder only appear in the model when fitting 2d data. |
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75 | On introducing "Orientational Distribution" in the angles, "distribution of theta" and "distribution of phi" parameters will |
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76 | appear. These are actually rotations about the axes $\delta_1$ and $\delta_2$ of the cylinder, which when $\theta = \phi = 0$ are parallel |
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77 | to the $Y$ and $X$ axes of the instrument respectively. Some experimentation may be required to understand the 2d patterns fully. |
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78 | (Earlier implementations had numerical integration issues in some circumstances when orientation distributions passed through 90 degrees, such |
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79 | situations, with very broad distributions, should still be approached with care.) |
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80 | |
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81 | Validation |
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82 | ---------- |
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83 | |
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84 | Validation of the code was done by comparing the output of the 1D model |
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85 | to the output of the software provided by the NIST (Kline, 2006). |
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86 | The implementation of the intensity for fully oriented cylinders was done |
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87 | by averaging over a uniform distribution of orientations using |
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88 | |
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89 | .. math:: |
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90 | |
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91 | P(q) = \int_0^{\pi/2} d\phi |
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92 | \int_0^\pi p(\theta) P_0(q,\theta) \sin \theta\ d\theta |
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93 | |
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94 | |
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95 | where $p(\theta,\phi) = 1$ is the probability distribution for the orientation |
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96 | and $P_0(q,\theta)$ is the scattering intensity for the fully oriented |
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97 | system, and then comparing to the 1D result. |
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98 | |
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99 | References |
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100 | ---------- |
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101 | |
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102 | J. S. Pedersen, Adv. Colloid Interface Sci. 70, 171-210 (1997). |
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103 | G. Fournet, Bull. Soc. Fr. Mineral. Cristallogr. 74, 39-113 (1951). |
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104 | """ |
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105 | |
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106 | import numpy as np # type: ignore |
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107 | from numpy import pi, inf # type: ignore |
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108 | |
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109 | name = "cylinder" |
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110 | title = "Right circular cylinder with uniform scattering length density." |
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111 | description = """ |
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112 | f(q,alpha) = 2*(sld - sld_solvent)*V*sin(qLcos(alpha)/2)) |
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113 | /[qLcos(alpha)/2]*J1(qRsin(alpha))/[qRsin(alpha)] |
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114 | |
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115 | P(q,alpha)= scale/V*f(q,alpha)^(2)+background |
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116 | V: Volume of the cylinder |
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117 | R: Radius of the cylinder |
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118 | L: Length of the cylinder |
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119 | J1: The bessel function |
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120 | alpha: angle between the axis of the |
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121 | cylinder and the q-vector for 1D |
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122 | :the ouput is P(q)=scale/V*integral |
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123 | from pi/2 to zero of... |
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124 | f(q,alpha)^(2)*sin(alpha)*dalpha + background |
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125 | """ |
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126 | category = "shape:cylinder" |
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127 | |
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128 | # [ "name", "units", default, [lower, upper], "type", "description"], |
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129 | parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "sld", |
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130 | "Cylinder scattering length density"], |
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131 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
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132 | "Solvent scattering length density"], |
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133 | ["radius", "Ang", 20, [0, inf], "volume", |
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134 | "Cylinder radius"], |
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135 | ["length", "Ang", 400, [0, inf], "volume", |
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136 | "Cylinder length"], |
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137 | ["theta", "degrees", 60, [-360, 360], "orientation", |
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138 | "cylinder axis to beam angle"], |
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139 | ["phi", "degrees", 60, [-360, 360], "orientation", |
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140 | "rotation about beam"], |
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141 | ] |
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142 | |
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143 | source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "cylinder.c"] |
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144 | |
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145 | def ER(radius, length): |
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146 | """ |
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147 | Return equivalent radius (ER) |
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148 | """ |
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149 | ddd = 0.75 * radius * (2 * radius * length + (length + radius) * (length + pi * radius)) |
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150 | return 0.5 * (ddd) ** (1. / 3.) |
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151 | |
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152 | def random(): |
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153 | import numpy as np |
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154 | V = 10**np.random.uniform(5, 12) |
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155 | length = 10**np.random.uniform(-2, 2)*V**0.333 |
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156 | radius = np.sqrt(V/length/np.pi) |
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157 | pars = dict( |
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158 | #scale=1, |
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159 | #background=0, |
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160 | length=length, |
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161 | radius=radius, |
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162 | ) |
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163 | return pars |
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164 | |
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165 | |
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166 | # parameters for demo |
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167 | demo = dict(scale=1, background=0, |
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168 | sld=6, sld_solvent=1, |
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169 | radius=20, length=300, |
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170 | theta=60, phi=60, |
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171 | radius_pd=.2, radius_pd_n=9, |
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172 | length_pd=.2, length_pd_n=10, |
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173 | theta_pd=10, theta_pd_n=5, |
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174 | phi_pd=10, phi_pd_n=5) |
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175 | |
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176 | qx, qy = 0.2 * np.cos(2.5), 0.2 * np.sin(2.5) |
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177 | # After redefinition of angles, find new tests values. Was 10 10 in old coords |
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178 | tests = [[{}, 0.2, 0.042761386790780453], |
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179 | [{}, [0.2], [0.042761386790780453]], |
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180 | # new coords |
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181 | [{'theta':80.1534480601659, 'phi':10.1510817110481}, (qx, qy), 0.03514647218513852], |
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182 | [{'theta':80.1534480601659, 'phi':10.1510817110481}, [(qx, qy)], [0.03514647218513852]], |
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183 | # old coords [{'theta':10.0, 'phi':10.0}, (qx, qy), 0.03514647218513852], |
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184 | # [{'theta':10.0, 'phi':10.0}, [(qx, qy)], [0.03514647218513852]], |
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185 | ] |
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186 | del qx, qy # not necessary to delete, but cleaner |
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187 | # ADDED by: RKH ON: 18Mar2016 renamed sld's etc |
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