1 | r""" |
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2 | Definitions |
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3 | ----------- |
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4 | |
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5 | Calculates the scattering from a cylinder with spherical section end-caps. |
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6 | Like :ref:`barbell`, this is a sphereocylinder with end caps that have a |
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7 | radius larger than that of the cylinder, but with the center of the end cap |
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8 | radius lying within the cylinder. This model simply becomes a convex |
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9 | lens when the length of the cylinder $L=0$. See the diagram for the details |
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10 | of the geometry and restrictions on parameter values. |
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11 | |
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12 | .. figure:: img/capped_cylinder_geometry.jpg |
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13 | |
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14 | Capped cylinder geometry, where $r$ is *radius*, $R$ is *bell_radius* and |
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15 | $L$ is *length*. Since the end cap radius $R \geq r$ and by definition |
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16 | for this geometry $h < 0$, $h$ is then defined by $r$ and $R$ as |
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17 | $h = - \sqrt{R^2 - r^2}$ |
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18 | |
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19 | The scattered intensity $I(q)$ is calculated as |
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20 | |
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21 | .. math:: |
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22 | |
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23 | I(q) = \frac{\Delta \rho^2}{V} \left<A^2(q,\alpha).sin(\alpha)\right> |
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24 | |
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25 | where the amplitude $A(q,\alpha)$ with the rod axis at angle $\alpha$ to $q$ is given as |
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26 | |
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27 | .. math:: |
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28 | |
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29 | A(q) =&\ \pi r^2L |
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30 | \frac{\sin\left(\tfrac12 qL\cos\alpha\right)} |
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31 | {\tfrac12 qL\cos\alpha} |
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32 | \frac{2 J_1(qr\sin\alpha)}{qr\sin\alpha} \\ |
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33 | &\ + 4 \pi R^3 \int_{-h/R}^1 dt |
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34 | \cos\left[ q\cos\alpha |
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35 | \left(Rt + h + {\tfrac12} L\right)\right] |
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36 | \times (1-t^2) |
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37 | \frac{J_1\left[qR\sin\alpha \left(1-t^2\right)^{1/2}\right]} |
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38 | {qR\sin\alpha \left(1-t^2\right)^{1/2}} |
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39 | |
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40 | The $\left<\ldots\right>$ brackets denote an average of the structure over |
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41 | all orientations. $\left< A^2(q)\right>$ is then the form factor, $P(q)$. |
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42 | The scale factor is equivalent to the volume fraction of cylinders, each of |
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43 | volume, $V$. Contrast $\Delta\rho$ is the difference of scattering length |
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44 | densities of the cylinder and the surrounding solvent. |
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45 | |
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46 | The volume of the capped cylinder is (with $h$ as a positive value here) |
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47 | |
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48 | .. math:: |
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49 | |
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50 | V = \pi r_c^2 L + \tfrac{2\pi}{3}(R-h)^2(2R + h) |
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51 | |
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52 | |
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53 | and its radius of gyration is |
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54 | |
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55 | .. math:: |
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56 | |
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57 | R_g^2 =&\ \left[ \tfrac{12}{5}R^5 |
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58 | + R^4\left(6h+\tfrac32 L\right) |
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59 | + R^2\left(4h^2 + L^2 + 4Lh\right) |
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60 | + R^2\left(3Lh^2 + \tfrac32 L^2h\right) \right. \\ |
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61 | &\ \left. + \tfrac25 h^5 - \tfrac12 Lh^4 - \tfrac12 L^2h^3 |
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62 | + \tfrac14 L^3r^2 + \tfrac32 Lr^4 \right] |
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63 | \left( 4R^3 6R^2h - 2h^3 + 3r^2L \right)^{-1} |
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64 | |
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65 | |
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66 | .. note:: |
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67 | |
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68 | The requirement that $R \geq r$ is not enforced in the model! |
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69 | It is up to you to restrict this during analysis. |
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70 | |
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71 | The 2D scattering intensity is calculated similar to the 2D cylinder model. |
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72 | |
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73 | .. figure:: img/cylinder_angle_definition.png |
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74 | |
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75 | Definition of the angles for oriented 2D cylinders. |
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76 | |
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77 | |
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78 | References |
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79 | ---------- |
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80 | |
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81 | .. [#] H Kaya, *J. Appl. Cryst.*, 37 (2004) 223-230 |
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82 | .. [#] H Kaya and N-R deSouza, *J. Appl. Cryst.*, 37 (2004) 508-509 (addenda |
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83 | and errata) |
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84 | L. Onsager, Ann. New York Acad. Sci. 51, 627-659 (1949). |
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85 | |
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86 | Authorship and Verification |
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87 | ---------------------------- |
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88 | |
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89 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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90 | * **Last Modified by:** Paul Butler **Date:** September 30, 2016 |
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91 | * **Last Reviewed by:** Richard Heenan **Date:** January 4, 2017 |
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92 | """ |
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93 | |
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94 | import numpy as np |
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95 | from numpy import inf, sin, cos, pi |
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96 | |
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97 | name = "capped_cylinder" |
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98 | title = "Right circular cylinder with spherical end caps and uniform SLD" |
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99 | description = """That is, a sphereocylinder |
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100 | with end caps that have a radius larger than |
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101 | that of the cylinder and the center of the |
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102 | end cap radius lies within the cylinder. |
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103 | Note: As the length of cylinder -->0, |
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104 | it becomes a Convex Lens. |
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105 | It must be that radius <(=) radius_cap. |
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106 | [Parameters]; |
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107 | scale: volume fraction of spheres, |
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108 | background:incoherent background, |
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109 | radius: radius of the cylinder, |
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110 | length: length of the cylinder, |
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111 | radius_cap: radius of the semi-spherical cap, |
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112 | sld: SLD of the capped cylinder, |
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113 | sld_solvent: SLD of the solvent. |
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114 | """ |
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115 | category = "shape:cylinder" |
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116 | # pylint: disable=bad-whitespace, line-too-long |
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117 | # ["name", "units", default, [lower, upper], "type", "description"], |
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118 | parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "sld", "Cylinder scattering length density"], |
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119 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Solvent scattering length density"], |
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120 | ["radius", "Ang", 20, [0, inf], "volume", "Cylinder radius"], |
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121 | |
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122 | # TODO: use an expression for cap radius with fixed bounds. |
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123 | # The current form requires cap radius R bigger than cylinder radius r. |
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124 | # Could instead use R/r in [1,inf], r/R in [0,1], or the angle between |
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125 | # cylinder and cap in [0,90]. The problem is similar for the barbell |
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126 | # model. Propose r/R in [0,1] in both cases, with the model specifying |
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127 | # cylinder radius in the capped cylinder model and sphere radius in the |
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128 | # barbell model. This leads to the natural value of zero for no cap |
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129 | # in the capped cylinder, and zero for no bar in the barbell model. In |
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130 | # both models, one would be a pill. |
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131 | ["radius_cap", "Ang", 20, [0, inf], "volume", "Cap radius"], |
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132 | ["length", "Ang", 400, [0, inf], "volume", "Cylinder length"], |
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133 | ["theta", "degrees", 60, [-360, 360], "orientation", "cylinder axis to beam angle"], |
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134 | ["phi", "degrees", 60, [-360, 360], "orientation", "rotation about beam"], |
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135 | ] |
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136 | # pylint: enable=bad-whitespace, line-too-long |
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137 | |
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138 | source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "capped_cylinder.c"] |
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139 | have_Fq = True |
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140 | effective_radius_type = [ |
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141 | "equivalent cylinder excluded volume", "equivalent volume sphere", |
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142 | "radius", "half length", "half total length", |
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143 | ] |
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144 | |
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145 | def random(): |
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146 | """Return a random parameter set for the model.""" |
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147 | # TODO: increase volume range once problem with bell radius is fixed |
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148 | # The issue is that bell radii of more than about 200 fail at high q |
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149 | volume = 10**np.random.uniform(7, 9) |
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150 | bar_volume = 10**np.random.uniform(-4, -1)*volume |
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151 | bell_volume = volume - bar_volume |
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152 | bell_radius = (bell_volume/6)**0.3333 # approximate |
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153 | min_bar = bar_volume/np.pi/bell_radius**2 |
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154 | bar_length = 10**np.random.uniform(0, 3)*min_bar |
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155 | bar_radius = np.sqrt(bar_volume/bar_length/np.pi) |
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156 | if bar_radius > bell_radius: |
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157 | bell_radius, bar_radius = bar_radius, bell_radius |
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158 | pars = dict( |
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159 | #background=0, |
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160 | radius_cap=bell_radius, |
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161 | radius=bar_radius, |
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162 | length=bar_length, |
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163 | ) |
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164 | return pars |
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165 | |
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166 | |
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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=260, radius_cap=290, length=290, |
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170 | theta=30, phi=15, |
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171 | radius_pd=.2, radius_pd_n=1, |
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172 | radius_cap_pd=.2, radius_cap_pd_n=1, |
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173 | length_pd=.2, length_pd_n=10, |
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174 | theta_pd=15, theta_pd_n=45, |
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175 | phi_pd=15, phi_pd_n=1) |
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176 | q = 0.1 |
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177 | # april 6 2017, rkh add unit tests, NOT compared with any other calc method, assume correct! |
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178 | qx = q*cos(pi/6.0) |
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179 | qy = q*sin(pi/6.0) |
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180 | tests = [ |
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181 | [{}, 0.075, 26.0698570695], |
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182 | [{'theta':80., 'phi':10.}, (qx, qy), 0.561811990502], |
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183 | ] |
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184 | del qx, qy # not necessary to delete, but cleaner |
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