1 | r""" |
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2 | Definition |
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3 | ---------- |
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4 | |
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5 | The output of the 2D scattering intensity function for oriented core-shell |
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6 | cylinders is given by (Kline, 2006 [#kline]_). The form factor is normalized |
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7 | by the particle volume. Note that in this model the shell envelops the entire |
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8 | core so that besides a "sleeve" around the core, the shell also provides two |
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9 | flat end caps of thickness = shell thickness. In other words the length of the |
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10 | total cyclinder is the length of the core cylinder plus twice the thickness of |
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11 | the shell. If no end caps are desired one should use the |
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12 | :ref:`core-shell-bicelle` and set the thickness of the end caps (in this case |
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13 | the "thick_face") to zero. |
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14 | |
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15 | .. math:: |
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16 | |
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17 | I(q,\alpha) = \frac{\text{scale}}{V_s} F^2(q,\alpha).sin(\alpha) + \text{background} |
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18 | |
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19 | where |
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20 | |
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21 | .. math:: |
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22 | |
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23 | F(q,\alpha) = &\ (\rho_c - \rho_s) V_c |
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24 | \frac{\sin \left( q \tfrac12 L\cos\alpha \right)} |
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25 | {q \tfrac12 L\cos\alpha} |
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26 | \frac{2 J_1 \left( qR\sin\alpha \right)} |
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27 | {qR\sin\alpha} \\ |
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28 | &\ + (\rho_s - \rho_\text{solv}) V_s |
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29 | \frac{\sin \left( q \left(\tfrac12 L+T\right) \cos\alpha \right)} |
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30 | {q \left(\tfrac12 L +T \right) \cos\alpha} |
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31 | \frac{ 2 J_1 \left( q(R+T)\sin\alpha \right)} |
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32 | {q(R+T)\sin\alpha} |
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33 | |
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34 | and |
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35 | |
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36 | .. math:: |
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37 | |
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38 | V_s = \pi (R + T)^2 (L + 2T) |
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39 | |
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40 | and $\alpha$ is the angle between the axis of the cylinder and $\vec q$, |
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41 | $V_s$ is the total volume (i.e. including both the core and the outer shell), |
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42 | $V_c$ is the volume of the core, $L$ is the length of the core, |
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43 | $R$ is the radius of the core, $T$ is the thickness of the shell, $\rho_c$ |
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44 | is the scattering length density of the core, $\rho_s$ is the scattering |
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45 | length density of the shell, $\rho_\text{solv}$ is the scattering length |
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46 | density of the solvent, and *background* is the background level. The outer |
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47 | radius of the shell is given by $R+T$ and the total length of the outer |
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48 | shell is given by $L+2T$. $J1$ is the first order Bessel function. |
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49 | |
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50 | .. _core-shell-cylinder-geometry: |
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51 | |
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52 | .. figure:: img/core_shell_cylinder_geometry.jpg |
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53 | |
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54 | Core shell cylinder schematic. |
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55 | |
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56 | To provide easy access to the orientation of the core-shell cylinder, we |
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57 | define the axis of the cylinder using two angles $\theta$ and $\phi$. |
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58 | (see :ref:`cylinder model <cylinder-angle-definition>`) |
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59 | |
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60 | NB: The 2nd virial coefficient of the cylinder is calculated based on |
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61 | the radius and 2 length values, and used as the effective radius for |
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62 | $S(q)$ when $P(q) \cdot S(q)$ is applied. |
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63 | |
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64 | The $\theta$ and $\phi$ parameters are not used for the 1D output. |
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65 | |
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66 | Reference |
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67 | --------- |
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68 | |
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69 | .. [#] see, for example, Ian Livsey J. Chem. Soc., Faraday Trans. 2, 1987,83, |
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70 | 1445-1452 |
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71 | .. [#kline] S R Kline, *J Appl. Cryst.*, 39 (2006) 895 |
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72 | L. Onsager, Ann. New York Acad. Sci. 51, 627-659 (1949). |
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73 | |
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74 | Authorship and Verification |
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75 | ---------------------------- |
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76 | |
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77 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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78 | * **Last Modified by:** Paul Kienzle **Date:** Aug 8, 2016 |
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79 | * **Last Reviewed by:** Richard Heenan **Date:** March 18, 2016 |
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80 | """ |
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81 | |
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82 | import numpy as np |
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83 | from numpy import pi, inf, sin, cos |
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84 | |
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85 | name = "core_shell_cylinder" |
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86 | title = "Right circular cylinder with a core-shell scattering length density profile." |
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87 | description = """ |
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88 | P(q,alpha)= scale/Vs*f(q)^(2) + background, |
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89 | where: f(q)= 2(sld_core - solvant_sld) |
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90 | * Vc*sin[qLcos(alpha/2)] |
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91 | /[qLcos(alpha/2)]*J1(qRsin(alpha)) |
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92 | /[qRsin(alpha)]+2(sld_shell-sld_solvent) |
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93 | *Vs*sin[q(L+T)cos(alpha/2)][[q(L+T) |
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94 | *cos(alpha/2)]*J1(q(R+T)sin(alpha)) |
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95 | /q(R+T)sin(alpha)] |
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96 | |
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97 | alpha:is the angle between the axis of |
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98 | the cylinder and the q-vector |
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99 | Vs: the volume of the outer shell |
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100 | Vc: the volume of the core |
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101 | L: the length of the core |
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102 | sld_shell: the scattering length density of the shell |
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103 | sld_solvent: the scattering length density of the solvent |
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104 | background: the background |
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105 | T: the thickness |
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106 | R+T: is the outer radius |
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107 | L+2T: The total length of the outershell |
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108 | J1: the first order Bessel function |
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109 | theta: axis_theta of the cylinder |
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110 | phi: the axis_phi of the cylinder |
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111 | """ |
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112 | category = "shape:cylinder" |
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113 | |
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114 | # ["name", "units", default, [lower, upper], "type", "description"], |
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115 | parameters = [["sld_core", "1e-6/Ang^2", 4, [-inf, inf], "sld", |
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116 | "Cylinder core scattering length density"], |
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117 | ["sld_shell", "1e-6/Ang^2", 4, [-inf, inf], "sld", |
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118 | "Cylinder shell scattering length density"], |
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119 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
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120 | "Solvent scattering length density"], |
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121 | ["radius", "Ang", 20, [0, inf], "volume", |
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122 | "Cylinder core radius"], |
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123 | ["thickness", "Ang", 20, [0, inf], "volume", |
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124 | "Cylinder shell thickness"], |
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125 | ["length", "Ang", 400, [0, inf], "volume", |
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126 | "Cylinder length"], |
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127 | ["theta", "degrees", 60, [-360, 360], "orientation", |
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128 | "cylinder axis to beam angle"], |
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129 | ["phi", "degrees", 60, [-360, 360], "orientation", |
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130 | "rotation about beam"], |
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131 | ] |
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132 | |
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133 | source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "core_shell_cylinder.c"] |
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134 | have_Fq = True |
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135 | effective_radius_type = [ |
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136 | "excluded volume", "equivalent volume sphere", "outer radius", "half outer length", |
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137 | "half min outer dimension", "half max outer dimension", "half outer diagonal", |
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138 | ] |
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139 | |
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140 | def random(): |
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141 | """Return a random parameter set for the model.""" |
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142 | outer_radius = 10**np.random.uniform(1, 4.7) |
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143 | # Use a distribution with a preference for thin shell or thin core |
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144 | # Avoid core,shell radii < 1 |
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145 | radius = np.random.beta(0.5, 0.5)*(outer_radius-2) + 1 |
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146 | thickness = outer_radius - radius |
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147 | length = np.random.uniform(1, 4.7) |
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148 | pars = dict( |
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149 | radius=radius, |
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150 | thickness=thickness, |
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151 | length=length, |
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152 | ) |
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153 | return pars |
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154 | |
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155 | demo = dict(scale=1, background=0, |
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156 | sld_core=6, sld_shell=8, sld_solvent=1, |
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157 | radius=45, thickness=25, length=340, |
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158 | theta=30, phi=15, |
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159 | radius_pd=.2, radius_pd_n=1, |
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160 | length_pd=.2, length_pd_n=10, |
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161 | thickness_pd=.2, thickness_pd_n=10, |
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162 | theta_pd=15, theta_pd_n=45, |
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163 | phi_pd=15, phi_pd_n=1) |
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164 | q = 0.1 |
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165 | # april 6 2017, rkh add unit tests, NOT compared with any other calc method, assume correct! |
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166 | qx = q*cos(pi/6.0) |
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167 | qy = q*sin(pi/6.0) |
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168 | tests = [ |
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169 | [{}, 0.075, 10.8552692237], |
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170 | [{}, (qx, qy), 0.444618752741], |
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171 | ] |
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172 | del qx, qy # not necessary to delete, but cleaner |
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