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 scattering from a barbell-shaped cylinder. Like |
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6 | :ref:`capped-cylinder`, this is a sphereocylinder with spherical end |
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7 | caps that have a radius larger than that of the cylinder, but with the center |
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8 | of the end cap radius lying outside of the cylinder. See the diagram for |
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9 | the details of the geometry and restrictions on parameter values. |
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10 | |
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11 | .. figure:: img/barbell_geometry.jpg |
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12 | |
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13 | Barbell geometry, where $r$ is *radius*, $R$ is *radius_bell* and |
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14 | $L$ is *length*. Since the end cap radius $R \geq r$ and by definition |
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15 | for this geometry $h < 0$, $h$ is then defined by $r$ and $R$ as |
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16 | $h = - \sqrt{R^2 - r^2}$ |
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17 | |
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18 | The scattered intensity $I(q)$ is calculated as |
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19 | |
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20 | .. math:: |
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21 | |
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22 | I(q) = \frac{\Delta \rho^2}{V} \left<A^2(q,\alpha).sin(\alpha)\right> |
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23 | |
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24 | where the amplitude $A(q,\alpha)$ with the rod axis at angle $\alpha$ to $q$ is given as |
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25 | |
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26 | .. math:: |
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27 | |
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28 | A(q) =&\ \pi r^2L |
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29 | \frac{\sin\left(\tfrac12 qL\cos\alpha\right)} |
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30 | {\tfrac12 qL\cos\alpha} |
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31 | \frac{2 J_1(qr\sin\alpha)}{qr\sin\alpha} \\ |
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32 | &\ + 4 \pi R^3 \int_{-h/R}^1 dt |
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33 | \cos\left[ q\cos\alpha |
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34 | \left(Rt + h + {\tfrac12} L\right)\right] |
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35 | \times (1-t^2) |
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36 | \frac{J_1\left[qR\sin\alpha \left(1-t^2\right)^{1/2}\right]} |
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37 | {qR\sin\alpha \left(1-t^2\right)^{1/2}} |
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38 | |
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39 | The $\left<\ldots\right>$ brackets denote an average of the structure over |
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40 | all orientations. $\left<A^2(q,\alpha)\right>$ is then the form factor, $P(q)$. |
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41 | The scale factor is equivalent to the volume fraction of cylinders, each of |
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42 | volume, $V$. Contrast $\Delta\rho$ is the difference of scattering length |
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43 | densities of the cylinder and the surrounding solvent. |
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44 | |
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45 | The volume of the barbell is |
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46 | |
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47 | .. math:: |
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48 | |
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49 | V = \pi r_c^2 L + 2\pi\left(\tfrac23R^3 + R^2h-\tfrac13h^3\right) |
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50 | |
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51 | |
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52 | and its radius of gyration is |
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53 | |
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54 | .. math:: |
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55 | |
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56 | R_g^2 =&\ \left[ \tfrac{12}{5}R^5 |
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57 | + R^4\left(6h+\tfrac32 L\right) |
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58 | + R^2\left(4h^2 + L^2 + 4Lh\right) |
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59 | + R^2\left(3Lh^2 + \tfrac32 L^2h\right) \right. \\ |
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60 | &\ \left. + \tfrac25 h^5 - \tfrac12 Lh^4 - \tfrac12 L^2h^3 |
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61 | + \tfrac14 L^3r^2 + \tfrac32 Lr^4 \right] |
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62 | \left( 4R^3 6R^2h - 2h^3 + 3r^2L \right)^{-1} |
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63 | |
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64 | .. note:: |
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65 | The requirement that $R \geq r$ is not enforced in the model! It is |
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66 | up to you to restrict this during analysis. |
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67 | |
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68 | The 2D scattering intensity is calculated similar to the 2D cylinder model. |
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69 | |
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70 | .. figure:: img/cylinder_angle_definition.png |
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71 | |
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72 | Definition of the angles for oriented 2D barbells. |
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73 | |
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74 | |
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75 | References |
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76 | ---------- |
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77 | |
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78 | .. [#] H Kaya, *J. Appl. Cryst.*, 37 (2004) 37 223-230 |
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79 | .. [#] H Kaya and N R deSouza, *J. Appl. Cryst.*, 37 (2004) 508-509 (addenda |
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80 | and errata) |
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81 | L. Onsager, Ann. New York Acad. Sci. 51, 627-659 (1949). |
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82 | |
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83 | Authorship and Verification |
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84 | ---------------------------- |
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85 | |
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86 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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87 | * **Last Modified by:** Paul Butler **Date:** March 20, 2016 |
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88 | * **Last Reviewed by:** Richard Heenan **Date:** January 4, 2017 |
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89 | """ |
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90 | |
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91 | import numpy as np |
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92 | from numpy import inf, sin, cos, pi |
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93 | |
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94 | name = "barbell" |
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95 | title = "Cylinder with spherical end caps" |
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96 | description = """ |
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97 | Calculates the scattering from a barbell-shaped cylinder. |
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98 | That is a sphereocylinder with spherical end caps that have a radius larger |
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99 | than that of the cylinder and the center of the end cap radius lies outside |
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100 | of the cylinder. |
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101 | Note: As the length of cylinder(bar) -->0,it becomes a dumbbell. And when |
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102 | rad_bar = rad_bell, it is a spherocylinder. |
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103 | It must be that rad_bar <(=) rad_bell. |
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104 | """ |
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105 | category = "shape:cylinder" |
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106 | # pylint: disable=bad-whitespace, line-too-long |
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107 | # ["name", "units", default, [lower, upper], "type","description"], |
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108 | parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "sld", "Barbell scattering length density"], |
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109 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Solvent scattering length density"], |
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110 | ["radius_bell", "Ang", 40, [0, inf], "volume", "Spherical bell radius"], |
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111 | ["radius", "Ang", 20, [0, inf], "volume", "Cylindrical bar radius"], |
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112 | ["length", "Ang", 400, [0, inf], "volume", "Cylinder bar length"], |
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113 | ["theta", "degrees", 60, [-360, 360], "orientation", "Barbell axis to beam angle"], |
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114 | ["phi", "degrees", 60, [-360, 360], "orientation", "Rotation about beam"], |
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115 | ] |
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116 | # pylint: enable=bad-whitespace, line-too-long |
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117 | |
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118 | source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "barbell.c"] |
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119 | have_Fq = True |
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120 | effective_radius_type = [ |
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121 | "equivalent cylinder excluded volume", "equivalent volume sphere", |
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122 | "radius", "half length", "half total length", |
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123 | ] |
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124 | |
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125 | def random(): |
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126 | """Return a random parameter set for the model.""" |
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127 | # TODO: increase volume range once problem with bell radius is fixed |
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128 | # The issue is that bell radii of more than about 200 fail at high q |
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129 | volume = 10**np.random.uniform(7, 9) |
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130 | bar_volume = 10**np.random.uniform(-4, -1)*volume |
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131 | bell_volume = volume - bar_volume |
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132 | bell_radius = (bell_volume/6)**0.3333 # approximate |
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133 | min_bar = bar_volume/np.pi/bell_radius**2 |
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134 | bar_length = 10**np.random.uniform(0, 3)*min_bar |
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135 | bar_radius = np.sqrt(bar_volume/bar_length/np.pi) |
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136 | if bar_radius > bell_radius: |
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137 | bell_radius, bar_radius = bar_radius, bell_radius |
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138 | pars = dict( |
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139 | #background=0, |
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140 | radius_bell=bell_radius, |
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141 | radius=bar_radius, |
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142 | length=bar_length, |
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143 | ) |
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144 | return pars |
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145 | |
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146 | # parameters for demo |
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147 | demo = dict(scale=1, background=0, |
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148 | sld=6, sld_solvent=1, |
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149 | radius_bell=40, radius=20, length=400, |
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150 | theta=60, phi=60, |
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151 | radius_pd=.2, radius_pd_n=5, |
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152 | length_pd=.2, length_pd_n=5, |
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153 | theta_pd=15, theta_pd_n=0, |
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154 | phi_pd=15, phi_pd_n=0, |
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155 | ) |
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156 | q = 0.1 |
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157 | # april 6 2017, rkh add unit tests, NOT compared with any other calc method, assume correct! |
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158 | qx = q*cos(pi/6.0) |
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159 | qy = q*sin(pi/6.0) |
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160 | tests = [ |
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161 | [{}, 0.075, 25.5691260532], |
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162 | [{'theta':80., 'phi':10.}, (qx, qy), 3.04233067789], |
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163 | ] |
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164 | del qx, qy # not necessary to delete, but cleaner |
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