1 | r""" |
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2 | Calculates the scattering from a cylinder with spherical section end-caps. |
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3 | This model simply becomes the a convex lens when the length of the cylinder |
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4 | $L=0$, that is, a sphereocylinder with end caps that have a radius larger |
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5 | than that of the cylinder and the center of the end cap radius lies within |
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6 | the cylinder. See the diagram for the details of the geometry and |
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7 | restrictions on parameter values. |
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8 | |
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9 | Definitions |
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10 | ----------- |
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11 | |
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12 | The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. |
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13 | |
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14 | The capped cylinder geometry is defined as |
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15 | |
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16 | .. image:: img/capped_cylinder_geometry.jpg |
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17 | |
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18 | where $r$ is the radius of the cylinder. All other parameters are as defined |
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19 | in the diagram. Since the end cap radius $R \ge r$ and by definition for this |
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20 | geometry $h < 0$, $h$ is then defined by $r$ and $R$ as |
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21 | |
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22 | .. math:: |
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23 | |
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24 | h = - \sqrt{R^2 - r^2} |
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25 | |
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26 | The scattered intensity $I(Q)$ is calculated as |
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27 | |
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28 | .. math:: |
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29 | |
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30 | I(Q) = \frac{(\Delta \rho)^2}{V} \left< A^2(Q)\right> |
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31 | |
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32 | where the amplitude $A(Q)$ is given as |
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33 | |
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34 | .. math:: |
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35 | |
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36 | A(Q) =&\ \pi r^2L |
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37 | {\sin\left(\tfrac12 QL\cos\theta\right) |
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38 | \over \tfrac12 QL\cos\theta} |
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39 | {2 J_1(Qr\sin\theta) \over Qr\sin\theta} \\ |
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40 | &\ + 4 \pi R^3 \int_{-h/R}^1 dt |
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41 | \cos\left[ Q\cos\theta |
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42 | \left(Rt + h + {\tfrac12} L\right)\right] |
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43 | \times (1-t^2) |
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44 | {J_1\left[QR\sin\theta \left(1-t^2\right)^{1/2}\right] |
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45 | \over QR\sin\theta \left(1-t^2\right)^{1/2}} |
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46 | |
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47 | The $\left< \ldots \right>$ brackets denote an average of the structure over |
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48 | all orientations. $\left< A^2(Q)\right>$ is then the form factor, $P(Q)$. |
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49 | The scale factor is equivalent to the volume fraction of cylinders, each of |
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50 | volume, $V$. Contrast is the difference of scattering length densities of |
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51 | the cylinder and the surrounding solvent. |
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52 | |
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53 | The volume of the capped cylinder is (with $h$ as a positive value here) |
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54 | |
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55 | .. math:: |
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56 | |
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57 | V = \pi r_c^2 L + \tfrac{2\pi}{3}(R-h)^2(2R + h) |
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58 | |
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59 | |
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60 | and its radius-of-gyration is |
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61 | |
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62 | .. math:: |
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63 | |
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64 | R_g^2 =&\ \left[ \tfrac{12}{5}R^5 |
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65 | + R^4\left(6h+\tfrac32 L\right) |
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66 | + R^2\left(4h^2 + L^2 + 4Lh\right) |
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67 | + R^2\left(3Lh^2 + \tfrac32 L^2h\right) \right. \\ |
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68 | &\ \left. + \tfrac25 h^5 - \tfrac12 Lh^4 - \tfrac12 L^2h^3 |
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69 | + \tfrac14 L^3r^2 + \tfrac32 Lr^4 \right] |
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70 | \left( 4R^3 6R^2h - 2h^3 + 3r^2L \right)^{-1} |
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71 | |
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72 | |
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73 | .. note:: |
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74 | |
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75 | The requirement that $R \ge r$ is not enforced in the model! |
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76 | It is up to you to restrict this during analysis. |
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77 | |
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78 | :num:`Figure #capped-cylinder-1d` shows the output produced by |
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79 | a running the 1D capped cylinder model, using *qmin* = 0.001 |Ang^-1|, |
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80 | *qmax* = 0.7 |Ang^-1| and the default values of the parameters. |
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81 | |
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82 | .. _capped-cylinder-1d: |
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83 | |
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84 | .. figure:: img/capped_cylinder_1d.jpg |
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85 | |
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86 | 1D plot using the default values (w/256 data point). |
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87 | |
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88 | The 2D scattering intensity is calculated similar to the 2D cylinder model. |
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89 | :num:`Figure #capped-cylinder-2d` shows the output for $\theta=45^\circ$ |
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90 | and $\phi=0^\circ$ with default values for the other parameters. |
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91 | |
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92 | .. _capped-cylinder-2d: |
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93 | |
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94 | .. figure:: img/capped_cylinder_2d.jpg |
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95 | |
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96 | 2D plot (w/(256X265) data points). |
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97 | |
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98 | .. figure:: img/orientation.jpg |
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99 | |
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100 | Definition of the angles for oriented 2D cylinders. |
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101 | |
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102 | .. figure:: img/orientation2.jpg |
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103 | |
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104 | Examples of the angles for oriented pp against the detector plane. |
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105 | |
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106 | REFERENCE |
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107 | |
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108 | H Kaya, *J. Appl. Cryst.*, 37 (2004) 223-230 |
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109 | |
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110 | H Kaya and N-R deSouza, *J. Appl. Cryst.*, 37 (2004) 508-509 (addenda and errata) |
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111 | """ |
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112 | from numpy import inf |
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113 | |
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114 | name = "capped_cylinder" |
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115 | title = "Right circular cylinder with spherical end caps and uniform SLD" |
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116 | description = """That is, a sphereocylinder |
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117 | with end caps that have a radius larger than |
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118 | that of the cylinder and the center of the |
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119 | end cap radius lies within the cylinder. |
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120 | Note: As the length of cylinder -->0, |
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121 | it becomes a ConvexLens. |
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122 | It must be that radius <(=) cap_radius. |
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123 | [Parameters]; |
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124 | scale: volume fraction of spheres, |
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125 | background:incoherent background, |
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126 | radius: radius of the cylinder, |
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127 | length: length of the cylinder, |
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128 | cap_radius: radius of the semi-spherical cap, |
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129 | sld: SLD of the capped cylinder, |
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130 | solvent_sld: SLD of the solvent. |
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131 | """ |
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132 | category = "shape:cylinder" |
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133 | |
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134 | # ["name", "units", default, [lower, upper], "type", "description"], |
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135 | parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "", |
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136 | "Cylinder scattering length density"], |
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137 | ["solvent_sld", "1e-6/Ang^2", 1, [-inf, inf], "", |
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138 | "Solvent scattering length density"], |
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139 | ["radius", "Ang", 20, [0, inf], "volume", "Cylinder radius"], |
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140 | # TODO: use an expression for cap radius with fixed bounds. |
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141 | # The current form requires cap radius R bigger than cylinder radius r. |
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142 | # Could instead use R/r in [1,inf], r/R in [0,1], or the angle between |
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143 | # cylinder and cap in [0,90]. The problem is similar for the barbell |
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144 | # model. Propose r/R in [0,1] in both cases, with the model specifying |
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145 | # cylinder radius in the capped cylinder model and sphere radius in the |
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146 | # barbell model. This leads to the natural value of zero for no cap |
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147 | # in the capped cylinder, and zero for no bar in the barbell model. In |
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148 | # both models, one would be a pill. |
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149 | ["cap_radius", "Ang", 20, [0, inf], "volume", "Cap radius"], |
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150 | ["length", "Ang", 400, [0, inf], "volume", "Cylinder length"], |
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151 | ["theta", "degrees", 60, [-inf, inf], "orientation", "In plane angle"], |
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152 | ["phi", "degrees", 60, [-inf, inf], "orientation", "Out of plane angle"], |
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153 | ] |
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154 | |
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155 | source = ["lib/J1.c", "lib/gauss76.c", "capped_cylinder.c"] |
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156 | |
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157 | demo = dict(scale=1, background=0, |
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158 | sld=6, solvent_sld=1, |
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159 | radius=260, cap_radius=290, length=290, |
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160 | theta=30, phi=15, |
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161 | radius_pd=.2, radius_pd_n=1, |
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162 | cap_radius_pd=.2, cap_radius_pd_n=1, |
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163 | length_pd=.2, length_pd_n=10, |
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164 | theta_pd=15, theta_pd_n=45, |
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165 | phi_pd=15, phi_pd_n=1) |
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166 | oldname = 'CappedCylinderModel' |
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167 | oldpars = dict(sld='sld_capcyl', solvent_sld='sld_solv', |
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168 | length='len_cyl', radius='rad_cyl', cap_radius='rad_cap') |
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