1 | #barbell model |
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2 | # cylinder model |
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3 | # Note: model title and parameter table are inserted automatically |
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4 | r""" |
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5 | |
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6 | Calculates the scattering from a barbell-shaped cylinder (This model simply becomes the DumBellModel when the length of |
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7 | the cylinder, *L*, is set to zero). That is, a sphereocylinder with spherical end caps that have a radius larger than |
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8 | that of the cylinder and the center of the end cap radius lies outside of the cylinder. All dimensions of the BarBell |
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9 | are considered to be monodisperse. See the diagram for the details of the geometry and restrictions on parameter values. |
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10 | |
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11 | Definition |
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12 | ---------- |
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13 | |
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14 | The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. |
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15 | |
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16 | The barbell geometry is defined as |
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17 | |
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18 | .. image:: img/image105.jpg |
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19 | |
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20 | where *r* is the radius of the cylinder. All other parameters are as defined in the diagram. |
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21 | |
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22 | Since the end cap radius |
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23 | *R* >= *r* and by definition for this geometry *h* < 0, *h* is then defined by *r* and *R* as |
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24 | |
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25 | *h* = -1 \* sqrt(*R*\ :sup:`2` - *r*\ :sup:`2`) |
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26 | |
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27 | The scattered intensity *I(q)* is calculated as |
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28 | |
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29 | .. math:: |
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30 | |
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31 | I(Q) = \frac{(\Delta \rho)^2}{V} \left< A^2(Q)\right> |
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32 | |
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33 | where the amplitude *A(q)* is given as |
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34 | |
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35 | .. math:: |
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36 | |
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37 | A(Q) =&\ \pi r^2L |
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38 | {\sin\left(\tfrac12 QL\cos\theta\right) |
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39 | \over \tfrac12 QL\cos\theta} |
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40 | {2 J_1(Qr\sin\theta) \over Qr\sin\theta} \\ |
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41 | &\ + 4 \pi R^3 \int_{-h/R}^1 dt |
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42 | \cos\left[ Q\cos\theta |
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43 | \left(Rt + h + {\tfrac12} L\right)\right] |
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44 | \times (1-t^2) |
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45 | {J_1\left[QR\sin\theta \left(1-t^2\right)^{1/2}\right] |
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46 | \over QR\sin\theta \left(1-t^2\right)^{1/2}} |
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47 | |
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48 | The < > brackets denote an average of the structure over all orientations. <*A* :sup:`2`\ *(q)*> is then the form |
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49 | factor, *P(q)*. The scale factor is equivalent to the volume fraction of cylinders, each of volume, *V*. Contrast is |
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50 | the difference of scattering length densities of the cylinder and the surrounding solvent. |
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51 | |
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52 | The volume of the barbell is |
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53 | |
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54 | .. math:: |
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55 | |
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56 | V = \pi r_c^2 L + 2\pi\left(\tfrac23R^3 + R^2h-\tfrac13h^3\right) |
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57 | |
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58 | |
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59 | and its radius-of-gyration is |
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60 | |
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61 | .. math:: |
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62 | |
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63 | R_g^2 =&\ \left[ \tfrac{12}{5}R^5 |
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64 | + R^4\left(6h+\tfrac32 L\right) |
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65 | + R^2\left(4h^2 + L^2 + 4Lh\right) |
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66 | + R^2\left(3Lh^2 + \tfrac32 L^2h\right) \right. \\ |
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67 | &\ \left. + \tfrac25 h^5 - \tfrac12 Lh^4 - \tfrac12 L^2h^3 |
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68 | + \tfrac14 L^3r^2 + \tfrac32 Lr^4 \right] |
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69 | \left( 4R^3 6R^2h - 2h^3 + 3r^2L \right)^{-1} |
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70 | |
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71 | **The requirement that** *R* >= *r* **is not enforced in the model!** It is up to you to restrict this during analysis. |
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72 | |
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73 | This example dataset is produced by running the Macro PlotBarbell(), using 200 data points, *qmin* = 0.001 |Ang^-1|, |
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74 | *qmax* = 0.7 |Ang^-1| and the following default values |
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75 | |
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76 | ============== ======== ============= |
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77 | Parameter name Units Default value |
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78 | ============== ======== ============= |
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79 | scale None 1.0 |
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80 | len_bar |Ang| 400.0 |
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81 | rad_bar |Ang| 20.0 |
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82 | rad_bell |Ang| 40.0 |
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83 | sld_barbell |Ang^-2| 1.0e-006 |
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84 | sld_solv |Ang^-2| 6.3e-006 |
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85 | background |cm^-1| 0 |
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86 | ============== ======== ============= |
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87 | |
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88 | .. image:: img/image110.jpg |
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89 | |
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90 | *Figure. 1D plot using the default values (w/256 data point).* |
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91 | |
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92 | For 2D data: The 2D scattering intensity is calculated similar to the 2D cylinder model. For example, for |
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93 | |theta| = 45 deg and |phi| = 0 deg with default values for other parameters |
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94 | |
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95 | .. image:: img/image111.jpg |
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96 | |
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97 | *Figure. 2D plot (w/(256X265) data points).* |
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98 | |
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99 | .. image:: img/orientation.jpg |
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100 | |
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101 | Figure. Definition of the angles for oriented 2D barbells. |
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102 | |
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103 | .. image:: img/orientation2.jpg |
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104 | |
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105 | *Figure. Examples of the angles for oriented pp against the detector plane.* |
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106 | |
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107 | REFERENCE |
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108 | |
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109 | H Kaya, *J. Appl. Cryst.*, 37 (2004) 37 223-230 |
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110 | |
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111 | H Kaya and N R deSouza, *J. Appl. Cryst.*, 37 (2004) 508-509 (addenda and errata) |
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112 | |
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113 | """ |
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114 | from numpy import pi, inf |
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115 | |
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116 | name = "barbell" |
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117 | title = "Cylinder with spherical end caps" |
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118 | description = """ |
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119 | Calculates the scattering from a barbell-shaped cylinder. That is a sphereocylinder with spherical end caps that have a radius larger than that of the cylinder and the center of the end cap |
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120 | radius lies outside of the cylinder. |
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121 | Note: As the length of cylinder(bar) -->0,it becomes a dumbbell. And when rad_bar = rad_bell, it is a spherocylinder. |
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122 | It must be that rad_bar <(=) rad_bell. |
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123 | """ |
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124 | |
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125 | parameters = [ |
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126 | # [ "name", "units", default, [lower, upper], "type","description" ], |
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127 | [ "sld", "1e-6/Ang^2", 4, [-inf,inf], "", "Barbell scattering length density" ], |
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128 | [ "solvent_sld", "1e-6/Ang^2", 1, [-inf,inf], "","Solvent scattering length density" ], |
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129 | [ "bell_radius", "Ang", 40, [0, inf], "volume","Spherical bell radius" ], |
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130 | [ "radius", "Ang", 20, [0, inf], "volume","Cylindrical bar radius" ], |
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131 | [ "length", "Ang", 400, [0, inf], "volume","Cylinder bar length" ], |
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132 | [ "theta", "degrees", 60, [-inf, inf], "orientation","In plane angle" ], |
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133 | [ "phi", "degrees", 60, [-inf, inf], "orientation","Out of plane angle" ], |
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134 | ] |
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135 | |
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136 | source = [ "lib/J1.c", "lib/gauss76.c", "barbell.c" ] |
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137 | |
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138 | def ER(radius, length): |
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139 | return 1.0 |
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140 | |
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141 | # parameters for demo |
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142 | demo = dict( |
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143 | scale=1, background=0, |
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144 | sld=6, solvent_sld=1, |
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145 | bell_radius = 40, radius=20, length=400, |
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146 | theta=60, phi=60, |
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147 | radius_pd=.2, radius_pd_n=0, |
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148 | length_pd=.2,length_pd_n=0, |
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149 | theta_pd=15, theta_pd_n=0, |
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150 | phi_pd=15, phi_pd_n=0, |
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151 | ) |
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152 | |
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153 | # For testing against the old sasview models, include the converted parameter |
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154 | # names and the target sasview model name. |
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155 | oldname='BarBellModel' |
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156 | oldpars=dict(sld='sld_barbell', |
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157 | solvent_sld='sld_solv', bell_radius='rad_bell', |
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158 | radius='rad_bar',length='len_bar') |
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