1 | .. model_functions.rst |
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2 | |
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3 | .. This is a port of the original SasView model_functions.html to ReSTructured text |
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4 | .. by S King, ISIS, during and after SasView CodeCamp-II in April 2014. |
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5 | |
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6 | .. Thanks are due to A Jackson & P Kienzle for advice on RST! |
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7 | |
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8 | .. The CoreShellEllipsoidXTModel was ported and documented by R K Heenan, ISIS, Apr 2014 |
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9 | .. The RectangularPrism models were coded and documented by M A Gonzalez, ILL, Apr 2014 |
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10 | |
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11 | .. To do: |
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12 | .. Add example parameters/plots for the CoreShellEllipsoidXTModel |
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13 | .. Add example parameters/plots for the RectangularPrism models |
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14 | .. Check the content against the NIST Igor Help File |
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15 | .. Wordsmith the content for consistency of style, etc |
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16 | |
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17 | |
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18 | |
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19 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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20 | |
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21 | |
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22 | .. note:: The contents of this document are presented in good faith and are |
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23 | believed to be mostly correct and accurate, however they have not |
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24 | yet been rigorously checked for errors. June2015 |
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25 | |
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26 | |
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27 | .. Set up some substitutions to make life easier... |
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28 | |
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29 | |
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30 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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31 | |
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32 | |
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33 | |
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34 | .. Actual document starts here... |
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35 | |
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36 | .. _SasView_model_functions: |
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37 | |
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38 | SasView Model Functions |
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39 | ======================= |
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40 | |
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41 | .. _Background: |
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42 | |
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43 | 1. Background |
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44 | --------------- |
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45 | |
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46 | Many of our models use the form factor calculations implemented in a c-library provided by the NIST Center for Neutron |
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47 | Research and thus some content and figures in this document are originated from or shared with the NIST SANS Igor-based |
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48 | analysis package. |
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49 | |
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50 | This software provides form factors for various particle shapes. After giving a mathematical definition of each model, |
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51 | we show the list of parameters available to the user. Validation plots for each model are also presented. |
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52 | |
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53 | Instructions on how to use SasView itself are available separately. |
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54 | |
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55 | To easily compare to the scattering intensity measured in experiments, we normalize the form factors by the volume of |
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56 | the particle |
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57 | |
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58 | .. image:: ..\img\olddocs\image001.PNG |
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59 | |
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60 | with |
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61 | |
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62 | .. image:: ..\img\olddocs\image002.PNG |
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63 | |
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64 | where |P0|\ *(q)* is the un-normalized form factor, |rho|\ *(r)* is the scattering length density at a given |
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65 | point in space and the integration is done over the volume *V* of the scatterer. |
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66 | |
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67 | For systems without inter-particle interference, the form factors we provide can be related to the scattering intensity |
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68 | by the particle volume fraction |
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69 | |
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70 | .. image:: ..\img\olddocs\image003.PNG |
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71 | |
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72 | Our so-called 1D scattering intensity functions provide *P(q)* for the case where the scatterer is randomly oriented. In |
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73 | that case, the scattering intensity only depends on the length of *q* . The intensity measured on the plane of the SAS |
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74 | detector will have an azimuthal symmetry around *q*\ =0 . |
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75 | |
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76 | Our so-called 2D scattering intensity functions provide *P(q,* |phi| *)* for an oriented system as a function of a |
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77 | q-vector in the plane of the detector. We define the angle |phi| as the angle between the q vector and the horizontal |
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78 | (x) axis of the plane of the detector. |
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79 | |
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80 | For information about polarised and magnetic scattering, click here_. |
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81 | |
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82 | .. _here: polar_mag_help.html |
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83 | |
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84 | |
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85 | |
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86 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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87 | |
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88 | |
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89 | |
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90 | .. _Model: |
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91 | |
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92 | 2. Model functions |
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93 | ------------------ |
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94 | |
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95 | .. _Shape-based: |
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96 | |
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97 | 2.1 Shape-based Functions |
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98 | ------------------------- |
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99 | |
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100 | Sphere-based |
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101 | ------------ |
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102 | |
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103 | - SphereModel_ (including magnetic 2D version) |
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104 | - BinaryHSModel_ |
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105 | - FuzzySphereModel_ |
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106 | - RaspBerryModel_ |
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107 | - CoreShellModel_ (including magnetic 2D version) |
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108 | - MicelleSphCoreModel_ |
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109 | - CoreMultiShellModel_ (including magnetic 2D version) |
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110 | - Core2ndMomentModel_ |
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111 | - MultiShellModel_ |
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112 | - OnionExpShellModel_ |
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113 | - VesicleModel_ |
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114 | - SphericalSLDModel_ |
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115 | - LinearPearlsModel_ |
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116 | - PearlNecklaceModel_ |
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117 | |
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118 | Cylinder-based |
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119 | -------------- |
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120 | |
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121 | - CylinderModel_ (including magnetic 2D version) |
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122 | - HollowCylinderModel_ |
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123 | - CappedCylinderModel_ |
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124 | - CoreShellCylinderModel_ |
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125 | - EllipticalCylinderModel_ |
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126 | - FlexibleCylinderModel_ |
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127 | - FlexCylEllipXModel_ |
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128 | - CoreShellBicelleModel_ |
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129 | - BarBellModel_ |
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130 | - StackedDisksModel_ |
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131 | - PringleModel_ |
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132 | |
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133 | Ellipsoid-based |
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134 | --------------- |
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135 | |
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136 | - EllipsoidModel_ |
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137 | - CoreShellEllipsoidModel_ |
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138 | - CoreShellEllipsoidXTModel_ |
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139 | - TriaxialEllipsoidModel_ |
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140 | |
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141 | Lamellae |
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142 | -------- |
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143 | |
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144 | - LamellarModel_ |
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145 | - LamellarFFHGModel_ |
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146 | - LamellarPSModel_ |
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147 | - LamellarPSHGModel_ |
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148 | |
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149 | Paracrystals |
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150 | ------------ |
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151 | |
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152 | - LamellarPCrystalModel_ |
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153 | - SCCrystalModel_ |
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154 | - FCCrystalModel_ |
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155 | - BCCrystalModel_ |
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156 | |
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157 | Parallelpipeds |
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158 | -------------- |
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159 | |
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160 | - ParallelepipedModel_ (including magnetic 2D version) |
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161 | - CSParallelepipedModel_ |
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162 | - RectangularPrismModel_ |
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163 | - RectangularHollowPrismModel_ |
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164 | - RectangularHollowPrismInfThinWallsModel_ |
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165 | |
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166 | .. _Shape-independent: |
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167 | |
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168 | 2.2 Shape-Independent Functions |
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169 | ------------------------------- |
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170 | |
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171 | (In alphabetical order) |
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172 | |
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173 | - AbsolutePower_Law_ |
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174 | - BEPolyelectrolyte_ |
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175 | - BroadPeakModel_ |
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176 | - CorrLength_ |
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177 | - DABModel_ |
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178 | - Debye_ |
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179 | - FractalModel_ |
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180 | - FractalCoreShell_ |
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181 | - GaussLorentzGel_ |
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182 | - GelFitModel_ |
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183 | - Guinier_ |
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184 | - GuinierPorod_ |
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185 | - LineModel_ |
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186 | - Lorentz_ |
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187 | - MassFractalModel_ |
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188 | - MassSurfaceFractal_ |
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189 | - PeakGaussModel_ |
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190 | - PeakLorentzModel_ |
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191 | - Poly_GaussCoil_ |
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192 | - PolyExclVolume_ |
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193 | - PorodModel_ |
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194 | - RPA10Model_ |
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195 | - StarPolymer_ |
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196 | - SurfaceFractalModel_ |
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197 | - TeubnerStrey_ |
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198 | - TwoLorentzian_ |
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199 | - TwoPowerLaw_ |
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200 | - UnifiedPowerRg_ |
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201 | - ReflectivityModel_ |
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202 | - ReflectivityIIModel_ |
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203 | |
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204 | .. _Structure-factor: |
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205 | |
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206 | 2.3 Structure Factor Functions |
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207 | ------------------------------ |
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208 | |
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209 | - HardSphereStructure_ |
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210 | - SquareWellStructure_ |
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211 | - HayterMSAStructure_ |
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212 | - StickyHSStructure_ |
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213 | |
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214 | .. _Customised: |
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215 | |
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216 | 2.4 Customized Functions |
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217 | ------------------------ |
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218 | |
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219 | - testmodel_ |
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220 | - testmodel_2_ |
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221 | - sum_p1_p2_ |
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222 | - sum_Ap1_1_Ap2_ |
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223 | - polynomial5_ |
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224 | - sph_bessel_jn_ |
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225 | |
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226 | Also see the documentation on :ref:`Adding_your_own_models` under Fitting Data. |
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227 | |
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228 | |
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229 | |
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230 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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231 | |
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232 | |
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233 | |
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234 | .. _References: |
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235 | |
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236 | 3. References |
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237 | ------------- |
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238 | |
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239 | *Small-Angle Scattering of X-Rays* |
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240 | A Guinier and G Fournet |
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241 | John Wiley & Sons, New York (1955) |
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242 | |
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243 | P Stckel, R May, I Strell, Z Cejka, W Hoppe, H Heumann, W Zillig and H Crespi |
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244 | *Eur. J. Biochem.*, 112, (1980), 411-417 |
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245 | |
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246 | G Porod |
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247 | in *Small Angle X-ray Scattering* |
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248 | (editors) O Glatter and O Kratky |
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249 | Academic Press (1982) |
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250 | |
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251 | *Structure Analysis by Small-Angle X-Ray and Neutron Scattering* |
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252 | L.A Feigin and D I Svergun |
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253 | Plenum Press, New York (1987) |
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254 | |
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255 | S Hansen |
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256 | *J. Appl. Cryst.* 23, (1990), 344-346 |
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257 | |
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258 | S J Henderson |
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259 | *Biophys. J.* 70, (1996), 1618-1627 |
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260 | |
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261 | B C McAlister and B P Grady |
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262 | *J. Appl. Cryst.* 31, (1998), 594-599 |
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263 | |
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264 | S R Kline |
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265 | *J Appl. Cryst.* 39(6), (2006), 895 |
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266 | |
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267 | **Also see the references at the end of the each model function descriptions.** |
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268 | |
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269 | |
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270 | |
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271 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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272 | |
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273 | |
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274 | |
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275 | Model Definitions |
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276 | ----------------- |
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277 | |
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278 | .. _SphereModel: |
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279 | |
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280 | **2.1.1. SphereModel** |
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281 | |
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282 | This model provides the form factor, *P(q)*, for a monodisperse spherical particle with uniform scattering length |
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283 | density. The form factor is normalized by the particle volume as described below. |
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284 | |
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285 | For information about polarised and magnetic scattering, click here_. |
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286 | |
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287 | .. _here: polar_mag_help.html |
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288 | |
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289 | *2.1.1.1. Definition* |
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290 | |
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291 | The 1D scattering intensity is calculated in the following way (Guinier, 1955) |
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292 | |
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293 | .. image:: ..\img\olddocs\image004.PNG |
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294 | |
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295 | where *scale* is a volume fraction, *V* is the volume of the scatterer, *r* is the radius of the sphere, *bkg* is |
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296 | the background level and *sldXXX* is the scattering length density (SLD) of the scatterer or the solvent. |
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297 | |
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298 | Note that if your data is in absolute scale, the *scale* should represent the volume fraction (which is unitless) if |
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299 | you have a good fit. If not, it should represent the volume fraction \* a factor (by which your data might need to be |
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300 | rescaled). |
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301 | |
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302 | The 2D scattering intensity is the same as above, regardless of the orientation of the q vector. |
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303 | |
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304 | The returned value is scaled to units of |cm^-1| and the parameters of the SphereModel are the following: |
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305 | |
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306 | ============== ======== ============= |
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307 | Parameter name Units Default value |
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308 | ============== ======== ============= |
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309 | scale None 1 |
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310 | radius |Ang| 60 |
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311 | sldSph |Ang^-2| 2.0e-6 |
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312 | sldSolv |Ang^-2| 1.0e-6 |
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313 | background |cm^-1| 0 |
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314 | ============== ======== ============= |
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315 | |
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316 | Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron |
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317 | Research (Kline, 2006). |
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318 | |
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319 | REFERENCE |
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320 | |
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321 | A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*, John Wiley and Sons, New York, (1955) |
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322 | |
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323 | *2.1.1.2. Validation of the SphereModel* |
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324 | |
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325 | Validation of our code was done by comparing the output of the 1D model to the output of the software provided by the |
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326 | NIST (Kline, 2006). Figure 1 shows a comparison of the output of our model and the output of the NIST software. |
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327 | |
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328 | .. image:: ..\img\olddocs\image005.jpg |
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329 | |
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330 | Figure 1: Comparison of the DANSE scattering intensity for a sphere with the output of the NIST SANS analysis software. |
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331 | The parameters were set to: Scale=1.0, Radius=60 |Ang|, Contrast=1e-6 |Ang^-2|, and Background=0.01 |cm^-1|. |
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332 | |
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333 | *2013/09/09 and 2014/01/06 - Description reviewed by S King and P Parker.* |
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334 | |
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335 | |
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336 | |
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337 | .. _BinaryHSModel: |
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338 | |
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339 | **2.1.2. BinaryHSModel** |
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340 | |
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341 | *2.1.2.1. Definition* |
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342 | |
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343 | This model (binary hard sphere model) provides the scattering intensity, for binary mixture of spheres including hard |
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344 | sphere interaction between those particles. Using Percus-Yevick closure, the calculation is an exact multi-component |
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345 | solution |
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346 | |
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347 | .. image:: ..\img\olddocs\image006.PNG |
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348 | |
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349 | where *Sij* are the partial structure factors and *fi* are the scattering amplitudes of the particles. The subscript 1 |
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350 | is for the smaller particle and 2 is for the larger. The number fraction of the larger particle, (*x* = n2/(n1+n2), |
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351 | where *n* = the number density) is internally calculated based on |
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352 | |
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353 | .. image:: ..\img\olddocs\image007.PNG |
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354 | |
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355 | The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as |
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356 | |
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357 | .. image:: ..\img\olddocs\image008.PNG |
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358 | |
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359 | The parameters of the BinaryHSModel are the following (in the names, *l* (or *ls*\ ) stands for larger spheres |
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360 | while *s* (or *ss*\ ) for the smaller spheres). |
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361 | |
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362 | ============== ======== ============= |
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363 | Parameter name Units Default value |
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364 | ============== ======== ============= |
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365 | background |cm^-1| 0.001 |
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366 | l_radius |Ang| 100.0 |
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367 | ss_sld |Ang^-2| 0.0 |
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368 | ls_sld |Ang^-2| 3e-6 |
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369 | solvent_sld |Ang^-2| 6e-6 |
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370 | s_radius |Ang| 25.0 |
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371 | vol_frac_ls None 0.1 |
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372 | vol_frac_ss None 0.2 |
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373 | ============== ======== ============= |
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374 | |
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375 | .. image:: ..\img\olddocs\image009.jpg |
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376 | |
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377 | *Figure. 1D plot using the default values above (w/200 data point).* |
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378 | |
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379 | Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron |
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380 | Research (Kline, 2006). |
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381 | |
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382 | See the reference for details. |
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383 | |
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384 | REFERENCE |
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385 | |
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386 | N W Ashcroft and D C Langreth, *Physical Review*, 156 (1967) 685-692 |
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387 | [Errata found in *Phys. Rev.* 166 (1968) 934] |
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388 | |
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389 | |
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390 | |
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391 | .. _FuzzySphereModel: |
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392 | |
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393 | **2.1.3. FuzzySphereModel** |
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394 | |
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395 | This model is to calculate the scattering from spherical particles with a "fuzzy" interface. |
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396 | |
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397 | *2.1.3.1. Definition* |
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398 | |
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399 | The scattering intensity *I(q)* is calculated as: |
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400 | |
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401 | .. image:: ..\img\olddocs\image010.PNG |
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402 | |
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403 | where the amplitude *A(q)* is given as the typical sphere scattering convoluted with a Gaussian to get a gradual |
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404 | drop-off in the scattering length density |
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405 | |
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406 | .. image:: ..\img\olddocs\image011.PNG |
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407 | |
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408 | Here |A2|\ *(q)* is the form factor, *P(q)*. The scale is equivalent to the volume fraction of spheres, each of |
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409 | volume, *V*\. Contrast (|drho|) is the difference of scattering length densities of the sphere and the surrounding |
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410 | solvent. |
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411 | |
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412 | Poly-dispersion in radius and in fuzziness is provided for. |
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413 | |
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414 | The returned value is scaled to units of |cm^-1|\ |sr^-1|; ie, absolute scale. |
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415 | |
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416 | From the reference |
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417 | |
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418 | The "fuzziness" of the interface is defined by the parameter |sigma| :sub:`fuzzy`\ . The particle radius *R* |
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419 | represents the radius of the particle where the scattering length density profile decreased to 1/2 of the core |
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420 | density. The |sigma| :sub:`fuzzy`\ is the width of the smeared particle surface; i.e., the standard deviation |
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421 | from the average height of the fuzzy interface. The inner regions of the microgel that display a higher density |
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422 | are described by the radial box profile extending to a radius of approximately *Rbox* ~ *R* - 2\ |sigma|\ . The |
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423 | profile approaches zero as *Rsans* ~ *R* + 2\ |sigma|\ . |
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424 | |
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425 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
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426 | |
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427 | .. image:: ..\img\olddocs\image008.PNG |
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428 | |
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429 | This example dataset is produced by running the FuzzySphereModel, using 200 data points, *qmin* = 0.001 -1, |
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430 | *qmax* = 0.7 |Ang^-1| and the default values |
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431 | |
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432 | ============== ======== ============= |
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433 | Parameter name Units Default value |
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434 | ============== ======== ============= |
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435 | scale None 1.0 |
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436 | radius |Ang| 60 |
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437 | fuzziness |Ang| 10 |
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438 | sldSolv |Ang^-2| 3e-6 |
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439 | sldSph |Ang^-2| 1e-6 |
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440 | background |cm^-1| 0.001 |
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441 | ============== ======== ============= |
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442 | |
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443 | .. image:: ..\img\olddocs\image012.jpg |
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444 | |
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445 | *Figure. 1D plot using the default values (w/200 data point).* |
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446 | |
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447 | REFERENCE |
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448 | |
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449 | M Stieger, J. S Pedersen, P Lindner, W Richtering, *Langmuir*, 20 (2004) 7283-7292 |
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450 | |
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451 | |
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452 | |
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453 | .. _RaspBerryModel: |
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454 | |
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455 | **2.1.4. RaspBerryModel** |
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456 | |
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457 | Calculates the form factor, *P(q)*, for a "Raspberry-like" structure where there are smaller spheres at the surface |
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458 | of a larger sphere, such as the structure of a Pickering emulsion. |
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459 | |
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460 | *2.1.4.1. Definition* |
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461 | |
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462 | The structure is: |
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463 | |
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464 | .. image:: ..\img\olddocs\raspberry_pic.jpg |
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465 | |
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466 | where *Ro* = the radius of the large sphere, *Rp* = the radius of the smaller sphere on the surface, |delta| = the |
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467 | fractional penetration depth, and surface coverage = fractional coverage of the large sphere surface (0.9 max). |
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468 | |
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469 | The large and small spheres have their own SLD, as well as the solvent. The surface coverage term is a fractional |
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470 | coverage (maximum of approximately 0.9 for hexagonally-packed spheres on a surface). Since not all of the small |
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471 | spheres are necessarily attached to the surface, the excess free (small) spheres scattering is also included in the |
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472 | calculation. The function calculated follows equations (8)-(12) of the reference below, and the equations are not |
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473 | reproduced here. |
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474 | |
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475 | The returned value is scaled to units of |cm^-1|. No inter-particle scattering is included in this model. |
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476 | |
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477 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
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478 | |
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479 | .. image:: ..\img\olddocs\image008.PNG |
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480 | |
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481 | This example dataset is produced by running the RaspBerryModel, using 2000 data points, *qmin* = 0.0001 |Ang^-1|, |
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482 | *qmax* = 0.2 |Ang^-1| and the default values below, where *Ssph/Lsph* stands for smaller or larger sphere, respectively, |
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483 | and *surfrac_Ssph* is the surface fraction of the smaller spheres. |
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484 | |
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485 | ============== ======== ============= |
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486 | Parameter name Units Default value |
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487 | ============== ======== ============= |
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488 | delta_Ssph None 0 |
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489 | radius_Lsph |Ang| 5000 |
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490 | radius_Ssph |Ang| 100 |
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491 | sld_Lsph |Ang^-2| -4e-07 |
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492 | sld_Ssph |Ang^-2| 3.5e-6 |
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493 | sld_solv |Ang^-2| 6.3e-6 |
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494 | surfrac_Ssph None 0.4 |
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495 | volf_Lsph None 0.05 |
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496 | volf_Lsph None 0.005 |
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497 | background |cm^-1| 0 |
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498 | ============== ======== ============= |
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499 | |
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500 | .. image:: ..\img\olddocs\raspberry_plot.jpg |
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501 | |
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502 | *Figure. 1D plot using the values of /2000 data points.* |
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503 | |
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504 | REFERENCE |
---|
505 | |
---|
506 | K Larson-Smith, A Jackson, and D C Pozzo, *Small angle scattering model for Pickering emulsions and raspberry* |
---|
507 | *particles*, *Journal of Colloid and Interface Science*, 343(1) (2010) 36-41 |
---|
508 | |
---|
509 | |
---|
510 | |
---|
511 | .. _CoreShellModel: |
---|
512 | |
---|
513 | **2.1.5. CoreShellModel** |
---|
514 | |
---|
515 | This model provides the form factor, *P(q)*, for a spherical particle with a core-shell structure. The form factor is |
---|
516 | normalized by the particle volume. |
---|
517 | |
---|
518 | For information about polarised and magnetic scattering, click here_. |
---|
519 | |
---|
520 | *2.1.5.1. Definition* |
---|
521 | |
---|
522 | The 1D scattering intensity is calculated in the following way (Guinier, 1955) |
---|
523 | |
---|
524 | .. image:: ..\img\olddocs\image013.PNG |
---|
525 | |
---|
526 | where *scale* is a scale factor, *Vs* is the volume of the outer shell, *Vc* is the volume of the core, *rs* is the |
---|
527 | radius of the shell, *rc* is the radius of the core, *c* is the scattering length density of the core, *s* is the |
---|
528 | scattering length density of the shell, *solv* is the scattering length density of the solvent, and *bkg* is the |
---|
529 | background level. |
---|
530 | |
---|
531 | The 2D scattering intensity is the same as *P(q)* above, regardless of the orientation of the *q* vector. |
---|
532 | |
---|
533 | NB: The outer most radius (ie, = *radius* + *thickness*) is used as the effective radius for *S(Q)* when |
---|
534 | *P(Q)* \* *S(Q)* is applied. |
---|
535 | |
---|
536 | The returned value is scaled to units of |cm^-1| and the parameters of the CoreShellModel are the following |
---|
537 | |
---|
538 | ============== ======== ============= |
---|
539 | Parameter name Units Default value |
---|
540 | ============== ======== ============= |
---|
541 | scale None 1.0 |
---|
542 | (core) radius |Ang| 60 |
---|
543 | thickness |Ang| 10 |
---|
544 | core_sld |Ang^-2| 1e-6 |
---|
545 | shell_sld |Ang^-2| 2e-6 |
---|
546 | solvent_sld |Ang^-2| 3e-6 |
---|
547 | background |cm^-1| 0.001 |
---|
548 | ============== ======== ============= |
---|
549 | |
---|
550 | Here, *radius* = the radius of the core and *thickness* = the thickness of the shell. |
---|
551 | |
---|
552 | Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron |
---|
553 | Research (Kline, 2006). |
---|
554 | |
---|
555 | REFERENCE |
---|
556 | |
---|
557 | A Guinier and G Fournet, *Small-Angle Scattering of X-Rays*, John Wiley and Sons, New York, (1955) |
---|
558 | |
---|
559 | *2.1.5.2. Validation of the core-shell sphere model* |
---|
560 | |
---|
561 | Validation of our code was done by comparing the output of the 1D model to the output of the software provided by |
---|
562 | NIST (Kline, 2006). Figure 1 shows a comparison of the output of our model and the output of the NIST software. |
---|
563 | |
---|
564 | .. image:: ..\img\olddocs\image014.jpg |
---|
565 | |
---|
566 | Figure 1: Comparison of the SasView scattering intensity for a core-shell sphere with the output of the NIST SANS |
---|
567 | analysis software. The parameters were set to: *Scale* = 1.0, *Radius* = 60 , *Contrast* = 1e-6 |Ang^-2|, and |
---|
568 | *Background* = 0.001 |cm^-1|. |
---|
569 | |
---|
570 | |
---|
571 | |
---|
572 | .. _CoreMultiShellModel: |
---|
573 | |
---|
574 | **2.1.6. CoreMultiShellModel** |
---|
575 | |
---|
576 | This model provides the scattering from a spherical core with 1 to 4 concentric shell structures. The SLDs of the core |
---|
577 | and each shell are individually specified. |
---|
578 | |
---|
579 | For information about polarised and magnetic scattering, click here_. |
---|
580 | |
---|
581 | *2.1.6.1. Definition* |
---|
582 | |
---|
583 | This model is a trivial extension of the CoreShell function to a larger number of shells. See the CoreShell function |
---|
584 | for a diagram and documentation. |
---|
585 | |
---|
586 | The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. |
---|
587 | |
---|
588 | Be careful! The SLDs and scale can be highly correlated. Hold as many of these parameters fixed as possible. |
---|
589 | |
---|
590 | The 2D scattering intensity is the same as P(q) of 1D, regardless of the orientation of the q vector. |
---|
591 | |
---|
592 | NB: The outer most radius (ie, = *radius* + 4 *thicknesses*) is used as the effective radius for *S(Q)* when |
---|
593 | *P(Q)* \* *S(Q)* is applied. |
---|
594 | |
---|
595 | The returned value is scaled to units of |cm^-1| and the parameters of the CoreMultiShell model are the following |
---|
596 | |
---|
597 | ============== ======== ============= |
---|
598 | Parameter name Units Default value |
---|
599 | ============== ======== ============= |
---|
600 | scale None 1.0 |
---|
601 | rad_core |Ang| 60 |
---|
602 | sld_core |Ang^-2| 6.4e-6 |
---|
603 | sld_shell1 |Ang^-2| 1e-6 |
---|
604 | sld_shell2 |Ang^-2| 2e-6 |
---|
605 | sld_shell3 |Ang^-2| 3e-6 |
---|
606 | sld_shell4 |Ang^-2| 4e-6 |
---|
607 | sld_solv |Ang^-2| 6.4e-6 |
---|
608 | thick_shell1 |Ang| 10 |
---|
609 | thick_shell2 |Ang| 10 |
---|
610 | thick_shell3 |Ang| 10 |
---|
611 | thick_shell4 |Ang| 10 |
---|
612 | background |cm^-1| 0.001 |
---|
613 | ============== ======== ============= |
---|
614 | |
---|
615 | NB: Here, *rad_core* = the radius of the core, *thick_shelli* = the thickness of the shell *i* and |
---|
616 | *sld_shelli* = the SLD of the shell *i*. *sld_core* and the *sld_solv* are the SLD of the core and the solvent, |
---|
617 | respectively. |
---|
618 | |
---|
619 | Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron |
---|
620 | Research (Kline, 2006). |
---|
621 | |
---|
622 | This example dataset is produced by running the CoreMultiShellModel using 200 data points, *qmin* = 0.001 -1, |
---|
623 | *qmax* = 0.7 -1 and the above default values. |
---|
624 | |
---|
625 | .. image:: ..\img\olddocs\image015.jpg |
---|
626 | |
---|
627 | *Figure: 1D plot using the default values (w/200 data point).* |
---|
628 | |
---|
629 | The scattering length density profile for the default sld values (w/ 4 shells). |
---|
630 | |
---|
631 | .. image:: ..\img\olddocs\image016.jpg |
---|
632 | |
---|
633 | *Figure: SLD profile against the radius of the sphere for default SLDs.* |
---|
634 | |
---|
635 | REFERENCE |
---|
636 | |
---|
637 | See the CoreShellModel_ documentation. |
---|
638 | |
---|
639 | |
---|
640 | |
---|
641 | .. _Core2ndMomentModel: |
---|
642 | |
---|
643 | **2.1.7. Core2ndMomentModel** |
---|
644 | |
---|
645 | This model describes the scattering from a layer of surfactant or polymer adsorbed on spherical particles under the |
---|
646 | conditions that (i) the particles (cores) are contrast-matched to the dispersion medium, (ii) *S(Q)* ~ 1 (ie, the |
---|
647 | particle volume fraction is dilute), (iii) the particle radius is >> layer thickness (ie, the interface is locally |
---|
648 | flat), and (iv) scattering from excess unadsorbed adsorbate in the bulk medium is absent or has been corrected for. |
---|
649 | |
---|
650 | Unlike a core-shell model, this model does not assume any form for the density distribution of the adsorbed species |
---|
651 | normal to the interface (cf, a core-shell model which assumes the density distribution to be a homogeneous |
---|
652 | step-function). For comparison, if the thickness of a (core-shell like) step function distribution is *t*, the second |
---|
653 | moment, |sigma| = sqrt((*t* :sup:`2` )/12). The |sigma| is the second moment about the mean of the density distribution |
---|
654 | (ie, the distance of the centre-of-mass of the distribution from the interface). |
---|
655 | |
---|
656 | *2.1.7.1. Definition* |
---|
657 | |
---|
658 | The *I* :sub:`0` is calculated in the following way (King, 2002) |
---|
659 | |
---|
660 | .. image:: ..\img\olddocs\secondmeq1.jpg |
---|
661 | |
---|
662 | where *scale* is a scale factor, *poly* is the sld of the polymer (or surfactant) layer, *solv* is the sld of the |
---|
663 | solvent/medium and cores, |phi|\ :sub:`cores` is the volume fraction of the core paraticles, and |biggamma| and |
---|
664 | |delta| are the adsorbed amount and the bulk density of the polymers respectively. The |sigma| is the second moment |
---|
665 | of the thickness distribution. |
---|
666 | |
---|
667 | Note that all parameters except the |sigma| are correlated for fitting so that fitting those with more than one |
---|
668 | parameter will generally fail. Also note that unlike other shape models, no volume normalization is applied to this |
---|
669 | model (the calculation is exact). |
---|
670 | |
---|
671 | The returned value is scaled to units of |cm^-1| and the parameters are the following |
---|
672 | |
---|
673 | ============== ======== ============= |
---|
674 | Parameter name Units Default value |
---|
675 | ============== ======== ============= |
---|
676 | scale None 1.0 |
---|
677 | density_poly g/cm2 0.7 |
---|
678 | radius_core |Ang| 500 |
---|
679 | ads_amount mg/m 2 1.9 |
---|
680 | second_moment |Ang| 23.0 |
---|
681 | volf_cores None 0.14 |
---|
682 | sld_poly |Ang^-2| 1.5e-6 |
---|
683 | sld_solv |Ang^-2| 6.3e-6 |
---|
684 | background |cm^-1| 0.0 |
---|
685 | ============== ======== ============= |
---|
686 | |
---|
687 | .. image:: ..\img\olddocs\secongm_fig1.jpg |
---|
688 | |
---|
689 | REFERENCE |
---|
690 | |
---|
691 | S King, P Griffiths, J. Hone, and T Cosgrove, *SANS from Adsorbed Polymer Layers*, |
---|
692 | *Macromol. Symp.*, 190 (2002) 33-42 |
---|
693 | |
---|
694 | |
---|
695 | |
---|
696 | .. _MultiShellModel: |
---|
697 | |
---|
698 | **2.1.8. MultiShellModel** |
---|
699 | |
---|
700 | This model provides the form factor, *P(q)*, for a multi-lamellar vesicle with *N* shells where the core is filled with |
---|
701 | solvent and the shells are interleaved with layers of solvent. For *N* = 1, this returns the VesicleModel (above). |
---|
702 | |
---|
703 | .. image:: ..\img\olddocs\image020.jpg |
---|
704 | |
---|
705 | The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as |
---|
706 | |
---|
707 | .. image:: ..\img\olddocs\image008.PNG |
---|
708 | |
---|
709 | NB: The outer most radius (= *core_radius* + *n_pairs* \* *s_thickness* + (*n_pairs* - 1) \* *w_thickness*) is used |
---|
710 | as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. |
---|
711 | |
---|
712 | The returned value is scaled to units of |cm^-1| and the parameters of the MultiShellModel are the following |
---|
713 | |
---|
714 | ============== ======== ============= |
---|
715 | Parameter name Units Default value |
---|
716 | ============== ======== ============= |
---|
717 | scale None 1.0 |
---|
718 | core_radius |Ang| 60.0 |
---|
719 | n_pairs None 2.0 |
---|
720 | core_sld |Ang^-2| 6.3e-6 |
---|
721 | shell_sld |Ang^-2| 0.0 |
---|
722 | background |cm^-1| 0.0 |
---|
723 | s_thickness |Ang| 10 |
---|
724 | w_thickness |Ang| 10 |
---|
725 | ============== ======== ============= |
---|
726 | |
---|
727 | NB: *s_thickness* is the shell thickness while the *w_thickness* is the solvent thickness, and *n_pair* |
---|
728 | is the number of shells. |
---|
729 | |
---|
730 | .. image:: ..\img\olddocs\image021.jpg |
---|
731 | |
---|
732 | *Figure. 1D plot using the default values (w/200 data point).* |
---|
733 | |
---|
734 | Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron |
---|
735 | Research (Kline, 2006). |
---|
736 | |
---|
737 | REFERENCE |
---|
738 | |
---|
739 | B Cabane, *Small Angle Scattering Methods*, in *Surfactant Solutions: New Methods of Investigation*, Ch.2, |
---|
740 | Surfactant Science Series Vol. 22, Ed. R Zana and M Dekker, New York, (1987). |
---|
741 | |
---|
742 | |
---|
743 | |
---|
744 | .. _OnionExpShellModel: |
---|
745 | |
---|
746 | **2.1.9. OnionExpShellModel** |
---|
747 | |
---|
748 | This model provides the form factor, *P(q)*, for a multi-shell sphere where the scattering length density (SLD) of the |
---|
749 | each shell is described by an exponential (linear, or flat-top) function. The form factor is normalized by the volume |
---|
750 | of the sphere where the SLD is not identical to the SLD of the solvent. We currently provide up to 9 shells with this |
---|
751 | model. |
---|
752 | |
---|
753 | *2.1.9.1. Definition* |
---|
754 | |
---|
755 | The 1D scattering intensity is calculated in the following way |
---|
756 | |
---|
757 | .. image:: ..\img\olddocs\image022.gif |
---|
758 | |
---|
759 | .. image:: ..\img\olddocs\image023.gif |
---|
760 | |
---|
761 | where, for a spherically symmetric particle with a particle density |rho|\ *(r)* |
---|
762 | |
---|
763 | .. image:: ..\img\olddocs\image024.gif |
---|
764 | |
---|
765 | so that |
---|
766 | |
---|
767 | .. image:: ..\img\olddocs\image025.gif |
---|
768 | |
---|
769 | .. image:: ..\img\olddocs\image026.gif |
---|
770 | |
---|
771 | .. image:: ..\img\olddocs\image027.gif |
---|
772 | |
---|
773 | Here we assumed that the SLDs of the core and solvent are constant against *r*. |
---|
774 | |
---|
775 | Now lets consider the SLD of a shell, *r*\ :sub:`shelli`, defined by |
---|
776 | |
---|
777 | .. image:: ..\img\olddocs\image028.gif |
---|
778 | |
---|
779 | An example of a possible SLD profile is shown below where *sld_in_shelli* (|rho|\ :sub:`in`\ ) and |
---|
780 | *thick_shelli* (|bigdelta|\ *t* :sub:`shelli`\ ) stand for the SLD of the inner side of the *i*\ th shell and the |
---|
781 | thickness of the *i*\ th shell in the equation above, respectively. |
---|
782 | |
---|
783 | For \| *A* \| > 0, |
---|
784 | |
---|
785 | .. image:: ..\img\olddocs\image029.gif |
---|
786 | |
---|
787 | For *A* ~ 0 (eg., *A* = -0.0001), this function converges to that of the linear SLD profile (ie, |
---|
788 | |rho|\ :sub:`shelli`\ *(r)* = *A*\ :sup:`'` ( *r* - *r*\ :sub:`shelli` - 1) / |bigdelta|\ *t* :sub:`shelli`) + *B*\ :sup:`'`), |
---|
789 | so this case is equivalent to |
---|
790 | |
---|
791 | .. image:: ..\img\olddocs\image030.gif |
---|
792 | |
---|
793 | .. image:: ..\img\olddocs\image031.gif |
---|
794 | |
---|
795 | .. image:: ..\img\olddocs\image032.gif |
---|
796 | |
---|
797 | .. image:: ..\img\olddocs\image033.gif |
---|
798 | |
---|
799 | For *A* = 0, the exponential function has no dependence on the radius (so that *sld_out_shell* (|rho|\ :sub:`out`) is |
---|
800 | ignored this case) and becomes flat. We set the constant to |rho|\ :sub:`in` for convenience, and thus the form |
---|
801 | factor contributed by the shells is |
---|
802 | |
---|
803 | .. image:: ..\img\olddocs\image034.gif |
---|
804 | |
---|
805 | .. image:: ..\img\olddocs\image035.gif |
---|
806 | |
---|
807 | In the equation |
---|
808 | |
---|
809 | .. image:: ..\img\olddocs\image036.gif |
---|
810 | |
---|
811 | Finally, the form factor can be calculated by |
---|
812 | |
---|
813 | .. image:: ..\img\olddocs\image037.gif |
---|
814 | |
---|
815 | where |
---|
816 | |
---|
817 | .. image:: ..\img\olddocs\image038.gif |
---|
818 | |
---|
819 | and |
---|
820 | |
---|
821 | .. image:: ..\img\olddocs\image039.gif |
---|
822 | |
---|
823 | The 2D scattering intensity is the same as *P(q)* above, regardless of the orientation of the *q* vector which is |
---|
824 | defined as |
---|
825 | |
---|
826 | .. image:: ..\img\olddocs\image040.gif |
---|
827 | |
---|
828 | NB: The outer most radius is used as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. |
---|
829 | |
---|
830 | The returned value is scaled to units of |cm^-1| and the parameters of this model (for only one shell) are the following |
---|
831 | |
---|
832 | ============== ======== ============= |
---|
833 | Parameter name Units Default value |
---|
834 | ============== ======== ============= |
---|
835 | A_shell1 None 1 |
---|
836 | scale None 1.0 |
---|
837 | rad_core |Ang| 200 |
---|
838 | thick_shell1 |Ang| 50 |
---|
839 | sld_core |Ang^-2| 1.0e-06 |
---|
840 | sld_in_shell1 |Ang^-2| 1.7e-06 |
---|
841 | sld_out_shell1 |Ang^-2| 2.0e-06 |
---|
842 | sld_solv |Ang^-2| 6.4e-06 |
---|
843 | background |cm^-1| 0.0 |
---|
844 | ============== ======== ============= |
---|
845 | |
---|
846 | NB: *rad_core* represents the core radius (*R1*) and *thick_shell1* (*R2* - *R1*) is the thickness of the shell1, etc. |
---|
847 | |
---|
848 | .. image:: ..\img\olddocs\image041.jpg |
---|
849 | |
---|
850 | *Figure. 1D plot using the default values (w/400 point).* |
---|
851 | |
---|
852 | .. image:: ..\img\olddocs\image042.jpg |
---|
853 | |
---|
854 | *Figure. SLD profile from the default values.* |
---|
855 | |
---|
856 | REFERENCE |
---|
857 | |
---|
858 | L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, |
---|
859 | Plenum Press, New York, (1987). |
---|
860 | |
---|
861 | |
---|
862 | |
---|
863 | .. _VesicleModel: |
---|
864 | |
---|
865 | **2.1.10. VesicleModel** |
---|
866 | |
---|
867 | This model provides the form factor, *P(q)*, for an unilamellar vesicle. The form factor is normalized by the volume |
---|
868 | of the shell. |
---|
869 | |
---|
870 | *2.1.10.1. Definition* |
---|
871 | |
---|
872 | The 1D scattering intensity is calculated in the following way (Guinier, 1955) |
---|
873 | |
---|
874 | .. image:: ..\img\olddocs\image017.PNG |
---|
875 | |
---|
876 | where *scale* is a scale factor, *Vshell* is the volume of the shell, *V1* is the volume of the core, *V2* is the total |
---|
877 | volume, *R1* is the radius of the core, *R2* is the outer radius of the shell, |rho|\ :sub:`1` is the scattering |
---|
878 | length density of the core and the solvent, |rho|\ :sub:`2` is the scattering length density of the shell, *bkg* is |
---|
879 | the background level, and *J1* = (sin\ *x*- *x* cos\ *x*)/ *x* :sup:`2`\ . The functional form is identical to a |
---|
880 | "typical" core-shell structure, except that the scattering is normalized by the volume that is contributing to the |
---|
881 | scattering, namely the volume of the shell alone. Also, the vesicle is best defined in terms of a core radius (= *R1*) |
---|
882 | and a shell thickness, *t*. |
---|
883 | |
---|
884 | .. image:: ..\img\olddocs\image018.jpg |
---|
885 | |
---|
886 | The 2D scattering intensity is the same as *P(q)* above, regardless of the orientation of the *q* vector which is |
---|
887 | defined as |
---|
888 | |
---|
889 | .. image:: ..\img\olddocs\image008.PNG |
---|
890 | |
---|
891 | NB: The outer most radius (= *radius* + *thickness*) is used as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* |
---|
892 | is applied. |
---|
893 | |
---|
894 | The returned value is scaled to units of |cm^-1| and the parameters of the VesicleModel are the following |
---|
895 | |
---|
896 | ============== ======== ============= |
---|
897 | Parameter name Units Default value |
---|
898 | ============== ======== ============= |
---|
899 | scale None 1.0 |
---|
900 | radius |Ang| 100 |
---|
901 | thickness |Ang| 30 |
---|
902 | core_sld |Ang^-2| 6.3e-6 |
---|
903 | shell_sld |Ang^-2| 0 |
---|
904 | background |cm^-1| 0.0 |
---|
905 | ============== ======== ============= |
---|
906 | |
---|
907 | NB: *radius* represents the core radius (*R1*) and the *thickness* (*R2* - *R1*) is the shell thickness. |
---|
908 | |
---|
909 | .. image:: ..\img\olddocs\image019.jpg |
---|
910 | |
---|
911 | *Figure. 1D plot using the default values (w/200 data point).* |
---|
912 | |
---|
913 | Our model uses the form factor calculations implemented in a c-library |
---|
914 | provided by the NIST Center for Neutron Research (Kline, 2006). |
---|
915 | |
---|
916 | REFERENCE |
---|
917 | |
---|
918 | A Guinier and G. Fournet, *Small-Angle Scattering of X-Rays*, John Wiley and Sons, New York, (1955) |
---|
919 | |
---|
920 | |
---|
921 | |
---|
922 | .. _SphericalSLDModel: |
---|
923 | |
---|
924 | **2.1.11. SphericalSLDModel** |
---|
925 | |
---|
926 | Similarly to the OnionExpShellModel, this model provides the form factor, *P(q)*, for a multi-shell sphere, where the |
---|
927 | interface between the each neighboring shells can be described by one of a number of functions including error, |
---|
928 | power-law, and exponential functions. This model is to calculate the scattering intensity by building a continuous |
---|
929 | custom SLD profile against the radius of the particle. The SLD profile is composed of a flat core, a flat solvent, |
---|
930 | a number (up to 9 ) flat shells, and the interfacial layers between the adjacent flat shells (or core, and solvent) |
---|
931 | (see below). Unlike the OnionExpShellModel (using an analytical integration), the interfacial layers here are |
---|
932 | sub-divided and numerically integrated assuming each of the sub-layers are described by a line function. The number |
---|
933 | of the sub-layer can be given by users by setting the integer values of *npts_inter* in the GUI. The form factor is |
---|
934 | normalized by the total volume of the sphere. |
---|
935 | |
---|
936 | *2.1.11.1. Definition* |
---|
937 | |
---|
938 | The 1D scattering intensity is calculated in the following way: |
---|
939 | |
---|
940 | .. image:: ..\img\olddocs\image022.gif |
---|
941 | |
---|
942 | .. image:: ..\img\olddocs\image043.gif |
---|
943 | |
---|
944 | where, for a spherically symmetric particle with a particle density |rho|\ *(r)* |
---|
945 | |
---|
946 | .. image:: ..\img\olddocs\image024.gif |
---|
947 | |
---|
948 | so that |
---|
949 | |
---|
950 | .. image:: ..\img\olddocs\image044.gif |
---|
951 | |
---|
952 | .. image:: ..\img\olddocs\image045.gif |
---|
953 | |
---|
954 | .. image:: ..\img\olddocs\image046.gif |
---|
955 | |
---|
956 | .. image:: ..\img\olddocs\image047.gif |
---|
957 | |
---|
958 | .. image:: ..\img\olddocs\image048.gif |
---|
959 | |
---|
960 | .. image:: ..\img\olddocs\image027.gif |
---|
961 | |
---|
962 | Here we assumed that the SLDs of the core and solvent are constant against *r*. The SLD at the interface between |
---|
963 | shells, |rho|\ :sub:`inter_i`, is calculated with a function chosen by an user, where the functions are |
---|
964 | |
---|
965 | 1) Exp |
---|
966 | |
---|
967 | .. image:: ..\img\olddocs\image049.gif |
---|
968 | |
---|
969 | 2) Power-Law |
---|
970 | |
---|
971 | .. image:: ..\img\olddocs\image050.gif |
---|
972 | |
---|
973 | 3) Erf |
---|
974 | |
---|
975 | .. image:: ..\img\olddocs\image051.gif |
---|
976 | |
---|
977 | The functions are normalized so that they vary between 0 and 1, and they are constrained such that the SLD is |
---|
978 | continuous at the boundaries of the interface as well as each sub-layers. Thus *B* and *C* are determined. |
---|
979 | |
---|
980 | Once |rho|\ :sub:`rinter_i` is found at the boundary of the sub-layer of the interface, we can find its contribution |
---|
981 | to the form factor *P(q)* |
---|
982 | |
---|
983 | .. image:: ..\img\olddocs\image052.gif |
---|
984 | |
---|
985 | .. image:: ..\img\olddocs\image053.gif |
---|
986 | |
---|
987 | .. image:: ..\img\olddocs\image054.gif |
---|
988 | |
---|
989 | where we assume that |rho|\ :sub:`inter_i`\ *(r)* can be approximately linear within a sub-layer *j*. |
---|
990 | |
---|
991 | In the equation |
---|
992 | |
---|
993 | .. image:: ..\img\olddocs\image055.gif |
---|
994 | |
---|
995 | Finally, the form factor can be calculated by |
---|
996 | |
---|
997 | .. image:: ..\img\olddocs\image037.gif |
---|
998 | |
---|
999 | where |
---|
1000 | |
---|
1001 | .. image:: ..\img\olddocs\image038.gif |
---|
1002 | |
---|
1003 | and |
---|
1004 | |
---|
1005 | .. image:: ..\img\olddocs\image056.gif |
---|
1006 | |
---|
1007 | The 2D scattering intensity is the same as *P(q)* above, regardless of the orientation of the *q* vector which is |
---|
1008 | defined as |
---|
1009 | |
---|
1010 | .. image:: ..\img\olddocs\image040.gif |
---|
1011 | |
---|
1012 | NB: The outer most radius is used as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. |
---|
1013 | |
---|
1014 | The returned value is scaled to units of |cm^-1| and the parameters of this model (for just one shell) are the following |
---|
1015 | |
---|
1016 | ============== ======== ============= |
---|
1017 | Parameter name Units Default value |
---|
1018 | ============== ======== ============= |
---|
1019 | background |cm^-1| 0.0 |
---|
1020 | npts_inter None 35 |
---|
1021 | scale None 1 |
---|
1022 | sld_solv |Ang^-2| 1e-006 |
---|
1023 | func_inter1 None Erf |
---|
1024 | nu_inter None 2.5 |
---|
1025 | thick_inter1 |Ang| 50 |
---|
1026 | sld_flat1 |Ang^-2| 4e-006 |
---|
1027 | thick_flat1 |Ang| 100 |
---|
1028 | func_inter0 None Erf |
---|
1029 | nu_inter0 None 2.5 |
---|
1030 | rad_core0 |Ang| 50 |
---|
1031 | sld_core0 |Ang^-2| 2.07e-06 |
---|
1032 | thick_core0 |Ang| 50 |
---|
1033 | ============== ======== ============= |
---|
1034 | |
---|
1035 | NB: *rad_core0* represents the core radius (*R1*). |
---|
1036 | |
---|
1037 | .. image:: ..\img\olddocs\image057.jpg |
---|
1038 | |
---|
1039 | *Figure. 1D plot using the default values (w/400 point).* |
---|
1040 | |
---|
1041 | .. image:: ..\img\olddocs\image058.jpg |
---|
1042 | |
---|
1043 | *Figure. SLD profile from the default values.* |
---|
1044 | |
---|
1045 | REFERENCE |
---|
1046 | |
---|
1047 | L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, |
---|
1048 | Plenum Press, New York, (1987) |
---|
1049 | |
---|
1050 | |
---|
1051 | |
---|
1052 | .. _LinearPearlsModel: |
---|
1053 | |
---|
1054 | **2.1.12. LinearPearlsModel** |
---|
1055 | |
---|
1056 | This model provides the form factor for *N* spherical pearls of radius *R* linearly joined by short strings (or segment |
---|
1057 | length or edge separation) *l* (= *A* - 2\ *R*)). *A* is the center-to-center pearl separation distance. The thickness |
---|
1058 | of each string is assumed to be negligible. |
---|
1059 | |
---|
1060 | .. image:: ..\img\olddocs\linearpearls.jpg |
---|
1061 | |
---|
1062 | *2.1.12.1. Definition* |
---|
1063 | |
---|
1064 | The output of the scattering intensity function for the LinearPearlsModel is given by (Dobrynin, 1996) |
---|
1065 | |
---|
1066 | .. image:: ..\img\olddocs\linearpearl_eq1.gif |
---|
1067 | |
---|
1068 | where the mass *m*\ :sub:`p` is (SLD\ :sub:`pearl` - SLD\ :sub:`solvent`) \* (volume of *N* pearls). V is the total |
---|
1069 | volume. |
---|
1070 | |
---|
1071 | The 2D scattering intensity is the same as *P(q)* above, regardless of the orientation of the *q* vector. |
---|
1072 | |
---|
1073 | The returned value is scaled to units of |cm^-1| and the parameters of the LinearPearlsModel are the following |
---|
1074 | |
---|
1075 | =============== ======== ============= |
---|
1076 | Parameter name Units Default value |
---|
1077 | =============== ======== ============= |
---|
1078 | scale None 1.0 |
---|
1079 | radius |Ang| 80.0 |
---|
1080 | edge_separation |Ang| 350.0 |
---|
1081 | num_pearls None 3 |
---|
1082 | sld_pearl |Ang^-2| 1e-6 |
---|
1083 | sld_solv |Ang^-2| 6.3e-6 |
---|
1084 | background |cm^-1| 0.0 |
---|
1085 | =============== ======== ============= |
---|
1086 | |
---|
1087 | NB: *num_pearls* must be an integer. |
---|
1088 | |
---|
1089 | .. image:: ..\img\olddocs\linearpearl_plot.jpg |
---|
1090 | |
---|
1091 | REFERENCE |
---|
1092 | |
---|
1093 | A V Dobrynin, M Rubinstein and S P Obukhov, *Macromol.*, 29 (1996) 2974-2979 |
---|
1094 | |
---|
1095 | |
---|
1096 | |
---|
1097 | .. _PearlNecklaceModel: |
---|
1098 | |
---|
1099 | **2.1.13. PearlNecklaceModel** |
---|
1100 | |
---|
1101 | This model provides the form factor for a pearl necklace composed of two elements: *N* pearls (homogeneous spheres |
---|
1102 | of radius *R*) freely jointed by *M* rods (like strings - with a total mass *Mw* = *M* \* *m*\ :sub:`r` + *N* \* *m*\ :sub:`s`, |
---|
1103 | and the string segment length (or edge separation) *l* (= *A* - 2\ *R*)). *A* is the center-to-center pearl separation |
---|
1104 | distance. |
---|
1105 | |
---|
1106 | .. image:: ..\img\olddocs\pearl_fig.jpg |
---|
1107 | |
---|
1108 | *2.1.13.1. Definition* |
---|
1109 | |
---|
1110 | The output of the scattering intensity function for the PearlNecklaceModel is given by (Schweins, 2004) |
---|
1111 | |
---|
1112 | .. image:: ..\img\olddocs\pearl_eq1.gif |
---|
1113 | |
---|
1114 | where |
---|
1115 | |
---|
1116 | .. image:: ..\img\olddocs\pearl_eq2.gif |
---|
1117 | |
---|
1118 | .. image:: ..\img\olddocs\pearl_eq3.gif |
---|
1119 | |
---|
1120 | .. image:: ..\img\olddocs\pearl_eq4.gif |
---|
1121 | |
---|
1122 | .. image:: ..\img\olddocs\pearl_eq5.gif |
---|
1123 | |
---|
1124 | .. image:: ..\img\olddocs\pearl_eq6.gif |
---|
1125 | |
---|
1126 | and |
---|
1127 | |
---|
1128 | .. image:: ..\img\olddocs\pearl_eq7.gif |
---|
1129 | |
---|
1130 | where the mass *m*\ :sub:`i` is (SLD\ :sub:`i` - SLD\ :sub:`solvent`) \* (volume of the *N* pearls/rods). *V* is the |
---|
1131 | total volume of the necklace. |
---|
1132 | |
---|
1133 | The 2D scattering intensity is the same as *P(q)* above, regardless of the orientation of the *q* vector. |
---|
1134 | |
---|
1135 | The returned value is scaled to units of |cm^-1| and the parameters of the PearlNecklaceModel are the following |
---|
1136 | |
---|
1137 | =============== ======== ============= |
---|
1138 | Parameter name Units Default value |
---|
1139 | =============== ======== ============= |
---|
1140 | scale None 1.0 |
---|
1141 | radius |Ang| 80.0 |
---|
1142 | edge_separation |Ang| 350.0 |
---|
1143 | num_pearls None 3 |
---|
1144 | sld_pearl |Ang^-2| 1e-6 |
---|
1145 | sld_solv |Ang^-2| 6.3e-6 |
---|
1146 | sld_string |Ang^-2| 1e-6 |
---|
1147 | thick_string |
---|
1148 | (=rod diameter) |Ang| 2.5 |
---|
1149 | background |cm^-1| 0.0 |
---|
1150 | =============== ======== ============= |
---|
1151 | |
---|
1152 | NB: *num_pearls* must be an integer. |
---|
1153 | |
---|
1154 | .. image:: ..\img\olddocs\pearl_plot.jpg |
---|
1155 | |
---|
1156 | REFERENCE |
---|
1157 | |
---|
1158 | R Schweins and K Huber, *Particle Scattering Factor of Pearl Necklace Chains*, *Macromol. Symp.* 211 (2004) 25-42 2004 |
---|
1159 | |
---|
1160 | |
---|
1161 | |
---|
1162 | .. _CylinderModel: |
---|
1163 | |
---|
1164 | **2.1.14. CylinderModel** |
---|
1165 | |
---|
1166 | This model provides the form factor for a right circular cylinder with uniform scattering length density. The form |
---|
1167 | factor is normalized by the particle volume. |
---|
1168 | |
---|
1169 | For information about polarised and magnetic scattering, click here_. |
---|
1170 | |
---|
1171 | *2.1.14.1. Definition* |
---|
1172 | |
---|
1173 | The output of the 2D scattering intensity function for oriented cylinders is given by (Guinier, 1955) |
---|
1174 | |
---|
1175 | .. image:: ..\img\olddocs\image059.PNG |
---|
1176 | |
---|
1177 | where |
---|
1178 | |
---|
1179 | .. image:: ..\img\olddocs\image060.PNG |
---|
1180 | |
---|
1181 | and |alpha| is the angle between the axis of the cylinder and the *q*-vector, *V* is the volume of the cylinder, |
---|
1182 | *L* is the length of the cylinder, *r* is the radius of the cylinder, and |drho| (contrast) is the |
---|
1183 | scattering length density difference between the scatterer and the solvent. *J1* is the first order Bessel function. |
---|
1184 | |
---|
1185 | To provide easy access to the orientation of the cylinder, we define the axis of the cylinder using two angles |theta| |
---|
1186 | and |phi|. Those angles are defined in Figure 1. |
---|
1187 | |
---|
1188 | .. image:: ..\img\olddocs\image061.jpg |
---|
1189 | |
---|
1190 | *Figure 1. Definition of the angles for oriented cylinders.* |
---|
1191 | |
---|
1192 | .. image:: ..\img\olddocs\image062.jpg |
---|
1193 | |
---|
1194 | *Figure 2. Examples of the angles for oriented pp against the detector plane.* |
---|
1195 | |
---|
1196 | NB: The 2nd virial coefficient of the cylinder is calculated based on the radius and length values, and used as the |
---|
1197 | effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. |
---|
1198 | |
---|
1199 | The returned value is scaled to units of |cm^-1| and the parameters of the CylinderModel are the following: |
---|
1200 | |
---|
1201 | ============== ======== ============= |
---|
1202 | Parameter name Units Default value |
---|
1203 | ============== ======== ============= |
---|
1204 | scale None 1.0 |
---|
1205 | radius |Ang| 20.0 |
---|
1206 | length |Ang| 400.0 |
---|
1207 | contrast |Ang^-2| 3.0e-6 |
---|
1208 | background |cm^-1| 0.0 |
---|
1209 | cyl_theta degree 60 |
---|
1210 | cyl_phi degree 60 |
---|
1211 | ============== ======== ============= |
---|
1212 | |
---|
1213 | The output of the 1D scattering intensity function for randomly oriented cylinders is then given by |
---|
1214 | |
---|
1215 | .. image:: ..\img\olddocs\image063.PNG |
---|
1216 | |
---|
1217 | The *cyl_theta* and *cyl_phi* parameter are not used for the 1D output. Our implementation of the scattering kernel |
---|
1218 | and the 1D scattering intensity use the c-library from NIST. |
---|
1219 | |
---|
1220 | *2.1.14.2. Validation of the CylinderModel* |
---|
1221 | |
---|
1222 | Validation of our code was done by comparing the output of the 1D model to the output of the software provided by the |
---|
1223 | NIST (Kline, 2006). Figure 3 shows a comparison of the 1D output of our model and the output of the NIST software. |
---|
1224 | |
---|
1225 | .. image:: ..\img\olddocs\image065.jpg |
---|
1226 | |
---|
1227 | *Figure 3: Comparison of the SasView scattering intensity for a cylinder with the output of the NIST SANS analysis* |
---|
1228 | *software.* The parameters were set to: *Scale* = 1.0, *Radius* = 20 |Ang|, *Length* = 400 |Ang|, |
---|
1229 | *Contrast* = 3e-6 |Ang^-2|, and *Background* = 0.01 |cm^-1|. |
---|
1230 | |
---|
1231 | In general, averaging over a distribution of orientations is done by evaluating the following |
---|
1232 | |
---|
1233 | .. image:: ..\img\olddocs\image064.PNG |
---|
1234 | |
---|
1235 | where *p(*\ |theta|,\ |phi|\ *)* is the probability distribution for the orientation and |P0|\ *(q,*\ |alpha|\ *)* is |
---|
1236 | the scattering intensity for the fully oriented system. Since we have no other software to compare the implementation |
---|
1237 | of the intensity for fully oriented cylinders, we can compare the result of averaging our 2D output using a uniform |
---|
1238 | distribution *p(*\ |theta|,\ |phi|\ *)* = 1.0. Figure 4 shows the result of such a cross-check. |
---|
1239 | |
---|
1240 | .. image:: ..\img\olddocs\image066.jpg |
---|
1241 | |
---|
1242 | *Figure 4: Comparison of the intensity for uniformly distributed cylinders calculated from our 2D model and the* |
---|
1243 | *intensity from the NIST SANS analysis software.* The parameters used were: *Scale* = 1.0, *Radius* = 20 |Ang|, |
---|
1244 | *Length* = 400 |Ang|, *Contrast* = 3e-6 |Ang^-2|, and *Background* = 0.0 |cm^-1|. |
---|
1245 | |
---|
1246 | |
---|
1247 | |
---|
1248 | .. _HollowCylinderModel: |
---|
1249 | |
---|
1250 | **2.1.15. HollowCylinderModel** |
---|
1251 | |
---|
1252 | This model provides the form factor, *P(q)*, for a monodisperse hollow right angle circular cylinder (tube) where the |
---|
1253 | form factor is normalized by the volume of the tube |
---|
1254 | |
---|
1255 | *P(q)* = *scale* \* *<F*\ :sup:`2`\ *>* / *V*\ :sub:`shell` + *background* |
---|
1256 | |
---|
1257 | where the averaging < > is applied only for the 1D calculation. |
---|
1258 | |
---|
1259 | The inside and outside of the hollow cylinder are assumed have the same SLD. |
---|
1260 | |
---|
1261 | *2.1.15.1 Definition* |
---|
1262 | |
---|
1263 | The 1D scattering intensity is calculated in the following way (Guinier, 1955) |
---|
1264 | |
---|
1265 | .. image:: ..\img\olddocs\image072.PNG |
---|
1266 | |
---|
1267 | where *scale* is a scale factor, *J1* is the 1st order Bessel function, *J1(x)* = (sin *x* - *x* cos *x*)/ *x*\ :sup:`2`. |
---|
1268 | |
---|
1269 | To provide easy access to the orientation of the core-shell cylinder, we define the axis of the cylinder using two |
---|
1270 | angles |theta| and |phi|\ . As for the case of the cylinder, those angles are defined in Figure 2 of the CylinderModel. |
---|
1271 | |
---|
1272 | NB: The 2nd virial coefficient of the cylinder is calculated based on the radius and 2 length values, and used as the |
---|
1273 | effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. |
---|
1274 | |
---|
1275 | In the parameters, the contrast represents SLD :sub:`shell` - SLD :sub:`solvent` and the *radius* = *R*\ :sub:`shell` |
---|
1276 | while *core_radius* = *R*\ :sub:`core`. |
---|
1277 | |
---|
1278 | ============== ======== ============= |
---|
1279 | Parameter name Units Default value |
---|
1280 | ============== ======== ============= |
---|
1281 | scale None 1.0 |
---|
1282 | radius |Ang| 30 |
---|
1283 | length |Ang| 400 |
---|
1284 | core_radius |Ang| 20 |
---|
1285 | sldCyl |Ang^-2| 6.3e-6 |
---|
1286 | sldSolv |Ang^-2| 5e-06 |
---|
1287 | background |cm^-1| 0.01 |
---|
1288 | ============== ======== ============= |
---|
1289 | |
---|
1290 | .. image:: ..\img\olddocs\image074.jpg |
---|
1291 | |
---|
1292 | *Figure. 1D plot using the default values (w/1000 data point).* |
---|
1293 | |
---|
1294 | Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research |
---|
1295 | (Kline, 2006). |
---|
1296 | |
---|
1297 | .. image:: ..\img\olddocs\image061.jpg |
---|
1298 | |
---|
1299 | *Figure. Definition of the angles for the oriented HollowCylinderModel.* |
---|
1300 | |
---|
1301 | .. image:: ..\img\olddocs\image062.jpg |
---|
1302 | |
---|
1303 | *Figure. Examples of the angles for oriented pp against the detector plane.* |
---|
1304 | |
---|
1305 | REFERENCE |
---|
1306 | |
---|
1307 | L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum Press, |
---|
1308 | New York, (1987) |
---|
1309 | |
---|
1310 | |
---|
1311 | |
---|
1312 | .. _CappedCylinderModel: |
---|
1313 | |
---|
1314 | **2.1.16 CappedCylinderModel** |
---|
1315 | |
---|
1316 | Calculates the scattering from a cylinder with spherical section end-caps. This model simply becomes the ConvexLensModel |
---|
1317 | when the length of the cylinder *L* = 0, that is, a sphereocylinder with end caps that have a radius larger than that |
---|
1318 | of the cylinder and the center of the end cap radius lies within the cylinder. See the diagram for the details |
---|
1319 | of the geometry and restrictions on parameter values. |
---|
1320 | |
---|
1321 | *2.1.16.1. Definition* |
---|
1322 | |
---|
1323 | The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. |
---|
1324 | |
---|
1325 | The Capped Cylinder geometry is defined as |
---|
1326 | |
---|
1327 | .. image:: ..\img\olddocs\image112.jpg |
---|
1328 | |
---|
1329 | where *r* is the radius of the cylinder. All other parameters are as defined in the diagram. Since the end cap radius |
---|
1330 | *R* >= *r* and by definition for this geometry *h* < 0, *h* is then defined by *r* and *R* as |
---|
1331 | |
---|
1332 | *h* = -1 \* sqrt(*R*\ :sup:`2` - *r*\ :sup:`2`) |
---|
1333 | |
---|
1334 | The scattered intensity *I(q)* is calculated as |
---|
1335 | |
---|
1336 | .. image:: ..\img\olddocs\image113.jpg |
---|
1337 | |
---|
1338 | where the amplitude *A(q)* is given as |
---|
1339 | |
---|
1340 | .. image:: ..\img\olddocs\image114.jpg |
---|
1341 | |
---|
1342 | The < > brackets denote an average of the structure over all orientations. <\ *A*\ :sup:`2`\ *(q)*> is then the form |
---|
1343 | factor, *P(q)*. The scale factor is equivalent to the volume fraction of cylinders, each of volume, *V*. Contrast is the |
---|
1344 | difference of scattering length densities of the cylinder and the surrounding solvent. |
---|
1345 | |
---|
1346 | The volume of the Capped Cylinder is (with *h* as a positive value here) |
---|
1347 | |
---|
1348 | .. image:: ..\img\olddocs\image115.jpg |
---|
1349 | |
---|
1350 | and its radius-of-gyration |
---|
1351 | |
---|
1352 | .. image:: ..\img\olddocs\image116.jpg |
---|
1353 | |
---|
1354 | **The requirement that** *R* >= *r* **is not enforced in the model! It is up to you to restrict this during analysis.** |
---|
1355 | |
---|
1356 | This following example dataset is produced by running the MacroCappedCylinder(), using 200 data points, |
---|
1357 | *qmin* = 0.001 |Ang^-1|, *qmax* = 0.7 |Ang^-1| and the default values |
---|
1358 | |
---|
1359 | ============== ======== ============= |
---|
1360 | Parameter name Units Default value |
---|
1361 | ============== ======== ============= |
---|
1362 | scale None 1.0 |
---|
1363 | len_cyl |Ang| 400.0 |
---|
1364 | rad_cap |Ang| 40.0 |
---|
1365 | rad_cyl |Ang| 20.0 |
---|
1366 | sld_capcyl |Ang^-2| 1.0e-006 |
---|
1367 | sld_solv |Ang^-2| 6.3e-006 |
---|
1368 | background |cm^-1| 0 |
---|
1369 | ============== ======== ============= |
---|
1370 | |
---|
1371 | .. image:: ..\img\olddocs\image117.jpg |
---|
1372 | |
---|
1373 | *Figure. 1D plot using the default values (w/256 data point).* |
---|
1374 | |
---|
1375 | For 2D data: The 2D scattering intensity is calculated similar to the 2D cylinder model. For example, for |
---|
1376 | |theta| = 45 deg and |phi| =0 deg with default values for other parameters |
---|
1377 | |
---|
1378 | .. image:: ..\img\olddocs\image118.jpg |
---|
1379 | |
---|
1380 | *Figure. 2D plot (w/(256X265) data points).* |
---|
1381 | |
---|
1382 | .. image:: ..\img\olddocs\image061.jpg |
---|
1383 | |
---|
1384 | *Figure. Definition of the angles for oriented 2D cylinders.* |
---|
1385 | |
---|
1386 | .. image:: ..\img\olddocs\image062.jpg |
---|
1387 | |
---|
1388 | *Figure. Examples of the angles for oriented pp against the detector plane.* |
---|
1389 | |
---|
1390 | REFERENCE |
---|
1391 | |
---|
1392 | H Kaya, *J. Appl. Cryst.*, 37 (2004) 223-230 |
---|
1393 | |
---|
1394 | H Kaya and N-R deSouza, *J. Appl. Cryst.*, 37 (2004) 508-509 (addenda and errata) |
---|
1395 | |
---|
1396 | |
---|
1397 | |
---|
1398 | .. _CoreShellCylinderModel: |
---|
1399 | |
---|
1400 | **2.1.17. CoreShellCylinderModel** |
---|
1401 | |
---|
1402 | This model provides the form factor for a circular cylinder with a core-shell scattering length density profile. The |
---|
1403 | form factor is normalized by the particle volume. |
---|
1404 | |
---|
1405 | *2.1.17.1. Definition* |
---|
1406 | |
---|
1407 | The output of the 2D scattering intensity function for oriented core-shell cylinders is given by (Kline, 2006) |
---|
1408 | |
---|
1409 | .. image:: ..\img\olddocs\image067.PNG |
---|
1410 | |
---|
1411 | where |
---|
1412 | |
---|
1413 | .. image:: ..\img\olddocs\image068.PNG |
---|
1414 | |
---|
1415 | .. image:: ..\img\olddocs\image239.PNG |
---|
1416 | |
---|
1417 | and |alpha| is the angle between the axis of the cylinder and the *q*\ -vector, *Vs* is the volume of the outer shell |
---|
1418 | (i.e. the total volume, including the shell), *Vc* is the volume of the core, *L* is the length of the core, *r* is the |
---|
1419 | radius of the core, *t* is the thickness of the shell, |rho|\ :sub:`c` is the scattering length density of the core, |
---|
1420 | |rho|\ :sub:`s` is the scattering length density of the shell, |rho|\ :sub:`solv` is the scattering length density of |
---|
1421 | the solvent, and *bkg* is the background level. The outer radius of the shell is given by *r+t* and the total length of |
---|
1422 | the outer shell is given by *L+2t*. *J1* is the first order Bessel function. |
---|
1423 | |
---|
1424 | .. image:: ..\img\olddocs\image069.jpg |
---|
1425 | |
---|
1426 | To provide easy access to the orientation of the core-shell cylinder, we define the axis of the cylinder using two |
---|
1427 | angles |theta| and |phi|\ . As for the case of the cylinder, those angles are defined in Figure 2 of the CylinderModel. |
---|
1428 | |
---|
1429 | NB: The 2nd virial coefficient of the cylinder is calculated based on the radius and 2 length values, and used as the |
---|
1430 | effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. |
---|
1431 | |
---|
1432 | The returned value is scaled to units of |cm^-1| and the parameters of the core-shell cylinder model are the following |
---|
1433 | |
---|
1434 | ============== ======== ============= |
---|
1435 | Parameter name Units Default value |
---|
1436 | ============== ======== ============= |
---|
1437 | scale None 1.0 |
---|
1438 | radius |Ang| 20.0 |
---|
1439 | thickness |Ang| 10.0 |
---|
1440 | length |Ang| 400.0 |
---|
1441 | core_sld |Ang^-2| 1e-6 |
---|
1442 | shell_sld |Ang^-2| 4e-6 |
---|
1443 | solvent_sld |Ang^-2| 1e-6 |
---|
1444 | background |cm^-1| 0.0 |
---|
1445 | axis_theta degree 90 |
---|
1446 | axis_phi degree 0.0 |
---|
1447 | ============== ======== ============= |
---|
1448 | |
---|
1449 | The output of the 1D scattering intensity function for randomly oriented cylinders is then given by the equation above. |
---|
1450 | |
---|
1451 | The *axis_theta* and *axis_phi* parameters are not used for the 1D output. Our implementation of the scattering kernel |
---|
1452 | and the 1D scattering intensity use the c-library from NIST. |
---|
1453 | |
---|
1454 | *2.1.17.2. Validation of the CoreShellCylinderModel* |
---|
1455 | |
---|
1456 | Validation of our code was done by comparing the output of the 1D model to the output of the software provided by the |
---|
1457 | NIST (Kline, 2006). Figure 1 shows a comparison of the 1D output of our model and the output of the NIST software. |
---|
1458 | |
---|
1459 | .. image:: ..\img\olddocs\image070.jpg |
---|
1460 | |
---|
1461 | *Figure 1: Comparison of the SasView scattering intensity for a core-shell cylinder with the output of the NIST SANS* |
---|
1462 | *analysis software.* The parameters were set to: *Scale* = 1.0, *Radius* = 20 |Ang|, *Thickness* = 10 |Ang|, |
---|
1463 | *Length* = 400 |Ang|, *Core_sld* = 1e-6 |Ang^-2|, *Shell_sld* = 4e-6 |Ang^-2|, *Solvent_sld* = 1e-6 |Ang^-2|, |
---|
1464 | and *Background* = 0.01 |cm^-1|. |
---|
1465 | |
---|
1466 | Averaging over a distribution of orientation is done by evaluating the equation above. Since we have no other software |
---|
1467 | to compare the implementation of the intensity for fully oriented cylinders, we can compare the result of averaging our |
---|
1468 | 2D output using a uniform distribution *p(*\ |theta|,\ |phi|\ *)* = 1.0. Figure 2 shows the result of such a cross-check. |
---|
1469 | |
---|
1470 | .. image:: ..\img\olddocs\image071.jpg |
---|
1471 | |
---|
1472 | *Figure 2: Comparison of the intensity for uniformly distributed core-shell cylinders calculated from our 2D model and* |
---|
1473 | *the intensity from the NIST SANS analysis software.* The parameters used were: *Scale* = 1.0, *Radius* = 20 |Ang|, |
---|
1474 | *Thickness* = 10 |Ang|, *Length* =400 |Ang|, *Core_sld* = 1e-6 |Ang^-2|, *Shell_sld* = 4e-6 |Ang^-2|, |
---|
1475 | *Solvent_sld* = 1e-6 |Ang^-2|, and *Background* = 0.0 |cm^-1|. |
---|
1476 | |
---|
1477 | .. image:: ..\img\olddocs\image061.jpg |
---|
1478 | |
---|
1479 | *Figure. Definition of the angles for oriented core-shell cylinders.* |
---|
1480 | |
---|
1481 | .. image:: ..\img\olddocs\image062.jpg |
---|
1482 | |
---|
1483 | *Figure. Examples of the angles for oriented pp against the detector plane.* |
---|
1484 | |
---|
1485 | 2013/11/26 - Description reviewed by Heenan, R. |
---|
1486 | |
---|
1487 | |
---|
1488 | |
---|
1489 | .. _EllipticalCylinderModel: |
---|
1490 | |
---|
1491 | **2.1.18 EllipticalCylinderModel** |
---|
1492 | |
---|
1493 | This function calculates the scattering from an elliptical cylinder. |
---|
1494 | |
---|
1495 | *2.1.18.1 Definition for 2D (orientated system)* |
---|
1496 | |
---|
1497 | The angles |theta| and |phi| define the orientation of the axis of the cylinder. The angle |bigpsi| is defined as the |
---|
1498 | orientation of the major axis of the ellipse with respect to the vector *Q*\ . A gaussian polydispersity can be added |
---|
1499 | to any of the orientation angles, and also for the minor radius and the ratio of the ellipse radii. |
---|
1500 | |
---|
1501 | .. image:: ..\img\olddocs\image098.gif |
---|
1502 | |
---|
1503 | *Figure.* *a* = *r_minor* and |nu|\ :sub:`n` = *r_ratio* (i.e., *r_major* / *r_minor*). |
---|
1504 | |
---|
1505 | The function calculated is |
---|
1506 | |
---|
1507 | .. image:: ..\img\olddocs\image099.PNG |
---|
1508 | |
---|
1509 | with the functions |
---|
1510 | |
---|
1511 | .. image:: ..\img\olddocs\image100.PNG |
---|
1512 | |
---|
1513 | and the angle |bigpsi| is defined as the orientation of the major axis of the ellipse with respect to the vector *q*\ . |
---|
1514 | |
---|
1515 | *2.1.18.2 Definition for 1D (no preferred orientation)* |
---|
1516 | |
---|
1517 | The form factor is averaged over all possible orientation before normalized by the particle volume |
---|
1518 | |
---|
1519 | *P(q)* = *scale* \* <*F*\ :sup:`2`> / *V* |
---|
1520 | |
---|
1521 | The returned value is scaled to units of |cm^-1|. |
---|
1522 | |
---|
1523 | To provide easy access to the orientation of the elliptical cylinder, we define the axis of the cylinder using two |
---|
1524 | angles |theta|, |phi| and |bigpsi|. As for the case of the cylinder, the angles |theta| and |phi| are defined on |
---|
1525 | Figure 2 of CylinderModel. The angle |bigpsi| is the rotational angle around its own long_c axis against the *q* plane. |
---|
1526 | For example, |bigpsi| = 0 when the *r_minor* axis is parallel to the *x*\ -axis of the detector. |
---|
1527 | |
---|
1528 | All angle parameters are valid and given only for 2D calculation; ie, an oriented system. |
---|
1529 | |
---|
1530 | .. image:: ..\img\olddocs\image101.jpg |
---|
1531 | |
---|
1532 | *Figure. Definition of angles for 2D* |
---|
1533 | |
---|
1534 | .. image:: ..\img\olddocs\image062.jpg |
---|
1535 | |
---|
1536 | *Figure. Examples of the angles for oriented elliptical cylinders against the detector plane.* |
---|
1537 | |
---|
1538 | NB: The 2nd virial coefficient of the cylinder is calculated based on the averaged radius (= sqrt(*r_minor*\ :sup:`2` \* *r_ratio*)) |
---|
1539 | and length values, and used as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. |
---|
1540 | |
---|
1541 | ============== ======== ============= |
---|
1542 | Parameter name Units Default value |
---|
1543 | ============== ======== ============= |
---|
1544 | scale None 1.0 |
---|
1545 | r_minor |Ang| 20.0 |
---|
1546 | r_ratio |Ang| 1.5 |
---|
1547 | length |Ang| 400.0 |
---|
1548 | sldCyl |Ang^-2| 4e-06 |
---|
1549 | sldSolv |Ang^-2| 1e-06 |
---|
1550 | background |cm^-1| 0 |
---|
1551 | ============== ======== ============= |
---|
1552 | |
---|
1553 | .. image:: ..\img\olddocs\image102.jpg |
---|
1554 | |
---|
1555 | *Figure. 1D plot using the default values (w/1000 data point).* |
---|
1556 | |
---|
1557 | *2.1.18.3 Validation of the EllipticalCylinderModel* |
---|
1558 | |
---|
1559 | Validation of our code was done by comparing the output of the 1D calculation to the angular average of the output of |
---|
1560 | the 2D calculation over all possible angles. The figure below shows the comparison where the solid dot refers to |
---|
1561 | averaged 2D values while the line represents the result of the 1D calculation (for the 2D averaging, values of 76, 180, |
---|
1562 | and 76 degrees are taken for the angles of |theta|, |phi|, and |bigpsi| respectively). |
---|
1563 | |
---|
1564 | .. image:: ..\img\olddocs\image103.gif |
---|
1565 | |
---|
1566 | *Figure. Comparison between 1D and averaged 2D.* |
---|
1567 | |
---|
1568 | In the 2D average, more binning in the angle |phi| is necessary to get the proper result. The following figure shows |
---|
1569 | the results of the averaging by varying the number of angular bins. |
---|
1570 | |
---|
1571 | .. image:: ..\img\olddocs\image104.gif |
---|
1572 | |
---|
1573 | *Figure. The intensities averaged from 2D over different numbers of bins and angles.* |
---|
1574 | |
---|
1575 | REFERENCE |
---|
1576 | |
---|
1577 | L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum, |
---|
1578 | New York, (1987) |
---|
1579 | |
---|
1580 | |
---|
1581 | |
---|
1582 | .. _FlexibleCylinderModel: |
---|
1583 | |
---|
1584 | **2.1.19. FlexibleCylinderModel** |
---|
1585 | |
---|
1586 | This model provides the form factor, *P(q)*, for a flexible cylinder where the form factor is normalized by the volume |
---|
1587 | of the cylinder. **Inter-cylinder interactions are NOT provided for.** |
---|
1588 | |
---|
1589 | *P(q)* = *scale* \* <*F*\ :sup:`2`> / *V* + *background* |
---|
1590 | |
---|
1591 | where the averaging < > is applied over all orientations for 1D. |
---|
1592 | |
---|
1593 | The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as |
---|
1594 | |
---|
1595 | .. image:: ..\img\olddocs\image040.gif |
---|
1596 | |
---|
1597 | *2.1.19.1. Definition* |
---|
1598 | |
---|
1599 | .. image:: ..\img\olddocs\image075.jpg |
---|
1600 | |
---|
1601 | The chain of contour length, *L*, (the total length) can be described as a chain of some number of locally stiff |
---|
1602 | segments of length *l*\ :sub:`p`\ , the persistence length (the length along the cylinder over which the flexible |
---|
1603 | cylinder can be considered a rigid rod). The Kuhn length (*b* = 2 \* *l* :sub:`p`) is also used to describe the |
---|
1604 | stiffness of a chain. |
---|
1605 | |
---|
1606 | The returned value is in units of |cm^-1|, on absolute scale. |
---|
1607 | |
---|
1608 | In the parameters, the sldCyl and sldSolv represent the SLD of the chain/cylinder and solvent respectively. |
---|
1609 | |
---|
1610 | ============== ======== ============= |
---|
1611 | Parameter name Units Default value |
---|
1612 | ============== ======== ============= |
---|
1613 | scale None 1.0 |
---|
1614 | radius |Ang| 20 |
---|
1615 | length |Ang| 1000 |
---|
1616 | sldCyl |Ang^-2| 1e-06 |
---|
1617 | sldSolv |Ang^-2| 6.3e-06 |
---|
1618 | background |cm^-1| 0.01 |
---|
1619 | kuhn_length |Ang| 100 |
---|
1620 | ============== ======== ============= |
---|
1621 | |
---|
1622 | .. image:: ..\img\olddocs\image076.jpg |
---|
1623 | |
---|
1624 | *Figure. 1D plot using the default values (w/1000 data point).* |
---|
1625 | |
---|
1626 | Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research |
---|
1627 | (Kline, 2006). |
---|
1628 | |
---|
1629 | From the reference |
---|
1630 | |
---|
1631 | "Method 3 With Excluded Volume" is used. The model is a parametrization of simulations of a discrete representation |
---|
1632 | of the worm-like chain model of Kratky and Porod applied in the pseudocontinuous limit. See equations (13,26-27) in |
---|
1633 | the original reference for the details. |
---|
1634 | |
---|
1635 | REFERENCE |
---|
1636 | |
---|
1637 | J S Pedersen and P Schurtenberger. *Scattering functions of semiflexible polymers with and without excluded volume* |
---|
1638 | *effects*. *Macromolecules*, 29 (1996) 7602-7612 |
---|
1639 | |
---|
1640 | Correction of the formula can be found in |
---|
1641 | |
---|
1642 | W R Chen, P D Butler and L J Magid, *Incorporating Intermicellar Interactions in the Fitting of SANS Data from* |
---|
1643 | *Cationic Wormlike Micelles*. *Langmuir*, 22(15) 2006 6539ââ¬â6548 |
---|
1644 | |
---|
1645 | |
---|
1646 | |
---|
1647 | .. _FlexCylEllipXModel: |
---|
1648 | |
---|
1649 | **2.1.20 FlexCylEllipXModel** |
---|
1650 | |
---|
1651 | This model calculates the form factor for a flexible cylinder with an elliptical cross section and a uniform scattering |
---|
1652 | length density. The non-negligible diameter of the cylinder is included by accounting for excluded volume interactions |
---|
1653 | within the walk of a single cylinder. The form factor is normalized by the particle volume such that |
---|
1654 | |
---|
1655 | *P(q)* = *scale* \* <*F*\ :sup:`2`> / *V* + *background* |
---|
1656 | |
---|
1657 | where < > is an average over all possible orientations of the flexible cylinder. |
---|
1658 | |
---|
1659 | *2.1.20.1. Definition* |
---|
1660 | |
---|
1661 | The function calculated is from the reference given below. From that paper, "Method 3 With Excluded Volume" is used. |
---|
1662 | The model is a parameterization of simulations of a discrete representation of the worm-like chain model of Kratky and |
---|
1663 | Porod applied in the pseudo-continuous limit. See equations (13, 26-27) in the original reference for the details. |
---|
1664 | |
---|
1665 | NB: there are several typos in the original reference that have been corrected by WRC. Details of the corrections are |
---|
1666 | in the reference below. Most notably |
---|
1667 | |
---|
1668 | - Equation (13): the term (1 - w(QR)) should swap position with w(QR) |
---|
1669 | |
---|
1670 | - Equations (23) and (24) are incorrect; WRC has entered these into Mathematica and solved analytically. The results |
---|
1671 | were then converted to code. |
---|
1672 | |
---|
1673 | - Equation (27) should be q0 = max(a3/sqrt(RgSquare),3) instead of max(a3*b/sqrt(RgSquare),3) |
---|
1674 | |
---|
1675 | - The scattering function is negative for a range of parameter values and q-values that are experimentally accessible. A correction function has been added to give the proper behavior. |
---|
1676 | |
---|
1677 | .. image:: ..\img\olddocs\image077.jpg |
---|
1678 | |
---|
1679 | The chain of contour length, *L*, (the total length) can be described as a chain of some number of locally stiff |
---|
1680 | segments of length *l*\ :sub:`p`\ , the persistence length (the length along the cylinder over which the flexible |
---|
1681 | cylinder can be considered a rigid rod). The Kuhn length (*b* = 2 \* *l* :sub:`p`) is also used to describe the |
---|
1682 | stiffness of a chain. |
---|
1683 | |
---|
1684 | The cross section of the cylinder is elliptical, with minor radius *a*\ . The major radius is larger, so of course, |
---|
1685 | **the axis ratio (parameter 4) must be greater than one.** Simple constraints should be applied during curve fitting to |
---|
1686 | maintain this inequality. |
---|
1687 | |
---|
1688 | The returned value is in units of |cm^-1|, on absolute scale. |
---|
1689 | |
---|
1690 | In the parameters, *sldCyl* and *sldSolv* represent the SLD of the chain/cylinder and solvent respectively. The |
---|
1691 | *scale*, and the contrast are both multiplicative factors in the model and are perfectly correlated. One or both of |
---|
1692 | these parameters must be held fixed during model fitting. |
---|
1693 | |
---|
1694 | If the scale is set equal to the particle volume fraction, |phi|, the returned value is the scattered intensity per |
---|
1695 | unit volume, *I(q)* = |phi| \* *P(q)*. |
---|
1696 | |
---|
1697 | **No inter-cylinder interference effects are included in this calculation.** |
---|
1698 | |
---|
1699 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
1700 | |
---|
1701 | .. image:: ..\img\olddocs\image008.PNG |
---|
1702 | |
---|
1703 | This example dataset is produced by running the Macro FlexCylEllipXModel, using 200 data points, *qmin* = 0.001 |Ang^-1|, |
---|
1704 | *qmax* = 0.7 |Ang^-1| and the default values below |
---|
1705 | |
---|
1706 | ============== ======== ============= |
---|
1707 | Parameter name Units Default value |
---|
1708 | ============== ======== ============= |
---|
1709 | axis_ratio None 1.5 |
---|
1710 | background |cm^-1| 0.0001 |
---|
1711 | Kuhn_length |Ang| 100 |
---|
1712 | Contour length |Ang| 1e+3 |
---|
1713 | radius |Ang| 20.0 |
---|
1714 | scale None 1.0 |
---|
1715 | sldCyl |Ang^-2| 1e-6 |
---|
1716 | sldSolv |Ang^-2| 6.3e-6 |
---|
1717 | ============== ======== ============= |
---|
1718 | |
---|
1719 | .. image:: ..\img\olddocs\image078.jpg |
---|
1720 | |
---|
1721 | *Figure. 1D plot using the default values (w/200 data points).* |
---|
1722 | |
---|
1723 | REFERENCE |
---|
1724 | |
---|
1725 | J S Pedersen and P Schurtenberger. *Scattering functions of semiflexible polymers with and without excluded volume* |
---|
1726 | *effects*. *Macromolecules*, 29 (1996) 7602-7612 |
---|
1727 | |
---|
1728 | Correction of the formula can be found in |
---|
1729 | |
---|
1730 | W R Chen, P D Butler and L J Magid, *Incorporating Intermicellar Interactions in the Fitting of SANS Data from* |
---|
1731 | *Cationic Wormlike Micelles*. *Langmuir*, 22(15) 2006 6539ââ¬â6548 |
---|
1732 | |
---|
1733 | |
---|
1734 | |
---|
1735 | .. _CoreShellBicelleModel: |
---|
1736 | |
---|
1737 | **2.1.21 CoreShellBicelleModel** |
---|
1738 | |
---|
1739 | This model provides the form factor for a circular cylinder with a core-shell scattering length density profile. The |
---|
1740 | form factor is normalized by the particle volume. |
---|
1741 | |
---|
1742 | This model is a more general case of core-shell cylinder model (see above and reference below) in that the parameters |
---|
1743 | of the shell are separated into a face-shell and a rim-shell so that users can set different values of the thicknesses |
---|
1744 | and SLDs. |
---|
1745 | |
---|
1746 | .. image:: ..\img\olddocs\image240.png |
---|
1747 | |
---|
1748 | *(Graphic from DOI: 10.1039/C0NP00002G)* |
---|
1749 | |
---|
1750 | The returned value is scaled to units of |cm^-1| and the parameters of the CoreShellBicelleModel are the following |
---|
1751 | |
---|
1752 | ============== ======== ============= |
---|
1753 | Parameter name Units Default value |
---|
1754 | ============== ======== ============= |
---|
1755 | scale None 1.0 |
---|
1756 | radius |Ang| 20.0 |
---|
1757 | rim_thick |Ang| 10.0 |
---|
1758 | face_thick |Ang| 10.0 |
---|
1759 | length |Ang| 400.0 |
---|
1760 | core_sld |Ang^-2| 1e-6 |
---|
1761 | rim_sld |Ang^-2| 4e-6 |
---|
1762 | face_sld |Ang^-2| 4e-6 |
---|
1763 | solvent_sld |Ang^-2| 1e-6 |
---|
1764 | background |cm^-1| 0.0 |
---|
1765 | axis_theta degree 90 |
---|
1766 | axis_phi degree 0.0 |
---|
1767 | ============== ======== ============= |
---|
1768 | |
---|
1769 | The output of the 1D scattering intensity function for randomly oriented cylinders is then given by the equation above. |
---|
1770 | |
---|
1771 | The *axis_theta* and *axis_phi* parameters are not used for the 1D output. Our implementation of the scattering kernel |
---|
1772 | and the 1D scattering intensity use the c-library from NIST. |
---|
1773 | |
---|
1774 | .. image:: ..\img\olddocs\cscylbicelle_pic.jpg |
---|
1775 | |
---|
1776 | *Figure. 1D plot using the default values (w/200 data point).* |
---|
1777 | |
---|
1778 | .. image:: ..\img\olddocs\image061.jpg |
---|
1779 | |
---|
1780 | *Figure. Definition of the angles for the oriented CoreShellBicelleModel.* |
---|
1781 | |
---|
1782 | .. image:: ..\img\olddocs\image062.jpg |
---|
1783 | |
---|
1784 | *Figure. Examples of the angles for oriented pp against the detector plane.* |
---|
1785 | |
---|
1786 | REFERENCE |
---|
1787 | |
---|
1788 | L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum Press, |
---|
1789 | New York, (1987) |
---|
1790 | |
---|
1791 | |
---|
1792 | |
---|
1793 | .. _BarBellModel: |
---|
1794 | |
---|
1795 | **2.1.22. BarBellModel** |
---|
1796 | |
---|
1797 | Calculates the scattering from a barbell-shaped cylinder (This model simply becomes the DumBellModel when the length of |
---|
1798 | the cylinder, *L*, is set to zero). That is, a sphereocylinder with spherical end caps that have a radius larger than |
---|
1799 | that of the cylinder and the center of the end cap radius lies outside of the cylinder. All dimensions of the BarBell |
---|
1800 | are considered to be monodisperse. See the diagram for the details of the geometry and restrictions on parameter values. |
---|
1801 | |
---|
1802 | *2.1.22.1. Definition* |
---|
1803 | |
---|
1804 | The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. |
---|
1805 | |
---|
1806 | The barbell geometry is defined as |
---|
1807 | |
---|
1808 | .. image:: ..\img\olddocs\image105.jpg |
---|
1809 | |
---|
1810 | where *r* is the radius of the cylinder. All other parameters are as defined in the diagram. |
---|
1811 | |
---|
1812 | Since the end cap radius |
---|
1813 | *R* >= *r* and by definition for this geometry *h* < 0, *h* is then defined by *r* and *R* as |
---|
1814 | |
---|
1815 | *h* = -1 \* sqrt(*R*\ :sup:`2` - *r*\ :sup:`2`) |
---|
1816 | |
---|
1817 | The scattered intensity *I(q)* is calculated as |
---|
1818 | |
---|
1819 | .. image:: ..\img\olddocs\image106.PNG |
---|
1820 | |
---|
1821 | where the amplitude *A(q)* is given as |
---|
1822 | |
---|
1823 | .. image:: ..\img\olddocs\image107.PNG |
---|
1824 | |
---|
1825 | The < > brackets denote an average of the structure over all orientations. <*A* :sup:`2`\ *(q)*> is then the form |
---|
1826 | factor, *P(q)*. The scale factor is equivalent to the volume fraction of cylinders, each of volume, *V*. Contrast is |
---|
1827 | the difference of scattering length densities of the cylinder and the surrounding solvent. |
---|
1828 | |
---|
1829 | The volume of the barbell is |
---|
1830 | |
---|
1831 | .. image:: ..\img\olddocs\image108.jpg |
---|
1832 | |
---|
1833 | |
---|
1834 | and its radius-of-gyration is |
---|
1835 | |
---|
1836 | .. image:: ..\img\olddocs\image109.jpg |
---|
1837 | |
---|
1838 | **The requirement that** *R* >= *r* **is not enforced in the model!** It is up to you to restrict this during analysis. |
---|
1839 | |
---|
1840 | This example dataset is produced by running the Macro PlotBarbell(), using 200 data points, *qmin* = 0.001 |Ang^-1|, |
---|
1841 | *qmax* = 0.7 |Ang^-1| and the following default values |
---|
1842 | |
---|
1843 | ============== ======== ============= |
---|
1844 | Parameter name Units Default value |
---|
1845 | ============== ======== ============= |
---|
1846 | scale None 1.0 |
---|
1847 | len_bar |Ang| 400.0 |
---|
1848 | rad_bar |Ang| 20.0 |
---|
1849 | rad_bell |Ang| 40.0 |
---|
1850 | sld_barbell |Ang^-2| 1.0e-006 |
---|
1851 | sld_solv |Ang^-2| 6.3e-006 |
---|
1852 | background |cm^-1| 0 |
---|
1853 | ============== ======== ============= |
---|
1854 | |
---|
1855 | .. image:: ..\img\olddocs\image110.jpg |
---|
1856 | |
---|
1857 | *Figure. 1D plot using the default values (w/256 data point).* |
---|
1858 | |
---|
1859 | For 2D data: The 2D scattering intensity is calculated similar to the 2D cylinder model. For example, for |
---|
1860 | |theta| = 45 deg and |phi| = 0 deg with default values for other parameters |
---|
1861 | |
---|
1862 | .. image:: ..\img\olddocs\image111.jpg |
---|
1863 | |
---|
1864 | *Figure. 2D plot (w/(256X265) data points).* |
---|
1865 | |
---|
1866 | .. image:: ..\img\olddocs\image061.jpg |
---|
1867 | |
---|
1868 | *Figure. Examples of the angles for oriented pp against the detector plane.* |
---|
1869 | |
---|
1870 | .. image:: ..\img\olddocs\image062.jpg |
---|
1871 | |
---|
1872 | Figure. Definition of the angles for oriented 2D barbells. |
---|
1873 | |
---|
1874 | REFERENCE |
---|
1875 | |
---|
1876 | H Kaya, *J. Appl. Cryst.*, 37 (2004) 37 223-230 |
---|
1877 | |
---|
1878 | H Kaya and N R deSouza, *J. Appl. Cryst.*, 37 (2004) 508-509 (addenda and errata) |
---|
1879 | |
---|
1880 | |
---|
1881 | |
---|
1882 | .. _StackedDisksModel: |
---|
1883 | |
---|
1884 | **2.1.23. StackedDisksModel** |
---|
1885 | |
---|
1886 | This model provides the form factor, *P(q)*, for stacked discs (tactoids) with a core/layer structure where the form |
---|
1887 | factor is normalized by the volume of the cylinder. Assuming the next neighbor distance (d-spacing) in a stack of |
---|
1888 | parallel discs obeys a Gaussian distribution, a structure factor *S(q)* proposed by Kratky and Porod in 1949 is used |
---|
1889 | in this function. |
---|
1890 | |
---|
1891 | Note that the resolution smearing calculation uses 76 Gauss quadrature points to properly smear the model since the |
---|
1892 | function is HIGHLY oscillatory, especially around the *q*-values that correspond to the repeat distance of the layers. |
---|
1893 | |
---|
1894 | The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as |
---|
1895 | |
---|
1896 | .. image:: ..\img\olddocs\image008.PNG |
---|
1897 | |
---|
1898 | The returned value is in units of |cm^-1| |sr^-1|, on absolute scale. |
---|
1899 | |
---|
1900 | *2.1.23.1 Definition* |
---|
1901 | |
---|
1902 | .. image:: ..\img\olddocs\image079.gif |
---|
1903 | |
---|
1904 | The scattering intensity *I(q)* is |
---|
1905 | |
---|
1906 | .. image:: ..\img\olddocs\image081.PNG |
---|
1907 | |
---|
1908 | where the contrast |
---|
1909 | |
---|
1910 | .. image:: ..\img\olddocs\image082.PNG |
---|
1911 | |
---|
1912 | and *N* is the number of discs per unit volume, |alpha| is the angle between the axis of the disc and *q*, and *Vt* |
---|
1913 | and *Vc* are the total volume and the core volume of a single disc, respectively. |
---|
1914 | |
---|
1915 | .. image:: ..\img\olddocs\image083.PNG |
---|
1916 | |
---|
1917 | where *d* = thickness of the layer (*layer_thick*), 2\ *h* = core thickness (*core_thick*), and *R* = radius of the |
---|
1918 | disc (*radius*). |
---|
1919 | |
---|
1920 | .. image:: ..\img\olddocs\image084.PNG |
---|
1921 | |
---|
1922 | where *n* = the total number of the disc stacked (*n_stacking*), *D* = the next neighbor center-to-center distance |
---|
1923 | (*d-spacing*), and |sigma|\ D= the Gaussian standard deviation of the d-spacing (*sigma_d*). |
---|
1924 | |
---|
1925 | To provide easy access to the orientation of the stacked disks, we define the axis of the cylinder using two angles |
---|
1926 | |theta| and |phi|. These angles are defined on Figure 2 of CylinderModel. |
---|
1927 | |
---|
1928 | NB: The 2nd virial coefficient of the cylinder is calculated based on the *radius* and *length* = *n_stacking* \* |
---|
1929 | (*core_thick* + 2 \* *layer_thick*) values, and used as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. |
---|
1930 | |
---|
1931 | ============== ======== ============= |
---|
1932 | Parameter name Units Default value |
---|
1933 | ============== ======== ============= |
---|
1934 | background |cm^-1| 0.001 |
---|
1935 | core_sld |Ang^-2| 4e-006 |
---|
1936 | core_thick |Ang| 10 |
---|
1937 | layer_sld |Ang^-2| 0 |
---|
1938 | layer_thick |Ang| 15 |
---|
1939 | n_stacking None 1 |
---|
1940 | radius |Ang| 3e+03 |
---|
1941 | scale None 0.01 |
---|
1942 | sigma_d |Ang| 0 |
---|
1943 | solvent_sld |Ang^-2| 5e-06 |
---|
1944 | ============== ======== ============= |
---|
1945 | |
---|
1946 | .. image:: ..\img\olddocs\image085.jpg |
---|
1947 | |
---|
1948 | *Figure. 1D plot using the default values (w/1000 data point).* |
---|
1949 | |
---|
1950 | .. image:: ..\img\olddocs\image086.jpg |
---|
1951 | |
---|
1952 | *Figure. Examples of the angles for oriented stackeddisks against the detector plane.* |
---|
1953 | |
---|
1954 | .. image:: ..\img\olddocs\image062.jpg |
---|
1955 | |
---|
1956 | *Figure. Examples of the angles for oriented pp against the detector plane.* |
---|
1957 | |
---|
1958 | Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research |
---|
1959 | (Kline, 2006) |
---|
1960 | |
---|
1961 | REFERENCE |
---|
1962 | |
---|
1963 | A Guinier and G Fournet, *Small-Angle Scattering of X-Rays*, John Wiley and Sons, New York, 1955 |
---|
1964 | |
---|
1965 | O Kratky and G Porod, *J. Colloid Science*, 4, (1949) 35 |
---|
1966 | |
---|
1967 | J S Higgins and H C Benoit, *Polymers and Neutron Scattering*, Clarendon, Oxford, 1994 |
---|
1968 | |
---|
1969 | |
---|
1970 | |
---|
1971 | .. _PringleModel: |
---|
1972 | |
---|
1973 | **2.1.24. PringleModel** |
---|
1974 | |
---|
1975 | This model provides the form factor, *P(q)*, for a 'pringle' or 'saddle-shaped' object (a hyperbolic paraboloid). |
---|
1976 | |
---|
1977 | .. image:: ..\img\olddocs\image241.png |
---|
1978 | |
---|
1979 | *(Graphic from Matt Henderson, matt@matthen.com)* |
---|
1980 | |
---|
1981 | The returned value is in units of |cm^-1|, on absolute scale. |
---|
1982 | |
---|
1983 | The form factor calculated is |
---|
1984 | |
---|
1985 | .. image:: ..\img\olddocs\pringle_eqn_1.jpg |
---|
1986 | |
---|
1987 | where |
---|
1988 | |
---|
1989 | .. image:: ..\img\olddocs\pringle_eqn_2.jpg |
---|
1990 | |
---|
1991 | The parameters of the model and a plot comparing the pringle model with the equivalent cylinder are shown below. |
---|
1992 | |
---|
1993 | ============== ======== ============= |
---|
1994 | Parameter name Units Default value |
---|
1995 | ============== ======== ============= |
---|
1996 | background |cm^-1| 0.0 |
---|
1997 | alpha None 0.001 |
---|
1998 | beta None 0.02 |
---|
1999 | radius |Ang| 60 |
---|
2000 | scale None 1 |
---|
2001 | sld_pringle |Ang^-2| 1e-06 |
---|
2002 | sld_solvent |Ang^-2| 6.3e-06 |
---|
2003 | thickness |Ang| 10 |
---|
2004 | ============== ======== ============= |
---|
2005 | |
---|
2006 | .. image:: ..\img\olddocs\pringle-vs-cylinder.png |
---|
2007 | |
---|
2008 | *Figure. 1D plot using the default values (w/150 data point).* |
---|
2009 | |
---|
2010 | REFERENCE |
---|
2011 | |
---|
2012 | S Alexandru Rautu, Private Communication. |
---|
2013 | |
---|
2014 | |
---|
2015 | |
---|
2016 | .. _EllipsoidModel: |
---|
2017 | |
---|
2018 | **2.1.25. EllipsoidModel** |
---|
2019 | |
---|
2020 | This model provides the form factor for an ellipsoid (ellipsoid of revolution) with uniform scattering length density. |
---|
2021 | The form factor is normalized by the particle volume. |
---|
2022 | |
---|
2023 | *2.1.25.1. Definition* |
---|
2024 | |
---|
2025 | The output of the 2D scattering intensity function for oriented ellipsoids is given by (Feigin, 1987) |
---|
2026 | |
---|
2027 | .. image:: ..\img\olddocs\image059.PNG |
---|
2028 | |
---|
2029 | where |
---|
2030 | |
---|
2031 | .. image:: ..\img\olddocs\image119.PNG |
---|
2032 | |
---|
2033 | and |
---|
2034 | |
---|
2035 | .. image:: ..\img\olddocs\image120.PNG |
---|
2036 | |
---|
2037 | |alpha| is the angle between the axis of the ellipsoid and the *q*\ -vector, *V* is the volume of the ellipsoid, *Ra* |
---|
2038 | is the radius along the rotational axis of the ellipsoid, *Rb* is the radius perpendicular to the rotational axis of |
---|
2039 | the ellipsoid and |drho| (contrast) is the scattering length density difference between the scatterer and |
---|
2040 | the solvent. |
---|
2041 | |
---|
2042 | To provide easy access to the orientation of the ellipsoid, we define the rotation axis of the ellipsoid using two |
---|
2043 | angles |theta| and |phi|\ . These angles are defined on Figure 2 of the CylinderModel_. For the ellipsoid, |theta| |
---|
2044 | is the angle between the rotational axis and the *z*\ -axis. |
---|
2045 | |
---|
2046 | NB: The 2nd virial coefficient of the solid ellipsoid is calculated based on the *radius_a* and *radius_b* values, and |
---|
2047 | used as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. |
---|
2048 | |
---|
2049 | The returned value is scaled to units of |cm^-1| and the parameters of the EllipsoidModel are the following |
---|
2050 | |
---|
2051 | ================ ======== ============= |
---|
2052 | Parameter name Units Default value |
---|
2053 | ================ ======== ============= |
---|
2054 | scale None 1.0 |
---|
2055 | radius_a (polar) |Ang| 20.0 |
---|
2056 | radius_b (equat) |Ang| 400.0 |
---|
2057 | sldEll |Ang^-2| 4.0e-6 |
---|
2058 | sldSolv |Ang^-2| 1.0e-6 |
---|
2059 | background |cm^-1| 0.0 |
---|
2060 | axis_theta degree 90 |
---|
2061 | axis_phi degree 0.0 |
---|
2062 | ================ ======== ============= |
---|
2063 | |
---|
2064 | The output of the 1D scattering intensity function for randomly oriented ellipsoids is then given by the equation |
---|
2065 | above. |
---|
2066 | |
---|
2067 | .. image:: ..\img\olddocs\image121.jpg |
---|
2068 | |
---|
2069 | The *axis_theta* and *axis_phi* parameters are not used for the 1D output. Our implementation of the scattering |
---|
2070 | kernel and the 1D scattering intensity use the c-library from NIST. |
---|
2071 | |
---|
2072 | .. image:: ..\img\olddocs\image122.jpg |
---|
2073 | |
---|
2074 | *Figure. The angles for oriented ellipsoid.* |
---|
2075 | |
---|
2076 | *2.1.25.1. Validation of the EllipsoidModel* |
---|
2077 | |
---|
2078 | Validation of our code was done by comparing the output of the 1D model to the output of the software provided by the |
---|
2079 | NIST (Kline, 2006). Figure 1 below shows a comparison of the 1D output of our model and the output of the NIST |
---|
2080 | software. |
---|
2081 | |
---|
2082 | .. image:: ..\img\olddocs\image123.jpg |
---|
2083 | |
---|
2084 | *Figure 1: Comparison of the SasView scattering intensity for an ellipsoid with the output of the NIST SANS analysis* |
---|
2085 | *software.* The parameters were set to: *Scale* = 1.0, *Radius_a* = 20, *Radius_b* = 400, *Contrast* = 3e-6 |Ang^-2|, |
---|
2086 | and *Background* = 0.01 |cm^-1|. |
---|
2087 | |
---|
2088 | Averaging over a distribution of orientation is done by evaluating the equation above. Since we have no other software |
---|
2089 | to compare the implementation of the intensity for fully oriented ellipsoids, we can compare the result of averaging |
---|
2090 | our 2D output using a uniform distribution *p(*\ |theta|,\ |phi|\ *)* = 1.0. Figure 2 shows the result of such a |
---|
2091 | cross-check. |
---|
2092 | |
---|
2093 | .. image:: ..\img\olddocs\image124.jpg |
---|
2094 | |
---|
2095 | *Figure 2: Comparison of the intensity for uniformly distributed ellipsoids calculated from our 2D model and the* |
---|
2096 | *intensity from the NIST SANS analysis software.* The parameters used were: *Scale* = 1.0, *Radius_a* = 20, |
---|
2097 | *Radius_b* = 400, *Contrast* = 3e-6 |Ang^-2|, and *Background* = 0.0 |cm^-1|. |
---|
2098 | |
---|
2099 | The discrepancy above *q* = 0.3 |cm^-1| is due to the way the form factors are calculated in the c-library provided by |
---|
2100 | NIST. A numerical integration has to be performed to obtain *P(q)* for randomly oriented particles. The NIST software |
---|
2101 | performs that integration with a 76-point Gaussian quadrature rule, which will become imprecise at high q where the |
---|
2102 | amplitude varies quickly as a function of *q*. The SasView result shown has been obtained by summing over 501 |
---|
2103 | equidistant points in . Our result was found to be stable over the range of *q* shown for a number of points higher |
---|
2104 | than 500. |
---|
2105 | |
---|
2106 | REFERENCE |
---|
2107 | |
---|
2108 | L A Feigin and D I Svergun. *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum, |
---|
2109 | New York, 1987. |
---|
2110 | |
---|
2111 | |
---|
2112 | |
---|
2113 | .. _CoreShellEllipsoidModel: |
---|
2114 | |
---|
2115 | **2.1.26. CoreShellEllipsoidModel** |
---|
2116 | |
---|
2117 | This model provides the form factor, *P(q)*, for a core shell ellipsoid (below) where the form factor is normalized by |
---|
2118 | the volume of the cylinder. |
---|
2119 | |
---|
2120 | *P(q)* = *scale* \* <*f*\ :sup:`2`> / *V* + *background* |
---|
2121 | |
---|
2122 | where the volume *V* = (4/3)\ |pi| (*r*\ :sub:`maj` *r*\ :sub:`min`\ :sup:`2`) and the averaging < > is applied over |
---|
2123 | all orientations for 1D. |
---|
2124 | |
---|
2125 | .. image:: ..\img\olddocs\image125.gif |
---|
2126 | |
---|
2127 | The returned value is in units of |cm^-1|, on absolute scale. |
---|
2128 | |
---|
2129 | *2.1.26.1. Definition* |
---|
2130 | |
---|
2131 | The form factor calculated is |
---|
2132 | |
---|
2133 | .. image:: ..\img\olddocs\image126.PNG |
---|
2134 | |
---|
2135 | To provide easy access to the orientation of the core-shell ellipsoid, we define the axis of the solid ellipsoid using |
---|
2136 | two angles |theta| and |phi|\ . These angles are defined on Figure 2 of the CylinderModel_. The contrast is defined as |
---|
2137 | SLD(core) - SLD(shell) and SLD(shell) - SLD(solvent). |
---|
2138 | |
---|
2139 | In the parameters, *equat_core* = equatorial core radius, *polar_core* = polar core radius, *equat_shell* = |
---|
2140 | *r*\ :sub:`min` (or equatorial outer radius), and *polar_shell* = = *r*\ :sub:`maj` (or polar outer radius). |
---|
2141 | |
---|
2142 | NB: The 2nd virial coefficient of the solid ellipsoid is calculated based on the *radius_a* (= *polar_shell*) and |
---|
2143 | *radius_b* (= *equat_shell*) values, and used as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. |
---|
2144 | |
---|
2145 | ============== ======== ============= |
---|
2146 | Parameter name Units Default value |
---|
2147 | ============== ======== ============= |
---|
2148 | background |cm^-1| 0.001 |
---|
2149 | equat_core |Ang| 200 |
---|
2150 | equat_shell |Ang| 250 |
---|
2151 | sld_solvent |Ang^-2| 6e-06 |
---|
2152 | ploar_shell |Ang| 30 |
---|
2153 | ploar_core |Ang| 20 |
---|
2154 | scale None 1 |
---|
2155 | sld_core |Ang^-2| 2e-06 |
---|
2156 | sld_shell |Ang^-2| 1e-06 |
---|
2157 | ============== ======== ============= |
---|
2158 | |
---|
2159 | .. image:: ..\img\olddocs\image127.jpg |
---|
2160 | |
---|
2161 | *Figure. 1D plot using the default values (w/1000 data point).* |
---|
2162 | |
---|
2163 | .. image:: ..\img\olddocs\image122.jpg |
---|
2164 | |
---|
2165 | *Figure. The angles for oriented CoreShellEllipsoid.* |
---|
2166 | |
---|
2167 | Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research |
---|
2168 | (Kline, 2006). |
---|
2169 | |
---|
2170 | REFERENCE |
---|
2171 | |
---|
2172 | M Kotlarchyk, S H Chen, *J. Chem. Phys.*, 79 (1983) 2461 |
---|
2173 | |
---|
2174 | S J Berr, *Phys. Chem.*, 91 (1987) 4760 |
---|
2175 | |
---|
2176 | |
---|
2177 | |
---|
2178 | .. _CoreShellEllipsoidXTModel: |
---|
2179 | |
---|
2180 | **2.1.27. CoreShellEllipsoidXTModel** |
---|
2181 | |
---|
2182 | An alternative version of *P(q)* for the core-shell ellipsoid (see CoreShellEllipsoidModel), having as parameters the |
---|
2183 | core axial ratio *X* and a shell thickness, which are more often what we would like to determine. |
---|
2184 | |
---|
2185 | This model is also better behaved when polydispersity is applied than the four independent radii in |
---|
2186 | CoreShellEllipsoidModel. |
---|
2187 | |
---|
2188 | *2.1.27.1. Definition* |
---|
2189 | |
---|
2190 | .. image:: ..\img\olddocs\image125.gif |
---|
2191 | |
---|
2192 | The geometric parameters of this model are |
---|
2193 | |
---|
2194 | *equat_core* = equatorial core radius = *Rminor_core* |
---|
2195 | |
---|
2196 | *X_core* = *polar_core* / *equat_core* = *Rmajor_core* / *Rminor_core* |
---|
2197 | |
---|
2198 | *T_shell* = *equat_outer* - *equat_core* = *Rminor_outer* - *Rminor_core* |
---|
2199 | |
---|
2200 | *XpolarShell* = *Tpolar_shell* / *T_shell* = (*Rmajor_outer* - *Rmajor_core*)/(*Rminor_outer* - *Rminor_core*) |
---|
2201 | |
---|
2202 | In terms of the original radii |
---|
2203 | |
---|
2204 | *polar_core* = *equat_core* \* *X_core* |
---|
2205 | |
---|
2206 | *equat_shell* = *equat_core* + *T_shell* |
---|
2207 | |
---|
2208 | *polar_shell* = *equat_core* \* *X_core* + *T_shell* \* *XpolarShell* |
---|
2209 | |
---|
2210 | (where we note that "shell" perhaps confusingly, relates to the outer radius) |
---|
2211 | |
---|
2212 | When *X_core* < 1 the core is oblate; when *X_core* > 1 it is prolate. *X_core* = 1 is a spherical core. |
---|
2213 | |
---|
2214 | For a fixed shell thickness *XpolarShell* = 1, to scale the shell thickness pro-rata with the radius |
---|
2215 | *XpolarShell* = *X_core*. |
---|
2216 | |
---|
2217 | When including an *S(q)*, the radius in *S(q)* is calculated to be that of a sphere with the same 2nd virial |
---|
2218 | coefficient of the **outer** surface of the ellipsoid. This may have some undesirable effects if the aspect ratio of |
---|
2219 | the ellipsoid is large (ie, if *X* << 1 or *X* >> 1), when the *S(q)* - which assumes spheres - will not in any case |
---|
2220 | be valid. |
---|
2221 | |
---|
2222 | If SAS data are in absolute units, and the SLDs are correct, then *scale* should be the total volume fraction of the |
---|
2223 | "outer particle". When *S(q)* is introduced this moves to the *S(q)* volume fraction, and *scale* should then be 1.0, |
---|
2224 | or contain some other units conversion factor (for example, if you have SAXS data). |
---|
2225 | |
---|
2226 | ============== ======== ============= |
---|
2227 | Parameter name Units Default value |
---|
2228 | ============== ======== ============= |
---|
2229 | background |cm^-1| 0.001 |
---|
2230 | equat_core |Ang| 20 |
---|
2231 | scale None 0.05 |
---|
2232 | sld_core |Ang^-2| 2.0e-6 |
---|
2233 | sld_shell |Ang^-2| 1.0e-6 |
---|
2234 | sld_solv |Ang^-2| 6.3e-6 |
---|
2235 | T_shell |Ang| 30 |
---|
2236 | X_core None 3.0 |
---|
2237 | XpolarShell None 1.0 |
---|
2238 | ============== ======== ============= |
---|
2239 | |
---|
2240 | REFERENCE |
---|
2241 | |
---|
2242 | R K Heenan, Private communication |
---|
2243 | |
---|
2244 | |
---|
2245 | |
---|
2246 | .. _TriaxialEllipsoidModel: |
---|
2247 | |
---|
2248 | **2.1.28. TriaxialEllipsoidModel** |
---|
2249 | |
---|
2250 | This model provides the form factor, *P(q)*, for an ellipsoid (below) where all three axes are of different lengths, |
---|
2251 | i.e., *Ra* =< *Rb* =< *Rc*\ . **Users should maintain this inequality for all calculations**. |
---|
2252 | |
---|
2253 | *P(q)* = *scale* \* <*f*\ :sup:`2`> / *V* + *background* |
---|
2254 | |
---|
2255 | where the volume *V* = (4/3)\ |pi| (*Ra* *Rb* *Rc*), and the averaging < > is applied over all orientations for 1D. |
---|
2256 | |
---|
2257 | .. image:: ..\img\olddocs\image128.jpg |
---|
2258 | |
---|
2259 | The returned value is in units of |cm^-1|, on absolute scale. |
---|
2260 | |
---|
2261 | *2.1.28.1. Definition* |
---|
2262 | |
---|
2263 | The form factor calculated is |
---|
2264 | |
---|
2265 | .. image:: ..\img\olddocs\image129.PNG |
---|
2266 | |
---|
2267 | To provide easy access to the orientation of the triaxial ellipsoid, we define the axis of the cylinder using the |
---|
2268 | angles |theta|, |phi| and |bigpsi|. These angles are defined on Figure 2 of the CylinderModel_. The angle |bigpsi| is |
---|
2269 | the rotational angle around its own *semi_axisC* axis against the *q* plane. For example, |bigpsi| = 0 when the |
---|
2270 | *semi_axisA* axis is parallel to the *x*-axis of the detector. |
---|
2271 | |
---|
2272 | The radius-of-gyration for this system is *Rg*\ :sup:`2` = (*Ra*\ :sup:`2` *Rb*\ :sup:`2` *Rc*\ :sup:`2`)/5. |
---|
2273 | |
---|
2274 | The contrast is defined as SLD(ellipsoid) - SLD(solvent). In the parameters, *semi_axisA* = *Ra* (or minor equatorial |
---|
2275 | radius), *semi_axisB* = *Rb* (or major equatorial radius), and *semi_axisC* = *Rc* (or polar radius of the ellipsoid). |
---|
2276 | |
---|
2277 | NB: The 2nd virial coefficient of the triaxial solid ellipsoid is calculated based on the |
---|
2278 | *radius_a* (= *semi_axisC*\ ) and *radius_b* (= sqrt(*semi_axisA* \* *semi_axisB*)) values, and used as the effective |
---|
2279 | radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. |
---|
2280 | |
---|
2281 | ============== ======== ============= |
---|
2282 | Parameter name Units Default value |
---|
2283 | ============== ======== ============= |
---|
2284 | background |cm^-1| 0.0 |
---|
2285 | semi_axisA |Ang| 35 |
---|
2286 | semi_axisB |Ang| 100 |
---|
2287 | semi_axisC |Ang| 400 |
---|
2288 | scale None 1 |
---|
2289 | sldEll |Ang^-2| 1.0e-06 |
---|
2290 | sldSolv |Ang^-2| 6.3e-06 |
---|
2291 | ============== ======== ============= |
---|
2292 | |
---|
2293 | .. image:: ..\img\olddocs\image130.jpg |
---|
2294 | |
---|
2295 | *Figure. 1D plot using the default values (w/1000 data point).* |
---|
2296 | |
---|
2297 | *2.1.28.2.Validation of the TriaxialEllipsoidModel* |
---|
2298 | |
---|
2299 | Validation of our code was done by comparing the output of the 1D calculation to the angular average of the output of |
---|
2300 | 2D calculation over all possible angles. The Figure below shows the comparison where the solid dot refers to averaged |
---|
2301 | 2D while the line represents the result of 1D calculation (for 2D averaging, 76, 180, and 76 points are taken for the |
---|
2302 | angles of |theta|, |phi|, and |psi| respectively). |
---|
2303 | |
---|
2304 | .. image:: ..\img\olddocs\image131.gif |
---|
2305 | |
---|
2306 | *Figure. Comparison between 1D and averaged 2D.* |
---|
2307 | |
---|
2308 | .. image:: ..\img\olddocs\image132.jpg |
---|
2309 | |
---|
2310 | *Figure. The angles for oriented ellipsoid.* |
---|
2311 | |
---|
2312 | Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research |
---|
2313 | (Kline, 2006) |
---|
2314 | |
---|
2315 | REFERENCE |
---|
2316 | |
---|
2317 | L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and Neutron Scattering*, Plenum, |
---|
2318 | New York, 1987. |
---|
2319 | |
---|
2320 | |
---|
2321 | |
---|
2322 | .. _LamellarModel: |
---|
2323 | |
---|
2324 | **2.1.29. LamellarModel** |
---|
2325 | |
---|
2326 | This model provides the scattering intensity, *I(q)*, for a lyotropic lamellar phase where a uniform SLD and random |
---|
2327 | distribution in solution are assumed. Polydispersity in the bilayer thickness can be applied from the GUI. |
---|
2328 | |
---|
2329 | *2.1.29.1. Definition* |
---|
2330 | |
---|
2331 | The scattering intensity *I(q)* is |
---|
2332 | |
---|
2333 | .. image:: ..\img\olddocs\image133.PNG |
---|
2334 | |
---|
2335 | The form factor is |
---|
2336 | |
---|
2337 | .. image:: ..\img\olddocs\image134.PNG |
---|
2338 | |
---|
2339 | where |delta| = bilayer thickness. |
---|
2340 | |
---|
2341 | The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
2342 | |
---|
2343 | .. image:: ..\img\olddocs\image040.gif |
---|
2344 | |
---|
2345 | The returned value is in units of |cm^-1|, on absolute scale. In the parameters, *sld_bi* = SLD of the bilayer, |
---|
2346 | *sld_sol* = SLD of the solvent, and *bi_thick* = thickness of the bilayer. |
---|
2347 | |
---|
2348 | ============== ======== ============= |
---|
2349 | Parameter name Units Default value |
---|
2350 | ============== ======== ============= |
---|
2351 | background |cm^-1| 0.0 |
---|
2352 | sld_bi |Ang^-2| 1e-06 |
---|
2353 | bi_thick |Ang| 50 |
---|
2354 | sld_sol |Ang^-2| 6e-06 |
---|
2355 | scale None 1 |
---|
2356 | ============== ======== ============= |
---|
2357 | |
---|
2358 | .. image:: ..\img\olddocs\image135.jpg |
---|
2359 | |
---|
2360 | *Figure. 1D plot using the default values (w/1000 data point).* |
---|
2361 | |
---|
2362 | Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research |
---|
2363 | (Kline, 2006). |
---|
2364 | |
---|
2365 | REFERENCE |
---|
2366 | |
---|
2367 | F Nallet, R Laversanne, and D Roux, J. Phys. II France, 3, (1993) 487-502 |
---|
2368 | |
---|
2369 | also in J. Phys. Chem. B, 105, (2001) 11081-11088 |
---|
2370 | |
---|
2371 | |
---|
2372 | |
---|
2373 | .. _LamellarFFHGModel: |
---|
2374 | |
---|
2375 | **2.1.30. LamellarFFHGModel** |
---|
2376 | |
---|
2377 | This model provides the scattering intensity, *I(q)*, for a lyotropic lamellar phase where a random distribution in |
---|
2378 | solution are assumed. The SLD of the head region is taken to be different from the SLD of the tail region. |
---|
2379 | |
---|
2380 | *2.1.31.1. Definition* |
---|
2381 | |
---|
2382 | The scattering intensity *I(q)* is |
---|
2383 | |
---|
2384 | .. image:: ..\img\olddocs\image136.PNG |
---|
2385 | |
---|
2386 | The form factor is |
---|
2387 | |
---|
2388 | .. image:: ..\img\olddocs\image137.jpg |
---|
2389 | |
---|
2390 | where |delta|\ T = tail length (or *t_length*), |delta|\ H = head thickness (or *h_thickness*), |
---|
2391 | |drho|\ H = SLD(headgroup) - SLD(solvent), and |drho|\ T = SLD(tail) - SLD(solvent). The total thickness is 2(H+T). |
---|
2392 | |
---|
2393 | The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
2394 | |
---|
2395 | .. image:: ..\img\olddocs\image040.gif |
---|
2396 | |
---|
2397 | The returned value is in units of |cm^-1|, on absolute scale. In the parameters, *sld_tail* = SLD of the tail group, |
---|
2398 | and *sld_head* = SLD of the head group. |
---|
2399 | |
---|
2400 | ============== ======== ============= |
---|
2401 | Parameter name Units Default value |
---|
2402 | ============== ======== ============= |
---|
2403 | background |cm^-1| 0.0 |
---|
2404 | sld_head |Ang^-2| 3e-06 |
---|
2405 | scale None 1 |
---|
2406 | sld_solvent |Ang^-2| 6e-06 |
---|
2407 | h_thickness |Ang| 10 |
---|
2408 | t_length |Ang| 15 |
---|
2409 | sld_tail |Ang^-2| 0 |
---|
2410 | ============== ======== ============= |
---|
2411 | |
---|
2412 | .. image:: ..\img\olddocs\image138.jpg |
---|
2413 | |
---|
2414 | *Figure. 1D plot using the default values (w/1000 data point).* |
---|
2415 | |
---|
2416 | Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research |
---|
2417 | (Kline, 2006). |
---|
2418 | |
---|
2419 | REFERENCE |
---|
2420 | |
---|
2421 | F Nallet, R Laversanne, and D Roux, J. Phys. II France, 3, (1993) 487-502 |
---|
2422 | |
---|
2423 | also in J. Phys. Chem. B, 105, (2001) 11081-11088 |
---|
2424 | |
---|
2425 | *2014/04/17 - Description reviewed by S King and P Butler.* |
---|
2426 | |
---|
2427 | |
---|
2428 | |
---|
2429 | .. _LamellarPSModel: |
---|
2430 | |
---|
2431 | **2.1.31. LamellarPSModel** |
---|
2432 | |
---|
2433 | This model provides the scattering intensity, *I(q)* = *P(q)* \* *S(q)*, for a lyotropic lamellar phase where a random |
---|
2434 | distribution in solution are assumed. |
---|
2435 | |
---|
2436 | *2.1.31.1. Definition* |
---|
2437 | |
---|
2438 | The scattering intensity *I(q)* is |
---|
2439 | |
---|
2440 | .. image:: ..\img\olddocs\image139.PNG |
---|
2441 | |
---|
2442 | The form factor is |
---|
2443 | |
---|
2444 | .. image:: ..\img\olddocs\image134.PNG |
---|
2445 | |
---|
2446 | and the structure factor is |
---|
2447 | |
---|
2448 | .. image:: ..\img\olddocs\image140.PNG |
---|
2449 | |
---|
2450 | where |
---|
2451 | |
---|
2452 | .. image:: ..\img\olddocs\image141.PNG |
---|
2453 | |
---|
2454 | Here *d* = (repeat) spacing, |delta| = bilayer thickness, the contrast |drho| = SLD(headgroup) - SLD(solvent), |
---|
2455 | K = smectic bending elasticity, B = compression modulus, and N = number of lamellar plates (*n_plates*). |
---|
2456 | |
---|
2457 | NB: **When the Caille parameter is greater than approximately 0.8 to 1.0, the assumptions of the model are incorrect.** |
---|
2458 | And due to a complication of the model function, users are responsible for making sure that all the assumptions are |
---|
2459 | handled accurately (see the original reference below for more details). |
---|
2460 | |
---|
2461 | The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
2462 | |
---|
2463 | .. image:: ..\img\olddocs\image040.gif |
---|
2464 | |
---|
2465 | The returned value is in units of |cm^-1|, on absolute scale. |
---|
2466 | |
---|
2467 | ============== ======== ============= |
---|
2468 | Parameter name Units Default value |
---|
2469 | ============== ======== ============= |
---|
2470 | background |cm^-1| 0.0 |
---|
2471 | contrast |Ang^-2| 5e-06 |
---|
2472 | scale None 1 |
---|
2473 | delta |Ang| 30 |
---|
2474 | n_plates None 20 |
---|
2475 | spacing |Ang| 400 |
---|
2476 | caille |Ang^-2| 0.1 |
---|
2477 | ============== ======== ============= |
---|
2478 | |
---|
2479 | .. image:: ..\img\olddocs\image142.jpg |
---|
2480 | |
---|
2481 | *Figure. 1D plot using the default values (w/6000 data point).* |
---|
2482 | |
---|
2483 | Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research |
---|
2484 | (Kline, 2006). |
---|
2485 | |
---|
2486 | REFERENCE |
---|
2487 | |
---|
2488 | F Nallet, R Laversanne, and D Roux, J. Phys. II France, 3, (1993) 487-502 |
---|
2489 | |
---|
2490 | also in J. Phys. Chem. B, 105, (2001) 11081-11088 |
---|
2491 | |
---|
2492 | |
---|
2493 | |
---|
2494 | .. _LamellarPSHGModel: |
---|
2495 | |
---|
2496 | **2.1.32. LamellarPSHGModel** |
---|
2497 | |
---|
2498 | This model provides the scattering intensity, *I(q)* = *P(q)* \* *S(q)*, for a lyotropic lamellar phase where a random |
---|
2499 | distribution in solution are assumed. The SLD of the head region is taken to be different from the SLD of the tail |
---|
2500 | region. |
---|
2501 | |
---|
2502 | *2.1.32.1. Definition* |
---|
2503 | |
---|
2504 | The scattering intensity *I(q)* is |
---|
2505 | |
---|
2506 | .. image:: ..\img\olddocs\image139.PNG |
---|
2507 | |
---|
2508 | The form factor is |
---|
2509 | |
---|
2510 | .. image:: ..\img\olddocs\image143.PNG |
---|
2511 | |
---|
2512 | The structure factor is |
---|
2513 | |
---|
2514 | .. image:: ..\img\olddocs\image140.PNG |
---|
2515 | |
---|
2516 | where |
---|
2517 | |
---|
2518 | .. image:: ..\img\olddocs\image141.PNG |
---|
2519 | |
---|
2520 | where |delta|\ T = tail length (or *t_length*), |delta|\ H = head thickness (or *h_thickness*), |
---|
2521 | |drho|\ H = SLD(headgroup) - SLD(solvent), and |drho|\ T = SLD(tail) - SLD(headgroup). |
---|
2522 | Here *d* = (repeat) spacing, *K* = smectic bending elasticity, *B* = compression modulus, and N = number of lamellar |
---|
2523 | plates (*n_plates*). |
---|
2524 | |
---|
2525 | NB: **When the Caille parameter is greater than approximately 0.8 to 1.0, the assumptions of the model are incorrect.** |
---|
2526 | And due to a complication of the model function, users are responsible for making sure that all the assumptions are |
---|
2527 | handled accurately (see the original reference below for more details). |
---|
2528 | |
---|
2529 | The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
2530 | |
---|
2531 | .. image:: ..\img\olddocs\image040.gif |
---|
2532 | |
---|
2533 | The returned value is in units of |cm^-1|, on absolute scale. In the parameters, *sld_tail* = SLD of the tail group, |
---|
2534 | *sld_head* = SLD of the head group, and *sld_solvent* = SLD of the solvent. |
---|
2535 | |
---|
2536 | ============== ======== ============= |
---|
2537 | Parameter name Units Default value |
---|
2538 | ============== ======== ============= |
---|
2539 | background |cm^-1| 0.001 |
---|
2540 | sld_head |Ang^-2| 2e-06 |
---|
2541 | scale None 1 |
---|
2542 | sld_solvent |Ang^-2| 6e-06 |
---|
2543 | deltaH |Ang| 2 |
---|
2544 | deltaT |Ang| 10 |
---|
2545 | sld_tail |Ang^-2| 0 |
---|
2546 | n_plates None 30 |
---|
2547 | spacing |Ang| 40 |
---|
2548 | caille |Ang^-2| 0.001 |
---|
2549 | ============== ======== ============= |
---|
2550 | |
---|
2551 | .. image:: ..\img\olddocs\image144.jpg |
---|
2552 | |
---|
2553 | *Figure. 1D plot using the default values (w/6000 data point).* |
---|
2554 | |
---|
2555 | Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research |
---|
2556 | (Kline, 2006). |
---|
2557 | |
---|
2558 | REFERENCE |
---|
2559 | |
---|
2560 | F Nallet, R Laversanne, and D Roux, J. Phys. II France, 3, (1993) 487-502 |
---|
2561 | |
---|
2562 | also in J. Phys. Chem. B, 105, (2001) 11081-11088 |
---|
2563 | |
---|
2564 | |
---|
2565 | |
---|
2566 | .. _LamellarPCrystalModel: |
---|
2567 | |
---|
2568 | **2.1.33. LamellarPCrystalModel** |
---|
2569 | |
---|
2570 | This model calculates the scattering from a stack of repeating lamellar structures. The stacks of lamellae (infinite |
---|
2571 | in lateral dimension) are treated as a paracrystal to account for the repeating spacing. The repeat distance is further |
---|
2572 | characterized by a Gaussian polydispersity. **This model can be used for large multilamellar vesicles.** |
---|
2573 | |
---|
2574 | *2.1.33.1. Definition* |
---|
2575 | |
---|
2576 | The scattering intensity *I(q)* is calculated as |
---|
2577 | |
---|
2578 | .. image:: ..\img\olddocs\image145.jpg |
---|
2579 | |
---|
2580 | The form factor of the bilayer is approximated as the cross section of an infinite, planar bilayer of thickness *t* |
---|
2581 | |
---|
2582 | .. image:: ..\img\olddocs\image146.jpg |
---|
2583 | |
---|
2584 | Here, the scale factor is used instead of the mass per area of the bilayer (*G*). The scale factor is the volume |
---|
2585 | fraction of the material in the bilayer, *not* the total excluded volume of the paracrystal. *Z*\ :sub:`N`\ *(q)* |
---|
2586 | describes the interference effects for aggregates consisting of more than one bilayer. The equations used are (3-5) |
---|
2587 | from the Bergstrom reference below. |
---|
2588 | |
---|
2589 | Non-integer numbers of stacks are calculated as a linear combination of the lower and higher values |
---|
2590 | |
---|
2591 | .. image:: ..\img\olddocs\image147.jpg |
---|
2592 | |
---|
2593 | The 2D scattering intensity is the same as 1D, regardless of the orientation of the *q* vector which is defined as |
---|
2594 | |
---|
2595 | .. image:: ..\img\olddocs\image040.gif |
---|
2596 | |
---|
2597 | The parameters of the model are *Nlayers* = no. of layers, and *pd_spacing* = polydispersity of spacing. |
---|
2598 | |
---|
2599 | ============== ======== ============= |
---|
2600 | Parameter name Units Default value |
---|
2601 | ============== ======== ============= |
---|
2602 | background |cm^-1| 0 |
---|
2603 | scale None 1 |
---|
2604 | Nlayers None 20 |
---|
2605 | pd_spacing None 0.2 |
---|
2606 | sld_layer |Ang^-2| 1e-6 |
---|
2607 | sld_solvent |Ang^-2| 6.34e-6 |
---|
2608 | spacing |Ang| 250 |
---|
2609 | thickness |Ang| 33 |
---|
2610 | ============== ======== ============= |
---|
2611 | |
---|
2612 | .. image:: ..\img\olddocs\image148.jpg |
---|
2613 | |
---|
2614 | *Figure. 1D plot using the default values above (w/20000 data point).* |
---|
2615 | |
---|
2616 | Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research |
---|
2617 | (Kline, 2006). |
---|
2618 | |
---|
2619 | REFERENCE |
---|
2620 | |
---|
2621 | M Bergstrom, J S Pedersen, P Schurtenberger, S U Egelhaaf, *J. Phys. Chem. B*, 103 (1999) 9888-9897 |
---|
2622 | |
---|
2623 | |
---|
2624 | |
---|
2625 | .. _SCCrystalModel: |
---|
2626 | |
---|
2627 | **2.1.34. SCCrystalModel** |
---|
2628 | |
---|
2629 | Calculates the scattering from a **simple cubic lattice** with paracrystalline distortion. Thermal vibrations are |
---|
2630 | considered to be negligible, and the size of the paracrystal is infinitely large. Paracrystalline distortion is assumed |
---|
2631 | to be isotropic and characterized by a Gaussian distribution. |
---|
2632 | |
---|
2633 | The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. |
---|
2634 | |
---|
2635 | *2.1.34.1. Definition* |
---|
2636 | |
---|
2637 | The scattering intensity *I(q)* is calculated as |
---|
2638 | |
---|
2639 | .. image:: ..\img\olddocs\image149.jpg |
---|
2640 | |
---|
2641 | where *scale* is the volume fraction of spheres, *Vp* is the volume of the primary particle, *V(lattice)* is a volume |
---|
2642 | correction for the crystal structure, *P(q)* is the form factor of the sphere (normalized), and *Z(q)* is the |
---|
2643 | paracrystalline structure factor for a simple cubic structure. |
---|
2644 | |
---|
2645 | Equation (16) of the 1987 reference is used to calculate *Z(q)*, using equations (13)-(15) from the 1987 paper for |
---|
2646 | *Z1*\ , *Z2*\ , and *Z3*\ . |
---|
2647 | |
---|
2648 | The lattice correction (the occupied volume of the lattice) for a simple cubic structure of particles of radius *R* |
---|
2649 | and nearest neighbor separation *D* is |
---|
2650 | |
---|
2651 | .. image:: ..\img\olddocs\image150.jpg |
---|
2652 | |
---|
2653 | The distortion factor (one standard deviation) of the paracrystal is included in the calculation of *Z(q)* |
---|
2654 | |
---|
2655 | .. image:: ..\img\olddocs\image151.jpg |
---|
2656 | |
---|
2657 | where *g* is a fractional distortion based on the nearest neighbor distance. |
---|
2658 | |
---|
2659 | The simple cubic lattice is |
---|
2660 | |
---|
2661 | .. image:: ..\img\olddocs\image152.jpg |
---|
2662 | |
---|
2663 | For a crystal, diffraction peaks appear at reduced *q*\ -values given by |
---|
2664 | |
---|
2665 | .. image:: ..\img\olddocs\image153.jpg |
---|
2666 | |
---|
2667 | where for a simple cubic lattice any *h*\ , *k*\ , *l* are allowed and none are forbidden. Thus the peak positions |
---|
2668 | correspond to (just the first 5) |
---|
2669 | |
---|
2670 | .. image:: ..\img\olddocs\image154.jpg |
---|
2671 | |
---|
2672 | **NB: The calculation of** *Z(q)* **is a double numerical integral that must be carried out with a high density of** |
---|
2673 | **points to properly capture the sharp peaks of the paracrystalline scattering.** So be warned that the calculation is |
---|
2674 | SLOW. Go get some coffee. Fitting of any experimental data must be resolution smeared for any meaningful fit. This |
---|
2675 | makes a triple integral. Very, very slow. Go get lunch! |
---|
2676 | |
---|
2677 | ============== ======== ============= |
---|
2678 | Parameter name Units Default value |
---|
2679 | ============== ======== ============= |
---|
2680 | background |cm^-1| 0 |
---|
2681 | dnn |Ang| 220 |
---|
2682 | scale None 1 |
---|
2683 | sldSolv |Ang^-2| 6.3e-06 |
---|
2684 | radius |Ang| 40 |
---|
2685 | sld_Sph |Ang^-2| 3e-06 |
---|
2686 | d_factor None 0.06 |
---|
2687 | ============== ======== ============= |
---|
2688 | |
---|
2689 | This example dataset is produced using 200 data points, *qmin* = 0.01 |Ang^-1|, *qmax* = 0.1 |Ang^-1| and the above |
---|
2690 | default values. |
---|
2691 | |
---|
2692 | .. image:: ..\img\olddocs\image155.jpg |
---|
2693 | |
---|
2694 | *Figure. 1D plot in the linear scale using the default values (w/200 data point).* |
---|
2695 | |
---|
2696 | The 2D (Anisotropic model) is based on the reference below where *I(q)* is approximated for 1d scattering. Thus the |
---|
2697 | scattering pattern for 2D may not be accurate. Note that we are not responsible for any incorrectness of the 2D model |
---|
2698 | computation. |
---|
2699 | |
---|
2700 | .. image:: ..\img\olddocs\image156.jpg |
---|
2701 | |
---|
2702 | .. image:: ..\img\olddocs\image157.jpg |
---|
2703 | |
---|
2704 | *Figure. 2D plot using the default values (w/200X200 pixels).* |
---|
2705 | |
---|
2706 | REFERENCE |
---|
2707 | |
---|
2708 | Hideki Matsuoka et. al. *Physical Review B*, 36 (1987) 1754-1765 |
---|
2709 | (Original Paper) |
---|
2710 | |
---|
2711 | Hideki Matsuoka et. al. *Physical Review B*, 41 (1990) 3854 -3856 |
---|
2712 | (Corrections to FCC and BCC lattice structure calculation) |
---|
2713 | |
---|
2714 | |
---|
2715 | |
---|
2716 | .. _FCCrystalModel: |
---|
2717 | |
---|
2718 | **2.1.35. FCCrystalModel** |
---|
2719 | |
---|
2720 | Calculates the scattering from a **face-centered cubic lattice** with paracrystalline distortion. Thermal vibrations |
---|
2721 | are considered to be negligible, and the size of the paracrystal is infinitely large. Paracrystalline distortion is |
---|
2722 | assumed to be isotropic and characterized by a Gaussian distribution. |
---|
2723 | |
---|
2724 | The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. |
---|
2725 | |
---|
2726 | *2.1.35.1. Definition* |
---|
2727 | |
---|
2728 | The scattering intensity *I(q)* is calculated as |
---|
2729 | |
---|
2730 | .. image:: ..\img\olddocs\image158.jpg |
---|
2731 | |
---|
2732 | where *scale* is the volume fraction of spheres, *Vp* is the volume of the primary particle, *V(lattice)* is a volume |
---|
2733 | correction for the crystal structure, *P(q)* is the form factor of the sphere (normalized), and *Z(q)* is the |
---|
2734 | paracrystalline structure factor for a face-centered cubic structure. |
---|
2735 | |
---|
2736 | Equation (1) of the 1990 reference is used to calculate *Z(q)*, using equations (23)-(25) from the 1987 paper for |
---|
2737 | *Z1*\ , *Z2*\ , and *Z3*\ . |
---|
2738 | |
---|
2739 | The lattice correction (the occupied volume of the lattice) for a face-centered cubic structure of particles of radius |
---|
2740 | *R* and nearest neighbor separation *D* is |
---|
2741 | |
---|
2742 | .. image:: ..\img\olddocs\image159.jpg |
---|
2743 | |
---|
2744 | The distortion factor (one standard deviation) of the paracrystal is included in the calculation of *Z(q)* |
---|
2745 | |
---|
2746 | .. image:: ..\img\olddocs\image160.jpg |
---|
2747 | |
---|
2748 | where *g* is a fractional distortion based on the nearest neighbor distance. |
---|
2749 | |
---|
2750 | The face-centered cubic lattice is |
---|
2751 | |
---|
2752 | .. image:: ..\img\olddocs\image161.jpg |
---|
2753 | |
---|
2754 | For a crystal, diffraction peaks appear at reduced q-values given by |
---|
2755 | |
---|
2756 | .. image:: ..\img\olddocs\image162.jpg |
---|
2757 | |
---|
2758 | where for a face-centered cubic lattice *h*\ , *k*\ , *l* all odd or all even are allowed and reflections where |
---|
2759 | *h*\ , *k*\ , *l* are mixed odd/even are forbidden. Thus the peak positions correspond to (just the first 5) |
---|
2760 | |
---|
2761 | .. image:: ..\img\olddocs\image163.jpg |
---|
2762 | |
---|
2763 | **NB: The calculation of** *Z(q)* **is a double numerical integral that must be carried out with a high density of** |
---|
2764 | **points to properly capture the sharp peaks of the paracrystalline scattering.** So be warned that the calculation is |
---|
2765 | SLOW. Go get some coffee. Fitting of any experimental data must be resolution smeared for any meaningful fit. This |
---|
2766 | makes a triple integral. Very, very slow. Go get lunch! |
---|
2767 | |
---|
2768 | ============== ======== ============= |
---|
2769 | Parameter name Units Default value |
---|
2770 | ============== ======== ============= |
---|
2771 | background |cm^-1| 0 |
---|
2772 | dnn |Ang| 220 |
---|
2773 | scale None 1 |
---|
2774 | sldSolv |Ang^-2| 6.3e-06 |
---|
2775 | radius |Ang| 40 |
---|
2776 | sld_Sph |Ang^-2| 3e-06 |
---|
2777 | d_factor None 0.06 |
---|
2778 | ============== ======== ============= |
---|
2779 | |
---|
2780 | This example dataset is produced using 200 data points, *qmin* = 0.01 |Ang^-1|, *qmax* = 0.1 |Ang^-1| and the above |
---|
2781 | default values. |
---|
2782 | |
---|
2783 | .. image:: ..\img\olddocs\image164.jpg |
---|
2784 | |
---|
2785 | *Figure. 1D plot in the linear scale using the default values (w/200 data point).* |
---|
2786 | |
---|
2787 | The 2D (Anisotropic model) is based on the reference below where *I(q)* is approximated for 1d scattering. Thus the |
---|
2788 | scattering pattern for 2D may not be accurate. Note that we are not responsible for any incorrectness of the 2D model |
---|
2789 | computation. |
---|
2790 | |
---|
2791 | .. image:: ..\img\olddocs\image165.gif |
---|
2792 | |
---|
2793 | .. image:: ..\img\olddocs\image166.jpg |
---|
2794 | |
---|
2795 | *Figure. 2D plot using the default values (w/200X200 pixels).* |
---|
2796 | |
---|
2797 | REFERENCE |
---|
2798 | |
---|
2799 | Hideki Matsuoka et. al. *Physical Review B*, 36 (1987) 1754-1765 |
---|
2800 | (Original Paper) |
---|
2801 | |
---|
2802 | Hideki Matsuoka et. al. *Physical Review B*, 41 (1990) 3854 -3856 |
---|
2803 | (Corrections to FCC and BCC lattice structure calculation) |
---|
2804 | |
---|
2805 | |
---|
2806 | |
---|
2807 | .. _BCCrystalModel: |
---|
2808 | |
---|
2809 | **2.1.36. BCCrystalModel** |
---|
2810 | |
---|
2811 | Calculates the scattering from a **body-centered cubic lattice** with paracrystalline distortion. Thermal vibrations |
---|
2812 | are considered to be negligible, and the size of the paracrystal is infinitely large. Paracrystalline distortion is |
---|
2813 | assumed to be isotropic and characterized by a Gaussian distribution. |
---|
2814 | |
---|
2815 | The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale. |
---|
2816 | |
---|
2817 | *2.1.36.1. Definition** |
---|
2818 | |
---|
2819 | The scattering intensity *I(q)* is calculated as |
---|
2820 | |
---|
2821 | .. image:: ..\img\olddocs\image167.jpg |
---|
2822 | |
---|
2823 | where *scale* is the volume fraction of spheres, *Vp* is the volume of the primary particle, *V(lattice)* is a volume |
---|
2824 | correction for the crystal structure, *P(q)* is the form factor of the sphere (normalized), and *Z(q)* is the |
---|
2825 | paracrystalline structure factor for a body-centered cubic structure. |
---|
2826 | |
---|
2827 | Equation (1) of the 1990 reference is used to calculate *Z(q)*, using equations (29)-(31) from the 1987 paper for |
---|
2828 | *Z1*\ , *Z2*\ , and *Z3*\ . |
---|
2829 | |
---|
2830 | The lattice correction (the occupied volume of the lattice) for a body-centered cubic structure of particles of radius |
---|
2831 | *R* and nearest neighbor separation *D* is |
---|
2832 | |
---|
2833 | .. image:: ..\img\olddocs\image159.jpg |
---|
2834 | |
---|
2835 | The distortion factor (one standard deviation) of the paracrystal is included in the calculation of *Z(q)* |
---|
2836 | |
---|
2837 | .. image:: ..\img\olddocs\image160.jpg |
---|
2838 | |
---|
2839 | where *g* is a fractional distortion based on the nearest neighbor distance. |
---|
2840 | |
---|
2841 | The body-centered cubic lattice is |
---|
2842 | |
---|
2843 | .. image:: ..\img\olddocs\image168.jpg |
---|
2844 | |
---|
2845 | For a crystal, diffraction peaks appear at reduced q-values given by |
---|
2846 | |
---|
2847 | .. image:: ..\img\olddocs\image162.jpg |
---|
2848 | |
---|
2849 | where for a body-centered cubic lattice, only reflections where (\ *h* + *k* + *l*\ ) = even are allowed and |
---|
2850 | reflections where (\ *h* + *k* + *l*\ ) = odd are forbidden. Thus the peak positions correspond to (just the first 5) |
---|
2851 | |
---|
2852 | .. image:: ..\img\olddocs\image169.jpg |
---|
2853 | |
---|
2854 | **NB: The calculation of** *Z(q)* **is a double numerical integral that must be carried out with a high density of** |
---|
2855 | **points to properly capture the sharp peaks of the paracrystalline scattering.** So be warned that the calculation is |
---|
2856 | SLOW. Go get some coffee. Fitting of any experimental data must be resolution smeared for any meaningful fit. This |
---|
2857 | makes a triple integral. Very, very slow. Go get lunch! |
---|
2858 | |
---|
2859 | ============== ======== ============= |
---|
2860 | Parameter name Units Default value |
---|
2861 | ============== ======== ============= |
---|
2862 | background |cm^-1| 0 |
---|
2863 | dnn |Ang| 220 |
---|
2864 | scale None 1 |
---|
2865 | sldSolv |Ang^-2| 6.3e-006 |
---|
2866 | radius |Ang| 40 |
---|
2867 | sld_Sph |Ang^-2| 3e-006 |
---|
2868 | d_factor None 0.06 |
---|
2869 | ============== ======== ============= |
---|
2870 | |
---|
2871 | This example dataset is produced using 200 data points, *qmin* = 0.001 |Ang^-1|, *qmax* = 0.1 |Ang^-1| and the above |
---|
2872 | default values. |
---|
2873 | |
---|
2874 | .. image:: ..\img\olddocs\image170.jpg |
---|
2875 | |
---|
2876 | *Figure. 1D plot in the linear scale using the default values (w/200 data point).* |
---|
2877 | |
---|
2878 | The 2D (Anisotropic model) is based on the reference below where *I(q)* is approximated for 1d scattering. Thus the |
---|
2879 | scattering pattern for 2D may not be accurate. Note that we are not responsible for any incorrectness of the 2D model |
---|
2880 | computation. |
---|
2881 | |
---|
2882 | .. image:: ..\img\olddocs\image165.gif |
---|
2883 | |
---|
2884 | .. image:: ..\img\olddocs\image171.jpg |
---|
2885 | |
---|
2886 | *Figure. 2D plot using the default values (w/200X200 pixels).* |
---|
2887 | |
---|
2888 | REFERENCE |
---|
2889 | |
---|
2890 | Hideki Matsuoka et. al. *Physical Review B*, 36 (1987) 1754-1765 |
---|
2891 | (Original Paper) |
---|
2892 | |
---|
2893 | Hideki Matsuoka et. al. *Physical Review B*, 41 (1990) 3854 -3856 |
---|
2894 | (Corrections to FCC and BCC lattice structure calculation) |
---|
2895 | |
---|
2896 | |
---|
2897 | |
---|
2898 | .. _ParallelepipedModel: |
---|
2899 | |
---|
2900 | **2.1.37. ParallelepipedModel** |
---|
2901 | |
---|
2902 | This model provides the form factor, *P(q)*, for a rectangular cylinder (below) where the form factor is normalized by |
---|
2903 | the volume of the cylinder. If you need to apply polydispersity, see the RectangularPrismModel_. |
---|
2904 | |
---|
2905 | *P(q)* = *scale* \* <*f*\ :sup:`2`> / *V* + *background* |
---|
2906 | |
---|
2907 | where the volume *V* = *A B C* and the averaging < > is applied over all orientations for 1D. |
---|
2908 | |
---|
2909 | For information about polarised and magnetic scattering, click here_. |
---|
2910 | |
---|
2911 | .. image:: ..\img\olddocs\image087.jpg |
---|
2912 | |
---|
2913 | *2.1.37.1. Definition* |
---|
2914 | |
---|
2915 | **The edge of the solid must satisfy the condition that** *A* < *B*. Then, assuming *a* = *A* / *B* < 1, |
---|
2916 | *b* = *B* / *B* = 1, and *c* = *C* / *B* > 1, the form factor is |
---|
2917 | |
---|
2918 | .. image:: ..\img\olddocs\image088.PNG |
---|
2919 | |
---|
2920 | and the contrast is defined as |
---|
2921 | |
---|
2922 | .. image:: ..\img\olddocs\image089.PNG |
---|
2923 | |
---|
2924 | The scattering intensity per unit volume is returned in units of |cm^-1|; ie, *I(q)* = |phi| *P(q)*\ . |
---|
2925 | |
---|
2926 | NB: The 2nd virial coefficient of the parallelpiped is calculated based on the the averaged effective radius |
---|
2927 | (= sqrt(*short_a* \* *short_b* / |pi|)) and length(= *long_c*) values, and used as the effective radius for |
---|
2928 | *S(Q)* when *P(Q)* \* *S(Q)* is applied. |
---|
2929 | |
---|
2930 | To provide easy access to the orientation of the parallelepiped, we define the axis of the cylinder using three angles |
---|
2931 | |theta|, |phi| and |bigpsi|. These angles are defined on Figure 2 of the CylinderModel_. The angle |bigpsi| is the |
---|
2932 | rotational angle around the *long_c* axis against the *q* plane. For example, |bigpsi| = 0 when the *short_b* axis is |
---|
2933 | parallel to the *x*-axis of the detector. |
---|
2934 | |
---|
2935 | .. image:: ..\img\olddocs\image090.jpg |
---|
2936 | |
---|
2937 | *Figure. Definition of angles for 2D*. |
---|
2938 | |
---|
2939 | .. image:: ..\img\olddocs\image091.jpg |
---|
2940 | |
---|
2941 | *Figure. Examples of the angles for oriented pp against the detector plane.* |
---|
2942 | |
---|
2943 | ============== ======== ============= |
---|
2944 | Parameter name Units Default value |
---|
2945 | ============== ======== ============= |
---|
2946 | background |cm^-1| 0.0 |
---|
2947 | contrast |Ang^-2| 5e-06 |
---|
2948 | long_c |Ang| 400 |
---|
2949 | short_a |Ang^-2| 35 |
---|
2950 | short_b |Ang| 75 |
---|
2951 | scale None 1 |
---|
2952 | ============== ======== ============= |
---|
2953 | |
---|
2954 | .. image:: ..\img\olddocs\image092.jpg |
---|
2955 | |
---|
2956 | *Figure. 1D plot using the default values (w/1000 data point).* |
---|
2957 | |
---|
2958 | *2.1.37.2. Validation of the parallelepiped 2D model* |
---|
2959 | |
---|
2960 | Validation of our code was done by comparing the output of the 1D calculation to the angular average of the output of |
---|
2961 | a 2D calculation over all possible angles. The Figure below shows the comparison where the solid dot refers to averaged |
---|
2962 | 2D while the line represents the result of the 1D calculation (for the averaging, 76, 180, 76 points are taken for the |
---|
2963 | angles of |theta|, |phi|, and |psi| respectively). |
---|
2964 | |
---|
2965 | .. image:: ..\img\olddocs\image093.gif |
---|
2966 | |
---|
2967 | *Figure. Comparison between 1D and averaged 2D.* |
---|
2968 | |
---|
2969 | Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research |
---|
2970 | (Kline, 2006). |
---|
2971 | |
---|
2972 | REFERENCE |
---|
2973 | |
---|
2974 | P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211 |
---|
2975 | Equations (1), (13-14). (in German) |
---|
2976 | |
---|
2977 | |
---|
2978 | |
---|
2979 | .. _CSParallelepipedModel: |
---|
2980 | |
---|
2981 | **2.1.38. CSParallelepipedModel** |
---|
2982 | |
---|
2983 | Calculates the form factor for a rectangular solid with a core-shell structure. **The thickness and the scattering** |
---|
2984 | **length density of the shell or "rim" can be different on all three (pairs) of faces.** |
---|
2985 | |
---|
2986 | The form factor is normalized by the particle volume *V* such that |
---|
2987 | |
---|
2988 | *P(q)* = *scale* \* <*f*\ :sup:`2`> / *V* + *background* |
---|
2989 | |
---|
2990 | where < > is an average over all possible orientations of the rectangular solid. |
---|
2991 | |
---|
2992 | An instrument resolution smeared version of the model is also provided. |
---|
2993 | |
---|
2994 | *2.1.38.1. Definition* |
---|
2995 | |
---|
2996 | The function calculated is the form factor of the rectangular solid below. The core of the solid is defined by the |
---|
2997 | dimensions *A*, *B*, *C* such that *A* < *B* < *C*. |
---|
2998 | |
---|
2999 | .. image:: ..\img\olddocs\image087.jpg |
---|
3000 | |
---|
3001 | There are rectangular "slabs" of thickness *tA* that add to the *A* dimension (on the *BC* faces). There are similar |
---|
3002 | slabs on the *AC* (= *tB*) and *AB* (= *tC*) faces. The projection in the *AB* plane is then |
---|
3003 | |
---|
3004 | .. image:: ..\img\olddocs\image094.jpg |
---|
3005 | |
---|
3006 | The volume of the solid is |
---|
3007 | |
---|
3008 | .. image:: ..\img\olddocs\image095.PNG |
---|
3009 | |
---|
3010 | **meaning that there are "gaps" at the corners of the solid.** |
---|
3011 | |
---|
3012 | The intensity calculated follows the ParallelepipedModel_, with the core-shell intensity being calculated as the |
---|
3013 | square of the sum of the amplitudes of the core and shell, in the same manner as a CoreShellModel_. |
---|
3014 | |
---|
3015 | **For the calculation of the form factor to be valid, the sides of the solid MUST be chosen such that** *A* < *B* < *C*. |
---|
3016 | **If this inequality is not satisfied, the model will not report an error, and the calculation will not be correct.** |
---|
3017 | |
---|
3018 | FITTING NOTES |
---|
3019 | If the scale is set equal to the particle volume fraction, |phi|, the returned value is the scattered intensity per |
---|
3020 | unit volume; ie, *I(q)* = |phi| *P(q)*\ . However, **no interparticle interference effects are included in this** |
---|
3021 | **calculation.** |
---|
3022 | |
---|
3023 | There are many parameters in this model. Hold as many fixed as possible with known values, or you will certainly end |
---|
3024 | up at a solution that is unphysical. |
---|
3025 | |
---|
3026 | Constraints must be applied during fitting to ensure that the inequality *A* < *B* < *C* is not violated. The |
---|
3027 | calculation will not report an error, but the results will not be correct. |
---|
3028 | |
---|
3029 | The returned value is in units of |cm^-1|, on absolute scale. |
---|
3030 | |
---|
3031 | NB: The 2nd virial coefficient of the CSParallelpiped is calculated based on the the averaged effective radius |
---|
3032 | (= sqrt((*short_a* + 2 *rim_a*) \* (*short_b* + 2 *rim_b*) / |pi|)) and length(= *long_c* + 2 *rim_c*) values, and |
---|
3033 | used as the effective radius for *S(Q)* when *P(Q)* \* *S(Q)* is applied. |
---|
3034 | |
---|
3035 | To provide easy access to the orientation of the parallelepiped, we define the axis of the cylinder using three angles |
---|
3036 | |theta|, |phi| and |bigpsi|. These angles are defined on Figure 2 of the CylinderModel_. The angle |bigpsi| is the |
---|
3037 | rotational angle around the *long_c* axis against the *q* plane. For example, |bigpsi| = 0 when the *short_b* axis is |
---|
3038 | parallel to the *x*-axis of the detector. |
---|
3039 | |
---|
3040 | .. image:: ..\img\olddocs\image090.jpg |
---|
3041 | |
---|
3042 | *Figure. Definition of angles for 2D*. |
---|
3043 | |
---|
3044 | .. image:: ..\img\olddocs\image091.jpg |
---|
3045 | |
---|
3046 | *Figure. Examples of the angles for oriented cspp against the detector plane.* |
---|
3047 | |
---|
3048 | This example dataset was produced by running the Macro Plot_CSParallelepiped(), using 100 data points, |
---|
3049 | *qmin* = 0.001 |Ang^-1|, *qmax* = 0.7 |Ang^-1| and the default values |
---|
3050 | |
---|
3051 | ============== ======== ============= |
---|
3052 | Parameter name Units Default value |
---|
3053 | ============== ======== ============= |
---|
3054 | background |cm^-1| 0.06 |
---|
3055 | sld_pcore |Ang^-2| 1e-06 |
---|
3056 | sld_rimA |Ang^-2| 2e-06 |
---|
3057 | sld_rimB |Ang^-2| 4e-06 |
---|
3058 | sld_rimC |Ang^-2| 2e-06 |
---|
3059 | sld_solv |Ang^-2| 6e-06 |
---|
3060 | rimA |Ang| 10 |
---|
3061 | rimB |Ang| 10 |
---|
3062 | rimC |Ang| 10 |
---|
3063 | longC |Ang| 400 |
---|
3064 | shortA |Ang| 35 |
---|
3065 | midB |Ang| 75 |
---|
3066 | scale None 1 |
---|
3067 | ============== ======== ============= |
---|
3068 | |
---|
3069 | .. image:: ..\img\olddocs\image096.jpg |
---|
3070 | |
---|
3071 | *Figure. 1D plot using the default values (w/256 data points).* |
---|
3072 | |
---|
3073 | .. image:: ..\img\olddocs\image097.jpg |
---|
3074 | |
---|
3075 | *Figure. 2D plot using the default values (w/(256X265) data points).* |
---|
3076 | |
---|
3077 | Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research |
---|
3078 | (Kline, 2006). |
---|
3079 | |
---|
3080 | REFERENCE |
---|
3081 | |
---|
3082 | P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211 |
---|
3083 | Equations (1), (13-14). (in German) |
---|
3084 | |
---|
3085 | |
---|
3086 | |
---|
3087 | .. _RectangularPrismModel: |
---|
3088 | |
---|
3089 | **2.1.39. RectangularPrismModel** |
---|
3090 | |
---|
3091 | This model provides the form factor, *P(q)*, for a rectangular prism. |
---|
3092 | |
---|
3093 | Note that this model is almost totally equivalent to the existing ParallelepipedModel_. The only difference is that the |
---|
3094 | way the relevant parameters are defined here (*a*, *b/a*, *c/a* instead of *a*, *b*, *c*) allows to use polydispersity |
---|
3095 | with this model while keeping the shape of the prism (e.g. setting *b/a* = 1 and *c/a* = 1 and applying polydispersity |
---|
3096 | to *a* will generate a distribution of cubes of different sizes). |
---|
3097 | |
---|
3098 | *2.1.39.1. Definition* |
---|
3099 | |
---|
3100 | The 1D scattering intensity for this model was calculated by Mittelbach and Porod (Mittelbach, 1961), but the |
---|
3101 | implementation here is closer to the equations given by Nayuk and Huber (Nayuk, 2012). |
---|
3102 | |
---|
3103 | The scattering from a massive parallelepiped with an orientation with respect to the scattering vector given by |theta| |
---|
3104 | and |phi| is given by |
---|
3105 | |
---|
3106 | .. math:: |
---|
3107 | A_P\,(q) = \frac{\sin \bigl( q \frac{C}{2} \cos\theta \bigr)}{q \frac{C}{2} \cos\theta} \, \times \, |
---|
3108 | \frac{\sin \bigl( q \frac{A}{2} \sin\theta \sin\phi \bigr)}{q \frac{A}{2} \sin\theta \sin\phi} \, \times \, |
---|
3109 | \frac{\sin \bigl( q \frac{B}{2} \sin\theta \cos\phi \bigr)}{q \frac{B}{2} \sin\theta \cos\phi} |
---|
3110 | |
---|
3111 | where *A*, *B* and *C* are the sides of the parallelepiped and must fulfill :math:`A \le B \le C`, |theta| is the angle |
---|
3112 | between the *z* axis and the longest axis of the parallelepiped *C*, and |phi| is the angle between the scattering |
---|
3113 | vector (lying in the *xy* plane) and the *y* axis. |
---|
3114 | |
---|
3115 | The normalized form factor in 1D is obtained averaging over all possible orientations |
---|
3116 | |
---|
3117 | .. math:: |
---|
3118 | P(q) = \frac{2}{\pi} \times \, \int_0^{\frac{\pi}{2}} \, \int_0^{\frac{\pi}{2}} A_P^2(q) \, \sin\theta \, d\theta \, d\phi |
---|
3119 | |
---|
3120 | The 1D scattering intensity is then calculated as |
---|
3121 | |
---|
3122 | .. math:: |
---|
3123 | I(q) = \mbox{scale} \times V \times (\rho_{\mbox{pipe}} - \rho_{\mbox{solvent}})^2 \times P(q) |
---|
3124 | |
---|
3125 | where *V* is the volume of the rectangular prism, :math:`\rho_{\mbox{pipe}}` is the scattering length of the |
---|
3126 | parallelepiped, :math:`\rho_{\mbox{solvent}}` is the scattering length of the solvent, and (if the data are in absolute |
---|
3127 | units) *scale* represents the volume fraction (which is unitless). |
---|
3128 | |
---|
3129 | **The 2D scattering intensity is not computed by this model.** |
---|
3130 | |
---|
3131 | The returned value is scaled to units of |cm^-1| and the parameters of the RectangularPrismModel are the following |
---|
3132 | |
---|
3133 | ============== ======== ============= |
---|
3134 | Parameter name Units Default value |
---|
3135 | ============== ======== ============= |
---|
3136 | scale None 1 |
---|
3137 | short_side |Ang| 35 |
---|
3138 | b2a_ratio None 1 |
---|
3139 | c2a_ratio None 1 |
---|
3140 | sldPipe |Ang^-2| 6.3e-6 |
---|
3141 | sldSolv |Ang^-2| 1.0e-6 |
---|
3142 | background |cm^-1| 0 |
---|
3143 | ============== ======== ============= |
---|
3144 | |
---|
3145 | *2.1.39.2. Validation of the RectangularPrismModel* |
---|
3146 | |
---|
3147 | Validation of the code was conducted by comparing the output of the 1D model to the output of the existing |
---|
3148 | parallelepiped model. |
---|
3149 | |
---|
3150 | REFERENCES |
---|
3151 | |
---|
3152 | P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211 |
---|
3153 | |
---|
3154 | R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854 |
---|
3155 | |
---|
3156 | |
---|
3157 | |
---|
3158 | .. _RectangularHollowPrismModel: |
---|
3159 | |
---|
3160 | **2.1.40. RectangularHollowPrismModel** |
---|
3161 | |
---|
3162 | This model provides the form factor, *P(q)*, for a hollow rectangular parallelepiped with a wall thickness |bigdelta|. |
---|
3163 | |
---|
3164 | *2.1.40.1. Definition* |
---|
3165 | |
---|
3166 | The 1D scattering intensity for this model is calculated by forming the difference of the amplitudes of two massive |
---|
3167 | parallelepipeds differing in their outermost dimensions in each direction by the same length increment 2 |bigdelta| |
---|
3168 | (Nayuk, 2012). |
---|
3169 | |
---|
3170 | As in the case of the massive parallelepiped, the scattering amplitude is computed for a particular orientation of the |
---|
3171 | parallelepiped with respect to the scattering vector and then averaged over all possible orientations, giving |
---|
3172 | |
---|
3173 | .. math:: |
---|
3174 | P(q) = \frac{1}{V^2} \frac{2}{\pi} \times \, \int_0^{\frac{\pi}{2}} \, \int_0^{\frac{\pi}{2}} A_{P\Delta}^2(q) \, |
---|
3175 | \sin\theta \, d\theta \, d\phi |
---|
3176 | |
---|
3177 | where |theta| is the angle between the *z* axis and the longest axis of the parallelepiped, |phi| is the angle between |
---|
3178 | the scattering vector (lying in the *xy* plane) and the *y* axis, and |
---|
3179 | |
---|
3180 | .. math:: |
---|
3181 | A_{P\Delta}\,(q) = A \, B \, C \, \times |
---|
3182 | \frac{\sin \bigl( q \frac{C}{2} \cos\theta \bigr)}{q \frac{C}{2} \cos\theta} \, |
---|
3183 | \frac{\sin \bigl( q \frac{A}{2} \sin\theta \sin\phi \bigr)}{q \frac{A}{2} \sin\theta \sin\phi} \, |
---|
3184 | \frac{\sin \bigl( q \frac{B}{2} \sin\theta \cos\phi \bigr)}{q \frac{B}{2} \sin\theta \cos\phi} - |
---|
3185 | 8 \, \bigl( \frac{A}{2} - \Delta \bigr) \, \bigl( \frac{B}{2} - \Delta \bigr) \, |
---|
3186 | \bigl( \frac{C}{2} - \Delta \bigr) \, \times |
---|
3187 | \frac{\sin \bigl[ q \bigl(\frac{C}{2}-\Delta\bigr) \cos\theta \bigr]} |
---|
3188 | {q \bigl(\frac{C}{2}-\Delta\bigr) \cos\theta} \, |
---|
3189 | \frac{\sin \bigl[ q \bigl(\frac{A}{2}-\Delta\bigr) \sin\theta \sin\phi \bigr]} |
---|
3190 | {q \bigl(\frac{A}{2}-\Delta\bigr) \sin\theta \sin\phi} \, |
---|
3191 | \frac{\sin \bigl[ q \bigl(\frac{B}{2}-\Delta\bigr) \sin\theta \cos\phi \bigr]} |
---|
3192 | {q \bigl(\frac{B}{2}-\Delta\bigr) \sin\theta \cos\phi} \, |
---|
3193 | |
---|
3194 | where *A*, *B* and *C* are the external sides of the parallelepiped fulfilling :math:`A \le B \le C`, and the volume *V* |
---|
3195 | of the parallelepiped is |
---|
3196 | |
---|
3197 | .. math:: |
---|
3198 | V = A B C \, - \, (A - 2\Delta) (B - 2\Delta) (C - 2\Delta) |
---|
3199 | |
---|
3200 | The 1D scattering intensity is then calculated as |
---|
3201 | |
---|
3202 | .. math:: |
---|
3203 | I(q) = \mbox{scale} \times V \times (\rho_{\mbox{pipe}} - \rho_{\mbox{solvent}})^2 \times P(q) |
---|
3204 | |
---|
3205 | where :math:`\rho_{\mbox{pipe}}` is the scattering length of the parallelepiped, :math:`\rho_{\mbox{solvent}}` is the |
---|
3206 | scattering length of the solvent, and (if the data are in absolute units) *scale* represents the volume fraction (which |
---|
3207 | is unitless). |
---|
3208 | |
---|
3209 | **The 2D scattering intensity is not computed by this model.** |
---|
3210 | |
---|
3211 | The returned value is scaled to units of |cm^-1| and the parameters of the RectangularHollowPrismModel are the |
---|
3212 | following |
---|
3213 | |
---|
3214 | ============== ======== ============= |
---|
3215 | Parameter name Units Default value |
---|
3216 | ============== ======== ============= |
---|
3217 | scale None 1 |
---|
3218 | short_side |Ang| 35 |
---|
3219 | b2a_ratio None 1 |
---|
3220 | c2a_ratio None 1 |
---|
3221 | thickness |Ang| 1 |
---|
3222 | sldPipe |Ang^-2| 6.3e-6 |
---|
3223 | sldSolv |Ang^-2| 1.0e-6 |
---|
3224 | background |cm^-1| 0 |
---|
3225 | ============== ======== ============= |
---|
3226 | |
---|
3227 | *2.1.40.2. Validation of the RectangularHollowPrismModel* |
---|
3228 | |
---|
3229 | Validation of the code was conducted by qualitatively comparing the output of the 1D model to the curves shown in |
---|
3230 | (Nayuk, 2012). |
---|
3231 | |
---|
3232 | REFERENCES |
---|
3233 | |
---|
3234 | R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854 |
---|
3235 | |
---|
3236 | |
---|
3237 | |
---|
3238 | .. _RectangularHollowPrismInfThinWallsModel: |
---|
3239 | |
---|
3240 | **2.1.41. RectangularHollowPrismInfThinWallsModel** |
---|
3241 | |
---|
3242 | This model provides the form factor, *P(q)*, for a hollow rectangular prism with infinitely thin walls. |
---|
3243 | |
---|
3244 | *2.1.41.1. Definition* |
---|
3245 | |
---|
3246 | The 1D scattering intensity for this model is calculated according to the equations given by Nayuk and Huber |
---|
3247 | (Nayuk, 2012). |
---|
3248 | |
---|
3249 | Assuming a hollow parallelepiped with infinitely thin walls, edge lengths :math:`A \le B \le C` and presenting an |
---|
3250 | orientation with respect to the scattering vector given by |theta| and |phi|, where |theta| is the angle between the |
---|
3251 | *z* axis and the longest axis of the parallelepiped *C*, and |phi| is the angle between the scattering vector |
---|
3252 | (lying in the *xy* plane) and the *y* axis, the form factor is given by |
---|
3253 | |
---|
3254 | .. math:: |
---|
3255 | P(q) = \frac{1}{V^2} \frac{2}{\pi} \times \, \int_0^{\frac{\pi}{2}} \, \int_0^{\frac{\pi}{2}} [A_L(q)+A_T(q)]^2 |
---|
3256 | \, \sin\theta \, d\theta \, d\phi |
---|
3257 | |
---|
3258 | where |
---|
3259 | |
---|
3260 | .. math:: |
---|
3261 | V = 2AB + 2AC + 2BC |
---|
3262 | |
---|
3263 | .. math:: |
---|
3264 | A_L\,(q) = 8 \times \frac{ \sin \bigl( q \frac{A}{2} \sin\phi \sin\theta \bigr) |
---|
3265 | \sin \bigl( q \frac{B}{2} \cos\phi \sin\theta \bigr) |
---|
3266 | \cos \bigl( q \frac{C}{2} \cos\theta \bigr) } |
---|
3267 | {q^2 \, \sin^2\theta \, \sin\phi \cos\phi} |
---|
3268 | |
---|
3269 | .. math:: |
---|
3270 | A_T\,(q) = A_F\,(q) \times \frac{2 \, \sin \bigl( q \frac{C}{2} \cos\theta \bigr)}{q \, \cos\theta} |
---|
3271 | |
---|
3272 | and |
---|
3273 | |
---|
3274 | .. math:: |
---|
3275 | A_F\,(q) = 4 \frac{ \cos \bigl( q \frac{A}{2} \sin\phi \sin\theta \bigr) |
---|
3276 | \sin \bigl( q \frac{B}{2} \cos\phi \sin\theta \bigr) } |
---|
3277 | {q \, \cos\phi \, \sin\theta} + |
---|
3278 | 4 \frac{ \sin \bigl( q \frac{A}{2} \sin\phi \sin\theta \bigr) |
---|
3279 | \cos \bigl( q \frac{B}{2} \cos\phi \sin\theta \bigr) } |
---|
3280 | {q \, \sin\phi \, \sin\theta} |
---|
3281 | |
---|
3282 | The 1D scattering intensity is then calculated as |
---|
3283 | |
---|
3284 | .. math:: |
---|
3285 | I(q) = \mbox{scale} \times V \times (\rho_{\mbox{pipe}} - \rho_{\mbox{solvent}})^2 \times P(q) |
---|
3286 | |
---|
3287 | where *V* is the volume of the rectangular prism, :math:`\rho_{\mbox{pipe}}` is the scattering length of the |
---|
3288 | parallelepiped, :math:`\rho_{\mbox{solvent}}` is the scattering length of the solvent, and (if the data are in absolute |
---|
3289 | units) *scale* represents the volume fraction (which is unitless). |
---|
3290 | |
---|
3291 | **The 2D scattering intensity is not computed by this model.** |
---|
3292 | |
---|
3293 | The returned value is scaled to units of |cm^-1| and the parameters of the RectangularHollowPrismInfThinWallModel |
---|
3294 | are the following |
---|
3295 | |
---|
3296 | ============== ======== ============= |
---|
3297 | Parameter name Units Default value |
---|
3298 | ============== ======== ============= |
---|
3299 | scale None 1 |
---|
3300 | short_side |Ang| 35 |
---|
3301 | b2a_ratio None 1 |
---|
3302 | c2a_ratio None 1 |
---|
3303 | sldPipe |Ang^-2| 6.3e-6 |
---|
3304 | sldSolv |Ang^-2| 1.0e-6 |
---|
3305 | background |cm^-1| 0 |
---|
3306 | ============== ======== ============= |
---|
3307 | |
---|
3308 | *2.1.41.2. Validation of the RectangularHollowPrismInfThinWallsModel* |
---|
3309 | |
---|
3310 | Validation of the code was conducted by qualitatively comparing the output of the 1D model to the curves shown in |
---|
3311 | (Nayuk, 2012). |
---|
3312 | |
---|
3313 | REFERENCES |
---|
3314 | |
---|
3315 | R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854 |
---|
3316 | |
---|
3317 | |
---|
3318 | |
---|
3319 | .. _MicelleSphCoreModel: |
---|
3320 | |
---|
3321 | **2.1.42. MicelleSphCoreModel** |
---|
3322 | |
---|
3323 | This model provides the form factor, *P(q)*, for a micelle with a spherical core |
---|
3324 | and Gaussian polymer chains attached to the surface. |
---|
3325 | |
---|
3326 | *2.1.42.1. Definition* |
---|
3327 | |
---|
3328 | The 1D scattering intensity for this model is calculated according to the equations given by Pedersen |
---|
3329 | (Pedersen, 2000). |
---|
3330 | |
---|
3331 | *2.1.42.2. Validation of the MicelleSphCoreModel* |
---|
3332 | |
---|
3333 | This model has not yet been validated. Feb2015 |
---|
3334 | |
---|
3335 | REFERENCES |
---|
3336 | |
---|
3337 | J Pedersen, *J. Appl. Cryst.*, 33 (2000) 637-640 |
---|
3338 | |
---|
3339 | |
---|
3340 | |
---|
3341 | 2.2 Shape-independent Functions |
---|
3342 | ------------------------------- |
---|
3343 | |
---|
3344 | The following are models used for shape-independent SAS analysis. |
---|
3345 | |
---|
3346 | .. _Debye: |
---|
3347 | |
---|
3348 | **2.2.1. Debye (Gaussian Coil Model)** |
---|
3349 | |
---|
3350 | The Debye model is a form factor for a linear polymer chain obeying Gaussian statistics (ie, it is in the theta state). |
---|
3351 | In addition to the radius-of-gyration, *Rg*, a scale factor *scale*, and a constant background term are included in the |
---|
3352 | calculation. **NB: No size polydispersity is included in this model, use the** Poly_GaussCoil_ **Model instead** |
---|
3353 | |
---|
3354 | .. image:: ..\img\olddocs\image172.PNG |
---|
3355 | |
---|
3356 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
3357 | |
---|
3358 | .. image:: ..\img\olddocs\image040.gif |
---|
3359 | |
---|
3360 | ============== ======== ============= |
---|
3361 | Parameter name Units Default value |
---|
3362 | ============== ======== ============= |
---|
3363 | scale None 1.0 |
---|
3364 | rg |Ang| 50.0 |
---|
3365 | background |cm^-1| 0.0 |
---|
3366 | ============== ======== ============= |
---|
3367 | |
---|
3368 | .. image:: ..\img\olddocs\image173.jpg |
---|
3369 | |
---|
3370 | *Figure. 1D plot using the default values (w/200 data point).* |
---|
3371 | |
---|
3372 | REFERENCE |
---|
3373 | |
---|
3374 | R J Roe, *Methods of X-Ray and Neutron Scattering in Polymer Science*, Oxford University Press, New York (2000) |
---|
3375 | |
---|
3376 | |
---|
3377 | |
---|
3378 | .. _BroadPeakModel: |
---|
3379 | |
---|
3380 | **2.2.2. BroadPeakModel** |
---|
3381 | |
---|
3382 | This model calculates an empirical functional form for SAS data characterized by a broad scattering peak. Many SAS |
---|
3383 | spectra are characterized by a broad peak even though they are from amorphous soft materials. For example, soft systems |
---|
3384 | that show a SAS peak include copolymers, polyelectrolytes, multiphase systems, layered structures, etc. |
---|
3385 | |
---|
3386 | The d-spacing corresponding to the broad peak is a characteristic distance between the scattering inhomogeneities (such |
---|
3387 | as in lamellar, cylindrical, or spherical morphologies, or for bicontinuous structures). |
---|
3388 | |
---|
3389 | The returned value is scaled to units of |cm^-1|, absolute scale. |
---|
3390 | |
---|
3391 | *2.2.2.1. Definition* |
---|
3392 | |
---|
3393 | The scattering intensity *I(q)* is calculated as |
---|
3394 | |
---|
3395 | .. image:: ..\img\olddocs\image174.jpg |
---|
3396 | |
---|
3397 | Here the peak position is related to the d-spacing as *Q0* = 2|pi| / *d0*. |
---|
3398 | |
---|
3399 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
3400 | |
---|
3401 | .. image:: ..\img\olddocs\image040.gif |
---|
3402 | |
---|
3403 | ================== ======== ============= |
---|
3404 | Parameter name Units Default value |
---|
3405 | ================== ======== ============= |
---|
3406 | scale_l (=C) None 10 |
---|
3407 | scale_p (=A) None 1e-05 |
---|
3408 | length_l (= |xi| ) |Ang| 50 |
---|
3409 | q_peak (=Q0) |Ang^-1| 0.1 |
---|
3410 | exponent_p (=n) None 2 |
---|
3411 | exponent_l (=m) None 3 |
---|
3412 | Background (=B) |cm^-1| 0.1 |
---|
3413 | ================== ======== ============= |
---|
3414 | |
---|
3415 | .. image:: ..\img\olddocs\image175.jpg |
---|
3416 | |
---|
3417 | *Figure. 1D plot using the default values (w/200 data point).* |
---|
3418 | |
---|
3419 | REFERENCE |
---|
3420 | |
---|
3421 | None. |
---|
3422 | |
---|
3423 | *2013/09/09 - Description reviewed by King, S and Parker, P.* |
---|
3424 | |
---|
3425 | |
---|
3426 | |
---|
3427 | .. _CorrLength: |
---|
3428 | |
---|
3429 | **2.2.3. CorrLength (Correlation Length Model)** |
---|
3430 | |
---|
3431 | Calculates an empirical functional form for SAS data characterized by a low-Q signal and a high-Q signal. |
---|
3432 | |
---|
3433 | The returned value is scaled to units of |cm^-1|, absolute scale. |
---|
3434 | |
---|
3435 | *2.2.3. Definition* |
---|
3436 | |
---|
3437 | The scattering intensity *I(q)* is calculated as |
---|
3438 | |
---|
3439 | .. image:: ..\img\olddocs\image176.jpg |
---|
3440 | |
---|
3441 | The first term describes Porod scattering from clusters (exponent = n) and the second term is a Lorentzian function |
---|
3442 | describing scattering from polymer chains (exponent = *m*). This second term characterizes the polymer/solvent |
---|
3443 | interactions and therefore the thermodynamics. The two multiplicative factors *A* and *C*, the incoherent |
---|
3444 | background *B* and the two exponents *n* and *m* are used as fitting parameters. The final parameter |xi| is a |
---|
3445 | correlation length for the polymer chains. Note that when *m*\ =2 this functional form becomes the familiar Lorentzian |
---|
3446 | function. |
---|
3447 | |
---|
3448 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
3449 | |
---|
3450 | .. image:: ..\img\olddocs\image040.gif |
---|
3451 | |
---|
3452 | ==================== ======== ============= |
---|
3453 | Parameter name Units Default value |
---|
3454 | ==================== ======== ============= |
---|
3455 | scale_l (=C) None  10 |
---|
3456 | scale_p (=A) None  1e-06 |
---|
3457 | length_l (= |xi| ) |Ang| 50 |
---|
3458 | exponent_p (=n) None  2 |
---|
3459 | exponent_l (=m) None 3 |
---|
3460 | Background (=B) |cm^-1| 0.1 |
---|
3461 | ==================== ======== ============= |
---|
3462 | |
---|
3463 | .. image:: ..\img\olddocs\image177.jpg |
---|
3464 | |
---|
3465 | *Figure. 1D plot using the default values (w/500 data points).* |
---|
3466 | |
---|
3467 | REFERENCE |
---|
3468 | |
---|
3469 | B Hammouda, D L Ho and S R Kline, *Insight into Clustering in Poly(ethylene oxide) Solutions*, *Macromolecules*, 37 |
---|
3470 | (2004) 6932-6937 |
---|
3471 | |
---|
3472 | *2013/09/09 - Description reviewed by King, S and Parker, P.* |
---|
3473 | |
---|
3474 | |
---|
3475 | |
---|
3476 | .. _Lorentz: |
---|
3477 | |
---|
3478 | **2.2.4. Lorentz (Ornstein-Zernicke Model)** |
---|
3479 | |
---|
3480 | *2.2.4.1. Definition* |
---|
3481 | |
---|
3482 | The Ornstein-Zernicke model is defined by |
---|
3483 | |
---|
3484 | .. image:: ..\img\olddocs\image178.PNG |
---|
3485 | |
---|
3486 | The parameter *L* is the screening length. |
---|
3487 | |
---|
3488 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
3489 | |
---|
3490 | .. image:: ..\img\olddocs\image040.gif |
---|
3491 | |
---|
3492 | ============== ======== ============= |
---|
3493 | Parameter name Units Default value |
---|
3494 | ============== ======== ============= |
---|
3495 | scale None 1.0 |
---|
3496 | length |Ang| 50.0 |
---|
3497 | background |cm^-1| 0.0 |
---|
3498 | ============== ======== ============= |
---|
3499 | |
---|
3500 | .. image:: ..\img\olddocs\image179.jpg |
---|
3501 | |
---|
3502 | *Â Figure. 1D plot using the default values (w/200 data point).* |
---|
3503 | |
---|
3504 | REFERENCE |
---|
3505 | |
---|
3506 | None. |
---|
3507 | |
---|
3508 | |
---|
3509 | |
---|
3510 | .. _DABModel: |
---|
3511 | |
---|
3512 | **2.2.5. DABModel (Debye-Anderson-Brumberger Model)** |
---|
3513 | |
---|
3514 | Calculates the scattering from a randomly distributed, two-phase system based on the Debye-Anderson-Brumberger (DAB) |
---|
3515 | model for such systems. The two-phase system is characterized by a single length scale, the correlation length, which |
---|
3516 | is a measure of the average spacing between regions of phase 1 and phase 2. **The model also assumes smooth interfaces** |
---|
3517 | **between the phases** and hence exhibits Porod behavior (I ~ *q*\ :sup:`-4`) at large *q* (*QL* >> 1). |
---|
3518 | |
---|
3519 | The DAB model is ostensibly a development of the earlier Debye-Bueche model. |
---|
3520 | |
---|
3521 | *2.2.5.1. Definition* |
---|
3522 | |
---|
3523 | .. image:: ..\img\olddocs\image180_corrected.PNG |
---|
3524 | |
---|
3525 | The parameter *L* is the correlation length. |
---|
3526 | |
---|
3527 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
3528 | |
---|
3529 | .. image:: ..\img\olddocs\image040.gif |
---|
3530 | |
---|
3531 | ============== ======== ============= |
---|
3532 | Parameter name Units Default value |
---|
3533 | ============== ======== ============= |
---|
3534 | scale None 1.0 |
---|
3535 | length |Ang| 50.0 |
---|
3536 | background |cm^-1| 0.0 |
---|
3537 | ============== ======== ============= |
---|
3538 | |
---|
3539 | .. image:: ..\img\olddocs\image181.jpg |
---|
3540 | |
---|
3541 | *Â Figure. 1D plot using the default values (w/200 data point).* |
---|
3542 | |
---|
3543 | REFERENCE |
---|
3544 | |
---|
3545 | P Debye, H R Anderson, H Brumberger, *Scattering by an Inhomogeneous Solid. II. The Correlation Function* |
---|
3546 | *and its Application*, *J. Appl. Phys.*, 28(6) (1957) 679 |
---|
3547 | |
---|
3548 | P Debye, A M Bueche, *Scattering by an Inhomogeneous Solid*, *J. Appl. Phys.*, 20 (1949) 518 |
---|
3549 | |
---|
3550 | *2013/09/09 - Description reviewed by King, S and Parker, P.* |
---|
3551 | |
---|
3552 | |
---|
3553 | |
---|
3554 | .. _AbsolutePower_Law: |
---|
3555 | |
---|
3556 | **2.2.6. AbsolutePower_Law** |
---|
3557 | |
---|
3558 | This model describes a simple power law with background. |
---|
3559 | |
---|
3560 | .. image:: ..\img\olddocs\image182.PNG |
---|
3561 | |
---|
3562 | Note the minus sign in front of the exponent. The parameter *m* should therefore be entered as a **positive** number. |
---|
3563 | |
---|
3564 | ============== ======== ============= |
---|
3565 | Parameter name Units Default value |
---|
3566 | ============== ======== ============= |
---|
3567 | Scale None 1.0 |
---|
3568 | m None 4 |
---|
3569 | Background |cm^-1| 0.0 |
---|
3570 | ============== ======== ============= |
---|
3571 | |
---|
3572 | .. image:: ..\img\olddocs\image183.jpg |
---|
3573 | |
---|
3574 | *Figure. 1D plot using the default values (w/200 data point).* |
---|
3575 | |
---|
3576 | REFERENCE |
---|
3577 | |
---|
3578 | None. |
---|
3579 | |
---|
3580 | |
---|
3581 | |
---|
3582 | .. _TeubnerStrey: |
---|
3583 | |
---|
3584 | **2.2.7. TeubnerStrey (Model)** |
---|
3585 | |
---|
3586 | This function calculates the scattered intensity of a two-component system using the Teubner-Strey model. Unlike the |
---|
3587 | DABModel_ this function generates a peak. |
---|
3588 | |
---|
3589 | *2.2.7.1. Definition* |
---|
3590 | |
---|
3591 | .. image:: ..\img\olddocs\image184.PNG |
---|
3592 | |
---|
3593 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
3594 | |
---|
3595 | .. image:: ..\img\olddocs\image040.gif |
---|
3596 | |
---|
3597 | ============== ======== ============= |
---|
3598 | Parameter name Units Default value |
---|
3599 | ============== ======== ============= |
---|
3600 | scale None 0.1 |
---|
3601 | c1 None -30.0 |
---|
3602 | c2 None 5000.0 |
---|
3603 | background |cm^-1| 0.0 |
---|
3604 | ============== ======== ============= |
---|
3605 | |
---|
3606 | .. image:: ..\img\olddocs\image185.jpg |
---|
3607 | |
---|
3608 | *Figure. 1D plot using the default values (w/200 data point).* |
---|
3609 | |
---|
3610 | REFERENCE |
---|
3611 | |
---|
3612 | M Teubner, R Strey, *J. Chem. Phys.*, 87 (1987) 3195 |
---|
3613 | |
---|
3614 | K V Schubert, R Strey, S R Kline and E W Kaler, *J. Chem. Phys.*, 101 (1994) 5343 |
---|
3615 | |
---|
3616 | |
---|
3617 | |
---|
3618 | .. _FractalModel: |
---|
3619 | |
---|
3620 | **2.2.8. FractalModel** |
---|
3621 | |
---|
3622 | Calculates the scattering from fractal-like aggregates built from spherical building blocks following the Texiera |
---|
3623 | reference. |
---|
3624 | |
---|
3625 | The value returned is in |cm^-1|\ . |
---|
3626 | |
---|
3627 | *2.2.8.1. Definition* |
---|
3628 | |
---|
3629 | .. image:: ..\img\olddocs\image186.PNG |
---|
3630 | |
---|
3631 | The *scale* parameter is the volume fraction of the building blocks, *R0* is the radius of the building block, *Df* is |
---|
3632 | the fractal dimension, |xi| is the correlation length, |rho|\ *solvent* is the scattering length density of the |
---|
3633 | solvent, and |rho|\ *block* is the scattering length density of the building blocks. |
---|
3634 | |
---|
3635 | **Polydispersity on the radius is provided for.** |
---|
3636 | |
---|
3637 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
3638 | |
---|
3639 | .. image:: ..\img\olddocs\image040.gif |
---|
3640 | |
---|
3641 | ============== ======== ============= |
---|
3642 | Parameter name Units Default value |
---|
3643 | ============== ======== ============= |
---|
3644 | scale None 0.05 |
---|
3645 | radius |Ang| 5.0 |
---|
3646 | fractal_dim None 2 |
---|
3647 | corr_length |Ang| 100.0 |
---|
3648 | block_sld |Ang^-2| 2e-6 |
---|
3649 | solvent_sld |Ang^-2| 6e-6 |
---|
3650 | background |cm^-1| 0.0 |
---|
3651 | ============== ======== ============= |
---|
3652 | |
---|
3653 | .. image:: ..\img\olddocs\image187.jpg |
---|
3654 | |
---|
3655 | *Figure. 1D plot using the default values (w/200 data point).* |
---|
3656 | |
---|
3657 | REFERENCE |
---|
3658 | |
---|
3659 | J Teixeira, *J. Appl. Cryst.*, 21 (1988) 781-785 |
---|
3660 | |
---|
3661 | |
---|
3662 | |
---|
3663 | .. _MassFractalModel: |
---|
3664 | |
---|
3665 | **2.2.9. MassFractalModel** |
---|
3666 | |
---|
3667 | Calculates the scattering from fractal-like aggregates based on the Mildner reference. |
---|
3668 | |
---|
3669 | *2.2.9.1. Definition* |
---|
3670 | |
---|
3671 | .. image:: ..\img\olddocs\mass_fractal_eq1.jpg |
---|
3672 | |
---|
3673 | where *R* is the radius of the building block, *Dm* is the **mass** fractal dimension, |zeta| is the cut-off length, |
---|
3674 | |rho|\ *solvent* is the scattering length density of the solvent, and |rho|\ *particle* is the scattering length |
---|
3675 | density of particles. |
---|
3676 | |
---|
3677 | Note: Â The mass fractal dimension *Dm* is only valid if 1 < mass_dim < 6. It is also only valid over a limited |
---|
3678 | *q* range (see the reference for details). |
---|
3679 | |
---|
3680 | ============== ======== ============= |
---|
3681 | Parameter name Units Default value |
---|
3682 | ============== ======== ============= |
---|
3683 | scale None 1 |
---|
3684 | radius |Ang| 10.0 |
---|
3685 | mass_dim None 1.9 |
---|
3686 | co_length |Ang| 100.0 |
---|
3687 | background |cm^-1| 0.0 |
---|
3688 | ============== ======== ============= |
---|
3689 | |
---|
3690 | .. image:: ..\img\olddocs\mass_fractal_fig1.jpg |
---|
3691 | |
---|
3692 | *Figure. 1D plot using default values.* |
---|
3693 | |
---|
3694 | REFERENCE |
---|
3695 | |
---|
3696 | D Mildner and P Hall, *J. Phys. D: Appl. Phys.*, 19 (1986) 1535-1545 |
---|
3697 | Equation(9) |
---|
3698 | |
---|
3699 | *2013/09/09 - Description reviewed by King, S and Parker, P.* |
---|
3700 | |
---|
3701 | |
---|
3702 | |
---|
3703 | .. _SurfaceFractalModel: |
---|
3704 | |
---|
3705 | **2.2.10. SurfaceFractalModel** |
---|
3706 | |
---|
3707 | Calculates the scattering from fractal-like aggregates based on the Mildner reference. |
---|
3708 | |
---|
3709 | *2.2.10.1. Definition* |
---|
3710 | |
---|
3711 | .. image:: ..\img\olddocs\surface_fractal_eq1.gif |
---|
3712 | |
---|
3713 | where *R* is the radius of the building block, *Ds* is the **surface** fractal dimension, |zeta| is the cut-off length, |
---|
3714 | |rho|\ *solvent* is the scattering length density of the solvent, and |rho|\ *particle* is the scattering length |
---|
3715 | density of particles. |
---|
3716 | |
---|
3717 | Note: Â The surface fractal dimension *Ds* is only valid if 1 < surface_dim < 3. It is also only valid over a limited |
---|
3718 | *q* range (see the reference for details). |
---|
3719 | |
---|
3720 | ============== ======== ============= |
---|
3721 | Parameter name Units Default value |
---|
3722 | ============== ======== ============= |
---|
3723 | scale None 1 |
---|
3724 | radius |Ang| 10.0 |
---|
3725 | surface_dim None 2.0 |
---|
3726 | co_length |Ang| 500.0 |
---|
3727 | background |cm^-1| 0.0 |
---|
3728 | ============== ======== ============= |
---|
3729 | |
---|
3730 | .. image:: ..\img\olddocs\surface_fractal_fig1.jpg |
---|
3731 | |
---|
3732 | *Figure. 1D plot using default values.* |
---|
3733 | |
---|
3734 | REFERENCE |
---|
3735 | |
---|
3736 | D Mildner and P Hall, *J. Phys. D: Appl. Phys.*, 19 (1986) 1535-1545 |
---|
3737 | Equation(13) |
---|
3738 | |
---|
3739 | |
---|
3740 | |
---|
3741 | .. _MassSurfaceFractal: |
---|
3742 | |
---|
3743 | **2.2.11. MassSurfaceFractal (Model)** |
---|
3744 | |
---|
3745 | A number of natural and commercial processes form high-surface area materials as a result of the vapour-phase |
---|
3746 | aggregation of primary particles. Examples of such materials include soots, aerosols, and fume or pyrogenic silicas. |
---|
3747 | These are all characterised by cluster mass distributions (sometimes also cluster size distributions) and internal |
---|
3748 | surfaces that are fractal in nature. The scattering from such materials displays two distinct breaks in log-log |
---|
3749 | representation, corresponding to the radius-of-gyration of the primary particles, *rg*, and the radius-of-gyration of |
---|
3750 | the clusters (aggregates), *Rg*. Between these boundaries the scattering follows a power law related to the mass |
---|
3751 | fractal dimension, *Dm*, whilst above the high-Q boundary the scattering follows a power law related to the surface |
---|
3752 | fractal dimension of the primary particles, *Ds*. |
---|
3753 | |
---|
3754 | *2.2.11.1. Definition* |
---|
3755 | |
---|
3756 | The scattered intensity *I(q)* is calculated using a modified Ornstein-Zernicke equation |
---|
3757 | |
---|
3758 | .. image:: ..\img\olddocs\masssurface_fractal_eq1.jpg |
---|
3759 | |
---|
3760 | where *Rg* is the size of the cluster, *rg* is the size of the primary particle, *Ds* is the surface fractal dimension, |
---|
3761 | *Dm* is the mass fractal dimension, |rho|\ *solvent* is the scattering length density of the solvent, and |rho|\ *p* is |
---|
3762 | the scattering length density of particles. |
---|
3763 | |
---|
3764 | Note: Â The surface (*Ds*) and mass (*Dm*) fractal dimensions are only valid if 0 < *surface_dim* < 6, |
---|
3765 | 0 < *mass_dim* < 6, and (*surface_dim*+*mass_dim*) < 6. |
---|
3766 | |
---|
3767 | ============== ======== ============= |
---|
3768 | Parameter name Units Default value |
---|
3769 | ============== ======== ============= |
---|
3770 | scale None 1 |
---|
3771 | primary_rg |Ang| 4000.0 |
---|
3772 | cluster_rg |Ang| Â 86.7 |
---|
3773 | surface_dim None 2.3 |
---|
3774 | mass_dim None  1.8 |
---|
3775 | background |cm^-1| Â 0.0 |
---|
3776 | ============== ======== ============= |
---|
3777 | |
---|
3778 | .. image:: ..\img\olddocs\masssurface_fractal_fig1.jpg |
---|
3779 | |
---|
3780 | *Figure. 1D plot using default values.* |
---|
3781 | |
---|
3782 | REFERENCE |
---|
3783 | |
---|
3784 | P Schmidt, *J Appl. Cryst.*, 24 (1991) 414-435 |
---|
3785 | Equation(19) |
---|
3786 | |
---|
3787 | A J Hurd, D W Schaefer, J E Martin, *Phys. Rev. A*, 35 (1987) 2361-2364 |
---|
3788 | Equation(2) |
---|
3789 | |
---|
3790 | |
---|
3791 | |
---|
3792 | .. _FractalCoreShell: |
---|
3793 | |
---|
3794 | **2.2.12. FractalCoreShell (Model)** |
---|
3795 | |
---|
3796 | Calculates the scattering from a fractal structure with a primary building block of core-shell spheres, as opposed to |
---|
3797 | just homogeneous spheres in the FractalModel_. This model could find use for aggregates of coated particles, or |
---|
3798 | aggregates of vesicles. |
---|
3799 | |
---|
3800 | The returned value is scaled to units of |cm^-1|, absolute scale. |
---|
3801 | |
---|
3802 | *2.2.12.1. Definition* |
---|
3803 | |
---|
3804 | .. image:: ..\img\olddocs\fractcore_eq1.gif |
---|
3805 | |
---|
3806 | The form factor *P(q)* is that from CoreShellModel_ with *bkg* = 0 |
---|
3807 | |
---|
3808 | .. image:: ..\img\olddocs\image013.PNG |
---|
3809 | |
---|
3810 | while the fractal structure factor S(q) is |
---|
3811 | |
---|
3812 | .. image:: ..\img\olddocs\fractcore_eq3.gif |
---|
3813 | |
---|
3814 | where *Df* = frac_dim, |xi| = cor_length, *rc* = (core) radius, and *scale* = volume fraction. |
---|
3815 | |
---|
3816 | The fractal structure is as documented in the FractalModel_. Polydispersity of radius and thickness is provided for. |
---|
3817 | |
---|
3818 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
3819 | |
---|
3820 | .. image:: ..\img\olddocs\image040.gif |
---|
3821 | |
---|
3822 | ============== ======== ============= |
---|
3823 | Parameter name Units Default value |
---|
3824 | ============== ======== ============= |
---|
3825 | volfraction None  0.05 |
---|
3826 | frac_dim None  2 |
---|
3827 | thickness |Ang| 5.0 |
---|
3828 | radius  |Ang| 20.0 |
---|
3829 | cor_length |Ang| 100.0 |
---|
3830 | core_sld |Ang^-2| 3.5e-6 |
---|
3831 | shell_sld |Ang^-2| 1e-6 |
---|
3832 | solvent_sld |Ang^-2| 6.35e-6 |
---|
3833 | background |cm^-1| 0.0 |
---|
3834 | ============== ======== ============= |
---|
3835 | |
---|
3836 | .. image:: ..\img\olddocs\image188.jpg |
---|
3837 | |
---|
3838 | *Figure. 1D plot using the default values (w/500 data points).* |
---|
3839 | |
---|
3840 | REFERENCE |
---|
3841 | |
---|
3842 | See the CoreShellModel_ and FractalModel_ descriptions. |
---|
3843 | |
---|
3844 | |
---|
3845 | |
---|
3846 | .. _GaussLorentzGel: |
---|
3847 | |
---|
3848 | **2.2.13. GaussLorentzGel(Model)** |
---|
3849 | |
---|
3850 | Calculates the scattering from a gel structure, but typically a physical rather than chemical network. It is modeled as |
---|
3851 | a sum of a low-*q* exponential decay plus a lorentzian at higher *q*-values. |
---|
3852 | |
---|
3853 | Also see the GelFitModel_. |
---|
3854 | |
---|
3855 | The returned value is scaled to units of |cm^-1|, absolute scale. |
---|
3856 | |
---|
3857 | *2.2.13.1. Definition* |
---|
3858 | |
---|
3859 | The scattering intensity *I(q)* is calculated as (eqn 5 from the reference) |
---|
3860 | |
---|
3861 | .. image:: ..\img\olddocs\image189.jpg |
---|
3862 | |
---|
3863 | |bigzeta| is the length scale of the static correlations in the gel, which can be attributed to the "frozen-in" |
---|
3864 | crosslinks. |xi| is the dynamic correlation length, which can be attributed to the fluctuating polymer chains between |
---|
3865 | crosslinks. *I*\ :sub:`G`\ *(0)* and *I*\ :sub:`L`\ *(0)* are the scaling factors for each of these structures. **Think carefully about how** |
---|
3866 | **these map to your particular system!** |
---|
3867 | |
---|
3868 | NB: The peaked structure at higher *q* values (Figure 2 from the reference) is not reproduced by the model. Peaks can |
---|
3869 | be introduced into the model by summing this model with the PeakGaussModel_ function. |
---|
3870 | |
---|
3871 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
3872 | |
---|
3873 | .. image:: ..\img\olddocs\image040.gif |
---|
3874 | |
---|
3875 | =================================== ======== ============= |
---|
3876 | Parameter name Units Default value |
---|
3877 | =================================== ======== ============= |
---|
3878 | dyn_colength (=dynamic corr length) |Ang| 20.0 |
---|
3879 | scale_g (=Gauss scale factor) None  100 |
---|
3880 | scale_l (=Lorentzian scale factor) None 50 |
---|
3881 | stat_colength (=static corr length) |Ang| 100.0 |
---|
3882 | background |cm^-1| 0.0 |
---|
3883 | =================================== ======== ============= |
---|
3884 | |
---|
3885 | .. image:: ..\img\olddocs\image190.jpg |
---|
3886 | |
---|
3887 | *Figure. 1D plot using the default values (w/500 data points).* |
---|
3888 | |
---|
3889 | REFERENCE |
---|
3890 | |
---|
3891 | G Evmenenko, E Theunissen, K Mortensen, H Reynaers, *Polymer*, 42 (2001) 2907-2913 |
---|
3892 | |
---|
3893 | |
---|
3894 | |
---|
3895 | .. _BEPolyelectrolyte: |
---|
3896 | |
---|
3897 | **2.2.14. BEPolyelectrolyte (Model)** |
---|
3898 | |
---|
3899 | Calculates the structure factor of a polyelectrolyte solution with the RPA expression derived by Borue and Erukhimovich. |
---|
3900 | |
---|
3901 | The value returned is in |cm^-1|. |
---|
3902 | |
---|
3903 | *2.2.14.1. Definition* |
---|
3904 | |
---|
3905 | .. image:: ..\img\olddocs\image191.PNG |
---|
3906 | |
---|
3907 | where *K* is the contrast factor for the polymer, *Lb* is the Bjerrum length, *h* is the virial parameter, *b* is the |
---|
3908 | monomer length, *Cs* is the concentration of monovalent salt, |alpha| is the ionization degree, *Ca* is the polymer |
---|
3909 | molar concentration, and *background* is the incoherent background. |
---|
3910 | |
---|
3911 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
3912 | |
---|
3913 | .. image:: ..\img\olddocs\image040.gif |
---|
3914 | |
---|
3915 | ============== ======== ============= |
---|
3916 | Parameter name Units Default value |
---|
3917 | ============== ======== ============= |
---|
3918 | K barns 10 |
---|
3919 | Lb |Ang| 7.1 |
---|
3920 | h |Ang^-3| 12 |
---|
3921 | b |Ang| 10 |
---|
3922 | Cs mol/L 0 |
---|
3923 | alpha None 0.05 |
---|
3924 | Ca mol/L 0.7 |
---|
3925 | background |cm^-1| 0.0 |
---|
3926 | ============== ======== ============= |
---|
3927 | |
---|
3928 | NB: 1 barn = 10\ :sup:`-24` |cm^2| |
---|
3929 | |
---|
3930 | REFERENCE |
---|
3931 | |
---|
3932 | V Y Borue, I Y Erukhimovich, *Macromolecules*, 21 (1988) 3240 |
---|
3933 | |
---|
3934 | J F Joanny, L Leibler, *Journal de Physique*, 51 (1990) 545 |
---|
3935 | |
---|
3936 | A Moussaid, F Schosseler, J P Munch, S Candau, *J. Journal de Physique II France*, 3 (1993) 573 |
---|
3937 | |
---|
3938 | E Raphael, J F Joanny, *Europhysics Letters*, 11 (1990) 179 |
---|
3939 | |
---|
3940 | |
---|
3941 | |
---|
3942 | .. _Guinier: |
---|
3943 | |
---|
3944 | **2.2.15. Guinier (Model)** |
---|
3945 | |
---|
3946 | This model fits the Guinier function |
---|
3947 | |
---|
3948 | .. image:: ..\img\olddocs\image192.PNG |
---|
3949 | |
---|
3950 | to the data directly without any need for linearisation (*cf*. Ln *I(q)* vs *q*\ :sup:`2`). |
---|
3951 | |
---|
3952 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
3953 | |
---|
3954 | .. image:: ..\img\olddocs\image040.gif |
---|
3955 | |
---|
3956 | ============== ======== ============= |
---|
3957 | Parameter name Units Default value |
---|
3958 | ============== ======== ============= |
---|
3959 | scale |cm^-1| 1.0 |
---|
3960 | Rg |Ang| 0.1 |
---|
3961 | ============== ======== ============= |
---|
3962 | |
---|
3963 | REFERENCE |
---|
3964 | |
---|
3965 | A Guinier and G Fournet, *Small-Angle Scattering of X-Rays*, John Wiley & Sons, New York (1955) |
---|
3966 | |
---|
3967 | |
---|
3968 | |
---|
3969 | .. _GuinierPorod: |
---|
3970 | |
---|
3971 | **2.2.16. GuinierPorod (Model)** |
---|
3972 | |
---|
3973 | Calculates the scattering for a generalized Guinier/power law object. This is an empirical model that can be used to |
---|
3974 | determine the size and dimensionality of scattering objects, including asymmetric objects such as rods or platelets, and |
---|
3975 | shapes intermediate between spheres and rods or between rods and platelets. |
---|
3976 | |
---|
3977 | The result is in the units of |cm^-1|, absolute scale. |
---|
3978 | |
---|
3979 | *2.2.16.1 Definition* |
---|
3980 | |
---|
3981 | The following functional form is used |
---|
3982 | |
---|
3983 | .. image:: ..\img\olddocs\image193.jpg |
---|
3984 | |
---|
3985 | This is based on the generalized Guinier law for such elongated objects (see the Glatter reference below). For 3D |
---|
3986 | globular objects (such as spheres), *s* = 0 and one recovers the standard Guinier_ formula. For 2D symmetry (such as |
---|
3987 | for rods) *s* = 1, and for 1D symmetry (such as for lamellae or platelets) *s* = 2. A dimensionality parameter (3-*s*) |
---|
3988 | is thus defined, and is 3 for spherical objects, 2 for rods, and 1 for plates. |
---|
3989 | |
---|
3990 | Enforcing the continuity of the Guinier and Porod functions and their derivatives yields |
---|
3991 | |
---|
3992 | .. image:: ..\img\olddocs\image194.jpg |
---|
3993 | |
---|
3994 | and |
---|
3995 | |
---|
3996 | .. image:: ..\img\olddocs\image195.jpg |
---|
3997 | |
---|
3998 | Note that |
---|
3999 | |
---|
4000 | the radius-of-gyration for a sphere of radius *R* is given by *Rg* = *R* sqrt(3/5) |
---|
4001 | |
---|
4002 |  the cross-sectional radius-of-gyration for a randomly oriented cylinder of radius *R* is given by *Rg* = *R* / sqrt(2) |
---|
4003 | |
---|
4004 | the cross-sectional radius-of-gyration of a randomly oriented lamella of thickness *T* is given by *Rg* = *T* / sqrt(12) |
---|
4005 | |
---|
4006 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
4007 | |
---|
4008 | .. image:: ..\img\olddocs\image008.PNG |
---|
4009 | |
---|
4010 | ============================== ======== ============= |
---|
4011 | Parameter name Units Default value |
---|
4012 | ============================== ======== ============= |
---|
4013 | scale (=Guinier scale, G) |cm^-1| 1.0 |
---|
4014 | rg |Ang| 100 |
---|
4015 | dim (=dimensional variable, s) None  1 |
---|
4016 | m (=Porod exponent) None  3 |
---|
4017 | background |cm^-1|Â 0.1 |
---|
4018 | ============================== ======== ============= |
---|
4019 | |
---|
4020 | .. image:: ..\img\olddocs\image196.jpg |
---|
4021 | |
---|
4022 | *Figure. 1D plot using the default values (w/500 data points).* |
---|
4023 | |
---|
4024 | REFERENCE |
---|
4025 | |
---|
4026 | A Guinier, G Fournet, *Small-Angle Scattering of X-Rays*, John Wiley and Sons, New York, (1955) |
---|
4027 | |
---|
4028 | O Glatter, O Kratky, *Small-Angle X-Ray Scattering*, Academic Press (1982) |
---|
4029 | Check out Chapter 4 on Data Treatment, pages 155-156. |
---|
4030 | |
---|
4031 | |
---|
4032 | |
---|
4033 | .. _PorodModel: |
---|
4034 | |
---|
4035 | **2.2.17. PorodModel** |
---|
4036 | |
---|
4037 | This model fits the Porod function |
---|
4038 | |
---|
4039 | .. image:: ..\img\olddocs\image197_corrected.PNG |
---|
4040 | |
---|
4041 | to the data directly without any need for linearisation (*cf*. Log *I(q)* vs Log *q*). |
---|
4042 | |
---|
4043 | Here *C* is the scale factor and *Sv* is the specific surface area (ie, surface area / volume) of the sample, and |
---|
4044 | |drho| is the contrast factor. |
---|
4045 | |
---|
4046 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
4047 | |
---|
4048 | .. image:: ..\img\olddocs\image040.gif |
---|
4049 | |
---|
4050 | ============== ======== ============= |
---|
4051 | Parameter name Units Default value |
---|
4052 | ============== ======== ============= |
---|
4053 | scale |Ang^-4| 0.1 |
---|
4054 | background |cm^-1| 0 |
---|
4055 | ============== ======== ============= |
---|
4056 | |
---|
4057 | REFERENCE |
---|
4058 | |
---|
4059 | None. |
---|
4060 | |
---|
4061 | |
---|
4062 | |
---|
4063 | .. _PeakGaussModel: |
---|
4064 | |
---|
4065 | **2.2.18. PeakGaussModel** |
---|
4066 | |
---|
4067 | This model describes a Gaussian shaped peak on a flat background |
---|
4068 | |
---|
4069 | .. image:: ..\img\olddocs\image198.PNG |
---|
4070 | |
---|
4071 | with the peak having height of *I0* centered at *q0* and having a standard deviation of *B*. The FWHM (full-width |
---|
4072 | half-maximum) is 2.354 B.  |
---|
4073 | |
---|
4074 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
4075 | |
---|
4076 | .. image:: ..\img\olddocs\image040.gif |
---|
4077 | |
---|
4078 | ============== ======== ============= |
---|
4079 | Parameter name Units Default value |
---|
4080 | ============== ======== ============= |
---|
4081 | scale |cm^-1| 100 |
---|
4082 | q0 |Ang^-1| 0.05 |
---|
4083 | B Â |Ang^-1| 0.005 |
---|
4084 | background |cm^-1|Â 1 |
---|
4085 | ============== ======== ============= |
---|
4086 | |
---|
4087 | .. image:: ..\img\olddocs\image199.jpg |
---|
4088 | |
---|
4089 | *Figure. 1D plot using the default values (w/500 data points).* |
---|
4090 | |
---|
4091 | REFERENCE |
---|
4092 | |
---|
4093 | None. |
---|
4094 | |
---|
4095 | |
---|
4096 | |
---|
4097 | .. _PeakLorentzModel: |
---|
4098 | |
---|
4099 | **2.2.19. PeakLorentzModel** |
---|
4100 | |
---|
4101 | This model describes a Lorentzian shaped peak on a flat background |
---|
4102 | |
---|
4103 | .. image:: ..\img\olddocs\image200.PNG |
---|
4104 | |
---|
4105 | with the peak having height of *I0* centered at *q0* and having a HWHM (half-width half-maximum) of B. |
---|
4106 | |
---|
4107 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
4108 | |
---|
4109 | .. image:: ..\img\olddocs\image040.gif |
---|
4110 | |
---|
4111 | ============== ======== ============= |
---|
4112 | Parameter name Units Default value |
---|
4113 | ============== ======== ============= |
---|
4114 | scale |cm^-1| 100 |
---|
4115 | q0 |Ang^-1| 0.05 |
---|
4116 | B Â |Ang^-1| 0.005 |
---|
4117 | background |cm^-1|Â 1 |
---|
4118 | ============== ======== ============= |
---|
4119 | |
---|
4120 | .. image:: ..\img\olddocs\image201.jpg |
---|
4121 | |
---|
4122 | *Figure. 1D plot using the default values (w/500 data points).* |
---|
4123 | |
---|
4124 | REFERENCE |
---|
4125 | |
---|
4126 | None. |
---|
4127 | |
---|
4128 | |
---|
4129 | |
---|
4130 | .. _Poly_GaussCoil: |
---|
4131 | |
---|
4132 | **2.2.20. Poly_GaussCoil (Model)** |
---|
4133 | |
---|
4134 | This model calculates an empirical functional form for the scattering from a **polydisperse** polymer chain in the |
---|
4135 | theta state assuming a Schulz-Zimm type molecular weight distribution. Polydispersity on the radius-of-gyration is also |
---|
4136 | provided for. |
---|
4137 | |
---|
4138 | The returned value is scaled to units of |cm^-1|, absolute scale. |
---|
4139 | |
---|
4140 | *2.2.20.1. Definition* |
---|
4141 | |
---|
4142 | The scattering intensity *I(q)* is calculated as |
---|
4143 | |
---|
4144 | .. image:: ..\img\olddocs\image202.PNG |
---|
4145 | |
---|
4146 | where the dimensionless chain dimension is |
---|
4147 | |
---|
4148 | .. image:: ..\img\olddocs\image203.PNG |
---|
4149 | |
---|
4150 | and the polydispersity is |
---|
4151 | |
---|
4152 | .. image:: ..\img\olddocs\image204.PNG |
---|
4153 | |
---|
4154 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
4155 | |
---|
4156 | .. image:: ..\img\olddocs\image040.gif |
---|
4157 | |
---|
4158 | This example dataset is produced using 200 data points, using 200 data points, |
---|
4159 | *qmin* = 0.001 |Ang^-1|, *qmax* = 0.7 |Ang^-1| and the default values |
---|
4160 | |
---|
4161 | ============== ======== ============= |
---|
4162 | Parameter name Units Default value |
---|
4163 | ============== ======== ============= |
---|
4164 | scale None 1.0 |
---|
4165 | rg |Ang| 60.0 |
---|
4166 | poly_m (Mw/Mn) None 2 |
---|
4167 | background |cm^-1| 0.001 |
---|
4168 | ============== ======== ============= |
---|
4169 | |
---|
4170 | .. image:: ..\img\olddocs\image205.jpg |
---|
4171 | |
---|
4172 | *Figure. 1D plot using the default values (w/200 data point).* |
---|
4173 | |
---|
4174 | REFERENCE |
---|
4175 | |
---|
4176 | O Glatter and O Kratky (editors), *Small Angle X-ray Scattering*, Academic Press, (1982) |
---|
4177 | Page 404 |
---|
4178 | |
---|
4179 | J S Higgins, and H C Benoit, Polymers and Neutron Scattering, Oxford Science Publications (1996) |
---|
4180 | |
---|
4181 | |
---|
4182 | |
---|
4183 | .. _PolyExclVolume: |
---|
4184 | |
---|
4185 | **2.2.21. PolymerExclVolume (Model)** |
---|
4186 | |
---|
4187 | This model describes the scattering from polymer chains subject to excluded volume effects, and has been used as a |
---|
4188 | template for describing mass fractals. |
---|
4189 | |
---|
4190 | The returned value is scaled to units of |cm^-1|, absolute scale. |
---|
4191 | |
---|
4192 | *2.2.21.1 Definition* |
---|
4193 | |
---|
4194 | The form factor was originally presented in the following integral form (Benoit, 1957) |
---|
4195 | |
---|
4196 | .. image:: ..\img\olddocs\image206.jpg |
---|
4197 | |
---|
4198 | where |nu| is the excluded volume parameter (which is related to the Porod exponent *m* as |nu| = 1 / *m*), *a* is the |
---|
4199 | statistical segment length of the polymer chain, and *n* is the degree of polymerization. This integral was later put |
---|
4200 | into an almost analytical form as follows (Hammouda, 1993) |
---|
4201 | |
---|
4202 | .. image:: ..\img\olddocs\image207.jpg |
---|
4203 | |
---|
4204 | where |gamma|\ *(x,U)* is the incomplete gamma function |
---|
4205 | |
---|
4206 | .. image:: ..\img\olddocs\image208.jpg |
---|
4207 | |
---|
4208 | and the variable *U* is given in terms of the scattering vector *Q* as |
---|
4209 | |
---|
4210 | .. image:: ..\img\olddocs\image209.jpg |
---|
4211 | |
---|
4212 | The square of the radius-of-gyration is defined as |
---|
4213 | |
---|
4214 | .. image:: ..\img\olddocs\image210.jpg |
---|
4215 | |
---|
4216 | Note that this model applies only in the mass fractal range (ie, 5/3 <= *m* <= 3) and **does not** apply to surface |
---|
4217 | fractals (3 < *m* <= 4). It also does not reproduce the rigid rod limit (*m* = 1) because it assumes chain flexibility |
---|
4218 | from the outset. It may cover a portion of the semi-flexible chain range (1 < *m* < 5/3). |
---|
4219 | |
---|
4220 | A low-*Q* expansion yields the Guinier form and a high-*Q* expansion yields the Porod form which is given by |
---|
4221 | |
---|
4222 | .. image:: ..\img\olddocs\image211.jpg |
---|
4223 | |
---|
4224 | Here |biggamma|\ *(x)* = |gamma|\ *(x,inf)* is the gamma function. |
---|
4225 | |
---|
4226 | The asymptotic limit is dominated by the first term |
---|
4227 | |
---|
4228 | .. image:: ..\img\olddocs\image212.jpg |
---|
4229 | |
---|
4230 | The special case when |nu| = 0.5 (or *m* = 1/|nu| = 2) corresponds to Gaussian chains for which the form factor is given |
---|
4231 | by the familiar Debye_ function. |
---|
4232 | |
---|
4233 | .. image:: ..\img\olddocs\image213.jpg |
---|
4234 | |
---|
4235 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
4236 | |
---|
4237 | .. image:: ..\img\olddocs\image040.gif |
---|
4238 | |
---|
4239 | This example dataset is produced using 200 data points, *qmin* = 0.001 |Ang^-1|, *qmax* = 0.2 |Ang^-1| and the default |
---|
4240 | values |
---|
4241 | |
---|
4242 | =================== ======== ============= |
---|
4243 | Parameter name Units Default value |
---|
4244 | =================== ======== ============= |
---|
4245 | scale None 1.0 |
---|
4246 | rg |Ang| 60.0 |
---|
4247 | m (=Porod exponent) None  3 |
---|
4248 | background |cm^-1| 0.0 |
---|
4249 | =================== ======== ============= |
---|
4250 | |
---|
4251 | .. image:: ..\img\olddocs\image214.jpg |
---|
4252 | |
---|
4253 | *Figure. 1D plot using the default values (w/500 data points).* |
---|
4254 | |
---|
4255 | REFERENCE |
---|
4256 | |
---|
4257 | H Benoit, *Comptes Rendus*, 245 (1957) 2244-2247 |
---|
4258 | |
---|
4259 | B Hammouda, *SANS from Homogeneous Polymer Mixtures  A Unified Overview*, *Advances in Polym. Sci.*, 106 (1993) 87-133 |
---|
4260 | |
---|
4261 | |
---|
4262 | |
---|
4263 | .. _RPA10Model: |
---|
4264 | |
---|
4265 | **2.2.22. RPA10Model** |
---|
4266 | |
---|
4267 | Calculates the macroscopic scattering intensity (units of |cm^-1|) for a multicomponent homogeneous mixture of polymers |
---|
4268 | using the Random Phase Approximation. This general formalism contains 10 specific cases |
---|
4269 | |
---|
4270 | Case 0: C/D binary mixture of homopolymers |
---|
4271 | |
---|
4272 | Case 1: C-D diblock copolymer |
---|
4273 | |
---|
4274 | Case 2: B/C/D ternary mixture of homopolymers |
---|
4275 | |
---|
4276 | Case 3: C/C-D mixture of a homopolymer B and a diblock copolymer C-D |
---|
4277 | |
---|
4278 | Case 4: B-C-D triblock copolymer |
---|
4279 | |
---|
4280 | Case 5: A/B/C/D quaternary mixture of homopolymers |
---|
4281 | |
---|
4282 | Case 6: A/B/C-D mixture of two homopolymers A/B and a diblock C-D |
---|
4283 | |
---|
4284 | Case 7: A/B-C-D mixture of a homopolymer A and a triblock B-C-D |
---|
4285 | |
---|
4286 | Case 8: A-B/C-D mixture of two diblock copolymers A-B and C-D |
---|
4287 | |
---|
4288 | Case 9: A-B-C-D tetra-block copolymer |
---|
4289 | |
---|
4290 | **NB: these case numbers are different from those in the NIST SANS package!** |
---|
4291 | |
---|
4292 | Only one case can be used at any one time. |
---|
4293 | |
---|
4294 | The returned value is scaled to units of |cm^-1|, absolute scale. |
---|
4295 | |
---|
4296 | The RPA (mean field) formalism only applies only when the multicomponent polymer mixture is in the homogeneous |
---|
4297 | mixed-phase region. |
---|
4298 | |
---|
4299 | **Component D is assumed to be the "background" component (ie, all contrasts are calculated with respect to** |
---|
4300 | **component D).** So the scattering contrast for a C/D blend = [SLD(component C) - SLD(component D)]\ :sup:`2`. |
---|
4301 | |
---|
4302 | Depending on which case is being used, the number of fitting parameters - the segment lengths (ba, bb, etc) and |chi| |
---|
4303 | parameters (Kab, Kac, etc) - vary. The *scale* parameter should be held equal to unity. |
---|
4304 | |
---|
4305 | The input parameters are the degrees of polymerization, the volume fractions, the specific volumes, and the neutron |
---|
4306 | scattering length densities for each component. |
---|
4307 | |
---|
4308 | Fitting parameters for a Case 0 Model |
---|
4309 | |
---|
4310 | ======================= ======== ============= |
---|
4311 | Parameter name Units Default value |
---|
4312 | ======================= ======== ============= |
---|
4313 | background |cm^-1| 0.0 |
---|
4314 | scale  None 1 |
---|
4315 | bc (=segment Length_bc) **unit** 5 |
---|
4316 | bd (=segment length_bd) **unit** 5 |
---|
4317 | Kcd (=chi_cd) **unit** -0.0004 |
---|
4318 | ======================= ======== ============= |
---|
4319 | |
---|
4320 | Fixed parameters for a Case 0 Model |
---|
4321 | |
---|
4322 | ======================= ======== ============= |
---|
4323 | Parameter name Units Default value |
---|
4324 | ======================= ======== ============= |
---|
4325 | Lc (=scatter. length_c) **unit** 1e-12 |
---|
4326 | Ld (=scatter. length_d) **unit** 0 |
---|
4327 | Nc (=degree polym_c) None 1000 |
---|
4328 | Nd (=degree polym_d) None  1000 |
---|
4329 | Phic (=vol. fraction_c) None  0.25 |
---|
4330 | Phid (=vol. fraction_d) None  0.25 |
---|
4331 | vc (=specific volume_c) **unit** 100 |
---|
4332 | vd (=specific volume_d) **unit** 100 |
---|
4333 | ======================= ======== ============= |
---|
4334 | |
---|
4335 | .. image:: ..\img\olddocs\image215.jpg |
---|
4336 | |
---|
4337 | *Figure. 1D plot using the default values (w/500 data points).* |
---|
4338 | |
---|
4339 | REFERENCE |
---|
4340 | |
---|
4341 | A Z Akcasu, R Klein and B Hammouda, *Macromolecules*, 26 (1993) 4136 |
---|
4342 | |
---|
4343 | |
---|
4344 | |
---|
4345 | .. _TwoLorentzian: |
---|
4346 | |
---|
4347 | **2.2.23. TwoLorentzian (Model)** |
---|
4348 | |
---|
4349 | This model calculates an empirical functional form for SAS data characterized by two Lorentzian-type functions. |
---|
4350 | |
---|
4351 | The returned value is scaled to units of |cm^-1|, absolute scale. |
---|
4352 | |
---|
4353 | *2.2.23.1. Definition* |
---|
4354 | |
---|
4355 | The scattering intensity *I(q)* is calculated as |
---|
4356 | |
---|
4357 | .. image:: ..\img\olddocs\image216.jpg |
---|
4358 | |
---|
4359 | where *A* = Lorentzian scale factor #1, *C* = Lorentzian scale #2, |xi|\ :sub:`1` and |xi|\ :sub:`2` are the |
---|
4360 | corresponding correlation lengths, and *n* and *m* are the respective power law exponents (set *n* = *m* = 2 for |
---|
4361 | Ornstein-Zernicke behaviour). |
---|
4362 | |
---|
4363 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
4364 | |
---|
4365 | .. image:: ..\img\olddocs\image040.gif |
---|
4366 | |
---|
4367 | =============================== ======== ============= |
---|
4368 | Parameter name Units Default value |
---|
4369 | =============================== ======== ============= |
---|
4370 | scale_1 (=A) None  10 |
---|
4371 | scale_2 (=C) None  1 |
---|
4372 | 1ength_1 (=correlation length1) |Ang| 100 |
---|
4373 | 1ength_2 (=correlation length2) |Ang| 10 |
---|
4374 | exponent_1 (=n) None  3 |
---|
4375 | exponent_2 (=m) None  2 |
---|
4376 | background (=B) |cm^-1| 0.1 |
---|
4377 | =============================== ======== ============= |
---|
4378 | |
---|
4379 | .. image:: ..\img\olddocs\image217.jpg |
---|
4380 | |
---|
4381 | *Figure. 1D plot using the default values (w/500 data points).* |
---|
4382 | |
---|
4383 | REFERENCE |
---|
4384 | |
---|
4385 | None. |
---|
4386 | |
---|
4387 | |
---|
4388 | |
---|
4389 | .. _TwoPowerLaw: |
---|
4390 | |
---|
4391 | **2.2.24. TwoPowerLaw (Model)** |
---|
4392 | |
---|
4393 | This model calculates an empirical functional form for SAS data characterized by two power laws. |
---|
4394 | |
---|
4395 | The returned value is scaled to units of |cm^-1|, absolute scale. |
---|
4396 | |
---|
4397 | *2.2.24.1. Definition* |
---|
4398 | |
---|
4399 | The scattering intensity *I(q)* is calculated as |
---|
4400 | |
---|
4401 | .. image:: ..\img\olddocs\image218.jpg |
---|
4402 | |
---|
4403 | where *qc* is the location of the crossover from one slope to the other. The scaling *coef_A* sets the overall |
---|
4404 | intensity of the lower *q* power law region. The scaling of the second power law region is then automatically scaled to |
---|
4405 | match the first. |
---|
4406 | |
---|
4407 | **NB: Be sure to enter the power law exponents as positive values!** |
---|
4408 | |
---|
4409 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
4410 | |
---|
4411 | .. image:: ..\img\olddocs\image040.gif |
---|
4412 | |
---|
4413 | ============== ======== ============= |
---|
4414 | Parameter name Units Default value |
---|
4415 | ============== ======== ============= |
---|
4416 | coef_A Â None 1.0 |
---|
4417 | qc |Ang^-1| 0.04 |
---|
4418 | power_1 (=m1) None  4 |
---|
4419 | power_2 (=m2) None  4 |
---|
4420 | background |cm^-1| 0.0 |
---|
4421 | ============== ======== ============= |
---|
4422 | |
---|
4423 | .. image:: ..\img\olddocs\image219.jpg |
---|
4424 | |
---|
4425 | *Figure. 1D plot using the default values (w/500 data points).* |
---|
4426 | |
---|
4427 | REFERENCE |
---|
4428 | |
---|
4429 | None. |
---|
4430 | |
---|
4431 | |
---|
4432 | |
---|
4433 | .. _UnifiedPowerRg: |
---|
4434 | |
---|
4435 | **2.2.25. UnifiedPowerRg (Beaucage Model)** |
---|
4436 | |
---|
4437 | This model deploys the empirical multiple level unified Exponential/Power-law fit method developed by G Beaucage. Four |
---|
4438 | functions are included so that 1, 2, 3, or 4 levels can be used. In addition a 0 level has been added which simply |
---|
4439 | calculates |
---|
4440 | |
---|
4441 | *I(q)* = *scale* / *q* + *background* |
---|
4442 | |
---|
4443 | The returned value is scaled to units of |cm^-1|, absolute scale. |
---|
4444 | |
---|
4445 | The Beaucage method is able to reasonably approximate the scattering from many different types of particles, including |
---|
4446 | fractal clusters, random coils (Debye equation), ellipsoidal particles, etc. |
---|
4447 | |
---|
4448 | *2.2.25.1 Definition* |
---|
4449 | |
---|
4450 | The empirical fit function is |
---|
4451 | |
---|
4452 | .. image:: ..\img\olddocs\image220.jpg |
---|
4453 | |
---|
4454 | For each level, the four parameters *Gi*, *Rg,i*, *Bi* and *Pi* must be chosen. |
---|
4455 | |
---|
4456 | For example, to approximate the scattering from random coils (Debye_ equation), set *Rg,i* as the Guinier radius, |
---|
4457 | *Pi* = 2, and *Bi* = 2 *Gi* / *Rg,i*Â |
---|
4458 | |
---|
4459 | See the references for further information on choosing the parameters. |
---|
4460 | |
---|
4461 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
4462 | |
---|
4463 | .. image:: ..\img\olddocs\image040.gif |
---|
4464 | |
---|
4465 | ============== ======== ============= |
---|
4466 | Parameter name Units Default value |
---|
4467 | ============== ======== ============= |
---|
4468 | scale  None 1.0 |
---|
4469 | Rg2 |Ang| 21 |
---|
4470 | power2 Â None 2 |
---|
4471 | G2 |cm^-1| 3 |
---|
4472 | B2 |cm^-1| 0.0006 |
---|
4473 | Rg1 |Ang| 15.8 |
---|
4474 | power1 Â None 4 |
---|
4475 | G1 |cm^-1| 400 |
---|
4476 | B1 |cm^-1| 4.5e-6 | |
---|
4477 | background |cm^-1| 0.0 |
---|
4478 | ============== ======== ============= |
---|
4479 | |
---|
4480 | .. image:: ..\img\olddocs\image221.jpg |
---|
4481 | |
---|
4482 | *Figure. 1D plot using the default values (w/500 data points).* |
---|
4483 | |
---|
4484 | REFERENCE |
---|
4485 | |
---|
4486 | G Beaucage, *J. Appl. Cryst.*, 28 (1995) 717-728 |
---|
4487 | |
---|
4488 | G Beaucage, *J. Appl. Cryst.*, 29 (1996) 134-146 |
---|
4489 | |
---|
4490 | |
---|
4491 | |
---|
4492 | .. _LineModel: |
---|
4493 | |
---|
4494 | **2.2.26. LineModel** |
---|
4495 | |
---|
4496 | This calculates the simple linear function |
---|
4497 | |
---|
4498 | .. image:: ..\img\olddocs\image222.PNG |
---|
4499 | |
---|
4500 | **NB: For 2D plots,** *I(q)* = *I(qx)*\ *\ *I(qy)*, **which is a different definition to other shape independent models.** |
---|
4501 | |
---|
4502 | ============== ============== ============= |
---|
4503 | Parameter name Units Default value |
---|
4504 | ============== ============== ============= |
---|
4505 | A |cm^-1| 1.0 |
---|
4506 | B |Ang|\ |cm^-1| 1.0 |
---|
4507 | ============== ============== ============= |
---|
4508 | |
---|
4509 | REFERENCE |
---|
4510 | |
---|
4511 | None. |
---|
4512 | |
---|
4513 | |
---|
4514 | |
---|
4515 | .. _GelFitModel: |
---|
4516 | |
---|
4517 | **2.2.27. GelFitModel** |
---|
4518 | |
---|
4519 | *This model was implemented by an interested user!* |
---|
4520 | |
---|
4521 | Unlike a concentrated polymer solution, the fine-scale polymer distribution in a gel involves at least two |
---|
4522 | characteristic length scales, a shorter correlation length (*a1*) to describe the rapid fluctuations in the position |
---|
4523 | of the polymer chains that ensure thermodynamic equilibrium, and a longer distance (denoted here as *a2*) needed to |
---|
4524 | account for the static accumulations of polymer pinned down by junction points or clusters of such points. The latter |
---|
4525 | is derived from a simple Guinier function. |
---|
4526 | |
---|
4527 | Also see the GaussLorentzGel_ Model. |
---|
4528 | |
---|
4529 | *2.2.27.1. Definition* |
---|
4530 | |
---|
4531 | The scattered intensity *I(q)* is calculated as |
---|
4532 | |
---|
4533 | .. image:: ..\img\olddocs\image233.gif |
---|
4534 | |
---|
4535 | where |
---|
4536 | |
---|
4537 | .. image:: ..\img\olddocs\image234.gif |
---|
4538 | |
---|
4539 | Note that the first term reduces to the Ornstein-Zernicke equation when *D* = 2; ie, when the Flory exponent is 0.5 |
---|
4540 | (theta conditions). In gels with significant hydrogen bonding *D* has been reported to be ~2.6 to 2.8. |
---|
4541 | |
---|
4542 | ============================ ======== ============= |
---|
4543 | Parameter name Units Default value |
---|
4544 | ============================ ======== ============= |
---|
4545 | Background |cm^-1| 0.01 |
---|
4546 | Guinier scale (= *I(0)G*) |cm^-1| 1.7 |
---|
4547 | Lorentzian scale (= *I(0)L*) |cm^-1| 3.5 |
---|
4548 | Radius of gyration (= *Rg*) |Ang| 104 |
---|
4549 | Fractal exponent (= *D*) None  2 |
---|
4550 | Correlation length (= *a1*) |Ang| 16 |
---|
4551 | ============================ ======== ============= |
---|
4552 | |
---|
4553 | .. image:: ..\img\olddocs\image235.gif |
---|
4554 | |
---|
4555 | *Figure. 1D plot using the default values (w/300 data points).* |
---|
4556 | |
---|
4557 | REFERENCE |
---|
4558 | |
---|
4559 | Mitsuhiro Shibayama, Toyoichi Tanaka, Charles C Han, J. Chem. Phys. 1992, 97 (9), 6829-6841 |
---|
4560 | |
---|
4561 | Simon Mallam, Ferenc Horkay, Anne-Marie Hecht, Adrian R Rennie, Erik Geissler, Macromolecules 1991, 24, 543-548 |
---|
4562 | |
---|
4563 | |
---|
4564 | |
---|
4565 | .. _StarPolymer: |
---|
4566 | |
---|
4567 | **2.2.28. Star Polymer with Gaussian Statistics** |
---|
4568 | |
---|
4569 | This model is also known as the Benoit Star model. |
---|
4570 | |
---|
4571 | *2.2.28.1. Definition* |
---|
4572 | |
---|
4573 | For a star with *f* arms: |
---|
4574 | |
---|
4575 | .. image:: ..\img\olddocs\star1.png |
---|
4576 | |
---|
4577 | where |
---|
4578 | |
---|
4579 | .. image:: ..\img\olddocs\star2.png |
---|
4580 | |
---|
4581 | and |
---|
4582 | |
---|
4583 | .. image:: ..\img\olddocs\star3.png |
---|
4584 | |
---|
4585 | is the square of the ensemble average radius-of-gyration of an arm. |
---|
4586 | |
---|
4587 | REFERENCE |
---|
4588 | |
---|
4589 | H Benoit,  J. Polymer Science., 11, 596-599 (1953) |
---|
4590 | |
---|
4591 | |
---|
4592 | |
---|
4593 | .. _ReflectivityModel: |
---|
4594 | |
---|
4595 | **2.2.29. ReflectivityModel** |
---|
4596 | |
---|
4597 | *This model was contributed by an interested user!* |
---|
4598 | |
---|
4599 | This model calculates **reflectivity** using the Parrett algorithm. |
---|
4600 | |
---|
4601 | Up to nine film layers are supported between Bottom(substrate) and Medium(Superstrate) where the neutron enters the |
---|
4602 | first top film. Each of the layers are composed of |
---|
4603 | |
---|
4604 | [œ of the interface (from the previous layer or substrate) + flat portion + œ of the interface (to the next layer or medium)] |
---|
4605 | |
---|
4606 | Two simple functions are provided to describe the interfacial density distribution; a linear function and an error |
---|
4607 | function. The interfacial thickness is equivalent to (-2.5 |sigma| to +2.5 |sigma| for the error function, where |
---|
4608 | |sigma| = roughness). |
---|
4609 | |
---|
4610 | Also see ReflectivityIIModel_. |
---|
4611 | |
---|
4612 | .. image:: ..\img\olddocs\image231.bmp |
---|
4613 | |
---|
4614 | *Figure. Comparison (using the SLD profile below) with the NIST web calculation (circles)* |
---|
4615 | http://www.ncnr.nist.gov/resources/reflcalc.html |
---|
4616 | |
---|
4617 | .. image:: ..\img\olddocs\image232.gif |
---|
4618 | |
---|
4619 | *Figure. SLD profile used for the calculation (above).* |
---|
4620 | |
---|
4621 | REFERENCE |
---|
4622 | |
---|
4623 | None. |
---|
4624 | |
---|
4625 | |
---|
4626 | |
---|
4627 | .. _ReflectivityIIModel: |
---|
4628 | |
---|
4629 | **2.2.30. ReflectivityIIModel** |
---|
4630 | |
---|
4631 | *This model was contributed by an interested user!* |
---|
4632 | |
---|
4633 | This **reflectivity** model is a more flexible version of ReflectivityModel_. More interfacial density |
---|
4634 | functions are supported, and the number of points (*npts_inter*) for each interface can be chosen. |
---|
4635 | |
---|
4636 | The SLD at the interface between layers, |rho|\ *inter_i*, is calculated with a function chosen by a user, where the |
---|
4637 | available functions are |
---|
4638 | |
---|
4639 | 1) Erf |
---|
4640 | |
---|
4641 | .. image:: ..\img\olddocs\image051.gif |
---|
4642 | |
---|
4643 | 2) Power-Law |
---|
4644 | |
---|
4645 | .. image:: ..\img\olddocs\image050.gif |
---|
4646 | |
---|
4647 | 3) Exp |
---|
4648 | |
---|
4649 | .. image:: ..\img\olddocs\image049.gif |
---|
4650 | |
---|
4651 | The constant *A* in the expressions above (but the parameter *nu* in the model!) is an input. |
---|
4652 | |
---|
4653 | REFERENCE |
---|
4654 | |
---|
4655 | None. |
---|
4656 | |
---|
4657 | |
---|
4658 | |
---|
4659 | 2.3 Structure-factor Functions |
---|
4660 | ------------------------------ |
---|
4661 | |
---|
4662 | The information in this section originated from NIST SANS package. |
---|
4663 | |
---|
4664 | .. _HardSphereStructure: |
---|
4665 | |
---|
4666 | **2.3.1. HardSphereStructure Factor** |
---|
4667 | |
---|
4668 | This calculates the interparticle structure factor for monodisperse spherical particles interacting through hard |
---|
4669 | sphere (excluded volume) interactions. |
---|
4670 | |
---|
4671 | The calculation uses the Percus-Yevick closure where the interparticle potential is |
---|
4672 | |
---|
4673 | .. image:: ..\img\olddocs\image223.PNG |
---|
4674 | |
---|
4675 | where *r* is the distance from the center of the sphere of a radius *R*. |
---|
4676 | |
---|
4677 | For a 2D plot, the wave transfer is defined as |
---|
4678 | |
---|
4679 | .. image:: ..\img\olddocs\image040.gif |
---|
4680 | |
---|
4681 | ============== ======== ============= |
---|
4682 | Parameter name Units Default value |
---|
4683 | ============== ======== ============= |
---|
4684 | effect_radius |Ang| 50.0 |
---|
4685 | volfraction None 0.2 |
---|
4686 | ============== ======== ============= |
---|
4687 | |
---|
4688 | .. image:: ..\img\olddocs\image224.jpg |
---|
4689 | |
---|
4690 | *Figure. 1D plot using the default values (in linear scale).* |
---|
4691 | |
---|
4692 | REFERENCE |
---|
4693 | |
---|
4694 | J K Percus, J Yevick, *J. Phys. Rev.*, 110, (1958) 1 |
---|
4695 | |
---|
4696 | |
---|
4697 | |
---|
4698 | .. _SquareWellStructure: |
---|
4699 | |
---|
4700 | **2.3.2. SquareWellStructure Factor** |
---|
4701 | |
---|
4702 | This calculates the interparticle structure factor for a square well fluid spherical particles. The mean spherical |
---|
4703 | approximation (MSA) closure was used for this calculation, and is not the most appropriate closure for an attractive |
---|
4704 | interparticle potential. This solution has been compared to Monte Carlo simulations for a square well fluid, showing |
---|
4705 | this calculation to be limited in applicability to well depths |epsilon| < 1.5 kT and volume fractions |phi| < 0.08. |
---|
4706 | |
---|
4707 | Positive well depths correspond to an attractive potential well. Negative well depths correspond to a potential |
---|
4708 | "shoulder", which may or may not be physically reasonable. |
---|
4709 | |
---|
4710 | The well width (*l*\ ) is defined as multiples of the particle diameter (2\*\ *R*\ ) |
---|
4711 | |
---|
4712 | The interaction potential is: |
---|
4713 | |
---|
4714 | .. image:: ..\img\olddocs\image225.PNG |
---|
4715 | |
---|
4716 | where *r* is the distance from the center of the sphere of a radius *R*. |
---|
4717 | |
---|
4718 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
4719 | |
---|
4720 | .. image:: ..\img\olddocs\image040.gif |
---|
4721 | |
---|
4722 | ============== ========= ============= |
---|
4723 | Parameter name Units Default value |
---|
4724 | ============== ========= ============= |
---|
4725 | effect_radius |Ang| 50.0 |
---|
4726 | volfraction None 0.04 |
---|
4727 | welldepth kT 1.5 |
---|
4728 | wellwidth diameters 1.2 |
---|
4729 | ============== ========= ============= |
---|
4730 | |
---|
4731 | .. image:: ..\img\olddocs\image226.jpg |
---|
4732 | |
---|
4733 | *Figure. 1D plot using the default values (in linear scale).* |
---|
4734 | |
---|
4735 | REFERENCE |
---|
4736 | |
---|
4737 | R V Sharma, K C Sharma, *Physica*, 89A (1977) 213 |
---|
4738 | |
---|
4739 | |
---|
4740 | |
---|
4741 | .. _HayterMSAStructure: |
---|
4742 | |
---|
4743 | **2.3.3. HayterMSAStructure Factor** |
---|
4744 | |
---|
4745 | This is an implementation of the Rescaled Mean Spherical Approximation which calculates the structure factor (the |
---|
4746 | Fourier transform of the pair correlation function *g(r)*) for a system of charged, spheroidal objects in a |
---|
4747 | dielectric medium. When combined with an appropriate form factor (such as sphere,core+shell, ellipsoid, etc), this |
---|
4748 | allows for inclusion of the interparticle interference effects due to screened coulomb repulsion between charged particles. |
---|
4749 | |
---|
4750 | **This routine only works for charged particles**. If the charge is set to zero the routine will self-destruct! |
---|
4751 | For non-charged particles use a hard sphere potential. |
---|
4752 | |
---|
4753 | The salt concentration is used to compute the ionic strength of the solution which in turn is used to compute the Debye |
---|
4754 | screening length. At present there is no provision for entering the ionic strength directly nor for use of any |
---|
4755 | multivalent salts. The counterions are also assumed to be monovalent. |
---|
4756 | |
---|
4757 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
4758 | |
---|
4759 | .. image:: ..\img\olddocs\image040.gif |
---|
4760 | |
---|
4761 | ============== ======== ============= |
---|
4762 | Parameter name Units Default value |
---|
4763 | ============== ======== ============= |
---|
4764 | effect_radius |Ang| 20.8 |
---|
4765 | charge *e* 19 |
---|
4766 | volfraction None 0.2 |
---|
4767 | temperature K 318 |
---|
4768 | salt conc M 0 |
---|
4769 | dielectconst None 71.1 |
---|
4770 | ============== ======== ============= |
---|
4771 | |
---|
4772 | .. image:: ..\img\olddocs\image227.jpg |
---|
4773 | |
---|
4774 | *Figure. 1D plot using the default values (in linear scale).* |
---|
4775 | |
---|
4776 | REFERENCE |
---|
4777 | |
---|
4778 | J B Hayter and J Penfold, *Molecular Physics*, 42 (1981) 109-118 |
---|
4779 | |
---|
4780 | J P Hansen and J B Hayter, *Molecular Physics*, 46 (1982) 651-656 |
---|
4781 | |
---|
4782 | |
---|
4783 | .. _StickyHSStructure: |
---|
4784 | |
---|
4785 | **2.3.4. StickyHSStructure Factor** |
---|
4786 | |
---|
4787 | This calculates the interparticle structure factor for a hard sphere fluid with a narrow attractive well. A perturbative |
---|
4788 | solution of the Percus-Yevick closure is used. The strength of the attractive well is described in terms of "stickiness" |
---|
4789 | as defined below. The returned value is a dimensionless structure factor, *S(q)*. |
---|
4790 | |
---|
4791 | The perturb (perturbation parameter), |epsilon|, should be held between 0.01 and 0.1. It is best to hold the |
---|
4792 | perturbation parameter fixed and let the "stickiness" vary to adjust the interaction strength. The stickiness, |tau|, |
---|
4793 | is defined in the equation below and is a function of both the perturbation parameter and the interaction strength. |
---|
4794 | |tau| and |epsilon| are defined in terms of the hard sphere diameter (|sigma| = 2\*\ *R*\ ), the width of the square |
---|
4795 | well, |bigdelta| (same units as *R*), and the depth of the well, *Uo*, in units of kT. From the definition, it is clear |
---|
4796 | that smaller |tau| means stronger attraction. |
---|
4797 | |
---|
4798 | .. image:: ..\img\olddocs\image228.PNG |
---|
4799 | |
---|
4800 | where the interaction potential is |
---|
4801 | |
---|
4802 | .. image:: ..\img\olddocs\image229.PNG |
---|
4803 | |
---|
4804 | The Percus-Yevick (PY) closure was used for this calculation, and is an adequate closure for an attractive interparticle |
---|
4805 | potential. This solution has been compared to Monte Carlo simulations for a square well fluid, with good agreement. |
---|
4806 | |
---|
4807 | The true particle volume fraction, |phi|, is not equal to *h*, which appears in most of the reference. The two are |
---|
4808 | related in equation (24) of the reference. The reference also describes the relationship between this perturbation |
---|
4809 | solution and the original sticky hard sphere (or adhesive sphere) model by Baxter. |
---|
4810 | |
---|
4811 | NB: The calculation can go haywire for certain combinations of the input parameters, producing unphysical solutions - in |
---|
4812 | this case errors are reported to the command window and the *S(q)* is set to -1 (so it will disappear on a log-log |
---|
4813 | plot). Use tight bounds to keep the parameters to values that you know are physical (test them) and keep nudging them |
---|
4814 | until the optimization does not hit the constraints. |
---|
4815 | |
---|
4816 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the *q* vector is defined as |
---|
4817 | |
---|
4818 | .. image:: ..\img\olddocs\image040.gif |
---|
4819 | |
---|
4820 | ============== ======== ============= |
---|
4821 | Parameter name Units Default value |
---|
4822 | ============== ======== ============= |
---|
4823 | effect_radius |Ang| 50 |
---|
4824 | perturb None 0.05 |
---|
4825 | volfraction None 0.1 |
---|
4826 | stickiness K 0.2 |
---|
4827 | ============== ======== ============= |
---|
4828 | |
---|
4829 | .. image:: ..\img\olddocs\image230.jpg |
---|
4830 | |
---|
4831 | *Figure. 1D plot using the default values (in linear scale).* |
---|
4832 | |
---|
4833 | REFERENCE |
---|
4834 | |
---|
4835 | S V G Menon, C Manohar, and K S Rao, *J. Chem. Phys.*, 95(12) (1991) 9186-9190 |
---|
4836 | |
---|
4837 | |
---|
4838 | |
---|
4839 | 2.4 Customised Functions |
---|
4840 | ------------------------------ |
---|
4841 | |
---|
4842 | |
---|
4843 | Customized model functions can be redefined or added to by users (See SansView tutorial for details). |
---|
4844 | |
---|
4845 | .. _testmodel: |
---|
4846 | |
---|
4847 | **2.4.1. testmodel** |
---|
4848 | |
---|
4849 | This function, as an example of a user defined function, calculates |
---|
4850 | |
---|
4851 | *I(q)* = *A* + *B* cos(2\ *q*\ ) + *C* sin(2\ *q*\ ) |
---|
4852 | |
---|
4853 | |
---|
4854 | |
---|
4855 | .. _testmodel_2: |
---|
4856 | |
---|
4857 | **2.4.2. testmodel_2** |
---|
4858 | |
---|
4859 | This function, as an example of a user defined function, calculates |
---|
4860 | |
---|
4861 | *I(q)* = *scale* * sin(*f*\ )/*f* |
---|
4862 | |
---|
4863 | where |
---|
4864 | |
---|
4865 | *f* = *A* + *Bq* + *Cq*\ :sup:`2` + *Dq*\ :sup:`3` + *Eq*\ :sup:`4` + *Fq*\ :sup:`5` |
---|
4866 | |
---|
4867 | |
---|
4868 | |
---|
4869 | .. _sum_p1_p2: |
---|
4870 | |
---|
4871 | **2.4.3. sum_p1_p2** |
---|
4872 | |
---|
4873 | This function, as an example of a user defined function, calculates |
---|
4874 | |
---|
4875 | *I(q)* = *scale_factor* \* (CylinderModel + PolymerExclVolumeModel) |
---|
4876 | |
---|
4877 | To make your own (*p1 + p2*) model, select 'Easy Custom Sum' from the Fitting menu, or modify and compile the file |
---|
4878 | named 'sum_p1_p2.py' from 'Edit Custom Model' in the 'Fitting' menu. |
---|
4879 | |
---|
4880 | NB: Summing models only works only for single functional models (ie, single shell models, two-component RPA models, etc). |
---|
4881 | |
---|
4882 | |
---|
4883 | |
---|
4884 | .. _sum_Ap1_1_Ap2: |
---|
4885 | |
---|
4886 | **2.4.4. sum_Ap1_1_Ap2** |
---|
4887 | |
---|
4888 | This function, as an example of a user defined function, calculates |
---|
4889 | |
---|
4890 | *I(q)* = (*scale_factor* \* CylinderModel + (1 - *scale_factor*\ ) \* PolymerExclVolume model) |
---|
4891 | |
---|
4892 | To make your own (*A*\ * *p1* + (1-*A*) \* *p2*) model, modify and compile the file named 'sum_Ap1_1_Ap2.py' from |
---|
4893 | 'Edit Custom Model' in the 'Fitting' menu. |
---|
4894 | |
---|
4895 | NB: Summing models only works only for single functional models (ie, single shell models, two-component RPA models, etc). |
---|
4896 | |
---|
4897 | |
---|
4898 | |
---|
4899 | .. _polynomial5: |
---|
4900 | |
---|
4901 | **2.4.5. polynomial5** |
---|
4902 | |
---|
4903 | This function, as an example of a user defined function, calculates |
---|
4904 | |
---|
4905 | *I(q)* = *A* + *Bq* + *Cq*\ :sup:`2` + *Dq*\ :sup:`3` + *Eq*\ :sup:`4` + *Fq*\ :sup:`5` |
---|
4906 | |
---|
4907 | This model can be modified and compiled from 'Edit Custom Model' in the 'Fitting' menu. |
---|
4908 | |
---|
4909 | |
---|
4910 | |
---|
4911 | .. _sph_bessel_jn: |
---|
4912 | |
---|
4913 | **2.4.6. sph_bessel_jn** |
---|
4914 | |
---|
4915 | This function, as an example of a user defined function, calculates |
---|
4916 | |
---|
4917 | *I(q)* = *C* \* *sph_jn(Ax+B)+D* |
---|
4918 | |
---|
4919 | where *sph_jn* is a spherical Bessel function of order *n*. |
---|
4920 | |
---|
4921 | This model can be modified and compiled from 'Edit Custom Model' in the 'Fitting' menu. |
---|