[d63cf11] | 1 | .. model_functions.rst |
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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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| 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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| 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 |
---|
| 478 | |
---|
| 479 | .. image:: ..\img\olddocs\image008.PNG |
---|
| 480 | |
---|
| 481 | This example dataset is produced by running the RaspBerryModel, using 2000 data points, *qmin* = 0.0001 |Ang^-1|, |
---|
| 482 | *qmax* = 0.2 |Ang^-1| and the default values below, where *Ssph/Lsph* stands for smaller or larger sphere, respectively, |
---|
| 483 | and *surfrac_Ssph* is the surface fraction of the smaller spheres. |
---|
| 484 | |
---|
| 485 | ============== ======== ============= |
---|
| 486 | Parameter name Units Default value |
---|
| 487 | ============== ======== ============= |
---|
| 488 | delta_Ssph None 0 |
---|
| 489 | radius_Lsph |Ang| 5000 |
---|
| 490 | radius_Ssph |Ang| 100 |
---|
| 491 | sld_Lsph |Ang^-2| -4e-07 |
---|
| 492 | sld_Ssph |Ang^-2| 3.5e-6 |
---|
| 493 | sld_solv |Ang^-2| 6.3e-6 |
---|
| 494 | surfrac_Ssph None 0.4 |
---|
| 495 | volf_Lsph None 0.05 |
---|
| 496 | volf_Lsph None 0.005 |
---|
| 497 | background |cm^-1| 0 |
---|
| 498 | ============== ======== ============= |
---|
| 499 | |
---|
| 500 | .. image:: ..\img\olddocs\raspberry_plot.jpg |
---|
| 501 | |
---|
| 502 | *Figure. 1D plot using the values of /2000 data points.* |
---|
| 503 | |
---|
| 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 | ============== ======== ============= |
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| 4687 | |
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| 4688 | .. image:: ..\img\olddocs\image224.jpg |
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| 4689 | |
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| 4690 | *Figure. 1D plot using the default values (in linear scale).* |
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| 4691 | |
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| 4692 | REFERENCE |
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| 4693 | |
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| 4694 | J K Percus, J Yevick, *J. Phys. Rev.*, 110, (1958) 1 |
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| 4695 | |
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| 4696 | |
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| 4697 | |
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| 4698 | .. _SquareWellStructure: |
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| 4699 | |
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| 4700 | **2.3.2. SquareWellStructure Factor** |
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| 4701 | |
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| 4702 | This calculates the interparticle structure factor for a square well fluid spherical particles. The mean spherical |
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| 4703 | approximation (MSA) closure was used for this calculation, and is not the most appropriate closure for an attractive |
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| 4704 | interparticle potential. This solution has been compared to Monte Carlo simulations for a square well fluid, showing |
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| 4705 | this calculation to be limited in applicability to well depths |epsilon| < 1.5 kT and volume fractions |phi| < 0.08. |
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| 4706 | |
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| 4707 | Positive well depths correspond to an attractive potential well. Negative well depths correspond to a potential |
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| 4708 | "shoulder", which may or may not be physically reasonable. |
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| 4709 | |
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| 4710 | The well width (*l*\ ) is defined as multiples of the particle diameter (2\*\ *R*\ ) |
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| 4711 | |
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| 4712 | The interaction potential is: |
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| 4713 | |
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| 4714 | .. image:: ..\img\olddocs\image225.PNG |
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| 4715 | |
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| 4716 | where *r* is the distance from the center of the sphere of a radius *R*. |
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| 4717 | |
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| 4718 | 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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| 4719 | |
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| 4720 | .. image:: ..\img\olddocs\image040.gif |
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| 4721 | |
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| 4722 | ============== ========= ============= |
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| 4723 | Parameter name Units Default value |
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| 4724 | ============== ========= ============= |
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| 4725 | effect_radius |Ang| 50.0 |
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| 4726 | volfraction None 0.04 |
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| 4727 | welldepth kT 1.5 |
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| 4728 | wellwidth diameters 1.2 |
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| 4729 | ============== ========= ============= |
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| 4730 | |
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| 4731 | .. image:: ..\img\olddocs\image226.jpg |
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| 4732 | |
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| 4733 | *Figure. 1D plot using the default values (in linear scale).* |
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| 4734 | |
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| 4735 | REFERENCE |
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| 4736 | |
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| 4737 | R V Sharma, K C Sharma, *Physica*, 89A (1977) 213 |
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| 4738 | |
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| 4739 | |
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| 4740 | |
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| 4741 | .. _HayterMSAStructure: |
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| 4742 | |
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| 4743 | **2.3.3. HayterMSAStructure Factor** |
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| 4744 | |
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| 4745 | This is an implementation of the Rescaled Mean Spherical Approximation which calculates the structure factor (the |
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| 4746 | Fourier transform of the pair correlation function *g(r)*) for a system of charged, spheroidal objects in a |
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| 4747 | dielectric medium. When combined with an appropriate form factor (such as sphere,core+shell, ellipsoid, etc), this |
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| 4748 | allows for inclusion of the interparticle interference effects due to screened coulomb repulsion between charged particles. |
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| 4749 | |
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| 4750 | **This routine only works for charged particles**. If the charge is set to zero the routine will self-destruct! |
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| 4751 | For non-charged particles use a hard sphere potential. |
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| 4752 | |
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| 4753 | The salt concentration is used to compute the ionic strength of the solution which in turn is used to compute the Debye |
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| 4754 | screening length. At present there is no provision for entering the ionic strength directly nor for use of any |
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| 4755 | multivalent salts. The counterions are also assumed to be monovalent. |
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| 4756 | |
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| 4757 | 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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| 4758 | |
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| 4759 | .. image:: ..\img\olddocs\image040.gif |
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| 4760 | |
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| 4761 | ============== ======== ============= |
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| 4762 | Parameter name Units Default value |
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| 4763 | ============== ======== ============= |
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| 4764 | effect_radius |Ang| 20.8 |
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| 4765 | charge *e* 19 |
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| 4766 | volfraction None 0.2 |
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| 4767 | temperature K 318 |
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| 4768 | salt conc M 0 |
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| 4769 | dielectconst None 71.1 |
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| 4770 | ============== ======== ============= |
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| 4771 | |
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| 4772 | .. image:: ..\img\olddocs\image227.jpg |
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| 4773 | |
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| 4774 | *Figure. 1D plot using the default values (in linear scale).* |
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| 4775 | |
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| 4776 | REFERENCE |
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| 4777 | |
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| 4778 | J B Hayter and J Penfold, *Molecular Physics*, 42 (1981) 109-118 |
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| 4779 | |
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| 4780 | J P Hansen and J B Hayter, *Molecular Physics*, 46 (1982) 651-656 |
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| 4781 | |
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| 4782 | |
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| 4783 | .. _StickyHSStructure: |
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| 4784 | |
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| 4785 | **2.3.4. StickyHSStructure Factor** |
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| 4786 | |
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| 4787 | This calculates the interparticle structure factor for a hard sphere fluid with a narrow attractive well. A perturbative |
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| 4788 | solution of the Percus-Yevick closure is used. The strength of the attractive well is described in terms of "stickiness" |
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| 4789 | as defined below. The returned value is a dimensionless structure factor, *S(q)*. |
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| 4790 | |
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| 4791 | The perturb (perturbation parameter), |epsilon|, should be held between 0.01 and 0.1. It is best to hold the |
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| 4792 | perturbation parameter fixed and let the "stickiness" vary to adjust the interaction strength. The stickiness, |tau|, |
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| 4793 | is defined in the equation below and is a function of both the perturbation parameter and the interaction strength. |
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| 4794 | |tau| and |epsilon| are defined in terms of the hard sphere diameter (|sigma| = 2\*\ *R*\ ), the width of the square |
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| 4795 | well, |bigdelta| (same units as *R*), and the depth of the well, *Uo*, in units of kT. From the definition, it is clear |
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| 4796 | that smaller |tau| means stronger attraction. |
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| 4797 | |
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| 4798 | .. image:: ..\img\olddocs\image228.PNG |
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| 4799 | |
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| 4800 | where the interaction potential is |
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| 4801 | |
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| 4802 | .. image:: ..\img\olddocs\image229.PNG |
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| 4803 | |
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| 4804 | The Percus-Yevick (PY) closure was used for this calculation, and is an adequate closure for an attractive interparticle |
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| 4805 | potential. This solution has been compared to Monte Carlo simulations for a square well fluid, with good agreement. |
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| 4806 | |
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| 4807 | The true particle volume fraction, |phi|, is not equal to *h*, which appears in most of the reference. The two are |
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| 4808 | related in equation (24) of the reference. The reference also describes the relationship between this perturbation |
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| 4809 | solution and the original sticky hard sphere (or adhesive sphere) model by Baxter. |
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| 4810 | |
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| 4811 | NB: The calculation can go haywire for certain combinations of the input parameters, producing unphysical solutions - in |
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| 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 |
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| 4813 | plot). Use tight bounds to keep the parameters to values that you know are physical (test them) and keep nudging them |
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| 4814 | until the optimization does not hit the constraints. |
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| 4815 | |
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| 4816 | 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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| 4817 | |
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| 4818 | .. image:: ..\img\olddocs\image040.gif |
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| 4819 | |
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| 4820 | ============== ======== ============= |
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| 4821 | Parameter name Units Default value |
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| 4822 | ============== ======== ============= |
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| 4823 | effect_radius |Ang| 50 |
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| 4824 | perturb None 0.05 |
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| 4825 | volfraction None 0.1 |
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| 4826 | stickiness K 0.2 |
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| 4827 | ============== ======== ============= |
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| 4828 | |
---|
| 4829 | .. image:: ..\img\olddocs\image230.jpg |
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| 4830 | |
---|
| 4831 | *Figure. 1D plot using the default values (in linear scale).* |
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| 4832 | |
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| 4833 | REFERENCE |
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| 4834 | |
---|
| 4835 | S V G Menon, C Manohar, and K S Rao, *J. Chem. Phys.*, 95(12) (1991) 9186-9190 |
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| 4836 | |
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| 4837 | |
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| 4838 | |
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| 4839 | 2.4 Customised Functions |
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| 4840 | ------------------------------ |
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| 4841 | |
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| 4842 | |
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| 4843 | Customized model functions can be redefined or added to by users (See SansView tutorial for details). |
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| 4844 | |
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| 4845 | .. _testmodel: |
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| 4846 | |
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| 4847 | **2.4.1. testmodel** |
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| 4848 | |
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| 4849 | This function, as an example of a user defined function, calculates |
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| 4850 | |
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| 4851 | *I(q)* = *A* + *B* cos(2\ *q*\ ) + *C* sin(2\ *q*\ ) |
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| 4852 | |
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| 4853 | |
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| 4854 | |
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| 4855 | .. _testmodel_2: |
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| 4856 | |
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| 4857 | **2.4.2. testmodel_2** |
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| 4858 | |
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| 4859 | This function, as an example of a user defined function, calculates |
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| 4860 | |
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| 4861 | *I(q)* = *scale* * sin(*f*\ )/*f* |
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| 4862 | |
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| 4863 | where |
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| 4864 | |
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| 4865 | *f* = *A* + *Bq* + *Cq*\ :sup:`2` + *Dq*\ :sup:`3` + *Eq*\ :sup:`4` + *Fq*\ :sup:`5` |
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| 4866 | |
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| 4867 | |
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| 4868 | |
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| 4869 | .. _sum_p1_p2: |
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| 4870 | |
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| 4871 | **2.4.3. sum_p1_p2** |
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| 4872 | |
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| 4873 | This function, as an example of a user defined function, calculates |
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| 4874 | |
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| 4875 | *I(q)* = *scale_factor* \* (CylinderModel + PolymerExclVolumeModel) |
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| 4876 | |
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| 4877 | To make your own (*p1 + p2*) model, select 'Easy Custom Sum' from the Fitting menu, or modify and compile the file |
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| 4878 | named 'sum_p1_p2.py' from 'Edit Custom Model' in the 'Fitting' menu. |
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| 4879 | |
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| 4880 | NB: Summing models only works only for single functional models (ie, single shell models, two-component RPA models, etc). |
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| 4881 | |
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| 4882 | |
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| 4883 | |
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| 4884 | .. _sum_Ap1_1_Ap2: |
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| 4885 | |
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| 4886 | **2.4.4. sum_Ap1_1_Ap2** |
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| 4887 | |
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| 4888 | This function, as an example of a user defined function, calculates |
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| 4889 | |
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| 4890 | *I(q)* = (*scale_factor* \* CylinderModel + (1 - *scale_factor*\ ) \* PolymerExclVolume model) |
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| 4891 | |
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| 4892 | To make your own (*A*\ * *p1* + (1-*A*) \* *p2*) model, modify and compile the file named 'sum_Ap1_1_Ap2.py' from |
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| 4893 | 'Edit Custom Model' in the 'Fitting' menu. |
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| 4894 | |
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| 4895 | NB: Summing models only works only for single functional models (ie, single shell models, two-component RPA models, etc). |
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| 4896 | |
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| 4897 | |
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| 4898 | |
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| 4899 | .. _polynomial5: |
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| 4900 | |
---|
| 4901 | **2.4.5. polynomial5** |
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| 4902 | |
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| 4903 | This function, as an example of a user defined function, calculates |
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| 4904 | |
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| 4905 | *I(q)* = *A* + *Bq* + *Cq*\ :sup:`2` + *Dq*\ :sup:`3` + *Eq*\ :sup:`4` + *Fq*\ :sup:`5` |
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| 4906 | |
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| 4907 | This model can be modified and compiled from 'Edit Custom Model' in the 'Fitting' menu. |
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| 4908 | |
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| 4909 | |
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| 4910 | |
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| 4911 | .. _sph_bessel_jn: |
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| 4912 | |
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| 4913 | **2.4.6. sph_bessel_jn** |
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| 4914 | |
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| 4915 | This function, as an example of a user defined function, calculates |
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| 4916 | |
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| 4917 | *I(q)* = *C* \* *sph_jn(Ax+B)+D* |
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| 4918 | |
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| 4919 | where *sph_jn* is a spherical Bessel function of order *n*. |
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| 4920 | |
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| 4921 | This model can be modified and compiled from 'Edit Custom Model' in the 'Fitting' menu. |
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