Changeset ee74edd in sasview for sansmodels
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sansmodels/src/sans/models/media/model_functions.html
r318b5bbb ree74edd 4 4 </head> 5 5 <body lang="EN-US"> 6 <p>< strong><span style="font-size: 16pt;">Model Functions</span></strong></p>6 <p><b><span style="font-size: 16pt;">Model Functions</span></b></p> 7 7 <ul style="margin-top: 0in;" type="disc"> 8 <li style="line-height: 115%;"><a href="#Introduction"><strong>Introduction</strong></a></li>9 <li style="line-height: 115%;"><a href="#Shapes"><strong>Shapes</strong></a>: <a href="#SphereModel">SphereModel (Magnetic 2D Model)</a>, <a href="#BinaryHSModel">BinaryHSModel</a>, <a href="#FuzzySphereModel">FuzzySphereModel</a>, <a href="#RaspBerryModel">RaspBerryModel</a>, <a href="#CoreShellModel">CoreShellModel (Magnetic 2D Model)</a>, <a href="#Core2ndMomentModel">Core2ndMomentModel</a>, <a href="#CoreMultiShellModel">CoreMultiShellModel (Magnetic 2D Model)</a>, <a href="#VesicleModel">VesicleModel</a>, <a href="#MultiShellModel">MultiShellModel</a>, <a href="#OnionExpShellModel">OnionExpShellModel</a>, <a href="#SphericalSLDModel">SphericalSLDModel</a>, <a href="#LinearPearlsModel">LinearPearlsModel</a>, <a href="#PearlNecklaceModel">PearlNecklaceModel</a> , <a href="#CylinderModel">CylinderModel (Magnetic 2D Model)</a>, <a href="#CoreShellCylinderModel">CoreShellCylinderModel</a>, <a href="#CoreShellBicelleModel">CoreShellBicelleModel</a>,<a href="#HollowCylinderModel">HollowCylinderModel</a>, <a href="#FlexibleCylinderModel">FlexibleCylinderModel</a>, <a href="#FlexibleCylinderModel">FlexCylEllipXModel</a>, <a href="#StackedDisksModel">StackedDisksModel</a>, <a href="#ParallelepipedModel">ParallelepipedModel (Magnetic 2D Model)</a>, <a href="#CSParallelepipedModel">CSParallelepipedModel</a>, <a href="#EllipticalCylinderModel">EllipticalCylinderModel</a>, <a href="#BarBellModel">BarBellModel</a>, <a href="#CappedCylinderModel">CappedCylinderModel</a>, <a href="#EllipsoidModel">EllipsoidModel</a>, <a href="#CoreShellEllipsoidModel">CoreShellEllipsoidModel</a>, <a href="#TriaxialEllipsoidModel">TriaxialEllipsoidModel</a>, <a href="#LamellarModel">LamellarModel</a>, <a href="#LamellarFFHGModel">LamellarFFHGModel</a>, <a href="#LamellarPSModel">LamellarPSModel</a>, <a href="#LamellarPSHGModel">LamellarPSHGModel</a>, <a href="#LamellarPCrystalModel">LamellarPCrystalModel</a>, <a href="#SCCrystalModel">SCCrystalModel</a>, <a href="#FCCrystalModel">FCCrystalModel</a>, <a href="#BCCrystalModel">BCCrystalModel</a>.</li>10 <li style="line-height: 115%;"><a href="#Shape-Independent"><strong>Shape-Independent</strong></a>: <a href="#Absolute%20Power_Law">AbsolutePower_Law</a>, <a href="#BEPolyelectrolyte">BEPolyelectrolyte</a>, <a href="#BroadPeakModel">BroadPeakModel,<span><span style="text-decoration: underline;"><span style="color: blue;">CorrLength</span></span></span><span>,</span></a> <a href="#DABModel">DABModel</a>, <a href="#Debye">Debye</a>, <a href="#Number_Density_Fractal">FractalModel</a>, <a href="#FractalCoreShell">FractalCoreShell</a>, <a href="#GaussLorentzGel">GaussLorentzGel</a>, <a href="#Guinier">Guinier</a>, <a href="#GuinierPorod">GuinierPorod</a>, <a href="#Lorentz">Lorentz</a>, <a href="#Mass_Fractal">MassFractalModel</a>, <a href="#MassSurface_Fractal">MassSurfaceFractal</a>, <a href="#Peak%20Gauss%20Model">PeakGaussModel</a>, <a href="#Peak%20Lorentz%20Model">PeakLorentzModel</a>, <a href="#Poly_GaussCoil">Poly_GaussCoil</a>, <a href="#PolymerExclVolume">PolyExclVolume</a>, <a href="#PorodModel">PorodModel</a>, <a href="#RPA10Model">RPA10Model</a>, <a href="#StarPolymer">StarPolymer</a>, <a href="#Surface_Fractal">SurfaceFractalModel</a>, <a href="#TeubnerStreyModel">Teubner Strey</a>, <a href="#TwoLorentzian">TwoLorentzian</a>, <a href="#TwoPowerLaw">TwoPowerLaw</a>, <a href="#UnifiedPowerRg">UnifiedPowerRg</a>, <a href="#LineModel">LineModel</a>, <a href="#ReflectivityModel">ReflectivityModel</a>, <a href="#ReflectivityIIModel">ReflectivityIIModel</a>, <a href="#GelFitModel">GelFitModel</a>.</li>11 <li style="line-height: 115%;"><a href="#Model"><strong>Customized Models</strong></a>: <a href="#testmodel">testmodel</a>, <a href="#testmodel_2">testmodel_2</a>, <a href="#sum_p1_p2">sum_p1_p2</a>, <a href="#sum_Ap1_1_Ap2">sum_Ap1_1_Ap2</a>, <a href="#polynomial5">polynomial5</a>, <a href="#sph_bessel_jn">sph_bessel_jn</a>.</li>12 <li style="line-height: 115%;"><a href="#Structure_Factors"><strong>Structure Factors</strong></a>: <a href="#HardsphereStructure">HardSphereStructure</a>, <a href="#SquareWellStructure">SquareWellStructure</a>, <a href="#HayterMSAStructure">HayterMSAStructure</a>, <a href="#StickyHSStructure">StickyHSStructure</a>.</li>13 <li style="line-height: 115%;"><a href="#References"><strong>References</strong></a></li>8 <li><a href="#Introduction"><b>Introduction</b></a></li> 9 <li><a href="#Shapes"><b>Shapes</b></a>: <a href="#SphereModel">SphereModel (Magnetic 2D Model)</a>, <a href="#BinaryHSModel">BinaryHSModel</a>, <a href="#FuzzySphereModel">FuzzySphereModel</a>, <a href="#RaspBerryModel">RaspBerryModel</a>, <a href="#CoreShellModel">CoreShellModel (Magnetic 2D Model)</a>, <a href="#Core2ndMomentModel">Core2ndMomentModel</a>, <a href="#CoreMultiShellModel">CoreMultiShellModel (Magnetic 2D Model)</a>, <a href="#VesicleModel">VesicleModel</a>, <a href="#MultiShellModel">MultiShellModel</a>, <a href="#OnionExpShellModel">OnionExpShellModel</a>, <a href="#SphericalSLDModel">SphericalSLDModel</a>, <a href="#LinearPearlsModel">LinearPearlsModel</a>, <a href="#PearlNecklaceModel">PearlNecklaceModel</a> , <a href="#CylinderModel">CylinderModel (Magnetic 2D Model)</a>, <a href="#CoreShellCylinderModel">CoreShellCylinderModel</a>, <a href="#CoreShellBicelleModel">CoreShellBicelleModel</a>,<a href="#HollowCylinderModel">HollowCylinderModel</a>, <a href="#FlexibleCylinderModel">FlexibleCylinderModel</a>, <a href="#FlexibleCylinderModel">FlexCylEllipXModel</a>, <a href="#StackedDisksModel">StackedDisksModel</a>, <a href="#ParallelepipedModel">ParallelepipedModel (Magnetic 2D Model)</a>, <a href="#CSParallelepipedModel">CSParallelepipedModel</a>, <a href="#EllipticalCylinderModel">EllipticalCylinderModel</a>, <a href="#BarBellModel">BarBellModel</a>, <a href="#CappedCylinderModel">CappedCylinderModel</a>, <a href="#EllipsoidModel">EllipsoidModel</a>, <a href="#CoreShellEllipsoidModel">CoreShellEllipsoidModel</a>, <a href="#TriaxialEllipsoidModel">TriaxialEllipsoidModel</a>, <a href="#LamellarModel">LamellarModel</a>, <a href="#LamellarFFHGModel">LamellarFFHGModel</a>, <a href="#LamellarPSModel">LamellarPSModel</a>, <a href="#LamellarPSHGModel">LamellarPSHGModel</a>, <a href="#LamellarPCrystalModel">LamellarPCrystalModel</a>, <a href="#SCCrystalModel">SCCrystalModel</a>, <a href="#FCCrystalModel">FCCrystalModel</a>, <a href="#BCCrystalModel">BCCrystalModel</a>.</li> 10 <li><a href="#Shape-Independent"><b>Shape-Independent</b></a>: <a href="#Absolute%20Power_Law">AbsolutePower_Law</a>, <a href="#BEPolyelectrolyte">BEPolyelectrolyte</a>, <a href="#BroadPeakModel">BroadPeakModel,<span><span style="text-decoration: underline;"><span style="color: blue;">CorrLength</span></span></span><span>,</span></a> <a href="#DABModel">DABModel</a>, <a href="#Debye">Debye</a>, <a href="#Number_Density_Fractal">FractalModel</a>, <a href="#FractalCoreShell">FractalCoreShell</a>, <a href="#GaussLorentzGel">GaussLorentzGel</a>, <a href="#Guinier">Guinier</a>, <a href="#GuinierPorod">GuinierPorod</a>, <a href="#Lorentz">Lorentz</a>, <a href="#Mass_Fractal">MassFractalModel</a>, <a href="#MassSurface_Fractal">MassSurfaceFractal</a>, <a href="#Peak%20Gauss%20Model">PeakGaussModel</a>, <a href="#Peak%20Lorentz%20Model">PeakLorentzModel</a>, <a href="#Poly_GaussCoil">Poly_GaussCoil</a>, <a href="#PolymerExclVolume">PolyExclVolume</a>, <a href="#PorodModel">PorodModel</a>, <a href="#RPA10Model">RPA10Model</a>, <a href="#StarPolymer">StarPolymer</a>, <a href="#Surface_Fractal">SurfaceFractalModel</a>, <a href="#TeubnerStreyModel">Teubner Strey</a>, <a href="#TwoLorentzian">TwoLorentzian</a>, <a href="#TwoPowerLaw">TwoPowerLaw</a>, <a href="#UnifiedPowerRg">UnifiedPowerRg</a>, <a href="#LineModel">LineModel</a>, <a href="#ReflectivityModel">ReflectivityModel</a>, <a href="#ReflectivityIIModel">ReflectivityIIModel</a>, <a href="#GelFitModel">GelFitModel</a>.</li> 11 <li><a href="#Model"><b>Customized Models</b></a>: <a href="#testmodel">testmodel</a>, <a href="#testmodel_2">testmodel_2</a>, <a href="#sum_p1_p2">sum_p1_p2</a>, <a href="#sum_Ap1_1_Ap2">sum_Ap1_1_Ap2</a>, <a href="#polynomial5">polynomial5</a>, <a href="#sph_bessel_jn">sph_bessel_jn</a>.</li> 12 <li><a href="#Structure_Factors"><b>Structure Factors</b></a>: <a href="#HardsphereStructure">HardSphereStructure</a>, <a href="#SquareWellStructure">SquareWellStructure</a>, <a href="#HayterMSAStructure">HayterMSAStructure</a>, <a href="#StickyHSStructure">StickyHSStructure</a>.</li> 13 <li><a href="#References"><b>References</b></a></li> 14 14 </ul> 15 <p style="margin-left: 0.25in; text-indent: -0.25in;">< strong><span style="font-size: 16pt;">1.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="Introduction"></a><strong><span style="font-size: 16pt;">Introduction </span></strong></p>15 <p style="margin-left: 0.25in; text-indent: -0.25in;"><b><span style="font-size: 16pt;">1.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="Introduction"></a><b><span style="font-size: 16pt;">Introduction </span></b></p> 16 16 17 17 <p> Many of our models use the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research and thus some content and figures in this document are originated from or shared with the NIST Igor analysis package.</p> 18 <p style="margin-left: 0.25in; text-indent: -0.25in;">< strong><span style="font-size: 16pt;">2.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="Shapes"></a><strong><span style="font-size: 16pt;">Shapes (Scattering Intensity Models)</span></strong></p>18 <p style="margin-left: 0.25in; text-indent: -0.25in;"><b><span style="font-size: 16pt;">2.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="Shapes"></a><b><span style="font-size: 16pt;">Shapes (Scattering Intensity Models)</span></b></p> 19 19 <p>This software provides form factors for various particle shapes. After giving a mathematical definition of each model, we draw the list of parameters available to the user. Validation plots for each model are also presented. Instructions on how to use the software is available with the source code.</p> 20 20 … … 23 23 <p>with</p> 24 24 <p style="text-align: center;" align="center"><span style="position: relative; top: 8pt;"><img src="img/image002.PNG" alt="" /></span> </p> 25 <p>where <em>P</em>0<em>(< strong>q</strong>)</em> is the un-normalized form factor, <em>ρ(<strong>r</strong>)</em> is the scattering length density at a given point in space and the integration is done over the volume <em>V</em> of the scatterer.</p>25 <p>where <em>P</em>0<em>(<b>q</b>)</em> is the un-normalized form factor, <em>ρ(<b>r</b>)</em> is the scattering length density at a given point in space and the integration is done over the volume <em>V</em> of the scatterer.</p> 26 26 <p>For systems without inter-particle interference, the form factors we provide can be related to the scattering intensity by the particle volume fraction:<span style="position: relative; top: 5pt;"> <img src="img/image003.PNG" alt="" /></span>.</p> 27 27 <p>Our so-called 1D scattering intensity functions provide <em>P(q) </em>for the case where the scatterer is randomly oriented. In that case, the scattering intensity only depends on the length of q. The intensity measured on the plane of the SANS detector will have an azimuthal symmetry around <em>q</em>=0.</p> 28 28 <p>Our so-called 2D scattering intensity functions provide <em>P(q, </em><em><span style="font-family: 'Arial','sans-serif';">φ</span>)</em> for an oriented system as a function of a q-vector in the plane of the detector. We define the angle <span style="font-family: 'Arial','sans-serif';">φ</span> as the angle between the q vector and the horizontal (x) axis of the plane of the detector.</p> 29 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.1.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="SphereModel"></a><strong><span style="font-size: 14pt;">Sphere Model (Magnetic 2D Model)</span></strong></p>29 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.1.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="SphereModel"></a><b><span style="font-size: 14pt;">Sphere Model (Magnetic 2D Model)</span></b></p> 30 30 <p>This model provides the form factor, P(q), for a monodisperse spherical particle with uniform scattering length density. The form factor is normalized by the particle volume as described below.</p> 31 31 For magnetic scattering, please see the '<a href="polar_mag_help.html">Polarization/Magnetic Scattering</a>' in Fitting Help. 32 <p style="margin-left: 0.85in; text-indent: -0.35in;">< strong>1.1.</strong><strong><span style="font-size: 7pt;"> </span>Definition</strong></p>32 <p style="margin-left: 0.85in; text-indent: -0.35in;"><b>1.1.</b><b><span style="font-size: 7pt;"> </span>Definition</b></p> 33 33 <p>The 1D scattering intensity is calculated in the following way (Guinier, 1955):</p> 34 34 <p style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="img/image004.PNG" alt="" /></span></p> … … 110 110 </div> 111 111 <p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p> 112 <p style="margin-left: 0.85in; text-indent: -0.35in;">< strong>2.1.</strong><strong><span style="font-size: 7pt;"> </span>Validation of the sphere model</strong></p>112 <p style="margin-left: 0.85in; text-indent: -0.35in;"><b>2.1.</b><b><span style="font-size: 7pt;"> </span>Validation of the sphere model</b></p> 113 113 <p>Validation of our code was done by comparing the output of the 1D model to the output of the software provided by the NIST (Kline, 2006). Figure 1 shows a comparison of the output of our model and the output of the NIST software.</p> 114 114 <p style="text-align: center; page-break-after: avoid;" align="center"><img src="img/image005.jpg" alt="sphere_1D_validation" width="573" height="315" /></p> 115 115 <p style="text-align: center; page-break-after: avoid;" align="center"> </p> 116 116 <p>Figure 1: Comparison of the DANSE scattering intensity for a sphere with the output of the NIST SANS analysis software. The parameters were set to: Scale=1.0, Radius=60 Å, Contrast=1e-6 Å -2, and Background=0.01 cm -1.</p> 117 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.2.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="BinaryHSModel"></a><strong><span style="font-size: 14pt;">BinaryHSModel</span></strong></p>117 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.2.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="BinaryHSModel"></a><b><span style="font-size: 14pt;">BinaryHSModel</span></b></p> 118 118 <p>This model (binary hard sphere model) provides the scattering intensity, for binary mixture of spheres including hard sphere interaction between those particles. Using Percus-Yevick closure, the calculation is an exact multi-component solution:</p> 119 119 <p style="text-align: center;" align="center"><span style="position: relative; top: 5pt;"><img src="img/image006.PNG" alt="" /></span></p> … … 228 228 </div> 229 229 <p style="text-align: center;" align="center"><img id="Picture 197" src="img/image009.jpg" alt="" width="484" height="361" /></p> 230 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values above (w/200 data point).</strong></p>230 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values above (w/200 data point).</b></p> 231 231 <p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p> 232 232 <p>See the reference for details.</p> … … 234 234 <p>N. W. Ashcroft and D. C. Langreth, Physical Review, v. 156 (1967) 685-692.</p> 235 235 <p>[Errata found in Phys. Rev. 166 (1968) 934.]</p> 236 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.3.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="FuzzySphereModel"></a><strong><span style="font-size: 14pt;">FuzzySphereModel</span></strong></p>237 <p>< strong> </strong>This model is to calculate the scattering from spherical particles with a "fuzzy" interface.</p>238 <p style="margin-left: 0.85in; text-indent: -0.35in;">< strong>1.1.</strong><strong><span style="font-size: 7pt;"> </span>Definition</strong></p>236 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.3.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="FuzzySphereModel"></a><b><span style="font-size: 14pt;">FuzzySphereModel</span></b></p> 237 <p><b> </b>This model is to calculate the scattering from spherical particles with a "fuzzy" interface.</p> 238 <p style="margin-left: 0.85in; text-indent: -0.35in;"><b>1.1.</b><b><span style="font-size: 7pt;"> </span>Definition</b></p> 239 239 <p>The 1D scattering intensity is calculated in the following way (Guinier, 1955):</p> 240 240 <p>The returned value is scaled to units of [cm-1 sr-1], absolute scale.</p> … … 336 336 </div> 337 337 <p style="text-align: center;" align="center"><img src="img/image012.jpg" alt="" width="442" height="275" /></p> 338 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/200 data point).</strong></p>339 <p> </p> 340 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.4.</span></strong> <a name="RaspBerryModel"></a><strong><span style="font-size: 14pt;">RaspBerryModel</span></strong></p>338 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/200 data point).</b></p> 339 <p> </p> 340 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.4.</span></b> <a name="RaspBerryModel"></a><b><span style="font-size: 14pt;">RaspBerryModel</span></b></p> 341 341 <p> </p> 342 342 <p>Calculates the form factor, P(q), for a "Raspberry-like" structure where there are smaller spheres at the surface of a larger sphere, such as the structure of a Pickering emulsion. </p> 343 <p style="margin-left: 0.85in; text-indent: -0.35in;">< strong>1.1.</strong><strong><span style="font-size: 7pt;"> </span>Definition</strong></p>343 <p style="margin-left: 0.85in; text-indent: -0.35in;"><b>1.1.</b><b><span style="font-size: 7pt;"> </span>Definition</b></p> 344 344 <p>The structure is:<br /> <img src="img/raspberry_pic.jpg" alt="" /></p> 345 345 <p><br /> Ro = the radius of the large sphere<br /> Rp = the radius of the smaller sphere on the surface<br /> delta = the fractional penetration depth<br /> surface coverage = fractional coverage of the large sphere surface (0.9 max)<br /> <br /> <br /> The large and small spheres have their own SLD, as well as the solvent. The surface coverage term is a fractional coverage (maximum of approximately 0.9 for hexagonally packed spheres on a surface). Since not all of the small spheres are necessarily attached to the surface, the excess free (small) spheres scattering is also included in the calculation. The function calculated follows equations (8)-(12) of the reference below, and the equations are not reproduced here.<br /> <br /> The returned value is scaled to units of [cm-1]. No interparticle scattering is included in this model.</p> … … 450 450 </div> 451 451 <p style="text-align: center;" align="center"><img src="img/raspberry_plot.jpg" alt="" /></p> 452 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the values of /2000 data points.</strong></p>452 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the values of /2000 data points.</b></p> 453 453 <p style="text-align: center;" align="center"> </p> 454 454 <p> </p> 455 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.5.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="CoreShellModel"></a><strong><span style="font-size: 14pt;">Core Shell (Sphere) Model (Magnetic 2D Model)</span></strong></p>455 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.5.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="CoreShellModel"></a><b><span style="font-size: 14pt;">Core Shell (Sphere) Model (Magnetic 2D Model)</span></b></p> 456 456 <p>This model provides the form factor, P(<em>q</em>), for a spherical particle with a core-shell structure. The form factor is normalized by the particle volume.</p> 457 457 For magnetic scattering, please see the '<a href="polar_mag_help.html">Polarization/Magnetic Scattering</a>' in Fitting Help. 458 <p style="margin-left: 0.85in; text-indent: -0.35in;">< strong>1.1.</strong><strong><span style="font-size: 7pt;"> </span>Definition</strong></p>458 <p style="margin-left: 0.85in; text-indent: -0.35in;"><b>1.1.</b><b><span style="font-size: 7pt;"> </span>Definition</b></p> 459 459 <p>The 1D scattering intensity is calculated in the following way (Guinier, 1955):</p> 460 460 <p style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="img/image013.PNG" alt="" /></span> </p> … … 563 563 <p>REFERENCE</p> 564 564 <p>Guinier, A. and G. Fournet, "Small-Angle Scattering of X-Rays", John Wiley and Sons, New York, (1955).</p> 565 <p style="margin-left: 0.85in; text-indent: -0.35in;">< strong>2.1.</strong><strong><span style="font-size: 7pt;"> </span>Validation of the core-shell sphere model</strong></p>565 <p style="margin-left: 0.85in; text-indent: -0.35in;"><b>2.1.</b><b><span style="font-size: 7pt;"> </span>Validation of the core-shell sphere model</b></p> 566 566 <p>Validation of our code was done by comparing the output of the 1D model to the output of the software provided by the NIST (Kline, 2006). Figure 1 shows a comparison of the output of our model and the output of the NIST software.</p> 567 567 <p style="text-align: center; page-break-after: avoid;" align="center"><img src="img/image014.jpg" alt="core_shell_sphere_1D_validation" width="573" height="315" /></p> 568 568 <p>Figure 7: Comparison of the DANSE scattering intensity for a core-shell sphere with the output of the NIST SANS analysis software. The parameters were set to: Scale=1.0, Radius=60 Å, Contrast=1e-6 Å -2, and Background=0.001 cm -1.</p> 569 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.6.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="Core2ndMomentModel"></a><strong><span style="font-size: 14pt;">Core2ndMomentModel</span></strong></p>569 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.6.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="Core2ndMomentModel"></a><b><span style="font-size: 14pt;">Core2ndMomentModel</span></b></p> 570 570 <p>This model describes the scattering from a layer of surfactant or polymer adsorbed on spherical particles under the conditions that (i) the <span style="font-weight: bold;">particles (cores) are contrast-matched to the dispersion medium,</span> (ii) <span style="font-weight: bold;">S(Q)~1 </span>(ie, the particle volume fraction is dilute), (iii)<span style="font-weight: bold;"> the particle radius is >> layer thickness</span> (ie, the interface is locally flat), and (iv) scattering from excess unadsorbed adsorbate in the bulk medium is absent or has been corrected for.</p> 571 571 <p>Unlike a core-shell model, this model does not assume any form for the density distribution of the adsorbed species normal to the interface (cf, a core-shell model which assumes the density distribution to be a homogeneous step-function). For comparison, if the thickness of a (core-shell like) step function distribution is t, the second moment, sigma = sqrt((t^2)/12). The sigma is the second moment about the mean of the density distribution (ie, the distance of the centre-of-mass of the distribution from the interface).</p> 572 <p style="margin-left: 0.85in; text-indent: -0.35in;">< strong>1.1.</strong><strong><span style="font-size: 7pt;"> </span>Definition</strong></p>572 <p style="margin-left: 0.85in; text-indent: -0.35in;"><b>1.1.</b><b><span style="font-size: 7pt;"> </span>Definition</b></p> 573 573 <p>The I0 is calculated in the following way (King, 2002):</p> 574 574 <p style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img style="width: 542px; height: 55px;" src="img/secondmeq1.jpg" alt="" /></span> </p> … … 687 687 <p> </p> 688 688 <p style="text-align: center; page-break-after: avoid;" align="center"><img style="width: 526px; height: 333px;" src="img/secongm_fig1.jpg" alt="core_scondmoment_1D_validation" /></p> 689 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.7.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="CoreMultiShellModel"></a><strong><span style="font-size: 14pt;">CoreMultiShell(Sphere)Model (Magnetic 2D Model)</span></strong></p>689 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.7.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="CoreMultiShellModel"></a><b><span style="font-size: 14pt;">CoreMultiShell(Sphere)Model (Magnetic 2D Model)</span></b></p> 690 690 <p>This model provides the scattering from spherical core with from 1 up to 4 shell structures. It has a core of a specified radius, with four shells. The SLDs of the core and each shell are individually specified. </p> 691 691 For magnetic scattering, please see the '<a href="polar_mag_help.html">Polarization/Magnetic Scattering</a>' in Fitting Help. 692 <p style="margin-left: 0.85in; text-indent: -0.35in;">< strong>1.1.</strong><strong><span style="font-size: 7pt;"> </span>Definition</strong></p>692 <p style="margin-left: 0.85in; text-indent: -0.35in;"><b>1.1.</b><b><span style="font-size: 7pt;"> </span>Definition</b></p> 693 693 <p>The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> 694 694 <p>This model is a trivial extension of the CoreShell function to a larger number of shells. See the CoreShell function for a diagram and documentation.</p> … … 866 866 <p>This example dataset is produced by running the CoreMultiShellModel using 200 data points, qmin = 0.001 Å-1, qmax = 0.7 Å-1 and the above default values.</p> 867 867 <p style="text-align: center;" align="center"><img id="Picture 131" src="img/image015.jpg" alt="" width="540" height="390" /></p> 868 <p style="text-align: center;" align="center">< strong>Figure: 1D plot using the default values (w/200 data point).</strong></p>868 <p style="text-align: center;" align="center"><b>Figure: 1D plot using the default values (w/200 data point).</b></p> 869 869 <p> The scattering length density profile for the default sld values (w/ 4 shells).</p> 870 870 <p style="text-align: center;" align="center"><img src="img/image016.jpg" alt="" width="504" height="351" /></p> 871 <p style="text-align: center;" align="center">< strong>Figure: SLD profile against the radius of the sphere for default SLDs.</strong></p>872 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.8.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="VesicleModel"></a><strong><span style="font-size: 14pt;">VesicleModel</span></strong></p>871 <p style="text-align: center;" align="center"><b>Figure: SLD profile against the radius of the sphere for default SLDs.</b></p> 872 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.8.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="VesicleModel"></a><b><span style="font-size: 14pt;">VesicleModel</span></b></p> 873 873 <p>This model provides the form factor, P(<em>q</em>), for an unilamellar vesicle. The form factor is normalized by the volume of the shell.</p> 874 874 <p>The 1D scattering intensity is calculated in the following way (Guinier, 1955):</p> … … 965 965 </div> 966 966 <p style="text-align: center;" align="center"><img id="Picture 158" src="img/image019.jpg" alt="" width="454" height="356" /></p> 967 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/200 data point).</strong></p>967 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/200 data point).</b></p> 968 968 <p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p> 969 969 <p>REFERENCE</p> 970 970 <p>Guinier, A. and G. Fournet, "Small-Angle Scattering of X-Rays", John Wiley and Sons, New York, (1955).</p> 971 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.9.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="MultiShellModel"></a><strong><span style="font-size: 14pt;">MultiShellModel</span></strong></p>971 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.9.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="MultiShellModel"></a><b><span style="font-size: 14pt;">MultiShellModel</span></b></p> 972 972 <p>This model provides the form factor, P(<em>q</em>), for a multi-lamellar vesicle with N shells where the core is filled with solvent and the shells are interleaved with layers of solvent. For N = 1, this return to the vesicle model (above).</p> 973 973 <p style="text-align: center;" align="center"><img id="Picture 32" src="img/image020.jpg" alt="" width="423" height="371" /></p> … … 1082 1082 </div> 1083 1083 <p style="text-align: center;" align="center"><img id="Picture 173" src="img/image021.jpg" alt="" width="484" height="368" /></p> 1084 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/200 data point).</strong></p>1084 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/200 data point).</b></p> 1085 1085 <p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p> 1086 1086 <p>REFERENCE</p> 1087 1087 <p>Cabane, B., Small Angle Scattering Methods, Surfactant Solutions: New Methods of Investigation, Ch.2, Surfactant Science Series Vol. 22, Ed. R. Zana, M. Dekker, New York, 1987.</p> 1088 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.10.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="OnionExpShellModel"></a><strong><span style="font-size: 14pt;">OnionExpShellModel</span></strong></p>1088 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.10.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="OnionExpShellModel"></a><b><span style="font-size: 14pt;">OnionExpShellModel</span></b></p> 1089 1089 <p> </p> 1090 1090 <p>This model provides the form factor, <em>P</em>(<em>q</em>), for a multi-shell sphere where the scattering length density (SLD) of the each shell is described by an exponential (linear, or flat-top) function. The form factor is normalized by the volume of the sphere where the SLD is not identical to the SLD of the solvent. We currently provide up to 9 shells with this model.</p> … … 1103 1103 <p>For |A|>0,</p> 1104 1104 <p style="text-align: center;" align="center"><span style="font-size: 12pt; font-family: 'Times New Roman','serif';"><img src="img/image029.gif" alt="" width="481" height="327" /></span></p> 1105 <p>For A < strong>~ </strong>0 (eg., A = - 0.0001), this function converges to that of the linear SLD profile (ie, <em><span style="font-family: Symbol;">r</span>shelli</em>(<em>r</em>) = <em>A<strong>’</strong></em>(<em>r</em> - <em>rshelli-1</em>) /<em><span style="font-family: Symbol;">D</span>tshelli</em>) + <em>B<strong>’</strong></em>), so this case it is equivalent to</p>1105 <p>For A <b>~ </b>0 (eg., A = - 0.0001), this function converges to that of the linear SLD profile (ie, <em><span style="font-family: Symbol;">r</span>shelli</em>(<em>r</em>) = <em>A<b>’</b></em>(<em>r</em> - <em>rshelli-1</em>) /<em><span style="font-family: Symbol;">D</span>tshelli</em>) + <em>B<b>’</b></em>), so this case it is equivalent to</p> 1106 1106 <p><span style="font-size: 12pt; font-family: 'Times New Roman','serif';"><img src="img/image030.gif" alt="" width="531" height="45" /></span></p> 1107 1107 <p><span style="font-size: 12pt; font-family: 'Times New Roman','serif';"><img src="img/image031.gif" alt="" width="483" height="45" /></span></p> … … 1243 1243 </div> 1244 1244 <p style="text-align: center;" align="center"><img src="img/image041.jpg" alt="" width="493" height="334" /></p> 1245 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/400 point).</strong></p>1245 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/400 point).</b></p> 1246 1246 <p style="text-align: center;" align="center"> <img src="img/image042.jpg" alt="" width="497" height="372" /></p> 1247 <p style="text-align: center;" align="center"> < strong>Figure. SLD profile from the default values.</strong></p>1247 <p style="text-align: center;" align="center"> <b>Figure. SLD profile from the default values.</b></p> 1248 1248 <p>REFERENCE</p> 1249 1249 <p>L.A.Feigin and D.I.Svergun, ‘Structure Analysis by Small-Angle X-Ray and Neutron Scattering’, Plenum Press, New York, 1987</p> 1250 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.11.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="SphericalSLDModel"></a><strong><span style="font-size: 14pt;">SphericalSLDModel</span></strong></p>1250 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.11.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="SphericalSLDModel"></a><b><span style="font-size: 14pt;">SphericalSLDModel</span></b></p> 1251 1251 <p> </p> 1252 1252 <p>Similarly to the OnionExpShellModel, this model provides the form factor, <em>P</em>(<em>q</em>), for a multi-shell sphere, where the interface between the each neighboring shells can be described by one of the functions including error, power-law, and exponential functions. This model is to calculate the scattering intensity by building a continuous custom SLD profile against the radius of the particle. The SLD profile is composed of a flat core, a flat solvent, a number (up to 9 shells) of flat shells, and the interfacial layers between the adjacent flat shells (or core, and solvent) (See below). Unlike OnionExpShellModel (using an analytical integration), the interfacial layers are sub-divided and numerically integrated assuming each sub-layers are described by a line function. The number of the sub-layer can be given by users by setting the integer values of ‘npts_inter#’ in GUI. The form factor is normalized by the total volume of the sphere.</p> … … 1454 1454 </div> 1455 1455 <p style="text-align: center;" align="center"><img src="img/image057.jpg" alt="" width="481" height="332" /></p> 1456 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/400 point).</strong></p>1456 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/400 point).</b></p> 1457 1457 <p style="text-align: center;" align="center"> <img src="img/image058.jpg" alt="" width="536" height="344" /></p> 1458 <p style="text-align: center;" align="center"> < strong>Figure. SLD profile from the default values.</strong></p>1458 <p style="text-align: center;" align="center"> <b>Figure. SLD profile from the default values.</b></p> 1459 1459 <p>REFERENCE</p> 1460 1460 <p>L.A.Feigin and D.I.Svergun, ‘Structure Analysis by Small-Angle X-Ray and Neutron Scattering’, Plenum Press, New York, 1987</p> 1461 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.12.</span></strong> <strong><a name="LinearPearlsModel"></a><span style="font-size: 14pt;">LinearPearlsModel</span></strong></p>1461 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.12.</span></b> <b><a name="LinearPearlsModel"></a><span style="font-size: 14pt;">LinearPearlsModel</span></b></p> 1462 1462 <p><a name="LinearPearlsModel"></a>This model provides the form factor for pearls linearly joined by short strings: N pearls (homogeneous spheres), the radius R and the string segment length (or edge separation) l (= A- 2R)). The A is the center to center pearl separation distance. The thickness of each string is assumed to be negligable.</p> 1463 1463 <p> </p> 1464 1464 <p><a name="LinearPearlsModel"></a><img src="img/linearpearls.jpg" alt="" /></p> 1465 <p style="margin-left: 0.85in; text-indent: -0.35in;"><a name="LinearPearlsModel"></a>< strong>1.1.</strong><strong><span style="font-size: 7pt;"> </span>Definition</strong></p>1465 <p style="margin-left: 0.85in; text-indent: -0.35in;"><a name="LinearPearlsModel"></a><b>1.1.</b><b><span style="font-size: 7pt;"> </span>Definition</b></p> 1466 1466 <p><a name="LinearPearlsModel"></a> </p> 1467 1467 <p><a name="LinearPearlsModel"></a>The output of the scattering intensity function for the linearpearls model is given by (Dobrynin, 1996):</p> … … 1568 1568 <p><a name="LinearPearlsModel"></a>REFERENCE</p> 1569 1569 <p><a name="LinearPearlsModel"></a>A. V. Dobrynin, M. Rubinstein and S. P. Obukhov, Macromol. 29, 2974-2979, 1996.</p> 1570 <p style="margin-left: 0.55in; text-indent: -0.3in;"><a name="LinearPearlsModel"></a>< strong><span style="font-size: 14pt;">2.13.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><strong><a name="PearlNecklaceModel"></a><span style="font-size: 14pt;">PearlNecklaceModel</span></strong></p>1570 <p style="margin-left: 0.55in; text-indent: -0.3in;"><a name="LinearPearlsModel"></a><b><span style="font-size: 14pt;">2.13.</span></b><b><span style="font-size: 7pt;"> </span></b><b><a name="PearlNecklaceModel"></a><span style="font-size: 14pt;">PearlNecklaceModel</span></b></p> 1571 1571 <p><a name="PearlNecklaceModel"></a>This model provides the form factor for a pearl necklace composed of two elements: N pearls (homogeneous spheres) freely jointed by M rods (like strings) (with a total mass Mw = M *mr + N * ms, the radius R and the string segment length (or edge separation) l (= A- 2R)). The A is the center to center pearl separation distance.</p> 1572 1572 <p> </p> 1573 1573 <p><a name="PearlNecklaceModel"></a><img src="img/pearl_fig.jpg" alt="" /></p> 1574 <p style="margin-left: 0.85in; text-indent: -0.35in;"><a name="PearlNecklaceModel"></a>< strong>1.1.</strong><strong><span style="font-size: 7pt;"> </span>Definition</strong></p>1574 <p style="margin-left: 0.85in; text-indent: -0.35in;"><a name="PearlNecklaceModel"></a><b>1.1.</b><b><span style="font-size: 7pt;"> </span>Definition</b></p> 1575 1575 <p><a name="PearlNecklaceModel"></a>The output of the scattering intensity function for the pearlnecklace model is given by (Schweins, 2004):</p> 1576 1576 <p style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><a name="PearlNecklaceModel"></a><img src="img/pearl_eq1.gif" alt="" /></span><a name="PearlNecklaceModel"></a> </p> … … 1709 1709 <p><a name="PearlNecklaceModel"></a>R. Schweins and K. Huber, ‘Particle Scattering Factor of Pearl Necklace Chains’, Macromol. Symp., 211, 25-42, 2004.</p> 1710 1710 <p><a name="PearlNecklaceModel"></a> </p> 1711 <p style="margin-left: 0.55in; text-indent: -0.3in;"><a name="PearlNecklaceModel"></a>< strong><span style="font-size: 14pt;">2.14.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="CylinderModel"></a><strong><span style="font-size: 14pt;">Cylinder Model (Magnetic 2D Model)</span></strong></p>1711 <p style="margin-left: 0.55in; text-indent: -0.3in;"><a name="PearlNecklaceModel"></a><b><span style="font-size: 14pt;">2.14.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="CylinderModel"></a><b><span style="font-size: 14pt;">Cylinder Model (Magnetic 2D Model)</span></b></p> 1712 1712 <p>This model provides the form factor for a right circular cylinder with uniform scattering length density. The form factor is normalized by the particle volume.</p> 1713 1713 For magnetic scattering, please see the '<a href="polar_mag_help.html">Polarization/Magnetic Scattering</a>' in Fitting Help. 1714 <p style="margin-left: 0.85in; text-indent: -0.35in;">< strong>1.1.</strong><strong><span style="font-size: 7pt;"> </span>Definition</strong></p>1714 <p style="margin-left: 0.85in; text-indent: -0.35in;"><b>1.1.</b><b><span style="font-size: 7pt;"> </span>Definition</b></p> 1715 1715 <p>The output of the 2D scattering intensity function for oriented cylinders is given by (Guinier, 1955):</p> 1716 1716 <p style="text-align: center;" align="center"><span style="position: relative; top: 12pt;"><img src="img/image059.PNG" alt="" /></span> </p> … … 1824 1824 <p style="text-align: center;" align="center"><a name="_Ref173306528"></a><a name="_Ref173306479"></a><span style="position: relative; top: 16pt;"><img src="img/image063.PNG" alt="" /></span> </p> 1825 1825 <p>The <em>cyl_theta</em> and <em>cyl_phi</em> parameter are not used for the 1D output. Our implementation of the scattering kernel and the 1D scattering intensity use the c-library from NIST.</p> 1826 <p style="margin-left: 0.85in; text-indent: -0.35in;">< strong>2.1.</strong><strong><span style="font-size: 7pt;"> </span>Validation of the cylinder model</strong></p>1826 <p style="margin-left: 0.85in; text-indent: -0.35in;"><b>2.1.</b><b><span style="font-size: 7pt;"> </span>Validation of the cylinder model</b></p> 1827 1827 <p>Validation of our code was done by comparing the output of the 1D model to the output of the software provided by the NIST (Kline, 2006). Figure 3 shows a comparison of the 1D output of our model and the output of the NIST software.</p> 1828 1828 <p>In general, averaging over a distribution of orientations is done by evaluating the following:</p> … … 1837 1837 <p><a name="_Ref173213305"></a>Figure 4: Comparison of the intensity for uniformly distributed cylinders calculated from our 2D model and the intensity from the NIST SANS analysis software. The parameters used were: Scale=1.0, Radius=20 Å, Length=400 Å, Contrast=3e-6 Å -2, and Background=0.0 cm -1.</p> 1838 1838 <p> </p> 1839 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.15.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="CoreShellCylinderModel"></a><strong><span style="font-size: 14pt;">Core-Shell Cylinder Model</span></strong></p>1839 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.15.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="CoreShellCylinderModel"></a><b><span style="font-size: 14pt;">Core-Shell Cylinder Model</span></b></p> 1840 1840 <p>This model provides the form factor for a circular cylinder with a core-shell scattering length density profile. The form factor is normalized by the particle volume.</p> 1841 <p style="margin-left: 0.85in; text-indent: -0.35in;">< strong>1.1.</strong><strong><span style="font-size: 7pt;"> </span>Definition</strong></p>1841 <p style="margin-left: 0.85in; text-indent: -0.35in;"><b>1.1.</b><b><span style="font-size: 7pt;"> </span>Definition</b></p> 1842 1842 <p>The output of the 2D scattering intensity function for oriented core-shell cylinders is given by (Kline, 2006):</p> 1843 1843 <p style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="img/image067.PNG" alt="" /></span> </p> … … 1977 1977 <p>The output of the 1D scattering intensity function for randomly oriented cylinders is then given by the equation above.</p> 1978 1978 <p>The <em>axis_theta</em> and axis<em>_phi</em> parameters are not used for the 1D output. Our implementation of the scattering kernel and the 1D scattering intensity use the c-library from NIST.</p> 1979 <p style="margin-left: 0.85in; text-indent: -0.35in;">< strong>2.1.</strong><strong><span style="font-size: 7pt;"> </span>Validation of the core-shell cylinder model</strong></p>1979 <p style="margin-left: 0.85in; text-indent: -0.35in;"><b>2.1.</b><b><span style="font-size: 7pt;"> </span>Validation of the core-shell cylinder model</b></p> 1980 1980 <p>Validation of our code was done by comparing the output of the 1D model to the output of the software provided by the NIST (Kline, 2006). Figure 8 shows a comparison of the 1D output of our model and the output of the NIST software.</p> 1981 1981 <p>Averaging over a distribution of orientation is done by evaluating the equation above. Since we have no other software to compare the implementation of the intensity for fully oriented core-shell cylinders, we can compare the result of averaging our 2D output using a uniform distribution <em>p(θ,</em><em><span style="font-family: 'Arial','sans-serif';">φ</span>)</em> = 1.0. Figure 9 shows the result of such a cross-check.</p> … … 1993 1993 <p style="text-align: center;" align="center">Figure. Examples of the angles for oriented pp against the detector plane.</p> 1994 1994 1995 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.16.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="CoreShellBicelleModel"></a><strong><span style="font-size: 14pt;">Core-Shell (Cylinder) Bicelle Model</span></strong></p>1995 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.16.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="CoreShellBicelleModel"></a><b><span style="font-size: 14pt;">Core-Shell (Cylinder) Bicelle Model</span></b></p> 1996 1996 <p>This model provides the form factor for a circular cylinder with a core-shell scattering length density profile. The form factor is normalized by the particle volume. This model is a more general case of <a href="#CoreShellCylinderModel">core-shell cylinder model </a> (see above and reference below) in that the parameters of the shell are separated into a face-shell and a rim-shell so that users can set different values of the thicknesses and slds. </p> 1997 1997 <p> </p> … … 2138 2138 <p> </p> 2139 2139 <p style="text-align: center;" align="center"><img id="cscylbicelle" style="width: 512px; height: 377px;" src="img/cscylbicelle_pic.jpg" alt="" /></p> 2140 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/200 data point).</strong></p>2140 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/200 data point).</b></p> 2141 2141 2142 2142 … … 2149 2149 2150 2150 <p> REFERENCE<br /> Feigin, L. A, and D. I. Svergun, "Structure Analysis by Small-Angle X-Ray and Neutron Scattering", Plenum Press, New York, (1987).</p> 2151 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.17.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="HollowCylinderModel"></a><strong><span style="font-size: 14pt;">HollowCylinderModel</span></strong></p>2151 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.17.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="HollowCylinderModel"></a><b><span style="font-size: 14pt;">HollowCylinderModel</span></b></p> 2152 2152 <p>This model provides the form factor, P(<em>q</em>), for a monodisperse hollow right angle circular cylinder (tube) where the form factor is normalized by the volume of the tube:</p> 2153 2153 <p>P(q) = scale*<f^2>/Vshell+background where the averaging < > id applied only for the 1D calculation. The inside and outside of the hollow cylinder have the same SLD.</p> … … 2256 2256 </div> 2257 2257 <p style="text-align: center;" align="center"><img id="Picture 220" src="img/image074.jpg" alt="" width="468" height="344" /></p> 2258 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/1000 data point).</strong></p>2258 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/1000 data point).</b></p> 2259 2259 <p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p> 2260 2260 … … 2267 2267 <p>REFERENCE</p> 2268 2268 <p>Feigin, L. A, and D. I. Svergun, "Structure Analysis by Small-Angle X-Ray and Neutron Scattering", Plenum Press, New York, (1987).</p> 2269 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.18.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="FlexibleCylinderModel"></a><strong><span style="font-size: 14pt;">FlexibleCylinderModel</span></strong></p>2269 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.18.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="FlexibleCylinderModel"></a><b><span style="font-size: 14pt;">FlexibleCylinderModel</span></b></p> 2270 2270 <p>This model provides the form factor, P(<em>q</em>), for a flexible cylinder where the form factor is normalized by the volume of the cylinder: Inter-cylinder interactions are NOT included. P(q) = scale*<f^2>/V+background where the averaging < > is applied over all orientation for 1D. The 2D scattering intensity is the same as 1D, regardless of the orientation of the <em>q</em> vector which is defined as<span style="font-size: 12pt; font-family: 'Times New Roman','serif'; position: relative; top: 4.5pt;"><img src="img/image040.gif" alt="" width="111" height="23" /></span><span style="font-size: 14pt;">. </span></p> 2271 2271 <p style="text-align: center;" align="center"><img id="Picture 35" src="img/image075.jpg" alt="" width="411" height="187" /></p> … … 2368 2368 </div> 2369 2369 <p style="text-align: center;" align="center"><img id="Picture 228" src="img/image076.jpg" alt="" width="465" height="345" /></p> 2370 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/1000 data point).</strong></p>2370 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/1000 data point).</b></p> 2371 2371 2372 2372 <p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> … … 2376 2376 <p>Correction of the formula can be found in:</p> 2377 2377 <p>Wei-Ren Chen, Paul D. Butler, and Linda J. Magid, "Incorporating Intermicellar Interactions in the Fitting of SANS Data from Cationic Wormlike Micelles" Langmuir, August 2006.</p> 2378 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.19.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><strong><span style="font-size: 14pt;">FlexCylEllipXMo<a name="FlexCylEllipXModel"></a>del</span></strong></p>2379 <p>< strong>Flexible Cylinder with Elliptical Cross-Section: </strong>Calculates the form factor for a flexible cylinder with an elliptical cross section and a uniform scattering length density. The non-negligible diameter of the cylinder is included by accounting for excluded volume interactions within the walk of a single cylinder. The form factor is normalized by the particle volume such that P(q) = scale*<f^2>/Vol + bkg, where < > is an average over all possible orientations of the flexible cylinder.</p>2380 <p style="margin-left: 0.85in; text-indent: -0.35in;">< strong>1.1.</strong><strong><span style="font-size: 7pt;"> </span>Definition</strong></p>2378 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.19.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;">FlexCylEllipXMo<a name="FlexCylEllipXModel"></a>del</span></b></p> 2379 <p><b>Flexible Cylinder with Elliptical Cross-Section: </b>Calculates the form factor for a flexible cylinder with an elliptical cross section and a uniform scattering length density. The non-negligible diameter of the cylinder is included by accounting for excluded volume interactions within the walk of a single cylinder. The form factor is normalized by the particle volume such that P(q) = scale*<f^2>/Vol + bkg, where < > is an average over all possible orientations of the flexible cylinder.</p> 2380 <p style="margin-left: 0.85in; text-indent: -0.35in;"><b>1.1.</b><b><span style="font-size: 7pt;"> </span>Definition</b></p> 2381 2381 <p>The function calculated is from the reference given below. From that paper, "Method 3 With Excluded Volume" is used. The model is a parameterization of simulations of a discrete representation of the worm-like chain model of Kratky and Porod applied in the pseudo-continuous limit. See equations (13, 26-27) in the original reference for the details.</p> 2382 2382 <p>NOTE: there are several typos in the original reference that have been corrected by WRC. Details of the corrections are in the reference below.</p> … … 2501 2501 </div> 2502 2502 <p style="text-align: center;" align="center"><img src="img/image078.jpg" alt="" width="440" height="300" /></p> 2503 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/200 data points).</strong></p>2504 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.20.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="StackedDisksModel"></a><strong><span style="font-size: 14pt;">StackedDisksModel </span></strong></p>2503 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/200 data points).</b></p> 2504 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.20.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="StackedDisksModel"></a><b><span style="font-size: 14pt;">StackedDisksModel </span></b></p> 2505 2505 <p>This model provides the form factor, P(<em>q</em>), for stacked discs (tactoids) with a core/layer structure where the form factor is normalized by the volume of the cylinder. Assuming the next neighbor distance (d-spacing) in a stack of parallel discs obeys a Gaussian distribution, a structure factor S(q) proposed by Kratky and Porod in 1949 is used in this function. Note that the resolution smearing calculation uses 76 Gauss quadrature points to properly smear the model since the function is HIGHLY oscillatory, especially around the q-values that correspond to the repeat distance of the layers.</p> 2506 2506 <p>The 2D scattering intensity is the same as 1D, regardless of the orientation of the <em>q</em> vector which is defined as<span style="font-size: 14pt; position: relative; top: 8pt;"><img src="img/image008.PNG" alt="" /></span><span style="font-size: 14pt;">.</span></p> … … 2644 2644 </div> 2645 2645 <p style="text-align: center;" align="center"><img src="img/image085.jpg" alt="" width="451" height="334" /></p> 2646 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/1000 data point).</strong></p>2646 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/1000 data point).</b></p> 2647 2647 <p style="text-align: center;" align="center"><img id="Picture 101" src="img/image086.jpg" /></p> 2648 2648 <p style="text-align: center;" align="center">Figure. Examples of the angles for oriented stackeddisks against the detector plane.</p> … … 2657 2657 <p>Kratky, O. and Porod, G., J. Colloid Science, 4, 35, 1949.</p> 2658 2658 <p>Higgins, J.S. and Benoit, H.C., "Polymers and Neutron Scattering", Clarendon, Oxford, 1994.</p> 2659 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.21.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="ParallelepipedModel"></a><strong><span style="font-size: 14pt;">ParallelepipedModel (Magnetic 2D Model) </span></strong></p>2659 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.21.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="ParallelepipedModel"></a><b><span style="font-size: 14pt;">ParallelepipedModel (Magnetic 2D Model) </span></b></p> 2660 2660 <p>This model provides the form factor, P(<em>q</em>), for a rectangular cylinder (below) where the form factor is normalized by the volume of the cylinder. P(q) = scale*<f^2>/V+background where the volume V= ABC and the averaging < > is applied over all orientation for 1D. </p> 2661 2661 For magnetic scattering, please see the '<a href="polar_mag_help.html">Polarization/Magnetic Scattering</a>' in Fitting Help. … … 2672 2672 <p>To provide easy access to the orientation of the parallelepiped, we define the axis of the cylinder using two angles θ , <span style="font-family: 'Arial','sans-serif';">φ </span>and<span style="font-family: Symbol;">Y</span>. Similarly to the case of the cylinder, those angles, θ and <span style="font-family: 'Arial','sans-serif';">φ,</span> are defined on Figure 2 of CylinderModel. The angle <span style="font-family: Symbol;">Y </span>is the rotational angle around its own long_c axis against the q plane. For example, <span style="font-family: Symbol;">Y </span>= 0 when the short_b axis is parallel to the x-axis of the detector.</p> 2673 2673 <p style="text-align: center;" align="center"><img src="img/image090.jpg"/></p> 2674 <p style="text-align: center;" align="center">< strong>Figure. Definition of angles for 2D</strong>.</p>2674 <p style="text-align: center;" align="center"><b>Figure. Definition of angles for 2D</b>.</p> 2675 2675 <p style="text-align: center;" align="center"><img src="img/image091.jpg" alt="" width="379" height="256" /></p> 2676 2676 <p style="text-align: center;" align="center">Figure. Examples of the angles for oriented pp against the detector plane.</p> … … 2757 2757 </div> 2758 2758 <p style="text-align: center;" align="center"><img id="Picture 492" src="img/image092.jpg" alt="" width="455" height="351" /></p> 2759 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/1000 data point).</strong></p>2760 <p style="margin-left: 1.35in; text-indent: -0.25in;"><span style="font-family: Symbol;">·</span><span style="font-size: 7pt;"> </span>< strong>Validation of the parallelepiped 2D model</strong></p>2759 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/1000 data point).</b></p> 2760 <p style="margin-left: 1.35in; text-indent: -0.25in;"><span style="font-family: Symbol;">·</span><span style="font-size: 7pt;"> </span><b>Validation of the parallelepiped 2D model</b></p> 2761 2761 <p>Validation of our code was done by comparing the output of the 1D calculation to the angular average of the output of 2 D calculation over all possible angles. The Figure below shows the comparison where the solid dot refers to averaged 2D while the line represents the result of 1D calculation (for the averaging, 76, 180, 76 points are taken over the angles of theta, phi, and psi respectively).</p> 2762 2762 <p style="text-align: center;" align="center"><img id="Picture 104" src="img/image093.gif" alt="" width="481" height="299" /></p> 2763 <p style="text-align: center;" align="center">< strong>Figure. Comparison between 1D and averaged 2D.</strong></p>2763 <p style="text-align: center;" align="center"><b>Figure. Comparison between 1D and averaged 2D.</b></p> 2764 2764 <p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 2765 2765 <p>REFERENCE</p> 2766 2766 <p>Mittelbach and Porod, Acta Physica Austriaca 14 (1961) 185-211.</p> 2767 2767 <p>Equations (1), (13-14). (in German)</p> 2768 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.22.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><strong><span style="font-size: 14pt;"><a name="CSParallelepipedModel"></a>CSParallelepipedModel</span></strong></p>2768 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.22.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;"><a name="CSParallelepipedModel"></a>CSParallelepipedModel</span></b></p> 2769 2769 <p>Calculates the form factor for a rectangular solid with a core-shell structure. The thickness and the scattering length density of the shell or "rim" can be different on all three (pairs) of faces. The form factor is normalized by the particle volume such that P(q) = scale*<f^2>/Vol + bkg, where < > is an average over all possible orientations of the rectangular solid. An instrument resolution smeared version is also provided.</p> 2770 2770 <p>The function calculated is the form factor of the rectangular solid below. The core of the solid is defined by the dimensions ABC such that A < B < C. </p> … … 2785 2785 <p>To provide easy access to the orientation of the CSparallelepiped, we define the axis of the cylinder using two angles θ , <span style="font-family: 'Arial','sans-serif';">φ </span>and<span style="font-family: Symbol;">Y</span>. Similarly to the case of the cylinder, those angles, θ and <span style="font-family: 'Arial','sans-serif';">φ,</span> are defined on Figure 2 of CylinderModel. The angle <span style="font-family: Symbol;">Y </span>is the rotational angle around its own long_c axis against the q plane. For example, <span style="font-family: Symbol;">Y </span>= 0 when the short_b axis is parallel to the x-axis of the detector.</p> 2786 2786 <p style="text-align: center;" align="center"><img id="Picture 102" src="img/image090.jpg" /></p> 2787 <p style="text-align: center;" align="center">< strong>Figure. Definition of angles for 2D</strong>.</p>2787 <p style="text-align: center;" align="center"><b>Figure. Definition of angles for 2D</b>.</p> 2788 2788 <p style="text-align: center;" align="center"><img id="Picture 103" src="img/image091.jpg" alt="" width="379" height="256" /></p> 2789 2789 <p style="text-align: center;" align="center">Figure. Examples of the angles for oriented cspp against the detector plane.</p> … … 2949 2949 </div> 2950 2950 <p style="text-align: center;" align="center"><img id="Picture 33" src="img/image096.jpg" alt="" width="450" height="338" /></p> 2951 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/256 data points).</strong></p>2952 <p style="text-align: center;" align="center">< strong> </strong></p>2951 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/256 data points).</b></p> 2952 <p style="text-align: center;" align="center"><b> </b></p> 2953 2953 <p style="text-align: center;" align="center"><img id="Picture 34" src="img/image097.jpg" alt="" width="451" height="339" /></p> 2954 <p style="text-align: center;" align="center">< strong>Figure. 2D plot using the default values (w/(256X265) data points).</strong></p>2954 <p style="text-align: center;" align="center"><b>Figure. 2D plot using the default values (w/(256X265) data points).</b></p> 2955 2955 2956 2956 <p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> … … 2958 2958 <p>see: Mittelbach and Porod, Acta Physica Austriaca 14 (1961) 185-211.</p> 2959 2959 <p>Equations (1), (13-14). (yes, it's in German) </p> 2960 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.23.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="EllipticalCylinderModel"></a><strong><span style="font-size: 14pt;">Elliptical Cylinder Model</span></strong></p>2960 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.23.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="EllipticalCylinderModel"></a><b><span style="font-size: 14pt;">Elliptical Cylinder Model</span></b></p> 2961 2961 <p>This function calculates the scattering from an oriented elliptical cylinder.</p> 2962 <p>< strong>For 2D (orientated system):</strong></p>2962 <p><b>For 2D (orientated system):</b></p> 2963 2963 <p>The angles theta and phi define the orientation of the axis of the cylinder. The angle psi is defined as the orientation of the major axis of the ellipse with respect to the vector Q. A gaussian poydispersity can be added to any of the orientation angles, and also for the minor radius and the ratio of the ellipse radii.</p> 2964 2964 <p style="text-align: center;" align="center"><img id="Picture 40" src="img/image098.gif" alt="" width="352" height="172" /></p> 2965 <p style="text-align: center;" align="center">< strong>Figure. a= r_minor and </strong><strong><span style="font-family: Symbol;">n</span>= r_ratio (i.e., r_major/r_minor).</strong></p>2965 <p style="text-align: center;" align="center"><b>Figure. a= r_minor and </b><b><span style="font-family: Symbol;">n</span>= r_ratio (i.e., r_major/r_minor).</b></p> 2966 2966 <p>The function calculated is:</p> 2967 2967 <p style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="img/image099.PNG" alt="" /></span> </p> … … 2971 2971 <p> </p> 2972 2972 <p>and the angle psi is defined as the orientation of the major axis of the ellipse with respect to the vector Q.</p> 2973 <p>< strong>For 1D (no preferred orientation):</strong></p>2973 <p><b>For 1D (no preferred orientation):</b></p> 2974 2974 <p>The form factor is averaged over all possible orientation before normalized by the particle volume: P(q) = scale*<f^2>/V .</p> 2975 2975 <p>The returned value is scaled to units of [cm-1].</p> … … 2977 2977 <p>All angle parameters are valid and given only for 2D calculation (Oriented system).</p> 2978 2978 <p style="text-align: center;" align="center"><img id="Picture 105" src="img/image101.jpg" /></p> 2979 <p style="text-align: center;" align="center">< strong>Figure. Definition of angels for 2D</strong>.</p>2979 <p style="text-align: center;" align="center"><b>Figure. Definition of angels for 2D</b>.</p> 2980 2980 <p style="text-align: center;" align="center"><img id="Picture 114" src="img/image062.jpg" alt="" width="379" height="256" /></p> 2981 2981 <p style="text-align: center;" align="center"><span style="font-size: 12pt;">Figure. Examples of the angles for oriented elliptical cylinders </span></p> 2982 2982 <p style="text-align: center;" align="center"><span style="font-size: 12pt;">against the detector plane.</span></p> 2983 <p>< strong>For P*S</strong>: The 2nd virial coefficient of the solid cylinder is calculate based on the averaged radius (=sqrt(r_minor^2*r_ratio)) and length values, and used as the effective radius toward S(Q) when P(Q)*S(Q) is applied.</p>2983 <p><b>For P*S</b>: The 2nd virial coefficient of the solid cylinder is calculate based on the averaged radius (=sqrt(r_minor^2*r_ratio)) and length values, and used as the effective radius toward S(Q) when P(Q)*S(Q) is applied.</p> 2984 2984 <div align="center"> 2985 2985 <table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0"> … … 3075 3075 </div> 3076 3076 <p style="text-align: center;" align="center"><img id="Picture 503" src="img/image102.jpg" alt="" width="443" height="328" /></p> 3077 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/1000 data point).</strong></p>3078 <p style="margin-left: 1.35in; text-indent: -0.25in;"><span style="font-family: Symbol;">·</span><span style="font-size: 7pt;"> </span>< strong>Validation of the elliptical cylinder 2D model</strong></p>3077 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/1000 data point).</b></p> 3078 <p style="margin-left: 1.35in; text-indent: -0.25in;"><span style="font-family: Symbol;">·</span><span style="font-size: 7pt;"> </span><b>Validation of the elliptical cylinder 2D model</b></p> 3079 3079 <p>Validation of our code was done by comparing the output of the 1D calculation to the angular average of the output of 2 D calculation over all possible angles. The Figure below shows the comparison where the solid dot refers to averaged 2D while the line represents the result of 1D calculation (for 2D averaging, 76, 180, 76 points are taken for the angles of theta, phi, and psi respectively).</p> 3080 3080 <p style="text-align: center;" align="center"><img id="Picture 106" src="img/image103.gif" alt="" width="448" height="278" /></p> 3081 <p style="text-align: center;" align="center">< strong>Figure. Comparison between 1D and averaged 2D.</strong></p>3082 <p>< strong> </strong></p>3081 <p style="text-align: center;" align="center"><b>Figure. Comparison between 1D and averaged 2D.</b></p> 3082 <p><b> </b></p> 3083 3083 <p>In the 2D average, more binning in the angle phi is necessary to get the proper result. The following figure shows the results of the averaging by varying the number of bin over angles.</p> 3084 3084 <p style="text-align: center;" align="center"><img id="Picture 107" src="img/image104.gif" alt="" width="409" height="303" /></p> 3085 <p style="text-align: center;" align="center">< strong>Figure. The intensities averaged from 2D over different number </strong></p>3086 <p style="text-align: center;" align="center">< strong>of points of binning of angles.</strong></p>3085 <p style="text-align: center;" align="center"><b>Figure. The intensities averaged from 2D over different number </b></p> 3086 <p style="text-align: center;" align="center"><b>of points of binning of angles.</b></p> 3087 3087 <p>REFERENCE</p> 3088 3088 <p style="text-indent: 0.25in;">L. A. Feigin and D. I. Svergun “Structure Analysis by Small-Angle X-Ray and Neutron Scattering”, Plenum, New York, (1987).</p> 3089 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.24.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="BarBellModel"></a><strong><span style="font-size: 14pt;">BarBell(/DumBell)Model</span></strong></p>3089 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.24.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="BarBellModel"></a><b><span style="font-size: 14pt;">BarBell(/DumBell)Model</span></b></p> 3090 3090 <p>Calculates the scattering from a barbell-shaped cylinder (This model simply becomes the DumBellModel when the length of the cylinder, L, is set to zero). That is, a sphereocylinder with spherical end caps that have a radius larger than that of the cylinder and the center of the end cap radius lies outside of the cylinder All dimensions of the barbell are considered to be monodisperse. See the diagram for the details of the geometry and restrictions on parameter values. </p> 3091 <p style="margin-left: 0.85in; text-indent: -0.35in;">< strong>1.1.</strong><strong><span style="font-size: 7pt;"> </span>Definition</strong></p>3091 <p style="margin-left: 0.85in; text-indent: -0.35in;"><b>1.1.</b><b><span style="font-size: 7pt;"> </span>Definition</b></p> 3092 3092 <p>The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> 3093 3093 <p>The barbell geometry is defined as:</p> … … 3205 3205 </div> 3206 3206 <p style="text-align: center;" align="center"><img id="Picture 59" src="img/image110.jpg" alt="" width="480" height="356" /></p> 3207 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/256 data point).</strong></p>3207 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/256 data point).</b></p> 3208 3208 <p>For 2D data: The 2D scattering intensity is calculated similar to the 2D cylinder model. At the theta = 45 deg and phi =0 deg with default values for other parameters,</p> 3209 3209 <p style="text-align: center;" align="center"><img id="Picture 66" src="img/image111.jpg" alt="" width="425" height="346" /></p> 3210 <p style="text-align: center;" align="center">< strong>Figure. 2D plot (w/(256X265) data points).</strong></p>3210 <p style="text-align: center;" align="center"><b>Figure. 2D plot (w/(256X265) data points).</b></p> 3211 3211 3212 3212 <p style="text-align: center; page-break-after: avoid;" align="center"><img src="img/image061.jpg" /></p> … … 3218 3218 3219 3219 3220 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.25.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="CappedCylinderModel"></a><strong><span style="font-size: 14pt;">CappedCylinder(/ConvexLens)Model</span></strong></p>3220 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.25.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="CappedCylinderModel"></a><b><span style="font-size: 14pt;">CappedCylinder(/ConvexLens)Model</span></b></p> 3221 3221 <p>Calculates the scattering from a cylinder with spherical section end-caps(This model simply becomes the ConvexLensModel when the length of the cylinder L = 0. That is, a sphereocylinder with end caps that have a radius larger than that of the cylinder and the center of the end cap radius lies within the cylinder. See the diagram for the details of the geometry and restrictions on parameter values.</p> 3222 3222 <p style="margin-left: 0.85in; text-indent: -0.35in;"> </p> 3223 <p style="margin-left: 0.85in; text-indent: -0.35in;">< strong>1.1.</strong><strong><span style="font-size: 7pt;"> </span>Definition</strong></p>3223 <p style="margin-left: 0.85in; text-indent: -0.35in;"><b>1.1.</b><b><span style="font-size: 7pt;"> </span>Definition</b></p> 3224 3224 <p>The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> 3225 3225 <p>The Capped Cylinder geometry is defined as:</p> … … 3336 3336 </div> 3337 3337 <p style="text-align: center;" align="center"><img id="Picture 72" src="img/image117.jpg" alt="" width="533" height="381" /></p> 3338 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/256 data point).</strong></p>3338 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/256 data point).</b></p> 3339 3339 <p>For 2D data: The 2D scattering intensity is calculated similar to the 2D cylinder model. At the theta = 45 deg and phi =0 deg with default values for other parameters,</p> 3340 3340 <p style="text-align: center;" align="center"><img id="Picture 71" src="img/image118.jpg" alt="" width="402" height="334" /></p> 3341 <p style="text-align: center;" align="center">< strong>Figure. 2D plot (w/(256X265) data points).</strong></p>3341 <p style="text-align: center;" align="center"><b>Figure. 2D plot (w/(256X265) data points).</b></p> 3342 3342 <p style="text-align: center; page-break-after: avoid;" align="center"><img src="img/image061.jpg" /></p> 3343 3343 <p style="text-align: center;" align="center"><a name="_Ref173213915"></a><a name="_Ref173306040"></a>Figure. Definition of the angles for oriented 2D cylinders.</p> … … 3347 3347 3348 3348 3349 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.26.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="EllipsoidModel"></a><strong><span style="font-size: 14pt;">Ellipsoid Model</span></strong></p>3349 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.26.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="EllipsoidModel"></a><b><span style="font-size: 14pt;">Ellipsoid Model</span></b></p> 3350 3350 <p>This model provides the form factor for an ellipsoid (ellipsoid of revolution) with uniform scattering length density. The form factor is normalized by the particle volume.</p> 3351 <p style="margin-left: 0.85in; text-indent: -0.35in;">< strong>1.1.</strong><strong><span style="font-size: 7pt;"> </span>Definition</strong></p>3351 <p style="margin-left: 0.85in; text-indent: -0.35in;"><b>1.1.</b><b><span style="font-size: 7pt;"> </span>Definition</b></p> 3352 3352 <p>The output of the 2D scattering intensity function for oriented ellipsoids is given by (Feigin, 1987):</p> 3353 3353 <p style="text-align: center;" align="center"><span style="position: relative; top: 12pt;"><img src="img/image059.PNG" alt="" /></span> </p> … … 3469 3469 <p style="text-align: center;" align="center"><span style="font-size: 12pt;">Figure. The angles for oriented ellipsoid </span></p> 3470 3470 3471 <p style="margin-left: 0.85in; text-indent: -0.35in;">< strong>2.1.</strong><strong><span style="font-size: 7pt;"> </span>Validation of the ellipsoid model</strong></p>3471 <p style="margin-left: 0.85in; text-indent: -0.35in;"><b>2.1.</b><b><span style="font-size: 7pt;"> </span>Validation of the ellipsoid model</b></p> 3472 3472 <p>Validation of our code was done by comparing the output of the 1D model to the output of the software provided by the NIST (Kline, 2006). Figure 5 shows a comparison of the 1D output of our model and the output of the NIST software.</p> 3473 3473 <p>Averaging over a distribution of orientation is done by evaluating the equation above. Since we have no other software to compare the implementation of the intensity for fully oriented ellipsoids, we can compare the result of averaging our 2D output using a uniform distribution <em>p(θ,</em><em><span style="font-family: 'Arial','sans-serif';">φ</span>)</em> = 1.0. Figure 6 shows the result of such a cross-check.</p> 3474 <p style="text-align: center;" align="center">< strong><em><span style="color: #3366ff;"> </span></em></strong></p>3474 <p style="text-align: center;" align="center"><b><em><span style="color: #3366ff;"> </span></em></b></p> 3475 3475 <p>The discrepancy above q=0.3 Å -1 is due to the way the form factors are calculated in the c-library provided by NIST. A numerical integration has to be performed to obtain P(q) for randomly oriented particles. The NIST software performs that integration with a 76-point Gaussian quadrature rule, which will become imprecise at high q where the amplitude varies quickly as a function of q. The DANSE result shown has been obtained by summing over 501 equidistant points in <span style="font-family: 'Arial','sans-serif';">α</span>. Our result was found to be stable over the range of q shown for a number of points higher than 500.</p> 3476 <p style="text-align: center; page-break-after: avoid;" align="center">< strong><img id="Picture 16" src="img/image123.jpg" alt="ellipsoid_1D_validation" width="573" height="315" /></strong></p>3476 <p style="text-align: center; page-break-after: avoid;" align="center"><b><img id="Picture 16" src="img/image123.jpg" alt="ellipsoid_1D_validation" width="573" height="315" /></b></p> 3477 3477 <p><a name="_Ref173222904"></a>Figure 5: Comparison of the DANSE scattering intensity for an ellipsoid with the output of the NIST SANS analysis software. The parameters were set to: Scale=1.0, Radius_a=20 Å, Radius_b=400 Å,</p> 3478 3478 <p>Contrast=3e-6 Å -2, and Background=0.01 cm -1.</p> … … 3481 3481 <p><a name="_Ref173223004"></a>Figure 6: Comparison of the intensity for uniformly distributed ellipsoids calculated from our 2D model and the intensity from the NIST SANS analysis software. The parameters used were: Scale=1.0, Radius_a=20 Å, Radius_b=400 Å, Contrast=3e-6 Å -2, and Background=0.0 cm -1.</p> 3482 3482 <p> </p> 3483 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.27.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="CoreShellEllipsoidModel"></a><strong><span style="font-size: 14pt;">CoreShellEllipsoidModel </span></strong></p>3483 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.27.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="CoreShellEllipsoidModel"></a><b><span style="font-size: 14pt;">CoreShellEllipsoidModel </span></b></p> 3484 3484 <p>This model provides the form factor, P(<em>q</em>), for a core shell ellipsoid (below) where the form factor is normalized by the volume of the cylinder. P(q) = scale*<f^2>/V+background where the volume V= 4pi/3*rmaj*rmin2 and the averaging < > is applied over all orientation for 1D. </p> 3485 3485 <p style="text-align: center;" align="center"> <img id="Picture 41" src="img/image125.gif" alt="" width="335" height="179" /></p> … … 3608 3608 </div> 3609 3609 <p style="text-align: center;" align="center"><img id="Picture 526" src="img/image127.jpg" alt="" width="426" height="333" /></p> 3610 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/1000 data point).</strong></p>3611 <p style="text-align: center;" align="center">< strong> </strong></p>3610 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/1000 data point).</b></p> 3611 <p style="text-align: center;" align="center"><b> </b></p> 3612 3612 <p style="text-align: center;" align="center"><img src="img/image122.jpg" alt="" width="379" height="256"/></p> 3613 3613 <p style="text-align: center;" align="center">Figure. The angles for oriented coreshellellipsoid .</p> … … 3616 3616 <p>Kotlarchyk, M.; Chen, S.-H. J. Chem. Phys., 1983, 79, 2461.</p> 3617 3617 <p>Berr, S. J. Phys. Chem., 1987, 91, 4760.</p> 3618 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.28.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="TriaxialEllipsoidModel"></a><strong><span style="font-size: 14pt;">TriaxialEllipsoidModel</span></strong></p>3618 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.28.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="TriaxialEllipsoidModel"></a><b><span style="font-size: 14pt;">TriaxialEllipsoidModel</span></b></p> 3619 3619 <p>This model provides the form factor, P(<em>q</em>), for an ellipsoid (below) where all three axes are of different lengths, i.e., Ra =< Rb =< Rc (Note that users should maintains this inequality for the all calculations). P(q) = scale*<f^2>/V+background where the volume V= 4pi/3*Ra*Rb*Rc, and the averaging < > is applied over all orientation for 1D. </p> 3620 3620 <p style="text-align: center;" align="center"> <img id="Picture 42" src="img/image128.jpg" alt="" width="376" height="226" /></p> … … 3721 3721 </div> 3722 3722 <p style="text-align: center;" align="center"><img id="Picture 545" src="img/image130.jpg" alt="" width="439" height="341" /></p> 3723 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/1000 data point).</strong></p>3724 <p style="margin-left: 1.35in; text-indent: -0.25in;"><span style="font-family: Symbol;">·</span><span style="font-size: 7pt;"> </span>< strong>Validation of the triaxialellipsoid 2D model</strong></p>3723 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/1000 data point).</b></p> 3724 <p style="margin-left: 1.35in; text-indent: -0.25in;"><span style="font-family: Symbol;">·</span><span style="font-size: 7pt;"> </span><b>Validation of the triaxialellipsoid 2D model</b></p> 3725 3725 <p>Validation of our code was done by comparing the output of the 1D calculation to the angular average of the output of 2 D calculation over all possible angles. The Figure below shows the comparison where the solid dot refers to averaged 2D while the line represents the result of 1D calculation (for 2D averaging, 76, 180, 76 points are taken for the angles of theta, phi, and psi respectively).</p> 3726 3726 <p style="text-align: center;" align="center"><img src="img/image131.gif" alt="" width="438" height="272" /></p> 3727 <p style="text-align: center;" align="center">< strong>Figure. Comparison between 1D and averaged 2D.</strong></p>3727 <p style="text-align: center;" align="center"><b>Figure. Comparison between 1D and averaged 2D.</b></p> 3728 3728 <p style="text-align: center;" align="center"><img src="img/image132.jpg" alt="" width="379" height="256" /></p> 3729 3729 <p style="text-align: center;" align="center">Figure. The angles for oriented ellipsoid.</p> … … 3731 3731 <p>REFERENCE</p> 3732 3732 <p>L. A. Feigin and D. I. Svergun “Structure Analysis by Small-Angle X-Ray and Neutron Scattering”, Plenum, New York, 1987.</p> 3733 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.29.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="LamellarModel"></a><strong><span style="font-size: 14pt;">LamellarModel</span></strong></p>3733 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.29.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="LamellarModel"></a><b><span style="font-size: 14pt;">LamellarModel</span></b></p> 3734 3734 <p>This model provides the scattering intensity, I(<em>q</em>), for a lyotropic lamellar phase where a uniform SLD and random distribution in solution are assumed. The ploydispersion in the bilayer thickness can be applied from the GUI.</p> 3735 3735 <p>The scattering intensity I(q) is:</p> … … 3813 3813 </div> 3814 3814 <p style="text-align: center;" align="center"><img id="Picture 571" src="img/image135.jpg" alt="" width="476" height="351" /></p> 3815 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/1000 data point).</strong></p>3815 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/1000 data point).</b></p> 3816 3816 <p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 3817 3817 <p>REFERENCE</p> 3818 3818 <p>Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993) 487-502.</p> 3819 3819 <p> also in J. Phys. Chem. B, 105, (2001) 11081-11088.</p> 3820 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.30.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="LamellarFFHGModel"></a><strong><span style="font-size: 14pt;">LamellarFFHGModel</span></strong></p>3820 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.30.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="LamellarFFHGModel"></a><b><span style="font-size: 14pt;">LamellarFFHGModel</span></b></p> 3821 3821 <p>This model provides the scattering intensity, I(<em>q</em>), for a lyotropic lamellar phase where a random distribution in solution are assumed. The SLD of the head region is taken to be different from the SLD of the tail region.</p> 3822 3822 <p>The scattering intensity I(q) is:</p> … … 3922 3922 </div> 3923 3923 <p style="text-align: center;" align="center"><img id="Picture 61" src="img/image138.jpg" alt="" width="505" height="344" /></p> 3924 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/1000 data point).</strong></p>3924 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/1000 data point).</b></p> 3925 3925 <p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 3926 3926 <p>REFERENCE</p> 3927 3927 <p>Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993) 487-502.</p> 3928 3928 <p> also in J. Phys. Chem. B, 105, (2001) 11081-11088.</p> 3929 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.31.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="LamellarPSModel"></a><strong><span style="font-size: 14pt;">LamellarPSModel</span></strong></p>3930 <p>This model provides the scattering intensity (< strong>form factor</strong> <strong>*</strong> <strong>structure factor</strong>), I(<em>q</em>), for a lyotropic lamellar phase where a random distribution in solution are assumed.</p>3929 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.31.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="LamellarPSModel"></a><b><span style="font-size: 14pt;">LamellarPSModel</span></b></p> 3930 <p>This model provides the scattering intensity (<b>form factor</b> <b>*</b> <b>structure factor</b>), I(<em>q</em>), for a lyotropic lamellar phase where a random distribution in solution are assumed.</p> 3931 3931 <p>The scattering intensity I(q) is:</p> 3932 3932 <p style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="img/image139.PNG" alt="" /></span></p> … … 4035 4035 </div> 4036 4036 <p style="text-align: center;" align="center"><img id="Picture 659" src="img/image142.jpg" alt="" width="439" height="348" /></p> 4037 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/6000 data point).</strong></p>4037 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/6000 data point).</b></p> 4038 4038 <p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 4039 4039 <p>REFERENCE</p> 4040 4040 <p>Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993) 487-502.</p> 4041 4041 <p> also in J. Phys. Chem. B, 105, (2001) 11081-11088.</p> 4042 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.32.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="LamellarPSHGModel"></a><strong><span style="font-size: 14pt;">LamellarPSHGModel</span></strong></p>4043 <p>This model provides the scattering intensity (< strong>form factor</strong> <strong>*</strong> <strong>structure factor</strong>), I(<em>q</em>), for a lyotropic lamellar phase where a random distribution in solution are assumed. The SLD of the head region is taken to be different from the SLD of the tail region.</p>4042 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.32.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="LamellarPSHGModel"></a><b><span style="font-size: 14pt;">LamellarPSHGModel</span></b></p> 4043 <p>This model provides the scattering intensity (<b>form factor</b> <b>*</b> <b>structure factor</b>), I(<em>q</em>), for a lyotropic lamellar phase where a random distribution in solution are assumed. The SLD of the head region is taken to be different from the SLD of the tail region.</p> 4044 4044 <p>The scattering intensity I(q) is:</p> 4045 4045 <p style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="img/image139.PNG" alt="" /></span></p> … … 4182 4182 </div> 4183 4183 <p style="text-align: center;" align="center"><img id="Picture 687" src="img/image144.jpg" alt="" width="463" height="360" /></p> 4184 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/6000 data point).</strong></p>4184 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/6000 data point).</b></p> 4185 4185 <p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 4186 4186 <p>REFERENCE</p> 4187 4187 <p>Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993) 487-502.</p> 4188 4188 <p> also in J. Phys. Chem. B, 105, (2001) 11081-11088.</p> 4189 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.33.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="LamellarPCrystalModel"></a><strong><span style="font-size: 14pt;">LamellarPCrystalModel</span></strong></p>4189 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.33.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="LamellarPCrystalModel"></a><b><span style="font-size: 14pt;">LamellarPCrystalModel</span></b></p> 4190 4190 <p>Lamella ParaCrystal Model: Calculates the scattering from a stack of repeating lamellar structures. The stacks of lamellae (infinite in lateral dimension) are treated as a paracrystal to account for the repeating spacing. The repeat distance is further characterized by a Gaussian polydispersity. This model can be used for large multilamellar vesicles.</p> 4191 4191 <p>The scattering intensity I(q) is calculated as:</p> … … 4298 4298 </div> 4299 4299 <p style="text-align: center;" align="center"><img src="img/image148.jpg" alt="" width="501" height="332" /></p> 4300 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values above (w/20000 data point).</strong></p>4300 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values above (w/20000 data point).</b></p> 4301 4301 <p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p> 4302 4302 <p>See the reference for details.</p> 4303 4303 <p>REFERENCE</p> 4304 4304 <p>M. Bergstrom, J. S. Pedersen, P. Schurtenberger, S. U. Egelhaaf, J. Phys. Chem. B, 103 (1999) 9888-9897.</p> 4305 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.34.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="SCCrystalModel"></a><strong><span style="font-size: 14pt;">SC(Simple Cubic Para-)CrystalModel</span></strong></p>4305 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.34.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="SCCrystalModel"></a><b><span style="font-size: 14pt;">SC(Simple Cubic Para-)CrystalModel</span></b></p> 4306 4306 <p>Calculates the scattering from a simple cubic lattice with paracrystalline distortion. Thermal vibrations are considered to be negligible, and the size of the paracrystal is infinitely large. Paracrystalline distortion is assumed to be isotropic and characterized by a Gaussian distribution.</p> 4307 4307 <p>The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> … … 4418 4418 <p>This example dataset is produced using 200 data points, qmin = 0.01 Å-1, qmax = 0.1 Å-1 and the above default values.</p> 4419 4419 <p style="text-align: center;" align="center"><img id="Picture 73" src="img/image155.jpg" alt="" width="515" height="403" /></p> 4420 <p style="text-align: center;" align="center">< strong>Figure. 1D plot in the linear scale using the default values (w/200 data point).</strong></p>4420 <p style="text-align: center;" align="center"><b>Figure. 1D plot in the linear scale using the default values (w/200 data point).</b></p> 4421 4421 <p> The 2D (Anisotropic model) is based on the reference (above) which I(q) is approximated for 1d scattering. Thus the scattering pattern for 2D may not be accurate. Note that we are not responsible for any incorrectness of the 2D model computation.</p> 4422 4422 <p style="text-align: center;" align="center"><img id="Object 23" src="img/image156.jpg" /></p> … … 4425 4425 <p> </p> 4426 4426 <p> </p> 4427 <p style="text-align: center;" align="center">< strong><img src="img/image157.jpg" alt="" width="447" height="322" /></strong></p>4428 <p style="text-align: center;" align="center">< strong>Figure. 2D plot using the default values (w/200X200 pixels).</strong></p>4429 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.35.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="FCCrystalModel"></a><strong><span style="font-size: 14pt;">FC(Face Centered Cubic Para-)CrystalModel</span></strong></p>4427 <p style="text-align: center;" align="center"><b><img src="img/image157.jpg" alt="" width="447" height="322" /></b></p> 4428 <p style="text-align: center;" align="center"><b>Figure. 2D plot using the default values (w/200X200 pixels).</b></p> 4429 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.35.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="FCCrystalModel"></a><b><span style="font-size: 14pt;">FC(Face Centered Cubic Para-)CrystalModel</span></b></p> 4430 4430 <p>Calculates the scattering from a face-centered cubic lattice with paracrystalline distortion. Thermal vibrations are considered to be negligible, and the size of the paracrystal is infinitely large. Paracrystalline distortion is assumed to be isotropic and characterized by a Gaussian distribution. </p> 4431 4431 <p>The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> … … 4543 4543 <p>This example dataset is produced using 200 data points, qmin = 0.01 Å-1, qmax = 0.1 Å-1 and the above default values.</p> 4544 4544 <p style="text-align: center;" align="center"><img src="img/image164.jpg" alt="" width="539" height="394" /></p> 4545 <p style="text-align: center;" align="center">< strong>Figure. 1D plot in the linear scale using the default values (w/200 data point).</strong></p>4545 <p style="text-align: center;" align="center"><b>Figure. 1D plot in the linear scale using the default values (w/200 data point).</b></p> 4546 4546 <p> The 2D (Anisotropic model) is based on the reference (above) in which I(q) is approximated for 1d scattering. Thus the scattering pattern for 2D may not be accurate. Note that we are not responsible for any incorrectness of the 2D model computation.</p> 4547 4547 <p style="text-align: center;" align="center"><img src="img/image165.gif" /></p> … … 4551 4551 <p> </p> 4552 4552 <p style="text-align: center;" align="center"><img src="img/image166.jpg" alt="" width="473" height="352" /></p> 4553 <p style="text-align: center;" align="center">< strong>Figure. 2D plot using the default values (w/200X200 pixels).</strong></p>4554 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">2.36.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="BCCrystalModel"></a><strong><span style="font-size: 14pt;">BC(Body Centered Cubic Para-)CrystalModel</span></strong></p>4553 <p style="text-align: center;" align="center"><b>Figure. 2D plot using the default values (w/200X200 pixels).</b></p> 4554 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.36.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="BCCrystalModel"></a><b><span style="font-size: 14pt;">BC(Body Centered Cubic Para-)CrystalModel</span></b></p> 4555 4555 <p>Calculates the scattering from a body-centered cubic lattice with paracrystalline distortion. Thermal vibrations are considered to be negligible, and the size of the paracrystal is infinitely large. Paracrystalline distortion is assumed to be isotropic and characterized by a Gaussian distribution.The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> 4556 4556 <p>The scattering intensity I(q) is calculated as:</p> … … 4667 4667 <p>This example dataset is produced using 200 data points, qmin = 0.001 Å-1, qmax = 0.1 Å-1 and the above default values.</p> 4668 4668 <p style="text-align: center;" align="center"><img src="img/image170.jpg" alt="" width="474" height="339" /></p> 4669 <p style="text-align: center;" align="center">< strong>Figure. 1D plot in the linear scale using the default values (w/200 data point).</strong></p>4669 <p style="text-align: center;" align="center"><b>Figure. 1D plot in the linear scale using the default values (w/200 data point).</b></p> 4670 4670 <p> The 2D (Anisotropic model) is based on the reference (1987) in which I(q) is approximated for 1d scattering. Thus the scattering pattern for 2D may not be accurate. Note that we are not responsible for any incorrectness of the 2D model computation.</p> 4671 4671 <p style="text-align: center;" align="center"><img id="Object 31" src="img/image165.gif" /></p> … … 4675 4675 <p> </p> 4676 4676 <p style="text-align: center;" align="center"><img src="img/image171.jpg" alt="" width="477" height="344" /></p> 4677 <p style="text-align: center;" align="center">< strong>Figure. 2D plot using the default values (w/200X200 pixels).</strong></p>4678 <p style="margin-left: 0.25in; text-indent: -0.25in;">< strong><span style="font-size: 16pt;">3.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="Shape-Independent"></a><strong><span style="font-size: 16pt;">Shape-Independent Models </span></strong></p>4677 <p style="text-align: center;" align="center"><b>Figure. 2D plot using the default values (w/200X200 pixels).</b></p> 4678 <p style="margin-left: 0.25in; text-indent: -0.25in;"><b><span style="font-size: 16pt;">3.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="Shape-Independent"></a><b><span style="font-size: 16pt;">Shape-Independent Models </span></b></p> 4679 4679 <p>The following are models used for shape-independent SANS analysis.</p> 4680 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.1.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><strong><span style="font-size: 14pt;"> <a name="Debye"></a>Debye (Model)</span></strong></p>4680 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.1.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;"> <a name="Debye"></a>Debye (Model)</span></b></p> 4681 4681 <p style="margin-left: 0.25in;">The Debye model is a form factor for a linear polymer chain. In addition to the radius of gyration, Rg, a scale factor "scale", and a constant background term are included in the calculation.</p> 4682 4682 <p style="text-align: center;" align="center"><span style="font-size: 14pt; position: relative; top: 27pt;"><img src="img/image172.PNG" alt="" /></span></p> … … 4736 4736 </div> 4737 4737 <p style="margin-left: 0.25in; text-align: center;" align="center"><img id="Picture 74" src="img/image173.jpg" alt="" width="423" height="273" /></p> 4738 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/200 data point).</strong></p>4738 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/200 data point).</b></p> 4739 4739 <p style="margin-left: 0.25in; text-align: center;" align="center"> </p> 4740 4740 <p style="margin-left: 0.25in;">Reference: Roe, R.-J., "Methods of X-Ray and Neutron Scattering in Polymer Science", Oxford University Press, New York (2000).</p> 4741 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.2.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><strong><span style="font-size: 14pt;"> <a name="BroadPeakModel"></a>BroadPeak Model</span></strong></p>4741 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.2.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;"> <a name="BroadPeakModel"></a>BroadPeak Model</span></b></p> 4742 4742 <p style="margin-left: 0.25in;">Calculate an empirical functional form for SANS data characterized by a broad scattering peak. Many SANS spectra are characterized by a broad peak even though they are from amorphous soft materials. The d-spacing corresponding to the broad peak is a characteristic distance between the scattering inhomogeneities (such as in lamellar, cylindrical, or spherical morphologies or for bicontinuous structures).</p> 4743 4743 <p style="margin-left: 0.25in;">The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> … … 4836 4836 </div> 4837 4837 <p style="margin-left: 0.25in; text-align: center;" align="center"><img src="img/image175.jpg" alt="" width="488" height="334" /></p> 4838 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/200 data point).</strong></p>4838 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/200 data point).</b></p> 4839 4839 <p style="margin-left: 0.25in; text-align: center;" align="center"> </p> 4840 4840 <p style="margin-left: 0.25in;">Reference: None.</p> 4841 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.3.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><strong><span style="font-size: 14pt;"> <a name="CorrLength"></a>CorrLength (CorrelationLengthModel)</span></strong></p>4841 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.3.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;"> <a name="CorrLength"></a>CorrLength (CorrelationLengthModel)</span></b></p> 4842 4842 <p style="margin-left: 0.25in;">Calculate an empirical functional form for SANS data characterized by a low-Q signal and a high-Q signal</p> 4843 4843 <p>The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> … … 4924 4924 </div> 4925 4925 <p style="margin-left: 0.25in; text-align: center;" align="center"><img id="Picture 80" src="img/image177.jpg" alt="" width="489" height="338" /></p> 4926 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/500 data points).</strong></p>4926 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/500 data points).</b></p> 4927 4927 <p style="margin-left: 0.25in; text-align: center;" align="center"> </p> 4928 4928 <p style="margin-left: 0.25in;">REFERENCE</p> 4929 4929 <p style="margin-left: 0.25in;">B. Hammouda, D.L. Ho and S.R. Kline, “Insight into Clustering in Poly(ethylene oxide) Solutions”, Macromolecules 37, 6932-6937 (2004).</p> 4930 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.4.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><strong><span style="font-size: 14pt;"> <a name="Lorentz"></a>(Ornstein-Zernicke) Lorentz (Model)</span></strong></p>4930 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.4.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;"> <a name="Lorentz"></a>(Ornstein-Zernicke) Lorentz (Model)</span></b></p> 4931 4931 <p style="text-indent: 0.25in;">The Ornstein-Zernicke model is defined by:</p> 4932 4932 <p><span style="font-size: 14pt;"> </span></p> … … 4988 4988 </table> 4989 4989 </div> 4990 <p style="text-align: center;" align="center">< strong><span style="font-size: 14pt;"><img id="Picture 75" src="img/image179.jpg" alt="" /></span></strong></p>4991 <p style="text-align: center;" align="center">< strong><span style="font-size: 14pt;"> </span>Figure. 1D plot using the default values (w/200 data point).</strong></p>4992 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.5.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><strong><span style="font-size: 14pt;"> <a name="DABModel"></a>DAB (Debye-Anderson-Brumberger)_Model</span></strong></p>4993 <p style="margin-left: 0.55in;">< strong><span style="font-size: 14pt;"> </span></strong></p>4990 <p style="text-align: center;" align="center"><b><span style="font-size: 14pt;"><img id="Picture 75" src="img/image179.jpg" alt="" /></span></b></p> 4991 <p style="text-align: center;" align="center"><b><span style="font-size: 14pt;"> </span>Figure. 1D plot using the default values (w/200 data point).</b></p> 4992 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.5.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;"> <a name="DABModel"></a>DAB (Debye-Anderson-Brumberger)_Model</span></b></p> 4993 <p style="margin-left: 0.55in;"><b><span style="font-size: 14pt;"> </span></b></p> 4994 4994 <p style="margin-left: 0.25in;">Calculates the scattering from a randomly distributed, two-phase system based on the Debye-Anderson-Brumberger (DAB) model for such systems. The two-phase system is characterized by a single length scale, the correlation length, which is a measure of the average spacing between regions of phase 1 and phase 2. The model also assumes smooth interfaces between the phases and hence exhibits Porod behavior (I ~ Q-4) at large Q (Q*correlation length >> 1).</p> 4995 4995 <p style="text-indent: 0.25in;"> </p> … … 5049 5049 </table> 5050 5050 </div> 5051 <p style="text-align: center;" align="center">< strong><span style="font-size: 14pt;"><img id="Picture 76" src="img/image181.jpg" alt="" /></span></strong></p>5052 <p style="text-align: center;" align="center">< strong><span style="font-size: 14pt;"> </span>Figure. 1D plot using the default values (w/200 data point).</strong></p>5051 <p style="text-align: center;" align="center"><b><span style="font-size: 14pt;"><img id="Picture 76" src="img/image181.jpg" alt="" /></span></b></p> 5052 <p style="text-align: center;" align="center"><b><span style="font-size: 14pt;"> </span>Figure. 1D plot using the default values (w/200 data point).</b></p> 5053 5053 <p style="margin-left: 0.25in;">References:</p> 5054 5054 <p style="margin-left: 0.5in;">Debye, Anderson, Brumberger, "Scattering by an Inhomogeneous Solid. II. The Correlation Function and its Application", J. Appl. Phys. 28 (6), 679 (1957).</p> 5055 5055 <p style="margin-left: 0.5in;"> </p> 5056 5056 <p style="margin-left: 0.5in;">Debye, Bueche, "Scattering by an Inhomogeneous Solid", J. Appl. Phys. 20, 518 (1949).</p> 5057 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.6.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><strong><span style="font-size: 14pt;"> <a name="Absolute Power_Law"></a>Absolute Power_Law </span></strong></p>5057 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.6.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;"> <a name="Absolute Power_Law"></a>Absolute Power_Law </span></b></p> 5058 5058 <p style="margin-left: 0.25in;">This model describes a power law with background.</p> 5059 5059 <p style="text-align: center;" align="center"><span style="font-size: 14pt; position: relative; top: 5pt;"><img src="img/image182.PNG" alt="" /></span></p> … … 5112 5112 </div> 5113 5113 <p style="text-align: center;" align="center"><img id="Picture 77" src="img/image183.jpg" alt="" /></p> 5114 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/200 data point).</strong></p>5115 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.7.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><strong><span style="font-size: 14pt;"> <a name="TeubnerStreyModel"></a>Teubner Strey (Model)</span></strong></p>5114 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/200 data point).</b></p> 5115 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.7.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;"> <a name="TeubnerStreyModel"></a>Teubner Strey (Model)</span></b></p> 5116 5116 <p style="margin-left: 0.25in;">This function calculates the scattered intensity of a two-component system using the Teubner-Strey model.</p> 5117 <p style="margin-left: 0.25in;">< strong><span style="font-size: 14pt;"> </span></strong></p>5117 <p style="margin-left: 0.25in;"><b><span style="font-size: 14pt;"> </span></b></p> 5118 5118 <p style="text-align: center;" align="center"><span style="font-size: 14pt; position: relative; top: 15pt;"><img src="img/image184.PNG" alt="" /></span></p> 5119 5119 <p style="text-align: center;" align="center"><span style="font-size: 14pt;"> </span></p> … … 5183 5183 </div> 5184 5184 <p style="text-align: center;" align="center"><img id="Picture 78" src="img/image185.jpg" alt="" /></p> 5185 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/200 data point).</strong></p>5185 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/200 data point).</b></p> 5186 5186 <p style="margin-left: 0.25in;">References:</p> 5187 5187 <p style="margin-left: 0.5in;">Teubner, M; Strey, R. J. Chem. Phys., 87, 3195 (1987).</p> 5188 5188 <p style="margin-left: 0.5in;"> </p> 5189 5189 <p style="margin-left: 0.5in;">Schubert, K-V., Strey, R., Kline, S. R. and E. W. Kaler, J. Chem. Phys., 101, 5343 (1994).</p> 5190 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.8.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><strong><span style="font-size: 14pt;"> <a name="Number_Density_Fractal"></a> <a name="FractalModel"></a>FractalModel</span></strong></p>5190 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.8.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;"> <a name="Number_Density_Fractal"></a> <a name="FractalModel"></a>FractalModel</span></b></p> 5191 5191 <p style="margin-left: 0.25in;">Calculates the scattering from fractal-like aggregates built from spherical building blocks following the Texiera reference. The value returned is in cm-1.</p> 5192 5192 <p> </p> … … 5194 5194 <p> </p> 5195 5195 <p style="margin-left: 0.25in;">The scale parameter is the volume fraction of the building blocks, R0 is the radius of the building block, Df is the fractal dimension, ξ is the correlation length, <em>ρsolvent</em> is the scattering length density of the solvent, and <em>ρblock</em> is the scattering length density of the building blocks.</p> 5196 <p style="margin-left: 0.25in;">< strong>The polydispersion in radius is provided.</strong></p>5196 <p style="margin-left: 0.25in;"><b>The polydispersion in radius is provided.</b></p> 5197 5197 <p style="margin-left: 0.25in;">For 2D plot, the wave transfer is defined as<span style="font-size: 12pt; font-family: 'Times New Roman','serif'; position: relative; top: 4.5pt;"><img src="img/image040.gif" alt="" /></span><span style="font-size: 14pt;">.</span></p> 5198 5198 <p style="text-align: center;" align="center"><span style="font-size: 14pt;"> </span></p> … … 5292 5292 </div> 5293 5293 <p style="margin-left: 0.25in; text-align: center;" align="center"><img id="Picture 79" src="img/image187.jpg" alt="" width="445" height="280" /></p> 5294 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/200 data point).</strong></p>5294 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/200 data point).</b></p> 5295 5295 <p style="margin-left: 0.25in; text-align: center;" align="center"> </p> 5296 5296 <p style="margin-left: 0.25in; text-align: center;" align="center"> </p> 5297 5297 <p style="margin-left: 0.25in;">References:</p> 5298 5298 <p style="margin-left: 0.25in; text-indent: 0.25in;">J. Teixeira, (1988) J. Appl. Cryst., vol. 21, p781-785</p> 5299 <p>< strong><span style="font-size: 14pt;"> </span></strong></p>5300 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.9.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><strong><span style="font-size: 14pt;"><a name="Mass_Fractal"></a><a name="MassFractalModel"></a>MassFractalModel</span></strong></p>5299 <p><b><span style="font-size: 14pt;"> </span></b></p> 5300 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.9.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;"><a name="Mass_Fractal"></a><a name="MassFractalModel"></a>MassFractalModel</span></b></p> 5301 5301 <p style="margin-left: 0.25in;">Calculates the scattering from fractal-like aggregates based on the Mildner reference (below). </p> 5302 5302 <p style="text-align: center;" align="center"><span style="font-size: 14pt; position: relative; top: 85pt;"><br /> </span></p> … … 5380 5380 </div> 5381 5381 <p style="margin-left: 0.25in; text-align: center;" align="center"><img id="Picture 79" style="width: 495px; height: 368px;" src="img/mass_fractal_fig1.jpg" alt="" /></p> 5382 <p style="text-align: center;" align="center">< strong>Figure. 1D plot</strong></p>5382 <p style="text-align: center;" align="center"><b>Figure. 1D plot</b></p> 5383 5383 <p style="margin-left: 0.25in; text-align: center;" align="center"> </p> 5384 5384 <p style="margin-left: 0.25in; text-align: center;" align="center"> </p> … … 5387 5387 <p> </p> 5388 5388 <p> </p> 5389 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.10.</span></strong><strong><span style="font-size: 7pt;"> </span></strong> <strong><span style="font-size: 14pt;"> <a name="Surface_Fractal"></a><a name="SurfaceFractalModel"></a>SurfaceFractalModel</span></strong></p>5389 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.10.</span></b><b><span style="font-size: 7pt;"> </span></b> <b><span style="font-size: 14pt;"> <a name="Surface_Fractal"></a><a name="SurfaceFractalModel"></a>SurfaceFractalModel</span></b></p> 5390 5390 <p style="margin-left: 0.25in;">Calculates the scattering based on the Mildner reference (below). </p> 5391 5391 <p style="text-align: center;" align="center"><span style="font-size: 14pt; position: relative; top: 85pt;"><br /> </span></p> … … 5469 5469 </div> 5470 5470 <p style="margin-left: 0.25in; text-align: center;" align="center"><img id="Picture 79" style="width: 507px; height: 375px;" src="img/surface_fractal_fig1.jpg" alt="" /></p> 5471 <p style="text-align: center;" align="center">< strong>Figure. 1D plot</strong></p>5471 <p style="text-align: center;" align="center"><b>Figure. 1D plot</b></p> 5472 5472 <p style="margin-left: 0.25in; text-align: center;" align="center"> </p> 5473 5473 <p style="margin-left: 0.25in; text-align: center;" align="center"> </p> … … 5476 5476 <p> </p> 5477 5477 <p> </p> 5478 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.11.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><strong><span style="font-size: 14pt;"><a name="MassSurface_Fractal"></a><a name="MassSurfaceFractal"></a>MassSurfaceFractal</span></strong></p>5478 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.11.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;"><a name="MassSurface_Fractal"></a><a name="MassSurfaceFractal"></a>MassSurfaceFractal</span></b></p> 5479 5479 <p> A number of natural and commercial processes form high-surface area materials as a result of the vapour-phase aggregation of primary particles. Examples of such materials include soots, aerosols, and ‘fume’ or pyrogenic silicas. These are all characterised by cluster mass distributions (sometimes also cluster size distributions) and internal surfaces that are fractal in nature. The scattering from such materials displays two distinct breaks in log-log representation, corresponding to the radius-of-gyration of the primary particles, rg, and the radius-of-gyration of the clusters (aggregates), Rg. Between these boundaries the scattering follows a power law related to the mass fractal dimension, Dm, whilst above the high-Q boundary the scattering follows a power law related to the surface fractal dimension of the primary particles, Ds.</p> 5480 5480 <p style="margin-left: 0.55in;">The scattered intensity I(Q) is then calculated using a modified Ornstein-Zernicke equation:</p> … … 5558 5558 </div> 5559 5559 <p style="margin-left: 0.25in; text-align: center;" align="center"><img id="Picture 79" style="width: 507px; height: 370px;" src="img/masssurface_fractal_fig1.jpg" alt="" /></p> 5560 <p style="text-align: center;" align="center">< strong>Figure. 1D plot</strong></p>5560 <p style="text-align: center;" align="center"><b>Figure. 1D plot</b></p> 5561 5561 <p style="margin-left: 0.25in; text-align: center;" align="center"> </p> 5562 5562 <p style="margin-left: 0.25in; text-align: center;" align="center"> </p> … … 5566 5566 <p> </p> 5567 5567 <p> </p> 5568 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.12.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><strong><span style="font-size: 14pt;"> <a name="FractalCoreShell"></a>FractalCoreShell(Model)</span></strong></p>5568 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.12.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;"> <a name="FractalCoreShell"></a>FractalCoreShell(Model)</span></b></p> 5569 5569 <p style="margin-left: 0.25in;">Calculates the scattering from a fractal structure with a primary building block of core-shell spheres.</p> 5570 5570 <p><img src="img/fractcore_eq1.gif"/></p> … … 5695 5695 </div> 5696 5696 <p style="margin-left: 0.25in; text-align: center;" align="center"><img id="Picture 81" src="img/image188.jpg" alt="" /></p> 5697 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/500 data points).</strong></p>5697 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/500 data points).</b></p> 5698 5698 <p style="margin-left: 0.25in; text-align: center;" align="center"> </p> 5699 5699 <p style="margin-left: 0.25in; text-align: center;" align="center"> </p> 5700 5700 <p style="margin-left: 0.25in;">References:</p> 5701 <p style="text-indent: 0.25in;">See the PolyCore and Fractal documentation.< strong><span style="font-size: 14pt;"> </span></strong></p>5702 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.13.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><strong><span style="font-size: 14pt;"> <a name="GaussLorentzGel"></a>GaussLorentzGel(Model)</span></strong></p>5701 <p style="text-indent: 0.25in;">See the PolyCore and Fractal documentation.<b><span style="font-size: 14pt;"> </span></b></p> 5702 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.13.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;"> <a name="GaussLorentzGel"></a>GaussLorentzGel(Model)</span></b></p> 5703 5703 <p style="margin-left: 0.25in;">Calculates the scattering from a gel structure, typically a physical network. It is modeled as a sum of a low-q exponential decay plus a lorentzian at higher q-values. It is generally applicable to gel structures.</p> 5704 5704 <p style="margin-left: 0.25in;">The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> … … 5779 5779 </div> 5780 5780 <p style="margin-left: 0.25in; text-align: center;" align="center"><img id="Picture 82" src="img/image190.jpg" alt="" width="471" height="320" /></p> 5781 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/500 data points).</strong></p>5781 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/500 data points).</b></p> 5782 5782 <p style="margin-left: 0.25in; text-align: center;" align="center"> </p> 5783 5783 <p style="margin-left: 0.25in; text-align: center;" align="center"> </p> 5784 5784 <p style="text-indent: 0.5in;">REFERENCE:</p> 5785 5785 <p style="text-indent: 0.5in;">G. Evmenenko, E. Theunissen, K. Mortensen, H. Reynaers, Polymer 42 (2001) 2907-2913.</p> 5786 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.14.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><strong><span style="font-size: 14pt;"> <a name="BEPolyelectrolyte"></a> BEPolyelectrolyte Model</span></strong></p>5786 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.14.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;"> <a name="BEPolyelectrolyte"></a> BEPolyelectrolyte Model</span></b></p> 5787 5787 <p style="margin-left: 0.25in;">Calculates the structure factor of a polyelectrolyte solution with the RPA expression derived by Borue and Erukhimovich. The value returned is in cm-1.</p> 5788 5788 <p> </p> … … 5902 5902 <p style="margin-left: 0.5in;">3, 573 (1993).</p> 5903 5903 <p style="margin-left: 0.5in;">Raphaël, E., Joanny, J.-F., Europhysics Letters 11, 179 (1990).</p> 5904 <p style="margin-left: 0.25in;">< strong><span style="font-size: 14pt;"> </span></strong></p>5905 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.15.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><strong><span style="font-size: 14pt;"> <a name="Guinier"></a>Guinier (Model)</span></strong></p>5904 <p style="margin-left: 0.25in;"><b><span style="font-size: 14pt;"> </span></b></p> 5905 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.15.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;"> <a name="Guinier"></a>Guinier (Model)</span></b></p> 5906 5906 <p style="margin-left: 0.25in;">A Guinier analysis is done by linearizing the data at low q by plotting it as log(I) versus Q2. The Guinier radius Rg can be obtained by fitting the following model:</p> 5907 5907 <p style="margin-left: 0.25in; text-align: center;" align="center"><span style="font-size: 14pt; position: relative; top: 7pt;"><img src="img/image192.PNG" alt="" /></span></p> 5908 5908 <p style="margin-left: 0.25in; text-align: center;" align="center"><span style="font-size: 14pt;"> </span></p> 5909 5909 <p style="margin-left: 0.25in;">For 2D plot, the wave transfer is defined as<span style="font-size: 12pt; font-family: 'Times New Roman','serif'; position: relative; top: 4.5pt;"><img src="img/image040.gif" alt="" /></span><span style="font-size: 14pt;">.</span></p> 5910 <p style="margin-left: 0.25in; text-align: center;" align="center">< strong><span style="font-size: 14pt;"> </span></strong></p>5910 <p style="margin-left: 0.25in; text-align: center;" align="center"><b><span style="font-size: 14pt;"> </span></b></p> 5911 5911 <div align="center"> 5912 5912 <table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0"> … … 5948 5948 </table> 5949 5949 </div> 5950 <p style="margin-left: 0.25in;">< strong><span style="font-size: 14pt;"> </span></strong></p>5951 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.16.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><strong><span style="font-size: 14pt;"> <a name="GuinierPorod"></a>GuinierPorod (Model)</span></strong></p>5950 <p style="margin-left: 0.25in;"><b><span style="font-size: 14pt;"> </span></b></p> 5951 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.16.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;"> <a name="GuinierPorod"></a>GuinierPorod (Model)</span></b></p> 5952 5952 <p style="margin-left: 0.25in;">Calculates the scattering for a generalized Guinier/power law object. This is an empirical model that can be used to determine the size and dimensionality of scattering objects.</p> 5953 5953 <p style="margin-left: 0.25in;">The returned value is P(Q) as written in equation (1), plus the incoherent background term. The result is in the units of [cm-1sr-1], absolute scale.</p> … … 5969 5969 <p style="margin-left: 0.25in;">[2] Glatter, O.; Kratky, O., “Small-Angle X-Ray Scattering”, Academic Press (1982). Check out Chapter 4 on Data Treatment, pages 155-156. </p> 5970 5970 <p style="margin-left: 0.25in;">For 2D plot, the wave transfer is defined as<span style="font-size: 14pt; position: relative; top: 8pt;"><img src="img/image008.PNG" alt="" /></span><span style="font-size: 14pt;">.</span></p> 5971 <p style="margin-left: 0.25in; text-align: center;" align="center">< strong><span style="font-size: 14pt;"> </span></strong></p>5971 <p style="margin-left: 0.25in; text-align: center;" align="center"><b><span style="font-size: 14pt;"> </span></b></p> 5972 5972 <div align="center"> 5973 5973 <table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0"> … … 6036 6036 </table> 6037 6037 </div> 6038 <p style="margin-left: 0.25in;">< strong><span style="font-size: 14pt;"> </span></strong></p>6039 <p style="margin-left: 0.25in; text-align: center;" align="center">< strong><span style="font-size: 14pt;"><img id="Picture 4" src="img/image196.jpg" alt="" /></span></strong></p>6040 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/500 data points).</strong></p>6041 <p style="text-align: center;" align="center">< strong> </strong></p>6042 <p style="margin-left: 0.25in;">< strong><span style="font-size: 14pt;"> </span></strong></p>6043 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.17.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><strong><span style="font-size: 14pt;"> <a name="PorodModel"></a>PorodModel</span></strong></p>6038 <p style="margin-left: 0.25in;"><b><span style="font-size: 14pt;"> </span></b></p> 6039 <p style="margin-left: 0.25in; text-align: center;" align="center"><b><span style="font-size: 14pt;"><img id="Picture 4" src="img/image196.jpg" alt="" /></span></b></p> 6040 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/500 data points).</b></p> 6041 <p style="text-align: center;" align="center"><b> </b></p> 6042 <p style="margin-left: 0.25in;"><b><span style="font-size: 14pt;"> </span></b></p> 6043 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.17.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;"> <a name="PorodModel"></a>PorodModel</span></b></p> 6044 6044 <p style="margin-left: 0.25in;">A Porod analysis is done by linearizing the data at high q by plotting it as log(I) versus log(Q). In the high q region we can fit the following model:</p> 6045 6045 <p style="margin-left: 0.25in; text-align: center;" align="center"><span style="font-size: 14pt; position: relative; top: 6pt;"><img src="img/image197.PNG" alt="" /></span></p> … … 6048 6048 <p style="margin-left: 0.25in;">The background term is added for data analysis.</p> 6049 6049 <p style="margin-left: 0.25in;">For 2D plot, the wave transfer is defined as<span style="font-size: 12pt; font-family: 'Times New Roman','serif'; position: relative; top: 4.5pt;"><img src="img/image040.gif" alt="" /></span><span style="font-size: 14pt;">.</span></p> 6050 <p style="margin-left: 0.25in; text-align: center;" align="center">< strong><span style="font-size: 14pt;"> </span></strong></p>6050 <p style="margin-left: 0.25in; text-align: center;" align="center"><b><span style="font-size: 14pt;"> </span></b></p> 6051 6051 <div align="center"> 6052 6052 <table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0"> … … 6088 6088 </table> 6089 6089 </div> 6090 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.18.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><strong><span style="font-size: 14pt;"> <a name="Peak Gauss Model"></a>PeakGaussModel</span></strong></p>6090 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.18.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;"> <a name="Peak Gauss Model"></a>PeakGaussModel</span></b></p> 6091 6091 <p style="margin-left: 0.25in;">Model describes a Gaussian shaped peak including a flat background,</p> 6092 6092 <p style="margin-left: 0.25in; text-align: center;" align="center"> </p> … … 6097 6097 <p style="margin-left: 0.25in;"> REFERENCE: None</p> 6098 6098 <p style="margin-left: 0.25in;">For 2D plot, the wave transfer is defined as<span style="font-size: 12pt; font-family: 'Times New Roman','serif'; position: relative; top: 4.5pt;"><img src="img/image040.gif" alt="" /></span><span style="font-size: 14pt;">.</span></p> 6099 <p style="margin-left: 0.25in; text-align: center;" align="center">< strong><span style="font-size: 14pt;"> </span></strong></p>6099 <p style="margin-left: 0.25in; text-align: center;" align="center"><b><span style="font-size: 14pt;"> </span></b></p> 6100 6100 <div align="center"> 6101 6101 <table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0"> … … 6155 6155 </table> 6156 6156 </div> 6157 <p style="margin-left: 0.25in;">< strong><span style="font-size: 14pt;"> </span></strong></p>6158 <p style="margin-left: 0.25in;">< strong><span style="font-size: 14pt;"> </span></strong></p>6159 <p style="margin-left: 0.25in; text-align: center;" align="center">< strong><span style="font-size: 14pt;"><img src="img/image199.jpg" alt="" /></span></strong></p>6160 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/500 data points).</strong></p>6161 <p style="text-align: center;" align="center">< strong> </strong></p>6162 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.19.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><strong><span style="font-size: 14pt;"> <a name="Peak Lorentz Model"></a>PeakLorentzModel</span></strong></p>6157 <p style="margin-left: 0.25in;"><b><span style="font-size: 14pt;"> </span></b></p> 6158 <p style="margin-left: 0.25in;"><b><span style="font-size: 14pt;"> </span></b></p> 6159 <p style="margin-left: 0.25in; text-align: center;" align="center"><b><span style="font-size: 14pt;"><img src="img/image199.jpg" alt="" /></span></b></p> 6160 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/500 data points).</b></p> 6161 <p style="text-align: center;" align="center"><b> </b></p> 6162 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.19.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;"> <a name="Peak Lorentz Model"></a>PeakLorentzModel</span></b></p> 6163 6163 <p style="margin-left: 0.25in;">Model describes a Lorentzian shaped peak including a flat background,</p> 6164 6164 <p> </p> … … 6169 6169 <p style="margin-left: 0.25in;"> REFERENCE: None</p> 6170 6170 <p style="margin-left: 0.25in;">For 2D plot, the wave transfer is defined as<span style="font-size: 12pt; font-family: 'Times New Roman','serif'; position: relative; top: 4.5pt;"><img src="img/image040.gif" alt="" /></span><span style="font-size: 14pt;">.</span></p> 6171 <p style="margin-left: 0.25in; text-align: center;" align="center">< strong><span style="font-size: 14pt;"> </span></strong></p>6171 <p style="margin-left: 0.25in; text-align: center;" align="center"><b><span style="font-size: 14pt;"> </span></b></p> 6172 6172 <div align="center"> 6173 6173 <table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0"> … … 6227 6227 </table> 6228 6228 </div> 6229 <p style="margin-left: 0.25in;">< strong><span style="font-size: 14pt;"> </span></strong></p>6230 <p style="margin-left: 0.25in;">< strong><span style="font-size: 14pt;"> </span></strong></p>6231 <p style="margin-left: 0.25in; text-align: center;" align="center">< strong><span style="font-size: 14pt;"><img src="img/image201.jpg" alt="" /></span></strong></p>6232 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/500 data points).</strong></p>6233 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.20. <a name="Poly_GaussCoil"></a>Poly_GaussCoil (Model)</span></strong></p>6229 <p style="margin-left: 0.25in;"><b><span style="font-size: 14pt;"> </span></b></p> 6230 <p style="margin-left: 0.25in;"><b><span style="font-size: 14pt;"> </span></b></p> 6231 <p style="margin-left: 0.25in; text-align: center;" align="center"><b><span style="font-size: 14pt;"><img src="img/image201.jpg" alt="" /></span></b></p> 6232 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/500 data points).</b></p> 6233 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.20. <a name="Poly_GaussCoil"></a>Poly_GaussCoil (Model)</span></b></p> 6234 6234 <p style="margin-left: 0.25in;">Polydisperse Gaussian Coil: Calculate an empirical functional form for scattering from a polydisperse polymer chain ina good solvent. The polymer is polydisperse with a Schulz-Zimm polydispersity of the molecular weight distribution. </p> 6235 6235 <p style="margin-left: 0.25in;">The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> … … 6308 6308 <p style="margin-left: 0.25in; text-align: center;" align="center"> </p> 6309 6309 <p style="margin-left: 0.25in; text-align: center;" align="center"><img id="Picture 69" src="img/image205.jpg" alt="" /></p> 6310 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/200 data point).</strong></p>6310 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/200 data point).</b></p> 6311 6311 <p style="margin-left: 0.25in; text-align: center;" align="center"> </p> 6312 6312 <p style="margin-left: 0.25in;">Reference:</p> … … 6314 6314 <p style="margin-left: 0.25in;">J.S. Higgins, and H.C. Benoit, “Polymers and Neutron Scattering”, Oxford Science</p> 6315 6315 <p style="margin-left: 0.25in;">Publications (1996).</p> 6316 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.21. <a name="PolymerExclVolume"></a>PolymerExclVolume (Model)</span></strong></p>6316 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.21. <a name="PolymerExclVolume"></a>PolymerExclVolume (Model)</span></b></p> 6317 6317 <p style="margin-left: 0.25in;">Calculates the scattering from polymers with excluded volume effects.</p> 6318 6318 <p style="margin-left: 0.25in;">The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> … … 6403 6403 <p style="margin-left: 0.25in; text-align: center;" align="center"> </p> 6404 6404 <p style="margin-left: 0.25in; text-align: center;" align="center"><img id="Picture 7" src="img/image214.jpg" alt="" width="479" height="333" /></p> 6405 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/500 data points).</strong></p>6405 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/500 data points).</b></p> 6406 6406 <p style="margin-left: 0.25in; text-align: center;" align="center"> </p> 6407 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.22.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><strong><span style="font-size: 14pt;"> <a name="RPA10Model"></a>RPA10Model</span></strong></p>6407 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.22.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;"> <a name="RPA10Model"></a>RPA10Model</span></b></p> 6408 6408 <p style="margin-left: 0.25in;">Calculates the macroscopic scattering intensity (units of cm^-1) for a multicomponent homogeneous mixture of polymers using the Random Phase Approximation. This general formalism contains 10 specific cases:</p> 6409 6409 <p style="margin-left: 0.25in;">Case 0: C/D Binary mixture of homopolymers</p> … … 6418 6418 <p style="margin-left: 0.25in;">Case 9: A-B-C-D Four-block copolymer</p> 6419 6419 <p style="margin-left: 0.25in;">Note: the case numbers are different from the IGOR/NIST SANS package.</p> 6420 <p style="margin-left: 0.25in;">< strong> </strong></p>6420 <p style="margin-left: 0.25in;"><b> </b></p> 6421 6421 <p style="margin-left: 0.25in;">Only one case can be used at any one time. Plotting a different case will overwrite the original parameter waves.</p> 6422 6422 <p style="margin-left: 0.25in;">The returned value is scaled to units of [cm-1].</p> … … 6494 6494 </table> 6495 6495 </div> 6496 <p style="margin-left: 0.25in;">< strong><span style="font-size: 14pt;"> </span></strong></p>6497 <p style="margin-left: 0.25in; text-align: center;" align="center">< strong><span style="font-size: 14pt;"> </span></strong></p>6496 <p style="margin-left: 0.25in;"><b><span style="font-size: 14pt;"> </span></b></p> 6497 <p style="margin-left: 0.25in; text-align: center;" align="center"><b><span style="font-size: 14pt;"> </span></b></p> 6498 6498 <p style="margin-left: 0.25in; text-align: center;" align="center">Fixed parameters for Case0 Model</p> 6499 6499 <div align="center"> … … 6586 6586 </table> 6587 6587 </div> 6588 <p style="margin-left: 0.25in;">< strong><span style="font-size: 14pt;"> </span></strong></p>6589 <p style="margin-left: 0.25in;">< strong><span style="font-size: 14pt;"> </span></strong></p>6588 <p style="margin-left: 0.25in;"><b><span style="font-size: 14pt;"> </span></b></p> 6589 <p style="margin-left: 0.25in;"><b><span style="font-size: 14pt;"> </span></b></p> 6590 6590 <p style="margin-left: 0.25in; text-align: center;" align="center"><img id="Picture 8" src="img/image215.jpg" alt="" /></p> 6591 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/500 data points).</strong></p>6592 <p style="margin-left: 0.25in;">< strong><span style="font-size: 14pt;"> </span></strong></p>6593 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.23.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><strong><span style="font-size: 14pt;"> <a name="TwoLorentzian"></a>TwoLorentzian(Model)</span></strong></p>6591 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/500 data points).</b></p> 6592 <p style="margin-left: 0.25in;"><b><span style="font-size: 14pt;"> </span></b></p> 6593 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.23.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;"> <a name="TwoLorentzian"></a>TwoLorentzian(Model)</span></b></p> 6594 6594 <p style="margin-left: 0.25in;">Calculate an empirical functional form for SANS data characterized by a two Lorentzian functions.</p> 6595 6595 <p style="margin-left: 0.25in;">The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> … … 6602 6602 <p style="margin-left: 0.25in;">The background term is added for data analysis.</p> 6603 6603 <p style="margin-left: 0.25in;">For 2D plot, the wave transfer is defined as<span style="font-size: 12pt; font-family: 'Times New Roman','serif'; position: relative; top: 4.5pt;"><img src="img/image040.gif" alt="" /></span><span style="font-size: 14pt;">.</span></p> 6604 <p style="margin-left: 0.25in; text-align: center;" align="center">< strong>Default input parameter values</strong></p>6604 <p style="margin-left: 0.25in; text-align: center;" align="center"><b>Default input parameter values</b></p> 6605 6605 <div align="center"> 6606 6606 <table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0"> … … 6692 6692 <p style="margin-left: 0.5in;"> </p> 6693 6693 <p style="margin-left: 0.5in; text-align: center;" align="center"><img id="Picture 9" src="img/image217.jpg" alt="" /></p> 6694 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/500 data points).</strong></p>6694 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/500 data points).</b></p> 6695 6695 <p style="margin-left: 0.5in; text-align: center;" align="center"> </p> 6696 <p style="text-indent: 0.25in;">< strong>REFERENCE: None</strong></p>6697 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.24.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><strong><span style="font-size: 14pt;"> <a name="TwoPowerLaw"></a>TwoPowerLaw(Model)</span></strong></p>6696 <p style="text-indent: 0.25in;"><b>REFERENCE: None</b></p> 6697 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.24.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;"> <a name="TwoPowerLaw"></a>TwoPowerLaw(Model)</span></b></p> 6698 6698 <p style="margin-left: 0.25in;">Calculate an empirical functional form for SANS data characterized by two power laws.</p> 6699 6699 <p style="margin-left: 0.25in;">The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> … … 6704 6704 <p style="margin-left: 0.25in;">qc is the location of the crossover from one slope to the other. The scaling A, sets the overall intensity of the lower Q power law region. The scaling of the second power law region is scaled to match the first. Be sure to enter the power law exponents as positive values.</p> 6705 6705 <p style="margin-left: 0.25in;">For 2D plot, the wave transfer is defined as<span style="font-size: 12pt; font-family: 'Times New Roman','serif'; position: relative; top: 4.5pt;"><img src="img/image040.gif" alt="" /></span><span style="font-size: 14pt;">.</span></p> 6706 <p style="margin-left: 0.25in; text-align: center;" align="center">< strong>Default input parameter values</strong></p>6706 <p style="margin-left: 0.25in; text-align: center;" align="center"><b>Default input parameter values</b></p> 6707 6707 <div align="center"> 6708 6708 <table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0"> … … 6774 6774 <p style="margin-left: 0.5in;"> </p> 6775 6775 <p style="margin-left: 0.5in; text-align: center;" align="center"><img id="Picture 10" src="img/image219.jpg" alt="" /></p> 6776 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/500 data points).</strong></p>6776 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/500 data points).</b></p> 6777 6777 <p style="margin-left: 0.5in; text-align: center;" align="center"> </p> 6778 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.25.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><strong><span style="font-size: 14pt;"> <a name="UnifiedPowerRg"></a>UnifiedPower(Law and)Rg(Model)</span></strong></p>6778 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.25.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;"> <a name="UnifiedPowerRg"></a>UnifiedPower(Law and)Rg(Model)</span></b></p> 6779 6779 <p style="margin-left: 0.25in;">The returned value is scaled to units of [cm-1sr-1], absolute scale. </p> 6780 6780 <p style="margin-left: 0.25in;">Note that the level 0 is an extra function that is the inverse function; I (q) = scale/q + background.</p> … … 6788 6788 <p style="margin-left: 0.25in;"> </p> 6789 6789 <p style="margin-left: 0.25in;">For 2D plot, the wave transfer is defined as<span style="font-size: 12pt; font-family: 'Times New Roman','serif'; position: relative; top: 4.5pt;"><img src="img/image040.gif" alt="" /></span><span style="font-size: 14pt;">.</span></p> 6790 <p style="margin-left: 0.25in; text-align: center;" align="center">< strong>Default input parameter values</strong></p>6790 <p style="margin-left: 0.25in; text-align: center;" align="center"><b>Default input parameter values</b></p> 6791 6791 <div align="center"> 6792 6792 <table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0"> … … 6913 6913 <p style="margin-left: 0.5in;"> </p> 6914 6914 <p style="margin-left: 0.5in; text-align: center;" align="center"><img id="Picture 11" src="img/image221.jpg" alt="" width="470" height="336" /></p> 6915 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/500 data points).</strong></p>6915 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/500 data points).</b></p> 6916 6916 <p style="margin-left: 0.5in; text-align: center;" align="center"> </p> 6917 6917 <p style="margin-left: 0.25in;"> REFERENCES</p> 6918 6918 <p style="margin-left: 0.25in;">G. Beaucage (1995). J. Appl. Cryst., vol. 28, p717-728.</p> 6919 6919 <p style="margin-left: 0.25in;">G. Beaucage (1996). J. Appl. Cryst., vol. 29, p134-146.</p> 6920 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.26.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><strong><span style="font-size: 14pt;"> <a name="LineModel"></a> LineModel</span></strong></p>6920 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.26.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;"> <a name="LineModel"></a> LineModel</span></b></p> 6921 6921 <p style="margin-left: 0.25in;">This is a linear function that calculates:</p> 6922 6922 <p style="margin-left: 0.25in; text-align: center;" align="center"><span style="font-size: 14pt; position: relative; top: 5pt;"><img src="img/image222.PNG" alt="" /></span></p> 6923 6923 <p style="margin-left: 0.25in; text-align: center;" align="center"><span style="font-size: 14pt;"> </span></p> 6924 6924 <p style="margin-left: 0.25in;">where A and B are the coefficients of the first and second order terms.</p> 6925 <p style="margin-left: 0.25in;">< strong>Note:</strong> For 2D plot, I(q) = I(qx)*I(qy) which is defined differently from other shape independent models.</p>6925 <p style="margin-left: 0.25in;"><b>Note:</b> For 2D plot, I(q) = I(qx)*I(qy) which is defined differently from other shape independent models.</p> 6926 6926 <div align="center"> 6927 6927 <table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0"> … … 6964 6964 </div> 6965 6965 <p style="margin-left: 0.55in; text-indent: -0.3in;"> </p> 6966 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.27.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="ReflectivityModel"></a><strong><span style="font-size: 14pt;">ReflectivityModel</span></strong></p>6966 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.27.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="ReflectivityModel"></a><b><span style="font-size: 14pt;">ReflectivityModel</span></b></p> 6967 6967 <p style="margin-left: 0.55in; text-indent: -0.3in;">This model calculates the reflectivity and uses the Parrett algorithm. Up to nine film layers are supported between Bottom(substrate) and Medium(Superstrate where the neutron enters the first top film). Each layers are composed of [ ½ of the interface(from the previous layer or substrate) + flat portion + ½ of the interface(to the next layer or medium)]. Only two simple interfacial functions are selectable, error function and linear function. The each interfacial thickness is equivalent to (- 2.5 sigma to +2.5 sigma for the error function, sigma=roughness).</p> 6968 6968 <p style="margin-left: 0.55in; text-indent: -0.3in;">Note: This model was contributed by an interested user.</p> 6969 6969 <p align="center"><img src="img/image231.bmp" alt="" /></p> 6970 <p style="text-align: center;" align="center">< strong>Figure. Comparison (using the SLD profile below) with NISTweb calculation (circles): http://www.ncnr.nist.gov/resources/reflcalc.html.</strong></p>6970 <p style="text-align: center;" align="center"><b>Figure. Comparison (using the SLD profile below) with NISTweb calculation (circles): http://www.ncnr.nist.gov/resources/reflcalc.html.</b></p> 6971 6971 <p align="center"><img src="img/image232.gif" alt="" /></p> 6972 <p style="text-align: center;" align="center">< strong>Figure. SLD profile used for the calculation(above).</strong></p>6973 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.28.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="ReflectivityIIModel"></a><strong><span style="font-size: 14pt;">ReflectivityIIModel</span></strong></p>6972 <p style="text-align: center;" align="center"><b>Figure. SLD profile used for the calculation(above).</b></p> 6973 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.28.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="ReflectivityIIModel"></a><b><span style="font-size: 14pt;">ReflectivityIIModel</span></b></p> 6974 6974 <p> Same as the ReflectivityModel except that the it is more customizable. More interfacial functions are supplied. The number of points (npts_inter) for each interface can be choosen. The constant (A below but 'nu' as a parameter name of the model) for exp, erf, or power-law is an input. The SLD at the interface between layers, <em><span style="font-family: Symbol;">r</span>inter_i</em>, is calculated with a function chosen by a user, where the functions are:</p> 6975 6975 <p style="margin-left: 0.55in;">1) Erf;</p> … … 6983 6983 <p> </p> 6984 6984 <p> Note: This model was implemented by an interested user.</p> 6985 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.29.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="GelFitModel"></a><strong><span style="font-size: 14pt;">GelFitModel</span></strong></p>6985 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.29.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="GelFitModel"></a><b><span style="font-size: 14pt;">GelFitModel</span></b></p> 6986 6986 <p> Unlike a concentrated polymer solution, the fine-scale polymer distribution in a gel involves at least two characteristic length scales, a shorter correlation length (a1) to describe the rapid fluctuations in the position of the polymer chains that ensure thermodynamic equilibrium, and a longer distance (denoted here as a2) needed to account for the static accumulations of polymer pinned down by junction points or clusters of such points. The letter is derived from a simple Guinier function.</p> 6987 6987 <p style="margin-left: 0.55in;">The scattered intensity I(Q) is then calculated as:</p> … … 6993 6993 <p> Note the first term reduces to the Ornstein-Zernicke equation when D=2; ie, when the Flory exponent is 0.5 (theta conditions). In gels with significant hydrogen bonding D has been reported to be ~2.6 to 2.8.</p> 6994 6994 <p> Note: This model was implemented by an interested user.</p> 6995 <p style="margin-left: 0.25in; text-align: center;" align="center">< strong>Default input parameter values</strong></p>6995 <p style="margin-left: 0.25in; text-align: center;" align="center"><b>Default input parameter values</b></p> 6996 6996 <div align="center"> 6997 6997 <table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0"> … … 7078 7078 <p style="margin-left: 0.5in;"> </p> 7079 7079 <p style="margin-left: 0.5in; text-align: center;" align="center"><img id="Picture 11" src="img/image235.gif" alt="" width="470" height="336" /></p> 7080 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (w/300 data points, qmin=0.001, and qmax=0.3).</strong></p>7080 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/300 data points, qmin=0.001, and qmax=0.3).</b></p> 7081 7081 <p style="margin-left: 0.5in; text-align: center;" align="center"> </p> 7082 7082 <p style="margin-left: 0.25in;"> REFERENCES</p> … … 7084 7084 <p style="margin-left: 0.25in;">Simon Mallam, Ferenc Horkay, Anne-Marie Hecht, Adrian R. Rennie, Erik Geissler, Macromolecules 1991, 24, 543-548.</p> 7085 7085 <p> </p> 7086 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">3.30.</span></strong> <strong> <span style="font-size: 7pt;"> </span> </strong> <strong> <span style="font-size: 14pt;"><a name="StarPolymer"></a><a name="StarPolymerModel"></a>Star Polymer with Gaussian Statistics</span> </strong></p>7086 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">3.30.</span></b> <b> <span style="font-size: 7pt;"> </span> </b> <b> <span style="font-size: 14pt;"><a name="StarPolymer"></a><a name="StarPolymerModel"></a>Star Polymer with Gaussian Statistics</span> </b></p> 7087 7087 <p style="margin-left: 0.25in;">For a star with <em>f</em> arms:</p> 7088 7088 <p style="text-align: center;" align="center"><span style="font-size: 12pt; font-family: 'Times New Roman','serif'; position: relative; top: 4.5pt;"><img src="img/star1.png" alt="" /></span></p> … … 7095 7095 <p> </p> 7096 7096 <p> </p> 7097 <p style="margin-left: 0.25in; text-indent: -0.25in;">< strong><span style="font-size: 16pt;">4.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="Model"></a><strong><span style="font-size: 16pt;">Customized Models </span></strong></p>7098 <p style="margin-left: 0.25in;">< strong><span style="font-size: 14pt;"> </span></strong></p>7097 <p style="margin-left: 0.25in; text-indent: -0.25in;"><b><span style="font-size: 16pt;">4.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="Model"></a><b><span style="font-size: 16pt;">Customized Models </span></b></p> 7098 <p style="margin-left: 0.25in;"><b><span style="font-size: 14pt;"> </span></b></p> 7099 7099 <p style="margin-left: 0.25in;">Customized model functions can be redefined or added by users (See SansView tutorial for details).</p> 7100 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">4.1.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="testmodel"></a><strong><span style="font-size: 14pt;">testmodel</span></strong></p>7101 <p style="margin-left: 0.55in;">< strong><span style="font-size: 14pt;"> </span></strong></p>7100 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">4.1.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="testmodel"></a><b><span style="font-size: 14pt;">testmodel</span></b></p> 7101 <p style="margin-left: 0.55in;"><b><span style="font-size: 14pt;"> </span></b></p> 7102 7102 <p>This function, as an example of a user defined function, calculates the intensity = A + Bcos(2q) + Csin(2q).</p> 7103 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">4.2.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="testmodel_2"></a><strong><span style="font-size: 14pt;">testmodel_2 </span></strong></p>7103 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">4.2.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="testmodel_2"></a><b><span style="font-size: 14pt;">testmodel_2 </span></b></p> 7104 7104 <p>This function, as an example of a user defined function, calculates the intensity = scale * sin(f)/f, where f = A + Bq + Cq2 + Dq3 + Eq4 + Fq5.</p> 7105 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">4.3.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="sum_p1_p2"></a><strong><span style="font-size: 14pt;">sum_p1_p2 </span></strong></p>7105 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">4.3.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="sum_p1_p2"></a><b><span style="font-size: 14pt;">sum_p1_p2 </span></b></p> 7106 7106 <p>This function, as an example of a user defined function, calculates the intensity = scale_factor * (CylinderModel + PolymerExclVolume model). To make your own sum(P1+P2) model, select 'Easy Custom Sum' from the Fitting menu, or modify and compile the file named 'sum_p1_p2.py' from 'Edit Custom Model' in the 'Fitting' menu. It works only for single functional models.</p> 7107 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">4.4.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="sum_Ap1_1_Ap2"></a><strong><span style="font-size: 14pt;">sum_Ap1_1_Ap2 </span></strong></p>7107 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">4.4.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="sum_Ap1_1_Ap2"></a><b><span style="font-size: 14pt;">sum_Ap1_1_Ap2 </span></b></p> 7108 7108 <p>This function, as an example of a user defined function, calculates the intensity = (scale_factor * CylinderModel + (1-scale_factor) * PolymerExclVolume model). To make your own A*p1+(1-A)*p2 model, modify and compile the file named 'sum_Ap1_1_Ap2.py' from 'Edit Custom Model' in the 'Fitting' menu. It works only for single functional models.</p> 7109 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">4.5.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="polynomial5"></a><strong><span style="font-size: 14pt;">polynomial5 </span></strong></p>7109 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">4.5.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="polynomial5"></a><b><span style="font-size: 14pt;">polynomial5 </span></b></p> 7110 7110 <p>This function, as an example of a user defined function, calculates the intensity = A + Bq + Cq2 + Dq3 + Eq4 + Fq5. This model can be modified and compiled from 'Edit Custom Model' in the 'Fitting' menu.</p> 7111 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">4.6.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="sph_bessel_jn"></a><strong><span style="font-size: 14pt;">sph_bessel_jn </span></strong></p>7111 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">4.6.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="sph_bessel_jn"></a><b><span style="font-size: 14pt;">sph_bessel_jn </span></b></p> 7112 7112 <p>This function, as an example of a user defined function, calculates the intensity = C*sph_jn(Ax+B)+D where the sph_jn is spherical Bessel function of the order n. This model can be modified and compiled from 'Edit Custom Model' in the 'Fitting' menu.</p> 7113 <p style="margin-left: 0.25in; text-indent: -0.25in;">< strong><span style="font-size: 16pt;">5.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="Structure_Factors"></a><strong><span style="font-size: 16pt;">Structure Factors</span></strong></p>7113 <p style="margin-left: 0.25in; text-indent: -0.25in;"><b><span style="font-size: 16pt;">5.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="Structure_Factors"></a><b><span style="font-size: 16pt;">Structure Factors</span></b></p> 7114 7114 <p style="margin-left: 0.25in;"> </p> 7115 7115 <p style="margin-left: 0.25in;">The information in this section is originated from NIST SANS IgorPro package.</p> 7116 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">5.1.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="HardsphereStructure"></a><strong><span style="font-size: 14pt;">HardSphere Structure </span></strong></p>7116 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">5.1.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="HardsphereStructure"></a><b><span style="font-size: 14pt;">HardSphere Structure </span></b></p> 7117 7117 <p>This calculates the interparticle structure factor for monodisperse spherical particles interacting through hard sphere (excluded volume) interactions. The calculation uses the Percus-Yevick closure where the interparticle potential is:</p> 7118 7118 <p style="margin-left: 0.25in; text-align: center;" align="center"><span style="font-size: 14pt; position: relative; top: 15pt;"><img src="img/image223.PNG" alt="" /></span></p> … … 7163 7163 </div> 7164 7164 <p style="text-align: center;" align="center"><img id="Picture 111" src="img/image224.jpg" alt="" /></p> 7165 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (in linear scale).</strong></p>7165 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (in linear scale).</b></p> 7166 7166 <p>References:</p> 7167 7167 <p style="margin-left: 0.5in;">Percus, J. K.; Yevick, J. Phys. Rev. 110, 1. (1958).</p> 7168 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">5.2.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="SquareWellStructure"></a><strong><span style="font-size: 14pt;"> SquareWell Structure </span></strong></p>7168 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">5.2.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="SquareWellStructure"></a><b><span style="font-size: 14pt;"> SquareWell Structure </span></b></p> 7169 7169 <p>This calculates the interparticle structure factor for a squar well fluid spherical particles The mean spherical approximation (MSA) closure was used for this calculation, and is not the most appropriate closure for an attractive interparticle potential. This solution has been compared to Monte Carlo simulations for a square well fluid, showing this calculation to be limited in applicability to well depths e < 1.5 kT and volume fractions f < 0.08.</p> 7170 7170 <p>Positive well depths correspond to an attractive potential well. Negative well depths correspond to a potential "shoulder", which may or may not be physically reasonable.</p> … … 7235 7235 </div> 7236 7236 <p style="text-align: center;" align="center"><img id="Picture 110" src="img/image226.jpg" alt="" /></p> 7237 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (in linear scale).</strong></p>7237 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (in linear scale).</b></p> 7238 7238 <p>References:</p> 7239 7239 <p style="margin-left: 0.5in;">Sharma, R. V.; Sharma, K. C. Physica, 89A, 213. (1977).</p> 7240 7240 <p style="margin-left: 0.5in;"> </p> 7241 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">5.3.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="HayterMSAStructure"></a><strong><span style="font-size: 14pt;"> HayterMSA Structure </span></strong></p>7241 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">5.3.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="HayterMSAStructure"></a><b><span style="font-size: 14pt;"> HayterMSA Structure </span></b></p> 7242 7242 <p>This calculates the Structure factor (the Fourier transform of the pair correlation function g(r)) for a system of charged, spheroidal objects in a dielectric medium. When combined with an appropriate form factor (such as sphere, core+shell, ellipsoid etc…), this allows for inclusion of the interparticle interference effects due to screened coulomb repulsion between charged particles. This routine only works for charged particles. If the charge is set to zero the routine will self destruct. For non-charged particles use a hard sphere potential.</p> 7243 7243 <p>The salt concentration is used to compute the ionic strength of the solution which in turn is used to compute the Debye screening length. At present there is no provision for entering the ionic strength directly nor for use of any multivalent salts. The counterions are also assumed to be monovalent.</p> … … 7321 7321 </div> 7322 7322 <p style="text-align: center;" align="center"><img id="Picture 112" src="img/image227.jpg" alt="" /></p> 7323 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (in linear scale).</strong></p>7323 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (in linear scale).</b></p> 7324 7324 <p>References:</p> 7325 7325 <p style="text-indent: 0.5in;">JP Hansen and JB Hayter, Molecular Physics 46, 651-656 (1982).</p> 7326 7326 <p> JB Hayter and J Penfold, Molecular Physics 42, 109-118 (1981).</p> 7327 <p style="margin-left: 0.55in; text-indent: -0.3in;">< strong><span style="font-size: 14pt;">5.4.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="StickyHSStructure"></a><strong><span style="font-size: 14pt;"> StickyHS Structure </span></strong></p>7327 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">5.4.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="StickyHSStructure"></a><b><span style="font-size: 14pt;"> StickyHS Structure </span></b></p> 7328 7328 <p>This calculates the interparticle structure factor for a hard sphere fluid with a narrow attractive well. A perturbative solution of the Percus-Yevick closure is used. The strength of the attractive well is described in terms of "stickiness" as defined below. The returned value is a dimensionless structure factor, S(q).</p> 7329 7329 <p>The perturb (perturbation parameter), epsilon, should be held between 0.01 and 0.1. It is best to hold the perturbation parameter fixed and let the "stickiness" vary to adjust the interaction strength. The stickiness, tau, is defined in the equation below and is a function of both the perturbation parameter and the interaction strength. Tau and epsilon are defined in terms of the hard sphere diameter (sigma = 2R), the width of the square well, delta (same units as R), and the depth of the well, uo, in units of kT. From the definition, it is clear that smaller tau mean stronger attraction.</p> … … 7396 7396 </div> 7397 7397 <p style="text-align: center;" align="center"><img id="Picture 113" src="img/image230.jpg" alt="" /></p> 7398 <p style="text-align: center;" align="center">< strong>Figure. 1D plot using the default values (in linear scale).</strong></p>7398 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (in linear scale).</b></p> 7399 7399 <p>References:</p> 7400 7400 <p style="text-indent: 0.5in;">Menon, S. V. G., Manohar, C. and K. Srinivas Rao J. Chem. Phys., 95(12), 9186-9190 (1991).</p> 7401 <p><a name="References"></a>< strong><span style="font-size: 14pt;">References</span></strong></p>7401 <p><a name="References"></a><b><span style="font-size: 14pt;">References</span></b></p> 7402 7402 <p>Feigin, L. A, and D. I. Svergun (1987) "Structure Analysis by Small-Angle X-Ray and Neutron Scattering", Plenum Press, New York.</p> 7403 7403 <p>Guinier, A. and G. Fournet (1955) "Small-Angle Scattering of X-Rays", John Wiley and Sons, New York.</p> 7404 <p>Kline, S. R. (2006) <em>J Appl. Cryst.</em> < strong>39</strong>(6), 895.</p>7404 <p>Kline, S. R. (2006) <em>J Appl. Cryst.</em> <b>39</b>(6), 895.</p> 7405 7405 <p>Hansen, S., (1990)<em> J. Appl. Cryst. </em>23, 344-346.</p> 7406 7406 <p>Henderson, S.J. (1996) <em>Biophys. J. </em>70, 1618-1627</p>
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