Changeset ee74edd in sasview for sansmodels/src


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Timestamp:
Feb 27, 2013 8:01:55 AM (12 years ago)
Author:
Jae Cho <jhjcho@…>
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master, ESS_GUI, ESS_GUI_Docs, ESS_GUI_batch_fitting, ESS_GUI_bumps_abstraction, ESS_GUI_iss1116, ESS_GUI_iss879, ESS_GUI_iss959, ESS_GUI_opencl, ESS_GUI_ordering, ESS_GUI_sync_sascalc, costrafo411, magnetic_scatt, release-4.1.1, release-4.1.2, release-4.2.2, release_4.0.1, ticket-1009, ticket-1094-headless, ticket-1242-2d-resolution, ticket-1243, ticket-1249, ticket885, unittest-saveload
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  • sansmodels/src/sans/models/media/model_functions.html

    r318b5bbb ree74edd  
    44</head> 
    55<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> 
    77<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>,&nbsp;<a href="#Core2ndMomentModel">Core2ndMomentModel</a>, <a href="#CoreMultiShellModel">CoreMultiShellModel (Magnetic 2D Model)</a>, <a href="#VesicleModel">VesicleModel</a>, <a href="#MultiShellModel">MultiShellModel</a>, &nbsp;<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>,&nbsp;<a href="#Core2ndMomentModel">Core2ndMomentModel</a>, <a href="#CoreMultiShellModel">CoreMultiShellModel (Magnetic 2D Model)</a>, <a href="#VesicleModel">VesicleModel</a>, <a href="#MultiShellModel">MultiShellModel</a>, &nbsp;<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> 
    1414</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;">&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp; </span></b><a name="Introduction"></a><b><span style="font-size: 16pt;">Introduction </span></b></p> 
    1616 
    1717<p>&nbsp;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;">&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp; </span></b><a name="Shapes"></a><b><span style="font-size: 16pt;">Shapes (Scattering Intensity Models)</span></b></p> 
    1919<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> 
    2020 
     
    2323<p>with</p> 
    2424<p style="text-align: center;" align="center"><span style="position: relative; top: 8pt;"><img src="img/image002.PNG" alt="" /></span>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</p> 
    25 <p>where <em>P</em>0<em>(<strong>q</strong>)</em> is the un-normalized form factor, <em>&rho;(<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>&rho;(<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> 
    2626<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> 
    2727<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> 
    2828<p>Our so-called 2D scattering intensity functions provide <em>P(q, </em><em><span style="font-family: 'Arial','sans-serif';">&phi;</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';">&phi;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="SphereModel"></a><b><span style="font-size: 14pt;">Sphere Model (Magnetic 2D Model)</span></b></p> 
    3030<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> 
    3131For 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;">&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p> 
    3333<p>The 1D scattering intensity is calculated in the following way (Guinier, 1955):</p> 
    3434<p style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="img/image004.PNG" alt="" /></span></p> 
     
    110110</div> 
    111111<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;">&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp; </span>Validation of the sphere model</b></p> 
    113113<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> 
    114114<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> 
    115115<p style="text-align: center; page-break-after: avoid;" align="center">&nbsp;</p> 
    116116<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 &Aring;, Contrast=1e-6 &Aring; -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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="BinaryHSModel"></a><b><span style="font-size: 14pt;">BinaryHSModel</span></b></p> 
    118118<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> 
    119119<p style="text-align: center;" align="center"><span style="position: relative; top: 5pt;"><img src="img/image006.PNG" alt="" /></span></p> 
     
    228228</div> 
    229229<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> 
    231231<p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p> 
    232232<p>See the reference for details.</p> 
     
    234234<p>N. W. Ashcroft and D. C. Langreth, Physical Review, v. 156 (1967) 685-692.</p> 
    235235<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><a name="FuzzySphereModel"></a><strong><span style="font-size: 14pt;">FuzzySphereModel</span></strong></p> 
    237 <p><strong>&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="FuzzySphereModel"></a><b><span style="font-size: 14pt;">FuzzySphereModel</span></b></p> 
     237<p><b>&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p> 
    239239<p>The 1D scattering intensity is calculated in the following way (Guinier, 1955):</p> 
    240240<p>The returned value is scaled to units of [cm-1 sr-1], absolute scale.</p> 
     
    336336</div> 
    337337<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>&nbsp;</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>&nbsp;</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> 
    341341<p>&nbsp;</p> 
    342342<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.&nbsp;</p> 
    343 <p style="margin-left: 0.85in; text-indent: -0.35in;"><strong>1.1.</strong><strong><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p> 
    344344<p>The structure is:<br /> <img src="img/raspberry_pic.jpg" alt="" /></p> 
    345345<p><br /> Ro = the radius of the&nbsp;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> 
     
    450450</div> 
    451451<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> 
    453453<p style="text-align: center;" align="center">&nbsp;</p> 
    454454<p>&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="CoreShellModel"></a><b><span style="font-size: 14pt;">Core Shell (Sphere) Model (Magnetic 2D Model)</span></b></p> 
    456456<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> 
    457457For 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;">&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p> 
    459459<p>The 1D scattering intensity is calculated in the following way (Guinier, 1955):</p> 
    460460<p style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="img/image013.PNG" alt="" /></span>&nbsp;</p> 
     
    563563<p>REFERENCE</p> 
    564564<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;">&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp; </span>Validation of the core-shell sphere model</b></p> 
    566566<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> 
    567567<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> 
    568568<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 &Aring;, Contrast=1e-6 &Aring; -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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="Core2ndMomentModel"></a><b><span style="font-size: 14pt;">Core2ndMomentModel</span></b></p> 
    570570<p>This model describes the scattering from a layer of surfactant or polymer adsorbed on spherical particles under the conditions that (i) the&nbsp;<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 &gt;&gt; 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> 
    571571<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&nbsp;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;">&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p> 
    573573<p>The I0 is calculated in the following way (King, 2002):</p> 
    574574<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>&nbsp;</p> 
     
    687687<p>&nbsp;</p> 
    688688<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="CoreMultiShellModel"></a><b><span style="font-size: 14pt;">CoreMultiShell(Sphere)Model (Magnetic 2D Model)</span></b></p> 
    690690<p>This model provides the scattering from spherical core with from 1 up to 4 shell structures. It&nbsp;has a core of a specified radius, with four shells. The SLDs of the core and each shell are individually specified.&nbsp;</p> 
    691691For 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;">&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p> 
    693693<p>The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> 
    694694<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> 
     
    866866<p>This example dataset is produced by running the CoreMultiShellModel using 200 data points, qmin = 0.001 &Aring;-1,&nbsp; qmax = 0.7 &Aring;-1 and the above default values.</p> 
    867867<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> 
    869869<p>&nbsp;The scattering length density profile for the default sld values (w/ 4 shells).</p> 
    870870<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="VesicleModel"></a><b><span style="font-size: 14pt;">VesicleModel</span></b></p> 
    873873<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> 
    874874<p>The 1D scattering intensity is calculated in the following way (Guinier, 1955):</p> 
     
    965965</div> 
    966966<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> 
    968968<p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p> 
    969969<p>REFERENCE</p> 
    970970<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="MultiShellModel"></a><b><span style="font-size: 14pt;">MultiShellModel</span></b></p> 
    972972<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> 
    973973<p style="text-align: center;" align="center"><img id="Picture 32" src="img/image020.jpg" alt="" width="423" height="371" /></p> 
     
    10821082</div> 
    10831083<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> 
    10851085<p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p> 
    10861086<p>REFERENCE</p> 
    10871087<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="OnionExpShellModel"></a><b><span style="font-size: 14pt;">OnionExpShellModel</span></b></p> 
    10891089<p>&nbsp;&nbsp;</p> 
    10901090<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> 
     
    11031103<p>For |A|&gt;0,</p> 
    11041104<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>&rsquo;</strong></em>(<em>r</em> - <em>rshelli-1</em>) /<em><span style="font-family: Symbol;">D</span>tshelli</em>) + <em>B<strong>&rsquo;</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>&rsquo;</b></em>(<em>r</em> - <em>rshelli-1</em>) /<em><span style="font-family: Symbol;">D</span>tshelli</em>) + <em>B<b>&rsquo;</b></em>), so this case it is equivalent to</p> 
    11061106<p><span style="font-size: 12pt; font-family: 'Times New Roman','serif';"><img src="img/image030.gif" alt="" width="531" height="45" /></span></p> 
    11071107<p><span style="font-size: 12pt; font-family: 'Times New Roman','serif';"><img src="img/image031.gif" alt="" width="483" height="45" /></span></p> 
     
    12431243</div> 
    12441244<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> 
    12461246<p style="text-align: center;" align="center">&nbsp;<img src="img/image042.jpg" alt="" width="497" height="372" /></p> 
    1247 <p style="text-align: center;" align="center">&nbsp;<strong>Figure. SLD profile from the default values.</strong></p> 
     1247<p style="text-align: center;" align="center">&nbsp;<b>Figure. SLD profile from the default values.</b></p> 
    12481248<p>REFERENCE</p> 
    12491249<p>L.A.Feigin and D.I.Svergun, &lsquo;Structure Analysis by Small-Angle X-Ray and Neutron Scattering&rsquo;, 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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="SphericalSLDModel"></a><b><span style="font-size: 14pt;">SphericalSLDModel</span></b></p> 
    12511251<p>&nbsp;&nbsp;</p> 
    12521252<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 &lsquo;npts_inter#&rsquo; in GUI. The form factor is normalized by the total volume of the sphere.</p> 
     
    14541454</div> 
    14551455<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> 
    14571457<p style="text-align: center;" align="center">&nbsp;<img src="img/image058.jpg" alt="" width="536" height="344" /></p> 
    1458 <p style="text-align: center;" align="center">&nbsp;<strong>Figure. SLD profile from the default values.</strong></p> 
     1458<p style="text-align: center;" align="center">&nbsp;<b>Figure. SLD profile from the default values.</b></p> 
    14591459<p>REFERENCE</p> 
    14601460<p>L.A.Feigin and D.I.Svergun, &lsquo;Structure Analysis by Small-Angle X-Ray and Neutron Scattering&rsquo;, 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> 
    14621462<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> 
    14631463<p>&nbsp;</p> 
    14641464<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;">&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p> 
    14661466<p><a name="LinearPearlsModel"></a>&nbsp;</p> 
    14671467<p><a name="LinearPearlsModel"></a>The output of the scattering intensity function for the linearpearls model is given by (Dobrynin, 1996):</p> 
     
    15681568<p><a name="LinearPearlsModel"></a>REFERENCE</p> 
    15691569<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b><a name="PearlNecklaceModel"></a><span style="font-size: 14pt;">PearlNecklaceModel</span></b></p> 
    15711571<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> 
    15721572<p>&nbsp;</p> 
    15731573<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;">&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p> 
    15751575<p><a name="PearlNecklaceModel"></a>The output of the scattering intensity function for the pearlnecklace model is given by (Schweins, 2004):</p> 
    15761576<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>&nbsp;</p> 
     
    17091709<p><a name="PearlNecklaceModel"></a>R. Schweins and K. Huber, &lsquo;Particle Scattering Factor of Pearl Necklace Chains&rsquo;, Macromol. Symp., 211, 25-42, 2004.</p> 
    17101710<p><a name="PearlNecklaceModel"></a>&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="CylinderModel"></a><b><span style="font-size: 14pt;">Cylinder Model (Magnetic 2D Model)</span></b></p> 
    17121712<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> 
    17131713For 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;">&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p> 
    17151715<p>The output of the 2D scattering intensity function for oriented cylinders is given by (Guinier, 1955):</p> 
    17161716<p style="text-align: center;" align="center"><span style="position: relative; top: 12pt;"><img src="img/image059.PNG" alt="" /></span>&nbsp;</p> 
     
    18241824<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>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</p> 
    18251825<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;">&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp; </span>Validation of the cylinder model</b></p> 
    18271827<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> 
    18281828<p>In general, averaging over a distribution of orientations is done by evaluating the following:</p> 
     
    18371837<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 &Aring;, Length=400 &Aring;, Contrast=3e-6 &Aring; -2, and Background=0.0 cm -1.</p> 
    18381838<p>&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="CoreShellCylinderModel"></a><b><span style="font-size: 14pt;">Core-Shell Cylinder Model</span></b></p> 
    18401840<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;">&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p> 
    18421842<p>The output of the 2D scattering intensity function for oriented core-shell cylinders is given by (Kline, 2006):</p> 
    18431843<p style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="img/image067.PNG" alt="" /></span>&nbsp;</p> 
     
    19771977<p>The output of the 1D scattering intensity function for randomly oriented cylinders is then given by the equation above.</p> 
    19781978<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;">&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp; </span>Validation of the core-shell cylinder model</b></p> 
    19801980<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> 
    19811981<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(&theta;,</em><em><span style="font-family: 'Arial','sans-serif';">&phi;</span>)</em> = 1.0.&nbsp; Figure 9 shows the result of such a cross-check.</p> 
     
    19931993<p style="text-align: center;" align="center">Figure. Examples of the angles for oriented pp against the detector plane.</p> 
    19941994 
    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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="CoreShellBicelleModel"></a><b><span style="font-size: 14pt;">Core-Shell (Cylinder) Bicelle Model</span></b></p> 
    19961996<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&nbsp;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.&nbsp;</p> 
    19971997<p>&nbsp;</p> 
     
    21382138<p>&nbsp;</p> 
    21392139<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> 
    21412141 
    21422142 
     
    21492149 
    21502150<p>&nbsp;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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="HollowCylinderModel"></a><b><span style="font-size: 14pt;">HollowCylinderModel</span></b></p> 
    21522152<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> 
    21532153<p>P(q) = scale*&lt;f^2&gt;/Vshell+background where the averaging &lt; &gt; id applied only for the 1D calculation. &nbsp;The inside and outside of the hollow cylinder have the same SLD.</p> 
     
    22562256</div> 
    22572257<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> 
    22592259<p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p> 
    22602260 
     
    22672267<p>REFERENCE</p> 
    22682268<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="FlexibleCylinderModel"></a><b><span style="font-size: 14pt;">FlexibleCylinderModel</span></b></p> 
    22702270<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*&lt;f^2&gt;/V+background where the averaging &lt; &gt;&nbsp; is applied over all orientation for 1D. &nbsp;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> 
    22712271<p style="text-align: center;" align="center"><img id="Picture 35" src="img/image075.jpg" alt="" width="411" height="187" /></p> 
     
    23682368</div> 
    23692369<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> 
    23712371 
    23722372<p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 
     
    23762376<p>Correction of the formula can be found in:</p> 
    23772377<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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: &nbsp;</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*&lt;f^2&gt;/Vol + bkg, where &lt; &gt; 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;">&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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: &nbsp;</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*&lt;f^2&gt;/Vol + bkg, where &lt; &gt; 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;">&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p> 
    23812381<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.&nbsp; See equations (13, 26-27) in the original reference for the details.</p> 
    23822382<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> 
     
    25012501</div> 
    25022502<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="StackedDisksModel"></a><b><span style="font-size: 14pt;">StackedDisksModel </span></b></p> 
    25052505<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. &nbsp;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> 
    25062506<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> 
     
    26442644</div> 
    26452645<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> 
    26472647<p style="text-align: center;" align="center"><img id="Picture 101" src="img/image086.jpg"  /></p> 
    26482648<p style="text-align: center;" align="center">Figure. Examples of the angles for oriented stackeddisks against the detector plane.</p> 
     
    26572657<p>Kratky, O. and Porod, G., J. Colloid Science, 4, 35, 1949.</p> 
    26582658<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="ParallelepipedModel"></a><b><span style="font-size: 14pt;">ParallelepipedModel (Magnetic 2D Model) </span></b></p> 
    26602660<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*&lt;f^2&gt;/V+background where the volume V= ABC and the averaging &lt; &gt;&nbsp; is applied over all orientation for 1D. &nbsp;</p> 
    26612661For magnetic scattering, please see the '<a href="polar_mag_help.html">Polarization/Magnetic Scattering</a>' in Fitting Help. 
     
    26722672<p>To provide easy access to the orientation of the parallelepiped, we define the axis of the cylinder using two angles &theta; , <span style="font-family: 'Arial','sans-serif';">&phi; </span>and<span style="font-family: Symbol;">Y</span>. Similarly to the case of the cylinder, those angles, &theta; &nbsp;and <span style="font-family: 'Arial','sans-serif';">&phi;,</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> 
    26732673<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> 
    26752675<p style="text-align: center;" align="center"><img src="img/image091.jpg" alt="" width="379" height="256" /></p> 
    26762676<p style="text-align: center;" align="center">Figure. Examples of the angles for oriented pp against the detector plane.</p> 
     
    27572757</div> 
    27582758<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;">&middot;</span><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&middot;</span><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><b>Validation of the parallelepiped 2D model</b></p> 
    27612761<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> 
    27622762<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> 
    27642764<p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 
    27652765<p>REFERENCE</p> 
    27662766<p>Mittelbach and Porod, Acta Physica Austriaca 14 (1961) 185-211.</p> 
    27672767<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b><span style="font-size: 14pt;"><a name="CSParallelepipedModel"></a>CSParallelepipedModel</span></b></p> 
    27692769<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.&nbsp; The form factor is normalized by the particle volume such that P(q) = scale*&lt;f^2&gt;/Vol + bkg, where &lt; &gt; is an average over all possible orientations of the rectangular solid. An instrument resolution smeared version is also provided.</p> 
    27702770<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 &lt; B &lt; C.&nbsp;</p> 
     
    27852785<p>To provide easy access to the orientation of the CSparallelepiped, we define the axis of the cylinder using two angles &theta; , <span style="font-family: 'Arial','sans-serif';">&phi; </span>and<span style="font-family: Symbol;">Y</span>. Similarly to the case of the cylinder, those angles, &theta; &nbsp;and <span style="font-family: 'Arial','sans-serif';">&phi;,</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> 
    27862786<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> 
    27882788<p style="text-align: center;" align="center"><img id="Picture 103" src="img/image091.jpg" alt="" width="379" height="256" /></p> 
    27892789<p style="text-align: center;" align="center">Figure. Examples of the angles for oriented cspp against the detector plane.</p> 
     
    29492949</div> 
    29502950<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>&nbsp;</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>&nbsp;</b></p> 
    29532953<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> 
    29552955 
    29562956<p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 
     
    29582958<p>see: Mittelbach and Porod, Acta Physica Austriaca 14 (1961) 185-211.</p> 
    29592959<p>Equations (1), (13-14). (yes, it's in German)&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="EllipticalCylinderModel"></a><b><span style="font-size: 14pt;">Elliptical Cylinder Model</span></b></p> 
    29612961<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> 
    29632963<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> 
    29642964<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> 
    29662966<p>The function calculated is:</p> 
    29672967<p style="text-align: center;" align="center"><span style="position: relative; top: 16pt;"><img src="img/image099.PNG" alt="" /></span>&nbsp;</p> 
     
    29712971<p>&nbsp;</p> 
    29722972<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> 
    29742974<p>The form factor is averaged over all possible orientation before normalized by the particle volume: P(q) = scale*&lt;f^2&gt;/V .</p> 
    29752975<p>The returned value is scaled to units of [cm-1].</p> 
     
    29772977<p>All angle parameters are valid and given only for 2D calculation (Oriented system).</p> 
    29782978<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> 
    29802980<p style="text-align: center;" align="center"><img id="Picture 114" src="img/image062.jpg" alt="" width="379" height="256" /></p> 
    29812981<p style="text-align: center;" align="center"><span style="font-size: 12pt;">Figure. Examples of the angles for oriented elliptical cylinders </span></p> 
    29822982<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))&nbsp; 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))&nbsp; and length values, and used as the effective radius toward S(Q) when P(Q)*S(Q) is applied.</p> 
    29842984<div align="center"> 
    29852985<table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0"> 
     
    30753075</div> 
    30763076<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;">&middot;</span><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&middot;</span><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><b>Validation of the elliptical cylinder 2D model</b></p> 
    30793079<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> 
    30803080<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>&nbsp;</strong></p> 
     3081<p style="text-align: center;" align="center"><b>Figure. Comparison between 1D and averaged 2D.</b></p> 
     3082<p><b>&nbsp;</b></p> 
    30833083<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> 
    30843084<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> 
    30873087<p>REFERENCE</p> 
    30883088<p style="text-indent: 0.25in;">L. A. Feigin and D. I. Svergun &ldquo;Structure Analysis by Small-Angle X-Ray and Neutron Scattering&rdquo;, 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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="BarBellModel"></a><b><span style="font-size: 14pt;">BarBell(/DumBell)Model</span></b></p> 
    30903090<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.&nbsp;</p> 
    3091 <p style="margin-left: 0.85in; text-indent: -0.35in;"><strong>1.1.</strong><strong><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p> 
    30923092<p>The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> 
    30933093<p>The barbell geometry is defined as:</p> 
     
    32053205</div> 
    32063206<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> 
    32083208<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> 
    32093209<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> 
    32113211 
    32123212<p style="text-align: center; page-break-after: avoid;" align="center"><img src="img/image061.jpg" /></p> 
     
    32183218 
    32193219 
    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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="CappedCylinderModel"></a><b><span style="font-size: 14pt;">CappedCylinder(/ConvexLens)Model</span></b></p> 
    32213221<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. &nbsp;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> 
    32223222<p style="margin-left: 0.85in; text-indent: -0.35in;">&nbsp;</p> 
    3223 <p style="margin-left: 0.85in; text-indent: -0.35in;"><strong>1.1.</strong><strong><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p> 
    32243224<p>The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> 
    32253225<p>The Capped Cylinder geometry is defined as:</p> 
     
    33363336</div> 
    33373337<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> 
    33393339<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> 
    33403340<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> 
    33423342<p style="text-align: center; page-break-after: avoid;" align="center"><img src="img/image061.jpg" /></p> 
    33433343<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> 
     
    33473347 
    33483348 
    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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="EllipsoidModel"></a><b><span style="font-size: 14pt;">Ellipsoid Model</span></b></p> 
    33503350<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;">&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp; </span>Definition</b></p> 
    33523352<p>The output of the 2D scattering intensity function for oriented ellipsoids is given by (Feigin, 1987):</p> 
    33533353<p style="text-align: center;" align="center"><span style="position: relative; top: 12pt;"><img src="img/image059.PNG" alt="" /></span>&nbsp;&nbsp;</p> 
     
    34693469<p style="text-align: center;" align="center"><span style="font-size: 12pt;">Figure. The angles for oriented ellipsoid </span></p> 
    34703470 
    3471 <p style="margin-left: 0.85in; text-indent: -0.35in;"><strong>2.1.</strong><strong><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp; </span>Validation of the ellipsoid model</b></p> 
    34723472<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> 
    34733473<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(&theta;,</em><em><span style="font-family: 'Arial','sans-serif';">&phi;</span>)</em> = 1.0.&nbsp; 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;">&nbsp;</span></em></strong></p> 
     3474<p style="text-align: center;" align="center"><b><em><span style="color: #3366ff;">&nbsp;</span></em></b></p> 
    34753475<p>The discrepancy above q=0.3 &Aring; -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';">&alpha;</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> 
    34773477<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 &Aring;, Radius_b=400 &Aring;,</p> 
    34783478<p>Contrast=3e-6 &Aring; -2, and Background=0.01 cm -1.</p> 
     
    34813481<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 &Aring;, Radius_b=400 &Aring;, Contrast=3e-6 &Aring; -2, and Background=0.0 cm -1.</p> 
    34823482<p>&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="CoreShellEllipsoidModel"></a><b><span style="font-size: 14pt;">CoreShellEllipsoidModel </span></b></p> 
    34843484<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*&lt;f^2&gt;/V+background where the volume V= 4pi/3*rmaj*rmin2 and the averaging &lt; &gt;&nbsp; is applied over all orientation for 1D. &nbsp;</p> 
    34853485<p style="text-align: center;" align="center">&nbsp;&nbsp;<img id="Picture 41" src="img/image125.gif" alt="" width="335" height="179" /></p> 
     
    36083608</div> 
    36093609<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>&nbsp;</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>&nbsp;</b></p> 
    36123612<p style="text-align: center;" align="center"><img src="img/image122.jpg" alt="" width="379" height="256"/></p> 
    36133613<p style="text-align: center;" align="center">Figure. The angles for oriented coreshellellipsoid .</p> 
     
    36163616<p>Kotlarchyk, M.; Chen, S.-H. J. Chem. Phys., 1983, 79, 2461.</p> 
    36173617<p>Berr, S.&nbsp; 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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="TriaxialEllipsoidModel"></a><b><span style="font-size: 14pt;">TriaxialEllipsoidModel</span></b></p> 
    36193619<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.,&nbsp; Ra =&lt; Rb =&lt; Rc (Note that users should maintains this inequality for the all calculations).&nbsp; P(q) = scale*&lt;f^2&gt;/V+background where the volume V= 4pi/3*Ra*Rb*Rc, and the averaging &lt; &gt;&nbsp; is applied over all orientation for 1D. &nbsp;</p> 
    36203620<p style="text-align: center;" align="center">&nbsp;&nbsp;<img id="Picture 42" src="img/image128.jpg" alt="" width="376" height="226" /></p> 
     
    37213721</div> 
    37223722<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;">&middot;</span><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&middot;</span><span style="font-size: 7pt;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><b>Validation of the triaxialellipsoid 2D model</b></p> 
    37253725<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> 
    37263726<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> 
    37283728<p style="text-align: center;" align="center"><img src="img/image132.jpg" alt="" width="379" height="256" /></p> 
    37293729<p style="text-align: center;" align="center">Figure. The angles for oriented ellipsoid.</p> 
     
    37313731<p>REFERENCE</p> 
    37323732<p>L. A. Feigin and D. I. Svergun &ldquo;Structure Analysis by Small-Angle X-Ray and Neutron Scattering&rdquo;, 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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="LamellarModel"></a><b><span style="font-size: 14pt;">LamellarModel</span></b></p> 
    37343734<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. &nbsp;The ploydispersion in the bilayer thickness can be applied from the GUI.</p> 
    37353735<p>The scattering intensity I(q) is:</p> 
     
    38133813</div> 
    38143814<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> 
    38163816<p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 
    38173817<p>REFERENCE</p> 
    38183818<p>Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993) 487-502.</p> 
    38193819<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="LamellarFFHGModel"></a><b><span style="font-size: 14pt;">LamellarFFHGModel</span></b></p> 
    38213821<p>This model provides the scattering intensity, I(<em>q</em>), for a lyotropic lamellar phase where a random distribution in solution are assumed. &nbsp;The SLD of the head region is taken to be different from the SLD of the tail region.</p> 
    38223822<p>The scattering intensity I(q) is:</p> 
     
    39223922</div> 
    39233923<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> 
    39253925<p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 
    39263926<p>REFERENCE</p> 
    39273927<p>Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993) 487-502.</p> 
    39283928<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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> 
    39313931<p>The scattering intensity I(q) is:</p> 
    39323932<p style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="img/image139.PNG" alt="" /></span></p> 
     
    40354035</div> 
    40364036<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> 
    40384038<p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 
    40394039<p>REFERENCE</p> 
    40404040<p>Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993) 487-502.</p> 
    40414041<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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. &nbsp;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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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. &nbsp;The SLD of the head region is taken to be different from the SLD of the tail region.</p> 
    40444044<p>The scattering intensity I(q) is:</p> 
    40454045<p style="text-align: center;" align="center"><span style="position: relative; top: 15pt;"><img src="img/image139.PNG" alt="" /></span></p> 
     
    41824182</div> 
    41834183<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> 
    41854185<p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 
    41864186<p>REFERENCE</p> 
    41874187<p>Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993) 487-502.</p> 
    41884188<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="LamellarPCrystalModel"></a><b><span style="font-size: 14pt;">LamellarPCrystalModel</span></b></p> 
    41904190<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> 
    41914191<p>The scattering intensity I(q) is calculated as:</p> 
     
    42984298</div> 
    42994299<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> 
    43014301<p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p> 
    43024302<p>See the reference for details.</p> 
    43034303<p>REFERENCE</p> 
    43044304<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="SCCrystalModel"></a><b><span style="font-size: 14pt;">SC(Simple Cubic Para-)CrystalModel</span></b></p> 
    43064306<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> 
    43074307<p>The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> 
     
    44184418<p>This example dataset is produced using 200 data points, qmin = 0.01 &Aring;-1,&nbsp; qmax = 0.1 &Aring;-1 and the above default values.</p> 
    44194419<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> 
    44214421<p>&nbsp;The 2D (Anisotropic model) is based on the reference (above) which I(q) &nbsp;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> 
    44224422<p style="text-align: center;" align="center"><img id="Object 23" src="img/image156.jpg" /></p> 
     
    44254425<p>&nbsp;</p> 
    44264426<p>&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="FCCrystalModel"></a><b><span style="font-size: 14pt;">FC(Face Centered Cubic Para-)CrystalModel</span></b></p> 
    44304430<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.&nbsp;</p> 
    44314431<p>The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> 
     
    45434543<p>This example dataset is produced using 200 data points, qmin = 0.01 &Aring;-1,&nbsp; qmax = 0.1 &Aring;-1 and the above default values.</p> 
    45444544<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> 
    45464546<p>&nbsp;The 2D (Anisotropic model) is based on the reference (above) in which I(q)&nbsp; 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> 
    45474547<p style="text-align: center;" align="center"><img src="img/image165.gif" /></p> 
     
    45514551<p>&nbsp;</p> 
    45524552<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="BCCrystalModel"></a><b><span style="font-size: 14pt;">BC(Body Centered Cubic Para-)CrystalModel</span></b></p> 
    45554555<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> 
    45564556<p>The scattering intensity I(q) is calculated as:</p> 
     
    46674667<p>This example dataset is produced using 200 data points, qmin = 0.001 &Aring;-1,&nbsp; qmax = 0.1 &Aring;-1 and the above default values.</p> 
    46684668<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> 
    46704670<p>&nbsp;The 2D (Anisotropic model) is based on the reference (1987) in which I(q) &nbsp;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> 
    46714671<p style="text-align: center;" align="center"><img id="Object 31" src="img/image165.gif" /></p> 
     
    46754675<p>&nbsp;</p> 
    46764676<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;">&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp; </span></b><a name="Shape-Independent"></a><b><span style="font-size: 16pt;">Shape-Independent Models </span></b></p> 
    46794679<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><strong><span style="font-size: 14pt;">&nbsp;<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b><span style="font-size: 14pt;">&nbsp;<a name="Debye"></a>Debye (Model)</span></b></p> 
    46814681<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> 
    46824682<p style="text-align: center;" align="center"><span style="font-size: 14pt; position: relative; top: 27pt;"><img src="img/image172.PNG" alt="" /></span></p> 
     
    47364736</div> 
    47374737<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> 
    47394739<p style="margin-left: 0.25in; text-align: center;" align="center">&nbsp;</p> 
    47404740<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><strong><span style="font-size: 14pt;">&nbsp;<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b><span style="font-size: 14pt;">&nbsp;<a name="BroadPeakModel"></a>BroadPeak Model</span></b></p> 
    47424742<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> 
    47434743<p style="margin-left: 0.25in;">The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> 
     
    48364836</div> 
    48374837<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> 
    48394839<p style="margin-left: 0.25in; text-align: center;" align="center">&nbsp;</p> 
    48404840<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><strong><span style="font-size: 14pt;">&nbsp;<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b><span style="font-size: 14pt;">&nbsp;<a name="CorrLength"></a>CorrLength (CorrelationLengthModel)</span></b></p> 
    48424842<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> 
    48434843<p>The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> 
     
    49244924</div> 
    49254925<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> 
    49274927<p style="margin-left: 0.25in; text-align: center;" align="center">&nbsp;</p> 
    49284928<p style="margin-left: 0.25in;">REFERENCE</p> 
    49294929<p style="margin-left: 0.25in;">B. Hammouda, D.L. Ho and S.R. Kline, &ldquo;Insight into Clustering in Poly(ethylene oxide) Solutions&rdquo;, 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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><strong><span style="font-size: 14pt;">&nbsp;<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b><span style="font-size: 14pt;">&nbsp;<a name="Lorentz"></a>(Ornstein-Zernicke) Lorentz (Model)</span></b></p> 
    49314931<p style="text-indent: 0.25in;">The Ornstein-Zernicke model is defined by:</p> 
    49324932<p><span style="font-size: 14pt;">&nbsp;</span></p> 
     
    49884988</table> 
    49894989</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;">&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><strong><span style="font-size: 14pt;">&nbsp;<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;">&nbsp;</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;">&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b><span style="font-size: 14pt;">&nbsp;<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;">&nbsp;</span></b></p> 
    49944994<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 &gt;&gt; 1).</p> 
    49954995<p style="text-indent: 0.25in;">&nbsp;</p> 
     
    50495049</table> 
    50505050</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;">&nbsp;</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;">&nbsp;</span>Figure. 1D plot using the default values (w/200 data point).</b></p> 
    50535053<p style="margin-left: 0.25in;">References:</p> 
    50545054<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> 
    50555055<p style="margin-left: 0.5in;">&nbsp;</p> 
    50565056<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><strong><span style="font-size: 14pt;">&nbsp; <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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b><span style="font-size: 14pt;">&nbsp; <a name="Absolute Power_Law"></a>Absolute Power_Law </span></b></p> 
    50585058<p style="margin-left: 0.25in;">This model describes a power law with background.</p> 
    50595059<p style="text-align: center;" align="center"><span style="font-size: 14pt; position: relative; top: 5pt;"><img src="img/image182.PNG" alt="" /></span></p> 
     
    51125112</div> 
    51135113<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><strong><span style="font-size: 14pt;">&nbsp;<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b><span style="font-size: 14pt;">&nbsp;<a name="TeubnerStreyModel"></a>Teubner Strey (Model)</span></b></p> 
    51165116<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;">&nbsp;</span></strong></p> 
     5117<p style="margin-left: 0.25in;"><b><span style="font-size: 14pt;">&nbsp;</span></b></p> 
    51185118<p style="text-align: center;" align="center"><span style="font-size: 14pt; position: relative; top: 15pt;"><img src="img/image184.PNG" alt="" /></span></p> 
    51195119<p style="text-align: center;" align="center"><span style="font-size: 14pt;">&nbsp;</span></p> 
     
    51835183</div> 
    51845184<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> 
    51865186<p style="margin-left: 0.25in;">References:</p> 
    51875187<p style="margin-left: 0.5in;">Teubner, M; Strey, R. J. Chem. Phys., 87, 3195 (1987).</p> 
    51885188<p style="margin-left: 0.5in;">&nbsp;</p> 
    51895189<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><strong><span style="font-size: 14pt;">&nbsp;<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b><span style="font-size: 14pt;">&nbsp;<a name="Number_Density_Fractal"></a> <a name="FractalModel"></a>FractalModel</span></b></p> 
    51915191<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> 
    51925192<p>&nbsp;</p> 
     
    51945194<p>&nbsp;</p> 
    51955195<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, &xi; is the correlation length, <em>&rho;solvent</em> is the scattering length density of the solvent, and <em>&rho;block</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> 
    51975197<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> 
    51985198<p style="text-align: center;" align="center"><span style="font-size: 14pt;">&nbsp;</span></p> 
     
    52925292</div> 
    52935293<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> 
    52955295<p style="margin-left: 0.25in; text-align: center;" align="center">&nbsp;</p> 
    52965296<p style="margin-left: 0.25in; text-align: center;" align="center">&nbsp;</p> 
    52975297<p style="margin-left: 0.25in;">References:</p> 
    52985298<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;">&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b><span style="font-size: 14pt;"><a name="Mass_Fractal"></a><a name="MassFractalModel"></a>MassFractalModel</span></b></p> 
    53015301<p style="margin-left: 0.25in;">Calculates the scattering from fractal-like aggregates based on the Mildner reference (below).&nbsp;</p> 
    53025302<p style="text-align: center;" align="center"><span style="font-size: 14pt; position: relative; top: 85pt;"><br /> </span></p> 
     
    53805380</div> 
    53815381<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> 
    53835383<p style="margin-left: 0.25in; text-align: center;" align="center">&nbsp;</p> 
    53845384<p style="margin-left: 0.25in; text-align: center;" align="center">&nbsp;</p> 
     
    53875387<p>&nbsp;</p> 
    53885388<p>&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b> <b><span style="font-size: 14pt;"> <a name="Surface_Fractal"></a><a name="SurfaceFractalModel"></a>SurfaceFractalModel</span></b></p> 
    53905390<p style="margin-left: 0.25in;">Calculates the scattering&nbsp; based on the Mildner reference (below).&nbsp;</p> 
    53915391<p style="text-align: center;" align="center"><span style="font-size: 14pt; position: relative; top: 85pt;"><br /> </span></p> 
     
    54695469</div> 
    54705470<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> 
    54725472<p style="margin-left: 0.25in; text-align: center;" align="center">&nbsp;</p> 
    54735473<p style="margin-left: 0.25in; text-align: center;" align="center">&nbsp;</p> 
     
    54765476<p>&nbsp;</p> 
    54775477<p>&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b><span style="font-size: 14pt;"><a name="MassSurface_Fractal"></a><a name="MassSurfaceFractal"></a>MassSurfaceFractal</span></b></p> 
    54795479<p>&nbsp;&nbsp;&nbsp; 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 &lsquo;fume&rsquo; or pyrogenic silicas. These are all characterised by cluster mass distributions (sometimes also cluster size distributions) and internal surfaces that are fractal in nature. &nbsp; 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> 
    54805480<p style="margin-left: 0.55in;">The scattered intensity I(Q) is then calculated using a modified Ornstein-Zernicke equation:</p> 
     
    55585558</div> 
    55595559<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> 
    55615561<p style="margin-left: 0.25in; text-align: center;" align="center">&nbsp;</p> 
    55625562<p style="margin-left: 0.25in; text-align: center;" align="center">&nbsp;</p> 
     
    55665566<p>&nbsp;</p> 
    55675567<p>&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><strong><span style="font-size: 14pt;">&nbsp; <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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b><span style="font-size: 14pt;">&nbsp; <a name="FractalCoreShell"></a>FractalCoreShell(Model)</span></b></p> 
    55695569<p style="margin-left: 0.25in;">Calculates the scattering from a fractal structure with a primary building block of core-shell spheres.</p> 
    55705570<p><img src="img/fractcore_eq1.gif"/></p> 
     
    56955695</div> 
    56965696<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> 
    56985698<p style="margin-left: 0.25in; text-align: center;" align="center">&nbsp;</p> 
    56995699<p style="margin-left: 0.25in; text-align: center;" align="center">&nbsp;</p> 
    57005700<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;">&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><strong><span style="font-size: 14pt;">&nbsp; <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;">&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b><span style="font-size: 14pt;">&nbsp; <a name="GaussLorentzGel"></a>GaussLorentzGel(Model)</span></b></p> 
    57035703<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> 
    57045704<p style="margin-left: 0.25in;">The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> 
     
    57795779</div> 
    57805780<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> 
    57825782<p style="margin-left: 0.25in; text-align: center;" align="center">&nbsp;</p> 
    57835783<p style="margin-left: 0.25in; text-align: center;" align="center">&nbsp;</p> 
    57845784<p style="text-indent: 0.5in;">REFERENCE:</p> 
    57855785<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><strong><span style="font-size: 14pt;">&nbsp;<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b><span style="font-size: 14pt;">&nbsp;<a name="BEPolyelectrolyte"></a> BEPolyelectrolyte Model</span></b></p> 
    57875787<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> 
    57885788<p>&nbsp;</p> 
     
    59025902<p style="margin-left: 0.5in;">3, 573 (1993).</p> 
    59035903<p style="margin-left: 0.5in;">Rapha&euml;l, E., Joanny, J.-F., Europhysics Letters 11, 179 (1990).</p> 
    5904 <p style="margin-left: 0.25in;"><strong><span style="font-size: 14pt;">&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><strong><span style="font-size: 14pt;">&nbsp;<a name="Guinier"></a>Guinier (Model)</span></strong></p> 
     5904<p style="margin-left: 0.25in;"><b><span style="font-size: 14pt;">&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b><span style="font-size: 14pt;">&nbsp;<a name="Guinier"></a>Guinier (Model)</span></b></p> 
    59065906<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> 
    59075907<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> 
    59085908<p style="margin-left: 0.25in; text-align: center;" align="center"><span style="font-size: 14pt;">&nbsp;</span></p> 
    59095909<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;">&nbsp;</span></strong></p> 
     5910<p style="margin-left: 0.25in; text-align: center;" align="center"><b><span style="font-size: 14pt;">&nbsp;</span></b></p> 
    59115911<div align="center"> 
    59125912<table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0"> 
     
    59485948</table> 
    59495949</div> 
    5950 <p style="margin-left: 0.25in;"><strong><span style="font-size: 14pt;">&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><strong><span style="font-size: 14pt;">&nbsp;<a name="GuinierPorod"></a>GuinierPorod (Model)</span></strong></p> 
     5950<p style="margin-left: 0.25in;"><b><span style="font-size: 14pt;">&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b><span style="font-size: 14pt;">&nbsp;<a name="GuinierPorod"></a>GuinierPorod (Model)</span></b></p> 
    59525952<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> 
    59535953<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> 
     
    59695969<p style="margin-left: 0.25in;">[2] Glatter, O.; Kratky, O., &ldquo;Small-Angle X-Ray Scattering&rdquo;, Academic Press (1982). Check out Chapter 4 on Data Treatment, pages 155-156.&nbsp;</p> 
    59705970<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;">&nbsp;</span></strong></p> 
     5971<p style="margin-left: 0.25in; text-align: center;" align="center"><b><span style="font-size: 14pt;">&nbsp;</span></b></p> 
    59725972<div align="center"> 
    59735973<table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0"> 
     
    60366036</table> 
    60376037</div> 
    6038 <p style="margin-left: 0.25in;"><strong><span style="font-size: 14pt;">&nbsp;</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>&nbsp;</strong></p> 
    6042 <p style="margin-left: 0.25in;"><strong><span style="font-size: 14pt;">&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><strong><span style="font-size: 14pt;">&nbsp;<a name="PorodModel"></a>PorodModel</span></strong></p> 
     6038<p style="margin-left: 0.25in;"><b><span style="font-size: 14pt;">&nbsp;</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>&nbsp;</b></p> 
     6042<p style="margin-left: 0.25in;"><b><span style="font-size: 14pt;">&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b><span style="font-size: 14pt;">&nbsp;<a name="PorodModel"></a>PorodModel</span></b></p> 
    60446044<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> 
    60456045<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> 
     
    60486048<p style="margin-left: 0.25in;">The background term is added for data analysis.</p> 
    60496049<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;">&nbsp;</span></strong></p> 
     6050<p style="margin-left: 0.25in; text-align: center;" align="center"><b><span style="font-size: 14pt;">&nbsp;</span></b></p> 
    60516051<div align="center"> 
    60526052<table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0"> 
     
    60886088</table> 
    60896089</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><strong><span style="font-size: 14pt;">&nbsp;<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b><span style="font-size: 14pt;">&nbsp;<a name="Peak Gauss Model"></a>PeakGaussModel</span></b></p> 
    60916091<p style="margin-left: 0.25in;">Model describes a Gaussian shaped peak including a flat background,</p> 
    60926092<p style="margin-left: 0.25in; text-align: center;" align="center">&nbsp;</p> 
     
    60976097<p style="margin-left: 0.25in;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; REFERENCE: None</p> 
    60986098<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;">&nbsp;</span></strong></p> 
     6099<p style="margin-left: 0.25in; text-align: center;" align="center"><b><span style="font-size: 14pt;">&nbsp;</span></b></p> 
    61006100<div align="center"> 
    61016101<table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0"> 
     
    61556155</table> 
    61566156</div> 
    6157 <p style="margin-left: 0.25in;"><strong><span style="font-size: 14pt;">&nbsp;</span></strong></p> 
    6158 <p style="margin-left: 0.25in;"><strong><span style="font-size: 14pt;">&nbsp;</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>&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><strong><span style="font-size: 14pt;">&nbsp;<a name="Peak Lorentz Model"></a>PeakLorentzModel</span></strong></p> 
     6157<p style="margin-left: 0.25in;"><b><span style="font-size: 14pt;">&nbsp;</span></b></p> 
     6158<p style="margin-left: 0.25in;"><b><span style="font-size: 14pt;">&nbsp;</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>&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b><span style="font-size: 14pt;">&nbsp;<a name="Peak Lorentz Model"></a>PeakLorentzModel</span></b></p> 
    61636163<p style="margin-left: 0.25in;">Model describes a Lorentzian shaped peak including a flat background,</p> 
    61646164<p>&nbsp;</p> 
     
    61696169<p style="margin-left: 0.25in;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; REFERENCE: None</p> 
    61706170<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;">&nbsp;</span></strong></p> 
     6171<p style="margin-left: 0.25in; text-align: center;" align="center"><b><span style="font-size: 14pt;">&nbsp;</span></b></p> 
    61726172<div align="center"> 
    61736173<table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0"> 
     
    62276227</table> 
    62286228</div> 
    6229 <p style="margin-left: 0.25in;"><strong><span style="font-size: 14pt;">&nbsp;</span></strong></p> 
    6230 <p style="margin-left: 0.25in;"><strong><span style="font-size: 14pt;">&nbsp;</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.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <a name="Poly_GaussCoil"></a>Poly_GaussCoil (Model)</span></strong></p> 
     6229<p style="margin-left: 0.25in;"><b><span style="font-size: 14pt;">&nbsp;</span></b></p> 
     6230<p style="margin-left: 0.25in;"><b><span style="font-size: 14pt;">&nbsp;</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.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <a name="Poly_GaussCoil"></a>Poly_GaussCoil (Model)</span></b></p> 
    62346234<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.&nbsp;</p> 
    62356235<p style="margin-left: 0.25in;">The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> 
     
    63086308<p style="margin-left: 0.25in; text-align: center;" align="center">&nbsp;</p> 
    63096309<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> 
    63116311<p style="margin-left: 0.25in; text-align: center;" align="center">&nbsp;</p> 
    63126312<p style="margin-left: 0.25in;">Reference:</p> 
     
    63146314<p style="margin-left: 0.25in;">J.S. Higgins, and H.C. Benoit, &ldquo;Polymers and Neutron Scattering&rdquo;, Oxford Science</p> 
    63156315<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.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <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.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <a name="PolymerExclVolume"></a>PolymerExclVolume (Model)</span></b></p> 
    63176317<p style="margin-left: 0.25in;">Calculates the scattering from polymers with excluded volume effects.</p> 
    63186318<p style="margin-left: 0.25in;">The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> 
     
    64036403<p style="margin-left: 0.25in; text-align: center;" align="center">&nbsp;</p> 
    64046404<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> 
    64066406<p style="margin-left: 0.25in; text-align: center;" align="center">&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><strong><span style="font-size: 14pt;">&nbsp; <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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b><span style="font-size: 14pt;">&nbsp; <a name="RPA10Model"></a>RPA10Model</span></b></p> 
    64086408<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> 
    64096409<p style="margin-left: 0.25in;">Case 0: C/D Binary mixture of homopolymers</p> 
     
    64186418<p style="margin-left: 0.25in;">Case 9: A-B-C-D Four-block copolymer</p> 
    64196419<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>&nbsp;</strong></p> 
     6420<p style="margin-left: 0.25in;"><b>&nbsp;</b></p> 
    64216421<p style="margin-left: 0.25in;">Only one case can be used at any one time.&nbsp; Plotting a different case will overwrite the original parameter waves.</p> 
    64226422<p style="margin-left: 0.25in;">The returned value is scaled to units of [cm-1].</p> 
     
    64946494</table> 
    64956495</div> 
    6496 <p style="margin-left: 0.25in;"><strong><span style="font-size: 14pt;">&nbsp;</span></strong></p> 
    6497 <p style="margin-left: 0.25in; text-align: center;" align="center"><strong><span style="font-size: 14pt;">&nbsp;</span></strong></p> 
     6496<p style="margin-left: 0.25in;"><b><span style="font-size: 14pt;">&nbsp;</span></b></p> 
     6497<p style="margin-left: 0.25in; text-align: center;" align="center"><b><span style="font-size: 14pt;">&nbsp;</span></b></p> 
    64986498<p style="margin-left: 0.25in; text-align: center;" align="center">Fixed parameters for Case0 Model</p> 
    64996499<div align="center"> 
     
    65866586</table> 
    65876587</div> 
    6588 <p style="margin-left: 0.25in;"><strong><span style="font-size: 14pt;">&nbsp;</span></strong></p> 
    6589 <p style="margin-left: 0.25in;"><strong><span style="font-size: 14pt;">&nbsp;</span></strong></p> 
     6588<p style="margin-left: 0.25in;"><b><span style="font-size: 14pt;">&nbsp;</span></b></p> 
     6589<p style="margin-left: 0.25in;"><b><span style="font-size: 14pt;">&nbsp;</span></b></p> 
    65906590<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;">&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><strong><span style="font-size: 14pt;">&nbsp;<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;">&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b><span style="font-size: 14pt;">&nbsp;<a name="TwoLorentzian"></a>TwoLorentzian(Model)</span></b></p> 
    65946594<p style="margin-left: 0.25in;">Calculate an empirical functional form for SANS data characterized by a two Lorentzian functions.</p> 
    65956595<p style="margin-left: 0.25in;">The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> 
     
    66026602<p style="margin-left: 0.25in;">The background term is added for data analysis.</p> 
    66036603<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> 
    66056605<div align="center"> 
    66066606<table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0"> 
     
    66926692<p style="margin-left: 0.5in;">&nbsp;</p> 
    66936693<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> 
    66956695<p style="margin-left: 0.5in; text-align: center;" align="center">&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><strong><span style="font-size: 14pt;">&nbsp;<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b><span style="font-size: 14pt;">&nbsp;<a name="TwoPowerLaw"></a>TwoPowerLaw(Model)</span></b></p> 
    66986698<p style="margin-left: 0.25in;">Calculate an empirical functional form for SANS data characterized by two power laws.</p> 
    66996699<p style="margin-left: 0.25in;">The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> 
     
    67046704<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> 
    67056705<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> 
    67076707<div align="center"> 
    67086708<table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0"> 
     
    67746774<p style="margin-left: 0.5in;">&nbsp;</p> 
    67756775<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> 
    67776777<p style="margin-left: 0.5in; text-align: center;" align="center">&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><strong><span style="font-size: 14pt;">&nbsp;<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b><span style="font-size: 14pt;">&nbsp;<a name="UnifiedPowerRg"></a>UnifiedPower(Law and)Rg(Model)</span></b></p> 
    67796779<p style="margin-left: 0.25in;">The returned value is scaled to units of [cm-1sr-1], absolute scale.&nbsp;</p> 
    67806780<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> 
     
    67886788<p style="margin-left: 0.25in;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</p> 
    67896789<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> 
    67916791<div align="center"> 
    67926792<table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0"> 
     
    69136913<p style="margin-left: 0.5in;">&nbsp;</p> 
    69146914<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> 
    69166916<p style="margin-left: 0.5in; text-align: center;" align="center">&nbsp;</p> 
    69176917<p style="margin-left: 0.25in;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; REFERENCES</p> 
    69186918<p style="margin-left: 0.25in;">G. Beaucage (1995).&nbsp; J. Appl. Cryst., vol. 28, p717-728.</p> 
    69196919<p style="margin-left: 0.25in;">G. Beaucage (1996).&nbsp; 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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><strong><span style="font-size: 14pt;">&nbsp;<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><b><span style="font-size: 14pt;">&nbsp;<a name="LineModel"></a> LineModel</span></b></p> 
    69216921<p style="margin-left: 0.25in;">This is a linear function that calculates:</p> 
    69226922<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> 
    69236923<p style="margin-left: 0.25in; text-align: center;" align="center"><span style="font-size: 14pt;">&nbsp;</span></p> 
    69246924<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)&nbsp; 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)&nbsp; which is defined differently from other shape independent models.</p> 
    69266926<div align="center"> 
    69276927<table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0"> 
     
    69646964</div> 
    69656965<p style="margin-left: 0.55in; text-indent: -0.3in;">&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="ReflectivityModel"></a><b><span style="font-size: 14pt;">ReflectivityModel</span></b></p> 
    69676967<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 [ &frac12; of the interface(from the previous layer or substrate) + flat portion + &frac12; 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> 
    69686968<p style="margin-left: 0.55in; text-indent: -0.3in;">Note: This model was contributed by an interested user.</p> 
    69696969<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> 
    69716971<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="ReflectivityIIModel"></a><b><span style="font-size: 14pt;">ReflectivityIIModel</span></b></p> 
    69746974<p>&nbsp;&nbsp;&nbsp; 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. &nbsp;&nbsp;&nbsp; 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> 
    69756975<p style="margin-left: 0.55in;">1) Erf;</p> 
     
    69836983<p>&nbsp;</p> 
    69846984<p>&nbsp;&nbsp;&nbsp; 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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="GelFitModel"></a><b><span style="font-size: 14pt;">GelFitModel</span></b></p> 
    69866986<p>&nbsp;&nbsp;&nbsp; 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> 
    69876987<p style="margin-left: 0.55in;">The scattered intensity I(Q) is then calculated as:</p> 
     
    69936993<p>&nbsp;&nbsp;&nbsp; Note the first term reduces to the Ornstein-Zernicke equation when D=2; ie, when the Flory exponent is 0.5 (theta conditions). &nbsp; In gels with significant hydrogen bonding D has been reported to be ~2.6 to 2.8.</p> 
    69946994<p>&nbsp;&nbsp;&nbsp; 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> 
    69966996<div align="center"> 
    69976997<table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0"> 
     
    70787078<p style="margin-left: 0.5in;">&nbsp;</p> 
    70797079<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> 
    70817081<p style="margin-left: 0.5in; text-align: center;" align="center">&nbsp;</p> 
    70827082<p style="margin-left: 0.25in;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; REFERENCES</p> 
     
    70847084<p style="margin-left: 0.25in;">Simon Mallam, Ferenc Horkay, Anne-Marie Hecht, Adrian R. Rennie, Erik Geissler, Macromolecules 1991, 24, 543-548.</p> 
    70857085<p>&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span> </b> <b> <span style="font-size: 14pt;"><a name="StarPolymer"></a><a name="StarPolymerModel"></a>Star Polymer with Gaussian Statistics</span> </b></p> 
    70877087<p style="margin-left: 0.25in;">For a star with <em>f</em> arms:</p> 
    70887088<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> 
     
    70957095<p>&nbsp;</p> 
    70967096<p>&nbsp;</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;">&nbsp;&nbsp;&nbsp; </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;">&nbsp;</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;">&nbsp;&nbsp;&nbsp; </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;">&nbsp;</span></b></p> 
    70997099<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</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;">&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</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;">&nbsp;</span></b></p> 
    71027102<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</span></b><a name="testmodel_2"></a><b><span style="font-size: 14pt;">testmodel_2 </span></b></p> 
    71047104<p>This function, as an example of a user defined function, calculates the intensity =&nbsp; 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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="sum_p1_p2"></a><b><span style="font-size: 14pt;">sum_p1_p2 </span></b></p> 
    71067106<p>This function, as an example of a user defined function, calculates the intensity =&nbsp; 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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</span></b><a name="sum_Ap1_1_Ap2"></a><b><span style="font-size: 14pt;">sum_Ap1_1_Ap2 </span></b></p> 
    71087108<p>This function, as an example of a user defined function, calculates the intensity =&nbsp; (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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="polynomial5"></a><b><span style="font-size: 14pt;">polynomial5 </span></b></p> 
    71107110<p>This function, as an example of a user defined function, calculates the intensity =&nbsp; 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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</span></b><a name="sph_bessel_jn"></a><b><span style="font-size: 14pt;">sph_bessel_jn </span></b></p> 
    71127112<p>This function, as an example of a user defined function, calculates the intensity =&nbsp; 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;">&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp; </span></b><a name="Structure_Factors"></a><b><span style="font-size: 16pt;">Structure Factors</span></b></p> 
    71147114<p style="margin-left: 0.25in;">&nbsp;</p> 
    71157115<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="HardsphereStructure"></a><b><span style="font-size: 14pt;">HardSphere Structure </span></b></p> 
    71177117<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> 
    71187118<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> 
     
    71637163</div> 
    71647164<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> 
    71667166<p>References:</p> 
    71677167<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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><a name="SquareWellStructure"></a><strong><span style="font-size: 14pt;">&nbsp;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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="SquareWellStructure"></a><b><span style="font-size: 14pt;">&nbsp;SquareWell Structure </span></b></p> 
    71697169<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 &lt; 1.5 kT and volume fractions f &lt; 0.08.</p> 
    71707170<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> 
     
    72357235</div> 
    72367236<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> 
    72387238<p>References:</p> 
    72397239<p style="margin-left: 0.5in;">Sharma, R. V.; Sharma, K. C. Physica, 89A, 213. (1977).</p> 
    72407240<p style="margin-left: 0.5in;">&nbsp;</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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><a name="HayterMSAStructure"></a><strong><span style="font-size: 14pt;">&nbsp;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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="HayterMSAStructure"></a><b><span style="font-size: 14pt;">&nbsp;HayterMSA Structure </span></b></p> 
    72427242<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.&nbsp; When combined with an appropriate form factor (such as sphere, core+shell, ellipsoid etc&hellip;), this allows for inclusion of the interparticle interference effects due to screened coulomb repulsion between charged particles. This routine only works for charged particles.&nbsp; If the charge is set to zero the routine will self destruct.&nbsp; For non-charged particles use a hard sphere potential.</p> 
    72437243<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.&nbsp; At present there is no provision for entering the ionic strength directly nor for use of any multivalent salts.&nbsp; The counterions are also assumed to be monovalent.</p> 
     
    73217321</div> 
    73227322<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> 
    73247324<p>References:</p> 
    73257325<p style="text-indent: 0.5in;">JP Hansen and JB Hayter, Molecular Physics 46, 651-656 (1982).</p> 
    73267326<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></strong><a name="StickyHSStructure"></a><strong><span style="font-size: 14pt;">&nbsp;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;">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></b><a name="StickyHSStructure"></a><b><span style="font-size: 14pt;">&nbsp;StickyHS Structure </span></b></p> 
    73287328<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> 
    73297329<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> 
     
    73967396</div> 
    73977397<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> 
    73997399<p>References:</p> 
    74007400<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> 
    74027402<p>Feigin, L. A, and D. I. Svergun (1987) "Structure Analysis by Small-Angle X-Ray and Neutron Scattering", Plenum Press, New York.</p> 
    74037403<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> 
    74057405<p>Hansen, S., (1990)<em> J. Appl. Cryst. </em>23, 344-346.</p> 
    74067406<p>Henderson, S.J. (1996) <em>Biophys. J. </em>70, 1618-1627</p> 
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