Changeset 318b5bbb in sasview for sansmodels/src/sans/models/media/model_functions.html
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sansmodels/src/sans/models/media/model_functions.html
r657e52c r318b5bbb 7 7 <ul style="margin-top: 0in;" type="disc"> 8 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 </a>, <a href="#BinaryHSModel">BinaryHSModel</a>, <a href="#FuzzySphereModel">FuzzySphereModel</a>, <a href="#RaspBerryModel">RaspBerryModel</a>, <a href="#CoreShellModel">CoreShellModel</a>, <a href="#Core2ndMomentModel">Core2ndMomentModel</a>, <a href="#CoreMultiShellModel">CoreMultiShellModel</a>, <a href="#VesicleModel">VesicleModel</a>, <a href="#MultiShellModel">MultiShellModel</a>, <a href="#OnionExpShellModel">OnionExpShellModel</a>, <a href="#SphericalSLDModel">SphericalSLDModel</a>, <a href="#LinearPearlsModel">LinearPearlsModel</a>, <a href="#PearlNecklaceModel">PearlNecklaceModel</a> , <a href="#CylinderModel">CylinderModel</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</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">DAB _Model</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>9 <li style="line-height: 115%;"><a href="#Shapes"><strong>Shapes</strong></a>: <a href="#SphereModel">SphereModel (Magnetic 2D Model)</a>, <a href="#BinaryHSModel">BinaryHSModel</a>, <a href="#FuzzySphereModel">FuzzySphereModel</a>, <a href="#RaspBerryModel">RaspBerryModel</a>, <a href="#CoreShellModel">CoreShellModel (Magnetic 2D Model)</a>, <a href="#Core2ndMomentModel">Core2ndMomentModel</a>, <a href="#CoreMultiShellModel">CoreMultiShellModel (Magnetic 2D Model)</a>, <a href="#VesicleModel">VesicleModel</a>, <a href="#MultiShellModel">MultiShellModel</a>, <a href="#OnionExpShellModel">OnionExpShellModel</a>, <a href="#SphericalSLDModel">SphericalSLDModel</a>, <a href="#LinearPearlsModel">LinearPearlsModel</a>, <a href="#PearlNecklaceModel">PearlNecklaceModel</a> , <a href="#CylinderModel">CylinderModel (Magnetic 2D Model)</a>, <a href="#CoreShellCylinderModel">CoreShellCylinderModel</a>, <a href="#CoreShellBicelleModel">CoreShellBicelleModel</a>,<a href="#HollowCylinderModel">HollowCylinderModel</a>, <a href="#FlexibleCylinderModel">FlexibleCylinderModel</a>, <a href="#FlexibleCylinderModel">FlexCylEllipXModel</a>, <a href="#StackedDisksModel">StackedDisksModel</a>, <a href="#ParallelepipedModel">ParallelepipedModel (Magnetic 2D Model)</a>, <a href="#CSParallelepipedModel">CSParallelepipedModel</a>, <a href="#EllipticalCylinderModel">EllipticalCylinderModel</a>, <a href="#BarBellModel">BarBellModel</a>, <a href="#CappedCylinderModel">CappedCylinderModel</a>, <a href="#EllipsoidModel">EllipsoidModel</a>, <a href="#CoreShellEllipsoidModel">CoreShellEllipsoidModel</a>, <a href="#TriaxialEllipsoidModel">TriaxialEllipsoidModel</a>, <a href="#LamellarModel">LamellarModel</a>, <a href="#LamellarFFHGModel">LamellarFFHGModel</a>, <a href="#LamellarPSModel">LamellarPSModel</a>, <a href="#LamellarPSHGModel">LamellarPSHGModel</a>, <a href="#LamellarPCrystalModel">LamellarPCrystalModel</a>, <a href="#SCCrystalModel">SCCrystalModel</a>, <a href="#FCCrystalModel">FCCrystalModel</a>, <a href="#BCCrystalModel">BCCrystalModel</a>.</li> 10 <li style="line-height: 115%;"><a href="#Shape-Independent"><strong>Shape-Independent</strong></a>: <a href="#Absolute%20Power_Law">AbsolutePower_Law</a>, <a href="#BEPolyelectrolyte">BEPolyelectrolyte</a>, <a href="#BroadPeakModel">BroadPeakModel,<span><span style="text-decoration: underline;"><span style="color: blue;">CorrLength</span></span></span><span>,</span></a> <a href="#DABModel">DABModel</a>, <a href="#Debye">Debye</a>, <a href="#Number_Density_Fractal">FractalModel</a>, <a href="#FractalCoreShell">FractalCoreShell</a>, <a href="#GaussLorentzGel">GaussLorentzGel</a>, <a href="#Guinier">Guinier</a>, <a href="#GuinierPorod">GuinierPorod</a>, <a href="#Lorentz">Lorentz</a>, <a href="#Mass_Fractal">MassFractalModel</a>, <a href="#MassSurface_Fractal">MassSurfaceFractal</a>, <a href="#Peak%20Gauss%20Model">PeakGaussModel</a>, <a href="#Peak%20Lorentz%20Model">PeakLorentzModel</a>, <a href="#Poly_GaussCoil">Poly_GaussCoil</a>, <a href="#PolymerExclVolume">PolyExclVolume</a>, <a href="#PorodModel">PorodModel</a>, <a href="#RPA10Model">RPA10Model</a>, <a href="#StarPolymer">StarPolymer</a>, <a href="#Surface_Fractal">SurfaceFractalModel</a>, <a href="#TeubnerStreyModel">Teubner Strey</a>, <a href="#TwoLorentzian">TwoLorentzian</a>, <a href="#TwoPowerLaw">TwoPowerLaw</a>, <a href="#UnifiedPowerRg">UnifiedPowerRg</a>, <a href="#LineModel">LineModel</a>, <a href="#ReflectivityModel">ReflectivityModel</a>, <a href="#ReflectivityIIModel">ReflectivityIIModel</a>, <a href="#GelFitModel">GelFitModel</a>.</li> 11 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 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> … … 17 17 <p> Many of our models use the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research and thus some content and figures in this document are originated from or shared with the NIST Igor analysis package.</p> 18 18 <p style="margin-left: 0.25in; text-indent: -0.25in;"><strong><span style="font-size: 16pt;">2.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="Shapes"></a><strong><span style="font-size: 16pt;">Shapes (Scattering Intensity Models)</span></strong></p> 19 <p>This software provides form factors for various particle shapes. After giving a mathematical definition of each model, we draw the list of parameters available to the user. Validation plots for each model are also presented. Instructions on how to use the software is available with the source code , available from SVN:</p>20 <p style="text-indent: 0.25in;"><em>https://sansviewproject.svn.sourceforge.net/svnroot/sansviewproject/</em></p> 19 <p>This software provides form factors for various particle shapes. After giving a mathematical definition of each model, we draw the list of parameters available to the user. Validation plots for each model are also presented. Instructions on how to use the software is available with the source code.</p> 20 21 21 <p>To easily compare to the scattering intensity measured in experiments, we normalize the form factors by the volume of the particle:</p> 22 22 <p style="text-align: center;" align="center"><span style="position: relative; top: 12pt;"><img src="img/image001.PNG" alt="" /></span> </p> … … 27 27 <p>Our so-called 1D scattering intensity functions provide <em>P(q) </em>for the case where the scatterer is randomly oriented. In that case, the scattering intensity only depends on the length of q. The intensity measured on the plane of the SANS detector will have an azimuthal symmetry around <em>q</em>=0.</p> 28 28 <p>Our so-called 2D scattering intensity functions provide <em>P(q, </em><em><span style="font-family: 'Arial','sans-serif';">φ</span>)</em> for an oriented system as a function of a q-vector in the plane of the detector. We define the angle <span style="font-family: 'Arial','sans-serif';">φ</span> as the angle between the q vector and the horizontal (x) axis of the plane of the detector.</p> 29 <p style="margin-left: 0.55in; text-indent: -0.3in;"><strong><span style="font-size: 14pt;">2.1.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="SphereModel"></a><strong><span style="font-size: 14pt;">Sphere Model </span></strong></p>29 <p style="margin-left: 0.55in; text-indent: -0.3in;"><strong><span style="font-size: 14pt;">2.1.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="SphereModel"></a><strong><span style="font-size: 14pt;">Sphere Model (Magnetic 2D Model)</span></strong></p> 30 30 <p>This model provides the form factor, P(q), for a monodisperse spherical particle with uniform scattering length density. The form factor is normalized by the particle volume as described below.</p> 31 For magnetic scattering, please see the '<a href="polar_mag_help.html">Polarization/Magnetic Scattering</a>' in Fitting Help. 31 32 <p style="margin-left: 0.85in; text-indent: -0.35in;"><strong>1.1.</strong><strong><span style="font-size: 7pt;"> </span>Definition</strong></p> 32 33 <p>The 1D scattering intensity is calculated in the following way (Guinier, 1955):</p> … … 242 243 <p>where the amplitude A(q) is given as the typical sphere scattering convoluted with a Gaussian to get a gradual drop-off in the scattering length density:</p> 243 244 <p style="text-align: center;" align="center"><span style="position: relative; top: 18pt;"><img src="img/image011.PNG" alt="" /></span></p> 244 <br>245 245 <p>Here A2(q) is the form factor, P(q). The ‘scale’ is equivalent to the volume fraction of spheres, each of volume, V. Contrast (<em><span style="font-family: 'Arial','sans-serif';">Δ</span>ρ</em> ) is the difference of scattering length densities of the sphere and the surrounding solvent.</p> 246 246 <p>The poly-dispersion in radius and in fuzziness is provided.</p> … … 453 453 <p style="text-align: center;" align="center"> </p> 454 454 <p> </p> 455 <p style="margin-left: 0.55in; text-indent: -0.3in;"><strong><span style="font-size: 14pt;">2.5.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="CoreShellModel"></a><strong><span style="font-size: 14pt;">Core Shell (Sphere) Model </span></strong></p>455 <p style="margin-left: 0.55in; text-indent: -0.3in;"><strong><span style="font-size: 14pt;">2.5.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="CoreShellModel"></a><strong><span style="font-size: 14pt;">Core Shell (Sphere) Model (Magnetic 2D Model)</span></strong></p> 456 456 <p>This model provides the form factor, P(<em>q</em>), for a spherical particle with a core-shell structure. The form factor is normalized by the particle volume.</p> 457 For magnetic scattering, please see the '<a href="polar_mag_help.html">Polarization/Magnetic Scattering</a>' in Fitting Help. 457 458 <p style="margin-left: 0.85in; text-indent: -0.35in;"><strong>1.1.</strong><strong><span style="font-size: 7pt;"> </span>Definition</strong></p> 458 459 <p>The 1D scattering intensity is calculated in the following way (Guinier, 1955):</p> … … 686 687 <p> </p> 687 688 <p style="text-align: center; page-break-after: avoid;" align="center"><img style="width: 526px; height: 333px;" src="img/secongm_fig1.jpg" alt="core_scondmoment_1D_validation" /></p> 688 <p style="margin-left: 0.55in; text-indent: -0.3in;"><strong><span style="font-size: 14pt;">2.7.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="CoreMultiShellModel"></a><strong><span style="font-size: 14pt;">CoreMultiShell(Sphere)Model </span></strong></p>689 <p style="margin-left: 0.55in; text-indent: -0.3in;"><strong><span style="font-size: 14pt;">2.7.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="CoreMultiShellModel"></a><strong><span style="font-size: 14pt;">CoreMultiShell(Sphere)Model (Magnetic 2D Model)</span></strong></p> 689 690 <p>This model provides the scattering from spherical core with from 1 up to 4 shell structures. It has a core of a specified radius, with four shells. The SLDs of the core and each shell are individually specified. </p> 691 For magnetic scattering, please see the '<a href="polar_mag_help.html">Polarization/Magnetic Scattering</a>' in Fitting Help. 690 692 <p style="margin-left: 0.85in; text-indent: -0.35in;"><strong>1.1.</strong><strong><span style="font-size: 7pt;"> </span>Definition</strong></p> 691 693 <p>The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> … … 1707 1709 <p><a name="PearlNecklaceModel"></a>R. Schweins and K. Huber, ‘Particle Scattering Factor of Pearl Necklace Chains’, Macromol. Symp., 211, 25-42, 2004.</p> 1708 1710 <p><a name="PearlNecklaceModel"></a> </p> 1709 <p style="margin-left: 0.55in; text-indent: -0.3in;"><a name="PearlNecklaceModel"></a><strong><span style="font-size: 14pt;">2.14.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="CylinderModel"></a><strong><span style="font-size: 14pt;">Cylinder Model </span></strong></p>1711 <p style="margin-left: 0.55in; text-indent: -0.3in;"><a name="PearlNecklaceModel"></a><strong><span style="font-size: 14pt;">2.14.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="CylinderModel"></a><strong><span style="font-size: 14pt;">Cylinder Model (Magnetic 2D Model)</span></strong></p> 1710 1712 <p>This model provides the form factor for a right circular cylinder with uniform scattering length density. The form factor is normalized by the particle volume.</p> 1713 For magnetic scattering, please see the '<a href="polar_mag_help.html">Polarization/Magnetic Scattering</a>' in Fitting Help. 1711 1714 <p style="margin-left: 0.85in; text-indent: -0.35in;"><strong>1.1.</strong><strong><span style="font-size: 7pt;"> </span>Definition</strong></p> 1712 1715 <p>The output of the 2D scattering intensity function for oriented cylinders is given by (Guinier, 1955):</p> … … 1715 1718 <p>where <span style="font-family: 'Arial','sans-serif';">α</span> is the angle between the axis of the cylinder and the q-vector, V is the volume of the cylinder, L is the length of the cylinder, r is the radius of the cylinder, and <em><span style="font-family: 'Arial','sans-serif';">Δ</span>ρ</em> (contrast) is the scattering length density difference between the scatterer and the solvent. J1 is the first order Bessel function.</p> 1716 1719 <p>To provide easy access to the orientation of the cylinder, we define the axis of the cylinder using two angles theta and phi. Those angles are defined on Figure 2.</p> 1717 <p style="text-align: center; page-break-after: avoid;" align="center"><img src="img/image061.jpg" alt="cylinderangles.gif" width="478" height="258" /></p> 1718 <p style="text-align: center;" align="center"><a name="_Ref173213915"></a><a name="_Ref173306040"></a>Figure 2a. Definition of the angles for oriented cylinders.</p> 1719 <p style="text-align: center;" align="center"><img src="img/image062.jpg" alt="cylinderangles2.gif" width="464" height="313" /></p> 1720 <p style="text-align: center;" align="center">Figure 2b. Examples of the angles for oriented cylinders against the detector plane.</p> 1720 <p style="text-align: center; page-break-after: avoid;" align="center"><img src="img/image061.jpg" /></p> 1721 <p style="text-align: center;" align="center"><a name="_Ref173213915"></a><a name="_Ref173306040"></a>Figure 2. Definition of the angles for oriented cylinders.</p> 1722 1723 <p style="text-align: center;" align="center"><img src="img/image062.jpg" alt="" width="379" height="256" /></p> 1724 <p style="text-align: center;" align="center">Figure. Examples of the angles for oriented pp against the detector plane.</p> 1725 1726 1721 1727 <p>For P*S: The 2nd virial coefficient of the cylinder is calculate based on the radius and length values, and used as the effective radius toward S(Q) when P(Q)*S(Q) is applied.</p> 1722 1728 <p>The returned value is scaled to units of [cm-1] and the parameters of the cylinder model are the following:</p> … … 1981 1987 <p style="text-align: center; page-break-after: avoid;" align="center"> </p> 1982 1988 <p><a name="_Ref173307204"></a>Figure 9: Comparison of the intensity for uniformly distributed core-shell cylinders calculated from our 2D model and the intensity from the NIST SANS analysis software. The parameters used were: Scale=1.0, Radius=20 Å, Thickness=10 Å, Length=400 Å, Core_sld=1e-6 Å -2, Shell_sld=4e-6 Å -2, Solvent_sld=1e-6 Å -2, and Background=0.0 cm -1.</p> 1989 1990 <p style="text-align: center; page-break-after: avoid;" align="center"><img src="img/image061.jpg" /></p> 1991 <p style="text-align: center;" align="center"><a name="_Ref173213915"></a><a name="_Ref173306040"></a>Figure. Definition of the angles for oriented core-shell cylinders.</p> 1992 <p style="text-align: center;" align="center"><img src="img/image062.jpg" alt="" width="379" height="256" /></p> 1993 <p style="text-align: center;" align="center">Figure. Examples of the angles for oriented pp against the detector plane.</p> 1994 1983 1995 <p style="margin-left: 0.55in; text-indent: -0.3in;"><strong><span style="font-size: 14pt;">2.16.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="CoreShellBicelleModel"></a><strong><span style="font-size: 14pt;">Core-Shell (Cylinder) Bicelle Model</span></strong></p> 1984 1996 <p>This model provides the form factor for a circular cylinder with a core-shell scattering length density profile. The form factor is normalized by the particle volume. This model is a more general case of <a href="#CoreShellCylinderModel">core-shell cylinder model </a> (see above and reference below) in that the parameters of the shell are separated into a face-shell and a rim-shell so that users can set different values of the thicknesses and slds. </p> … … 2127 2139 <p style="text-align: center;" align="center"><img id="cscylbicelle" style="width: 512px; height: 377px;" src="img/cscylbicelle_pic.jpg" alt="" /></p> 2128 2140 <p style="text-align: center;" align="center"><strong>Figure. 1D plot using the default values (w/200 data point).</strong></p> 2141 2142 2143 <p style="text-align: center; page-break-after: avoid;" align="center"><img src="img/image061.jpg" /></p> 2144 <p style="text-align: center;" align="center"><a name="_Ref173213915"></a><a name="_Ref173306040"></a>Figure. Definition of the angles for the 2145 oriented Core-Shell Cylinder Bicelle Model.</p> 2146 <p style="text-align: center;" align="center"><img src="img/image062.jpg" alt="" width="379" height="256" /></p> 2147 <p style="text-align: center;" align="center">Figure. Examples of the angles for oriented pp against the detector plane.</p> 2148 2149 2129 2150 <p> REFERENCE<br /> Feigin, L. A, and D. I. Svergun, "Structure Analysis by Small-Angle X-Ray and Neutron Scattering", Plenum Press, New York, (1987).</p> 2130 2151 <p style="margin-left: 0.55in; text-indent: -0.3in;"><strong><span style="font-size: 14pt;">2.17.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="HollowCylinderModel"></a><strong><span style="font-size: 14pt;">HollowCylinderModel</span></strong></p> … … 2237 2258 <p style="text-align: center;" align="center"><strong>Figure. 1D plot using the default values (w/1000 data point).</strong></p> 2238 2259 <p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006).</p> 2260 2261 <p style="text-align: center; page-break-after: avoid;" align="center"><img src="img/image061.jpg" /></p> 2262 <p style="text-align: center;" align="center"><a name="_Ref173213915"></a><a name="_Ref173306040"></a>Figure. Definition of the angles for the 2263 oriented HollowCylinderModel.</p> 2264 <p style="text-align: center;" align="center"><img src="img/image062.jpg" alt="" width="379" height="256" /></p> 2265 <p style="text-align: center;" align="center">Figure. Examples of the angles for oriented pp against the detector plane.</p> 2266 2239 2267 <p>REFERENCE</p> 2240 2268 <p>Feigin, L. A, and D. I. Svergun, "Structure Analysis by Small-Angle X-Ray and Neutron Scattering", Plenum Press, New York, (1987).</p> … … 2341 2369 <p style="text-align: center;" align="center"><img id="Picture 228" src="img/image076.jpg" alt="" width="465" height="345" /></p> 2342 2370 <p style="text-align: center;" align="center"><strong>Figure. 1D plot using the default values (w/1000 data point).</strong></p> 2371 2343 2372 <p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 2344 2373 <p>From the reference, "Method 3 With Excluded Volume" is used. The model is a parametrization of simulations of a discrete representation of the worm-like chain model of Kratky and Porod applied in the pseudocontinuous limit. See equations (13,26-27) in the original reference for the details.</p> … … 2616 2645 <p style="text-align: center;" align="center"><img src="img/image085.jpg" alt="" width="451" height="334" /></p> 2617 2646 <p style="text-align: center;" align="center"><strong>Figure. 1D plot using the default values (w/1000 data point).</strong></p> 2618 <p style="text-align: center;" align="center"><img id="Picture 101" src="img/image086.jpg" alt="" width="377" height="215"/></p>2647 <p style="text-align: center;" align="center"><img id="Picture 101" src="img/image086.jpg" /></p> 2619 2648 <p style="text-align: center;" align="center">Figure. Examples of the angles for oriented stackeddisks against the detector plane.</p> 2649 2650 <p style="text-align: center;" align="center"><img src="img/image062.jpg" alt="" width="379" height="256" /></p> 2651 <p style="text-align: center;" align="center">Figure. Examples of the angles for oriented pp against the detector plane.</p> 2652 2653 2620 2654 <p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 2621 2655 <p>REFERENCE</p> … … 2623 2657 <p>Kratky, O. and Porod, G., J. Colloid Science, 4, 35, 1949.</p> 2624 2658 <p>Higgins, J.S. and Benoit, H.C., "Polymers and Neutron Scattering", Clarendon, Oxford, 1994.</p> 2625 <p style="margin-left: 0.55in; text-indent: -0.3in;"><strong><span style="font-size: 14pt;">2.21.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="ParallelepipedModel"></a><strong><span style="font-size: 14pt;">ParallelepipedModel </span></strong></p>2659 <p style="margin-left: 0.55in; text-indent: -0.3in;"><strong><span style="font-size: 14pt;">2.21.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="ParallelepipedModel"></a><strong><span style="font-size: 14pt;">ParallelepipedModel (Magnetic 2D Model) </span></strong></p> 2626 2660 <p>This model provides the form factor, P(<em>q</em>), for a rectangular cylinder (below) where the form factor is normalized by the volume of the cylinder. P(q) = scale*<f^2>/V+background where the volume V= ABC and the averaging < > is applied over all orientation for 1D. </p> 2661 For magnetic scattering, please see the '<a href="polar_mag_help.html">Polarization/Magnetic Scattering</a>' in Fitting Help. 2627 2662 <p><span style="font-size: 14pt;"> </span></p> 2628 2663 <p style="text-align: center;" align="center"><img src="img/image087.jpg" alt="" width="326" height="247" /></p> … … 2636 2671 <p>For P*S: The 2nd virial coefficient of the solid cylinder is calculate based on the averaged effective radius (= sqrt(short_a*short_b/pi)) and length( = long_c) values, and used as the effective radius toward S(Q) when P(Q)*S(Q) is applied.</p> 2637 2672 <p>To provide easy access to the orientation of the parallelepiped, we define the axis of the cylinder using two angles θ , <span style="font-family: 'Arial','sans-serif';">φ </span>and<span style="font-family: Symbol;">Y</span>. Similarly to the case of the cylinder, those angles, θ and <span style="font-family: 'Arial','sans-serif';">φ,</span> are defined on Figure 2 of CylinderModel. The angle <span style="font-family: Symbol;">Y </span>is the rotational angle around its own long_c axis against the q plane. For example, <span style="font-family: Symbol;">Y </span>= 0 when the short_b axis is parallel to the x-axis of the detector.</p> 2638 <p style="text-align: center;" align="center"><img src="img/image090.jpg" alt="" width="352" height="264"/></p>2673 <p style="text-align: center;" align="center"><img src="img/image090.jpg"/></p> 2639 2674 <p style="text-align: center;" align="center"><strong>Figure. Definition of angles for 2D</strong>.</p> 2640 2675 <p style="text-align: center;" align="center"><img src="img/image091.jpg" alt="" width="379" height="256" /></p> … … 2749 2784 <p>For P*S: The 2nd virial coefficient of this CSPP is calculate based on the averaged effective radius (= sqrt((short_a+2*rim_a)*(short_b+2*rim_b)/pi)) and length( = long_c+2*rim_c) values, and used as the effective radius toward S(Q) when P(Q)*S(Q) is applied.</p> 2750 2785 <p>To provide easy access to the orientation of the CSparallelepiped, we define the axis of the cylinder using two angles θ , <span style="font-family: 'Arial','sans-serif';">φ </span>and<span style="font-family: Symbol;">Y</span>. Similarly to the case of the cylinder, those angles, θ and <span style="font-family: 'Arial','sans-serif';">φ,</span> are defined on Figure 2 of CylinderModel. The angle <span style="font-family: Symbol;">Y </span>is the rotational angle around its own long_c axis against the q plane. For example, <span style="font-family: Symbol;">Y </span>= 0 when the short_b axis is parallel to the x-axis of the detector.</p> 2751 <p style="text-align: center;" align="center"><img id="Picture 102" src="img/image090.jpg" alt="" width="352" height="264"/></p>2786 <p style="text-align: center;" align="center"><img id="Picture 102" src="img/image090.jpg" /></p> 2752 2787 <p style="text-align: center;" align="center"><strong>Figure. Definition of angles for 2D</strong>.</p> 2753 2788 <p style="text-align: center;" align="center"><img id="Picture 103" src="img/image091.jpg" alt="" width="379" height="256" /></p> … … 2918 2953 <p style="text-align: center;" align="center"><img id="Picture 34" src="img/image097.jpg" alt="" width="451" height="339" /></p> 2919 2954 <p style="text-align: center;" align="center"><strong>Figure. 2D plot using the default values (w/(256X265) data points).</strong></p> 2955 2920 2956 <p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 2921 2957 <p> REFERENCE</p> … … 2940 2976 <p>To provide easy access to the orientation of the elliptical, we define the axis of the cylinder using two angles θ , <span style="font-family: 'Arial','sans-serif';">φ </span>and<span style="font-family: Symbol;">Y</span>. Similarly to the case of the cylinder, those angles, θ and <span style="font-family: 'Arial','sans-serif';">φ,</span> are defined on Figure 2 of CylinderModel. The angle <span style="font-family: Symbol;">Y </span>is the rotational angle around its own long_c axis against the q plane. For example, <span style="font-family: Symbol;">Y </span>= 0 when the r_minor axis is parallel to the x-axis of the detector.</p> 2941 2977 <p>All angle parameters are valid and given only for 2D calculation (Oriented system).</p> 2942 <p style="text-align: center;" align="center"><img id="Picture 105" src="img/image101.jpg" alt="" width="370" height="277"/></p>2978 <p style="text-align: center;" align="center"><img id="Picture 105" src="img/image101.jpg" /></p> 2943 2979 <p style="text-align: center;" align="center"><strong>Figure. Definition of angels for 2D</strong>.</p> 2944 <p style="text-align: center;" align="center"><img id="Picture 114" src="img/image0 91.jpg" alt="" width="379" height="256" /></p>2980 <p style="text-align: center;" align="center"><img id="Picture 114" src="img/image062.jpg" alt="" width="379" height="256" /></p> 2945 2981 <p style="text-align: center;" align="center"><span style="font-size: 12pt;">Figure. Examples of the angles for oriented elliptical cylinders </span></p> 2946 2982 <p style="text-align: center;" align="center"><span style="font-size: 12pt;">against the detector plane.</span></p> … … 3173 3209 <p style="text-align: center;" align="center"><img id="Picture 66" src="img/image111.jpg" alt="" width="425" height="346" /></p> 3174 3210 <p style="text-align: center;" align="center"><strong>Figure. 2D plot (w/(256X265) data points).</strong></p> 3211 3212 <p style="text-align: center; page-break-after: avoid;" align="center"><img src="img/image061.jpg" /></p> 3213 <p style="text-align: center;" align="center"><img src="img/image062.jpg" alt="" width="379" height="256" /></p> 3214 <p style="text-align: center;" align="center">Figure. Examples of the angles for oriented pp against the detector plane.</p> 3215 3216 3217 <p style="text-align: center;" align="center"><a name="_Ref173213915"></a><a name="_Ref173306040"></a>Figure. Definition of the angles for oriented 2D barbells.</p> 3218 3219 3175 3220 <p style="margin-left: 0.55in; text-indent: -0.3in;"><strong><span style="font-size: 14pt;">2.25.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="CappedCylinderModel"></a><strong><span style="font-size: 14pt;">CappedCylinder(/ConvexLens)Model</span></strong></p> 3176 3221 <p>Calculates the scattering from a cylinder with spherical section end-caps(This model simply becomes the ConvexLensModel when the length of the cylinder L = 0. That is, a sphereocylinder with end caps that have a radius larger than that of the cylinder and the center of the end cap radius lies within the cylinder. See the diagram for the details of the geometry and restrictions on parameter values.</p> … … 3295 3340 <p style="text-align: center;" align="center"><img id="Picture 71" src="img/image118.jpg" alt="" width="402" height="334" /></p> 3296 3341 <p style="text-align: center;" align="center"><strong>Figure. 2D plot (w/(256X265) data points).</strong></p> 3342 <p style="text-align: center; page-break-after: avoid;" align="center"><img src="img/image061.jpg" /></p> 3343 <p style="text-align: center;" align="center"><a name="_Ref173213915"></a><a name="_Ref173306040"></a>Figure. Definition of the angles for oriented 2D cylinders.</p> 3344 <p style="text-align: center;" align="center"><img src="img/image062.jpg" alt="" width="379" height="256" /></p> 3345 <p style="text-align: center;" align="center">Figure. Examples of the angles for oriented pp against the detector plane.</p> 3346 3347 3348 3297 3349 <p style="margin-left: 0.55in; text-indent: -0.3in;"><strong><span style="font-size: 14pt;">2.26.</span></strong><strong><span style="font-size: 7pt;"> </span></strong><a name="EllipsoidModel"></a><strong><span style="font-size: 14pt;">Ellipsoid Model</span></strong></p> 3298 3350 <p>This model provides the form factor for an ellipsoid (ellipsoid of revolution) with uniform scattering length density. The form factor is normalized by the particle volume.</p> … … 3414 3466 <p>The output of the 1D scattering intensity function for randomly oriented ellipsoids is then given by the equation above.</p> 3415 3467 <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> 3416 <p style="text-align: center;" align="center"><img src="img/image122.jpg" alt="" width="3 96" height="297"/></p>3417 <p style="text-align: center;" align="center"><span style="font-size: 12pt;">Figure. Examples of the angles for oriented ellipsoid </span></p>3418 <p style="text-align: center;" align="center"><span style="font-size: 12pt;">against the detector plane</span>.</p> 3468 <p style="text-align: center;" align="center"><img src="img/image122.jpg" alt="" width="379" height="256"/></p> 3469 <p style="text-align: center;" align="center"><span style="font-size: 12pt;">Figure. The angles for oriented ellipsoid </span></p> 3470 3419 3471 <p style="margin-left: 0.85in; text-indent: -0.35in;"><strong>2.1.</strong><strong><span style="font-size: 7pt;"> </span>Validation of the ellipsoid model</strong></p> 3420 3472 <p>Validation of our code was done by comparing the output of the 1D model to the output of the software provided by the NIST (Kline, 2006). Figure 5 shows a comparison of the 1D output of our model and the output of the NIST software.</p> … … 3558 3610 <p style="text-align: center;" align="center"><strong>Figure. 1D plot using the default values (w/1000 data point).</strong></p> 3559 3611 <p style="text-align: center;" align="center"><strong> </strong></p> 3560 <p style="text-align: center;" align="center"><img src="img/image122.jpg" alt="" width="3 96" height="297"/></p>3561 <p style="text-align: center;" align="center">Figure. Examples of the angles for oriented coreshellellipsoid against the detector plane where a =polar axis.</p>3612 <p style="text-align: center;" align="center"><img src="img/image122.jpg" alt="" width="379" height="256"/></p> 3613 <p style="text-align: center;" align="center">Figure. The angles for oriented coreshellellipsoid .</p> 3562 3614 <p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 3563 3615 <p>REFERENCE</p> … … 3674 3726 <p style="text-align: center;" align="center"><img src="img/image131.gif" alt="" width="438" height="272" /></p> 3675 3727 <p style="text-align: center;" align="center"><strong>Figure. Comparison between 1D and averaged 2D.</strong></p> 3676 <p style="text-align: center;" align="center"><img src="img/image132.jpg" alt="" width="3 96" height="297" /></p>3677 <p style="text-align: center;" align="center">Figure. Examples of the angles for oriented ellipsoid against the detector plane.</p>3728 <p style="text-align: center;" align="center"><img src="img/image132.jpg" alt="" width="379" height="256" /></p> 3729 <p style="text-align: center;" align="center">Figure. The angles for oriented ellipsoid.</p> 3678 3730 <p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 3679 3731 <p>REFERENCE</p> … … 4368 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> 4369 4421 <p> The 2D (Anisotropic model) is based on the reference (above) which I(q) is approximated for 1d scattering. Thus the scattering pattern for 2D may not be accurate. Note that we are not responsible for any incorrectness of the 2D model computation.</p> 4370 <p style="text-align: center;" align="center"><img id="Object 23" src="img/image156. gif" alt="" width="304" height="321" /></p>4422 <p style="text-align: center;" align="center"><img id="Object 23" src="img/image156.jpg" /></p> 4371 4423 <p style="text-align: center;" align="center"> </p> 4372 4424 <p> </p> … … 4493 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> 4494 4546 <p> The 2D (Anisotropic model) is based on the reference (above) in which I(q) is approximated for 1d scattering. Thus the scattering pattern for 2D may not be accurate. Note that we are not responsible for any incorrectness of the 2D model computation.</p> 4495 <p style="text-align: center;" align="center"><img src="img/image165.gif" alt="" width="304" height="321"/></p>4547 <p style="text-align: center;" align="center"><img src="img/image165.gif" /></p> 4496 4548 <p style="text-align: center;" align="center"> </p> 4497 4549 <p> </p> … … 4617 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> 4618 4670 <p> The 2D (Anisotropic model) is based on the reference (1987) in which I(q) is approximated for 1d scattering. Thus the scattering pattern for 2D may not be accurate. Note that we are not responsible for any incorrectness of the 2D model computation.</p> 4619 <p style="text-align: center;" align="center"><img id="Object 31" src="img/image165.gif" alt="" width="304" height="321"/></p>4671 <p style="text-align: center;" align="center"><img id="Object 31" src="img/image165.gif" /></p> 4620 4672 <p style="text-align: center;" align="center"> </p> 4621 4673 <p> </p>
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